The Pennsylvania State University The Graduate School College of Engineering SIGNAL ANALYSIS USING RAISED COSINE EMPIRICAL MODE DECOMPOSITION

Transcription

1 The Pennsylvania State University The Graduate School College of Engineering SIGNAL ANALYSIS USING RAISED COSINE EMPIRICAL MODE DECOMPOSITION A Dissertation in Electrical Engineering by Arnab Roy c 2011 Arnab Roy Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2011

2 The dissertation of Arnab Roy was reviewed and approved by the following: John F. Doherty Professor of Electrical Engineering Dissertation Advisor, Chair of Committee John D. Mathews Professor of Electrical Engineering Ram M. Narayanan Professor of Electrical Engineering Karl M. Reichard Assistant Professor of Acoustics W. Kenneth Jenkins Head of the Department of Electrical Engineering Professor of Electrical Engineering Signatures are on file in the Graduate School.

3 Abstract The inherent nonstationarity of signals in nature imparts their usefulness. This suggests the use of time-frequency methods to study these signals. The empirical mode decomposition (EMD) and the Hilbert-Huang transform (HHT) provide an adaptive and efficient method to analyze such signals. The EMD technique, being based on the local characteristic time scale of the signal, also works as a time-frequency filter to isolate nonstationary signal components. The rapidly growing list of applications points to its capability. This dissertation s approach towards the EMD technique revolves around enhancing its performance while simultaneously leveraging its unique capabilities in practical applications. The original contributions of this dissertation are two-fold: firstly, a new signal-analysis technique based on EMD is developed. This new technique, called raised cosine empirical mode decomposition (RCEMD), possesses several desirable qualities: enhanced frequency resolution, computational efficiency and lower sampling rate requirement. A theoretical framework is developed to compare the performances of the original and proposed techniques. A pre-emphasis and de-emphasis based technique to improve the frequency resolution of the EMD family of algorithms is also developed. The second substantial contribution of this dissertation concerns novel applications of signal analysis techniques including RCEMD. An overlay communication technique that utilizes the unique instantaneous frequency based signal decomposition property of RCEMD is developed. A modification of this technique that is suitable for interference rejection in broadband communications is also described. Finally, two applications of signal analysis techniques concerning atmospheric remote sensing are explored. First, an RCEMD-based technique to isolate both persistent and sporadic signal features in atmospheric pressure measurements is developed. Secondly, a genetic algorithm method to resolve and estimate the parameters of fragmenting meteoroids observed using radar measurements is presented. iii

4 Table of Contents List of Figures List of Tables Acknowledgments viii xii xiii Chapter 1 Introduction Background and Motivation Hilbert Spectrum of Simple Signals Hilbert Spectrum of Combination of Signals Contributions of this Dissertation and Summary of Publications Dissertation Outline Chapter 2 Time-Frequency Analysis of Signals Signal Analysis: Concepts Analytical Signal Instantaneous Frequency Monocomponent and Multicomponent Signals Signal Analysis: Methods Fourier Analysis Fourier Series Fourier Transform Short-Time Fourier Transform Wavelet Transform Continuous Wavelet Transform (CWT) Discrete Wavelet Transform (DWT) Bilinear Time-Frequency Distribution The Wigner-Ville Distribution Reduced Interference Distributions Time-Frequency Distribution Illustration iv

5 2.4.1 Hilbert-Huang Transform (HHT) Empirical Mode Decomposition Procedure Algorithmic Variations Theoretical Developments Applications I Signal Analysis using Empirical Mode Decomposition: Theoretical Developments and Communication Examples via Mathematical Modeling 38 Chapter 3 Raised Cosine Empirical Mode Decomposition Introduction Raised Cosine Interpolation Raised Cosine Empirical Mode Decomposition Signal Decomposition Quality of RCEMD Algorithm Combination of tones Two frequency modulated components Bicomponent trigonometric function Multicomponent signal Tidal component extraction EMD: Computational Complexity Finding the extrema Finding the cubic spline coefficients Complexity of the raised cosine filter approach Complexity of windowed RCEMD Low Sampling Rate Performance of RCEMD Timing jitter at low sampling rates Performance Comparison Conclusions Chapter 4 Pre-emphasis and De-emphasis Introduction Optimum choice of stopping criterion for sifting Pre-Emphasis and De-Emphasis Conclusion v

6 Chapter 5 Overlay Communications using Raised Cosine Empirical Mode Decomposition Introduction Signal Design Performance Analysis Choice of decomposition level Performance approximation Simulation Results Effect on primary users Operations on the Complex Baseband Signal Covert Communications using Empirical Mode Decomposition Simulation Results Communication Range Determination Conclusions Chapter 6 Wideband Interference Removal using Raised Cosine Empirical Mode Decomposition Introduction Signal Design and Excision Procedure Simulation Results Multiple tone interference Tone modulated FM interference Filtered noise modulation of FM interferer Conclusions II Signal Analysis of Sensor Data 123 Chapter 7 Atmospheric Pressure Signal Analysis using Raised Cosine Empirical Mode Decomposition Introduction Data analysis using HHT and wavelets Signal Feature Extraction Signal Feature Extraction using RCEMD Conclusion Chapter 8 Genetic Algorithm based Parameter Estimation Technique for Fragmenting Radar Meteor Head-echoes Introduction Coarse parameter estimation of meteoroid fragments vi

7 8.3 Fine parameter estimation for individual fragments using GA Conclusions Chapter 9 Summary and Open Problems Research Summary Open Problems Appendix Derivation of t u δ 157 Bibliography 159 vii

8 List of Figures 1.1 Time series representation of linear frequency modulated signal Wavelet spectrum of linear frequency modulated signal Hilbert spectrum of linear frequency modulated signal Hilbert spectrum of multicomponent signal Time-frequency representation of three-component signal Hilbert spectrum of three-component signal after wavelet decomposition Hilbert spectrum of three-component signal after EMD Spectral representations for a monocomponent signal Spectral representations for a multicomponent signal Pictorial illustration of EMD steps for synthetic two-tone signal Time-and frequency-domain raised cosine pulses for several roll-off factors Locations of local maxima of the component signal relative to HF component maxima Error between ideal and actual maxima sampling points Comparison of simulation results with theory for raised cosine interpolation based on transient value of performance metric Ω k Comparison of simulation results with theory for raised cosine interpolation based on minimum iteration count for desired signal decomposition quality Signal decomposition quality of RCEMD for a wide range of constituent signal amplitude and frequency ratios Direct comparison of signal decomposition quality of EMD and RCEMD algorithms for combination of tones Instantaneous frequencies of the synthetically generated frequency modulated signal components Direct comparison of signal decomposition quality of EMD and RCEMD algorithms for frequency modulated signal components viii

9 3.10 Comparison of signal decomposition quality of EMD and RCEMD algorithms for frequency modulated signal components based on steadystate value Comparison of signal decomposition quality of EMD and RCEMD algorithms for frequency modulated signal components based on convergence rate Signal decomposition quality comparison between RCEMD and EMD algorithms for bicomponent trigonometric function Frequency-domain signals for the multicomponent signal example Time-domain signals for the multicomponent signal example Demonstration of application of RCEMD technique to sea level data Computational complexity comparison for frequency modulated signal components Effect of sampling rate on interpolation quality Signal analysis performance of the RCEMD, EMD and hybrid techniques for a combination of tones at different sampling rates Signal analysis performance of the RCEMD, EMD and hybrid techniques for a combination of frequency modulated signals at different sampling rates Effect of stopping criterion threshold on EMD signal separation quality Effect of maximum iteration limit on EMD signal separation quality EMD signal separation quality for two tones with unequal strengths Performance improvement using pre-emphasis and de-emphasis method for two tones of unequal strengths Frequency domain representation of the performance improvement using pre-emphasis and de-emphasis method for two tones of unequal strengths Block diagram of the secondary receiver Extraction of the secondary signal from the composite received signal using EMD Instantaneous frequencies of the primary signal (FM) and secondary signal (FSK) showing crossings BER performance of proposed technique compared with some other signal extraction techniques Normalized mean square error (NMSE) between the actual FM signal and the first extracted IMF versus E b /N Cross-validation of theoretical and simulation results for system BER Block diagram of the receiver using remodulation technique ix

10 5.8 Cross-validation of BER results obtained from simulations and semianalytical method for PLL based signal detection technique Performance improvement offered by complex EMD Frequency domain representation of the primary (FM) and frequencyhopped covert (FSK) signals shown here. The primary signal power is 26 db larger than that of the FSK signal in this illustration Covert communication error rate performance with FSK modulation Covert communication error rate performance with QPSK modulation Cross-validation of error rate performance derived from simple numerical model and computer simulation output for QPSK modulated covert signal Illustration of signal analysis quality of several techniques Numerical comparison of decomposition quality for several techniques Maximum achievable range for covert communication technique Time-domain signal for the overlay communications technique and signal decomposition results Block diagram of the interference excision by resynthesis technique E b /N 0 -vs-ber plots for various interference cancelation techniques for multiple tone interference Frequency domain representation of the spread spectrum signal and an interfering tone modulated FM signal Instantaneous frequency of the tone-modulated FM signal E b /N 0 -vs-ber plots for various interference cancelation techniques for tone modulated FM interference E b /N 0 -vs-ber plots for various interference cancelation techniques for filtered noise modulated FM interference EMD output and Hilbert spectrum of microbarograph signal Fourier transform, short-time Fourier transform spectrum and wavelet spectrum of microbarograph signal Complex mother wavelet used for wavelet analysis Illustration of instantaneous frequency overlap of consecutive IMFs produced by EMD Frequency thresholds for diurnal tide extraction and instantaneous frequencies of the first three IMFs from the microbarograph observations Results of semidiurnal and diurnal tide extraction using EMD-based feature extraction technique Illustration of IMF combining technique for isolated feature extraction using EMD x

11 7.8 Performance comparison of isolated feature extraction using EMD- and wavelet-based methods Results of semidiurnal and diurnal tide extraction using RCEMD-based feature extraction technique Frequency thresholds for diurnal tide extraction and instantaneous frequencies of the first three IMFs using RCEMD Performance of RCEMD-based feature extraction technique in isolated feature extraction Range-Time-Intensity (RTI) and Signal-to-Noise Ratio of three meteor events observed with the Poker Flat Incoherent Scatter Radar (PFISR) Illustration of signal pre-processing step to estimate and remove antenna pattern Illustration of radar complex voltages and output of the model using parameters estimated by the GA technique Fast Fourier transform of actual signal and output of the model using parameters estimated by the GA technique Comparison of Range-time-intensity (RTI) of the actual radar signal after pre-processing and reconstructed RTI plot using estimated parameters from our technique xi

12 List of Tables 3.1 No. of computations to find extrema points Modeled parameters of meteor event Modeled parameters of meteor event Modeled parameters of meteor event xii

13 Acknowledgments I would like to thank my family for their love and support during my long academic pursuit, my dissertation advisor Professor John F. Doherty for his continued support, financial assistance, and academic and professional guidance, and my committee members, Professor John D. Mathews, Professor Ram M. Narayanan and Professor Karl M. Reichard for their technical feedback and comments. Thanks are also due to Professor Mathews for collaboration on the remote sensing project. I would also like to acknowledge the contributions of my colleagues and friends for their help and company over the years: Dr. Prashant Bansal, Dr. Glenn Carl, Mr. Stephane Caron, Dr. Arnab Das, Ms. Priya Fotedar-Khorana, Mr. Nitin Kamat, Dr. Ming-Wei Liu, Mr. Vishal Mody, Dr. Azin Neishaboori, Mr. Nipun Patel, Mr. Ashu Sabharwal, Dr. Sanjeev Tavathia, Mr. Shashi Udyavar, Dr. Chun-Hsien Wen and Dr. Qina Zhou. Parts of this work were supported by the National Science Foundation through NSF grant no. ITR/AP to The Pennsylvania State University. xiii

14 Chapter 1 Introduction This chapter serves as an introduction to the thesis. A brief discussion on the motivation for this study is followed by a list of research contributions of this work. Finally, a concise outline of the succeeding chapters concludes this chapter. 1.1 Background and Motivation A signal is a physical carrier of some information. It can originate from a variety of sources (acoustic, biological, mechanical, optical, seismic, etc.). Beyond this diversity, however, the main object of interest is the observation of a time-varying quantity, which is collected at one or more sensors. An important class of signal processing problem deals with signal analysis, which is often the initial step to realize forecasting, data compression, automatic extraction of features and interpretation of seismic data, radar, speech or images. The choice of signal analysis technique is crucial for the ultimate task of processing data, which often comprises several consecutive steps of solving a statistical decision problem (detection, estimation, classification, recognition, etc.). The pertinence of an appropriate technique is rooted in its capability to provide well-suited descriptors of this task. Viewed from the perspective of signal analysis, the decomposed components should have a direct correspondence to the physical properties of the system that generated the signal. Signal analysis principles lead to rejection of narrowband interference from direct-sequence spread spectrum signals [1], efficient image compression [2], geophysical studies for oil exploration [3], to name a few applications. Generally, signal analysis techniques can be classified based on their operational

15 2 s(t) time Figure 1.1: Time series representation of linear frequency modulated signal. domain, namely, time, frequency or time-frequency, although in many cases these distinctions are merely implementational. The Fourier transform (FT) and its windowed version, the short-time Fourier transform (STFT) are signal analysis techniques applicable to signal components that are stationary or at least locally stationary. However, signals in many practical situations, such as electroencephalogram (EEG) signals, which are monitored to observe brain health, and speech signals are known to be nonstationary. More advanced techniques utilizing localized unit energy elementary functions (called time-frequency atoms) such as wavelets and chirplets have simple algorithmic structures and seem to address the problems associated with nonstationary signals [2]. However, optimum signal analysis using these techniques requires some a priori knowledge of signal components. There has been widespread agreement in the signal processing community over the steps that constitute a general signal analysis procedure [4]: 1. Determine if the signal is stationary or not, and whether the signal is monocomponent or multicomponent, 2. Break down the multicomponent signal into its subcomponents (usually using

16 3 frequency time Figure 1.2: Wavelet spectrum of linear frequency modulated signal. frequency time Figure 1.3: Hilbert spectrum of linear frequency modulated signal.

17 4 frequency time Figure 1.4: Result of direct application of Hilbert transform to multicomponent signal. The horizontal lines indicate the frequencies of the component tones. The time series data is shown later in Fig. 2.3a. windowing methods in the time-frequency domain), 3. Track the spectral variation of the components and indicate the energy concentration of the signal around its instantaneous frequency, 4. Model the signal. If each component of a multicomponent signal is defined in terms of its amplitude and phase, then the analysis problem is to find these parameters for each of the signal components. An accepted method of decomposing multicomponent signals with nonstationary components is via time-frequency processing techniques involving wavelets and chirplets amongst others Hilbert Spectrum of Simple Signals An accurate and unambiguous frequency estimate of a sinusoidal signal is obtainable using the FT. The FT possesses several desirable qualities such as ease of computation

18 5 s(t) s 1 (t) frequency frequency time time s 2 (t) s 3 (t) frequency frequency time time Figure 1.5: Time-frequency representation of three-component signal used to test wavelet decomposition and empirical mode decomposition (EMD). Top left panel shows the time-frequency representation for the multicomponent signal. The remaining panels show individual components. The time-series for this example is shown in Fig and invertibility for stationary signals (signals whose frequency content do not change with time). However, due to lack of time resolution, it is not invertible for nonstationary signals (signals with time-varying frequency content). The STFT gains time resolution by performing FT on small data segments sequentially, thereby sacrificing some amount of frequency resolution. This trade-off between time and frequency resolutions is no accident, but a manifestation of the Heisenberg Uncertainty Principle. The wavelet transform adaptively adjusts to the Heisenberg Uncertainty Principle by delivering good resolution in time for large frequencies, and in frequency for small frequencies. Analogous to the concept of frequency for stationary signals, the notion of instantaneous frequency of a nonstationary signal follows naturally. This quantity, which is formally defined in Chapter 2, refers to the the number of oscillations per unit time as a function of time for a signal. The Hilbert transform presents a practical way to compute the instantaneous frequency of a signal. A two-dimensional representation of the instan-

19 6 frequency time Figure 1.6: Hilbert spectrum of the result of wavelet decomposition of multicomponent signal. Input is a three-component signal shown in Fig taneous frequency, with time along the horizontal axis, and frequency along the vertical axis is called the Hilbert spectrum. Compared to other time-frequency spectrums such as the wavelet spectrum, the excellent time-frequency properties of the Hilbert spectrum makes it a useful tool in the field of time-frequency analysis. The time-frequency localization quality of the Hilbert and wavelet spectrums is demonstrated next via an example. Consider a linear frequency modulated signal s(t). This refers to a sinusoid with linearly-varying frequency. The time-domain signal is shown in Fig The wavelet and Hilbert spectrums are shown in Figs. 1.2 and 1.3 respectively. The drawbacks of the wavelet spectrum are evident: poor frequency resolution for large frequencies and poor time resolution for small frequencies. The Hilbert spectrum, on the other hand, exhibits uniformly good time-frequency localization Hilbert Spectrum of Combination of Signals We saw above that the Hilbert spectrum exhibits good time-frequency resolution for simple signals called monocomponent signals, which refers to signals that have only one

20 7 frequency time Figure 1.7: Result of application of Hilbert transform to EMD components. Input is a three-component signal shown in Fig oscillatory mode at any time instant. However, this technique fails to provide meaningful time-frequency representation for more complex signals, called multicomponent signals. The example in Fig. 1.4 shows a multicomponent signal consisting of two tones and its Hilbert spectrum. Since the Hilbert spectrum is meaningful only for monocomponent signals, it fails to correctly identify the instantaneous frequencies of multicomponent signal constituents in this example. To obtain a meaningful value of instantaneous frequency using Hilbert transform the multicomponent signal should be decomposed into its constituents before applying Hilbert transform on each component. The superposition of the individual Hilbert spectrums gives the spectrum for the multicomponent signal. Any model-based nonadaptive decomposition procedure will be ineffective in separating the signal components, in general, due to fixed frequency boundaries. The empirical mode decomposition (EMD) technique, on the other hand, is fully data-driven, not model-based whose purpose is to adaptively decompose any signal into its oscillatory contributions. Therefore the resulting components admit meaningful instantaneous frequencies after Hilbert

21 8 transform. This concept is explained using an example. Consider a multicomponent signal with three components as shown in Fig The analyzed signal is the sum of two sinusoid frequency modulated components and a Gaussian wavepacket. The timefrequency analysis of the multicomponent signal (top left panel in the figure) reveals three time-frequency signatures that overlap in both time and frequency, thus forbidding the components to be separated by any nonadaptive filtering technique. The instantaneous frequencies derived from wavelet decomposition and EMD are shown in Figs. 1.6 and 1.7 respectively. While the wavelet decomposition does not result in meaningful instantaneous frequencies of the components, the EMD produces components with the correct instantaneous frequencies. The time-domain signals for this example appear later in Fig

22 1.2 Contributions of this Dissertation and Summary of Publications The following original contributions in signal analysis research are presented in this dissertation: 1. Development of a new version of the EMD algorithm using raised cosine interpolation with superior signal analysis properties (either more resolution or reduced sampling requirements) and reduced computation requirement. This technique is called raised cosine empirical mode decomposition (RCEMD). 2. Development of associated mathematical framework to study the signal analysis performance of EMD-like algorithms for simple signals. 3. Introduction of an overlay communications technique using RCEMD technique and its extension to covert communications. 4. Application of RCEMD technique for wideband interference rejection in wireless communications. 5. Development of an RCEMD-based technique for study of persistent and sporadic signal features in atmospheric pressure measurements using microbarographs with higher precision than existing techniques. 6. Development of a signal analysis and parameter estimation estimation technique for fragmenting radar meteor echoes using genetic algorithms. Parts of this dissertation work appear in the following publications: 9 Book Chapter 1. A. Roy, and J. F. Doherty, Nyquist Pulse based Empirical Mode Decomposition and its Applications to Remote Sensing Problems, in Signal and Image Processing for Remote Sensing, 2 nd Edition, CRC Press, to appear in 2011.

23 10 Journal Publications 1. A. Roy, and J. F. Doherty, Raised cosine filter-based empirical mode decomposition, IET Signal Processing, vol. 5, no. 2, pp , Apr A. Roy, and J. F. Doherty, Overlay communications using empirical mode decomposition, IEEE Systems Journal, vol. 5, no. 1, pp , Mar A. Roy, and J. F. Doherty, Covert communications using signal overlay, Advances in Adaptive Data Analysis, vol. 2, no. 3, pp , A. Roy, and J. F. Doherty, Improved signal analysis performance at low sampling rates using raised cosine empirical mode decomposition, Electronic Letters, vol. 46, no. 2, pp , Jan A. Roy, S.J. Briczinski, J.F. Doherty, and J. D. Mathews, Genetic algorithm based parameter estimation technique for fragmenting meteor head-echoes, IEEE Geoscience and Remote Sensing Letters, vol. 6, no. 3, pp July A. Roy, C.-H. Wen, J. F. Doherty, and J. D. Mathews, Signal feature extraction from microbarograph observations using the Hilbert-Huang transform (HHT), IEEE Transactions on Geoscience and Remote Sensing, vol. 46, no. 5, pp , May Conference Proceedings 1. A. Roy, and J. F. Doherty, Partial band jamming excision in WCDMA using raised cosine empirical mode decomposition, in Proc. Virginia Tech 2010 Symposium and Summer School, Blacksburg, VA, 2-4 Jun A. Roy, and J. F. Doherty, Covert communications using empirical mode decomposition, in Proc IEEE Sarnoff Symposium, Princeton, NJ, pp. 1-5, 30 Mar.-1 Apr A. Roy, and J. F. Doherty, Raised cosine interpolation for empirical mode decomposition, in Proc. 43rd Annual Conference on Information Sciences and Systems, 2009, CISS 2009, Baltimore, MD, pp , Mar

24 11 4. A. Roy, and J. F. Doherty, Empirical mode decomposition frequency resolution improvement using the pre-emphasis and de-emphasis method, in Proc. 42nd Annual Conference on Information Sciences and Systems,2008, CISS 2008, Princeton, NJ, pp , Mar Dissertation Outline This dissertation consists of two independent, yet related parts. The theoretical part of this dissertation comprising Chapters 3 through 6 involves introduction of a new signal analysis algorithm related to EMD followed by development of new applications and performance verification based on mathematical models. The next part, covering Chapters 7 and 8, describes remote sensing applications of signal analysis techniques using in-field measurements. Part I of this dissertation introduces a new signal analysis algorithm related to EMD that uses raised cosine interpolation called RCEMD. Theoretical development of this technique, development of mathematical tools to formalize the study of EMD performance, performance comparison of the two algorithms, and development of communications applications based on RCEMD including signal overlay, covert communications, and interference cancelation from spread spectrum signals are covered in this part. Chapter 2 introduces signal analysis concepts and techniques from a historical perspective. Techniques such as FT, STFT, wavelet decomposition and EMD are discussed. The EMD technique is described in some detail along with some algorithmic variations and applications. The chapter also includes a discussion on select time-frequency concepts that are used in later chapters. Chapter 3 introduces the RCEMD technique for signal analysis. In addition to algorithmic description of this new technique, a generalized mathematical framework is developed to study the performance of iterative signal analysis algorithms following the basic idea of EMD. Advantages of this new technique related to improved frequency resolution, relaxed sampling requirements and fewer computations are demonstrated using a combination of synthetically-generated and real-life signals.

