SPECTRAL ANALYSIS OF SHORT DATA RECORDSMU AIR FORCE 1/12 INST OF TECH &IRIGI4T-PATTERSON AiF9 0N SCHOOL OF ENGINEERING T E CARTER DEC 87

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1 SPECTRAL ANALYSIS OF SHORT DATA RECORDSMU AIR FORCE 1/12 INST OF TECH &IRIGI4T-PATTERSON AiF9 0N SCHOOL OF ENGINEERING T E CARTER DEC 87 AFIT/GE/ENG/BOO-9 7 MLSIF led WF/O 17/4 U

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3 11T1Il FIL E ai 4,,! Spetral Analysis of Short Data Reords THESIS Thorlough E. Carter Jr. First Lieutenant, USAF AFIT/GE/ENG/87D-9 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY * AIR FORCE INSTITUTE OF TECHNOLOGY.%ls~~TW datam Wright-Patterson b.w inem _ Air "sofore Base, Ohio i4 W&"=4 a.4..-w muma**4 1. * I. '4

4 AFIT/GE/ENG/87D-9 Spetral Analysis of Short Data Reords THESIS Thorlough E. Carter Jr. First Lieutenant, USAF.1 AFIT/GE/ENG/87D-9 I Approved for publi release; distribution unlimited

5 AFIT/GE/ENG/87D-9 Spetral Analysis.1 of Short Data Reords THESIS Presented to the Faulty of the Shool of Engineering of the Air Fore Institute of Tehnology Air University In Partial Fulfillment of the Requirements for the Degree of Master of Siene in Eletrial Engineering Thorlough E. Carter, B.S.E.E. *. First Lieutenant, USAF Deember 1987 Approved for publi release; distribution unlimited

6 Pre fae The purpose of this researh is to investigate two methods of spetral estimation for typial radar type signals. and Burg. The two methods investigated are Blakman-Tukey A omparison of the two methods is made based on their statistial properties and omputational effiieny. Valuable guidane was provided by Maj Glenn Presott of the Air Fore Institute of Tehnology (AFIT) during the ourse of this work. His help is gratefully appreiated. My deepest gratitude is expressed to my dear wife, Alfreda, for her onstant support and enouragement. Her help in preparing the manusript has enable me to aomplish a major V. goal in my life. Another major goal that we both aomplished during my stay at AFIT was the birth of our beautiful daughter Ashlee. A speial thanks is given to our divine reator. Aession For NTIS GRA&I DTIC TAB da Unannouned 0 S Justifiatio WAvailability Distribution/ Codes -- vai andlor. Dist Speial II - ii

7 Table of Contents Page Prefae List of Figures...v Absrtat...viii I. overview... Introdution Bakground...2 Problem Statement...7 Sope...7 Assumptions...7 Presentation...8 II. ISPX Software Pakage and Previous Researh.9 Introdution..*..*...9 ISPX Software Pakage...0 Previous Researh...13 III. Detailed Theory...16 IV. Introdution Conventional Spetral Estimation - Via the BT Method...18 Modern Spetral Estimation - Via the AR/Burg Method...30 Comparison of the BT and Burg Methods Analysis...44 Introdution...44 Problem One...46 Problem Two...55 Problem Three...63 Problem Four...66 Threshold Detetion Routine V. Conlusions and Reommendations...78 Conlusions...78 Reommendations Z

8 Appendix A: Subroutine Used for Spetral Estimation Appendix B. Threshold Detetion Routine...88 Bibliogjraphy...97 Vita iv N

