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PAGE 1 SIGNAL ANALYSIS USING THE WARP ED DISCRETE FOURIER TRANSFORM By OHBONG KWON A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORID A IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 1 PAGE 2 2010 Ohbong Kwon 2 PAGE 3 To my Family 3 PAGE 4 ACKNOWLEDGMENTS This dissertation is the result of many years of work whereby I have been accompanied and supported by many people. Therefore, their appreciation is welldeserved. First and foremost, I would like to sincerely thank my advisor Dr. Fred J. Taylor for his excellent support, encour agement and consideration in guiding the research. I would also like to thank all the member s of my advisory committee: Dr. Herman Lam, Dr. Janise McNair and Dr. Douglas Cenzer for their val uable time and interest in serving on my supervisory committee as we ll as their advice and comments, which helped improve the quality of this dissertation. Lastly, I could not thank enough to my par ents who gently offered unconditional support, and encouragements at eac h turn of the road. Thei r love has enabled me to complete this process as well as grow from it. 4 PAGE 5 TABLE OF CONTENTS page ACKNOWLEDG MENTS..................................................................................................4LIST OF TABLES............................................................................................................7LIST OF FI GURES..........................................................................................................8ABSTRACT .....................................................................................................................9 CHAPTER 1 INTRODUC TION....................................................................................................101.1 Background .......................................................................................................101.2 Research Contributi ons....................................................................................111.3 Organization of th e Dissertat ion........................................................................12 2 OVERVIEW OF SPECTRAL ANAL YSIS................................................................142.1 Spectral analysis of sinusoidal signals ..............................................................142.2 Limitations: Spectral leak age and uniform re solution ........................................19 3 TRANSFORMS FOR SIGNAL ANAL YSIS..............................................................303.1 Discrete Fourie r Transform (DFT).....................................................................303.2 Nonuniform Discrete F ourier Transfo rm (NDFT).............................................313.3 Warped Discrete Fourie r Transform (W DFT)....................................................33 4 WDFT USING FIRSTORDE R ALLPASS FILTER.................................................414.1 Warping Parameter Cases Usi ng FirstOrder Allpass filter...............................414.2 MATLB Simu lation ............................................................................................46 5 FILTER BANKS USING A LLPASS TRANSF ORMATI ON.......................................485.1 Uniform DFT Filter Bank...................................................................................485.2 Nonuniform DFT Filter Bank ............................................................................505.3 MATLAB Simu lation..........................................................................................54 5 PAGE 6 6 PRELIMINARY STUD Y AND RESU LTS................................................................566.1 FREQUENCY DI SCRIMINA TION.....................................................................566.2 OPTIMIZATION OF FR EQUENCY RESO LUTION...........................................576.3 RESULTS AND COMPARIS ON.......................................................................587 CONCLUSIONS AND FUTURE WORK.................................................................687.1 Conclu sions ......................................................................................................687.2 Future work .......................................................................................................69LIST OF RE FERENCES...............................................................................................70BIOGRAPHICAL SKETCH ............................................................................................72 6 PAGE 7 LIST OF TABLES Table page 21 Properties of some fixed window functi ons.........................................................2931 Some properti es of th e WDFT ............................................................................4032 Comparison of number of operations.................................................................4061 Comparison of the warping parameter...............................................................67 7 PAGE 8 LIST OF FIGURES Figure page 21 Magnitude of a 32point DFT of a sinus oid of frequency 10 Hz ..........................2322 Magnitude of a 32point DFT of a sinusoi d of frequency 11 Hz ..........................2323 DTFT of a sinusoid sequence windowed by a rect angular window....................2424 DFTbased spectral analysis of a sum of two finitel ength sinusoi dal.................2525 Example of tw o tones s eparated ........................................................................2731 Scheme for computing the magni tude spectrum with unequal resolution...........3932 WDFT r ealizatio n................................................................................................3941 Frequen cy mapping............................................................................................4342 Location of frequency samples for D FT and WD FT............................................4443 Magnitude spectrum for two tones with 64 point WD FT.....................................4651 Typical subband decomposition system ............................................................5152 Uniform DFT filter bank magnitude frequency response for M =8.......................5153 A) DFT filter bank and B) DFT filter bank wit h decimat ors..................................5254 Nonuniform DFT filter bank...............................................................................5355 A) 16channel uniform DFT filter bank, B) 16channel nonuniform....................5461 Magnitude spectrum for two t ones separated A) by one harm onic.....................6062 Magnitude spectrum for two tones separated by 1.6 harm onics........................6163 Magnitude spectrum for two tones with 64point WD FT.....................................6264 Magnitude spectrum for two tones separated by 1.6 harm onics........................6465 Magnitude spectrum for two tone detection with 64point WDFT.......................65 8 PAGE 9 ABSTRACT OF DISSERTATION PRESEN TED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY SPECTRAL ANALSYS USING THE WARPED DISCRETE FOURIER TRANSFORM By Ohbong Kwon May 2010 Chair: Fred J. Taylor Major: Electrical and Computer Engineering In this dissertation, we present a mult itone signal processi ng paradigm that is based on the use of the warped discrete Fourier transform (WDFT). The WDFT evaluates a discretetime signal in the c ontext of a nonuniform frequency spectrum, a process called warping. Compared to a conventional DFT or FFT, which produces a spectrum having uniform frequency resoluti on across the entire baseband, the WDFTs frequency resolution is both nonuniform and progr ammable. This feature is exploited for use in analyzing multitone signals whic h are problematic to the DFT/FFT. This dissertation focuses on optimizing frequency discrimination by determining the best warping strategy and control by using the in telligent search algorit hms and criteria of optimization, or cost functional. T he system developed and tested focuses on maximizing the SDFT frequency resolution over those frequencies that exhibit a localized concentration of spectral energy a nd, implicitly, diminishing the importance of other frequency ranges. This dissertation demonstrates that by externally controlling the frequency resolution of the WDFT in an inte lligent manner, multitone signals can be more readily detected and classified. Furt hermore, the WDFT can be built upon an FFT enabled framework, insuring high efficiency and bandwidths. 