25 12 Chapter 4 describes the pre-emphasis and de-emphasis technique to enhance the signal analysis quality of iterative algorithms like EMD. Improved frequency resolution for a specific configuration of constituent signal components is demonstrated using synthetic signal examples. Further, the effect of certain algorithmic parameters on signal analysis performance is demonstrated. Chapter 5 presents a new signal overlay technique using RCEMD. This technique frequency spectrum utilization for wireless communications by enabling opportunistic communications by a secondary user on the same frequencies as an existing primary user. Feasibility of this technique is demonstrated using computer simulations based on mathematical models of wireless channels and transceivers. A covert version of this technique using frequency-hopping (FH) technique is also described. Chapter 6 describes a new application of RCEMD to wideband interference suppression in wireless communications. Here, the problem of nonstationary interference affecting a widely used communication standard is considered and a solution based on RCEMD is formulated. Simulation study results analyzing the effectiveness of the technique are presented. Part II of this dissertation introduces new signal processing techniques for remote sensing applications. This includes development of periodic and sporadic feature isolation techniques using the RCEMD procedure developed in Part I for microbarograph observations, and a genetic algorithm based technique for accurate meteoroid fragment parameter estimation based on radar meteor head-echoes. Chapter 7 describes a novel feature extraction procedure using RCEMD applied to data recorded using sensors deployed to measure atmospheric pressure. The ability to isolate hurricane signature and extract diurnal and semi-diurnal atmospheric tide signals from the noisy raw data with greater precision than existing techniques is demonstrated. Chapter 8 introduces a new method to study meteoroid fragmentation using genetic algorithms to radar measurements. Radar returns from multiple, closelyspaced traveling particles result in an interference pattern, rendering signal analysis necessary for study of individual particle behavior. A method using genetic

26 13 algorithms is developed to estimate orbital parameters of such fragmenting meteoroids in this chapter. Chapter 9 discusses the important findings and results of this work and highlights some relevant open problems.

27 Chapter 2 Time-Frequency Analysis of Signals This chapter provides a brief presentation of the basic concepts related to time-frequency analysis of signals. It begins with a review of important time-frequency concepts such as analytic signals, monocomponent and multicomponent signals and instantaneous frequency. Next, various tools available to analyze a nonstationary signal are studied and their relative merits are compared. This is followed by a description of the EMD algorithm for signal analysis. Finally, an overview of the the developments in the field of EMD and its applications concludes this chapter. 2.1 Signal Analysis: Concepts In this section some basic concepts related to time-frequency analysis of signals are presented. We start by defining an analytical signal, then move on to instantaneous frequency of a signal, and finally discuss the classification of signals as monocomponent and multicomponent Analytical Signal The phase of a signal may be required in some cases, for example to determine its instantaneous frequency, a concept that will discussed in Section Thus, a proper definition of the phase is required. To properly define the phase ϕ(t) for a real signal f (t), Gabor [5] proposed an approach to suppress the amplitudes belonging to negative frequencies and multiply the amplitudes of positive frequencies by two. Following this

28 15 approach, the Gabor s time domain complex signal can be defined as follows where This yields z(t) = 2 1 F(ω)e ıωt dt (2.1) 2π F(ω) = 1 2π 0 z(t) = f (t) + i π P where P denotes the principal value integral defined as [6] β P α f (u)du = lim ε 0 + ξ ε α f (t)e ıωt dt. (2.2) f (τ) dτ, (2.3) t τ f (u)du + β ξ +ε f (u)du. (2.4) This class of complex functions satisfy the Cauchy-Riemann conditions for differentiation and are called analytic functions [7, 8] and thus z(t) is called analytical signal [9] Instantaneous Frequency The frequency of a stationary signal is well-defined following the Fourier approach. Generally, the frequency is defined as the number of oscillations per unit time of a physical field parameter such as displacement, current or electromagnetic waves. But for nonstationary signals commonly encountered in radar, seismic and communications applications this definition becomes ambiguous [9] due to the time-varying nature of the spectral characteristics of the signal. This leads to the notion of instantaneous frequency of a signal. Gabor [5] was the first to introduce a complex analytic signal, which was later employed to define instantaneous frequency as the time derivative of the phase of a signal by Ville [10]. This definition works well for monocomponent signals. However, it fails to produce physically reasonable results for multicomponent signals. Cohen [11] also used the concept of instantaneous frequency, as well as instantaneous bandwidth to explain what a multicomponent signal is. He defined the instantaneous frequency of a

29 16 monocomponent signal as an average of the frequencies that exist at a particular time, and the instantaneous bandwidth as the spread of the frequencies about the average for that time. For a mathematical definition we reconsider (2.3) where the imaginary part is the Hilbert transform f (t) of the signal f (t) [12]. Then (2.3) can be written as z(t) = f (t) + f (t) (2.5) or in the exponential form z(t) = a(t)e ıϕ(t), (2.6) where amplitude a(t), and the phase ϕ(t) are defined as a(t) = f 2 (t) + f 2 (t), and ϕ(t) = arctan f (t) f (t) (2.7) respectively. Therefore, the instantaneous frequency of the signal x(t) is IF(t) = dϕ(t) dt = f (t) f (t) ḟ (t) f (t) f 2 (t) f 2. (2.8) (t) The above definition captures the notion of instantaneousness in nature and fits our intuitive expectation of the instantaneous frequency concept. It is encouraging that when the definition is applied to a sinusoidal signal, the obtained instantaneous frequency is exactly the frequency of the signal Monocomponent and Multicomponent Signals Although several definitions of a multicomponent signal exist in literature, the one proposed by Boashash [4] is the most widely accepted and therefore adopted in this work. Accordingly, an analytical signal is referred to as a monocomponent signal if its instantaneous frequency accurately represents the frequency modulation of the signal, and if the signal is single-valued and invertible (so that the inverse function of the instantaneous frequency exists). An asymptotic signal z(t) is referred to as multicomponent if there exists a finite number N of monocomponent signals z i (t), i = 1, 2,...,N, such that the relation z(t) = N z i (t) holds for all values of t for which z(t) is defined, and this i=1

30 17 decomposition is meaningful. 2.2 Signal Analysis: Methods In studying time series, several methods have been developed and used by researchers and practitioners. The ones that are frequently used include FT, STFT, wavelet transform, Wigner-Ville representation, adaptive chirplet decomposition and EMD (which is a part of the Hilbert-Huang transform). In the following, basic information about these methods is presented. Each method has its own advantages and disadvantages depending on the application at hand Fourier Analysis The most commonly used method has been Fourier analysis. It reveals the frequency content of a signal by decomposing it into sinusoids of different frequencies. Fourier series is used for periodic signals, whereas for nonperiodic signals there is FT Fourier Series Fourier stated that any periodic signal f (t) of period T (i.e., f (t) = f (t + T )) can be expressed as f (t) = a k=1 a k cos(kω 0 t) + k=1 b k sin(kω 0 t) (2.9) where ω 0 = 2π T is the fundamental angular frequency in radians per second. The coefficients of the sine and cosine terms (Fourier coefficients) are obtained as follows: a 0 = 2 T a k = 2 T b k = 2 T T /2 T /2 T /2 T /2 T /2 T /2 f (t)dt f (t)cos(kω 0 t)dt (2.10) f (t)sin(kω 0 t)dt, k = 1, 2,...,.

31 Fourier Transform While FT of a square integrable function f ( f L 2 (R)) 1 has already been defined (2.2), its inverse can be written as follows f (t) = 1 2π F(ω)e ıωt. (2.11) Analyzing signals by FT, called spectral analysis, is a standard technique to obtain information about a periodic signal. The discrete Fourier transform (DFT) extends the use of FT to sampled time series data. DFT can be computed in a fast way using an algorithm called the butterfly algorithm [13] that computes the coefficients recursively. While FT gives valuable information about frequencies in a seismogram, it is not possible to have any information on temporal location of those frequencies. Therefore, it is suitable only for stationary signals. To overcome this problem STFT was proposed Short-Time Fourier Transform The idea behind STFT is to cut the original signal into segments of smaller duration and applying FT to obtain the frequency components of each slice. The functions obtained by this crude slicing are not periodic in general and FT will interpret the jumps at the boundaries as discontinuities and will introduce higher order harmonics to fit the waveform. To avoid these, the concept of windowing has been introduced. Instead of localizing by means of rectangular function, a smooth window function, which is close to unity near origin and decays towards zero at the edges, is used. For this reason STFT is sometimes called windowed FT. Any square integrable function may be used as a window, but certain criteria should be met for good performance. The main property of a good window is its good localization in both time and frequency domains. Some windows are favorable such as Hamming, Hanning, Bartlett, Blackman, Kaiser, Gaussian 2 [14], and the discrete prolate spheroid [15]. The reason for the use of these windows is that they have functional forms and their FT is concentrated around ω = 0. The window 1 In mathematics, a square integrable function is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite, i.e., f (x) 2 dx <. 2 Note that STFT using Gaussian window has the special name Gabor transform, and is known to optimize the Heisenberg s uncertainty principle.

32 19 in the time-domain is referred to as the time window and its FT as the spectral window. The signal is multiplied by one of the window functions g(t b), where g(t) represents the functional form of the window and is nonzero only in a finite region around time b. Then the FT of f (t)g(t b) is taken, and the window is moved to a different location to repeat the operation. The method can therefore be represented by S f (ω,b) = 1 2π f (t)g(t b)e iωt dt. (2.12) The signal can be reconstructed from its transform by the formula f (t) = 1 2π S f (ω,b)g(t b)e iωt. (2.13) The fundamental problem with STFT is that it has fixed resolution along both time and frequency axes. As argued by Chui [16], since frequency is directly proportional to number of cycles in a specific time interval, a narrow time window is required to locate high-frequency phenomena and a wide time-window is necessary for more thorough investigation of low frequency phenomena. As a result, the STFT is not well suited for analysis of signals that may have both low and high frequency components Wavelet Transform Wavelet analysis has emerged as a powerful tool to analyze a signal with particular effectiveness for nonstationary signals. A wavelet is a small wave with finite energy, which has its energy concentrated in time or frequency to serve as a basis function for the analysis of transient phenomena. While being similar to Fourier analysis as far as complex expansions are concerned, it differs by decomposing a signal into a series of local basis functions called wavelets. Each wavelet is located at a different position of the time axis and is local in the sense that it decays to zero away from its center. The terminology wavelet was first introduced, in the context of a mathematical transform by Grossmann and Morlet [17]. The wavelet transform is a two-parameter expansion of a signal in terms of a particular wavelet basis function or mother wavelet. Temporal analysis is performed with a contracted high frequency version of the prototype wavelet,

33 20 while frequency analysis is performed with a dilated, low frequency version of the same wavelet Continuous Wavelet Transform (CWT) In spite of its name, the continuous wavelet transform (CWT) is a discrete process in implementation. Its continuity comes from the flexibility of the set of scales and positions on which it operates. Unlike the discrete wavelet transform the CWT can operate at every scale. The CWT is also continuous in terms of shifting: during computation, the analyzing wavelet is shifted smoothly over the full domain of the analyzed function. Let ψ(t) be the mother wavelet. All other wavelets are obtained by scaling and translating ψ(t) as follows [18]: ψ a,b = 1 a ψ ( t b a ). (2.14) Let f (t) be a square integral function of time t. The CWT of f (t) is defined as W ψ f a,b = f (t)ψa,b (t)dt = 1 a ( ) t b f (t)ψ dt (2.15) a where a, b R, a 0 and. denotes complex conjugate. The normalizing factor 1 a is used to keep the energy level the same for different values of a and b. In CWT nomenclature a is called scale parameter and b is called translation parameter. When a is increased the wavelet ψ a,b (t) is dilated and when b is varied, the signal is translated in time. After the parameters a and b are selected, the basis or mother wavelet is stretched or dilated according to the as and translated according to the bs to produce a family of wavelets ψ a,b (t). The wavelets ψ a,b (t) are multiplied by f (t) at different scales and different translations. The CWT coefficients are obtained by summing the product showing the correlation between the signal and the wavelet functions. The original time domain signal can be reconstructed through the inverse wavelet transform f (t) = 1 2πC ψ W ψ f a,b a 2 ψ a,b (t)dadb (2.16)

34 21 where C ψ = ˆψ(ω) 2 dω (2.17) ω and ˆψ(ω) is the FT of ψ(t). A wavelet analysis is often called a time-scale analysis rather than a time-frequency analysis because the analysis function ψ(t) is scaled by a. Among these wavelets are orthogonal, biorthogonal and harmonic wavelet systems. Orthogonal wavelets decompose signals into well-behaved orthogonal signal spaces. In 1988, Daubechies introduced a class of compactly-supported orthogonal wavelets with growing smoothness for increasing support. Mallat [19] and Meyer [20] presented the theory of multiresolution analysis Discrete Wavelet Transform (DWT) The discrete wavelet transform (DWT) is more efficient in terms of computational effort than CWT because of the dyadic nature of the scales and positions. In contrast to CWT that uses a flexible frequency range, DWT uses frequency only in the octave band. Although this later method is computationally less expensive, it does not give a very precise result to interpret, and is used mostly in signal compression. Let f [n] be the discrete signal obtained by a low-pass filtering of a continuous time signal and uniform sampling at intervals N 1. Its DWT can only be calculated at scales N 1 < s < 1. It is calculated for s = a j, with a = 2 1/v, which provides v intermediate scales in each octave [2 j,2 j+1 ). Let ψ(t) be a wavelet with a support included in [ K/2,K/2]. For 1 a j NK 1, a discrete wavelet scaled by a j is defined by ψ j [n] = 1 a j ψ ( n a j ). (2.18) To avoid border problems we treat f [n] and the wavelets ψ j [n] as periodic signals of period N. The discrete wavelet transform can then be written as a circular convolution with ψ j [n] = ψ j [ n] [21]: W f [n,a j ] = N 1 f [m]ψ j [m n] = f ψ j [n]. (2.19) m=0

35 22 This circular convolution is calculated with the fast Fourier transform algorithm that requires O(Nlog 2 N) operations. An advantage of the wavelet transform is that although there are numerous timefrequency transformations available, the wavelet transform is uniquely capable of adaptively adjusting to the Heisenberg s uncertainty principle. In essence, the wavelet transform concedes that arbitrarily good resolution in both time and frequency is impossible. Thus, the transform optimizes its resolution as needed. It provides good resolution at high dilations or low frequencies, while sacrificing time resolution to satisfy the uncertainty principle. In the time domain, the transform has good resolution at high frequencies in order to identify signal singularities or discontinuities. A major disadvantage of wavelet transform, as compared to EMD, is that its performance depends upon the choice of mother wavelet. Although there are wavelets that have good time or frequency resolution, there is no wavelet that has uniformly superior performance for all applications. The choice of mother wavelet depends on a priori knowledge of the frequency content of signal to be analyzed. 2.3 Bilinear Time-Frequency Distribution The STFT and CWT are based on the concept of finding the similarity between the signal and the analyzing functions and have the disadvantage that Heisenberg s uncertainty principle restrains their time-frequency resolution. Another approach which in several cases gives significantly better results is the bilinear (quadratic) time-frequency analysis by means of time-frequency distributions. A comprehensive review [22] provides an overview of time-frequency distributions. This section addresses a specific subset of t-f distributions belonging to Cohen s class. These are the time-shift and frequency-shift invariant t-f distributions. For these distributions, a time shift in the signal is reflected as an equivalent time shift in the t-f distribution, and a shift in the frequency of the signal is reflected as an equivalent frequency shift in the t-f distribution. The spectrogram, the Wigner-Ville distribution (WVD) and the reduced interference distributions (RID) all have this property. Different distributions can be obtained by selecting different kernel functions in the Cohen s class. Performance comparison of several time-frequency distributions in terms of resolution is provided in [4].

36 The Wigner-Ville Distribution The WVD has been employed as an alternative to overcome the liabilities and limitations of the spectrogram. It was first introduced in the context of quantum mechanics [23] and revived for signal analysis by Ville [10]. It provides a high-resolution representation in time and in frequency for a nonstationary signal such as chirp. In addition, WVD has the important property of satisfying the time and frequency marginals in terms of the instantaneous power in time and energy spectrum in frequency. However, its energy distribution is not nonnegative and it often possesses severe cross-terms, or interference terms, between components in different t-f regions, potentially leading to confusion and misinterpretation. The WVD of real signal f (t) is defined as W z (t,ω) = ( z t + τ ) ( z t τ ) e jωτ dτ (2.20) 2 2 where z(t) is the analytic associate of f (t) (see Section 2.1.1). This process is the correlation of the signal with itself. We may interpret this equation as the computation of a local autocorrelation function at each time instant, t, followed by the evaluation of its Fourier transform. This leads to a local power spectral density at each time instant. In practice, only one realization of the process is available and this forces us to ignore the implicit expectation operation in autocorrelation. The Wigner-Ville transform is optimally localized in the time domain for Dirac signals, and in the frequency domain for linear chirps. Equation 2.20 defines time-frequency distributions that are quadratic (bilinear) in the signal z(t). This implies that if z(t) consists of two components z 1 (t) and z 2 (t), then the quadratic time-frequency representation will not only include the two components but also their cross product z 1 (t)z 2 (t). The extra terms are known as cross-terms, or artifacts and they are a major drawback of multicomponent signal time-frequency analysis using quadratic time-frequency distributions. It can be shown that the WVD of the signal z(t) = z 1 (t) + z 2 (t) is given by W {z1 (t)+z 2 (t)}(t,ω) = W {z1 (t)}(t,ω) +W {z2 (t)} (2.21)

37 24 + 2Re ( z 1 t + τ ) ( z 2 t τ ) e jωτ dτ 2 2 (2.22) The last term in this expression is the cross-term. The cross-terms in the WVD are oscillating contributions located midway between the components. Note that for a multicomponent signal with N components, there will be N(N 1)/2 cross-terms in the signal WVD. As the number of cross-terms increases quadratically their interpretation becomes impossible. Although this cross-term interference is a good indication that the signal is multicomponent, it reduces the resolution of the WVD and thus is in general undesirable Reduced Interference Distributions In order to suppress the cross-terms in the WVD of multicomponent signals, the WVD can be convolved with a smoothing function g(t,ω) which is commonly referred to as kernel. That is, C z (t,ω) = W z (t,ω) g(t,ω) (2.23) where the symbol denotes two-dimensional convolution. The set of all bilinear distributions of 2.23 is called Cohen s class. With this general approach an infinite number of time-frequency representations can be generated by appropriately selecting a kernel g(τ, θ). Obviously, the kernel for WVD is g WV D (τ,θ) = 1. (2.24) Then the Cohen s class distributions for a kernel g(τ,θ) can be written as C z (t,ω) = 1 4π 2 A(τ,θ)G(τ,θ)e j(τt+θω) dτdθ, (2.25) where G(τ,θ) is the Fourier transform of the kernel g(t,ω) and A(τ,θ) is the symmetrical ambiguity function defined as the Fourier transform of the WVD A(τ,θ) = ( z t + τ ) ( z t τ ) e jθτ dτ. (2.26) 2 2

38 25 These distributions are also called reduced interference distributions (RIDs). One of the first RIDs was the Choi-Williams distribution (CWD). Choi and Williams [24] defined a two-dimensional Gaussian-shaped kernel in Doppler-lag domain as: g(θ,τ) = e (θτ)2 /σ (2.27) where σ is a smoothing parameter that controls the kernel spread in the ambiguity domain, and so controls the amount of cross-terms suppression in the time-frequency domain. The σ parameter may be varied over a range of values to obtain different tradeoffs between cross-term suppression and auto-term time-frequency resolution, since the kernel, while reducing the cross-term, increases smearing in the time-frequency domain. Large computational requirement is another serious drawback of this approach. 2.4 Time-Frequency Distribution Illustration Before moving on to the EMD technique we study the performance of the different spectral and time-frequency techniques considered here using two examples: Example 1: Monocomponent signal We again go back to the linear frequency modulated signal first studied in Chapter 1. Figure 2.1 shows the Fourier spectrum, spectrogram, wavelet spectrum and the WVD spectrum for the monocomponent signal. While no information about frequency variation of the signal is available from the Fourier transform, the other spectrums are successful in conveying the time varying frequency of the signal to varying degrees. The uniform resolution of the STFT (spectrogram) has been mentioned before. Moreover, this example clearly shows the variable frequency resolution of the wavelet spectrum. Finally, the excellent resolution of the WVD is not surprising due to its optimality for linear frequency modulated signals. Example 2: Multicomponent signal Here we consider a multicomponent signal consisting of two linearly frequency modulated signals with intersecting frequencies. We show the corresponding spectral representations in Fig This example represents a particularly difficult problem due to

39 26 a. b. amplitude frequency frequency time c. d. frequency frequency time Figure 2.1: Spectral representations for a monocomponent signal introduced earlier in Chapter 1 (see Fig. 1.1 for time-series): a. Fourier spectrum, b. Spectrogram, c. Wavelet spectrum, and d. WVD spectrum. time intersecting frequencies. Notice that the amplitude spectrum from Fourier analysis is basically useless in identifying signal components. Due to the poor frequency resolution of the STFT, the two signals are virtually indistinguishable in the spectrogram. The trade-off between time and frequency resolutions in the wavelet spectrum is evident. Finally, the appearance of ghost frequencies or artifacts in the WVD spectrum can be directly attributed to the cross-terms due to multicomponent signal Hilbert-Huang Transform (HHT) To obtain meaningful instantaneous frequency, some restrictions should be applied to the data [5, 9, 25]. Essentially, the signal has to be monocomponent, meaning that there should be no riding waves. Therefore, the signals that can be studied by Hilbert transform are limited to simple free vibrations. The limitation of the data makes this transform non-applicable to multicomponent signals. However, pre-processing of the signal by band-pass filtering or other appropriate methods to separate the various com-

40 27 a. b. amplitude frequency frequency time c. d. frequency frequency time Figure 2.2: Spectral representations for a multicomponent signal: a. Fourier spectrum, b. Spectrogram, c. Wavelet spectrum, and d. WVD spectrum. time ponents [26] expands its applicability to multicomponent signals. Huang, et. al., [27] introduced the concept of empirical mode decomposition (EMD) to make the signal ready for Hilbert transform analysis. The EMD and Hilbert transform together are referred to as Hilbert-Huang transform (HHT). HHT is a relatively new technique that analyzes transient time-domain signals. It has shown great utility in time-frequency analysis of dispersive, nonlinear and nonstationary signals and systems. The transform uses the EMD, with which the signal is decomposed into a series of constituents. By applying the Hilbert transform to each of the constituents we get a set of analytical signals representing the input signal. The HHT calculates the instantaneous frequency of each constituent and presents the result as a time-frequency analysis in a Hilbert spectrum plot. The signal analysis step of HHT, the EMD, is the subject of this work and the following sections and chapters focus on this technique and its variants.