9 .List of Figures rigure eage 1.1 I neri k eeiver blok Diagram i. Continuous and Disrq-tL_ &,,iju Z A:.. nq tneir Respetive PSD's Plots for both Autoorrelation Estimators The Ideal PSD of a Strong and a Weak Signal Embedded in WGN The Simple PSD of a Weak and Strong Signal Example for Various Values of N A Funtional Desription of the BT Method The BT PSD of the Weak and Strong Signal Example Transform Interpolation by Zero Padding S3.8 An All-Pole Filter for Generating an AR " Proess Sixty-Four Samples of a Single Sinusoid in WGN, SNR = 10 db Sixty-Four Samples of a Single Sinusoid in WGN, SNR = 15 db Sixty-Four Samples of a Single Sinusoid in WGN, SNR = 20 db a BT Estimator of the Single Sinusoid in WGN, SNR = 10 db " 4.4b BT Estimator of the Single Sinusoid in WGN, SNR = 15 db BT Estimator of the Single Sinusoid in WGN, SNR = 21 db Superposition of the BT Estimator for SNR's 10, 15, and 20 db Superposition of the Burg Estimator for 0 SNR's 10, 15, and 20 db with P =

10 4.7 Sixty-Four Samples of a Single Sinusoid in WGN, SNR = 15 db BT Estimator of Two Sinusoids in WGN, SNR = 15 db a Burg Estimator of Two Sinusoids in WGN, SNR = 15 db with P = b Burg Estimator of Two Sinusoids in WGN, SNR = 15 db with P = Burg Estimator of Two Sinusoids in WGN, SNR = 15 db with P = Sixty-Four Samples of a Single Sinusoid in WGN, SNR = 15 db BT Estimator of Two Sinusoids in WGN, SNR = 15 db a Burg Estimator of Two Sinusoids in WGN, SNR = 15 db with P = b Burg Estimator of Two Sinusoids in WGN, SNR = 15 db with P = Burg Estimator of Two Sinusoids in WGN, SNR = 15 db with P = Sixty-Four Samples of a Single Sinusoid in WGN, SNR = 15 db BT Estimator of Four Sinusoids in WGN, SNR = 15 db Burg Estimator of Four Sinusoids in WGN, SNR = 15 db with P = Burg Estimator of Two Sinusoids in WGN, SNR = 15 db with P = B.1 Burg Estimator of a Single Sinusoid in WGN, SNR = 15 db with P = B.2 Burg Estimator of Two Sinusoids in WGN, SNR = 15 db with P = B.3 Burg Estimator of Three Sinusoids in WGN, SNR = 15 db with P vi 6W

11 A' B.4 Burg Estimator of Four Sinusoids in WGN, SNR = 15 db with P = B 5 Burg Estimator of Five Sinusoids in WGN, SNR = 15 db with P = B.6 Burg Estimator of Two Sinusoids in WGN, SNR = 15 db with P = B.7 Burg Estimator of Three Sinusoids in WGN, SNR = 15 db with P = Ot vii e A

12 Abstrat The purpose of this study was to examine the Blakman-Tukey (BT) and Burg methods of spetral estimation for typial eletroni warfare reeived signals. Suh signals are generally short in duration, resulting in short data reords. The BT method is a onventional spetral estimation sheme and is based on omputing the disrete Fourier transform of the autoorrelation sequene (ASC) derived from the data reord. An inherent problem of this approah is that of data windowing. Data windowing may result in poor frequeny resolution, partiularly for short data reords. The Burg method of spetral estimation, a modern approah, is apable of providing relatively good frequeny resolution for short data reords. However, this method requires suffiient input signal-to-noise ratio (SNR). The idea here is to extend the ACS by extrapolation (or predition) rather than windowing the data. * The Burg method was found to yield far superior performane for data reords onsisting of 64 data samples. Note, however, that a minimum SNR of 15 db was assumed. *# Using this method a "smart" routine was developed that automatially determines the atual frequeny omponents of the data reord. viii S

