Hybrid Frequency Estimation Method

Transcription

1 Hybrid Frequency Estimation Method Y. Vidolov Key Words: FFT; frequency estimator; fundamental frequencies. Abstract. The proposed frequency analysis method comprised Fast Fourier Transform and two consecutive frequency rectifications. Its application leads to a substantial error reduction when estimating the fundamental frequencies and amplitudes of harmonic components in the studied signal in comparison with the widespread spectral analysis methods. Problem Formulation Spectral analysis is the main function of many measuring and diagnostic devices and systems. It is based not only on classical spectroscopy, but on methods for indicator assessment, which can not be measured directly. Examples of this latter application are many and range from study of cardiac function in medicine, to purely technical solutions, such as determining the density of pulp in raw material mills. Spectral analysis is widely used in systems for diagnosis of various mechanizations and machinery. For example, the noise spectrum and vibrations can be used to detect slip on conveyor belts, the irregular support ring load of rotor excavators, working conditions and loading of large mining machinery, etc. When the frequencies of all harmonics forming the studied signal are multiples of the time sampling frequency (Fs), then after applying digital Fourier transformation (DFT), to each frequency corresponds a single harmonic from the spectrum, their respective amplitudes correspond to those of the signal s harmonics. In reality, signal frequencies are not known in advance. This makes the selection of an appropriate sampling rate impossible. When the signal contains a harmonic not coinciding with the one from the Fourier transformation (FT) spectrum, then its energy is distributed to harmonics near the real frequency, this forms the so-called bell of frequencies. The actual frequency amplitude is divided unevenly between its describing harmonics from the spectrum, as the nearest have the largest magnitude. The bell amplitude is always less than the actual one. This problem is called spectral leakage and has an adverse effect in spectral analysis. It occurs when the signal s end phase is different from its starting one, i.e. the signal is not synchronized. Spectral leakage is a serious problem with negative impact on spectral analysis accuracy, this is why different approaches are developed to resolve the problem. They can be divided into two main areas - improving the FT frequency resolution and detecting a frequency component in a noisy spectrum [14]. There are many methods for reducing the spectrum leakage and noice, but none can remove them completely. These include the use of window functions, Capon and APES methods for amplitude and phase-frequency estimation, power evaluation by Welch s method and others [6,10,11]. Another group represent interpolation methods known as Grandki, Quinn and Johnson methods, and a weighted linear approximation method based on Parseval s theory [3-5]. They have improved precision, but require signal pre-processing. Despite many developments in this area, searching and exploring spectrum leakage suppression methods continues to be a question of present interest. There are mainly two reasons: the fact that the problem has not been fully and finally solved and the desire to improve the quality characteristics of instrumentation, used in spectral analysis. Hybrid Frequency Estimation (HFE) Method General Algorithm Description The proposed method consists of three main phases: Initial spectrum evaluation by fast Fourier transformation (FFT) First frequency correction by Johnson s interpolation method. Second frequency correction through optimization procedure. Merger of similar frequencies. HFE is an iterative method. On each iteration one of the available frequency harmonics in the studied signal is calculated. The algorithm consists of the following steps: 1. Perform FFT on the studied signal. It serves as an estimate of the fundamental frequencies. 2. The resulting spectrum is examined to find the global extreme corresponding to the harmonic with maximum amplitude. 3. Johnson s quadratic frequency interpolation is used to find the fundamental frequency. 4.Perform a second frequency correction using the single dimension simplex method. 5. The newly established frequency, amplitude and phase angle are stores in an array of fundamental harmonics. 6. The last found harmonic is generated and subtracted from the signal. The algorithm is repeated from step 1 to find the next fundamental frequency. The search ends when a harmonic is detected with amplitude less than an initially set value, which is often comparable to that of the signal s noise amplitude. 7. The fundamental harmonics array undergoes merger operation, thus reducing part of the similar frequencies. Description of the Main Algorithm Phases Johnson s method [15] is applied for frequency correction. The actual frequency is given by f =ΔF( k real( δ )), where:

