AN ABSTRACT OF A THESIS COMPRESSED SENSING FOR ULTRA WIDEBAND (UWB) SYSTEMS. Daniel Zahonero Inesta. Master of Science in Electrical Engineering

Transcription

1 AN ABSTRACT OF A THESIS COMPRESSED SENSING FOR ULTRA WIDEBAND (UWB) SYSTEMS Daniel Zahonero Inesta Master of Science in Electrical Engineering The demanding characteristics of the UWB technology include extremely high sampling rates in the receiver. These sampling rates require sophisticated devices, sometimes out of the scope of the state-of-art technology. Among the different methods to make the reception possible, Compressed Sensing seems to be the one that presents better performance. It consists basically of compressing the information while this is sampled, avoiding processing a huge chunk of redundant data and lowering the sampling rate. In order to reconstruct the data after the compression, different methods have come up presenting different favorable features. However this methods also present a trade-off between sampling rate and processing time. Using real measurements of the channel, the performance in different environments has been analyzed for different frequencies. Thanks to the theoretical account and practical results this study will help to understand better the Compressed Sensing techniques applied to a real communication, and specifically, to Ultra Wideband.

2 COMPRESSED SENSING FOR ULTRA WIDEBAND (UWB) SYSTEMS A Thesis Presented to the Faculty of the Graduate School Tennessee Technological University by Daniel Zahonero Inesta In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE Electrical Engineering May 2009

3 Copyright c Daniel Zahonero Inesta, 2009 All rights reserved ii

4 CERTIFICATE OF APPROVAL OF THESIS COMPRESSED SENSING FOR ULTRA WIDEBAND (UWB) SYSTEMS by Daniel Zahonero Inesta Graduate Advisory Committee: Dr. Robert C. Qiu, Chairperson Date Dr. Periasamy K. Rajan Date Dr. Omar ElKeelany Date Approved for the Faculty: Francis Otuonye Associate Vice President for Research and Graduate Studies Date iii

5 STATEMENT OF PERMISSION TO USE In presenting this thesis in partial fulfillment of the requirements for a Master of Science degree at Tennessee Technological University, I agree that the University Library shall make it available to borrowers under rules of the Library. Brief quotations from this thesis are allowable without special permissions, provided that accurate acknowledgment of the source is made. Permission for extensive quotation from or reproduction of this thesis may by granted by my major professor when the proposed use of the material is for scholarly purposes. Any copying or use of the material in this thesis for financial gain shall not be allowed without my written permission. Signature Date iv

6 DEDICATION This thesis is dedicated to my parents who have given me invaluable educational opportunities. v

7 ACKNOWLEDGMENTS Appreciation is extended to Peng Zhang whose knowledge in the field and support has been vital for the fulfillment of this thesis. To Dr. Robert C. Qiu whose advices and recommendations have been tremendously helpful. I would also like to thank all my lab mates for their help and support throughout my stay here. Finally, I would also like to thank the Department of Electrical and Computer Engineering, and Center for Manufacturing Research for its invaluable help, putting at my disposal all the equipment and resources available. vi

8 TABLE OF CONTENTS Page List of Tables ix List of Figures x Chapter 1. INTRODUCTION Motivation Literature Survey Research Approach Organization of the Thesis UWB UWB Technology Definition Antennas Channel Modulations Challenges Pulse-Shape Distortion Multiple-Access Interference Channel Estimation High Sampling Rate Summary COMPRESSED SENSING Compressive Sensing Theory Isometry Measurements CS for UWB Communication Model Communication Model: Theoretical Approach Communication Model: Practical Approach vii

9 Table of Contents viii 3.4 Summary THE UWB CHANNEL Channel Measurement Hallway Metalic Box Inter-Vehicle Channel Sounding Results Summary CS ALGORITHMS COMPARISON Basis Pursuit (BP) Algorithm Definition Characteristics Orthogonal Matrix Pursuit (OMP) Algorithm Definition Characteristics Stagewise Orthogonal Matrix Pursuit (StOMP) Algorithm Definition Threshold False Alarm Rate (FAR) False Discovery Rate (FDR) Simulation in Matlab Measuring Matrix Simulation Results Accuracy on the Recovery Number of Measurements Processing Time Summary CONCLUSION CS applied to UWB Reconstruction Algorithms Future Work BIBLIOGRAPHY VITA

10 LIST OF TABLES Table Page 3.1 Bit-Rate due to the number of pulses Sampling Rate at the Receiver Sampling BP Constant ix

11 LIST OF FIGURES Figure Page 2.1 UWB Receiver Block Diagram Symbol. Pulse Position Modulation (PPM) UWB Communication System Based on Compressed Sensing Vector Network Analyzer (VNA) GHz UWB Antenna Hallway Environment Metallic Box Environment Inter-Car Environment Hallway Channel in Frequency Domain Hallway Channel Impulse Response Metallic Box Channel in Frequency Domain Metallic Box Channel Impulse Response Inter-Car Channel in Frequency Domain Inter-Car Channel Impulse Response Heuristic approximation to the minimization problem Atoms related to non-zero elements (OMP) Quasi-Toeplitz Matrix Error Rate of BP Recovery for different levels of sparsity Error Rate of BP Recovery (6-8 GHz) Error Rate BP Recovery Metallic Box (4-6 GHz) Error Rate BP Recovery Inter-Car (4-6 GHz) Error Rate OMP Recovery (4-6 GHz) Error Rate OMP Recovery (6-8 GHz) Error Rate OMP Inter-Car (4-6 GHz) Error Rate OMP Metallic Box (4-6 GHz) Error Rate StOMP Hallway (4-6 GHz) Error Rate StOMP Hallway (6-8 GHz) Error Rate StOMP Inter-Car (4-6 GHz) Error Rate StOMP Metallic Box (4-6 GHz) Error Rate OMP Inter-Car (4-6 GHz) Error Rate OMP Metallic Box (4-6 GHz) Error Rate OMP Inter-Car (4-6 GHz) Error Rate OMP Metallic Box (4-6 GHz) Error Rate OMP Inter-Car (4-6 GHz) Error Rate OMP Metallic Box (4-6 GHz) x

12 List of Figures xi 5.22 Sampling Rate (High Sparsity) Sampling Rate (Low Sparsity) Processing Time of BP Processing Time of OMP Processing Time of Stagewise OMP Processing Time (High Sparsity) Processing Time (Low Sparsity)

13 CHAPTER 1 INTRODUCTION 1.1 Motivation One of the latest breakthroughs made in wireless communications has been the Ultra Wideband Technology. The growing demand on wireless services has overcrowded the RF Spectrum over the last decade and new services face a problem of allocation. Since in 2002 the FCC authorized UWB operations over the top of other bands, it has become one of the most promising alternatives. However, its demanding features in terms of sampling and processing information require new methods and ways to set the transmission. It has been demonstrated theoretically that Compressive Sensing technique is a perfect suitable candidate to implement UWB. Following this premise, this study presents as a goal to demonstrate that CS can implement UWB, supporting the idea not only with theoretical background but with practical results. The study also analyzes its limitations dealing with frequency and different channels. The channels used in the study will vary from favorable line-of-sight indoor to hostile outdoor environments. The frequency will be modified in order to study the influence in the performance of the CS. 1