9 PAGE 10 CHAPTER 1 INTRODUCTION 1.1 Background Multitone signal detection and discrimination is a cont inuing signal processing problem. The applications include dualtone multifrequency (DTMF) systems, Doppler radar, electronic countermeasures, wirel ess communications, and OFDMbased radar exciters, to name but a few. Traditional mu ltitone detection systems are often based on a filter bank architecture that uses an arra y of product modulators to heterodyne signals down to DC, and then processes the array of downconverters that serve as a bank of lowpass filters [1]. The output s of the filter bank are t hen processed using a suite of energy detection operations to detect the pres ence of multiple input tones [2]. The capability of such a system to isolate and detect multiple narrowband signals is predicated on the choice of initial modulating frequencies and postprocessing algorithms. Other approaches to the signal di scrimination problem are based on multiple signal classification (MUSIC) algorithms, least meansquare (LMS) estimators, and windowed and unwindowed DFTs and their derivativ es, such as Goertzel algorithm [3]. The DFT or DFT variant decomposes a signal into a fixed set of frequency harmonics that are uniformly spaced over the normalized baseband. A variation of the DFT theme has been suggested to provide a more flexible choice of the frequencies. In the modulated DFT system, for exampl e, the input sequence is m odulated before the signal is presented to a conventional DFT [4]. As a result, all frequency samples are shifted by a fixed amount, giving uniformly spaced frequenc ies starting with any arbitrary value. Another instance is the notch Fourier transfo rm which assumes that the input consists of a few sinusoids of arbitr ary frequencies and then employs a few notch filters that are 10 PAGE 11 used to compute the corresponding Fourier c oefficients [5]. The target harmonics are, however, limited in number For applications demanding ex ponential frequency samples, digital frequency warping, wh ich provides unequal spectr al bandwidth has also been suggested [6]. The approach taken in this dissertation is to explore the use of a specific DFT derivative called the warped discrete Fourier transform or WDFT [7]. The WDFT will be shown to decompose a signal into a frequency domain signature having a nonuniform frequency spectrum. 1.2 Research Contributions The major objectives of the research are to explore the use of the WDFT as multitone signal detection technology, assess its capabilities, and compare its performance to other established techniques. It should be appreciated that a salient difference between a DFT and WDFT is frequency resolution. The traditional DFT possesses a uniform frequency resolution across the ent ire baseband. That means that the resolution of the DFT depends on the s pacing between two consecutive frequency harmonics and that an increase in the frequency resolution can be obtained by increasing the DFT length. However, an incr ease in the DFT length will also increase the computational complexity and thereby reduce the real time capabilities of the system. On the other hand, the frequency resolution of the WDFT is nonuniform and externally controlled (programmable). This feature will be used to overcome the multitone signal detection limitation of the uniform re solution DFT. By intelligently and locally controlling the frequency resolution of the WDFT it is anticipated that a high level of signal discrimination can be ac hieved. What needs to be accomplished is to develop an 11 PAGE 12 adaptive steering algorithm that will pl ace the highest WDFT resolution in the frequency range containing the signals of interest. The first contribution of this dissertation is to reduce spectral leakage due to the finite frequency resolution of the DFT withou t increasing the DFT length for multitone detection using the WDFT. As a matter of fac t, an increase in the DFT length improves the sampling accuracy by reducing the spectral separation of adjacent DFT samples, while it causes higher computational co mplexity and cost penalty. The WDFT can control the spectral separat ion through the warping parameter so as to minimize spectral leakage The second contribution is to obt ain higher and optimized local frequency resolution using the WDFT through finding the bes t warping control strategy. In general, a uniformly windowed DFT/FFT cannot determi ne if one tone or multiple tones are present locally about a harmonic fr equency for two tones separated by 1.6 (1.6 harmonics) or less. It will be shown t hat the WDFT can discriminate between two signals separated by as little as 1.3 harmonics. 1.3 Organization of the Dissertation The dissertation is organized as follows: In Chapter 2, we provide a general overview of spectral analysis and its limit ations. Chapter 3 introduces the Discrete Fourier Transform and the Warped Discrete F ourier Transform. Chapter 4 presents the simple WDFT example using the firstorder allpass filter and shows some cases according to the warping parameter using the MATLAB. In Chapter 5, we briefly summarize the traditional uniform DFT filt er banks and introduce the nonuniform DFT filter banks using an allpass filter transfo rmation. Chapter 6 reports the outcome of preliminary studies regarding WDFTbased mu ltitone signal detection and optimization 12 PAGE 13 of frequency discrimination. Chapter 7 presents conclusions and suggestions for future studies. 13 PAGE 14 CHAPTER 2 OVERVIEW OF SPECTRAL ANALYSIS An important application of digital signal processing methods is in determining in the frequency content of a discretetime signal using a process called spectral analysis [8]. More specifically, spectral analysis invo lves the determination of either the energy spectrum or the power spectrum of the signal as a function of frequency. Applications of digital spectral analysis can be found in many diverse fields. In general, spectral analysis often begins with a continuoustime signal x( t ) that is bandlimited, establishing what is called the Nyquist frequency and Nyqui st rate. The continuous time signal is presented to a continuoustime (a nalog) antialiasing filter t hat limits the signal or the baseband having a maximum passband frequency limited by the Nyquist rate. The filters output is then sampl ed above the Nyquist rate using a device called an analog to digital converter. The sampled or quantized sign is then presented to a spectral analysis device, such as a FFT. The resulting spec tral image produced by the spectral analysis device is a facsimile of the actual frequen cy response of the continuoustime signal. 