41 Empirical Mode Decomposition The EMD is an adaptive signal-dependent decomposition with which any complicated signal can be decomposed into a series of constituents. Adding all the extracted constituents together reconstructs the original signal without information loss or distortion. Many methods exist that analyze signals simultaneously in the time and frequency domains, some of which were highlighted in Section 2.2. These methods are based on the expansion of the signal into a set of basis functions that are defined by the method. The concept of EMD is to expand the signal into a set of functions defined by the signal itself. These decomposed constituents are called intrinsic mode functions (IMF). Signal adaptive decomposition by means of Principal Component Analysis (PCA) [28] also expands the signal into a basis defined by the signal itself. PCA differs from EMD in that it is based on the signal statistics, while EMD is deterministic and is based on local properties. The EMD process allows time-frequency analysis of transient signals for which Fourier based methods have been unsuccessful. Whenever we use the Fourier transform to represent frequencies we are limited by the uncertainty principle. For infinite signal length we can get exact information about the frequencies in the signal, but when we restrict ourselves to analyze a signal of finite length there is a bound on the precision of the frequencies that we can detect. The instantaneous frequency represents the frequency of the signal at one time, without any information of the signal at other times. A problem with using instantaneous frequency is that it provides a single value at each time. A multicomponent signal consists of many intrinsic frequencies and this is where the EMD is used, to decompose the signal into its IMFs, each with its own instantaneous frequency, so that multiple instantaneous frequencies of the signal components can be computed. Another advantage of EMD is that it results in an adaptive signal-dependent time-variant filtering procedure able to directly extract signal components which significantly overlap in time and frequency [29]. Moreover, the physical meaning of the intrinsic processes underlying the complex signal is often preserved in the decomposed signals. This is mainly due to the fact that the results are not influenced by predetermined bases and/or subband filtering processes. EMD represents a totally different approach to signal analysis. EMD is an adaptive decomposition with which any complicated signal can be decomposed into a series

42 29 of constituents. EMD is an analysis method that in many respects gives a better understanding of the physics behind the signals. Because of its ability to describe short time changes in frequencies that cannot be resolved by Fourier spectral analysis, it can be used for nonlinear and nonstationary time series analysis. Each extracted signal admits well-defined instantaneous frequency. The original purpose for the EMD was to find a decomposition that made it possible to use the instantaneous frequency for timefrequency analysis of nonstationary signals. In the following sections we explore this technique in more detail Procedure As discussed above, the elementary AM-FM-type signal components that are produced by the EMD procedure are called IMFs in literature. The original researchers outlined two conditions that must be satisfied by an extracted component to be declared an IMF [27]: 1. The number of extrema and the number of zero crossings must differ at most by one. 2. The mean value of the envelopes defined by the local maxima and the local minima should be zero at any point, meaning that the functions should be symmetric with respect to the local zero mean. Each of these IMFs is extracted by a process called sifting. The goal of sifting is to remove the higher frequency components until only the low frequency components remain. Given a signal x(t) the sifting procedure divides it into a high frequency detail, d(t), and the low frequency residual (or trend), m(t), so that x(t) = m(t) + d(t). This detail becomes the first IMF and the sifting process is repeated on the residual, m(t) = x(t) d(t). After K iterations of the sifting procedure the input signal can be represented as follows x(t) = K y k (t) + m K (t) (2.28) k=1 where y k (t), k = 1,...,K represent the IMFs and m K (t) is the residual, or the mean trend, after K sifting iterations. The effective algorithm of EMD can be summarized as follows [29]:

43 30 1. Identify all extrema of x(t). 2. Interpolate between minima (respectively maxima), resulting in the envelope e min (t) (respectively e max (t)). 3. Compute the mean m(t) = (e min (t) + e max (t))/2. 4. Extract the detail d(t) = x(t) m(t). 5. If d(t) satisfies all IMF conditions, then set y 1 (t) = d(t), the first IMF, else repeat above steps with d(t). 6. Evaluate the residual m 1 (t) = x(t) y 1 (t). 7. Iterate on the residual m 1 (t). Steps 1 through 4 may have to be repeated several times until the detail d(t) satisfies the IMF conditions. Practical methods to determine if d(t) satisfies the IMF conditions, also called stopping criteria, are discussed next. In the original work [27] the sifting procedure for a particular IMF stops when the normalized difference in the extracted signal between two consecutive iterations is smaller than a pre-determined threshold ε. A new stopping criterion was suggested in [30] where the iterations stop when the envelope mean signal is close enough to zero ( m(t) < ε, t). The reason for this choice is that forcing the envelope mean to zero will guarantee the symmetry of the envelope and the correct relation between the number of zero crossings and number of extremes that define the IMF. A modified version of this stopping criterion with two thresholds was introduced in [29], along with a discussion of typical threshold values. Yet another stopping criterion was introduced in [31] where sifting is stopped when the number of extrema and zeros crossings remains constant over some pre-determined number of iterations. The latter is the most commonly used criterion. An example is presented next to show the algorithmic steps pictorially. Example: Decomposition of Tones The EMD algorithm is demonstrated pictorially using a simple combination of tones of the form x(t) = s 1 (t) + s 2 (t) (2.29)

44 31 where the two tones are s 1 (t) = A 1 cos(2π f 1 t + ϕ 1 ), s 2 (t) = A 2 cos(2π f 2 t + ϕ 2 ) (2.30) and the symbols have their usual meanings with f 1 > f 2. Intermediate signals generated by the algorithm are shown in Fig Fig. 2.3a shows the original signal followed by the positions of the positive and negative extrema (also called maxima and minima) in Fig. 2.3b. Smooth envelopes are drawn through the identified maxima and minima using cubic spline interpolation. These curves, denoted by e max (t) and e min (t) in the algorithm listed above, are shown in Fig. 2.3c along with their mean, m(t). We omit the original signal in this figure for clarity. The mean of the two envelopes, represented by the dashed curve, is a slowlyvarying signal that resembles the smaller tone, s 1 (t). The mean signal and the detail, d(t), obtained by subtracting the mean from the original signal are shown in the two panels of Fig. 2.3d, superimposed on the two tones, s 2 (t) and s 1 (t), respectively. The resemblance between the mean and the detail signals and the original tones is obvious at this stage. The process of computing the envelopes, mean and detail signals continues by iterating on the detail signal. The result after five iterations, shown in Fig. 2.3e indicates good decomposition quality Algorithmic Variations Several variations of the original algorithm have been proposed by researchers either to improve the performance or to simplify the implementation. Some of the ways by which the algorithm has been modified include different interpolation methods, new ways of identifying IMFs, extending the algorithm to two dimensions and optimization-based decomposition. We discuss some of these modifications here. In order to construct smooth envelopes through the respective extrema, on each subinterval x(t), t k t t k+1, where the k th and k + 1 th extrema are located at t k, t k+1, an interpolant to the given values and certain slopes at the two end points is devised. Between any two neighboring end-points x(t k ) and x(t k+1 ), x(t) is a polynomial. Neighboring polynomials match in value, and derivatives across their common end-points. The interpolation to produce envelopes from the extrema points can be performed in dif-

45 32 2 x(t) = s 1 (t) + s 2 (t) (a) 2 Positive and negative extrema (b) Figure 2.3

46 33 2 e min, e max and m(t) (c) 1.5 IMF1 and s 1 (t) IMF2 and s 2 (t) (d) Figure 2.3

47 34 1 IMF1 and s 1 (t) IMF2 and s 2 (t) (e) Figure 2.3: The major EMD algorithmic steps are shown here for a synthetic two-tone signal. Starting from the top the sub-figures show (a) the original signal; (b) the maxima and minima locations; (c) smooth envelopes constructed through the maxima and minima, and the mean envelope; (d) the mean and detail signal after one iteration; (e) the same signals after five iterations. ferent ways. The original algorithm [27] uses the natural cubic spline. References [30] and [32] explore the use of Hermite interpolation in EMD and report performance improvement. The use of B-splines that leads to simpler analytical description of performance of the EMD algorithm was introduced in [33]. A new interpolant called rational spline that possesses variable, controllable tautness is discussed in [34,35] as a replacement for cubic splines. While guidance for appropriate parameter selection based on optimization criterion is provided, although with accompanying tradeoffs, no universal optimum parameter setting has been reported. Practical signals suffer from intermittency, where a component at a particular time scale either comes into existence or disappears from the signal completely. This leads to the situation called mode mixing where an IMF has components of different frequencies. This problem has been addressed in [36] based on a change in the choice of extrema and

48 35 in [37] where the use of masking signals is explored. Another solution has been introduced in [38] where the authors have introduced a noise assisted data analysis technique called ensemble EMD which is essentially a controlled repeated experiment to produce an ensemble mean for nonstationary data. A further variation in this direction has been introduced in [39], where the authors point out the problem of residual noise in ensemble EMD, and propose using pairs of complementary noise sequences to reduce the residual noise in the decomposition. Reference [31] introduced a confidence limit based stopping criterion choice to combat mode mixing. Several works have focussed on improving the performance of EMD by changing the extrema sampling points or the knots for interpolation [40 42]. A new extrema identification based on the derivative of the signal was introduced in [43], which is predicated on the notion that for more accurate filtering the signal should be sampled at the points where the fast oscillating signal has its extrema, and differentiating emphasizes the faster oscillating signal relative to the slower one, thereby reducing the error between the estimated and ideal extrema locations. It should be noted that the idea of differentiating the signal to improve decomposition quality, although in a slightly different fashion, has also been advocated in [44]. An appropriate definition of IMF that leads to particular benefits has been the focus of certain researchers. This has led to a bandwidth-based criterion for IMF [45] and the alternate definition in [46] where a particular constituent is accepted or rejected as IMF based on the cross-correlation with the original signal. Variations of the technique based on optimization include a parabolic partial differential equation-based method for mean envelope detection [47], an optimization technique that provides control over the resolution by tuning of certain parameters [48] and a new constrained optimization based technique that obviates the need for a stopping criterion [49]. Since a majority of interpolants operate upon global data, the effect of abrupt data termination at the boundary can propagate into the interior of the output signal. Therefore, a solution to the boundary effect is highly desirable. All published research on this topic involves data extension by different means. While [27,50] recommend adding characteristic or typical waves by deriving parameters from the actual signal ends, they differ slightly in their actual implementation. Signal mirrorizing is adopted in [29]. A linear extrapolation-like approach is proposed in [51]. Bivariate or complex EMD is another area of active research and has seen contributions by several researchers. Some implementations of bidimensional EMD include

49 36 decomposition using finite elements [52], that based on Delaunay triangulation [53], one based on properties of the complex field [54] and finally, one that adapts the the rationale underlying the EMD to the bivariate framework [55, 56]. Bivariate EMD applied to image compression is the subject of [57] while [58] describes image texture analysis using bidimensional EMD based on radial basis function for surface interpolation Theoretical Developments Although EMD essentially remains algorithmic in nature, recently some researchers have tried to explore the theoretical aspects of the technique. The authors of [59] were perhaps the first researchers to examine the theoretical aspects of EMD. They applied white noise to EMD and concluded that EMD is effectively a dyadic filter (a dyadic filter provides octave band frequency decomposition of the input). This has led to the filterbank interpretation of the algorithm [60]. Although the dyadic filter nature of the EMD algorithm has been quoted by several authors subsequently, it must be remembered that the algorithm behaves so only when presented with white noise-like broadband signal. A mathematical analysis of the signal decomposition performance of EMD for a combination of tones is developed in [41, 61]. A research problem that has received considerable attention recently is that of sampling rate. Various researchers have tried to answer questions regarding the minimum sampling rate required for successful decomposition, the effect of sampling on decomposition quality and finally, they have tried to improve the performance of the algorithm under low sampling rates. A sampling limit for the algorithm was derived in [62], both empirically and theoretically. The authors also concluded that the algorithm performs poorly at low sampling rates, near Nyquist rate. The effect of sampling on decomposition quality was studied in [63]. Finally, [40] introduces a technique based on Fourier interpolation to improve the low sampling rate performance of EMD Applications A representative list of applications of the EMD algorithm for signal analysis in the fields of geophysics, structural safety and visualization is presented in [64]. Many more applications have emerged recently. Some of these include newborn EEG seizure detection [65], discrimination between normal and laryngeal pathological speech sig-

50 37 nals [66] and detection of synchronization in EEG [67] in the medical field; speckle interferometry in optics [68]; line simplification in cartography using points of extreme curvature [69]; fusion of visual and thermal images for enhanced biometric authentication [70]; antijamming techniques for global positioning system (GPS) signals [71] and extraction of micro-doppler signature in Doppler radars [72] in the fields of communications and radar. Although the intent behind this short list of EMD applications is to convey the varied nature of its applications, it must be noted that it is by no means exhaustive.

51 Part I Signal Analysis using Empirical Mode Decomposition: Theoretical Developments and Communication Examples via Mathematical Modeling

52 Chapter 3 Raised Cosine Empirical Mode Decomposition The empirical mode decomposition (EMD) is a relatively new method to decompose multicomponent signals that requires no a priori knowledge about the components. In this chapter a modified algorithm using raised cosine interpolation is proposed with the associated title of raised cosine empirical mode decomposition (RCEMD). The decomposition quality of our developed technique is controllable via an adjustable parameter. This results in improved performance including faster convergence or lower final error, than the original technique, under different conditions. An efficient fast Fourier transform (FFT) based implementation of the proposed technique is presented. The signal decomposition performance of the new algorithm is demonstrated by application to a variety of synthetic and real-life multicomponent signals and a comparison with EMD algorithm is presented. Computational complexity of the two techniques is compared next. Finally, signal decomposition quality improvement at low sampling rates due to RCEMD is demonstrated in the final section. 3.1 Introduction In this chapter we describe a modified EMD technique that uses raised cosine interpolation. This algorithm, called raised cosine empirical mode decomposition (RCEMD), is introduced in Section 3.2. Here the cubic spline interpolation step of the original algorithm is replaced by raised cosine interpolation. The adjustable roll-off factor of

53 40 the raised cosine pulse allows the user to adapt the performance of the filter according to the nature of the composite signal. A windowed version of this technique is described in Section 3.3. In the proposed technique raised cosine interpolation using a large roll-off factor is applied to small data segments at a time. The interpolation filter design procedure based on local signal properties is described. The local operation of this technique results in improved interpolation quality than the cubic spline interpolation used in EMD, resulting in improved signal decomposition. The frequency domain implementation of the raised cosine filter is used for reduced computation complexity. Signal separation performance comparison between the original and proposed algorithms is studied in Section 3.4 for both synthetically generated and real-world signals. A variety of synthetic composite signals including combination of pure tones, frequency modulated components and trigonometric functions are considered. A performance metric described in Section is used to compare the performance of the algorithms where relevant. Section 3.5 addresses the issue of computational complexity of the original EMD algorithm and RCEMD applied to bicomponent signals. Section 3.6 describes the poor EMD signal separation quality at low sampling rates and the performance improvement due to RCEMD. Finally, a summary of the findings of this research and concluding remarks constitute Section Raised Cosine Interpolation The choice of cubic spline interpolation in EMD has been popular due to its reasonable performance and availability of computationally efficient software routines. Here a new algorithm using Nyquist pulse interpolation is introduced. In communications theory Nyquist s condition for distortionless transmission of a bandlimited signal is that [73] p(0) = 1 p(nt ) = 0, n = ±1,±2,... (3.1) where p(t) is a signalling pulse and T is the time duration between successive symbols. This condition guarantees that a sequence of pulses sampled at the optimum, uniformly spaced sampling instants, n = 0, ±1, ±2,... will have zero intersymbol interference (ISI). Nyquist showed that pulses satisfying a vestigial sideband criterion, namely, that the

54 41 pulse spectrum has odd symmetry about the corresponding ideally bandlimited spectrum band edge, will have this property. There are an infinite number of such pulses corresponding to different vestigial sidebands (see [74,75] for Nyquist pulse examples). Perhaps the most widely employed Nyquist pulse is the raised cosine pulse (of which the sinc pulse is a special case). Raised cosine interpolation has several advantages: 1. A finite impulse response (FIR) filter realization of the raised cosine filter simplifies hardware implementation. 2. Use of fast Fourier transform (FFT) ensures computationally efficient implementation. 3. Frequency resolution of the EMD technique can be controlled via external parameter. The time and frequency domain expressions of the raised cosine pulse are [76] h(t) = sinc ( πt ) cos T ( ) πβt T T ; f ( ))] 1 β 2T T H( f ) = 2 [1 + cos( πt β f 1 β 2T 0 ; otherwise 1 4β 2 t 2 T 2 (3.2) ; 1 β 2T < f 1+β 2T. (3.3) The roll-off factor, β, is a measure of the excess bandwidth of the filter, i.e. the bandwidth occupied beyond the Nyquist bandwidth of 1/2T. Its value varies between 0 and 1. As one increases the value of β, the pass-band in the frequency-domain increases and there is a corresponding decrease in the time-domain ripple level. This shows that the excess bandwidth of the filter can be reduced, but only at the expense of an elongated impulse response and this facilitates control over the performance of the interpolation scheme by the user. Figure 3.1 shows the raised cosine pulse in the time and frequency domains for several roll-off factors. The EMD algorithm uses cubic spline interpolation. This is replaced by the raised cosine interpolation here and the modified algorithm is called raised cosine empirical mode decomposition (RCEMD). To explain this procedure we refer to the EMD algorithm in Section Step two of the algorithm requires evaluating the envelope

55 β=0 β=0.1 β=0.5 β=1 0.4 h(t) T 2T T 0 T 2T 3T t (a) Time-domain RC pulse 1 β=0 β=0.1 β=0.5 β=1 H(f) /T 1/2T 0 1/2T 1/T f (b) Frequency-domain RC pulse Figure 3.1: Time-and frequency-domain raised cosine pulses for several roll-off factors.

56 43 values at intermediate points between successive extrema. In the original method the cubic spline constructed of piecewise third-order polynomials is used. In the modified method this is achieved by convolving the zero-padded sequence containing the extrema with the raised cosine pulse. This can be implemented using an FIR filter in hardware. The simplest member of this family, the sinc pulse (β = 0), has the added advantage of ideal low-pass frequency characteristics, for an infinite length sequence. However, a consequence of the ideal frequency behavior of the sinc pulse is that a filter implementing the sinc pulse is extremely sensitive to non-uniformity in sampling, resulting in poor performance relative to the cubic spline. The sinc filter is highly sensitive to non-uniform sampling points due to slow decay of the pulse. The rate of decay of the tails of the raised cosine pulse increases with the roll-off factor and this makes it less sensitive to sampling point errors [76]. However, increasing the roll-off factor also has the effect of increased filter bandwidth, thereby reducing its frequency resolution. So, to summarize, increasing the roll-off factor 0 through 1 has two conflicting consequences: it decreases the sensitivity of the filter to sampling point errors, but simultaneously reduces its frequency resolution. In section we introduce a two-tone signal model to study the performance of the two interpolating techniques, where it has been previously shown [61] that the uniformity of the spacing between the sampling points (local maxima or minima) is determined by the ratio A 2 A 1 f 2 f1 Γ, where A 1, A 2, f 1 and f 2 represent the amplitudes and frequencies of the two tones. When Γ 1, the sampling points are nearly equidistant and the choice of raised cosine pulse roll-off factor does not affect the frequency resolution of the algorithm. However, a pulse with a smaller roll-off factor (β close to 0) can resolve the signal components in fewer iterations and should therefore be preferred. When Γ 1 is not satisfied, the spacings between consecutive local maxima and minima are no longer approximately uniform causing a raised cosine interpolator with small β to fail in signal separation and a larger roll-off factor is required. FIR filter implementation is a major advantage of raised cosine interpolation. Filter coefficients are determined by two parameters: the roll-off factor, β and the sampling period, T. The roll-off factor, β, is a pre-defined system parameter affecting frequency resolution of the algorithm. The sampling period, T, on the other hand, which is the mean duration between consecutive maxima (or minima), is a signal-dependent parameter that is estimated on a block-by-block basis as described in Section 3.3. The interpo-

57 44 lated envelope is derived by convolving the zero-padded vector containing the maxima (or minima) with the filter tap values. Moreover, frequency-domain implementation reduces the computational complexity by replacing convolution by multiplication. So to reduce computational steps we perform convolution in the frequency domain by transforming the zero-padded time-domain signals into the frequency domain using FFT, filtering by the estimated interpolation filter coefficients and subsequent reconversion to time-domain via inverse FFT. 3.3 Raised Cosine Empirical Mode Decomposition Signal interpolation using the raised cosine pulse requires approximately uniform spacing between control points or knots. A large β results in faster decay of the interpolant tails, thereby reducing its sensitivity to non-uniform sampling intervals. However, nonstationary signals result in larger variations in sampling intervals which reduces the effectiveness of a raised cosine pulse with large β as an interpolant. To solve this problem, the original signal is split into small, overlapping segments and interpolation using the raised cosine pulse is applied to each of them individually. By careful choice of the interpolation window length relative to the maximum rate of change of the signal instantaneous frequency, approximately uniform sampling intervals can be ensured. We now enumerate the steps involved in the RCEMD algorithm. First, various parameters and variables are initialized. A. Set the roll-off factor β, window shape w (a rectangular window is used in our analysis) and window size K (corresponding to the number of successive extrema to be included in the window). Further, N and M refer to the data length and the number of maxima or minima in the signal respectively. B. Initialize ẽ max [n] and ẽ min [n] to N-length zero-vectors. Then the main loop of the algorithm is as follows: 1. Identify the extrema of x[n]. 2. For i = 1 : M K do the following

58 45 (a) Define index q = u i..u i+k, where u j represents the position of the j th maxima of x[n]. (b) Compute window coefficients w[q], according to the shape chosen in Step A. (c) Compute the windowed upper envelope x upper [q] = w[q] x[u i : u i+k ]. (d) Compute T = K 1 x[u k+1 ] x[u k ] k=1 K 1. (e) Compute raised cosine filter coefficients h[q] using pre-defined β and computed T. (f) Compute ẽ max [q] = ẽ max [q] + x upper [q] h[q]. (g) Compute ẽ min [q] similarly, by first computing x lower [q] (Repeat steps c)-f) for ẽ min [q]). 3. Compute m[n] = ẽmin[n]+ẽ max [n] Extract the detail d[n] = x[n] m[n]. 5. If d[n] satisfies all IMF conditions, then set y 1 [n] = d[n], the first IMF, else repeat above steps with d[n]. 6. Evaluate the residual x 1 [n] = x[n] y 1 [n]. 7. Iterate on the residual x 1 [n]. The proposed algorithm differs from the original EMD algorithm in two crucial ways: firstly, a new interpolant is used; and secondly, signal filtering is performed at the local level. In our experiments we have found that window spanning five consecutive extrema (either minima or maxima), i.e., K = 5 produces good signal resolution for a wide variety of cases and has therefore been used consistently in all simulations. Signal resolution performance and convergence rate comparison with EMD, which are considered in the following section, ignore boundary condition remediation operations such as signal mirrorizing. In general, the effect of signal boundary on the RCEMD algorithm was not found to be any worse than that for EMD, either in extent or severity, for the examples considered here, but a detailed analysis of this phenomenon is relegated to future research. Finally, β = 1 is used in all simulation results presented to minimize the effect of non-uniform sampling.