14 2 Finally a comparison of the main algorithms that perform CS will be made. The key features, advantages and disadvantages will be discussed easing the election of them according to the conditions and constraints. 1.2 Literature Survey Since 2002, there has been a noticeable increase in literature published about UWB. Many books have been written introducing the priciples[1][2] and fundamentals[3][4]. Some other came up presenting the applications[5], and finally a radio model system was introduced [6] [7][8][9]. In the last years, matematicians came up with a new revolucionary solution that applied Compressed Sensing to communications [10][11], and more specifically to UWB[12][13]. Numerous studies have been made about the CS alogrithms applied to communication, describing them in detail: Basis Pursuit[14][15][16][17], Orthogonal Matching Pursuit[18][19] and Stagewise Oothogonal Matching Pursuit[20]. Even some of them are compared in a given situation[21] [22]. However there is no study that goes that far and compare the three of them in different environments and frequency bands. 1.3 Research Approach Using Matlab, the CS algorithms will be applied to simulate a transmission using different channel data from real measurements. To implement the communica-

15 3 tion model, VNA measurements of the channel are made for two Frequency Bands. The frequency data is transformed into time domain using IFFT to obtain the impulse response. The surroundings where the channel is measured are a Hallway, a Metallic Box and an Inter-Vehicle environment. The two 2 GHz Bandwidth Channel will be centered in 5 GHz and 7 GHz. With the obtained data, three different featured algorithms are used for the same environments in order to achieve a reliable comparison. 1.4 Organization of the Thesis Chapter 2 presents a briefly introduction and the fundamentals of UWB focusing mainly in the Channel, the PPM modulation and the Challenges UWB entails. The Chapter 3 introduces the Compressed Sensing concept. The CS theoretical background is explained and its application to communication is discussed. Finally a communication model for UWB is proposed. In the Chapter 4 the Channel is examined for the three environments after explaining the measuring method. The Channel souning results are commented later on in the chapter. Chapter 5 focuses in three algorithms used for CS. The algorithms are explained in detail pointing out the strengths and weaknesses. The simulation of the

16 4 earlier proposed model is set out. Eventually, the results of the simulations are explained and critically analyzed. Chapter 6 gathers all the final conclusions and contributions for the thesis as well as points out the future work.

17 CHAPTER 2 UWB The principle of UWB is based on transmitting in a short-range with large bandwidth using low energy levels. This way the power is spread all over the spectrum and becomes immune to frequency flat fading[1]. The UWB Technology also offers a robust performance under multipath environments, but its most interesting feature is the possibility to reuse the frequencies. Spreading the power over the spectrum allows transmitting with low power levels making the signal noise-like so that it causes minimal or no interference to other signals that may be transmitted in the same frequency. Hence UWB is able to coexist with current radio systems and no need to be allocated in the spectrum. UWB technology was designed for military applications but new U.S. Federal regulations and the demand for higher data rates at short-range opened up UWB for commercial applications [23]. The centimeter accuracy in ranging and communications provides unique solutions to applications, including logistics, security applications, medical applications, control of home appliances, search-andrescue, family communications and supervision of children[2]. 5

18 6 2.1 UWB Technology Definition To be considered UWB, the signal bandwidth has to be greater than 500 MHz or having a fractional bandwidth 1 of larger than 20% and the radiated power cannot exceed dbm/mhz according to the Federal Communication Commission(FCC)[24]. UWB transmissions transmit information by generating radio energy at specific time instants with the shape of a pulse. These pulses are on the order of nano-seconds and are used as the elementary pulse shaping to carry the information[3]. The basic pulse used for UWB is a Gaussian monocycle in which the width determines the center frequency and the bandwidth. A typical UWB pulse is between 0.2 and 0.5 nanoseconds width. The monocycle itself contains no data, so that a long sequence of monocycles with data modulation is used for communication Antennas In UWB technology, power level is much more important than in standard narrowband system. Thus an effective UWB antenna is a critical part of an overall UWB system design. A wide variety of antennas is suitable for UWB applications. 1 2(f H f C ) (f H + f C )

19 7 They can be classified as directional and non-directional. High-gain antennas concentrate energy into a narrower solid angle than omnidirectional antennas. An isotropic antenna radiates equally in every direction. Differences between using a directional or non-directional is a tradeoff between Gain, the Field of view and the size of the antenna. Regulatory constraints limit the power to the same peak radiated emission limit, so basically what is reached by using the directional antenna is to reduce the emissions in undesired directions. These kind of antennas can be implemented in relatively compact planar designs. The design of the receiving antenna can affect dramatically to the link performance[5]. UWB antennas can be modeled as the frontend pulse shaping filters that affect the baseband detection[2]. Antennas act as a filter for the generated UWB signal, and only allows those signal components that radiate to be passed. The basic effect of antennas is that they produce the derivative of the transmitted or received pulse waveform. This also has the effect of extending the duration of the transmitted and received pulse. This extension of pulse duration decreases the time resolution of the system[25] Channel Propagation through the medium will attenuate and distort the incident pulsebased signals, but the most important feature is that when a short UWB pulse propagates through the channel, multiple pulses are received via multipath. These pulses

20 8 have shapes different from the incident short pulse[2]. If the channel is well characterized, the effect of disturbances and other sources of perturbation can be reduced by proper design of the transmitter and receiver. Detailed characterization of UWB radio propagation is required for successful design of UWB communication systems[4] and necessary for the algorithms to reconstruct the original signal from data sampled. The channel can be measured in the Frequency Domain (FD) using a frequency sweeping technique. Using a set of narrow-band signals a wide frequency band is swept thanks to a vector network analyzer (VNA). This measurement corresponds to S21 parameter measurement set-up, where the device under test (DUT) is the radio channel. There is another way to measure the channel in Time Domain (TD) using channel sounders that are based on impulse transmission or direct sequence spread spectrum signaling[25]. During the sweep the channel must be static to maintain the conditions during the soundings. For fast changing channels, other sounding techniques are needed. Another possible errors is the frequency shift caused by the propagation delay when long cables are used, or when the flight time of the sounding signal is long[25]. Signal Analysis using IFFT will provide the Time Domain signal from the Frequency Domain signal. This processing is possible since the receiver has a downconversion stage with a mixer device. To convert the signal into the time domain Hermitian Processing is used[25]. This method is based in reflecting the negative

21 9 conjugate to the negative frequencies. The result is transformed into time domain by using the IFFT[25]. Parameters related to penetration, reflection, path loss, and many other effects should be considered frequency-variant and investigated more carefully.[4] Modulations The different techniques to encode the information in UWB include amplitude, polarity and position. In order to choose a modulation, data rate, transceiver complexity, spectral characteristics, robustness against narrowband interference, intersymbol interference and error performance must be taken in account. They can be grouped as: On-Off Keying (OOK) Pulse-Amplitude Modulation (PAM) Pulse-Position (PPM) Biphase Modulation (BPM) Transmitted-Reference Modulation (TRM) In this thesis, the communication model will emulate a K-sparse PPM encoding transmission. Each symbol is K-sparse, meaning that for N possible positions there

22 10 will be just K (K<<N) pulses. These possible positions of the pulses will determine the symbol and provide the information. 2.2 Challenges Although it has been demonstrated that UWB is a great alternative for shortdistance low-power wireless applications[26], it has its weak points. These issues must be minimized in order to make UWB a reliable technology. The main disadvantages are based on the large frequency band that makes the pulses deform through propagation and the low-level power that makes multiple-user communication hard to implement as well as stress the distortion Pulse-Shape Distortion Unlike narrow-band communications, UWB uses a large portion of the Spectrum and the attenuation due to the propagation does not affect all the frequencies of the band the same way. The received power from higher frequencies of the band will be lower than the received from the lower frequencies originating a distortion of the pulse. Moreover the low-level power of the signal makes this distortion much more noticeable. To face these problems matched filters were used to correlate the pulses with predefined templates. Because of the changing conditions of the channel they are doomed to fail, so an equivalent distorted signal is used as a reference signal in