2.1 Spectral analysis of sinusoidal signals Generally it is assumed t hat the spectral analysis of sinusoidal signals can be characterized by the sinusoidal signals parameters amplitudes and phase for a constant frequency. For such cases the discr ete Fourier transform (DFT) of a discrete time x[ n ] is transformed into the DTFT ()j X ewhere 2/[](),01j kNXkXe kN (2.1) 14 PAGE 15 In practice, the discretetime signal x[ n ] maybe windowed by multiplying it with a lengthM window w [ n ] to create a new M sample discrete time windowed y[ n ] = x[ n ] w [ n ]. The spectral characteristics of t he windowed discrete time signal y[ n ] is obtained from its DTFT ()jYe ()j which is assumed to provide a reasonable estimate of the DTFT X e The DTFT ()jYe of the windowed finitelength segment y[ n ] is next evaluated at a set of N ( N M ) discrete angular frequencies equ ally spaced in the range 0 2 by computing its N point discrete Fourier transform (DFT) Yk. To provide sufficient resolution, the N point DFT is chosen to be greater than the window M by zeropadding the windowed sequence with N M zerovalued samples. The DFT is usually computed using an FFT algorithm [9]. []()jWe examine the above approach in more detail to understand its limitations so that we can properly make us e of the results obtained. In particular, we analyze here the effects of windowing and t he evaluation of the frequen cy samples of the DTFT via the DFT. Before we can interpret the spectral content of Ye from Yk, we need to reexamine the relations betw een these transforms and their corresponding frequencies. Now, the relation between the N point DFT Yk of y[ n ] and its DTFT Ye[] []()j is given by 2/[](),01j kNYkYe kN k. (2.2) The normalized discretetime angular frequency corresponding to the DFT bin number k (DFT frequency) is given by 2kk N (2.3) 15 PAGE 16 Likewise, the continuoustime angular frequency k corresponding to the DFT bin number k (DFT frequency) is given by 2 k NT k0[]cos(),01 xnnnM. (2.4) To interpret the results of the DFTbas ed spectral analysis correctly, we first consider the frequency domain analysis of a sinusoidal sequence. Now an infinitelength sinusoidal sequence x[ n ] of normalized angular frequency 0 is given by (2.5) By expressing the above sequence as 00()()1 []( ) 2jnjnxnee00()(2)(2)jj j lXee le l (2.6) we arrive at the expression for its DTFT as (2.7) Thus, the DTFT is a periodic function of with a period 2 containing two impulses in each period. In t he frequency range, there is an impulse at = 0 of complex amplitude ej and an impulse at =0 of complex amplitude ej. To analyze x[ n ] in the spectral domain using the DFT, the simple example of the computation of the DFT of a fi nitelength sinusoid has been in troduced. In this example, we computed the DFT of a length32 sinusoi d of frequency 10 Hz sampled at 64 Hz, as shown in Figure 21. As can be seen from th is figure, there are only two nonzero DFT samples, one at bin k = 5 and the other at bin k = 27. From Eq. (2.4), bin k = 5 corresponds to frequency 10 Hz, while bin k = 27 corresponds to frequency 54 Hz, or 16 PAGE 17 equivalently, 10 Hz. Thus, the DFT has correctly identified the frequency of the sinusoid. Next, we computed the 32point DFT of a length32 sinusoid of frequency 11 Hz sampled at 64 Hz, as shown in Figure 22. This figure shows two strong peaks at bin locations k = 5 and k = 6 with nonzero DFT samples at other bin locations in the positive half of the frequency range. Note that the bin locations 5 and 6 correspond to frequencies 10 Hz and 12 Hz, respectively, according to Eq. (2.4). Thus the frequency of the sinusoid being analyzed is exactly halfway between these two bin locations. The phenomenon of the spread of energy from a single frequency to many DFT frequency locations as demonstrated by this figure is called spectral leakage [10]. To understand the cause of this ef fect, we recall that the DFT Yk of a lengthM sequence y[ n ] is given by the samples of its di scretetime Fourier transform (DTFT) Ye[]()j evaluated at = 2 k/N, k = 0, 1, ... N 1. Figure 23 shows the DTFT of the length32 sinusoidal sequence of frequency 11 Hz sampled at 64 Hz. It can be seen that the DFT samples shown in Figure 22 are indeed obtai ned by the frequency samples of the plot of Figure 23. To understand the shape of the DTFT shown in Figure 23 we observe that the sequence of Eq. (2.5) is also translated as a windowed version of the infinitelength sequence x [ n ] obtained using a rectangular window w [ n ] [11]: 1,01, [] 0,otherwise. nM wn()j (2.9) Hence, the DTFT Ye of y[ n ] is given by the frequencydomain convolution of the DTFT ()j()j Re of x[ n ] with the DTFT of the rectangular window w [ n ]: X e 17 PAGE 18 ()1 ()()() 2jj j RYeXeed (2.10) where (1)/2sin(/2) () sin(/2)jjM RM ee ()j. (2.11) from Eq. (2.7) into Eq. (2.10), we arrive at X eSubstituting 00() ()11 ()()() 22jj jj j RRYeeeee ()j. (2.12) As indicated by the above equation, the DTFT Ye of the windowed sequence y[ n ] is a sum of the frequency shifted and amplitude scaled DTFT ()j Re ()j Re of the window w [ n ] with the amount of frequen cy shifts given by 0. Now, for the length32 sinusoid of frequency 11 Hz sampled at 64 Hz the normalized frequency of the sinusoid is 11/64 = 0.172. Hence, its DTFT is obtained by frequency shifting the DTFT of a length32 rectangular window to the ri ght and to the left by the amount 0.172 2 = 0.344, adding both shifted versions, and then amplit ude scaling by a fact or 1/2. In the normalized angular frequency range 0 to 2 which is one period of the DTFT, there are two peaks, one at 0.344 and the other at 2 (1 0.172) = 1.656 as verified by Figure 23. A 32point DFT of this DTFT is precis ely the DFT shown in Figure 22. The two peaks of the DFT at bin locations k = 5 and k = 6 are frequency samples of the main lobe located on both sides of the peak at the normalized frequency 0.172. Likewise, the two peaks of the DFT at bin locations k = 26 and k = 27 are frequency samples of the main lobe located on both sides of the peak at the normalized frequen cy 0.828. All other DFT samples are given by the samples of the sidelobes of the DTFT of the window 18 PAGE 19 causing the leakage of the frequency components at 0 to other bin locations with the amount of leakage determined by the relative amplitude of the main lobe and the sidelobes. Since the relative sidelobe level Asl, defined by the ratio in dB of the amplitude of the main lobe to t hat of the largest sidelobe, of the rectangular window is very high, there is a considerable amount of leakage to the bin locations adjacent to the bins showing the peaks in Figure 22. 2.2 Limitations: Spectral l eakage and uniform resolution The limitation, as mentioned in the previous section, gets more complicated if the signal being analyzed has more than one sinusoid, as is typically the case. We illustrate the DFTbased spectral analysis approach by means of the next simulations. Through these simulations we examine the effects of the Npoint DFT, the type of window being used, and its length M on the results of spectral analysis. To show the limitation due to spectral l eakage, we compute the various length N of the DFT of the two lengt h16 sinusoidal sequences whos e frequencies are 0.22 and 0.34. Figure 24 (A) shows the magnitude of the DFT samples of the signal x[ n ] for N =16. From the Figure 24 (A) it is difficult to determine whether there are one or more sinusoids in the signal being examined, and the exact locations of the sinusoids. To increase the accuracy of the locations of the sinusoids, we increase the size of the DFT to 32 and recomputed the DFT as indicated in Figure 24 (B). In this plot there appears to be some concentrations around k= 7 and around k=11 in the normalized frequency range from 0 to 0.5. Figure 24 (C ) shows the DFT plot obtained for N =24. In this plot there are two clear peaks occurring at k=13 and k=22 that corresp ond to the normalized frequencies of 0.2031 and 0.3438, respectively. To improve further the accuracy of the 19 PAGE 20 peak location, we compute next a 128point D FT as shown in Figure24 (D) in which the peak occurs around k=27 and k=45 corresponding to the normalized frequencies of 0.2109 and 0.3516, respectively. However, th is plot also shows a number of minor peaks, and it is not clear by examining