59 3.4 Signal Decomposition Quality of RCEMD Algorithm Combination of tones Frequency resolution of RCEMD and EMD algorithms for a combination of two tones is studied here. Mathematically, these signals are defined as 46 x(t) = s 1 (t) + s 2 (t) (3.4) where s 1 (t) = A 1 sin(2π f 1 t + ϕ 1 ) s 2 (t) = A 2 sin(2π f 2 t + ϕ 2 ). (3.5) and the symbols have their usual meanings and f 1 > f 2. Two metrics that measure the similarity between the extracted IMFs and original tones are defined as Ω k 1 = s 1(t)y 1 (t) s 2 1 (t) Ω k 2 = s 2(t)y 2 (t) s 2 2 (t) (3.6) (3.7) where y 1 (t) and y 2 (t) represent the extracted IMFs after k iterations of the algorithm and. denotes time-averaging. These quantities assume values between 0 and 1 with large values indicating better signal decomposition quality. Signal separation performance of the two algorithms for bicomponent signals considered in the present and following examples is based on Ω k 2. This choice is based on the observation that due to the presence of only two signal components and two extracted IMFs, a strong match between one signal component and a particular IMF implies strong match between the other signal and the other IMF, thereby rendering Ω k 1, Ωk 2 or a combined metric (such as their mean) as equivalent measures of signal decomposition quality. Based on this decomposition quality metric we can state the following theorem for EMD performance using an arbitrary interpolating filter: Theorem 1. Suppose H( f ) denotes the frequency response of the interpolating filter. Suppose further that A 2 A 1 f 2 f1 1 is satisfied. Then, Ω k 2 = 1 (1 H( f 2)) k.

60 47 Proof. Starting from (3.5) we observe that in the EMD technique the composite signal is sampled at the local maxima and minima to obtain the upper and lower envelopes respectively. These sampling points should ideally coincide with the extrema of the high frequency (HF) component. We denote these ideal sampling instants by t u 1 and tl 1 for the local maxima and minima respectively t u 1 = (4m 3) 1 4 f 1 ϕ 1 2π f 1 (3.8) t l 1 = (4m 1) 1 4 f 1 ϕ 1 2π f 1 (3.9) for m = 1,2,3,... with the sub-script indicating that these time instants refer to the higher tone. Sampling the composite signal at these ideal sampling instants yields the lower frequency tone offset by a constant value (A 1 for the upper envelope and A 1 in case of the lower envelope). However, estimation of the local extrema from the composite signal by evaluating the derivative leads to errors resulting in actual sampling points ( t u 1, t 1 l ) that are different from the ideal sampling points (tu 1, tl 1 ). Figure 3.2 illustrates the sampling point error described here. Next we derive the expression for the upper envelope sampled at the points obtained by determining the location of sign change of the derivative of x(t). The technique described here can be adapted to derive the expression for the lower envelope also. We have for the upper envelope e u ( t 1 u ) = A 1sin(2π f 1 t 1 u + ϕ 1) + A 2 sin(2π f 2 t 1 u + ϕ 2) (3.10) = A 1 sin ( 2π f 1 (t1 u +tu δ ) + ϕ 1) + A2 sin(2π f 2 t 1 u + ϕ 2) (3.11) = A 1 sin(2π f 1 t1 u + ϕ 1)cos(2π f 1 t u δ ) + A 1 cos(2π f 1 t1 u + ϕ 1)sin(2π f 1 t u δ ) (3.12) + A 2 sin(2π f 2 t 1 u + ϕ 2) = A 1 cos(2π f 1 t u δ ) + A 2sin(2π f 2 t 1 u + ϕ 2) (3.13) ( ) A 1 1 4π2 f1 2tu 2 δ + A 2 sin(2π f 2 t 1 u 2 + ϕ 2) (3.14) A 1 A 1 2π 2 f1 2 t u2 δ + A 2 sin(2π f 2 t 1 u + ϕ 2). (3.15) The term within the big parentheses in (3.14) follows from the Taylor series expansion

61 s 1 (t) t 1 u t 1 u +tδ u s 1 (t) t (s) Figure 3.2: Positions of the HF component maxima and local maxima of the composite signal superimposed on the HF component. The maximum separation between the two depends on the relative amplitudes and frequencies of the signal components. of the cosine term in (3.13), followed by its truncation after the first two terms. Of course this approximation is valid only when 2π f 1 t u δ 1. The sampling error (tu δ ) is A 2 t u δ = 1 f 2 2π A 1 f1 2 cos(2π f 2 t1 u + ϕ m). (3.16) The proof for this formula is given in Appendix. The values of t u δ computed from (3.16) and those obtained from simulation are shown in Fig Further, we can write where f m = f 1 2 f 1 t u2 δ = 1 ( ) A2 f 2 2 4π 2 A 1 f1 2 cos 2 (2π f m t1 u + ϕ m). (3.17) 2 mod(2 f 2, f 1 ) is the possibly aliased version of 2 f 2 and ϕ m = ϕ 2.

62 t δ u (s) Simulation Theory t (s) Figure 3.3: The separation between the HF component maxima and composite signal local maxima is plotted as a function of time. The amplitude of this oscillatory quantity is determined by the relative amplitudes and frequencies of the signal components, and its frequency matches the LF component. So we have e u ( t u 1 ) A 1 A2 2 2A 1 ( f2 f 1 + A 2 sin(2π f 2 t u 1 + ϕ 2). A similar analysis for the lower envelope leads to ( e l ( t 1 l ) = A 1sin 2π f 1 t1 l A 1 + A2 2 2A 1 ) ( f2 f 1 + A 2 sin(2π f 2 t l 1 + ϕ 2). ) 2 cos 2 (2π f m t u 1 + ϕ m) ( ) + A 2 sin 2π f 2 t 1 l + ϕ 2 ) 2 cos 2 (2π f m t1 l + ϕ m) (3.18) (3.19) (3.20) The above expressions represent the upper and lower signal envelopes sampled at the approximate instants when the higher frequency tone attains its maxima and minima

63 50 respectively. The preceding derivation assumes that 2π f 1 t u δ 1 and 2π f 1t l δ 1. These conditions are satisfied when t u δ 1 2π f 1 and t l δ 1 2π f 1 respectively and from (3.16) we can see that fulfillment of the first condition requires that A 2 A 1 f 2 f1 1 for the upper envelope. A similar argument holds true for the lower signal envelope. It is reasonable to assume that the sampling points of the envelopes are approximately equidistant as long as the condition A 2 A 1 f 2 f1 1 holds. In that case, interpolation of the sparsely sampled envelopes leads to reconstruction of the continuous-time version of the signal envelopes with the different frequency components weighted by the gain derived from the frequency response of the interpolator, H( f ) ( ) 2 e u (t) H(0)A 1 H( f m ) A2 2 f2 sin 2 (2π f m t + ϕ m ) A 1 2 f 1 + H( f 2 )A 2 sin(2π f 2 t + ϕ 2 ) e l (t) H(0)A 1 + H( f m ) A2 2 A 1 ( f2 2 f 1 + H( f 2 )A 2 sin(2π f 2 t + ϕ 2 ). ) 2 sin 2 (2π f m t + ϕ m ) (3.21) (3.22) The mean (m 1 (t)) is simply the average of the two signal envelopes m 1 (t) = H( f 2 )A 2 sin(2π f 2 t + ϕ 2 ). (3.23) The detail (d 1 (t)) is then given by d 1 (t) = x(t) m 1 (t). So d 1 (t) = A 1 sin(2π f 1 t + ϕ 1 ) + (1 H( f 2 ))A 2 sin(2π f 2 t + ϕ 2 ) (3.24) where the superscript 1 in d 1 (t) and m 1 (t) indicates that these quantities refer to the detail and mean signals after the first EMD iteration. The next iteration of the EMD algorithm continues with d 1 (t) as the new starting signal and the process continues until some stopping criterion is fulfilled. Recalling that the original signal has only two components we can conclude that the detail signal (d k (t)) and the residual (r k (t) = x(t) d k (t)) after k iterations should be sufficient to represent the individual components and when d k (t) satisfies some stopping criterion, we stop iterating and d k (t) and r k (t)

64 51 are declared the first two IMFs. Then we can write IMF 1 d k (t) = A 1 sin(2π f 1 t + ϕ 1 ) + (1 H( f 2 )) k (3.25) A 2 sin(2π f 2 t + ϕ 2 ), ( IMF 2 r k (t) = 1 (1 H( f 2 )) k) A 2 sin(2π f 2 t + ϕ 2 ). (3.26) Then, following the definition of Ω k 2 (3.7) we have Ω k 2 = = A 2 sin(2π f 2 t + ϕ 2 ) A 2 sin(2π f 2 t + ϕ 2 ) (1 (1 H( f 2 )) k) A 2 2 sin 2 (2π f 2 t + ϕ 2 ) (3.27) A (1 H( f 2)) k A A 2. (3.28) 2 2 = 1 (1 H( f 2 )) k. (3.29) Although this result was first discussed in [61], the present proof provides mathematical justification for the condition A 2 A 1 f 2 f1 1. While deriving the above result, effects of insufficient sampling of the signal envelopes for f 2 f1 > 0.5 and end effects are not accounted. So, the practical value of this measure is usually somewhat smaller than the derived one, especially for f 2 f1 > 0.5. The convergence rate of the signal decomposition algorithm is studied next. For this, the minimum iteration count (k min ( f 2,Ω 2lim )) for which { Ω k 2 > Ω 2lim} is satisfied, is determined. Therefore, Ω k 2 > Ω 2lim 1 (1 H( f 2 )) k > Ω 2lim (3.30) 1 Ω 2lim > (1 H( f 2 )) k (3.31) k > log(1 Ω 2lim) log(1 H( f 2 )). (3.32) Further, we may denote k min ( f 2,Ω 2lim ) = log(1 Ω2lim ). (3.33) log(1 H( f 2 ))

65 Ω 2 k Theory, f 2 /f 1 =0.14 Simln., f 2 /f 1 =0.14 Theory, f 2 /f 1 =0.44 Simln., f 2 /f 1 = Iterations (k) Figure 3.4: Comparison of simulation results with theory (3.29) for raised cosine interpolation based on transient value of performance metric Ω k 2. Two frequency ratios ( f 2 f1 ) are considered here. This represents the minimum iteration count for a particular signal decomposition quality. Although these relations are applicable to EMD using any interpolation filter with known frequency response (H( f 2 )), we validate the results for raised cosine interpolation via comparison with simulation results using synthetic signals. First, the transient behavior of the performance metric Ω k 2 is plotted in Fig. 3.4 for two frequency ratios ( f 2 f1 ). Next, the convergence rate of the algorithm represented by (3.33) is compared with simulation results in Fig We next compare the signal separation performance of EMD and RCEMD algorithms for two-component signals consisting of tones based on Ω k 2. Monte-Carlo simulations are performed by generating synthetic signals according to (3.4) and (3.5), where A 1 = 1 ϕ 1 = 0 and ϕ 2 varies uniformly over [0,2π). The signal separation performance of the two techniques is studied here for a wide range of amplitude and frequency ratios, A 2 /A 1 and f 2 / f 1. Fig. 3.6 shows the results of this experiment in a format similar to that used in [61] for EMD. In short, the intensity values in Fig. 3.6 represent Ω 100 2, the value of the performance metric after 100 iterations of the RCEMD algorithm, with a

66 Simulation Theory k min (f 2,Ω 2lim ) f 2 /f 1 Figure 3.5: Comparison of simulation results with theory (3.33) for raised cosine interpolation based on minimum iteration count for desired signal decomposition quality. The selected threshold is Ω 2lim = 0.9. lighter shade representing a larger value (better signal separation quality). An important observation from Fig. 3.6 is that the demarcation between the regions of good and poor signal separation quality generally lies to the right of the curve representing the equation A 2 A 1 ( f2 f1 ) 2 = 1. The significance of the curve is that it was shown to be the theoretical limit for successful signal separation by EMD [61]. To directly compare the performance of the two techniques, the difference between the performance metric values of the two techniques after 100 iterations is shown in Fig The superior signal analysis performance of RCEMD for large amplitude and frequency ratios of signal components, conditions under which the EMD algorithm performs poorly, is a significant advantage of this technique Two frequency modulated components Previously, a combination of pure tones was considered to study the RCEMD algorithm. Here RCEMD performance is evaluated for a combination of FM signals. The following model allows signal separation quality comparison between EMD and RCEMD for non-

67 f 2 /f A 2 A 1 ( f2 f 1 ) 2 = log(a 2 /A 1 ) 0 Figure 3.6: Final value of the performance metric after 100 iterations(ω ) of the RCEMD algorithm plotted for a range of amplitude and frequency ratios. A curve representing the theoretical limit for successful signal separation by EMD [61] is also shown. stationary components. This is similar to the non-stationary signal model used in [32,61] EMD performance evaluation. The FM signals can be represented as s p (t) = A p cos(ω c t + k f t m p (τ)dτ + θ p ) (3.34) and m p (t) = A mp cos(ω p t + ϕ p ) + B mp (3.35) where p = 1,2. Then the instantaneous frequencies of the signals are given by ω Ip (t) = ω c + k f m p (t), p = 1,2. (3.36) In this signal model each signal component has four adjustable parameters affecting the instantaneous frequency: A mp, B mp, ω p and ϕ p. The relation between A mp and B mp

68 f 2 /f log(a 2 /A 1 ) Figure 3.7: Differences between the [ final values ] of the performance metric after 100 iterations of the two techniques, ω RCEMD [ ω2 100 ] EMD. The differences are ([ quantized ] to three values: 1, signifying that performance of RCEMD is better ω RCEMD [ ω2 100 ] EMD 0.05) ; -1, signifying that performance of EMD is better ([ ] ω EMD [ ω2 100 ] RCEMD 0.05) ; 0, signifying that the performance of both techniques is about the same ( [ ω2 100 ] RCEMD [ ω2 100 ] ) EMD < The regions representing the values 1 and 0 are in white and black respectively. The value -1 does not appear in this graph. and their effect on the instantaneous frequency are shown in Fig Moreover, ω p and ϕ p control the starting phase and the rate of change of the instantaneous frequencies of the two signals, respectively. So, while ω p determines the starting phase of the instantaneous frequencies in Fig. 3.8, their oscillation frequencies depend on ω p. From the figure it is clear that the parameters A mp and B mp determine the relative frequency separation between the components and, consequently, the level of difficulty for signal separation. In our experiments we fix the parameters of signal s 2 (t) and vary those of s 1 (t) to achieve different frequency compositions of the component signals. Moreover, ω p and ϕ p are identical for the two components to prevent crossing of instantaneous frequencies. Signal separation quality of the two algorithms is evaluated based on the metric Ω 100 2

69 56 A m1 frequency Bm1 Bm2 A m time Figure 3.8: Instantaneous frequencies of the two frequency modulated signal components. B m1 and B m2 represent the frequency offsets from the carrier frequency and A m1 and A m2 indicate the frequency spread around the offsets. that was introduced earlier. Here we set the signal parameters such that the instantaneous frequencies of the two components bear a constant ratio at all times. To achieve this, the parameters A m2, B m2, ω 2 and ϕ 2 corresponding to s 2 (t) are assigned values first. Then the instantaneous frequency of s 1 (t) is related to that for s 2 (t) as f I1 = η f I2, where η > 1 is a constant, that then modulates the carrier signal. Twenty trials of the experiment are performed with different uniformly distributed values of A m2 and B m2. The averaged results of this experiment are presented in Fig Similar to the previous experiment this figure shows the performance difference between EMD and RCEMD after 100 iterations of the algorithm. The advantage of the RCEMD technique for large instantaneous frequency ratios ( f I 2 f I1 ) and large amplitude ratios ( A 2 A 1 ) results from the reduced sensitivity of the raised cosine filter (with β = 1) to irregular sampling, which is a significant problem for large frequency and amplitude ratios. A second example of frequency modulated component signals involves signals with time-varying instantaneous frequency ratio. This case involves higher degree of irregularity of the extrema spacings. Simulation results are shown in Fig where variation

70 f I2 /f I log(a 2 /A 1 ) Figure 3.9: Similar to Fig. 3.7, except that the component signals are frequency modulated in this case. Here the regions representing the values 1, 0 and -1 are represented by white, gray and black colors respectively. of Ω as a function of the ratio B m 2 B m1 is presented. The results of Fig are averaged over five values each of A m1 and A m2, chosen randomly while ensuring no intersection of the instantaneous frequencies of the two signals. A similar trend as before is observed where the RCEMD algorithm s performance is superior when the frequencies of the signals are closer (large B m 2 B m1 ). Significant performance improvement for A 2 A 1 = 10 is evident and the gap is larger than the previous examples that had more regular extrema spacings. Results highlighting the convergence rates of the algorithms are shown in Fig where the minimum iterations necessary for Ω k 2 to exceed Ω 2lim = 0.9, denoted as k min (Ω 2lim ), are presented. Each algorithm was terminated after 100 iterations and saturation of the curves at the maximum iteration count in the graphs represents the condition Ω < Ω 2lim for the particular choice of parameters.

71 Ω Raised Cosine Cubic Spline B m2 /B m1 1 (a) A 2 A 1 = Ω Raised Cosine Cubic Spline B m2 /B m1 (b) A 2 A 1 = 0.1 Figure 3.10

72 Raised Cosine Cubic Spline 0.6 Ω B m2 /B m1 (c) A 2 A 1 = 10 Figure 3.10: Final values of the performance measure Ω for the RCEMD and EMD algorithms. Simulation results for three different amplitude ratios are shown Bicomponent trigonometric function Next we consider an example that has been previously examined in EMD literature. This involves identifying the components at frequencies f and 3 f in the signal cos 3 (2π ft). Here, the performance of the two algorithms in identifying and isolating the two frequency components of the signal cos 3 (2π ft) at frequencies f and 3 f is compared. The RCEMD and the EMD algorithms are applied to the signal and their outputs are shown in Fig The failure of the EMD algorithm to separate these signal components is unexpected considering that here A 2 A 1 ( f2 f1 ) 2 = 1/3. Our experiments indicate that the reason lies in the particular configuration of the starting phases of the components of this signal, that results in non-separation by EMD Multicomponent signal The validity of the RCEMD algorithm when the signal has more than two components is demonstrated here. In this example all signal components have overlapping, time-

73 k min (Ω 2lim ) 10 1 Raised Cosine Cubic Spline B m2 /B m (a) A 2 A 1 = 1 k min (Ω 2lim ) 10 1 Raised Cosine Cubic Spline B m2 /B m1 (b) A 2 A 1 = 0.1 Figure 3.11

74 k min (Ω 2lim ) 10 1 Raised Cosine Cubic Spline B m2 /B m1 (c) A 2 A 1 = 10 Figure 3.11: Minimum number of iterations necessary for Ω k 2 to exceed Ω 2lim = 0.9 for the windowed RCEMD and EMD algorithms. Simulation results for three different amplitude ratios are shown. varying instantaneous frequencies that are difficult to separate using traditional filtering techniques. Moreover, two of the signal components have time-varying amplitudes also, meaning that the amplitudes A p in (3.34) are now time-varying for two of the three signal components (here p = 1, 2, 3). Successful signal decomposition using EMD for a similar signal has been previously demonstrated [61]. Here we test the ability of the RCEMD algorithm to decompose the signals. The frequency-and time-domain signals are shown in Fig and Fig respectively. Although, the signals have been correctly separated into their respective IMFs, some signal-mixing is evident in regions of small instantaneous frequencies for this difficult signal separation problem. This is because there are fewer cycles of the signal over these intervals that results in an elongated RCEMD window, the length of which depends on inter-extrema spacing. The windowed signal no longer has constant instantaneous frequency over the extended interval, resulting in non-optimum filtering. This effect, which is seen when the signal components have small instantaneous frequencies, is not observed for EMD.

75 Amplitude time (s) Figure 3.12: Signal decomposition quality of RCEMD and EMD algorithms for cos 3 (2π ft) is shown here. First panel shows the original signal. The subsequent panels show the signal components at frequencies 3 f and f (solid lines), and the IMFs generated by RCEMD (dotted-dashed) and EMD (dotted) algorithms superimposed on them. The dotted-dashed line corresponding to RCEMD coincides with the corresponding signal components for the most part and deviates from the expected result only at the ends where it is visible. It is clear from this figure that EMD fails in resolving the signal components Tidal component extraction In this section we validate the new RCEMD algorithm by applying it to real-world data. In this example we apply the signal decomposition algorithm to sea level measurements and expect to see components corresponding to diurnal and semi-diurnal tides. Successful isolation of signal components with known physical interpretation is sought in this exercise. Sea level data obtained from the Intergovernmental Oceanographic Commission database at [77] was used in this study. We used tide gauge data from Honolulu, Hawaii, USA spanning approximately thirty days for signal decomposition. Some signal pre-processing steps were carried out to prepare the data for the subsequent step. First, the one-minute sampled data was downsampled by a factor of fifteen to reduce the data length. No useful information is lost in the process because the tidal phenomena occur at much longer time-scales. A carefully designed noise-filter that has a flat response at

76 63 a b frequency frequency c d time time Figure 3.13: Frequency-domain signals for the multicomponent signal example. Panels a, b, c, d correspond to the combined signal and the three extracted components using the RCEMD algorithm respectively a b c d time Figure 3.14: Time-domain signals for the multicomponent signal example. Panels a, b, c, d correspond to the combined signal and the three extracted components using the RCEMD algorithm respectively.

77 Water level (m) time (days) Figure 3.15: Demonstration of application of RCEMD technique to sea level data. The RCEMD algorithm is applied to the sub-sampled and noise-removed time-series data shown in the top panel and the generated components are shown in the subsequent panels. The first extracted component corresponds to a roughly 12-hour period signal and the other to a superimposed variation of period that is twice as long. The diurnal inequity is due to several reasons including inclination of the lunar orbit with respect to the earth s equator, some solar contribution and Pacific resonances. tidal frequencies is next applied to the data. The presence of noise causes mode-mixing in EMD and related algorithms, and should be minimized before decomposition. This requirement is related to the peak-finding step in the decomposition procedure and is common to both EMD and RCEMD. Finally, the RCEMD algorithm is applied to the filtered data and the results are shown in Fig Clean separation into two components - one with an approximately 12-hour period and the other with an approximately 24-hour period is observed with smooth amplitude variations in each case corresponding to the shifting configurations of the sun and the moon, the two major planetary bodies affecting sea levels. The diurnal variations arise due to the moon s declination effect (change in angle relative to the equator) and the diurnal variations are themselves amplitude modulated due to the roughly monthly cycle of movement of the moon between the two hemispheres of the earth.