23 11 the matched filter Multiple-Access Interference In the case of the presence of several users transmitting over the same channel, pulses originating in other transmission links may collide with pulses belonging to a reference transmission, giving rise to an interference noise called Multi-User Interference[7]. Fortunately as the channel can be modeled as pseudo-random(pn) noise code, by shifting each monocycle at a pseudo-random time interval, the pulses appear to be white background noise to users with a different PN code[8]. Hence UWB technology allows several users to transmit at the same time, using the same frequency range Channel Estimation The characteristic low level of energy transmitted in UWB limits the range of the transmission and makes it hard to implement in an outdoor environment. However UWB is very powerful when it comes to indoor scenario due to the rich-multipath. It only occurs if the channel is known since in UWB it is assumed stationary or quasistationary[2]. Unlike other technologies UWB channel needs to be considered in time domain rather than frequency domain due to its unique features. In opposition to narrowband communications, the signals are huge-bandwidth pulses instead of sine

24 12 waves so that selective fading is no longer a problem. The overlapping superposition of unresolved multipaths hardly affects the received signal so the time-resolvable multipath will carry great part of the energy of the transmission. The channel is considered quasi-static so the collected total energy is almost constant in each instant[9]. Due to this a close estimation of the channel can be obtained. Some studies consider propagation as the single most important issue in the success of UWB technology [4]. Thus, a well characterized channel is fundamental for a successful UWB communication. The attenuations, interferences and delays affect the design of the transmitter and receiver so having all the information in advance will result in a more efficient communication. Also, as shown later on, knowing the channel time-response will be essential for the recovering in compressed sensing. In the receiver, the transmitted signal will be correlated with a template signal which will reflect the effect of the channel so that it will be a reliable reference for the received signals to compare. The attenuations and propagation delays must be estimated to predict the shape of the template signal that matches the received signal[5]. However the characteristics of the UWB make the pulse vulnerable to distortion and hard to interpret the channel. The huge bandwidth of the pulse makes different frequency components attenuate unevenly causing distortion in an already low-power pulse. To deal with this drawback training sequences known by the receiver are used[27]. These predefined training sequences are transmitted and compared with the original, the difference between them

25 13 Figure 2.1: UWB Receiver Block Diagram will help to depict the channel response in terms of attenuation, resolvable multipaths and delay spread High Sampling Rate One of the most important challenges UWB has to face is how to manage the high speed transmissions, being the transformation from Analog domain to Digital domain one of the key parts of the UWB receiver model (Figure 2.1). The obtained analog signal is sampled at a 1/Ts, where Ts is the Time of the Symbol, in order to get a sequence of digital values which be able to work with. This sampler will limit the speed of the transmission, the information unable to sample due to higher speed than the sampler can handle will be lost. Finally the sampled sequence will be processed afterward by a Maximum Likelihood Sequential Estimator (MLSE) that will interpret the symbol detected. As in any other Telecommunication area, sampling is always a challenge. This task in UWB technology, due to its features, is even more complicated. Accurate

26 14 data detection for pulsed UWB systems is crucial to achieve good Bit Error Rate (BER), performance, capacity, throughput and network flexibility[2]. Most of the UWB communication applications are targeting between 100 Mbps and 500 Mbps transmissions, remarkably faster compared with current wireless standards [4]. The ultra-short pulse widths go from tens of picoseconds to few nanoseconds[28][29] requiring extremely high sampling rate in the receiver to collect them. In a UWB communication, every symbol is transmitted with a low duty cycle over a large number of frames gathering adequate symbol energy while maintaining low power density [28]. This low duty cycle makes it hard to calculate the Time Gating, time which the receiver and the detector have to remain turned on in order to match the signal as well as when they have to be turned on[6]. The duration time must be just the expected duration of the signal, if they remain open too long, excess energy noise will be collected affecting the signal-to-noise ratio. On the other hand, if the pulse is sampled less time than needed, since there is not enough energy some information may be lost. Another problem for synchronization is the dense multipath. Although it entails large diversity and can enhance the energy capture, this is challenging during synchronization phase because the channel and time information, not being available, are hard to figure out. However, the main challenge is to reduce the acquisition time which calls for a more sophisticated ADC, sometimes out of the scope of the state-of-art technology[30][31]. This limitation led to leave aside classical conception

27 15 of sampling (Nyquist) and seek for new techniques that allow more information rate using less sample requirements. In order to address this challenge, a revolutionary sampling theory appeared: Compressed Sensing. It permits to reconstruct a transmitted signal using just a few percentage of the original number of samples, opening a new range of possibilities in UWB. In broad strokes this technique is based on compress while sampling, freeing the receiver from having to sample in a high rate and to process a huge chunk of data due to the speed of the data. Thanks to this, impressive low sampling rates can be achieved related to the degrees of freedom of the information instead of to the Bandwidth of the signal. This thesis focuses its goal on giving a general idea of how Compressed Sensing is applied to UWB and analyze different reconstruction algorithms. 2.3 Summary This chapter introduces the fundamentals and applications of UWB technology. The advantages are pointed out, but also the challenges that UWB involves emphasizing the challenge of dealing with great high-frequency. Since UWB has very demanding sampling requirements, sometimes out of the scope of the state-of-art technology, the A/D converter will be the bottleneck. To address this challenge, the Compressed Sensing is proposed.

28 CHAPTER 3 COMPRESSED SENSING Classical compression methods throw away the undesired information after data acquisition. For few data this is not a problem, but when it comes to compress massive data, many resources are wasted acquiring and processing a chunk of unnecessary data. As an answer for this inefficiency appeared the Compressed Sensing (CS) which allows to compress the data while is sampled. In other words Compressed Sensing suggest ways to economically translate analog data into already compressed digital form [32][10]. This new way to sample achieves sampling rates below Nyquist rate. It originates from the idea that it is not necessary to invest a lot of power into observing the entries of a sparse signal in all coordinates when most of them are zero anyway. Rather it should be possible to collect only a small number of measurements that still allow for reconstruction[33]. CS requires a compressible signal with certain sparsity, considering sparse a signal that can be written either exactly or accurately as a superposition of a small number of vectors in some fixed basis. The procedure then, gets the most of the properties of some signals in which a small number of non-adaptive samples carries sufficient information to approximate the signal properly. This is potentially useful in applications where one cannot afford to collect or 16

29 17 transmit a lot of measurements[33] such as UWB is. Among the attractive features of CS is the ability to reconstruct any sparse (or nearly sparse) signal from a relatively small number of samples, even when the observations are corrupted by additive noise. Nevertheless, the potential of CS in other signal processing applications is still not fully known[12]. CS methods provide a robust framework for reducing the number of measurements required to summarize sparse signals. For this reason CS methods are useful in areas where analog-to-digital conversion costs are high.[34] Research in this area has two major components[35]: Sampling: How many samples are necessary to reconstruct signals to a specified precision? What type of samples? How can these sampling schemes be implemented in practice? Reconstruction: Given the compressive samples, what algorithms can efficiently construct a signal approximation? This thesis will address both areas presenting a study for different algorithms and different sampling rates analyzing which one fits better for each situation, environment and requirements.