2 ΔF is a FFT s frequency resolution; k is an integer index of the highest magnitude harmonic F[k] real (δ) is the real part of the frequency deviation δ, it is given by the following equation: (1) F [.] is a discrete spectrum of complex harmonics. When using the Johnson s method, the frequency can be corrected within the set (-ΔF/2; ΔF/2), with maximum error of 0.06, 0.04 and 0.03 for signal-noise ratio respectively 3, 6 and 9 db. Disadvantages: It is applicable only when the majority of a frequency s spectral energy is within the area ΔF between two harmonics. The need to find the global optimum of the spectrum, which requires array traversal, resulting from the FFT, but this requirement is also present in the rest interpolation methods. Advantages: Good correction characteristics. Fast computation - operations are limited to summation, subtraction, rotation and a single multiplication and division operation per iteration. Single dimension simplex optimization aims to maximize the magnitude obtained as a function of DFT with argument - the studied frequency. DFT can produce only one harmonic, which is not limited to the integer set. It is this property that allows maximizing the magnitude by small frequency changes. The initial scanning frequency used to start the optimization procedure is obtained as a result of Johnson s method. Scanning is terminated upon reaching a step commensurate with an acceptable frequency error, for the proposed algorithm its recommended value is around Δs = 10-6 Hz. For the located extreme the amplitude of the studied harmonic is determined as well as the phase angle. Through the proposed optimization procedure the quality of spectral analysis is improved, especially in the presence of confounding effects or close harmonics, which change their spectrum leakage bell forms. Operation merger is essentially a weighted averaging of similar frequencies, which are stored in the fundamental harmonics array. The following equation is used: (2) δ = f = p i = 1 p fi [] ai [] i= 1 ai [] F [k + 1] - F [k - 1] 2F [k] - F [k + 1]- F [k - 1] where f ' is the resultant frequency, a[i] is the amplitude of the harmonic with index i from the array, f[i] is the frequency of the i-th harmonic in the array, p is the number of harmonics to merge. Merging is performed separately for each group of similar frequencies. Their difference is less than a set margin step, which is directly proportional to the frequency s resolution ΔF. The resultant amplitude is calculated as the sum of amplitudes of harmonics falling in one group or by applying DFT on the resultant frequency. Fundamental frequencies obtained as a result of the operation merger form a set of the minimum number of harmonics, which are able to present the studied signal with certain accuracy. After executing the algorithm, the noise still remains in the signal s array, so it could be subjected to further analysis. Method Application Results The performance and operation principle of the proposed algorithm can be illustrated by the following example. A signal containing three harmonics is being analyzed, their frequencies are respectively 2, 3 and 7.77 Hz, their amplitudes are 1, 0.77 and The sampling frequency is 30 Hz, the sample size (n) is 200. The terminating ratio between the maximal and minimal amplitude that the algorithm is able to account is MaxAmplitudeRatio = 100, the simplex method minimum step (Δs) Hz, the minimum merging frequency Hz. The detection process of a frequency using the proposed method is illustrated in figure 1. Figure 1a and figure 1b show the determined spectrum on each iteration of the search, respectively, in linear and logarithmic scale. Solid vertical lines represent the frequency resolution. Figure 1c presents graphically the reduced signal obtained after removal of the detected harmonics. The set of the established fundamental frequencies and their errors are depicted in figure 2 and presented in table 1: The amount of noise in the signal and its influence on the frequency and amplitude errors is shown in table 2. Comparison of the results obtained for one signal using Johnson s method and the proposed hybrid method for Fs = Hz, n = , frequency range ( Hz), MaxAmplitudeRatio = 20. Frequency error for both approaches is presented graphically in figure 3. We see that in this case, the accuracy of the second method is improved about 266 times. Figure 4 presents the average frequency error obtained using the FFT, Johnson s method (J), the hybrid algorithm without frequency merging (H1) and the hybrid algorithm with frequency merging (H2). Results infer about the effectiveness of the proposed hybrid method. On the base of the experiments we conclude the following: A similar accuracy is obtained when working with relatively small (0.01) and large (100) amplitudes of the forming harmonics As noise increases the terminating amplitude ratio is reduced and so is the ability to identify the harmonics with relatively small amplitudes. In a spectrum with a single harmonic, the maximum frequency error at 1 Hz is up to and is reduced by 19 db/dec as frequency increases (figure 5). Significant increase in errors occurs in the presence of harmonics in the signal with close frequencies, spaced less than the frequency resolution ΔF = Fs / n. The results of processing such a signal are presented in figure 6. The maximum frequency error reaches values of 0.04 for frequency spacing (0 50) of ΔF, 0.01 for spacing (50 100)ΔF and for spacing ( )ΔF

3 Table 1 Frequency, Hz Frequency error, Amplitude error, Figure 1. Results from the hybrid frequency estimation method. Gradually eliminating the harmonic with the max amplitude

4 Figure 2. Spectrum of matched fundamental frequencies and their frequency and amplitude errors Figure 3 Table 2 Description Fs, Hz n A single harmonic with arbitrary frequency and amplitude Three harmonics with arbitrary frequencies and amplitudes Two similar harmonics with a magnitude of 1. Merg e coef. ΔF/100 Ampl. ratio Δs, Hz SNR, db Avg freq. error, Avg. amp l. error, Figure 4. Mean frequency error in of the real frequency