78 EMD: Computational Complexity In this section we study the computational complexity of the EMD and RCEMD algorithms. We adopt the following procedure to calculate computational complexity: first, computational complexity is calculated assuming the signal window length is identical to the entire data length, and then in the next step the result is modified by assuming a shorter window length. An incremental approach for overlapping windows is described that results in computational savings. As discussed previously, the EMD technique decomposes a composite signal into its constituents, referred to as IMFs, by an iterative process called sifting. Computing the IMFs by sifting involves the following steps: 1. Determining the local maxima and minima in the signal. 2. Computing the upper and lower envelopes of the signal passing through the extrema using interpolation. 3. Finding the mean value of the envelopes. This gives the mean or residual signal containing the low frequencies. 4. Subtracting the residual signal from the original signal to get the high frequency detail signal. These steps represent one iteration of the algorithm. The detail signal generated is tested to see if it satisfies the conditions for being an IMF. If yes, it is declared the first extracted component or IMF1. Else the sifting procedure is continued till the signal satisfies the conditions. In this section the number of computations needed for each sifting iteration is computed Finding the extrema Based on a discrete-time signal model the original multicomponent signal is represented by x[n]. Then the steps to be followed to extract the extrema can be represented mathematically as x 1 [n] = [ ] 1 1 x[n] (3.37) x 2 [n] = sign(x 1 [n]) (3.38) [ ] x 3 [n] = 1 1 x 2 [n] (3.39)

79 66 Table 3.1: No. of computations to find extrema points Equation Operation count 3.37 N 3.38 N 3.39 N 3.40a 2N 3.40b 2N 3.41a N 3.41b N Total 9N x 41 [n] = sign(x 3 [n]) + x 3 [n]/2 2 x 42 [n] = sign(x 3 [n]) + x 3 [n]/2 2 (3.40a) (3.40b) x 51 [n] = x 41 [n] x[n] x 52 [n] = x 42 [n] x[n] (3.41a) (3.41b) where denotes convolution and denotes vector element-by-element multiplication. Here in the first two steps we find the sign of the numerical derivative of the signal. In step three we find the derivative of the result, which gives us the location of the signal local maxima (where first derivative changes sign). Steps 4 and 5 find the locations of the signal local maxima and minima respectively. In the last two steps the function values at the local signal maxima and minima locations are evaluated. If there are N samples in the data-set then the number of computations required for each mathematical step above are listed in Table 3.1. The number of maxima (or minima) depends on the number of complete cycles of the highest frequency component present in the signal, M Finding the cubic spline coefficients A cubic spline is a spline constructed of piecewise third-order polynomials which pass through a set of M control points. The second derivative of each polynomial is commonly set to zero at the endpoints, since this provides a boundary condition that completes the system of M 1 equations. This produces a so-called natural cubic spline

80 67 and leads to a simple tridiagonal system which can be solved easily to give the coefficients of the polynomials. However, this choice is not the only one possible, and other boundary conditions can be used instead. We use the natural cubic spline system in our system as it leads to a tridiagonal system of equations that can be solved in O(M 1) operations instead of O((M 1) 3 ) required by Gaussian elimination [78]. Specifically, in our case, 18M 28 operations are required for computing the piecewise cubic spline coefficients for either of the two envelopes. Further, 21N operations are needed to find the residual and detail signals. So in all (9N) + 2(18M 28) + (21N) = 30N + 36M 56 computations are required for each sifting iteration of the EMD algorithm. Denoting by S i the number of sifting iterations needed to generate IMF i, the total number of operations needed to extract IMF i is simply obtained by multiplying the above expression by S i Complexity of the raised cosine filter approach In addition to some extremely efficient hardware implementations of the FFT available ( [79, 80]), the fastest software implementation of the FFT algorithm requires 2N log 2 N 3N +4 operations for real values [81]. A simple calculation of the operation count for the raised cosine interpolation follows from this result. This approach involves four steps: finding the mean of x 51 [n] and x 52 [n], computing the Fourier transform, multiplying by the frequency response of the filter and computing the inverse Fourier transform. So, in all 2N +2N log 2 N 3N +4+N +2N log 2 N 3N +4 = 4N log 2 N 3N +8 operations are needed to compute the mean signal using the raised cosine filter approach after evaluating x 51 [n] and x 52 [n] compared to 21N + 2(18M 28) = 21N + 36M 56 operations for the cubic spline technique for the corresponding steps. Finally, the overall operation count for separating two signals is obtained by multiplying the just-derived operation counts per iteration by S i, the number of iterations needed by each algorithm to achieve a certain signal separation quality. Moreover, it is seen that the operation counts derived above can be split nearly in half between the number of multiplication and divisions on one side and addition and subtraction on the other for both algorithms, with the number of additions and subtractions slightly larger than the other group. Finally, we observe that per iteration, the RCEMD algorithm is generally faster than EMD for shorter signal lengths.

81 Operation count 10 5 RC, A 2 /A 1 =1 CS, A 2 /A 1 =1 RC, A 2 /A 1 =10 CS, A 2 /A 1 = B m2 /B m1 Figure 3.16: Computational complexity comparison for frequency modulated signal components. Results for two amplitude ratios A 2 A 1 = 1 and A 2 A 1 = 10 are presented. Computational complexity when A 2 A 1 = 0.1 is similar to that for the equal amplitude case and is therefore omitted for clarity Complexity of windowed RCEMD Next, we compare the computational complexity of the RCEMD and the EMD algorithms applied to multicomponent signals with time-varying frequencies. In this case a segment of the multicomponent signal containing K consecutive extrema (maxima or minima) is decomposed at a time. So a window contains K M N L signal samples on an average. After operating on a particular segment of the signal the window is advanced to the next maxima or minima, while discarding the oldest data samples and the process repeats until the entire signal is decomposed. By realizing that the operation count for the first window being 4Llog 2 L 3L + 8, and that IMF segments for the following segments can be computed by incrementally performing K 2 times the number of operations for the first window (corresponding to the first and the last inter-extrema samples for each window), and that M K shifts of the window are required, the resulting total operation count per iteration is ( 2M K 1)(4Llog 2 L 3L + 8). The total operation count to decompose a multicomponent signal into two IMFs is arrived at by multiplying the per

82 69 iteration operation count by the number of iterations required to achieve desired signal analysis quality. The operation counts of the RCEMD and EMD algorithms are shown in Fig The plots saturate for large B m 2 B m1 due to the termination of the decomposition after 100 iterations. Rectangular window with K = 5 was used in the simulations. To summarize, assuming uniform distribution of the ratio B m 2 B m1 over the non-saturated region of Fig. 3.16, on an average, the computational complexity of the EMD algorithm is nearly twice and four times that of the RCEMD algorithm for comparable signal separation quality when A 2 A 1 = 1 and A 2 A 1 = 10 respectively. 3.6 Low Sampling Rate Performance of RCEMD Successful signal analysis using the empirical mode decomposition (EMD) algorithm requires high degree of oversampling. This requirement arises from the need to precisely identify the local extrema of the signal to recover the signal envelope using natural cubic spline interpolation. This problem is alleviated in RCEMD which allows high fidelity reconstruction of the signal envelope when the local extrema cannot be precisely identified due to low sampling rates. The advantage of this technique at low sampling rates is demonstrated using synthetic signals in this section. Reference [82] introduced a solution to the fast sampling requirement of the EMD algorithm by using a new extrema identification procedure based on Fourier interpolation. References [62] and [63] present further research on the effect of sampling on EMD. Here, we demonstrate the advantage of the raised cosine interpolation-based technique introduced in this chapter at low sampling rates. The timing jitter problem at low sampling rates is first introduced followed by some synthetic signal examples to demonstrate the the superior signal analysis performance of RCEMD algorithm at low sampling rates Timing jitter at low sampling rates The effect of timing jitter on signal analysis performance using EMD is examined here. For this, following a similar approach as our previous analysis in Section 3.4.1, we

83 70 consider an N-sample long two-tone signal, s(t p ) = A 1 sin(2π f 1 t p + ϕ 1 ) + A 2 sin(2π f 2 t p + ϕ 2 ), (3.42) where p = 1,...,N, f 1 > f 2 and the symbols have the usual meanings. In the EMD algorithm an estimate of the low frequency component is obtained by evaluating the average of the upper and lower signal envelopes of the sampled multicomponent signal by interpolating between successive local maxima and minima respectively. If we denote by t m, m = 1,2,... the locations of the sampled composite signal local maxima, then the f 1 -rate sampled upper envelope is given by e u (t u m) = A 1 sin(2π f 1 t u m + ϕ 1 ) + A 2 sin(2π f 2 t u m + ϕ 2 ). (3.43) If the envelopes are sampled at the exact peaks and troughs of the larger frequency component, denoted by t u m and t l m respectively, then e u ( t u m) = A 1 sin((m 1)2π + π/2) + A 2 sin(2π f 2 t u m + ϕ 2 ) = A 1 + A 2 sin(2π f 2 t u m + ϕ 2 ). (3.44) Similarly, e l ( t m) l = A1 +A 2 sin ( 2π f 2 t m l ) + ϕ 2, so that the average of the two envelopes interpolated to f s -rate closely approximates the smaller tone. However, finite sampling rate introduces timing jitter that causes the actual sampling points to deviate from the ideal ones. Denoting the timing error sequence by δm, u we find that δm u = tm u t m, u where δm u is uniformly distributed over ( 1/2 f s,1/2 f s ). The sampled upper envelope can then be represented as e u (tm) u = A 1 sin((m 1)2π + π/2 + 2π f 1 δm) u + A 2 sin(2π f 2 ( t + m u + δm) u + ϕ 2 ). (3.45) Therefore, we find that finite sampling not only causes irregular sampling of the envelope but also introduces a high frequency component at frequency f 1. Raised cosine interpolation using large β results in more accurate signal representation than natural cubic splines when irregular sampling results in introduction of spurious high frequency components as demonstrated by the following experiment. We first choose the frequencies f 1, f 2 and sampling rate f s such that f 2 < f 1 f s /2.

84 Raised cosine, β=1 Cubic spline NMSE Sampling frequency, multiples of f N Figure 3.17: Effect of sampling rate on interpolation quality Ω 2 k RC EMD, f s =1.2f N EMD, f s =1.2f N EMD, f s =10.0f N Hybrid, f s =3.0f N Iterations, k Figure 3.18: Signal analysis performance of the RCEMD, EMD and hybrid techniques for a combination of tones at different sampling rates.

85 Ω 2 k RC EMD, f s =3f N EMD, f s =3f N EMD, f s =25f N Hybrid, f s =15f N Iterations, k Figure 3.19: Signal analysis performance of the RCEMD, EMD and hybrid techniques for a combination of frequency modulated signals at different sampling rates. Our aim here is to recover the signal s 2 (t) = A 2 sin(2π f 2 t + ϕ 2 ) at sampling rate f s from the available irregularly-spaced samples at an average rate of f 1. This sampled signal also includes a higher frequency component at f 1 as described above. We compare the performances of two interpolation techniques: one using the raised cosine pulse with β = 1, and the other using the natural cubic spline. Normalized mean squared error (NMSE) is used as the performance criterion. The results are presented in Fig for a range of sampling rates. Here A 1 = A 2 = 1, f 2 = 0.5 f 1 and the results are averaged over several uniformly distributed starting phases and timing jitter realizations. It is clear from the results that not only is the interpolation quality using raised cosine pulse better than that using cubic splines, but the difference between them is larger at low sampling rates, close to the Nyquist rate ( f N = 2 f 1 ). This justifies the use of RCEMD procedure that uses the raised cosine pulse for interpolation. In the next section we will compare the signal analysis performance of this technique to EMD at low sampling rates.

86 Performance Comparison We evaluate the performances of the two techniques for two classes of signals: stationary and non-stationary. For the stationary case, we use the signal model consisting of tones as described above and perform 100 iterations of each algorithm. For RC-EMD we use β = 1 and estimate the time constant T, which is the mean duration between the local maxima (or minima) at each iteration from the actual signal, as explained in Section 3.2. Overlapping data blocks spanning five consecutive maxima (or minima) were used for the RCEMD algorithm. We also present the performance results of the hybrid technique introduced in [82] to improve the low sampling rate performance of EMD. We use the normalized cross-correlation (Ω k 2, where k represents the iteration count) between the extracted second IMF and s 2 (t) as the performance index. It is clear from the results in Fig that RCEMD successfully identifies the tones at a low sampling rate, while the other techniques fail to match the decomposition quality even at much finer sampling. Next we consider a combination of frequency-modulated signals of the form s(t) = cos t ω 1 (t)dt + ϕ 1 + 2cos t 0.6ω I (t) + ϕ 2 (3.46) where ω I (t) = 2π ( f c + k f (A m cos(2π f m t + ϕ m ) + B m ) ). Here f c and k f represent the center frequency and frequency-deviation constant and the results are averaged over several values of modulation parameters A m and B m. Figure 3.19 shows the minimum sampling requirements for the three techniques for successful signal analysis. Clearly, the sampling requirements for the hybrid technique and the original EMD algorithms far exceeds those for RCEMD for equivalent signal analysis performance. 3.7 Conclusions In this chapter a variation of the EMD algorithm, called RCEMD, that involves raised cosine interpolation was introduced. The signal separation performance of this new algorithm was studied and compared against EMD performance using both synthetically generated signals and real-life data. A performance metric based on the cross-correlation between the extracted signal and the original component was used. A variety of signal components including simple tones and frequency modulated signals were considered

87 74 for performance comparison. Relative superiority of the RCEMD technique for situations where signal separation is particularly difficult, such as when the instantaneous frequencies of the components are similar or when the high frequency signal is weaker, is a significant advantage of this technique. Further, successful signal decomposition using the developed technique for a cubic sinusoid as well as a multicomponent signal case with both amplitude- and frequency-modulated components was demonstrated. Reduced signal separation quality for very small signal instantaneous frequencies was observed as a potential drawback of this technique. Validity of this technique was further demonstrated by successful decomposition of real-world data into physically justifiable signal components. Thereafter, the two algorithms were compared based on computational complexity. To achieve equivalent signal separation quality, the proposed technique was shown to require fewer computations for particularly difficult signal decomposition problems. Finally, it was shown that the RCEMD algorithm requires much lower sampling rates than the EMD and an improved algorithm reported in literature to resolve two signal components. Performance comparison of RCEMD algorithm with other time-frequency analysis techniques, such as STFT and wavelets, follows in Chapter 7 for microbarograph data.

88 Chapter 4 Pre-emphasis and De-emphasis In this chapter we propose a new pre-emphasis and de-emphasis technique to improve the frequency resolution of EMD for a particular configuration of tonal signals with unequal amplitudes. Specifically, we present here a technique for improving the frequency resolution of the EMD technique when the lower tone has a larger amplitude by augmenting the higher tone before applying EMD and subsequently reversing the effect of frequency augmentation after the decomposition is completed. Practical filters to achieve pre-emphasis and de-emphasis are also introduced. Although this technique was developed to improve the performance of the original EMD algorithm, it can be applied to RCEMD also. 4.1 Introduction In this chapter we present results of our study on the frequency resolution of EMD. We demonstrate the effect of the choice of certain simulation control parameters on the performance of the algorithm and also suggest a new pre-processing step that improves its frequency resolution for simple tonal signals with unequal amplitudes. In Section 4.3 we focus on the inability of EMD to extract closely spaced tones when the lower tone is stronger. Although this problem is common to most time-frequency analysis techniques, the performance of EMD is especially bad in this case. This is due to the sequential nature of the technique where progressively lower frequency component signals are extracted consecutively. So, failure in separating the tones in the first step results in a single IMF to be generated by the procedure. To overcome this problem

89 76 we propose a technique where the original composite signal is pre-processed so that the strength of the higher tone is artificially increased in relation to the lower tone before performing EMD and the output from EMD is then compensated by a post-processing step that seeks to undo the distortion caused by the pre-processor. The effectiveness of this technique is demonstrated by simulations using synthetic signals. Many researchers have attempted to provide an analytical framework for this empirical technique [33, 47] and to gain better understanding about its working [59]. Some of these works include the study of the effect of sampling on EMD [62, 63] and its interpretation as a filter-bank [60]. Attempts to achieve better decomposition have led researchers to suggest using masking signals [37] and white noise at the signal boundaries [67]. But relatively few works (e.g., [29,37,67] in parts) have considered the issue of frequency resolution of the EMD algorithm which we aim to address in this chapter. In this work we have utilized the package available at [83] to implement the EMD technique. 4.2 Optimum choice of stopping criterion for sifting The sifting iterations are terminated when the residual signal satisfies the conditions for being an IMF. Two conditions are to be fulfilled in this respect [27]: the first one is that the number of extrema and the number of zero-crossings must differ at most by 1; and the second one is that the mean value of the upper and lower envelopes must be close to zero by some criterion. The authors of [29] have come up with a parameter that quantifies the second condition and lets the user set a pair of thresholds depending on the required level of adherence to the second condition above. Although there is no linear relation, generally large values for the thresholds result in fewer sifting iterations whereas lower threshold values cause more sifting iterations to occur. In this section we aim to demonstrate the effect of this stopping criterion as well as the choice of maximum iteration limit on the performance of the EMD technique. For this purpose we use a synthetically generated signal consisting of two closely spaced tones. We represent the original signal consisting of two superimposed tones as x(t) = s 1 (t) + s 2 (t) (4.1)

90 77 where s 1 (t) = A 1 sin(2π f 1 t + ϕ 1 ) s 2 (t) = A 2 sin(2π f 2 t + ϕ 2 ) (4.2) and f 1 > f 2. IMFs are represented by y 1,y 2,...,y N. We use the average correlation coefficient as the performance evaluation metric of the algorithm which is closely related to the error metric used in [29] r k = max i=1,...,n E{y i (t)s k (t)} E{y i (t) 2 }E{s k (t) 2 } (4.3) where k = 1,2 and the average correlation coefficient, r = E{r k }. The metric values lie between -1 and 1, with a large positive value indicating high correlation between the extracted IMFs and the original tones. So a larger average correlation coefficient value indicates successful separation of the tones by the EMD algorithm. The aim of our first set of experiments is to study the impact of the choice of stopping criterion on the frequency resolution of EMD using synthesized signals. To achieve this we create a composite signal consisting of two tones of equal strength (A 1 = A 2 ) and vary f 1 and f 2 over the entire frequency range from dc to Nyquist frequency in steps and evaluate the average correlation coefficient between the constituent tones and the extracted IMFs. The results of this experiment are presented in Fig. 4.1 as intensity plots for two different settings of the stopping criterion threshold: the default threshold value used by the software package (θ 1 = 0.05) and a reduced threshold (θ 1 = 0.001). Comparing the two plots it is clear that a lower threshold allows extraction of tones that are closer to each other than that possible with a higher threshold. The dark wedgeshaped regions just below the f 1 = f 2 line in the two plots represent frequency pairs for which tone separation is impossible. Following on the observation of the authors of [29] that the lower boundary of the wedge-shaped region is defined by the relation f 2 / f 1 = α, where α is some constant, our results indicate that the value of this constant is not fixed but dependent upon the value of the stopping criterion threshold chosen. Based on our observations θ 1 = 10 3 seems to be a reasonable choice for the case of two tones. Lower values result in more computational load with relatively little improvement in performance. Although the authors of [67] advocated a value of 10 5 for the threshold,

91 f 2 (Hz) f 1 (Hz) (a) θ 1 = 0.05 (default) f 2 (Hz) f 1 (Hz) (b) θ 1 = Figure 4.1: EMD of two equal-amplitude tones - Average correlation coefficient for two different stopping criterion thresholds are shown. The two axes are marked in units of normalized frequency and the gray-scale value represents the correlation.

92 79 Average correlation coefficient Default θ 1 = f 1 /f 2 Figure 4.2: EMD of two equal-amplitude tones - Average correlation coefficient is plotted against frequency ratio for different settings of maximum iteration limit and two different stopping criterion thresholds. while admitting that a value several orders of magnitude higher did not cause appreciable loss of performance, we believe that similar level of performance can be achieved for θ 1 = 10 3 at much lower computational effort. Another aspect of the results that needs mentioning is the fuzzy dark region in the right hand corner of the plots. To explain it must be remembered that for EMD to be able to decompose a composite signal sufficient amount of oversampling is required (a factor of five was suggested in [27]). So when the two tones are close to each other and close to the Nyquist frequency, the algorithm is unable to decompose them into separate IMFs resulting in a low correlation value. The next set of simulations provide further insight into the relation between stopping criterion and frequency resolution of EMD. The results presented in Fig. 4.2 show the variation of the average correlation coefficient with frequency ratio for different settings of maximum sifting iteration limit as well as stopping criterion threshold. It is evident from the plot that more number of sifting iterations results in higher correlation value for a particular frequency ratio thereby indicating better isolation of components. The improvement in performance achieved by using a lower threshold for the stopping

93 80 criterion compared to the default value is also evident. Another observation that can be made from the plot is that although fixing the maximum iteration limit to a large value allows separation of closely spaced tones, it results in wasted computation for tones that are farther apart (higher frequency ratios) that can be separated by fewer sifting iterations. So finding a way that automatically adjusts the number of sifting iterations depending on the closeness of the tones would seem advantageous. Fortunately, setting a low value for the stopping criterion threshold seems to achieve just that by allowing more iterations when the tones are closer and terminating the sifting process early when the tones are farther apart. Consequently, setting a low threshold value for the stopping criterion seems to offer a good combination of computational effort and performance relative to the difficulty of extraction. 4.3 Pre-Emphasis and De-Emphasis In this section we discuss a problem with the EMD technique when the two tones have unequal amplitudes. The inability of the EMD technique to extract closely spaced tones into separate IMFs when the lower tone is stronger is demonstrated in Fig The simulation set-up is similar to that described in Section 4.2 except that the amplitude of the lower tone is set to four times that of the higher tone in the present simulations. Upon comparing Fig. 4.1b and Fig. 4.3 it is clear that all other conditions remaining the same, the EMD algorithm has difficulty extracting two closely spaced tones when the lower tone is stronger as evidenced by the low correlation values for closely spaced tones in Fig The sequential nature of operation of the EMD algorithm means that the performance of the first stage extraction is the most important. If the first stage fails in extracting the higher tone then the result consists of only one IMF. A solution to this problem that seems natural is to attenuate the lower tone with respect to the higher one so that they have comparable amplitudes before letting the EMD algorithm operate on the modified signal. At the other end some compensation of the generated IMFs is required to reverse the effect of the pre-distortion. The frequency separation performance of this pre-emphasis/de-emphasis technique asymptotically approaches that for the original EMD operating on equal strength tones as better amplitude equalization is achieved. Some simulation results for this technique are presented in Fig. 4.4 and Fig. 4.5 where we compare the performance of the EMD with and

94 f 2 (Hz) f 1 (Hz) Figure 4.3: EMD of two tones of unequal strengths - Average correlation coefficient when the lower tone is stronger than the higher tone (A 1 = 1,A 2 = 4). The two axes are marked in units of normalized frequency and the gray-scale value represents the correlation. without our suggested modification for tones of different strengths. Pre-emphasis is performed using a differentiator and correspondingly an integrator is used at the other end for de-emphasis in our simulations. The two sub-figures in Fig. 4.4 show the average correlation coefficient for different frequency and amplitude ratios of the two tones for the original algorithm and our suggested improvement. Improved frequency resolution of the proposed technique is evident in these plots. Fig. 4.5 shows the Fourier transforms of a typical original composite signal and the first three IMFs generated by the original algorithm and our suggested improvement. Notice that the original algorithm is unable to separate the two tones and essentially produces just one IMF, whereas using our modified approach the two tones are captured in two separate IMFs. Moreover, the use of more sophisticated pre-emphasis and de-emphasis schemes is expected to yield even better results as far as extraction of closely spaced frequencies is considered.

95 A 2 /A f 1 /f 2 (a) Original EMD A 2 /A f 1 /f 2 (b) EMD + pre-emphasis/de-emphasis Figure 4.4: EMD of two tones of unequal strengths - Performance comparison between the original algorithm and the pre-emphasis and de-emphasis method. A differentiatorintegrator pair is used for pre-emphasis and de-emphasis in this case. The gray-scale values represent the average correlation coefficients for each amplitude and frequency ratio pairs.

96 Absolute FFT Absolute FFT f (Hz) (a) Original EMD f (Hz) (b) EMD + pre-emphasis/de-emphasis Figure 4.5: EMD of two tones of unequal strengths - Performance comparison between the original algorithm and the pre-emphasis and de-emphasis method. The figures show the Fourier transforms of a typical original composite signal and the first three IMFs generated by the respective techniques.