30 Compressive Sensing Theory The fundamental of Compressive Sensing is basically reconstruct a signal x from a downsampled one y. In order to demonstrate the mathematical background, the signal x and y will be consider as a superposition of spikes so that can be represented as vectors of n and m elements respectively, being m<<n. Mathematically: y = Φx Where Φ represents the downsampling matrix from the n elements into m. The number of samplings will indicate the degree of compression of the signal and will be given by the sampling rate. Thus instead of depending on the number of samples intended to transmit or the bandwidth of the signal, the sampling rate will depend on the degrees of freedom leaving the possibility to achieve surprisingly low sub-nyquist sampling rates. To achieve the sparse vector required for the CS, a basis that provide a k- sparse representation is needed. Hence x will be a linear combination of K vector chosen from the basis. Considering α as the k-sparse vector and Ψ the basis: x = N 1 n=0 ψ nα n = K l=1 ψ lα l Then the former problem can we rewritten in terms of a sparse vector: y = Φx = ΦΨα

31 19 Leading to a l 1 norm optimization problem: α = argmin α 1 so that y = ΦΨα Linear programming techniques like Basis Pursuit (BP) or greedy algorithms such Orthogonal Matching Pursuit(OMP and StOMP) can be used to solve this problem[13]. Throughout the thesis these algorithms will be compared in order to find out which one suits better in each situation depending on the sampling rate, the channel and the sparsity of the signal Isometry To succeed in this reconstruction, the Matrix used as downsampling basis must meet certain isometry requirements. Several studies about the isometry properties of the sensing matrix (Φ) [11][14][36], revealed that a matrix that follows the Restricted Isometry Hypotesis-also called Restricted Isometry Propierties (RIP)- can ensure better recovery results. To meet the Restricted Isometry Hypotesis is necessary that every set of columns with a number of columns less than K behaves approximately like an orthogonal system. Hence if the columns of the sensing matrix (Φ) are approximately orthogonal, then the exact recovery phenomenon occurs. In the same studies[14][36], the Uniform Uncertainty Principle is defined to set the isometry conditions that the

32 20 matrix has to meet. If a signal is just nearly sparse, keeping the largest coefficients and setting the rest to zero, is possible to achieve an accurate reconstruction if the sensing matrix undergoes the UUP (at K level). Or putted in another words, if the matrix meets certain characteristics, being x a vector in R N and x K its best K-sparse approximation, the recovery error will not be much worse than x x K l2. At this point, find a matrix that obeys the UUP is essential. This matrix should be designed as collection of N vectors in M dimensions so that any subset of columns of size about K be roughly orthogonal. Although it might be difficult to exhibit a matrix which probably obeys the UUP for very large values of K, is well known that randomized constructions can achieve so with high probability. The reason why this holds may be explained by some sort of blessing of high-dimensionality. Because the high-dimensional sphere is mostly empty, it is possible to pack many vectors while maintaining approximate orthogonality[11]. Being Φ a measurement matrix, it will obey the Uniform Uncertainty Principle with oversampling factor λ if for all subsets T such that T M/λ the following is true [36]: 1 M λ 2 N min (Φ T Φ T ) λ max (Φ T Φ T ) 3 M 2 N What this theory basically tries to ensure is the condition that the geometry of sparse signals should be preserved under the action of the sampling matrix. In

33 21 this case to build the sampling matrix a pre-coding matrix and a matrix representing the channel will be combined. For the pre-coding matrix a Gaussian ensemble can be picked: F (i, j) := 1 N X i,j, X i,j i.i.d.n (0, 1) It will be combined with another matrix representing the channel. The resultant matrix will be consider as the measurement ensemble and will have the rows corresponding with the number of measurements and the number of columns match the number of elements of the sparse vector. To keep meeting the properties of isometry and orthogonality both the random pre-coding matrix and the channel matrix should be incoherent. Since the channel is completely unpredictable, it is hard to ensure such a thing although some properties of the channel have been demonstrated empirically as necessaries to reach a good recovery. Among them a rich-multipath environment has been proved as a necessary condition due to the fact that preserves the orthogonality of the pre-coding matrix Measurements In the theory of Compressed Sensing, sample is to apply a linear function to a signal, being the process of collecting multiple samples the fact of applying a sampling matrix to the signal[35]. Hence down-sample a signal X of length N into

34 22 M samples demands a sampling matrix of N rows and M columns. Obviously the smaller the matrix the less measurements meaning slower sampling rate. Knowing in advance the minimum number of measurements will result in an improvement of the efficiency. Unfortunately although many accurate approximations can be made, is impossible to predict the exact minimum necessary number of measurements needed for reconstruction. Without taking in account the sampling matrix characteristics a broadly restrictions can be made. Several studies pointed out[37][38] some restrictions for the sparsity for a given number on measurements: K C(M/logN) limited to: being C a positive constant. Therefore the number of measurements can be M CKlog(N) However the sampling matrix and its characteristics do affect the number of measurements so they must be taken into account. There are few examples of random matrices in terms of their behavior but this study will focus on simulations made with Gaussian random matrices due to their general features. Considering this, the minimum number of measurements can be restricted to:

35 23 M CKlog( N K ) ɛ 2 δ K ɛ with epsilon being the Restricted Isometry Constant from the Uniform Uncertain Principle, also written as: (1 δ K ) x 2 2 Φx 2 2 (1 + δ K) x 2 2 where δ K := sup Λ =K δ (Φ Λ ). When δ K 1 imply that each collection of r columns from Φ is nonsingular, which is the minimum requirement for acquiring sparse signals. The last but not least consideration that may limit the number of measurements will be the algorithm used for reconstruction. These algorithms are very effective but some need more measurements than others to meet the same accuracy being the first the less complex and fastest. This study will analyze this last dependency and find out the reasons of the behavior of the algorithms and since channel affects the measuring matrix and the reconstruction process, different environments will do so.

36 CS for UWB In this thesis the architecture used to represent the CS recovering for UWB communication will be filter-based using a Finite Impulse Response (FIR) filter. To put in practice this idea is necessary to assume a linear time-invariant system in which the channel, once estimated will remain fixed during the transmission process. It will allow to fairly compare the results for signals with different sparsity levels, something impossible if the channel would be changing. Back to the main equation: y = Φx For communications, y will be the sampled signal while x the transmitted. The channel and the downsampling will be represented by a FIR filter so that the equation can be rewritten as: y (mt S ) = h (mt S ) x (mt S ) where T S is the sampling period and m the number of samples collected. Considering that h is a FIR filter: h (t) = L 1 i=0 h iδ (t it h )

37 25 to implement the system a Toeplitz matrix will be needed. However as the matrix has also to downsample will be necessary to define a quasi-toeplitz in which a each row is the row above, shifted the relation between Sampling Period and Tap Period of the filter q = T S /T h. In other words a Toeplitz matrix in which the rows that does not correspond to q multiples are removed. 3.3 Communication Model In this study Compressed Sensing will be applied to a UWB communication. The model will implement UWB transmission system including UWB transmitter, the channel and a low rate receiver Communication Model: Theoretical Approach The proposed architecture is based in a UWB series of pulses generated from a sparse bit sequence that represent the information as shown in Figure 3.1. It will be modulated with K-Pulse Position Modulation so there will be K pulses distributed along N positions (K<<N). In this study the position of the pulses will be randomized and the data obtained will be averaged for different combinations in order to get reliable results. These pulses pass through a FIR filter before being transmitted. This Precoding filter will be combined with the channel -another FIR filter- giving as a result

38 26 Figure 3.1: Symbol. Pulse Position Modulation (PPM) the φ matrix. It will be implemented by a PN sequence to ensure the matrix meet all the necessary conditions to make possible the recovering of the signal. In the receiver, the downsampling will be performed by deleting the undesired rows of the matrix. The number of remaining rows will represent the sampling rate or number of measurements per symbol. Thus the remaining matrix will have M rows by N columns. Along this study different number of measurements will be tested for different degrees of sparsity. After analyzing the theory, an empiric study will help to understand the practical problems and limitations of the methods proposed. As in every real-environment simulations, variations from theory due to irregularities of the channel are expected. Also new challenges can arise from real channel data and its combination with the pre-coding matrix. The results will help to understand the scope and limitations of real implementation of the Compressed Sensing methods applied to UWB.