this DFT plot whether additional sinusoids of lesser strengths are present in the original si gnal or not. As this simulations point out, in fact, an increase in the DFT length improves the sampling accuracy of the DTFT by reducing the spectral separati on of adjacent DFT samples. But it also increases the computational complexity and thereby reduces the realtime capabilit ies of the system The other limitation is the uniform resolu tion. In general, the DFT provides a uniform frequency resolution given by = 2 / N over the normalized baseband, The DFTs frequency resolution is uniform across the baseband. This fact historically has limited the role of the DFT in perfo rming acoustic and modal (vibration) signal analysis applications. These application areas prefer to interpret a signal spectrum in the context of logarithmic (octave) frequency dispersion. Another application area in which a fixed frequency resolution is a limiting factor is multitone signal detection and classification. It is generally assumed that if two tones are separated by 1.6 (1.6 harmonics) or less, then a uniformly windowed DFT/FFT cannot determine if one tone or multiple tones are present locally about a harmonic frequency. To show this limitation, we compute the DFT of a sum of two fini telength sinusoidal sequences with one of the sinusoids at a fixed frequency, while the frequency of the other sinusoid is varied. Specifically, we keep one freque ncy 0.34 and vary the other frequency from 0.28 to 0.31. Figure 25 shows the plots of the 256point DFT computed, along with the frequencies of the sinusoids. As can be seen from these pl ots, the two sinusoids are clearly resolved 20 PAGE 21 in Figure 25 (A) and (B), while they cannot be resolved in Figure 25 (C) and (D) as the frequencies are closer to each other. The reduc ed resolution occurs when the difference between the two frequencies becomes less than 0.04. In the case of a sum of two lengthM sinusoids of normalized angular frequencies 1 and 2, the DTFT is obtained by summing the DTFTs of the individua l sinusoids. As the difference between the tw o frequencies becomes smaller, the main lobes of the DTFTs of the individual sinusoids get clos er and eventually overlap. If there is a significant overlap, it will be difficult to re solve the peaks. It follo ws therefore that the frequency resolution is essentially determined by the width ML of the main lobe of the DTFT of the window. The main lobe width ML of a lengthM rectangular window is given by 4 / M from Table 21 [12] In terms of normalized frequency, the main lobe width of a length16 rectangular window is 0.125. Hence, two closely spaced sinusoids windowed with a rectangular wi ndow of length 16 can be clearly resolved if the difference in their frequencies is about half of the main lobe width, i.e., 0.0625. Even though the rectangular window ha s the smallest main lobe width, it has the largest relative sidelobe amplitude and, as a cons equence, causes considerable leakage. So the large amount of leakage results in minor peaks that may be fa lsely identified as sinusoids as shown in Figure 24 and 25. As a result, the performance of the DFTbased spectral analysis depends on several factors, the type of window being used and its length, and the size ( N ) of the DFT. To improve the frequency resolution, one must use a window with a very small main lobe width, and to reduce the leakage, t he window must have a very small relative sidelobe level. The main lobe width can be reduced by increasi ng the length of the 21 PAGE 22 window. Furthermore, an increase in the accu racy of locating the peaks is achieved by increasing the size of the DFT. To this end, it is preferable to use a DFT length that is a power of 2 so that very efficient FFT algor ithms can be employed to compute the DFT. An increase in the DFT size, however, also increases the computat ional complexity of the spectral analysis procedure which causes cost penalty. Therefore, to resolve two or more tones without increasing the DFT size the WDFT is introduced as an alternative to spectral analysis in the next chapter. 22 PAGE 23 18 16 14 12 10 8 6 4 2 0 0 5 10 15 20 25 30 kX[k] Figure 21. Magnitude of a 32point DFT of a sinusoid of frequency 10 Hz sampled at a 64Hz rate 18 16 14 12 10 8 6 4 2 0 0 5 10 15 kX[k]20 25 30 Figure 22. Magnitude of a 32point DFT of a sinusoid of frequency 11 Hz sampled at a 64Hz rate. 23 PAGE 24 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 Normalized frequencyD TFT M agnitude Figure 23. DTFT of a sinusoid se quence windowed by a rectangular window. 24 PAGE 25 0 5 10 15 0 1 2 3 4 5 6 kM agnitudeA 8 7 6 5 4 3 2 1 0 0 5 10 15 kM agnitude20 25 30B Figure 24. DFTbased spectral analysis of a sum of two finite length sinusoidal sequences of normalized frequencies 0.22 and 0.34, respectively, of length 16 each for A) N =16, B) N =32, C) N =64, and D) N =128 of DFT length. 25 PAGE 26 0 10 20 30 40 50 60 0 1 2 3 4 5 6 7 8 kM agnitudeC 0 20 40 60 80 100 120 0 1 2 3 4 5 6 7 8 kM agnitudeD Figure 24. Continued 26 PAGE 27 0 50 100 150 200 250 0 1 2 3 4 5 6 7 8 9 10 kM agnitudeA 0 50 100 150 200 250 0 1 2 3 4 5 6 7 8 9 10 kM agnitudeB Figure 25. Example of tw o tones separated between A) f1 = 0.28 and f2 = 0.34, B) f1 = 0.29 and f2 = 0.34, C) f1 = 0.3 and f2 = 0.34, and D) f1 = 0.31 and f2 = 0.34 with 256point DFT. 27 PAGE 28 0 50 100 150 200 250 0 1 2 3 4 5 6 7 8 9 10 kM agnitudeC 0 50 100 150 200 250 0 1 2 3 4 5 6 7 8 9 10 kMagnitudeD Figure 25. Continued 28 PAGE 29 Table 21. Properties of so me fixed window functions Type of window Main lobe width Relative sidelobe level ML sl A Min. stopband attenuation Transition bandwidth Rectangular 4 /(2M+1) 13.3 dB 20.9 dB 0.92 / M Hann 8 /(2M+1) 31.5 dB 43.9 dB 3.11 / M Hamming 8 /(2M+1) 42.7 dB 54.5 dB 3.32 / M Blackman 12 /(2M+1) 58.1 dB 75.3 dB 5.56 / M 29 PAGE 30 CHAPTER 3 TRANSFORMS FOR SIGNAL ANALYSIS 3.1 Discrete Fourier Transform (DFT) The venerable discrete Fourier transform (DFT) is an important signal analysis tool. Generally, the preferred inst antiation of the DFT is the CooleyTukey fast Fourier transform (FFT) algorithm. An N point DFT X [ k], for 0 k N 1, of a lengthN timeseries x[ n ], for 0 n N 1, is defined by: 1 0,][][1 0 /2 Nk enxkXN n NknjNj Ne/21 0,][][1 0 NkWnxkXN n kn N (3.1) Note that X [ k] is a length N sequence of complex harmonics corresponding to N /2 positive harmonics and N /2 negative harmonics. Using the socalled W notation (W), equation (3.1) can be rewritten as: (3.2) The inverse discrete Fourier tr ansform (IDFT) is given by: 1 0,][ 1 ][1 0 NnWkX N nxN k kn N[]. (3.3) There are applications where it is desirable to compute t he frequency components of a finitelength sequence having unequal or nonuniform frequency resolution. To this end, Oppenheim and Johnson proposed transformation x n [] yn () Yz [] yn [] yn into a new sequence by means of an allpass network [13]. T he magnitude of the frequency components of the z transform is equallyspaced points on the periphery of unit circle in the zdomain. Since the allpass netwo rk has an IIR transfer function, is an infinitelength sequence and hence, must be made into a finitelength sequence by using an 30 PAGE 31 appropriate window function bef ore its DFT can be computed. Figure 31 shows their scheme. Moreover, the initial conditions of the IIR allpa ss network will pr opagate to the output. As a result, this scheme provides an approximate estimate of the magnitudes of the DFT of [] x n () at unequal resolutions. 