97 Conclusion The newly developed empirical mode decomposition (EMD) technique has been applied to many fields for signal analysis without a proper understanding its capabilities and limitations. Further a clear understanding of how the simulation control parameters affect its performance has eluded so far. In this work we have tried to answer some of these questions. Using synthetic signals we have shown that more number of sifting iterations results in better frequency isolation and have shown that setting the stopping criterion threshold to a low value results in more sifting iterations and therefore better frequency separation. Further we have pointed out the inability of this technique to separate two tones when the lower tone is stronger and have offered a pre-emphasis/de-emphasis technique to solve this problem. Simulation studies demonstrating the improvement in performance due to the modification were presented. Although the poor frequency resolution of the original EMD algorithm for certain signal condition is highlighted here and a remedy introduced, the same improvement in performance can be realized in case of RCEMD that also suffers poor frequency resolution for the same signal configuration.

98 Chapter 5 Overlay Communications using Raised Cosine Empirical Mode Decomposition A signal overlay technique employing the empirical mode decomposition procedure is presented here. A weak narrow-band signal is added to the primary signal that shares the same frequency band. Careful signal design reduces interference caused to primary users while ensuring successful recovery of the added signal. At the receiver a stationary filtering approach is ineffective in separating the signals because a fixed filter designed to isolate one of the signals will also capture significant portion of the other signal energy due to overlapping spectrums. However, the empirical mode decomposition technique, that isolates signal components based on their instantaneous frequencies, is ideally suited to separate these time-varying signals with overlapping frequency components. The choice of overlay signal transmission frequencies relative to that of the primary signal is made in such a way that leads to greater resemblance of one of the extracted components to the original overlay signal. An application to commercial frequency modulation overlay is introduced initially with associated analysis and empirical performance results. In the latter part of the chapter a covert communication technique based on this overlay procedure is discussed and its performance analyzed. In addition to the unique signal overlay structure discussed above, the covert technique employs frequency-hopping and directional antenna to enhance signal covertness.

99 Introduction Conventional communication systems are designed such that users are separated either in the time, frequency, code or spatial domain. Traditional overlay systems involve overlap in the time and frequency domains, but the users are distinguishable due to distinct codes or spatial positions. Here we describe a signal overlay procedure that involves signal overlap in all four domains. Successful communication between different users is possible due to distinct instantaneous frequencies (IF). We use the EMD algorithm to separate users based on their IFs. The EMD technique [27] provides an adaptive and efficient method to analyze nonstationary signals. Here, overlay communications using the EMD technique is investigated. We note that the tremendous growth in deployment of wireless technologies in the past has led to a heavily utilized spectrum with most frequency bands already assigned to licensed (primary) users for specific services. Although cognitive radio technology [84] admits co-existence of multiple radio signals within the same frequency band, identifying and utilizing spectral holes are challenging issues. In this chapter a signal overlay technique employing EMD is presented to improve spectrum utilization. A primary, licensed user of the frequency band is considered here. In this work a frequency modulated (FM) signal is considered as the primary signal. A secondary transmitter that is not necessarily co-located with the primary transmitter transmits another signal in the same frequency band. The secondary signal is weak and narrow-band relative to the primary signal. Due to the weak nature of the secondary signal, primary users of the frequency band experience negligible interference. At the same time, due to careful signal design, the secondary receiver is able to extract the secondary signal using the EMD procedure from the composite signal present at its input. Traditionally, signal overlay involves spreading the transmission bandwidth of the secondary signal so that its power spectral density is less than that of noise to eliminate interference to the primary user. Strategies that make this possible include the directsequence spread spectrum (DSSS) and ultra-wideband (UWB) signalling techniques ( [85] describes a DSSS overlay system). Due to the large transmission bandwidths associated with these techniques, the primary user s signal appears as narrow-band interference (NBI) to the secondary user. For DSSS signals two classes of interference

100 87 rejection techniques are used for general NBI: those based upon least-mean square estimation techniques, and those following transform domain processing principles [86,87]. Further, [88 91] describe DSSS overlay systems where the NBI is an angle-modulated signal, specifically an FM signal. Narrowband interference rejection techniques in UWB systems generally follow space-time receiver strategies [92 94]. In contrast to the preceding techniques that require a wideband secondary signal, the proposed technique involves transmission of a narrowband secondary signal occupying a portion of the primary signal spectrum. Primary users experience negligible interference due to the added signal because of its small relative power. A specific configuration of the secondary signal relative to the primary, based on their frequencies, that leads to successful decomposition using EMD is described here. This choice of frequencies, that is crucial for successful signal extraction, exploits the unique properties of the EMD procedure. Moreover, traditional nonstationary filtering techniques are ineffective in separating the signals due to their overlapping IFs. To summarize, the essential attributes of the proposed technique are as follows: The primary and secondary users overlap in the spatial domain. Simultaneous overlap in the time and frequency domains is also assumed. The two signals are have distinct IFs most of the time and are therefore separable by an appropriate technique that exploits this property. Fixed filters are ineffective in separating the signals due to overlapping IFs, since they have fixed pass- and stop-bands. A time-frequency approach that isolates the signals based on their IFs is required for signal separation. Specific design considerations and performance results for this new technique are presented in the following sections. Section 5.2 describes this signal overlay technique in greater detail and also gives a brief overview of the EMD procedure that is crucial for successful signal extraction. Section 5.3 addresses performance issues of this communication scheme and introduces an analytical performance measure. Simulation results are discussed in Section 5.4. A complex-valued version of this algorithm is introduced in Section 5.5 while Section 5.7 presents some concluding remarks.

101 Signal Design The simultaneous transmission of a weak secondary signal in the same frequency band of a primary user and its subsequent extraction is the objective of this work. The small power level of the secondary signal guarantees negligible additional interference to the primary user. This new technique is illustrated using an example of a weak frequency shift keying (FSK) signal superimposed on a FM broadcast signal. Despite the apparent similarity to cognitive radio technology, the proposed technique has a crucial difference: the secondary transmitter in the proposed technique transmits continuously, and therefore does not require spectrum sensing to identify holes. The secondary receiver performs EMD on the received signal (FM + FSK) to generate a series of elementary signal components, one of which corresponds to the transmitted FSK signal with high probability. The EMD technique decomposes the multicomponent signal into its constituents solely based upon the IFs present at any particular time and the relative amplitudes of the components (an analysis of the signal separation abilities of EMD as a function of relative amplitudes of the components is presented in [61] for the specific case of twotone signals). When the signal components satisfy certain conditions on the ratio of their IFs and their relative amplitudes, they can be extracted into distinct IMFs. However, when the IFs of the components intersect, the signal decomposition algorithm fails to extract the components into distinct IMFs and portions of each signal component are spread over several IMFs. The effect of this phenomenon, that is sometimes referred to as mode splitting in literature, on our proposed technique is examined in Section The choice of secondary signal frequency relative to the primary is based on the properties of the EMD algorithm. The choice is mainly influenced by two factors: one, that the secondary signal has a smaller amplitude relative to the primary to reduce interference to the users of the primary signal, and, second, that signal decomposition quality using EMD depends on the amplitude ratio of the signal components; higher quality resulting when the stronger component has a larger IF [44, 61]. Consequently, the choice of secondary signal frequencies is such that the IF of the secondary signal is smaller than that of the primary signal with high probability. In practice, this is accomplished by choosing frequencies near the lower edge of the FM band to insert the secondary signal.

102 89 r p + r s + n r p + r s + n f F M f I LP F EMD ˆr p ˆr s ˆn Figure 5.1: Block diagram of the secondary receiver. Formally, the combined signal as seen by the primary and secondary receivers may be represented as r (t) = r p(t) + r s(t) + n (t) (5.1) where r p(t) and r s(t) represent the primary and the superimposed secondary signals respectively and n (t) is the additive white Gaussian noise (AWGN) with power spectral density N 0 Watts/Hz. The secondary transmitter and receiver are located within the FM broadcast region. The maximum separation between the two is determined by the maximum overlay signal strength that can be transmitted without causing noticeable interference to nearby FM receivers and by the maximum allowable bit-error-rate (BER) at the overlay receiver. A more thorough discussion on these constraints is presented in the following sections. At the secondary receiver, the combined FM broadcast plus FSK signal having a bandwidth of B Hz is first frequency shifted to a range of 0 B Hz, generating a complex-valued signal. The EMD procedure is then applied to the downconverted signal to extract the secondary signal. Figure 5.1 shows the block diagram of the secondary receiver. In this case, f FM is the center frequency of the commercial FM band and the intermediate frequency, f I = B/2 Hz. After down-conversion the complexvalued signals are denoted in the same fashion as in (5.1), but without the primes. The validity of the technique is demonstrated by applying real-valued EMD algorithm to the composite signal in the following two sections. Section 5.5 introduces the complexvalued algorithm. The reason for the frequency down-conversion step follows. As pointed out in [61, 95] in the context of separation of pure tones of frequencies f 1 and f 2 ( f 1 > f 2 ), the quality of extraction depends upon the ratio f 1 / f 2. A larger frequency ratio for a given amplitude ratio results in better signal decomposition quality. By down-converting the

103 (a) (b) (c) (d) t (ms) Figure 5.2: Time domain representation of the composite signal (FM + FSK) and the extracted components by the EMD algorithm for a time interval extending over 3 FSK symbol durations. In (c) the actual FSK signal is shown by the dashed curve. signals, we increase the ratio of the IFs of the two signals even though their absolute difference remains the same resulting in better decomposition quality. 5.3 Performance Analysis Typical extraction results of the EMD algorithm and IFs of the FM and FSK signals are presented in Fig. 5.2 and Fig. 5.3 respectively. Here only the real part of the complex baseband signal is considered. In general, the initial IMFs that EMD generates contain the high frequency components of the signal, including noise, and the subsequent IMFs contain the lower frequency components [96]. Figure 5.2(a) shows the composite signal (FM + FSK) as observed at the secondary receiver for a low noise case and the extracted components from the EMD algorithm are shown in Fig. 5.2(b)-(d). Clearly, the first IMF corresponds to the FM signal, the second resembles the overlaid FSK signal and the remaining IMFs (of which only the third is shown in the figure) contain the residual FSK signal energy. It is evident from the actual FSK waveform and the second IMF

104 Instantaneous frequency (khz) t (ms) Figure 5.3: Instantaneous frequencies of the FM signal (solid) and the FSK signal (dashed) for a time interval extending over 5 FSK symbol durations. The hatched area indicates the times when signal decomposition is not possible. in Fig. 5.2(c) that although the second IMF closely approximates the original signal over certain intervals, at other times it deviates from the original due to superimposed amplitude and frequency modulations. The superimposed amplitude modulated (AM) signal and the newly introduced frequencies in the second IMF can be explained based on the IFs of the two signals, shown in Fig For an FM signal represented by the IF is given by [97] r p (t) = A p cos 2π f I t + k f t m(τ)dτ (5.2) f inst = f I + k f m(t) (5.3) 2π where k f represents the frequency-deviation constant (bandwidth of the FM signal is directly proportional to k f ), m(t) is the modulating/information signal and A p is the

105 92 amplitude of the FM signal. This IF is shown in Fig. 5.3 for an arbitrary modulating signal. The IF of the FSK signal is also shown in the figure. Due to the intersecting IFs, common nonstationary signal decomposition techniques result in poor signal decomposition quality. However, using the EMD procedure, when the IF of the primary signal is significantly larger than that of the secondary signal, extraction is successful and high fidelity separation of the signal components into different IMFs occurs. When the primary signal s IF is either close to that of the secondary signal or smaller, the EMD extraction fails to produce an IMF that contains a useful copy of the FSK signal. The algorithm fills the interval with a signal having smoothly varying amplitude and random frequency to maintain phase continuity with the correctly extracted portion of the secondary signal. The affected intervals are shown by the hatched area in Fig To avoid this phenomenon, the overlay FSK frequencies are chosen such that they are smaller than the IFs of the primary signal with high probability. Thus, frequencies near the lower edge of frequency band of the primary signal are selected for transmitting the secondary signal. Moreover, the EMD algorithm is more effective in signal separation when the stronger signal has a larger IF [44] Choice of decomposition level The number of IMFs generated in the overlay receiver are in concordance with the signal overlay design presented in the previous section. For small E b /N 1 0, mode splitting, i.e., the distribution of the constituent signal components across multiple IMFs, is likely and reconstituting the secondary FSK signal for detection is intractable. In short, for small E b /N 0 the error rate is large regardless of the number of IMFs generated. For larger E b /N 0, the EMD reliably separates the overlay FSK signal into the second and subsequent IMFs. We see experimentally that the second IMF produces the highest fidelity representation of the FSK signal, however, smaller residuals of the FSK signal may still appear in higher order IMFs if they are generated. Thus, the overlay FSK receiver only generates the first two IMFs and always uses the second IMF for FSK detection. This approach has a two-fold advantage. First, it improves error rate performance over a higher-order decomposition because it does not permit the FSK signal to distribute among multiple IMFs. Second, it reduces the receiver complexity since only 1 E b /N 0 is a measure of the signal-to-noise-ratio (SNR), also known as SNR per bit, and is a commonly used metric in communication literature.

106 93 two IMFs are generated and IMF selection is not an issue. We can track the performance degradation when the overlay receiver generates and uses more than two IMFs from that presented in Section 5.4 (Fig. 5.11). Taking BER=10 2 as a reference, the overlay receiver loses 2 db if it generates all IMFs and uses the one with the largest decision statistic for detection. The overlay receiver loses 3 db if it generates all IMFs and always uses the second IMF for detection. Both of these alternate approaches only use a portion of the FSK signal for detection. This assertion is further validated when the overlay receiver attempts to collect the FSK signal by using the sum of the second and third IMFs, which results in small fractions of a decibel degradation from that shown in Fig Performance approximation Here we develop a model for the extracted secondary signal using EMD that leads to a simple formula for the resulting BER performance. Based on the above discussion, it is clear that second IMF generated by the algorithm is a high fidelity approximation of the original FSK signal over the interval when the IF of the FM signal exceeds that of the FSK signal, and has little resemblance to the transmitted signal when the order of the IFs gets reversed. So corresponding to the two conditions we model the second IMF either as an exact copy of the transmitted FSK signal or as a signal with random frequency and amplitude over the respective intervals. As a result, the cross-correlation coefficient between the segments of the second IMF containing random frequencies and the original FSK signal is zero, on average. To model this, we represent by T 1 and T 2 the average amount of time within an FSK bit interval, T, that the two FSK frequencies are larger than the FM IF, f inst. The secondary BER can then be derived by suitably adjusting the E b /N 0 value in the standard non-coherent FSK result to reflect the fraction of the total signal energy that is useful in FSK detection. Then the resulting BER for non-coherent FSK is P b = 1 2 e 1 2 ( Eb N 0 ) e f f (5.4) where ( Eb ) N 0 e f f = E b N 0 (1 ξ ) (5.5)

107 94 where ξ = T 1+T 2 T is the fraction of time when the FSK IF is larger than f inst. As a result, (1 ξ ) represents the fraction of the signal energy that contributes to successful symbol detection. The BER derived from this simple model and simulation results are shown in Section 5.4 (Fig. 5.6), where close agreement is observed. Analyzing the system performance based on the relative IFs of the two signals as described above provides insight into the working of the technique, thereby allowing a judicious choice of secondary signal transmission frequencies. It is clear that choosing the FSK transmission frequencies near the upper and lower bounds of the FM IF range results in maximum and minimum duration of unfavorable IF configuration, corresponding to the largest and smallest error in the extracted signals using EMD, respectively for a given E b /N 0 (See Fig. 5.3). Therefore, the minimum IF attained by the FM signal determines the smallest FSK transmission frequency in this technique. In practical situations, the FM modulating signal amplitude bears a linear relationship to its IF and ultimately determines the duration over which the EMD algorithm fails to correctly decompose the two signals. Due to the statistical nature of the modulating signal, it is simulated using a stochastic model for performance verification in Section Simulation Results FSK BER and FM normalized mean-squared-error (NMSE) for the overlay technique using EMD are presented in Fig and Fig. 5.5 respectively. The NMSE for FM signal, ε, is defined as ε = r p v 1 (t) r p (5.6) where v 1 (t) represents the first IMF and. indicates the L 2 norm defined for a function f (x) as follows f (x) = ( f (x) 2 dx) 1/2. Since only two IMFs are generated in our technique, the second IMF, v 2 (t) is used for FSK detection. The drop in BER and NMSE values at large E b /N 0 indicates that the two signals are successfully separated into the intended IMFs, as discussed in Section 5.3. For these results, the received FSK signal was 26 db weaker than the FM signal. Also, secondary data transmission rate of 5 kbps was simulated and an over-sampling

108 BER EMD WVD AF Remod E b /N 0 (db) Figure 5.4: E b /N 0 -vs-ber plots for the signal overlay scheme. Performance of the proposed technique is compared with that of the Wigner-Ville distribution (WVD)-based technique and the adaptive filter (AF) technique. factor of 10 was employed due to dense sampling requirement of EMD for accurate determination of local extrema, as discussed in Section 5.2. So, voice communications and low-rate data communications using error-correcting codes are possible applications of this overlay scheme. The FM modulating signal in our experiments is generated using a general stochastic time-varying model, namely a first-order auto-regressive (AR(1)) model. The choice of low-order filter-model is based on our experimental observation that the proposed technique is insensitive to the model-order used for signal generation for the same signal bandwidth. In Fig BER performance results for three alternate techniques are also presented. The first method is a two-step procedure where the IF of the FM signal is first estimated using the Wigner-Ville distribution (WVD) and then a short, time-varying finite impulse response (FIR) notch filter is designed to remove that frequency [88]. The time-varying nature of the primary signal requires a short length notch filter, which corresponds to a wide notch in the frequency domain, thereby significantly distorting the secondary signal. In the original scenario in [88] since the interference (FM) signal oc-

109 NMSE E b /N 0 (db) Figure 5.5: Normalized mean square error (NMSE) between the actual FM signal and the first extracted IMF versus E b /N 0. cupied a small fraction of the frequency band of the signal of interest (DSSS signal), distortion of the entire band of frequencies containing the interferer was acceptable. However, in the present case a wide notch filter, in addition to eliminating the primary signal, also severely degrades the secondary signal. Secondly, we study the performance of a simple adaptive filter (AF) based on the least-mean-square (LMS) algorithm at removing the primary signal at the secondary receiver. Poor signal separation quality results due to similarity of the autocorrelation functions of the constituent signals and due to the time-varying nature of the primary signal. The final method that we study here involves subtracting a resynthesized FM signal from the received signal to generate the FSK signal. We use a first-order phase locked loop (PLL) to demodulate the FM signal from the received composite signal (FM+FSK). The estimated modulating signal, thus derived, is then used to remodulate a carrier signal which when subtracted from the received signal produces an estimate of the secondary FSK signal. However, due to the noisy input to the PLL, the resynthesized FM signal is not identical to the original FM signal, resulting in the appearance of

110 FSK BER (Sim.) FSK BER (Anal.) BER E b /N 0 (db) Figure 5.6: E b /N 0 -vs-ber plot using the proposed signal extraction technique. BER values derived from theory (Equation 5.4) and those obtained from simulation for a range of E b /N 0 are shown. some FM signal energy in the difference. Figure 5.7 shows the block diagram for this receiver. It is observed from Fig. 5.8 that the BER for this remodulation technique saturates for large E b /N 0 because the residual phase error at PLL output due to noisy input (FM+FSK+thermal noise) is essentially limited by the FSK signal amplitude, which is independent of E b /N 0. To validate the performance result of the PLL technique obtained from simulations, we followed a semi-analytical method to study the performance. Here, the response of the PLL due to noisy input is modeled as random phase error of certain level [98]. The detected signal including the random phase error is then used to resynthesize the FM signal as described above. The BER of the FSK signal resulting from this semi-analytical method and that obtained from simulating a PLL are shown in Fig. 5.8 for a range of E b /N 0 values. The closeness of the two plots validates our results.

111 98 f F M r p + r s + n F M Demodulator (P LL) ˆm F M Modulator ˆr p + ˆr s + ˆn Delay Figure 5.7: Block diagram of the receiver using remodulation technique Effect on primary users To analyze the effect of the secondary signal on the users of the primary signal we note that at any instant the FSK signal appears as a tone interferer to the FM receiver. It has been shown that the output of an ideal frequency demodulator due to a tone interferer is given by [97] y s (t) = A s A p 2π( f I f k )cos(2π( f I f k )t) (5.7) where k = 1, 2 corresponding to the two FSK frequencies and A p and A s are the amplitudes of the primary and secondary signals respectively, with A s A p. Since the interference output is inversely proportional to the primary signal amplitude, the weak interference is suppressed and so the interference level must be at least 6 db weaker than the FM signal to avoid objectionable interference to the FM listener [97]. To quantify the distortion caused at the output of an ideal frequency demodulator due to the combined effect of the secondary (FSK) signal and background noise, we introduce a measure η = ˆm(t) m(t) 2 m(t) 2 (5.8) where. indicates the L 2 norm as discussed earlier and ˆm(t) is the estimate of modulating signal, m(t) produced by the ideal frequency demodulator. Simulation results indicate that the additional distortion introduced due to insertion of the secondary signal is less than 2 db more than that caused by background noise alone for the entire of range of secondary signal and noise levels shown in Fig This distortion is small

112 PLL (Order 1) Added phase noise BER E b /N 0 (db) Figure 5.8: Comparison of FSK BER derived from simulation of PLL and semianalytical method that models the phase error at the PLL output. Similar results from the two techniques is evident. at low values of E b /N 0 where the effect of background noise is strong, but increases at large values of E b /N 0 when the effect of the added secondary signal dominates. Also, our studies indicate that any possible aliasing due to frequency down-conversion has negligible effect on the performance, implying that this technique is not alias-limited. 5.5 Operations on the Complex Baseband Signal So far we have considered only real-valued signals for secondary signal detection using EMD. However, it is possible to further reduce the error rate by exploiting the information available in the in-phase and quadrature-phase components of the baseband signal as a whole. In this section we will quantify the performance improvement realized by utilizing the inherent symmetry of the complex signal components. A logical way to extend the EMD technique to the complex domain is to independently apply the algorithm to the two signal components, and thereafter computing the log-likelihood ratio (LLR) corresponding to each FSK bit. The final bit decisions are

113 BER, real BER, cmplx. BER, BEMD 10 2 BER E b /N 0 (db) Figure 5.9: E b /N 0 -vs-ber graph showing performance improvement offered by performing signal decomposition on the complex signal. based on the combined LLR. This represents an straight-forward way to apply the EMD algorithm to complex signals. The bivariate EMD algorithm [55] (which is a generalization of the algorithm introduced in [56]) is an extension of the original EMD algorithm to the complex domain. This algorithm operates on complex-valued signals, generating bivariate IMFs. Applied to complex-valued signals, slow oscillations are extracted as the mean of a three-dimensional envelope enclosing the signal. This envelope is computed by considering a fixed number of directions: the signal is projected in each direction and an envelope is computed for each of these real-valued signals using the original EMD procedure. Projecting these envelopes back into the complex domain defines the shape of the three-dimensional envelope at each time instant. Higher fidelity definition of the three-dimensional envelope due to more numerous projecting directions has been claimed [55]. The mean of this envelope is analogous to the mean signal derived using classical EMD. We next consider the achievable performance for the overlay technique when the complex-valued baseband signal is decomposed. Figure 5.9 shows these results. Two

114 101 Spectrum of primary and covert signals x 105 Frequency band of frequency hopped FSK (covert) signal Primary (FM) signal Covert (FSK) signal f (khz) Figure 5.10: Frequency domain representation of the primary (FM) and frequencyhopped covert (FSK) signals shown here. The primary signal power is 26 db larger than that of the FSK signal in this illustration. methods are analyzed: one is the simple application of classical EMD to the complexvalued signal components separately, and the other is the bivariate EMD algorithm. The performance of the two techniques is found to be approximately equivalent and performance improvement of about 2 db is observed at E b /N 0 = 10 4 over the classical approach operating on the real signal. The bivariate EMD algorithm employed here used 16 directions for projecting the signal. A drawback of the complex domain signal decomposition is the increased computational load: twice in case of the complex algorithm, and 16 times for bivariate EMD over the classical EMD approach. Moreover, results for competing techniques are not shown in Fig. 5.9 because the WVD and remodulation results in Fig already considered complex-valued signals. Further, no improvement in performance is observed for the LMS method when complex-valued signals are considered because of the similarly shaped autocorrelation functions of the two signals.