5 Figure 5. Frequency and amplitude errors for frequency estimation of a single harmonic signal without noise Table 3 Operation Count FFT n DFT nk Finding FFT spectrum s maximum n Johnson s method n Frequency merging 1 Operation Table 4 Proportional maximum execution time FFT 1 1 Finding the maximum 12.8 amplitude 0.17 Метод на Джонсън Johnson s method Frequency merging Proportional mean execution time

6 Figure 6. Frequency and amplitude errors for frequency estimation of a signal with two harmonics, spaced at (0 2ΔF)Hz Computational Costs when Implementing the Hybrid Algorithm The enhanced spectral analysis quality using the hybrid algorithm is achieved at the expense of using significantly greater computational resources. The types of operations to be implemented and their count for n-fundamental frequencies in the studied signal are presented in table 3. k marks the average number of DFTs performed at each iteration. The relative execution time of the key algorithm phases are given in table 4. The time required to calculate FFT was adopted for a base value of 1 and the time resources used by the other components are given proportional to it. The experiments indicate that the average execution time of the hybrid algorithm is 17.7 times higher compared to FFT, the major time loss is related to the simplex method implementation. Conclusion A hybrid frequency estimation algorithm is proposed based on FFT and DFT, Robert Bristou-Johnson frequency interpolation method, an additional correction for increased estimation accuracy by an optimization algorithm and the merging of similar frequencies. Experiments showed significant improvement in the frequency estimation s quality characteristics compared to FFT and Johnson s method, with frequency error reaching for frequencies above 10 Hz. The mean frequency error when analysing signals containing 3 harmonics varies within The low levels of errors remain when the number of harmonics in the studied signal increases. The amplitude error for frequencies exceeding 10 Hz varies in the range proportional to the actual frequency. Similar results are obtained when SNR is 0 db with compromise in an amplitude manner. The method shows very good performance when processing signals containing harmonics with close frequencies, frequency error reaches 0.05 for frequency spacing less than the frequency resolution. Due to the relatively lower performance of the proposed method compared to classical approaches to spectral analysis, a guideline for future research are methods and means for increasing computational speed, such efforts will be aimed primarily at using parallel processing

7 References 1. V. K. Jain et al.high-accuracy Analog Measurements via Interpolated FFT. - IEEE Trans.Instrumentation and Measurement, IM-28, June 1979, Rife, D. C. and R. R. Boorstyn. Single-Tone Parameter Estimation from Discrete-Time Observations. - IEEE Trans. Information Theory,. IT-20, September 1974, Quinn, B. G. and P. J. Kootsookos. Threshold Behaviour of the Maximum Likelihood Estimator of Frequency, - IEEE Trans. Signal Processing, 42, November 1994, Quinn, B. G. Estimating Frequency by Interpolation Using Fourier Coefficients, - IEEE Trans. Signal Processing, 42, May 1994, Grandke, Thomas. Interpolation Algorithms for Discrete Fourier Transforms of Weighted Signals, - IEEE Trans. Instrumentation and Measurement, IM-32, June 1983, Welch, P. D. A Direct Digital Method of Power Spectrum Estimation. - IBM J., April 1961, Brillinger, D. R. An Introduction to Polyspectra. - Annals of Mathematical Statistics, 36,, 1965, Kay, S. M. Modern Spectral Estimation Theory and Application. Prentice Hall, Jan Van Trees, H. L. Detection, Estimation and Modulation Theory Part- I. John Wiley, Jan Liu, Z. S., H. Li, J. Li. Efficient Implementation of Capon and APES for Spectral Estimation. - IEEE Trans. Aero. and Elec. Syst., 34 (4), 1998, Capon, J. Maximum-likelihood Spectral Estimation. Nonlinear Methods of Spectral Analysis, S. Haykin, Ed. New York: Springer-Verlag, Jakobsson, A. and P. Stoica. Combining Capon and APES for Estimation of Spectral Lines. - Circuits, Syst., Signal Process., 19, 2000, Stoica, P. and R. Moses. Spectral Analysis Of Signals. Upper Saddle River, NJ: Prentice-Hall, < windows.pdf> 15. Lyons, Richard G. Understanding Digital Signal Processing. Second Edition, ISBN , Pearson Education, Inc., 2004, , < ?_requestid=2155>. Manuscript received on Yasen Vidolov was born in 1980 in Sofia city. He graduated with Master s degree from the University of Mining and Geology St. Ivan Rilski, specialising in Automation, Information and Control Systems. He started working on his PhD thesis in 2008, which is focused on high-speed computation in smart sensors. His current research interests include sensors, spectral estimation, frequency interpolation approaches, parallel processing and FPGAs. Contacts: Sofia, Studentski Grad, Hristo Botev str. University of Mining and Geology St. Ivan Rilski nre@abv.bg