13 ,N. SPECTRAL ANALYSIS OF SHORT DATA RECORDS I. Overview Introdution The motivation for this study is the need for improved N detetion of hostile signals (i.e., signals that are radiated by hostile emitters). An emitter is any signal soure (i.e., a radar, a jammer, et). The signal radiated by the emitter is made up of many measurable harateristis. Wiley (33:8-12) provides a summary of the measurable signal harateristis and the emitter apabilities inferred from these harateristis. The most ommon measurable signal harateristis are: 1. Pulse repetition frequeny (PRF) 2. Transmitted signal power 3. Transmitted beamwidth 4. Operating frequeny The arrier (or operating) frequeny of the hostile signal is the signal harateristi of interest in this study. The intent of this study is to examine two methods (Blakman-Tukey and Burg) for determining the operating frequeny of one or more emitters, via the power spetral density (PSD)..,, " S2

14 Bakground The basi senario is an eletroni warfare (EW) reeiver deteting signals from hostile emitters. Figure 1.1 shows a generi blok diagram of a reeiver. No distintion is made between the reeiver and the digital proessor. However, note that all bloks to the right of the dashed line represent digital hardware. rt= FKT IF F SAMi1 L7 x(n) (t)+r(t) MIXER signat MIXER HOLD ALGORITHM t Sxwor) DI.TA FdPLOTTER J I CONVERTER COMPUTER SHORT-TERM PSD Figure 1.1. Generi Reeiver Blok Diagram The reeiver input signal r(t) operates within the radio frequeny (RF) band, 300 MHz to 30 GHz. Hene, the signal is termed the RF signal. For the purpose of this study, the RF signal is assumed to operate at the high end of the RF band (greater than say 1GHz). The RF signal is down-onverted (or translated) to an appropriate baseband frequeny and then sampled, as indiated in Figure 1.1. The purpose of sampling the baseband representation of r(t) is to allow for digital proessing. Generally, digital 2 4 N.-..."

15 7.proessing is more effiient and less ostly than analog (28:119). However, quantization error (or noise) is an inherent problem of digital proessing. This error results from the roundoff effet that ours when onverting a ontinuous signal into a set of finite disrete levels. Tong (31: ) provides a detailed disussion of quantization error and methods for reduing this error. The sampling theorem states that a baseband signal is ompletely reonstruted (no information loss) if the sampling rate is at least twie the highest frequeny * omponent of the baseband signal (9:230). Ideally, sampling is desirable at the RF bandpass signal - allowing for an all digital EW reeiver. Unfortunately, digital tehnology is n not apable of providing reasonable results when the RF signal is above 1 GHz. The bandpass sampling theorem requires that the sampling rate lie between 2 BHz and 4 BHz, inlusively. The variable B is the bandwidth of the RF signal whih an be on the order of 1000's of MHz. Current,: digital iruitry is unable to sample at suh fast rates * (27:55). One the baseband signal is sampled and onverted into a digital word, the digital omputer is used to determine the PSD of the signal. * OThe above disussion did not onsider the statistial nature of the bandpass signal r(t). The signal r(t) onsists of the transmitted hostile signal s(t) plus noise n(t). The noise is indued onto the hostile signal by the A. 3 0 " -,% " % ' "..-%, ",-.' --

16 hannel (medium of propagation). Thus, r(t) is atually a random signal (or random proess). A random proess is ompletely desribed statistially if all its N-th order joint density funtions are known (i.e., N approahes infinity), a pratial impossibility (9:482). For many pratial appliations, only the first and seond order statistis are used to desribe a random proess (24:208). The autoorrelation funtion, a seond order statisti, for a ontinuous time random proess is defined as R x(t,t+t) = E{x(t)x(t+T)} (1.1) where E{-) denotes the expeted value (or statistial average) of the produt x(t)x(t+t). If a random proess x(t) is wide sense stationary (WSS), then '. R (t,t+t) = R (T) x x (1.2) This implies that the autoorrelation funtion is independent of time shifts. The variable T is alled the lag variable, the amount of time by whih t lags behind t+t. If x(t) is also ergodi, then * E{x(t)x(t+T)} = <x(t)x(t+t)> (1.3) where <o> denote the time average (13:250). Thus, ergodiity requires that the result of the statistial (or * ensemble) average and the time average be idential. The reeiver input signal r(t) is assumed to be both WSS and ergodi. ". The above onsiders the ase of a ontinuous time 4 0