115 Covert Communications using Empirical Mode Decomposition Covert communication has traditionally involved either spreading the bandwidth over which the signal is transmitted so that its power spectral density is smaller than that of noise or changing the carrier frequency of the signal rapidly to avoid detection. Developments in this area have given rise to the whole field of transmission security (TRANSEC) that deals with signals possessing low probability of detection (LPD), low probability of interception (LPI), low probability of exploitation (LPE) and anti-jam (AJ) features [99]. In some situations it may be useful to hide the presence or existence of the communicator s signal. Traditionally, this has meant that at an unintended receiver, the communicated signal plus receiver noise and interference cannot be reliably distinguished from just receiver noise plus interference. Therefore, LPD signals refer to those that make it difficult for unintended receivers to detect them. Here we propose a LPD signal design strategy that aims to hide the signal under a stronger primary signal corresponding to an existing legitimate communication or broadcast service. The covert signal design ensures that it is undetectable by primary users while at the same time allows reliable recovery at the covert receiver. This concept is demonstrated here using an example of a covert transceiver on the ground communicating with an aerial vehicle using a weak narrow-band signal. This signal is superimposed on an existing primary signal that is analogous to a cover signal. The covert receiver, at the other end, receives the composite (primary + covert) signal and extracts the weak covert signal by performing signal decomposition using RCEMD [95]. Distinct spatial positions and codes between the primary and covert users can be additionally used to improve the performance of the technique in terms of achievable data rate or communication range. This is achieved by the use of directional antenna by the covert transmitter to reduce the observable covert signal power by the primary receivers on the ground. Frequency hopping (FH) is also introduced to make signal detection by the unintended receiver more difficult. Further, no cooperation is assumed between the primary and covert signal transmissions and this technique does not depend upon the presence of spectral nulls in the stronger primary signal for successful communication. The signal design for covert communications is similar to the earlier case of overlay communications, involving commercial FM broadcast signal as the primary (cover) sig-

116 103 nal. However, for the covert signal we consider two kinds of modulation for demonstration and performance evaluation: frequency shift keying (FSK) and quadrature phase shift keying (QPSK). The covert receiver performs the RCEMD procedure on the received signal (FM + FSK/QPSK) to generate a series of elementary signal components, one of which corresponds to the transmitted FSK or QPSK signal. The signal design in this case resembles the previous instance and is therefore omitted here. However, the frequency spectrums of the primary and secondary signals is shown in Fig to illustrate the relative frequencies and amplitudes of the concerned signals for a typical case. In the following the proposed technique is described based on the choice of FSK as the covert signal modulation. The same analysis applies to QPSK modulation also, and is therefore omitted. However achievable error rates for both modulation types obtained from computer simulations are presented separately. The mathematical expression derived for the overlay communication technique in Section applies to an FSK secondary signal. An analogous expression when the secondary signal is QPSK modulated is ( Eb N 0 )e f f ( P b = 1 (Eb ) 2 er f c N 0 e f f ) (5.9) where is calculated exactly as in (5.5), where in this case ξ = T 1 T is the time fraction for which the QPSK signal frequency is larger than the FM IF, f inst. For a given covert signal transmission frequency, the time fraction (ξ ) can be easily calculated. Error rate estimates from this simple model are compared with simulation results in Section (Fig. 5.13) Simulation Results To improve the degree of covertness of the inserted signal we employ the FH principle. In this example we use slow FH with 16 hop frequencies to reduce signal detection probability by an unintended receiver. The BER-vs-E b /N 0 results when the covert signal is FSK and QPSK modulated are shown in Fig and Fig respectively. For these results, the received covert signal was 26 db weaker than the FM signal. Hop frequencies were confined within a band stretching from 0.4B to 0.8B. Also, secondary data transmission rate of 5 kbps was simulated and an over-sampling factor of 10 was

117 BER RCEMD WVD AF Resynthesis E b /N 0 (db) Figure 5.11: E b /N 0 -vs-ber plots when covert transmitter uses FSK modulation. Frequency hopping is used with 16 hop frequencies BER RCEMD WVD AF Resynthesis E b /N 0 (db) Figure 5.12: E b /N 0 -vs-ber plots when covert transmitter uses QPSK modulation. Frequency hopping is used with 16 hop frequencies.

118 QPSK BER (Simul.) QPSK BER (Theor.) 10 3 BER E b /N 0 (db) Figure 5.13: Cross-validation of error rate performance derived from simple numerical model (Equation 5.9) and computer simulation output for QPSK modulated covert signal. Ratio of primary to covert signal power is 26 db here. employed. So, voice communications and low-rate data communications using errorcorrecting codes are possible applications of this overlay scheme. For our experiments the FM modulating signal is modeled as a filtered noise output of a first-order autoregressive (AR(1)) model. In Fig and Fig BER performance results for three alternate techniques are also presented. The first method is a two-step procedure where the IF of the FM signal is first estimated using the Wigner-Ville distribution (WVD) and then a short, time-varying finite impulse response (FIR) notch filter is designed to remove that frequency [88]. The time-varying nature of the primary signal requires a short length notch filter, which corresponds to a wide notch in the frequency domain, thereby significantly distorting the secondary signal. In the original scenario in [88] since the interference (FM) signal occupied a small fraction of the frequency band of the signal of interest (DSSS signal), distortion of the entire band of frequencies containing the interferer was negligible. However, in the present case a wide notch filter, in addition to eliminating the primary signal, also severely degrades the secondary signal. Secondly, we study

119 t (ms) a b c d Figure 5.14: Illustration of signal analysis quality of several techniques. The four techniques described in the text are considered: RCEMD, WVD, AF and Resynthesis, and their estimates for the covert signal superimposed on the actual signal. the performance of a simple adaptive filter (AF) based on the least-mean-square (LMS) algorithm at removing the primary signal at the secondary receiver. Poor signal separation quality results due to similarity of the autocorrelation functions of the constituent signals and due to the time-varying nature of the primary signal. The final method that we study involves subtracting a resynthesized FM signal from the received signal to generate the FSK signal. We use a first-order phase locked loop (PLL) to demodulate the FM signal from the received composite signal (FM+FSK). The estimated modulating signal, thus derived, is then used to remodulate a carrier signal which when subtracted from the received signal produces an estimate of the secondary FSK signal. However, due to the noisy input to the PLL, the resynthesized FM signal is not identical to the original FM signal, resulting in the appearance of some FM signal energy in the difference. It is observed from results that the BER for this resynthesis technique saturates for large E b /N 0 because the residual phase error at PLL output due to noisy input (FM+FSK+thermal noise) is essentially limited by the FSK signal amplitude, which is independent of E b /N 0. When QPSK is used for modulating the covert signal the BER saturates at large E b /N 0 due to the same reason as seen in Fig Finally, we demonstrate the relative decomposition quality of the different tech-

120 107 RCEMD WVD AF Resynthesis 10 0 NMSE E b /N 0 (db) Figure 5.15: Numerical comparison of decomposition quality for several techniques is presented here. The normalized mean squared error between the actual and estimated covert signal is shown for four algorithms described in text. niques introduced above. Figure 5.14 presents the estimate of the covert signal obtained using the four methods considered in this chapter, superimposed on the original covert signal. Although severe distortion due to the AF technique is evident, the RCEMD, WVD and Resynthesis results appear better, with the RCEMD output resembling the original signal the most. A quantitative measure of the decomposition quality is shown in Fig. 5.15, where the normalized mean-squared-error (NMSE) between the extracted components using different techniques and the original signal is presented for varying E b /N Communication Range Determination The maximum communication range for the covert users is limited by two constraints. The minimum SNR required at the covert receiver for reliable signal detection is one constraint. Large transmit power is desirable in this case. The other requirement is that users of the primary signal in the vicinity of the covert transmitter be unaffected by the inserted signal. This requires small transmit power. It is evident that the two

121 108 Maximum range of aerial vehicle (km) FSK QPSK Distance of covert transmitter from FM tower (km) Figure 5.16: Maximum achievable range for the covert communication technique is shown here. The plotted horizontal ranges of the aerial vehicle are the maximum possible to ensure BER < 10 5 for the respective modulation types. requirements place opposing constraints on the transmit signal power. To calculate the useful communication range for this technique we first find that largest allowable transmit power for the covert user that allows normal signal reception by nearby primary users and then find the maximum range at which the covert receiver can be located for reliable signal detection for this power level. To analyze the effect of the secondary (covert) signal on the users of the primary signal we note that at any instant the FSK signal appears as a tone interferer to the FM receiver. It has been shown that the output of an ideal frequency demodulator due to a tone interferer is given by [97] y s (t) = A s A p 2π( f I f k )cos(2π( f I f k )t) (5.10) where k = 1, 2 corresponding to the two FSK frequencies and A p and A s are the amplitudes of the primary and secondary (covert) signals respectively, with A s A p. Since the interference output is inversely proportional to the primary signal amplitude, the

122 109 weak interference is suppressed and so the interference level must be at least 6 db weaker than the FM signal to avoid objectionable interference to the FM listener [97]. A directional antenna pointed towards the aerial vehicle reduces the interference to FM receivers on the ground. Sidelobe power of 18 db is assumed in our studies. So although this limits the secondary transmit power to 12 db larger than the FM level, we assume that the covert transmitter transmits at only 6 db above the primary signal level for safety. Based on these values, the maximum distance of an aerial vehicle flying at an altitude of 33,000 ft. (10,000 m) to attain a BER of 10 5 (corresponding to E b /N 0 of approximately 18 db and 11 db from Fig and Fig for FSK and QPSK modulations respectively) is presented in Fig for various distances of the covert transmitter from the FM tower. The reason for the drop in range of the covert communication as the transmitter is placed farther away from the FM tower follows from our discussion above where it was pointed out that the power of the covert transmitter depends upon the FM signal power at the location. Due to decrease in primary signal power away from the FM tower, the covert transmitter is required to reduce its transmit power to prevent interference to neighboring primary users. This reduces its effective communication range. However, in the reverse direction (from aerial vehicle to ground) the power constraint is less stringent as the covert signal is required to be weaker than the primary signal only near ground level and so the aerial vehicle can transmit at a higher power level than the transmitter on ground. Therefore, the effective range of this system is decided by power requirements in the ground-to-air direction. 5.7 Conclusions A technique for signal overlay where a weak secondary signal is transmitted in the same frequency band as an existing primary signal was demonstrated in the first part of this chapter. The EMD algorithm was used to separate the primary signal from the secondary at the receiver. Although functionally similar to cognitive radio technology, the proposed technique involves continuous transmission by the secondary transmitter, without requiring spectrum sensing. The resulting hardware simplification is accompanied by a constraint on the secondary signal: it must be significantly weaker than the primary to reduce interference caused to primary users. The success of the proposed technique at acceptable secondary signal level was demonstrated in this chapter.

123 110 A practical example of an FSK signal superimposed on commercial FM signal was presented and the performance studied. The performance of the described technique was compared to some other techniques: one, involving estimating the IF of the primary and then filtering it from the received signal; second, involving an adaptive filter to separate the signals; and finally, subtracting a resynthesized primary signal from the received signal. The advantage of EMD over these techniques was demonstrated. Application of this technique to complex-valued baseband signals was shown to produce further performance improvement. Subsequently, a new covert communication technique based on the principle of signal overlay is introduced that uses RCEMD. In the proposed technique, probability of detection by an unintended receiver is reduced due to masking by the strong primary signal. Frequency hopping further enhances signal covertness. Moreover, the use of directional antenna by the covert transmitter to communicate with an aerial receiver reduces the interference caused to nearby terrestrial FM receivers. This technique permits covert voice communication and data transmission from unattended sensors in possibly hostile territories. This technique is inherently resistant to jamming due to the difficulty of blocking the covert signal without significantly degrading the primary signal. Performance of this technique in terms of achievable BER and communication ranges were studied.

124 Chapter 6 Wideband Interference Removal using Raised Cosine Empirical Mode Decomposition Nonstationary interference suppression in wireless communications is addressed in this chapter. Here a novel algorithm for partial band interference excision in direct sequence spread spectrum (DSSS) communication systems like the Wideband Code Division Multiple Access (WCDMA) air interface standard used in third generation mobile telecommunication networks is presented. The excision algorithm consists of two stages: signal decomposition using the RCEMD technique, followed by despreading of appropriate extracted component. An advantage of this technique is that it does not use an implicit parameterized model for the interference signal to perform excision and is applicable to a wide variety of interfering signals such as multiple tones and frequency modulated signals with complicated instantaneous frequencies. The bit-error-rate (BER) performance of this technique is studied and compared against some existing techniques for partial band interference excision. 6.1 Introduction Wireless communication systems are often faced with the problem of jamming interference that could severely distort the transmitted information. This interference could be intentional, such as jamming in military communication systems, or unintentional, such

125 112 as interference from electromagnetic energy emitters. Further, the interference may be narrowband or wideband; it may be nonlinear and time-varying if its spectral components are continuously changing with time in a nonlinear fashion. DSSS techniques involve spreading narrowband information using a pseudo-noise (PN) noise sequence before transmission over wireless channels. As the transmission signal is now wideband, it is less susceptible to unintentional or intentional jamming with a narrowband interference signal. Thus, resistance to jamming is an advantage of spreading techniques. As a result, when the processing gain of the system can accommodate high jamming-to-signal ratios (JSRs), no interference mitigation technique is required. However, for larger JSR values, efficient suppression techniques must be investigated. Moreover, development of interference suppression techniques allows coexistence of the DSSS signal and another relatively narrowband signal, thereby resulting in better spectrum utilization. A comprehensive overview of the early work on narrowband interference (NBI) rejection techniques can be found in [86]. Two classes of rejection schemes are described there: (a) those based on least-mean square (LMS) estimation techniques, and (b) those based on transform domain processing structures. The improvement achieved by these techniques is subject to the constraint that the interference be relatively narrowband with respect to the spread signal. An overview of NBI suppression in DSSS communications with focus on code division multiple access (CDMA) communications is given in [100]. There categorization into linear techniques, nonlinear estimation techniques and multiuser detection techniques is discussed. A more modern overview of the developments in the field can be found in [101]. A new technique for nonstationary interference suppression in DSSS communications, applicable to partial band interference is presented here. Time-frequency methods such as the Wigner-Ville distribution (WVD) have been applied to suppress timevarying interference such as linear frequency-modulated (LFM) chirp signals. One possible method to suppress LFM interference is by computing the WVD of the received signal, masking the WVD of the LFM interference in the two-dimensional (2-D) timefrequency plane, and then using the WVD least-squares synthesis technique to obtain an estimate of the interference. The estimated interference is then subtracted from the received signal before detection [90]. Other methods for wideband time-varying interference removal include time-frequency adaptive filtering, fractional Fourier trans-

126 113 form [102], matched signal transforms [103], use of discrete evolutionary and Hough transforms [104], decorrelating time-varying autoregressive model [105] and chirplet time-frequency decomposition [106]. Most of these techniques, however, require simple variation of the interference instantaneous frequency or assume constant interference amplitude, that may sometimes limit their utility. In this chapter we propose a new technique to suppress partial band interference for DSSS signals encountered in wideband code division multiple access (WCDMA) communications using the RCEMD technique [95, 107]. Here the RCEMD algorithm is applied to the interfered signal and at its output high fidelity approximations of the interfering and the spread spectrum signals are obtained. Signal despreading is then applied to the signal component corresponding to the spread spectrum signal. The performance of this technique is studied for different interference types: multiple tones jamming a portion of the spread spectrum band representing a stationary jammer, and a frequency-modulated (FM) jammer that represents time-varying jamming. Different modulating signals are considered for the FM jamming signal such as a pure tone and filtered noise. In Section 6.3 interference cancelation performance of other competing techniques such as adaptive filters, Wigner-Ville distribution-based filtering and signal resynthesis is also studied for identical interference conditions and compared against the proposed technique. Specific design considerations and performance results for this new technique are presented in the following sections. Section 6.2 presents the signal model adopted in this work and details of the interference removal technique. Simulation results are presented in Section 6.3 where a variety of cases are discussed. Section 6.4 presents some concluding remarks. 6.2 Signal Design and Excision Procedure The DSSS system model used here consists of a transmitter that generates a spread spectrum signal which in turn is transmitted over a communications channel as a binary phase shift keying (BPSK) modulated signal. Additive channel noise as well as jamming signal act on the transmitted signal. At the receiver, the noise and interference corrupted signal is first demodulated. The standard spread spectrum receiver correlates the baseband spread spectrum signal with the synchronized PN sequence, and the

127 rk sk, y (1) k ik, y (2) k time samples Figure 6.1: Representation of time-domain signals. Top panel shows the received signal comprising spread spectrum, interference and noise signals. Middle panel shows the first extracted component of the decomposition algorithm superimposed on the spread spectrum signal. The final panel shows the second extracted component superimposed on the FM interference signal. f F M r S + r J + n F M Demodulator (P LL) ˆm F M Modulator ˆr J + ˆr S + ˆn Delay Figure 6.2: Block diagram of the interference excision by resynthesis technique. Here r S, r J and n correspond to the spread spectrum, interference and noise signals, represented by s k, i k, and n k, respectively in text. In this technique the interfering signal is estimated by subtracting a resynthesized FM signal from the received signal. The FM modulating signal which is estimated using a phase-locked loop then re-modulates a carrier and this signal approximates the FM signal.

128 115 resulting signal is processed and input into a threshold detector to estimate the transmitted binary data sequence. Let b k = ±1 be the kth message symbol transmitted in a DSSS system such that s k = b k c k (6.1) where c k = [c k (0),...,c k (L 1)] T for {k = 1,2,...} is a PN sequence with a chip length L, c n = ±1 is the nth chip of the PN sequence, and s k is the DSSS signal. The received signal r k at the output of the BPSK demodulator will consist of the DSSS signal s k, additive white Gaussian noise (AWGN) term n k, and interference term i k such that r k = s k + n k + i k. (6.2) At the receiver, to estimate b k, we use the PN sequence c k to despread r k, and integrate the result to generate the test statistic Λ k : Λ k = r k,c k = c T k r k = Using the test statistic Λ k, we estimate the message symbols as bk = L 1 c k (n)r k (n). (6.3) n=0 { +1, if Λk 0, 1, if Λ k < 0. (6.4) In the proposed technique the received signal r k is decomposed into two components using the RCEMD algorithm, y (1) k and y (2) k. Analogous to wavelet decomposition, the generated components have a decreasing trend of instantaneous frequencies. However, unlike wavelets, the transition frequencies are not fixed; they are signal dependent. Since the on-center interference has a smaller IF relative to the DSSS signal for a large time fraction, it manifests itself in the second component, whereas the first generated component closely resembles the spread spectrum signal. Therefore, the first component, y (1) k is despread using the PN sequence to generate the test statistic Λ k in a similar fashion as represented in (6.3). The decomposition is stopped at the first level, after generation of two components because the signal component corresponding to the spread spectrum signal is generated first and the residue consists of interference and noise components

129 116 and is not useful in transmitted message symbol estimation. The result of the RCEMD decomposition applied to the signal described in (6.2) is shown in Fig The first panel shows the received signal, r k, the second shows the first decomposed component, y (1) k, superimposed on the spread spectrum signal, i k, and the bottom panel shows the interference signal (tone-modulated FM signal here) and the second component from the RCEMD algorithm superimposed on each other. While the first RCEMD generated component closely resembles the spread spectrum signal, s k, they are not identical due to the presence of residual noise and in-band interference signal in the decomposition result. 6.3 Simulation Results The interference excision performance of the proposed RCEMD-based technique is studied on the basis of achieved bit-error rate (BER) in this section. Two classes of interferers are considered here: static partial band interferers comprising multiple tones, and time-varying interferers modeled as FM signals. Further, we consider two kinds of modulating signals for the FM signals: (a) simple tones, and (b) filtered noise following an autoregressive (AR) model. These three classes of interferers are studied in this section. Some existing interference suppression techniques are also considered and their BER performances are compared with our proposed technique in this section. Here BER performance results for three alternate techniques are also presented. The first method is a two-step procedure where the IF of the interfering signal is first estimated using the Wigner-Ville distribution (WVD) and then a short, time-varying finite impulse response (FIR) notch filter is designed to remove that signal [88]. The time-varying nature of the FM interfering signal requires a short length notch filter, which corresponds to a wide notch in the frequency domain, thereby also distorting the spread spectrum signal. Secondly, we study the performance of a simple adaptive filter (AF) based on the least-mean-square (LMS) algorithm at removing the primary signal at the secondary receiver. A nine-tap adaptive FIR filter is considered for performance comparison in all three cases. To allow convergence of filter coefficients, the initial data frames are ignored while computing the BER for the adaptive filter technique. The final method that we will study here involves subtracting a resynthesized FM

130 BER 10 2 No preprocessing EMD WVD Adaptive filter No Jammer present SNR (db) Figure 6.3: BER for multiple tone interference. Here twenty tones occupy approximately 20% of the DSSS signal band. JSR = 2dB is used here. signal from the received signal to form an estimate of the spread spectrum signal. A firstorder phase locked loop (PLL) is used to demodulate the FM signal from the received signal. Due to the large amplitude of the FM signal relative to the spread spectrum signal, a good estimate of the FM modulating signal can be generated for small AWGN. The estimated modulating signal is then used to remodulate a carrier signal which when subtracted from the received signal produces an estimate of the noisy spread spectrum signal. However, due to the noisy input to the PLL, the resynthesized FM signal is not identical to the original, resulting in the appearance of some FM signal energy in the difference. Figure 6.2 shows the block diagram for this receiver. In the following study, we simulate DSSS signals using a PN sequence with 128 chips/bit (L = 128). This is a standard value for WCDMA uplink and downlink directions [108]. For a constant JSR, and for each signal-to-noise ratio (SNR) value 8000 independent realizations of 200-bit sequences are generated for subsequent signal processing and BER determination. The JSR per bit is the ratio of jammer energy to bit energy, i.e., JSR = E J /E b, while the SNR per bit is given by SNR = E b /N 0 = L/(2σn 2 )

131 118 Signal of interest Jamming signal Absolute Fourier transform Normalized frequency (Hz) Figure 6.4: Absolute Fourier transforms of the DSSS signal and the tone modulated FM interference corresponding to JSR = 4dB. Noise signal is not shown in this figure. where σ 2 n is the variance of the AWGN with single-sided spectral density of N Multiple tone interference We first consider a partial band interference scenario comprising multiple tones. This constitutes a static interference situation. The BER performance of the various techniques is presented in Fig The situation considered here is as follows: there are twenty tones with random frequencies, occupying approximately 20% of the spread spectrum band. The total interference power is equally distributed amongst all tones, thereby signifying equal amplitudes, and they have different phase offsets, uniformly distributed over [0,2π). In Fig. 6.3 BER performance for the three techniques RCEMD, WVD and LMS adaptive filter are presented for a range of SNR values, with JSR = 2dB. Also, BER results when no processing is done and when no jammer is present are also shown. These constitute the maximum and minimum attainable BERs, respectively. The BER

132 Normalized instantaneous frequency (Hz) time Figure 6.5: Instantaneous frequency of the tone-modulated FM signal. of the proposed technique is found to be the smallest of the presented techniques and there is an approximately 2dB gain over the WVD technique at BER = A five-tap filter was used to suppress the interference signal in the WVD technique. The adaptive filter performance is observed to saturate for large SNR values because the filter noise is essentially set by the interference level and frequencies and is unaffected by the additive noise level at large SNRs Tone modulated FM interference Here an FM interference signal is considered. The modulating signal is a sinusoid in this case. The frequency domain representation of the interference and spread spectrum signals is shown in Fig. 6.4 for JSR = 4dB. The IF of the interference signal for this case is shown in Fig Again, approximately 20% of the spread spectrum band is covered by FM interference. The BER performances of the different techniques when JSR = 4dB are shown in Fig Here resynthesis technique performance is also included. As before, the RCEMD technique produces smaller BER at large SNRs, with the gain being approximately 2dB at BER = 10 4 over both WVD and adaptive filter

133 No preprocessing EMD WVD Resynthesis Adaptive filter No Jammer present BER SNR (db) Figure 6.6: BER for frequency modulated interference. Here simple tone modulation of the FM signal is considered. JSR = 4dB is used in this case. techniques. The saturation of the resynthesis technique BER at large SNRs is because its performance is essentially limited by the signal energy relative to jammer energy (via JSR) and is unaffected by change in SNR for large values Filtered noise modulation of FM interferer Finally, filtered noise modulation of the FM interference signal is considered. The FM modulating signal in our experiments is generated using a general stochastic timevarying model, namely a first-order auto-regressive (AR(1)) model. The choice of loworder filter model is based on our experimental observation that the proposed technique is insensitive to the model-order used for signal generation for the same signal bandwidth. An approximately 20% spread spectrum band coverage by the FM interference is simulated. The performance results are shown in Fig Here, the performance advantage of the RCEMD technique over the next best, WVD, is approximately 1dB at the reference BER level of Here, results corresponding to JSR = 0dB are shown.