3.2 Nonuniform Discrete Fourier Transform (NDFT) The nonuniform DFT (NDFT) [14] is t he most general form of DFT that can be employed to evaluate the frequency components of X z at N arbitrary but distinct points in zplane. If k z denote N distinct frequency points in the zplane, the N point NDFT of the lengthN sequence 01 kN [] x n1 0[]()[],01.N n NDFT k k nXkXzxnzkN [1] [1]NDFT NDFT NX x X x is then given by (3.4) In matrix form, the above N equations can be written in the form [0] [0] [1] [1]NDFTXN xN D NDNN 11 00 11 11 11 111 1 1N N N N NNzz zz zz D k (3.5) where is the NDFT matrix given by (3.6) where DN is a Van der Monde matrix and is invertible if the points z are distinct. In general, the computation of an NDFT involves the multiplication of the NDFT matrix ND 31 PAGE 32 [] x with the lengthN vector composed of sample values of nk resulting in an algorithm of complexity O( N2), i.e., N2 complex multiplications. The NDFT computation can be made computat ionally efficient by imposing some restrictions on the locations of the points z in the z domain. For example, if the points are located equidistant on the unit circle at j 2/ kN kzek, then the NDFT reduces to the conventional DFT which can be computed very efficiently using fast Fourier transform (FFT) algorithms. Note that the resolution of the DFT de pends on the spacing between frequency points. An increase in frequency reso lution can be obtained by increasing the DFT length. However, an increase in the D FT length also increases the computational complexity. In this dissertation, we consider an al ternate structure to the location of the frequency points z by applying an allpass transforma tion to warp the frequency axis. Then, uniformly spaced points on the warped frequency axis are equivalent to a nonuniform spacing frequency points on the origi nal frequency axis. This has led to the concept of the warped D FT (WDFT) which evaluates the frequency signature of x[ n ] at unequally spaced points (frequencies) on the unit circle. By choosing a control variable called the warping parameter, the WDFT c an more densely space some of frequency and sparsely distribute frequency points elsewhere reducing s pectral precision without increasing the DFT length. An efficient rea lization of the WDFT is also proposed that can implement a WDFT. 32 PAGE 33 3.3 Warped Discrete Fourier Transform (WDFT) A. Definition and properties The WDFT is a derivative of the familiar DFT. The DFT is designed to evaluate a signal at frequencies that are uniformly distri buted along the periphery of the unitcircle in the zdomain at locations Unlike the DFT, the WDFT evaluates a signal at nonuniformly distributed locations on the periphery of the unit circle in the zdomain (i.e., critical frequencies). More specifically, an N point WDFT, denoted Nkjez/2 ][kX][ nx, of a lengthN timeseries is a modified ztransform ) (zX)( zX)(1zAz )( zA 1 0][)(N n nznxzXobtained from by applying the transformation: (3.7) where is an Mthorder real coefficient allpas s filter function. The allpass transformation warps the frequency axis and thereby associates uniformly spaced points on the unit circle with their nonuniformlyspaced counterparts in the zplane. Applying the mapping of equation (3.7) to: (3.8) one obtains: j ez n N n j ezzAnx zXkX )(][)(][1 0. (3.9) Using matrix notation, the above equat ion can be written in the form: X = DWDFT x (3.10) where x=[x[0],x[1],,x[N1]]T is the vector of the input time sequence, and DWDFT is the complex NN WDFT matrix given by: 33 PAGE 34 11 11 11 11 WDFT)()(1 )( )(1 1 11NN N N N N N NWA WA WA WA D. (3.11) Many, but not all, of the well known DFT properties such as linearity, periodicity, time or frequency shifting, symmetries etc., also hold fo r WDFT. A few key properties are reported in Table 31. B. Realization To realize the WDFT, we denote from equation (3.7) () () () Az Az Az ()(3.12)where A z is the mirrorimage polynomial of () z1()()M z zz ), resulting in A (i.e., 1 0() () ()[] ()()n N e n ePz Az Xzxn AzDz (3.13) where, 1()()N NnzAz[]1 0()[]e nPzxnA (3.14) x Pe( z) is a polynomial of degree M(N1) that is a function of n1()()NDzAz[] and (3.15) e x n. is another polynomial of degree M(N1) that is not a function of WDFT is defined as () z2/ j kNze X evaluated at 2/ 2/ 2/() []() ()jkN jkN jkNe ze ze e zePz XkXz Dz (3.16) To simplify equation (3.16), we define 34 PAGE 35 1 00,(1)()()modnmNN e N n e nmnmNMNPzPzz p z (3.17) where is the ith coefficient of and iep()ePz()()modN eDzDzz () Pz (). (3.18) D and z both have degree N 1. So the WDFT computation as shown in equation (3.16) is simplified to 2/ 2/() [] ()jkN jkNze zePz Xk Dz (3.19) [] PkLet and [] Dk() Pz () be the N point DFT of the length N sequences obtained from the co efficients of and D z 2/[]() j kNzePkPz (3.20) 2/[]() j kNzeDkDzThen we obtain () () Pz Xz P=Qx() Dz (3.21) Using matrix notation, we can define P as (3.22) where is the column vector formed by the coefficients of P() Pz and Q is an NN real matrix .Further, [] Pk is obtained as [0] [1] P PN W[1] P WQx (3.23) where is the NN DFT matrix. Finally, the WDFT coefficients are obtained as follows. 35 PAGE 36 [0] [1] X X D WQx (3.24) [1] XN where 1 00 [0] 1 00 1 00 [1] D DN [1] D D (3.25) Therefore, we obtain a factor ization of the WDFT matrix into the product of a diagonal matrix, the DFT matrix, and a real matrix. This realization of the WDFT, which follows directly from equation (3.24), is shown in Figure 32. C. Complexity Let x be an N dimensional complex input vector Direct computation of the WDFT coefficients from x requires multiplying x by an NN complex matrix, or 4 N2 real multiplications and 4 N22 N real additions (assuming that one complex multiplication involves four real multiplications and two real additions, and one complex addition involves two real additions). Using a reported factorization of the WDFT matrix [15], the total requirement is N ( N +2log2N +4) real multiplications and N (2 N +3log2N ) real additions while 2 N log2N real multiplications and 3 N log2N real additions for the N point DFT with the above assumption. Table 32 shows the r equired number of operations for some typical values of N 36 PAGE 37 D. Inverse WDFT An inverse DWDFT equation (3.11) can be computed with the complexity of the WDFT. Unfortunately, in certain in stances the condition number of DWDFT can easily exceed 1012 for transform lengths N above 50. Therefore an exact computation of an inverse WDFT can be compromised due to numerical instability. In order to stabilize the matrix inversion, we propose to compute a pseudoinverse of a DWDFT by employing only the r largest singular values i into inversion process [16]. In such cases the inverse WDFT can be expressed as: r i H ii i 11Xuvx. (3.27) Using the pseudoinverse, one can obtain r i i i H i 1v Xu x. (3.28) The l2 norm of the errorvector can then be estimated by: 2 2 1 2 2 2 2 2 2 2 1 2 2 2 1 2 21 1 1 X v Xu Xvu x r i i i H i r i i r i i H i i (3.29) with 12 2iv and 12 2iu. Equation (3.11) shows that a high error amplification is possible due to the large condition number of the DWDFT matrix (the smallest singular value is dominant). In the case of a conv entional DFT, such an effect would not be present because the condition number of the DWDFT matrix is equal to 1. The limit evokes only a small signal distortion but make s the matrix inversion unstable. A high results in a stable matrix inversion but is a ssociated with signal distortions, which can be 37 PAGE 38 considered as a filtering effect. We have obs erved that neglecting small singular values mostly affects the interpre tation of the high frequency co mponents of the spectra. To obtain an efficient computation of in verse WDFT, the factor ization of the WDFT matrix in equation (3.24) can be used. A ssuming the frequency samples of the WDFT to be distinct, the inverse WDFT is given by 111[0] [1] [1] X X XN DxQW Q1 1 (3.30) where is the NN IDFT matrix. Since the rows of are mirror image pairs or symmetric, Q is a matrix with real coefficients whose columns are mirror image pairs or symmetric. With an appropr iate column permutation, its rows can be made mirror image