39 Communication Model: Practical Approach The Model proposed to apply Compressed Sensing to UWB, as shown in Figure 3.2, is based on a measuring matrix (Φ) that combines the effect of the Pre-coding filter and the Channel. The number of measurements will determine the Sampling Rate at the A/D and will define the number of rows of the Matrix. The information will be applied to a UWB Pulse generator and Modulated with a Pulse Position Modulation (PPM). The pulses generated will have a 2 GHz bandwidth and will be centered in the frequencies 5 GHz and 7 GHz to study the bands 4-6 GHz and 6-8 GHz. Figure 3.2: UWB Communication System Based on Compressed Sensing This model intends to transmit a symbol of 256 different positions for the non-zero elements as portrait in Figure 3.1. Considering Pulse Position Modulation, the position of these non-zero elements will determine the symbol. This modulation, as described above makes the most of UWB by transmitting several pulses, carrying

40 28 the information in the order that the pulses are placed within the frame. Obviously, the more non-zero elements, the more information can be transmitted. However at a given point the signal is not sparse anymore and compressed sensing is no longer applicable. In the simulations the number of non-zero elements will be rose up to 128 in order to see how the density affects the performance of the different algorithms. The output of the Pre-coding Filter will be a PN sequence of 128ns. Taking in account the delay spread the length of the received signal will be over 256ns so a guard period will be added at the end of each symbol. The total length of the symbol will be of 512ns. Hence the link will be about 1.9 MSymbols/s. As depicted in the Table 3.1, the number of pulses (k) used to encode will define the final bit-rate. Nevertheless, the more non-zero elements will require raising the Sampling-Rate, something against one of the goals of UWB which is to reduce complexity in the receiver. This trade-off will be solved depending on the requirements of each case bearing always in mind that the signal has to have sparsity properties. The measuring matrix (P hi) will be the combination of the PN filter and the channel. Both the Pre-coding filter and the channel are modeled as Finite Impulse Response (FIR) filters. Earlier simulations[13] showed that if the chip rate of the PN sequence is equal to the bandwidth of the UWB pulse, the signal can be recovered using the proposed algorithms. Thus the necessary chip rate will be 2 GHz. The band conversion to the 4-6 GHz and 6-8 GHz will be done after the pre-coding filter.

41 29 Non-Zero Elements Bits Encoded Bit-Rate(Mbps) Table 3.1: Bit-Rate due to the number of pulses Number of Measurements Sampling Rate (Msps) % of Nyquist Sampling Rate Table 3.2: Sampling Rate at the Receiver Finally, the A/D Converter at the receiver does not need to down-convert but sample directly. This is the real achieving of Compressed Sensing, how with a very low sampling rate compared with the bandwidth, is possible to recover the transmitted signal. The Table 3.2 shows the impressive sampling rate that the model can achieve. Considering the Bandwidth of the signal of 2 GHz, these rates are at most a 10% of the Nyquist(4 Gsps).

42 Summary The Chapter describes the Compressed Sensing theory and gives mathematical support. It is key to underline the importance of the Matrix and its properties in order to reach a proper recovering after compression. At the end, a model to apply CS sensing to UWB communication is suggested. The simulations as stated above, will be based in this model.

43 CHAPTER 4 THE UWB CHANNEL To verify the feasibility of Compressed Sensing the method will be tested for different frequency bands and different environments. The environments chosen for the research are completely different from each other. Its performance through this different settings will show the reliability for changing conditions and will help to understand better the communication model proposed. 4.1 Channel Measurement The data from the channel has been obtained by a Vector Network Analyzer (VNA), shown in Figure 4.1. The device performs channel frequency measurements. These measurements include both frequency bands (4-6 and 6-8 GHz) and have 1 MHz frequency step with averages of 256 for the 4-5 GHz Band and 511 for the 6-8 GHz. Frequency measurements will be transformed into time domain thanks to the Fast Fourier Transform (FFT) in order to implement the Matrix Φ. The antennas employed are Azimuth Omni-directional and have linear phase across frequency. The frequency range of these antennas, since they were designed for UWB purposes, is from 3.1 to 10 GHz. In the simulations, these antennas are placed 1.5 meters high in different scenarios. They are connected to the VNA through a 31

44 32 Figure 4.1: Vector Network Analyzer (VNA) Figure 4.2: GHz UWB Antenna coaxial cable which losses will be subtracted to the final channel measurement.

45 33 Figure 4.3: Hallway Environment Hallway As one of real possible indoor environments to use UWB a hallway has been chosen for the simulations. This scenario will provide a realistic approach to a daily context. It will allow to demonstrate the positive contribution that multipath has in UWB. One Antenna will be placed at the end of the Hallway, right before a wall that will act as a multipath reflector. The other one will be placed on the other end, 20 meters apart. As seen in the Figure 4.3 the ceiling is not flat so it will not generate proper reflections. However, as all the doors are closed, the walls and the floor will make a perfect cavity for the rays to propagate.

46 Metalic Box This environment perfectly represents the inside of a Submarine or a Ship, were all the walls are metallic. The principal property of the Metallic Box fixed to ground, is that reflects everything. Thus the multipath will be extremely rich. Unlike most of wireless technologies, especially narrow-band, the multipath is a problem in these kind of situations. However in UWB it is used to collect larger quantity of energy. The box is a square of 2.44 meters(8 feet) side as shown in the Figure 4.4. Inside the box, the antennas will be placed perpendicular to one of the walls, 0.3 meters (1 foot) apart from the wall. Thus the distance between the antennas will be 2 meters (6 feet) Inter-Vehicle Since UWB has been proposed as a candidate for new upcoming Inter-vehicle communication, the feasibility of a car-to-car transmission will be studied. The challenge of this scenario is that it is outdoors and is hard to collect most of the energy for the receiver. Nevertheless the antennas have been placed inside the car, using it as a receiving cavity, making the most of the multipath by collecting as much energy as possible. One of the antennas will be placed inside the car in the backseat, close to the car roof while the other will be situated outside in the roof of the other car. The cars

47 35 Figure 4.4: Metallic Box Environment will be up to 8 meters away from each other and will simulate a urban environment. The Doppler Effect will not be considered as stationary channel is assumed. 4.2 Channel Sounding Results In this section, all the data obtained from Channel Measurements will be presented and analyzed. The data obtained from VNA measurements of the Channel in each one of the environments and its correspondent impulse respond. The frequency measurements made on the Hallway, show a relatively stability in terms of frequency. The several selective fading does not affect to the average signal level, which is around -20 db as shown in the Figure 4.6. For the 6 to 8 GHz interval,

48 36 Figure 4.5: Inter-Car Environment the channel behavior follows the same pattern, very stable along the frequency and a slightly lower level. The Impulse Response (Figure 4.7), obtained from the Frequency Data, shows that the multipath contribution is declining along the time with a slightly upturn at the end probably due to the reflections in the background wall. The first component is the greater one, this corroborates the fact that it is a line-of-sight communication since it will be the fastest path. The other contributions along the time are the multipaths, that will be collected from the receiver. While in narrowband this would lead to a synchronization problem or to a distortion of the signal, in UWB is used to collect more energy.