17 random proess. However, as eluded to earlier the ontinuous proess r(t) is down-onverted to a baseband frequeny and sampled. Therefore, a disrete implementation of the autoorrelation funtion is desired. A sampled ontinuous time random proess is a disrete time random proess. Papoulis (24:290) shows that if a ontinuous time random ~x proess x(t) is WSS with autoorrelation R (T), then x(n), a sampled version of x(t), has an autoorrelation given by R (k) = E{x(n)x(n+k)} (1.4) This is alled the disrete autoorrelation funtion (or autoorrelation sequene) and is a sampled version of 0The R (T). x ground work has been developed and it is now time to define the PSD. Aording to the Wiener-Khinthine theorem (14:181), the PSD of an ergodi disrete time random proess is given by P (f) = DTFT [R (k)] (1.5) X X This says that the PSD is the disrete-time Fourier transform (DTFT) of R x(k) and, onversely, R (k) = DTFT [P (f)] (1.6) The disrete autoorrelation funtion is the inverse disrete-time Fourier transform of the PSD. A more definitive explanation and expression of R x(k) and P x(f) are presented in Chapter III. It is important to understand the motivation for stating the disrete autoorrelation w. 5

18 '. funtion. The random signal proessed by the digital omputer for omputation of its PSD is not ontinuous but disrete. This is the signal x(n) that appears as the output of the sampler in Figure 1.1. Several methods have been developed for implementation on digital omputers, that determine the PSD of a disrete random signal based on its disrete autoorrelation funtion (18:1382). In order to be onsistent with urrent literature, the disrete random signal x(n) is hene forth referred to as a data reord (or disrete time series). The two methods onsidered in this study are Blakman-Tukey (BT) and Burg. The BT method is a fast Fourier transform (FFT) tehnique employed to determine the V j PSD of a data reord. This method, ommonly referred to as a onventional approah, is omputationally effiient and generally produes reasonable results. However, degrading frequeny resolution problems arise when the data reord being onsidered is short. The Burg method, referred to as a modern approah, attempts to overome the frequeny resolution problems of the BT method, partiularly for short data reords. The idea here is to onstrut a parametri linear disrete model that approximates the data reord. The PSD is omputed based on this model. The motivation for using the Burg method is its promise for better frequeny resolution. 6 %

19 The BT and Burg methods are disussed in detail in Chapter III. Problem Statement Several approahes are available to determine the PSD of a random proess. There are pros and ons assoiated with the seletion of any one of the approahes. andidate approahes are examined in this study. Two The two approahes are applied to several known data reords. A omparison of the two approahes is made based on their statistial properties and omputational effiieny. Sope This study proposes to examine the merits of the BT and Burg methods of spetral estimation for typial hostile emitter signals. The evaluation of the two methods is based solely on manual alulations and omputer simulations. No hardware prototypes are attempted. Also, a "smart routine" is introdued that attempts to automatially determine the spetral peaks orresponding to emitter frequenies. Assumptions The emitters are assumed to transmit at a single operating frequeny per pulse. Of ourse, eah emitter transmits at its own respetive frequeny. The reason for the restrition of a single frequeny per pulse is beause of software limitations. That is, the software pakage used, Interative Signal Proessing Exeutive (ISPX), to *1