pairs or symmetric. Therefore, 1 W Q can be written as a like matrix postmultiplied by a permutation matrix. C onsequently, the computation of an Npoint inverse WDFT is equivalent to that of an N point WDFT with an additional permutation involving no multiplication. Q 38 PAGE 39 Figure 31. Scheme for computing the magnitude spectrum with unequal resolution using the FFT. Figure 32. WDFT realization 39 PAGE 40 Table 31. Some properties of the WDFT Property Function WDFT Conjugate symmetry x[n] real X[Nk]=X*[k] Conjugation x*[n] X*[Nk] Table 32. Comparison of number of operations DFT WDFT N real multiplication real addition real multiplication real addition 4 64 56 48 56 8 256 240 144 200 16 1024 992 448 704 40 PAGE 41 CHAPTER 4 WDFT USING FIRSTORDER ALLPASS FILTER 4.1 Warping Parameter Cases Usi ng FirstOrder Allpass filter As noted in Chapter 3, the WDFT is defined in terms of an allpass filter. The allpass mapping includes a control parameter a, called the warping parameter To explain the warping parameter, a simple motivational exampl e is presented using a firstorder allpass filter 1 11 )( az za zA N n nznxzX0][)(. (4.1) The warping parameter a is real and a<1 for stability [17]. To showcase the warping parameter cases, it may be re called that the standard ztransform of an N sample input time series x[ n ], is given by: (4.2) From the X ( z), the conventional DFT of x[ n ] is obtained by: 1 0,)(][/2 Nk zXkXNkj ezNkje/21 z (4.3) where evaluates X ( z) at uniformly distributed point s located on the periphery of the unit circle in the zdomain. Referring to equation (3.9) and replacing with equation (4.1), then )(zX is given by: N n naz za nxzX0 1 11 ][)(. (4.4) The WDFT coefficient, then, ][kX are similarly obtained by sampling )(zX at nonuniform points on the periphery of the unit circle in the zdomain, namely 41 PAGE 42 1 0,)(][/2 Nk zXkXNkj ez jez Nkk/2. (4.5) A conventional uniform frequency resolution DFT, defined by has harmonic frequencies uniformly located at frequencies at The center frequencies of an N point WDFT spectrum are lo cated at the warped frequencies (to differentiate them from ), where jez. The frequencies are related to through the nonlinear frequency warping relationship 1 1 arctan2a a 2 tan. (4.6) Equation (4.6) establishes a nonlinear frequency warping relationship that is controlled by the real parameter a. The conventional DFT becomes a special case of the WDFT when a=0. The frequency warping property of equation (4.6) is graphically illustrated in Figure 41 over a range of a A positive a provides higher frequency resolution on the high frequency baseban d region. A negative value of a increases frequency resolution in the low frequency baseband region. Figure 42 illustrates the nonuniform sampling of the WDFT by showing the location of samples on the unit circle of the z plane for the DFT and the WDFT for N =32. 42 PAGE 43 Figure 41. Frequency mapping 43 PAGE 44 1 0.5 0 0.5 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Real PartImaginary PartA 1 0.5 0 0.5 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Real PartImaginary PartB Figure 42. Location of frequency samples for DFT and WDFT for N =32 with A).a = 0, B) a=0.5 and C) a=0.5. 44 PAGE 45 1 0.5 0 0.5 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Real PartImaginary PartC Figure 42. Continued 45 PAGE 46 4.2 MATLB Simulation 0.1 a The WDFT ( ) is compared to the DFT (a=0) in Figure 43. Observe that the frequency resolution is compressed as a decreases. This property can be used to perform nonuniform frequency discrimination. A B Figure 43. Magnitude spectrum for two tones with 64 point WDFT for A) a = 0, B) a = 0.23, and C) a = 0.4. 46 PAGE 47 C Figure 43. Continued 47 PAGE 48 CHAPTER 5 FILTER BANKS USING ALLPASS TRANSFORMATION 5.1 Uniform DFT Filter Bank Multirate systems often appear as a bank of filters where each filter maps the input into a baseband frequency subband [18]. T he filter bank is a set of bandpass filters with either a common input or a summed output as shown in Figure 51. The structure of Figure 51 (A) is an M band analysis filter bank with the subfilters known as the analysis filters. It is used to decompose the input signal x[ n ] into a set of M subband signals with each subband signal occupying a portion of the original frequency band. On the other hand, a set of subband signals occupying a portion of the original frequency band is combined into one signal y[ n ] as shown in Figure 51 (B), a synthesis filter bank, where each filter is a synthesis filter. )(zHk[n]kv[n]kv )(zFk)(zHk0H1 0),()(0An interesting application of this subband ar chitecture is called a uniform DFT filter bank. The kth filter of a uniform DFT filter bank, denoted in Figure 51 is defined in terms of called the prototype filter. Specifically MkzWHzHk M k1 0), ()()/2( 0 Mk eHeHMkj j k )(0zH Mfkfsk/. (5.1) The frequency response of the kth filter is given by (5.2) It can be seen that the frequency response envelope of filter is copied to new center frequencies located at as shown in Figure 52. An interesting case occurs when is defined as a multirate pol yphase filter represented as: )(0zH 48 PAGE 49 1 0,)( )(1 0 0 MmzPzzHM m M m m1 0,)( )(1 0 MkzPzWzHM m M m mim M k 1 0][ )(M j j ik M knyWnh)(nk)(zHk)(nyk)(zk. (5.3) Equation (5.2) implies that: (5.4) which has a DFT structure. That is: (5.5) where yj[ n ] is the output of the jth polyphase filter shown in Figure 53. Equivalently, it follows that Ym(z) = zmPm( zM). The polynomial outputs are combined using an M point DFT, as shown in Figure 53(A), and are used to synthesize the frequencyselective filtersh. The DFT filter bank can, therefor e, possess a number of known benefits including reduced arithmetic complexity w hen using a well designed DFT. To motivate this claim, consider that each filter is on the order N A filter bank would therefore require MN multiplies to complete a filter cycle. Each polyphase filter, however, is of order N/M which reduces the complexi ty of each filter cycle to MN/M=N multiplies. That is, in N multiply cycles, all the intermediate values of shown in Figure 53(A) can be computed. Adding decimat ors, as shown in Figure 53(B), the bandwidth requirement of each of the polyphase filters can also be reduced. The multiply count of an Mpoint DFT would have to be added to this count to complete the multiply audit for a filter cycle. The multip ly count of DFT is dependent of M and, for some choices, can be made extremely small. It can be noted that t he data moving through the polyphase filters is slowed by a factor of M through decimation. By placing H 49 PAGE 50 decimators in the filter bank at the loca tions suggested in Figure 53(B), a further complexity reduction can be realized. 5.2 Nonuniform DFT Filter Bank The nonuniform DFT filter bank can be built upon a uniform DFT filter bank infrastructure and an allpass filter transforma tion. The WDFT differentiates itself from the uniform DFT filter banks in that the signal to be transf ormed is preprocessed using the allpass filter A(z) prior to performing the DFT. T he polyphase multirate filter architecture shown in Figure 54 was used by Galijasevic and Kliewer [19] to implement a nonuniform filter bank. For the case where A ( z)=1, the design degenerates to a traditional uniform DFT filter bank [20, 21]. Fo r the case where the polyphase filter terms are Pi(z)=1 (see Figure 54), the system is degenerated to an M point DFT. When the analysis subband filter s are derived in terms of a prototype filter having an impulse response h ( n ) and a set of allpass filters, a bank of complex modulated filter result. The transfer function of the kth subband filter Hk(z) can be expressed as 1 ,)()( )1 0 MkzPzAWM m M m m im M0 (zHk. (5.6) The resulting frequency response of a 16channe l system is motivated in Figure 55. 