49 37 Figure 4.6: Hallway Channel in Frequency Domain Figure 4.7: Hallway Channel Impulse Response

50 38 Figure 4.8: Metallic Box Channel in Frequency Domain In order to simulate a marine environment, a box which walls are covered by metallic material is used as a background. For this channel, due to the reflection in the walls the level of the signal in for each frequency will be more irregular than other environments. As portrait in Figure 4.8 The level of the signal decreases along the frequency but is scarcely noticeable. And as there is much more multipath the selective fading is greater than in the Hallway due to destructive interferences. The Time Response (Figure 4.9), confirms the large number of multipath generated by the reflections on the walls. The energy decreases progressively due to propagation, because very few is absorbed by the surroundings. This environment, unlike for narrowband will not entail any challenge but it may help recovery. As the matrix that represent the channel will have more non-zero elements, will make the re-

51 39 Figure 4.9: Metallic Box Channel Impulse Response sulting measuring matrix having better isometry properties therefore easing recovery. However this recovery is not guaranteed. For Inter-Vehicle Environment, different measurements were made choosing the worst one in order to prove the feasibility of the Technology. As expected, it was the farthermost and without line-of-sight one. In these scenarios, the receiving antenna was placed inside the car to use this as a cavity to collect as much energy as possible for a proper transmission. This is shown in Figure 4.10, where the level of the signal is remarkably lower than the other settings. The main reason is the fact that there is no direct path and the antennas are placed further than the other simulations. Also the slope is significantly steeper than the others scenarios, so the pulse is sensitive with the frequency although is safe to say it is within an acceptable

52 40 Figure 4.10: Inter-Car Channel in Frequency Domain range. On the other side, the Impulse Response does not have the rich-multipath that other environments do. This is due to the open-air surroundings in which there are not many reflections. In fact the only reflections are the ones collected by the inside car structure. The sudden drop in the signal along the time displayed in Figure 4.11 is due to this lack of reflections. It can become an issue for further distances and will definitely limit the application of this technology to the Inter-Car applications. As seen in the channels, almost all the energy of the pulse is transmitted within the first 40 ns. Knowing the delay spread, and the length of the pre-coding filter, is safe to say that the signal will be spread out within the 256ns frame as expected.

53 41 Figure 4.11: Inter-Car Channel Impulse Response 4.3 Summary Throughout the Chapter, the different environments are presented in terms of frequency and Time. Their features are discussed stressing the ones that may affect the most to CS implementation such as multipath and the delay spread.

54 CHAPTER 5 CS ALGORITHMS COMPARISON Wireless Communications technology is moving forward dramatically in the last few years. However, sometimes the advances in hardware cannot fulfill the even higher demanding levels in all kinds of fields. Then more efficient software is needed to reach the expectations as well as improve the performance of the avant-garde hardware. Besides, while high technology on hardware will always imply more expensive devices and manufacturing process, software will try to simplify some of the requirements, reducing this way the prices. In our case the algorithms used for recovering in the Compressed Sensing will address de tradeoff between the high sampling rate which demands a very sophisticated A/D Converter and the great computational time inherent to the process of reconstruction which requires a complex receiver. Neither of the options is desirable but in some cases any of them could be less restrictive. This study will consider two main approaches to the reconstruction problem, once based on Linear Programming (BP) and another using Greedy Methods(OMP and StOMP). 42

55 Basis Pursuit (BP) Basis Pursuit itself is not exactly an algorithm but a principle that stands for finding signal representations in over-complete dictionaries by complex optimization[15]. By dictionary a collection of parameterized waveforms is meant so it can be Frequency dictionary using sinusoidal waveforms, Time dictionary based on wavelets or even a simply collection of waveforms that are zero except in one point. They will determine the complexity and the feasibility of the method. Applied to Compressed Sending, this dictionary will be given by the sensing matrix so it will contain information of the channel and the pre-coding matrix. Thus BP addresses perfectly the reconstruction problem on Compressed Sensing in which the sensing matrix (Φ) is an over-complete dictionary and the signal representation will be the recovered signal (x) from the original: y = Φx Broadly, the principle is to find a representation of the signal minimizing the coefficients, making them sparse. To reach this the sparsest solution is necessary to minimize in L 1 norm due to its singular property, with L 2 norm, the solution would never preserve the sparsity. This can be explained by the heuristics[39], the solution for the L 2 norm is the contact point where the smallest Euclidean ball and the subspace y meet, as it is shown in the Figure 5.1. In contrast, for L 1 norm minimization,

56 44 this ball becomes an octahedron so that the solution is the meeting point between the subspace and any of the vertex of it. Any of the vertices of the octahedron will be a sparse solution, hence the method guarantees the solution being the optimal, will be the sparse. All of this is due to the fact that for most large underdetermined systems of linear equations the minimal L 1 norm solution is also the sparsest solution. Figure 5.1: Heuristic approximation to the minimization problem Algorithm Definition Defining the problem as an optimization in L 1 norm min x 1 subject to Φx = y Becomes a convex, non-quadratic optimization problem so that a translation into Linear Programming problem is made: min c T X subject to AX = bx 0

57 45 where X R m, X := (u; v), c := (1; 1), A := (Φ, Φ), b := y in order to find the recovered signal as: x := u v During the process, the nonzero coefficients are associated with m columns of the matrix A, building a basis of R m. The solution will be given by this basis after optimizing through an iterative process. This process involves swapping columns of the basis in order to find the combination that minimizes the solution X. In other words, finding the solution to the Linear Programming is equivalent to a process of Basis Pursuit. Of all the Linear Programming algorithm, the most interesting ones for BP resolution are the Simplex and Interior due to their characteristics: The first one, BP-Simplex, starts from a linear independent columns of A for which the product with y is feasible to iteratively improving the basis swapping one term in the basis for one term is not. On each iteration the swap used will be the one that best improves the objective function. Studies[16][17] have shown how to select terms to guarantee convergence. The method achieves improvement in each swap except at the optimal solution and the speed will be given by the number of constraints, bounds on variables are implicitly handed and provide little computational cost[40].

58 46 On the other hand, Although BP-Interior works swapping columns, not always choose the optimal swapping. Considering all the feasible points as a convex Polyhedron, Simplex would be travelling around the border while Interior would find the solution travelling inside reaching the border in the last iteration. This method requires processing more information at every iteration so that In some cases some intermediate iterations may not be feasible and find the feasibility eventually[15]. The algorithm used for the simulations will be Interior, is called PDCO[41], and is a Primal-Dual interior method for Convex Objectives. It is the one that only requires matrix-vector products with A and A T [42] instead of implicit functions, something incompatible with Compressed Sensing for UWB purposes. Unfortunately it often requires many of these matrix-vector products to converge representing higher processing time, the main drawback of Basis Pursuit. This dead end situation is due to the impossibility to find a implicit matrix to emulate the channel and the precoding matrix even in a stationary environment Characteristics The unique properties of the Basis Pursuit, have made it one of the references of solving Compressed Sensing problems. Many studies and researches have come up with a large number of alternative algorithms trying to overcome BP with faster methods but none of them reach the accuracy of BP with such a few number of