20 simulate the emitter signals does not allow the user to vary the frequeny per pulse. Thus, r(t) is assumed to onsist of distint frequenies orresponding to eah emitter. The random proess r(t) is assumed WSS. many random proesses are not stritly WSS. Pratially, Hene, the PSD annot be defined as given by (1.5). However, if the duration of the observation interval (the number of data points) is small, the proess may be onsidered loally WSS (24:125). A loally WSS proess is one for whih the variations of the autoorrelation funtion is small over the duration of the observation interval. Most real world random proesses are onsidered loally WSS. Presentation Chapter II provides a brief disussion of the ISPX software pakage used and a disussion of previous researh. Chapter III provides the development of the theory for both the BT and the Burg methods of spetral estimation. Chapter IV disusses findings as they relate to several known input data reords for both methods. Chapter V provides onlusions and reommendations for further study. 8 I

21 trodii. ISPX Software Pakage and Previous Researh,, Introdution The purpose of this hapter is twofold - to briefly disuss the ISPX software pakage and to present previous researh on the BT and Burg methods of spetral estimation. The ISPX software pakage urrently resides on the Air Fore Wright Aeronautial Laboratories (AFWAL) VAX omputer in the diretory of Dr. James Tsui. The pakage allows the user to generate a variety of "real world" data reords. A number of different spetral estimation methods an be applied to eah data reord, and the results viewed graphially. However, as mentioned in the previous hapter only the BT and Burg methods are onsidered in this study. The ISPX pakage is a tool whih very niely allows for studying the harateristis of the BT and Burg methods. The findings presented in Chapter IV are a result of applying the appropriate routines of ISPX to several known data reords. The appropriate routines are provided in Appendix A. I Referene to eah routine is made in the next hapter. The data reord is traditionally proessed by using FFT based (or onventional) methods of spetral estimation. I There are inherent limitations assoiated with onventional methods. The most prominent limitations are poor frequeny resolution and leakage. As will be disussed in the following two hapters, these limitations beome more Ih pronouned for short data reords. I9 Short data reords are a

22 ommon ourrene for reeiver systems like that outlined in Chapter I. In an attempt to alleviate the inherent limitations of onventional methods, researhers have devised other methods of spetral estimation. The other methods are referred to in the literature as modern spetral estimation methods. The modern methods, however, are not a "fix-all" solution to spetral estimation for all data reords. They also have their limitations. The BT and Burg methods are respetively onsidered onventional and modern spetral estimation methods. Both methods are disussed in detailed in the next hapter. ISPX Software Pakage This setion is intended to provide a brief overview of the ISPX software pakage. The interested reader is referred to referene (2:1-20) for a more omplete disussion of ISPX. The ISPX pakage provides a number of signal proessing operations. However, only the operations required to analyze the previous two mentioned spetral estimation methods are disussed. The operations are: 1. Generate data arrays 2. Choose PSD method and exeute 3. Plot an array The generate data operation allows the user to generate a wide variety of "real world" data reords. Eah data reord generated an onsists of up to five independent *. sinusoids. The variable parameters of eah sinusoid are 10

23 frequeny, amplitude, delay, and duration. The values of these parameters are speified by the user; however, default values are provided. This multiple sinusoid feature allows the user to simulate a data reord orresponding to a reeived signal onsisting of sinusoids from different *, emitters. A pseudo random WGN proess an be embedded on the data reord to emulate the effets of the noisy hannel. The noise amplitude is the parameter the user hanges to vary the input signal-to-noise ratio (SNR) of the data reord. That is, for a desired input SNR, the noise amplitude required may be alulated by the following '? A - 1 SA NA = I (2.1) v SNR/20 Oi where NA and SA represent noise and signal amplitudes, respetively. One the data reord is established, the PSD operation is performed. The newly generated data reord is sampled, and the appropriate PSD hoie is made. The user speifies I the sampling frequeny, suh that the Nyquist baseband sampling theorem is obeyed. A unity sampling frequeny allows the user to deal in normalized (or frational) I frequeny. Of ourse, the frequenies ontained in the data reord must be hosen aordingly. For simpliity, the I.2'.several. examples presented in Chapter IV are all based on a unity sampling frequeny. ISPX allows the user to selet one of spetral estimation shemes, both onventional and 0 q1