50 PAGE 51 A B Figure 51. Typical subband dec omposition system showing A) analysis filter bank, and B) synthesis filter bank. Figure 52. Uniform DFT filter bank magnitude frequency response for M =8. 51 PAGE 52 A B Figure 53. A) DFT filter bank and B) DFT filter bank with decimators 52 PAGE 53 Figure 54. Nonuniform DFT filter bank 53 PAGE 54 5.3 MATLAB Simulation A B Figure 55. A) 16channel uniform DFT filt er bank, B) 16channel nonuniform DFT filter bank for a=0.3, and C) 16channel nonuniform DFT filter bank for a=0.3. 54 PAGE 55 C Figure 55. Continued 55 PAGE 56 CHAPTER 6 PRELIMINARY STUDY AND RESULTS 6.1 FREQUENCY DISCRIMINATION For spectral analysis applications, the DFT provides a uniform frequency resolution given by = 2 / N over the normalized baseband ].[ The DFTs frequency resolution is uniform across the baseband. This fact historically has limited the role of the DFT in performing acousti c and modal (vibration) signal analysis applications. These application ar eas prefer to interpret a si gnal spectrum in the context of logarithmic (octave) fr equency dispersion. Another application area in which a fixed frequency resolution is a limiting factor is mu ltitone signal detection and classification. It is generally assumed that if two tones are separated by 1.6 (1.6 harmonics) or less, then a uniformly windowed DFT/FFT cannot dete rmine if one tone or multiple tones are present locally about a harmonic frequency. This problem is exacerbated when data widows are employed ( e.g ., Hamming window). This conditi on is illustrated in Figure 61. In Figure 61(A), two tones s eparated by one harmonic (i.e., ) are transformed. The output spectrum is seen to consist of a si ngle peak, losing the identification of each individual input tone because their main lobes get closer and eventually overlap. In the case reported by Figure 61(B), the tw o tones being transformed are separated in frequency by two harmonics (i.e., 2 ). The presence of two distinct tones is now selfevident. In the case that two tones are not discriminated in Figure 61 (B), an increased resolution (increased N ) is used to locally analyze the multitone signal. In fact, an increase in the DFT length im proves the sampling accuracy by reducing the spectral separation of adjacent DFT samples, whil e it causes the higher computational complexity and cost penalty. In Figure 62 (A ) and (B), it has been seen that the two 56 PAGE 57 tones separated by 1.6 harmonics are unresolv ed with 64point DFT, but resolved with 256point DFT at the ex pense of increased complexity, respectively. So to resolve two tones without increasing the l ength of the DFT this dissertat ion proposes to exploit the WDFT as a frequency discrimination technology [22] that is operated using a control parameter a that locally defined frequency resoluti on as mentioned in Chapter 2. The effect of the warping relationship is demons trated in Figure 63 which compares a DFT (a=0) to WDFTs for a=0.071, a=0.23 and a=0.4 for the case where two tones are present separated by a single DFT harmonic. It is easily seen that by intelligently choosing the cont rol parameter a the locally imposed frequency resolution can be expanded or contracted. To enhance the systems frequency discrimination, the frequency resolution should be maximized in the local region containing the input signals. As such, an intelligent agent will ne ed to assign the best warping parameter a strategy, one that concentrates the highest fr equency resolution in the spectral region occupied by the multitone process. The next section describes the outcome of a preliminary study that compares two criteria and two search algorithms and develops an intelligent frequency resolution discrimination policy that can be used to improve multitone detection. 6.2 OPTIMIZATION OF FREQUENCY RESOLUTION To optimize the choice the warping parameter a, a<1, an intelligent search algorithm or agent is required. An initial search strategy is being evaluated and enabled using an optimal singlevariable Fibonacci s earch and the modified Golden Section search techniques [23]. The search proce ss is expected to iterate over a range of values of a that places a high local frequency resolution in the region occupied by multitone activity. To find the best warping parameter a, two criteria of optimization 57 PAGE 58 and cost functionals have been singled out for focused attention. The search methods iteratively restrict and shift t he search range so as to optimize spectral resolution within a convergent range. The direct ion of the search is decided by the value of the cost functional at two points in the range. Two criteria studied to date are developed as follows. A. Criterion #1 bkXkX][][max)(1 (6.1) where b is a frequency within the search interval and )(1 is designed to reward the local concentration of spectral energy and penalize more sparsely populated section of the spectrum. To obtain a locally optim al value of the warping parameter a, the difference from the maximum value to av erage values between two adjacent spectral lines are computed using equation (6.1). B. Criterion #2 ][ )(2kX (6.2) where is the threshold used to suppress leakage and )(2 is designed to reward the local concentration of spectral ener gy and minimize leakage determined by the relative amplitude of the main lobe and the side lobes to identify each individual input tone. 6.3 RESULTS AND COMPARISON The dissertation reports on a multit one signal discrimination study conducted using two search methods, namely a Fibonacci search and a modified Golden Section search algorithm. Both are iterative methods that restrict and sh ift the searching range so as to determine an optimal operating point within a frequency range. Studies based 58 PAGE 59 on these criteria involved presenting to the WDFT frequency discriminator two sinusoidal tones located at 0.157 and 0.314 r ad/s. The search method was charged to find the best warping parameter. The comparison of results is shown in Table 61 and the evidence of this activity can be seen in Figure 65. To co mpare the temporal efficiency of each case, Table 61 also shows elapsed time needed to execute a search using MATLAB. The two tones, separated by one harmonic, were unresolved with 64point DFT but resolved with 512point DFT at the expense of increased complexity in Figure 64 (A) and (B), respectively. In Figur e 65 (A)(D), however, the two tones are shown to be present using 64point WDFT. To calibrate the WDFT spectra, the locations of the actual two tones are also shown. Comparing the outcomes, a Fibonacci search was found to be the fastest and most effective in finding the best warping parameter using either search criteria. Criterion #1 resulted in frequency resolution with a bigger variation according to search methods, while Criterion #2 facilitated the optimization of the local resolution and ident ified the two tones cl oser to the actual locations of the tones. 59 PAGE 60 40 35 30 25 20 15 10 5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1A 40 35 30 25 20 15 10 5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 B Figure 61. Magnitude spectr um for two tones separated A) by one harmonic and B) by two harmonics with 64point DFT 60 PAGE 61 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 C 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 D Figure 62. Magnitude spec trum for two tones separated by 1.6 harmonics A) with 64point DFT and B) with 256point DFT 61 PAGE 62 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 A 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 B Figure 63. Magnitude spectrum for two tones with 64point WDFT with (A) a = 0, (B) a = 0.071, (C) a = 0.23, and (D) a = 0.40. 62 PAGE 63 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 C 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 D Figure 63. Continued 63 PAGE 64 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0. 