134 BER 10 2 No preprocessing EMD WVD Resynthesis Adaptive filter No Jammer present SNR (db) Figure 6.7: BER for frequency modulated interference. Here the modulating signal is filtered noise following an AR(1) model. JSR = 0dB is used here. 6.4 Conclusions A novel technique for partial band interference excision using the RCEMD technique, applicable to WCDMA communication system was presented in this chapter. Here the received spread spectrum signal affected by interference and background noise is decomposed into two components using the RCEMD technique. Due to the signaldependent decomposition property of the EMD-based technique, the two generated components closely resemble the spread spectrum plus noise and the interference signals respectively. The extracted component corresponding to the spread spectrum signal is despread using the spreading code to derive an estimate of the transmitted symbol. The BER performance of this technique was compared to some alternate interference excision techniques and improvement was observed. An advantage of an EMD-based technique in interference excision is that no tuning of the technique based on a priori knowledge of signal components is required unlike other techniques. Another advantage of this technique that its performance is robust with respect to interference type, whereas the performance of other techniques is observed to depend on the interference

135 122 type. However, the performance of this technique is seen to degrade when the interference is off-center due to inability of the decomposition technique to produce high fidelity representation of the spread spectrum and interference signals in separate components. This aspect of the problem needs further investigation.

136 Part II Signal Analysis of Sensor Data

137 Chapter 7 Atmospheric Pressure Signal Analysis using Raised Cosine Empirical Mode Decomposition The study of the atmospheric pressure is of interest to meteorologists in two ways: one directly, and secondly via the study of gravity waves that can be inferred from the pressure observations. A network of microbarographs is used by researchers to measure this quantity. The presence of seasonally-varying diurnal and semidiurnal tides, cyclonic and anticyclonic pressure variations, as well as sporadic events such as hurricanes often obscures the quantities of interest to these researchers. It therefore becomes imperative for the chosen data-processing method to effectively eliminate the effects of these features before a meaningful analysis of the underlying phenomena can be performed. Naturally, a technique based on the Fourier transform is inappropriate due to the poor time localization of its basis functions. An EMD-based approach to eliminate these time-varying quantities from microbarograph observations is presented here. Accurate estimation of the diurnal and semidiurnal tide signals using EMD is demonstrated and its performance compared to existing time-frequency techniques like wavelets and short-time Fourier transform. The use of the EMD-based technique to isolate a hurricane signature is also shown. Further, feature extraction using RCEMD algorithm is introduced in the final section of the chapter and its advantages over the EMD technique are highlighted.

138 Introduction Analysis of non-stationary and nonlinear processes poses serious challenges to traditional signal processing techniques. The transient nature of events in real-life situations essentially limits the utility of the simple Fourier analysis, which has no time resolution. The short time Fourier transform (STFT), or the spectrogram, allows time-frequency analysis using the familiar fast Fourier transform (FFT). However, it has the disadvantage of fixed time-frequency resolution and the implicit assumption of piecewise stationarity of the signal, which is not valid in general. The wavelet transform method overcomes some of the above limitations by allowing the decomposition of a signal into a set of basis functions that are localized both in time and frequency. The need for a priori knowledge about the kind of scale elements present in the signal and the corresponding choice of wavelet to isolate them is a serious drawback of the wavelet transform method. This has led to the widespread use of the EMD algorithm for analysis of nonstationary signals. A microbarograph operating at the Arecibo observatory (AO), Puerto Rico has been taking almost continuous measurements of the tropical surface atmospheric pressure since early 2003 (daily pressure plots updated at [109]). The microbarograph measures pressure continuously with a sampling interval of 1 s and resolution of 10 µbar. An interesting feature of atmospheric pressure data is the presence of tides of varying durations. The 12-hour duration (semidiurnal) tides and the 24-hour duration (diurnal) tides have their origins in the thermal heating of the atmosphere by the sun as well as the gravitational attraction of the sun and the moon. Further, the elliptical orbit of the earth around the sun results in seasonal variations in the amplitude of these tides. A thorough explanation of the physics behind atmospheric tides can be found in [110]. The semidiurnal tide, being the strongest, is plainly visible in a plot of the time-series signal from the microbarograph. However, superimposed variations of longer periods and other transient pressure changes make accurate measurement of the semidiurnal and diurnal tide amplitudes difficult to obtain. In this chapter we present a new EMD-based technique to extract the diurnal and semidiurnal tides from the pressure data. Results are presented for normal conditions as well as for an instance of severe pressure disturbance due to a passing hurricane. We also present results obtained from applying the wavelet transform technique to the same data for comparison. Finally, signal feature extraction

139 126 using RCEMD is introduced and its advantages highlighted. 7.2 Data analysis using HHT and wavelets We briefly describe here the important steps involved in the HHT technique (details can be found in Chapter 2). First step is to find the IMFs by a procedure termed sifting or empirical mode decomposition (EMD) in [27]. This involves projecting the signal onto basis functions that are implicitly defined and signal-dependent. Unlike traditional signal decomposition techniques such as wavelets and STFT that decompose the original signal into a series of constituents of fixed, pre-determined frequencies, the generated IMFs do not necessarily have constant frequency or amplitude and it is for this reason that it is often difficult to assign any physical meaning to them. Next step in this process involves computing the instantaneous frequency of each IMF by first evaluating its Hilbert transform, followed by evaluating the derivative of the phase. Mathematically, the Hilbert transform of a real-valued function x(t) is defined as (see [111]) and x(t) = θ(t) = arctan x(u) du, (7.1) π(t u) [ ] x(t), (7.2) x(t) f 0 (t) = ( ) 1 dθ(t) 2π dt (7.3) is the instantaneous frequency. The instantaneous frequency of each IMF is superimposed and presented as a color coded map, with optional smoothing applied. This is referred to as the Hilbert spectrum of the signal and is analogous to the wavelet spectrum. We next present the equations defining the continuous wavelet transform (CWT) and its inverse (see [112]): and X w (b,a) = 1 ( ) t b x(t)ψ a a (7.4)

140 127 where 1 x(t) = C ψ X w (b,a)ψ b,a (t) dadb a (7.5) and C ψ = ˆψ(ω) 2 dω < (7.6) ω ψ b,a (t) = 1 ( ) t b a ψ a where ˆψ(ω) is the Fourier transform of the mother wavelet ψ(t). (7.7) In this work we have utilized the code available at [83] to implement the HHT technique. A few pre-processing steps were carried out on the raw data before applying the time-frequency techniques. First, since we are interested in events with periods longer than 6 hours, we time averaged the data to get sampling interval of 3 hours. Some gaps were observed in the existing data due to equipment malfunction. Over the two-year period for which data is available loss of data spanning a period of about half a day was observed twelve times and on one occasion we do not have data for a two day period. These gaps in data were interpolated using sine waves. Finally, it was observed that the non-zero mean of the data causes considerable ringing to appear at the window edges in results using traditional techniques like wavelets and short time Fourier transform (STFT), whereas EMD remains unaffected by the non-zero mean of the data. Hence, mean removal was carried out prior to analysis by each of the techniques to get meaningful results in all cases. Figure 7.1 shows the time series, IMFs and the Hilbert spectrum computed for the pressure data for a two year period starting at 00:00 AST 1 January, Severe pressure fluctuations due to a passing hurricane event are visible during days in the time series, Hilbert spectrum and some of the IMFs. The semidiurnal and diurnal tides are also visible in the Hilbert spectrum. Results from some competing techniques are presented in Fig. 7.2 for the same record. In the wavelet spectrum presented in Fig. 7.2a, although the hurricane event is clearly defined, considerable smearing is evident in 24-hour tide while the 12-hour tide is practically undetectable. The mother wavelet used to analyze the signal was the analytic signal whose real part is the fourth derivative of the Gaussian with variance equal to one (see [112]). The real,

141 Pressure (mbar) Hurricane Time (days) (a) Pressure (mbar) Time (days) (b) Figure 7.1

142 129 x Frequency (Hz) Hurricane 12 hour tide 24 hour tide Time (days) (c) Figure 7.1: Microbarograph data 1 January December 2005: (a) time series, (b) IMFs computed using EMD, (c) Hilbert spectrum with 12-hour, 24-hour tides and hurricane events visible. imaginary and absolute values of the basic wavelet are shown in Fig Good time localization property of this mother wavelet assures efficient filtering of solitary waves or bumps that are intrinsically very localized in time. The second existing technique that we evaluated is the STFT. Results for the STFT technique using the Hamming window are shown in Fig. 7.2b. Signatures of the two tides and the hurricane are visible in the spectrum. The presence of a transient event as well as underlying periodic events in the data requires careful selection of window length. A window length of 60 samples, corresponding to 7.5 days offers a good compromise between requirements of high time resolution and high frequency resolution for optimum representation of the hurricane event and the persistent tides respectively. It must be noted here that the HHT procedure does not require similar signal-dependent tuning of parameters to optimize the decomposition quality. Finally, tides of different durations show up quite clearly in the FFT plot, which is expected because Fourier analysis is known to be an efficient representa-

143 130 tion of stationary signals. 7.3 Signal Feature Extraction Next we proceed with signal feature extraction using EMD. As mentioned earlier, the IMFs that are produced using the EMD procedure do not necessarily have constant frequency. In fact, when applied to real-life signals where irregular structure is common, the sifting process distributes signal components of a particular frequency among several IMFs. Figure 7.4 shows several instances of overlap of instantaneous frequencies of adjacent IMFs. This leads us to the conclusion that to extract a particular frequency component from a signal, it is not sufficient to select the IMF that produces that particular instantaneous frequency. Rather, a few neighboring IMFs that have instantaneous frequencies equal to the desired frequency at certain instants need to be included too. Further, in order to take into account the effect of noise and other irregularities in the signal, instead of a single frequency we consider a small band of frequencies centered on the desired frequency to decide which IMFs to combine to get the desired frequency component. We describe a practical way of implementing the signal extraction process here. The initial steps, viz., sifting, forming analytic signals, and finding instantaneous frequency are performed as usual. Next we form a small band of frequencies around the desired frequency according to the tolerance level desired. In this particular case placing the frequency band limits at 90% and 110% of the desired frequency appears to include most of the energy of the relevant IMF as indicated by the two horizontal lines in Fig In other practical cases the frequency band will need adjusting so that most of the energy of the desired IMF is contained within it. However, expanding the frequency band excessively will result in inclusion of spurious energy from adjacent IMFs leading to contamination of the extracted feature by other undesired features. Further research is required to develop a realizable mapping between features in the time domain and their corresponding projections in the IMF domain leading to an automatic selection of IMF sections. Next we form an inclusion matrix to decide which IMFs are combined together at each instant to form the signal of desired frequency. The number of rows of the inclusion matrix equals the number of computed IMFs and the column count equals the number of time samples in the signal. The matrix is initially set to all zeroes. Then

144 x Frequency (Hz) hour tide Hurricane 4.5 x Time (days) (a) Frequency (Hz) Hurricane 12 hour tide 24 hour tide Time (days) (b) Figure 7.2

145 132 Normalized absolute value hour tide hour tide hour tide 6 hour tide Frequency (Hz) x 10 5 (c) Figure 7.2: Microbarograph data 1 January December 2005: (a) wavelet spectrum using complex mother wavelet, (b) short time Fourier transform spectrum using Hamming window, (c) fast Fourier transform with frequency bins corresponding to 24, 12, 8, 6-hour tides showing large amplitudes. at each time instant a 1 is placed at the position corresponding to an IMF that has its instantaneous frequency within the frequency band set earlier. When this procedure is completed for the entire time duration, the matrix containing the IMFs as its rows is multiplied by the inclusion matrix and the components added up to get the desired frequency signal component. The concept of overlap of instantaneous frequencies of IMFs and the desired frequency thresholds is shown in Fig. 7.5 for extracting the 24- hour tide from the microbarograph observations. The results of this technique are shown in Fig. 7.6 where we present the 12 and 24-hour tides extracted from the microbarograph observations using the above procedure. Clear seasonal variations are observed in the extracted 12-hour tide signal. We now demonstrate a sporadic feature signature isolation technique using EMD. Effects of a passing hurricane were observed in the microbarograph data from September The presence of such a large energy event in the observations some-

146 Re Im Abs Figure 7.3: Real, imaginary and absolute values of the mother wavelet used to compute the wavelet spectrum. The real part is the fourth derivative of the Gaussian with unit variance. Pressure (mbar) Pressure (mbar) Normalized Frequency IMF 3 IMF Time (days) Figure 7.4: Instantaneous frequency overlap of adjoining IMFs. Top two panels show successive IMFs and the bottom panel shows their instantaneous frequencies.

147 Normalized frequency (Hz) Time (days) Figure 7.5: Frequency thresholds for 24-hour tide extraction and instantaneous frequencies of the first three IMFs from the microbarograph observations. The upper and lower thresholds are set at ±10% of the desired frequency. It is clear that IMF 2 (middle) contains most of the 24-hour tide energy since its oscillations are mostly confined within the desired band of frequencies. times severely degrades the performance of signal processing routines intended to study the periodic signal components. Consequently, it is desirable to remove such shortduration, large-amplitude disturbances while at the same time causing minimal distortion to the underlying signal of interest. Here we use a technique that differs from the above procedure to isolate the sporadic event signature. This is illustrated in Fig. 7.7 where the original mean-removed signal is shown along with the IMFs that contain portions of the hurricane signal. First the IMFs containing energy corresponding to the hurricane signal are identified and then portions of those IMFs containing the relevant signals are retained with the rest of the IMFs blanked out (This is illustrated by the portions of the IMFs enclosed within the curve in the figure). Although at present we perform manual selection of the region, automatic selection and elimination of interfering high-energy regions using these techniques is a distinct possibility. In the next step individual IMFs are combined together resulting in a reconstruction of the hurricane sig-

148 Pressure (mbar) hour tide hour tide Time (Days) Figure 7.6: 12-hour (semidiurnal) and 24-hour (diurnal) tides extracted from the original microbarograph observations (on top) using the EMD-based technique described in text. nal, which can then be subtracted from the original signal. Results of alternate feature extraction using this method are shown in Fig. 7.8a. For comparison we also present results using an analogous technique based on wavelets (see [112]) in Fig. 7.8b. Both figures show the original signal, the extracted signal feature and the residue. Some observations regarding these results are in order. First, the start and end of the extracted event is rather abrupt using the EMD-based technique while it is more gradual for the wavelet based one. This is a direct consequence of how the IMFs are combined in the inverse EMD process (where an IMF component is either included or not included in the reconstruction process) and the way the inverse CWT operation works (which is more like a convolution operation over the desired region). Secondly, from observing the residues using the two techniques it is clear that the effect of the EMD-based technique is to remove all frequency components from the affected period whereas the wavelet method attempts to maintain continuity as far as the low energy background signal is considered. Which of the two approaches is desirable depends upon the particular application since although removal of all frequency components in the EMD case may be unacceptable in some situations, the tendency of the wavelet approach to maintain continuity in the low energy signal may introduce severe distortions

149 136 Figure 7.7: The original month-long data (mean removed) in the top panel and the IMFs containing components of the hurricane signal in the subsequent panels. Portions of individual IMFs that include projections of the hurricane signal are enclosed by the approximate curve. in the residue as a consequence. Moreover, careful selection of the subset region from the spectrum remains a critical step in either technique. 7.4 Signal Feature Extraction using RCEMD After having demonstrated the use of the EMD algorithm to perform signal feature extraction, we now turn our attention to the RCEMD algorithm that was introduced Chapter 3. As discussed there, RCEMD uses the raised cosine pulse for interpolation. Due to the nonstationary nature of the microbarograph signal we will be using the windowed version of RCEMD. By applying the new algorithm to the same tasks as before we will demonstrate the superior feature extraction performance of this technique. First we use the RCEMD algorithm to isolate the diurnal and semidiurnal tides as described in Section 7.3. We simply replace the EMD algorithm by RCEMD and the results are presented in Fig Although the results look similar to that using EMD in Fig. 7.6, the advantage of the new technique becomes clear upon examination of Fig.

150 137 Pressure (mbar) Pressure (mbar) Time (Days) (a) Time (Days) (b) Figure 7.8: Results for alternate event extraction for (a) EMD-based technique; and (b) wavelet based technique, for a hurricane event observed in the microbarograph data in September The three panels show the original month-long data, the extracted hurricane event and the residue after event removal from top to bottom respectively.

151 Pressure (mbar) hour tide 2 24 hour tide Time (Days) Figure 7.9: Identical to Fig. 7.6 except that RCEMD algorithm is used here Normalized frequency (Hz) Time (days) Figure 7.10: Identical to Fig. 7.5 except that RCEMD algorithm is used here.

152 139 Pressure (mbar) Time (Days) Figure 7.11: Identical to Fig. 7.8 except that RCEMD is used here There, as before, we show the instantaneous frequencies of the first three IMFs generated by the RCEMD algorithm along with a pair of horizontal lines representing the 10% tolerance level around the expected diurnal tide frequency. Compared to Fig. 7.5, improved spectral purity of the second IMF, that contains the diurnal tide signal, is evident using the new technique. Next, we apply the RCEMD algorithm to hurricane signal extraction. Again, the steps described above are repeated, this time using the RCEMD algorithm in place of EMD. The results are presented in Fig Reduced pre- and post-event oscillations using RCEMD are apparent. More importantly, for similar quality of sporadic event signature isolation quality, the new technique requires 75% fewer iterations than the EMD algorithm. Therefore using the RCEMD algorithm leads to computational efficiency in this case. The choice of technique for sporadic signal extraction, either using wavelets, EMD or RCEMD, depends on the subsequent signal processing steps intended for the particular data-set, and cannot be generalized.

153 Conclusion In this chapter we have studied the EMD technique for signal event extraction, utilizing microbarograph observations from Arecibo Observatory. We compared the Hilbert spectrum derived using the HHT technique with the wavelet spectrum using a complex wavelet and demonstrated considerable improvement in clarity in the representation of tides by the Hilbert spectrum over the wavelet spectrum. Next we demonstrated a technique for signal feature extraction based on the EMD. Application of this signal feature extraction technique to extract diurnal and semidiurnal tides from the atmospheric pressure data was presented. High precision in the extracted signal allows observation of the seasonal variations in the semidiurnal tides quite clearly. Further, we adapted the signal feature extraction technique to isolate and remove large amplitude disturbances from time-series data. Performance of this technique is demonstrated via extraction of a hurricane event from the pressure observations. Further improvement in performance is demonstrated by the use of RCEMD. Performance of the newly developed feature extraction techniques is compared to that of a wavelet based technique using a complex wavelet. Relative advantages of each technique are highlighted and situations where either approach might find favor are pointed out.

154 Chapter 8 Genetic Algorithm based Parameter Estimation Technique for Fragmenting Radar Meteor Head-echoes Meteoroid fragmentation presents a serious problem for Doppler estimation using Fourier transform techniques. Radar returns from multiple, closely-spaced bodies traveling at nearly identical speeds result in an interference pattern which makes it difficult to estimate properties of individual bodies by traditional techniques. Here we present a genetic algorithm based procedure to determine the properties of the individual fragments, such as relative scattering cross-section, speed and deceleration. The radar meteor observations presented here were made using the Poker Flat (Alaska) Incoherent Scatter Radar (PFISR) operating at MHz. 8.1 Introduction The scientific community has been interested in observing sporadic radar meteors due to the role of meteoroids in understanding space weather, in the aeronomy of the meteor zone and in various aspects of plasma physics [113, 114]. Here we consider headecho observations in which radar returns are from the distribution of plasma immediately surrounding the meteoroid and that travels with the meteoroid itself. For details regarding scattering mechanism and models of head-echoes the reader is referred to [113, ]. For meteor events observed in more than four radar pulses, a fast

155 142 Figure 8.1: Range-Time-Intensity (RTI) and Signal-to-Noise Ratio (SNR-similar to optical meteor light curves) of three meteor events observed with the Poker Flat MHz Incoherent Scatter Radar (PFISR). The (a.) event (Event 1) is consistent with two meteoroids traveling along the same trajectory and each producing a headecho that result in the strong interference pattern. The (b.) event (Event 2) shows a more complex structure that is consistent with three meteoroid fragments. The (c.) event (Event 3) which shows strong frequency modulation of the SNR curve is otherwise similar to event 1. Fourier transform (FFT) based technique has been developed that provides estimates of the event altitude, signal-to-noise ratio (SNR) and speed as a function of time throughout the event [ ]. Many events having high enough SNR also yield deceleration estimates. Fragmented meteoroids present a problem for speed estimation using FFT techniques. Scattering from two nearby slowly separating point targets (relative to the wavelength) exhibit strong interference effects as the two signals add in-phase and out of phase. That is, two (or more) common-trajectory meteoroid fragments exist within the radar range resolution cell and, as the scattered electric fields exhibit nearly common Doppler phase effects, the net electric field at the receiver shows a strong interference pattern. Some researchers have previously tried to estimate the properties of fragmented meteoroids in the past [122,123]. Further, the authors of [124] reported evidence of fragmentation based on their observations but stopped short of analyzing them. We present here a genetic algorithm (GA) [125, 126] based optimization technique that searches the multidimensional fragment parameter space to find the parameter set that minimizes