24 modern. However, as eluded to earlier only the onventional BT and the modern Burg methods are onsidered in this study. In seleting the BT method, the user must speify the number of frequeny samples, the largest lag, and the lag window. Eah of these parameters are disussed fully in the next hapter. However, it is worth noting the different lag windows provided by ISPX. They are: 1. Cosine 2. Hamming 3. Kaiser-Bessel As will be disussed in the following hapters, the hamming lag window provides the overall best performane for the examples onsidered in this study. In seleting the Burg method, the user must speify the number of frequeny samples, and the model order number. Again, these parameters are disussed in the next hapter. The plotting operation is invoked after the user speifies the appropriate PSD method. The plotting operation allows the user to graphially illustrate the spetral estimation shemes. The user is provided the option of plotting the results of up to four spetral estimation shemes on a single graph. This feature is partiularly useful for making visual omparisons between different spetral estimation methods, or omparisons of the same method with differing parameters. '. 12 6

25 Previous Researh This setion provides a brief summary of urrent -researh in the area of spetral estimation. Sp-ifially, -. researh done over the past deade pertaining tu the BT and Burg methods of spetral estimation is presented. Cooper and Kaveh (7: ) onsiders the effets of noise on the spetral estimate provided by the Burg method. They note that as the noise inreases the resolution of the spetral estimation beomes progressively worse. In an attempt to mitigate the effets of the noise, they inreased the model order number of the estimate. Noting that the resolution improved; however, spurious (or unwanted) peaks began to appear. They also ompared the spetral estimates produed by both the BT and Burg methods for a given data reord. The data reord onsisted of two sinusoids orresponding to doppler shifts from two radar targets. They note that the number of lags required by the BT method p are generally greater than the model order number required r.1 by the Burg method to ahieve a given frequeny resolution. -6 Haykin (12: ) applies the Burg method to solve the problem of estimating the angle(s) of arrival of plane wave(s) impinging on a linear array antenna from unknown diretion(s). The loation of the spetral peaks indiates the diretions of the inident plane waves. For an input SNR of 18dB, he notes that for a single soure (one target) illuminating a linear vertial array omposed of eight 0.): 13 -p.1

26 equally spaed horn antennas: (1) the loation of the spetral peak oinides very losely with the atual diretion of the soure; (2) the resolution inreases with.5.. inreasing model order number; and (3) spurious spetral peaks appear as the model order number is inreased. In Z. fat, there may be as many as P peaks in the spetrum. The variable P is alled the model order number and will be disussed in the next hapter. Kunt (19: ) provides several examples illustrating the performane of the BT spetral estimator. '0 4j, He notes that as the data samples of a given data reord are dereased, the frequeny resolution of the estimator also dereases. He investigates the resolution-leakage phenomenon of several lag windows. Substantiating the well-known fat, that for a sinusoidal proess the frequeny 4.%" resolution of the estimator is largely determined by the main lobe and sidelobe harateristis of the lag window hosen. These harateristis tend to beome less favorable as the number of data samples dereases. For this reason, the BT spetral estimator typially provides poor results for short data reords. Partiularly, when the data reord onsists of two or more losely spaed sinusoids. Bishop and Ulryh (4: ) investigate the Akaike riterion (briefly disussed in the next hapter) for determining the model order number of the Burg spetral 0 estimator. Ideally, the orret model number is that number 14 '_4%

27 whih results in a minimum value of the Akaike riterion. They note that the Akaike riterion typially results in model order numbers that are too low. They emphasize that model order number seletion is in general a diffiult problem, and that the Akaike riterion and other proposed riteria serve only as guidelines. However, they state that for sinusoidal proesses the Akaike riterion yields the "best" results. A A 15