5 0 5 10 15 20 25 30 35 40 A 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 5 10 15 20 25 30 35 40 B Figure 64. Magnitude spec trum for two tones separated by 1.6 harmonics A) with 64point DFT (a=0) and B) with 512point DFT (a=0). 64 PAGE 65 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0. 5 0 5 10 15 20 25 30 35 40 WDFT Actual A 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0. 5 0 5 10 15 20 25 30 35 40 WDFT Actual B Figure 65. Magnitude spectrum for two tone detection with 64poi nt WDFT with A) a=0.1087 B) a=0.0721, C) a=0.0996, and D) a=0.1381. 65 PAGE 66 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0. 5 0 5 10 15 20 25 30 35 40 WDFT Actual C 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0. 5 0 5 10 15 20 25 30 35 40 WDFT Actual D Figure 65. Continued 66 PAGE 67 Table 61. Comparison of the warping parameter Criterion Criterion #1 Criterion # 2 Search Method Fibonacci search M. Golden section search Fibonacci search M. Golden section search Warping parameter a = 0.1087 a = 0.0721 a = 0.0996 a = 0.1381 Elapsed time (sec.) t = 0.749252s t = 0.888743s t = 0.736151s t = 0.751035s 67 PAGE 68 CHAPTER 7 CONCLUSIONS AND FUTURE WORK 7.1 Conclusions This dissertation aims at exploring spec tral analysis using the warped discrete Fourier transform (WDFT) compared to a c onventional discrete F ourier transform (DFT). It focuses on detecting multiple narrowband signa ls which are not able to be isolated with uniform frequency resolution and optimiz ing the local frequency resolution by finding the best warping control strategy. And the system developed and tested also focuses on maximizing the WDFT frequency re solution over those frequencies that exhibit a localized concentration of spectr al energy and, implicitly, diminishing the importance of other frequency ranges. This dissertation demonstrates that multitone signals are able to be more readily detect ed and discriminated by ex ternally controlling the frequency resolution of the WDFT in in telligent manners using optimal singlevariable search techniques, a Fibonacci search and a modified Golden Section search. Finally, this dissertation shows the bes t frequency resolution reducing spectral leakage which obscures the spectral separ ation between of adjacent DFT harmonics due to the finite frequency resolution of the DFT without increasing the DFT length for multitone detection using the WDFT. In fac t, an increase in the DFT length improves the sampling accuracy by reducing the spectral separation of adjacent DFT samples, while it brings up the higher computational complexity and cost penalty. In order to minimize and suppress spectral leakage the WD FT is exploited to control the spectral separation through the warping parameter. Moreover, the usage of the WDFT presents obtaining hi gher and optimized local frequency resolution through finding the best warping control strategy. In general, a 68 PAGE 69 uniformly windowed DFT/FFT cannot determine if one tone or mult iple tones are present locally about a harmonic fr equency for two tones separated by 1.6 (1.6 harmonics) or less. It shows that, however, the WDFT can discriminate between two signals separated by as little as 1.3 harmonics. Overall, the new spectral analysis technol ogy using the WDFT results in a higher local resolution, less computational comp lexity, more capability, and lower cost. 7.2 Future work Based on the frequency discrimination for multitone signal det ection using the WDFT, the proposed spectral ana lyzer design holds the promise of lower complexity. This can be translated into low power and/or high speed. Future work should focus on quantifying this advantage in a physical instan tiation. The design outcome can then be benchmarked against a commercial unit and performance advantages directly measured. Another area that may prove productive is developing autoconfiguration software or MATLAB attachments that will optimiz e a design outcome based on a set of specifications. 69 PAGE 70 LIST OF REFERENCES [1] A. V. Oppenheim and R. W. Schafer, DiscreteTime Si gnal Processing. Englewood Cliffs, NJ, PrenticeHall, 1989. [2] B. Evans, Dual Tone Mulitple Frequencies, U.C. Berkeley, http://ptolemy.eecs.berkeley. edu/papers/96/dtmf_ict/www/nodel.html [3] G. Goertzel, An algorithm for evaluatio n of finite trigonometric series. American Mathematical Monthly, vol. 65, Jan. 1958, pp. 3435. [4] E. A. Hoyer, R. F. Stork, T he zoom FFT using complex modulation, Proceedings IEEE ICASSP vol. 2, 1977, pp. 7881. [5] Y. Tadokoro, K. Abe, Notch Fourier transform, IEEE transactions Acoustics Speech Signal Processing vol. ASSP35, no. 9, Sep. 1987, pp.12821288. [6] C. Braccini, A. Oppenheim, Unequal Band width Spectral Analysis using Digital Frequency Warping, IEEE transactions Acoustics Speech Signal Processing vol. ASSP22, no. 4, A ug. 1974, pp. 236244. [7] A. Makur, S. K. Mitra, Warped discreteFourier tr ansform: Theory and applications. IEEE Transactions on Circuits Systems, vol. 48 Sep. 2001, pp. 10861093. [8] R. Kumaresan. Spectral analysis. In S.K. Mitra and J.F. Kaiser, editors, Handbook for Digital Signal Processing, chapter 16, pp. 11431242, WileyInterscience, New York NY, 1993. [9] J. G. Proakis, D. G. Manolakis, Digital Signal Processing: Principles, Algorithms and Applications, 3rd ed. New Jersey: Prentice Hall, 1995. [10] S. K. Mitra, Digital Signal Processing: A ComputerBased Approach, 2nd ed. New York: McGrawHill, 2001. [11] F. J. Harris, On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform, Proceedings of the I EEE, vol. 66, pp. 5183, January 1978. [12] T. Saramki, Finite impulse response filter design, In S. K. Mi tra and J. F. Kaiser, editors, Handbook for Digital Signal Processing, chapter 4, pp. 155277, WileyInterscience, New York NY, 1993. [13] A. Oppenheim and D. J ohnson, Computational of spectra with unequal resolution using the fast Fourier transform, Proc IEEE vol. 59, pp. 209301. [14] S. Baghci and S. K. Mitra, Nonunifo rm Discrete Fourier Transform and its signal processing Applications. No rwell, MA: Kluwer, 1999. 70 PAGE 71 [15] G. H. Golub, C. F. Van Loan, Matrix Computations, third edition, Johns Hopkins University Press, 1996 [16] S. Franz, S. K. Mitra, J. C. Sc hmidt, G. Doblinger, Wa rped discrete Fourier transform: a new concept in digital signal processing, Proceedings IEEE ICASSP vol. 2, pp.12051208, 2002 [17] A.C. Constantinides Spectral transformations for digital filters, Pro. Inst. Elect. Eng., vol. 117, no.8 1970 pp. 15851590 [18] P. P. Vaidyanathan, Multirate System s and Filter Banks, Englewood Cliffs, NJ: Prentice Hall, 1993. [19] E. Galijasevic, and J. Kiel, Design of AllpassBased NonUniform Oversampled DFT Filter Banks, IEEE ICASSP, Orlando, FL, May 2002, pp 11811184. [20] F. Taylor and J. Mello tt, HandsOn Digital Signal Pr ocessing, McGraw Hill, 1998. [21] A. Williams and F. Taylor, Electronic Filter Design Handbook, 4th Ed., McGraw Hill, 2006. [22] Ohbong Kwon, Fred Taylor, Multit one Detection Using the Warped Discrete Fourier Transform, MWSCAS, Knoxville, TN, Aug. 2008 [23] Ralph W. Pike, Optimization for E ngineering Systems, Van Nostrand Reinhold company, 2001, http://www.mpri.lsu.edu/bookindex.html 71 PAGE 72 BIOGRAPHICAL SKETCH Ohbong Kwon was born in Guelph, Canada and has grown up in Seoul, South Korea. He received both his B.S. degree and M.S. degree in electrical engineering from Hanyang University, South Korea in 1998 and 2000, respectively. He was in the High Speed Digital Architecture Labor atory (HSDAL) in the Electrical and Computer Engi neering Department at the University of Florida during his PhD study. His present research interests are in the areas of signal processing, digital filter design, and optimization. 72 