59 47 samples. BP is founded on a solid theoretical basis, so if a signal is strictly sparse in certain transform domain it can be exactly recovered. The number of samples needed and the processing time will be given by the sparsity of the signal and the properties of the matrix as stated above. This relationship and the way they are related will be studied throughout the simulations below in the thesis. Because is based in global optimization it can stably super-resolve in ways other methods cannot[15]. The way the iterations are made, enable to use significantly less measurements than any other method. Although it implies more complexity in the receiver, the sampling rate is reduced noticeably. In contrast with other algorithms BP is based on a linear programming approach to the sparse representation problem, where instead of minimizing the number of nonzero coefficients in the approximation, minimization of the sum of the absolute values of the coefficients. Since the channel cannot be implemented implicitly, the BP-algorithm that better suits the problem is the Interior and it can be easily obtained from the software package SparseLab[41]. It is slower than other algorithms but there is no formula to construct the combination of the channel matrix and the pre-coding matrix. BP is stable in presence of noise, in fact there is a variation of BP that still finds the optimal answer in presence of certain level of noise. It is called BP Denoising (BPDN) and is based in the same principle but relaxing the constraint of the

60 48 original problem[43]: min x 1 subject to Φx y 2 σ Where σ is an estimation of the noise level in the data. Although in this thesis the noise will not be considered it is important to point out that BP solves this kind of problems with high level of reliability. Unfortunately life is not all that beautiful and there are several drawbacks that are needed to be taken in account. Since BP is a relatively computationally expensive algorithm and the number of iterations is unbounded, it is hard to set a limit or make safe approximations about the processing time. Along the simulations will observe a tendency but there can always be cases in which, due to the characteristics of the channel, or the combination between the pre-coding matrix and the signal, the computational time suddenly increases. The properties of the Matrix will also affect the performance, and although the method is stable the matrix still have to meet the Isometry conditions. Good restricted Isometry constants are required to reach an acceptable performance. The isometry conditions are held with overwhelming probability if the matrix presents entries that are independent and identically distributed (iid). This is reached for dense matrices that represent high multipath channels. However, the more dense the matrix, the worse performance in terms of timing.

61 Orthogonal Matrix Pursuit (OMP) Due to the density of the Sampling Matrix a new alternative to Linear Programming was needed in terms of simplicity and speed. Dense matrices entail great computational burden that some real-time transmission cannot permit. A main alternative is Orthogonal Matching Pursuit (OMP) due to its speed and its ease of implementation. It is an alternative approach that is not based on optimization; it does not seek for any optimization goal but identify which components of the sampling matrix are related with the non-zero elements to build the sparse signal. OMP still maintain the property of recovering a k sparse signal when the number of measurements m is nearly proportional to k [18]. However, the sparsity level of the signal is needed in advance for the method to resolve the problem. OMP method starts from an empty model and builds up a smaller matrix with all the columns that are related with the non-zero elements, picking one column at each iteration. OMP is considered a greedy algorithm because selects the columns in a greedy fashion. At each iteration chooses the column that is more correlated to the sampled signal y to build this way the matrix with the chosen atoms as is shown in the Figure 5.2. After k iterations, the algorithm should have chosen all the columns related to the non-zero elements of the original sparse signal.

62 50 Figure 5.2: Atoms related to non-zero elements (OMP) Algorithm Definition Defining r as the residual, γ t as the number of the column with greater correlation level with the received signal y at iteration t and Γ t as the vector of the index of the selected columns from which the estimation of the signal ( x ) will be obtained, the matrix with the selected columns(λ) will be obtained after k iterations. In the first iteration, the data is initialized: r 0 = y, Γ = and t = 1 signal: The first column will be selected according to the correlation with the received γ t = arg max i=1,...,n r t 1, φ i If there is several columns with the highest correlation level, the method will choose one deterministically[18]. Then the column is added to the set of index Γ and to the matrix Λ:

63 51 Γ t = Γ t 1 γt, Λ t = [Λ t 1 φ γt ] Then it is just solving a least squares problem in order to get the signal estimation. This signal estimation will be an all non-zero elements that together with the index of the matrix will give the final solution after the last iteration: x t = arg min x y Λ t x 2 To solve this projects orthogonally y onto all selected atoms: x = Λ ty where Λ t is the pseudo-inverse calculated based on QR or Cholesky factorization. The coefficient vector is the orthogonal projection of the signal onto the dictionary elements selected up to this iteration. This property gives the method the name and ensures the algorithm selects a new element in each iteration. If it is not the last iteration, the residual is updated: r t = y Λ t x Then the correlation with the remaining columns is made again, repeating the same steps until the last iteration that corresponds with the number of non-zero elements of the original signal. As stated before, the signal estimation will be the elements of x placed in the positions of Γ.

64 Characteristics As an alternative for BP, Orthogonal Matching Pursuit is much faster, both theoretically and experimentally. According to several studies [44] [45] [19] OMP is observed to perform faster and is easier to implement than L1-minimization. OMP, iteratively selects the vectors from the sampling matrix that contain most of the energy of the measurement vector y. The selection at each iteration is made based on inner products between the columns of the matrix and a residual. The residual reflects the component of y that is orthogonal to the previously selected columns[46][47]. It takes k iterations, where each iteration amounts to a multiplication by a mxn matrix Φ and includes solving a least squares problem in dimensions at most mxk, yielding a strongly polynomial running time. Besides the simplicity of each iteration, the iterations are limited, it has been proven is that OMP selects a correct term at each iteration, and terminates with the correct solution after just k iterations. In fact studies [17] pointed out that the k-step solution property is not a necessary condition for OMP to succeed in the recovery of the sparsest solution although it is sufficient. The fact that the iterations are bounded makes OMP even simpler making the complexity of OMP significantly smaller than that of LP methods, especially when the signal sparsity level K is small[48].

65 53 It has been demonstrated OMP is simpler and faster, but its success in Compressed Sensing against other fast algorithms finding the sparsest solution is its orthogonality. Thanks to orthogonal projection used, the residual r n is always orthogonal to all previously selected elements and then these elements are not selected repeatedly[49]. Methods such Matrix Pursuit (MP) can converge to a solution that explains the data but it is not guaranteed that it is a sparse solution. Instead OMP additionally orthogonalizes the residual against all previously selected measurement vectors. Despite this step increases the complexity of the algorithm, it improves its performance and provides better reconstruction guarantees compared to plain old MP. Experiments[50] shown OMP as the algorithm with superior performance from the family of matching pursuits. As a trade-off, OMP lacks of stability where other algorithms do not. As demonstrated in some studies[35][21] if OMP selects a wrong element in some step it might never recover the right signal. Like other greedy algorithms OMP cannot provide uniform recovery guarantees as other methods like convex relaxation does. Being a heuristic mode, there is not solid theoretical foundation about the reliability of the methods but empirical experiences have shown it works in most of the cases. Perhaps building up an approximation one step at a time by making locally optimal choices at each step makes it more vulnerable to fail in certain scenarios than l 1 minimization which uses a global optimization.