28 'I III. Detailed Theory Introdution "-' The power spetral density (PSD) and its estimation are -* reeiving muh attention by researhers in many disiplines. The PSD is onsidered an effetive medium that allows for *. haraterizing a stationary random proess (6:27). Two ommon but distintly different shemes are usually onsidered when attempting to estimate the true PSD. They are: 1. The onventional method of PSD estimation 2. The modern method of PSD estimation The most notable onventional method is the method proposed by Blakman and Tukey (BT) - the BT method (5: )..- This method is based on omputing the DFT of a finite windowed (or tapered) autoorrelation sequene (ACS). Thus, the BT method produes a PSD estimate whih is the transform of the window funtion onvolved with the simple PSD estimator. The window funtion plays a very important role in determining the resolution and the sidelobe phenomena \ assoiated with the estimate of the BT method, partiularly for sinusoidal proesses embedded in additive WGN. Often, the seletion of the window funtion that provides the So- "optimal" ompromise between high resolution and low sidelobes is determined empirially (23:84--86). The BT method generally provides unaeptable results for short data reords - a smeared spetral estimate with very poor *5, o " '- \JC. - * ~. 16

29 'p..resolution (1:450). Short data reords our frequently in pratie, partiularly for radar systems in whih only a few data samples are available for eah reeived pulse (18:1381). A ommon error is that of appending a sequene of zeroes to the data reord in hope for improved resolution. This tehnique, ommonly referred to as zero padding, merely redues the spaing between adjaent spetral lines by interpolation (21:43). The motivation for modern methods is their promise for better frequeny resolution, partiularly for short data reords. The most notable modern method is the autoregressive (AR) method (11:56). The idea here is to model the data reord by an all-pole rational transfer funtion. This allows for extrapolation of data samples beyond the observation interval, not impliitly assuming the samples are zero as with onventional methods (7:321). Theoretially, the data samples an be extended indefinitely. More data samples allow for omputation of more autoorrelation lags, and therefore in general an improvement in resolution. Also, for all pratial purposes, the window funtion is removed; therefore, the degrading effets of sidelobes are alleviated. Of ourse, the auray of the PSD estimate prvided by the AR method depends diretly on the auray of the rational transfer funtion used to model the data reord. The approah for estimating the PSD, using the AR method, is drastially as 17 d% %

30 different from that of the BT method. The parameters of the assumed model are estimated from the available autoorrelation lags, and then the theoretial PSD implied by the model is alulated using the estimated parameters. There are several known tehniques for estimating the parameters of the assumed model (20:562). In this hapter, only the Burg approah is onsidered. The following setions provide a detailed disussion of the BT and Burg methods. 4 Conventional Spetral Estimation - Via the BT Method I The PSD simply represents the power distribution of the random proess along the frequeny axis (26:137). For a WSS ontinuous random proess x(t), the true PSD is given by the Fourier transform "0 ( J Rx('T)exp(-j2nf-T)dT (3.1) of the autoorrelation funtion R (T). Therefore, the PSD x of the WSS disrete random proess x(n) is given by the disrete-time Fourier transform (DTFT) (f) = Z R(kT)exp(-j2nfkT) (3.2) < k = - of the ACS R x(kt), whih orresponds to equal distant shifted versions of P (f) (see Figure 3.1) (6:291). x The I * 18 V! "4e" %

31 5, ( " B 0J B 0) B 2 B Figure 3.1. Continuous and Disrete Random Proesses and their Respetive PSD's (6:291) ACS (10:54), assuming a real and ergodi x(n), as defined by Eq (1.4) is given as M xr M x(n)x(n+k) (3.3) x M -* 2M1Z n =- M For simpliity in notation, the sampling interval T is implied (i.e., k = kt and n = nt). Properties of the ACS are R (k) = F l-k) x x (3.4) Rx(0) -> R(k) I These properties indiate_ that the ACS is a real even i 19...," 4 -'" 4 '.' ',.-V..''-. '.."". ''.""..'''..'" -""..-." '-.'" All" '-2-"''- "" "" "" ""6" '