66 54 Another drawbacks are the requirements of OMP needs to solve the problem. To implement the algorithm, the level of sparsity (k) is needed and it is used as the upper bound in the iterations. Also requires that the correlation between all pairs of columns of the matrix is at most 1/2k to operate successfully representing a more restrictive constraint than the Restricted Isometry Property [48]. Comparing again with MP, calculating the pseudo-inverse of the sub-matrix through QR or Cholesky factorization will require computationally more demanding than Matching Pursuit but ensuring that the algorithm selects a new element in each iteration and that the error is minimal for the currently selected set of elements[22]. These factorizations require additional storage, not very significative when for smallsized problems but when it comes to large problems, the storage requirements can became an issue and sometimes Λ cannot be stored. New studies [49] try to develop fast approximate OMP algorithms that require less storage. Finally, the vulnerability against noise is similar to algorithms such BP when the level noise is not too high as shown in some studies[21][35]. For higher level of noise, the performance of OMP gets worse due to the instability of the method although it has not been demonstrated theoretically. As stated before this study will not go deeper in the matter since the simulations are noiseless. OMP has been the reference of many other models that tried to make speed and simplicity their principal feature, achieving highly efficient computations comparable

67 55 to existing CS algorithms. 5.3 Stagewise Orthogonal Matrix Pursuit (StOMP) So far two main trucks has been portrayed to solve the recovery of sparse solutions problem with CS, the fast and simple option (OMP), and the accurate and less demanding (BP). However in the case where of large scale problems both methods become extremely slower, sometimes unacceptable. Addressing these cases, StOMP was implemented. The nomenclature Stage-wise OMP is due to the fact that the algorithm is able to select most of the columns related with the non-zero elements of the vector in one iteration or step. StOMP as an extension of OMP, is a fast greedy algorithm but its singular characteristics make it one of the state-of-the-art fast CS algorithms[51]. Its main difference with original OMP is the way to select the columns for the matrix that contains the non-zero atoms. Instead of selecting one column at a time, fixed a threshold, all the columns whose correlation value is over the threshold will be selected as matched columns. Hence with just few iterations all the columns can be selected. Nevertheless the proper performance of the algorithm will rely in the correct choice of the threshold. As shown below it can become an issue. Due to the demanding requirements of the method, it is not that efficient for high sparsity (few non-zero components), which is a contradiction because it should be

68 56 easy to set a threshold able to differentiate the columns corresponding with the nonzero elements. The values of the correlation in that case should be clearly different, but in most of the cases the columns of the matrix are not taken from the Uniform Spherical Ensemble (USE)[20] they are not Independent Identically Distributed (iid) as the method requires Algorithm Definition As in OMP, a matrix with the selected columns from non-zero elements will be build up to solve a system of equations. Using the same notation: r will represent the residual, γ t the number of the column selected at iteration t and Γ t as the vector of the index of the selected columns and Λ the actual matrix with the selected columns. Besides, with StOMP a vector of residual correlations will be saved (C t ). In the first iteration, the data is initialized: r 0 = y, Γ = and t = 1 The correlation vector is obtained by applying a matched filtering: C t = Φ T r t 1 In this case, instead of selecting one column, all the columns whose correlation is above the threshold will be selected:

69 57 Γ = j : C t (j) Then the set of selected columns is added to the the matrix Λ: Λ t = [Λ t 1 φ j ] : j Γ Again, the problem is reduced to solving a least squares problem: x t = arg min x y Λ t x 2 solution. where the elements of x are the values of the non-zero components of the Unlike OMP the x t approximation will be reached by: x = ( Λ T t Λ t ) Λ T t y At this point, the residual is updated to find the vector of correlations again: r t = y Λ t x If the none of the components of C t = Φ T r t 1 is above the threshold, the algorithm considers the process done. Otherwise the procedure will continue with the newly selected columns. As in OMP, the signal estimation will be the elements of x placed in the positions of Γ.

70 Threshold The main advantage of StOMP over the rest of methods is the capability of solving the problem within just few iterations due to the use of the threshold. The threshold allows to select several atoms at each iteration reducing the complexity of the algorithm and hence the computational time. However the choice of the threshold is not an easy task. A wrong election of the threshold can lead to a poor performance. Two criteria have been proposed to set the threshold. After analyzing both of them, this study will provide empirical values for a good performance in the concrete cases that are treated. The two ways of fixing the value of the threshold try to limit the number wrong selected columns also called False Alarms. Once fixed this limit, each step will try to maximize the number of detections to find the solution as fast as possible. The limits are controlled by setting either the False Alarm Rate(FAR) or the False Discovery Rate(FDR): False Alarm Rate (FAR). This rate measures the number of False Alarms over the total possible False Alarms that can occur: F AR = F alsealarms m k To find the threshold from this rate the inverse of the normal Cumulative Distribution Function (CDF) of 1 F AR/2 will be calculated. This is possible thanks to the Gaussian behavior of the vector of correlations. It is considered a non-zero

71 59 entries combined with large number of Gaussian Noise [20]. Furthermore it has been proved[20] that if the matrix belongs to Uniform Spherical Ensemble(USE)(iid points on the unit sphere S n 1 ), the residual Z = x x 0 has a histogram that is nearly Gaussian. Obviously, with this criterion is necessary to know in advance the sparsity of the signal in order to set it properly. Holding the threshold with a tight FAR can lead to miss any essential atom so the recovery will be unreachable. On the other hand, a very loose may affect in the timing performance of the method and some accuracy. The method still can reach the solution even selecting more columns than necessaries. The values of the solution that correspond with zeros will have very low -close to zero-, but not zero so some accuracy will be lost False Discovery Rate (FDR). It also measures the number of False Alarms but over the discoveries already made: F DR = F alsealarms k where k is the number of discoveries that not necessarily has to be the same as non-zero elements. FDR also assumes that the correlations vector have Gaussian behavior. However it does not consider the sparsity so, although it is a bit more complex than the FAR method, FDR can be used when the sparsity is unknown a priori.

72 60 To apply the criteria to obtain the threshold the Complementary Error Function values of the correlation vector are calculated and sorted. These values are compared with a curve defined by the FDR value. the values under this curve will define the threshold. The value of the threshold in both cases will have to be different depending on the percentage of non-zero elements of the signal. Even if the exact number is unknown, an approximated idea of the non-zero elements will be necessary in order to set the threshold accurately. StOMP, in theory is the fastest method to recover the signal but the threshold setting makes it unstable and dependent. Only by finding a pattern or a reliable criterion to define it, will make it efficient and competitive with the other suggested algorithms. Along this study different thresholds will be analyzed in order to find any trend that allows a stable and reliable setting for proper reconstruction. 5.4 Simulation in Matlab The Simulations in Matlab will show the processing that would take part in the receiver. The received signal will be processed assuming that the channel and the precoding information are known by the receiver. Then the three methods commented

73 61 above will be applied to recover the original signal and obtain results related to the Error Rate, Processing Time and Number of Missed Detections Measuring Matrix To obtain the measuring matrix the Pre-coding Filter and the Channel will be combined. The Pre-coding Filter as well as the Channel will be considered as FIR filters so they will be implemented with Toepliz matrices. So will be the resultant matrix Φ. However, since the receiver samples with lower rate than the transmitter does, the number of rows will be less than the columns. This downsampling makes the matrix a Quasi-Toepliz Matrix in which its rows are selected rows from the original Toepliz. In the new matrix, each row will be the row above shifted several positions. The number of positions the rows will be shifted will be given by the downsampling factor. For Example, if the 256 elements signal that is intended to transmit, is sampled with 64 measurements, the downsampling factor will be 4, so the rows will be shifted 4 positions from the row before. As shown in the Figure 5.3. The procedure will repeat transmission at the same environment changing the order of the non-zero elements for a given sparsity. This will offer a widest perspective of recovering, so that the results just will take in account the sparsity and not the particular combination of the signal and the measuring matrix. The methods will be tested in order to set a minimum number of measurements that are needed to recover

74 62 Figure 5.3: Quasi-Toeplitz Matrix a signal for a given sparsity. This number of measurements will be compared with former approximations that have been made in earlier studies. The sparsity of the signals will vary from one single spike to 128, being this, the half of the total elements. It is with 128 non-zero elements when the maximum degrees of freedom is reached. Nonetheless, to reduce the hardware complexity, one of the goals for the future, is necessary to reduce the sampling rate as much as possible. Thus, lowering the number of measurements will be the principal premise.