32 sequene (for real x(n)) with a maximum at the zero lag. The omputation of the ACS is pratially impossible, requiring an infinite number of data samples. An estimate of the ACS is the only alternative. Hene, the PSD as given by Eq (3.2) an at best be approximated. Two ommonly used autoorrelation estimates (17:73-74) are N-k I-I R x(k) N x(n)x(n+k) (3.5a) n=0 and N-1k -1 R x(k) - 1 -n=0 x(n)x(n+k) (3.5b) pwhere N is the total number of data samples (i.e., x(n) for n = 0 to n = N-i). Thus, a total of 2N-1 lags of the autoorrelation estimates are possible. denotes the biased estimator and symbol The symbol the unbiased estimator. A biased estimator is one in whih.ts expeted N value (or mean) deviates from the atual value being A estimated. The expeted value of an unbiased estimator is >k 0N the atual value (30:9). The expeted values of the biased and unbiased autoorrelation estimators are, respetively, E{R(k)) (1 Rk) (3.6a) and E (k) = Rx(k) (3.6b) 20 01o'

33 --- Observe that the expeted value of the biased autoorrelation estimator is equal to the atual value R x(k) weighted by the Bartlett (or triangular) window. The expeted value of the unbiased autoorrelation estimator is the atual value. Intuitively, it might be thought that the unbiased autoorrelation estimator is the best hoie for use in Eq (3.2). However, Chen (6:27-28), Kunt (19: ), and Marple (21: ) show that as the lag value k approahes its limit (i.e., k approahes N-i), the variane (or flutuation) of the unbiased autoorrelation estimator inreases dramatially as ompared with that of the biased autoorrelation estimator. This inreased variane auses the unbiased autoorrelation estimator to yield results not always satisfying the properties of Eq. (3.4). Figure 3.2 presents plots of both estimators for a data reord generated by a first-order digital filter, with a psuedo WGN input proess (19: ). The plots are for different values of k and N. A omparison of the plots indiates the variane of the unbiased estimator is greater for k : N. However, the variane of both estimators appears idential for k << N. The latter ours most often in typial radar type problems. Therefore, some researhers argue that the unbiased autoorrelation estimator is in fat the best hoie based solely on Eq (3.6). A more valid omparison, suggested by Jenkins and Watt (15: ), is to onsider the mean square error (mse) of 21

34 R x(k) Rx(k) N WUJTT01[ N 50 4 ~ L I 40.Ji o ,2 0 1, R x(k) R (k) fi N = 20 Sx R x(k) 10A10 dl 1W IL R (k),i j ~ xlk I IN = "0il, )0 I' Figure 3.2. 'I (19: ) Plots for both Autoorrelation Estimators 22 IJ

35 the two autoorrelation estimators. Jenkins and Watt onluded that the mse of the biased autoorrelation estimator R (k) is typially smaller for all values of k. Therefore, in this study the PSD is estimated using the biased autoorrelation estimator, namely N-1 Sx(f) = Rx(k)exp(-j2nfk) (3.7) k=-(n-l) defined for -1/2T : f : 1/2T. This equation represents a low-pass filtered approximation of P x(f) and hene an approximation of J x (f) (see Figure 3.1). The use of R x (k) in Eq (3.7) is not to suggest that the biased autoorrelation estimator always yields a superior estimate of the PSD. Jenkins and Watt only suggested that the PSD determined from the biased autoorrelation estimator is better on average, partiularly for large k. The PSD estimator given in Eq (3.7) is alled the simple (or periodogram) PSD estimator (29:238). This estimator, at first glane, might appear to approah that of Eq (3.2) for large N. If this happened, then the simple PSD estimator is said to be onsistent. A onsistent estimator is one in whih its bias and variane both tend to zero as the number of data samples inreases (32:71). Kay (17:80-82) and Kunt (19: ) show that the simple PSD estimator is not onsistent, as a onsequene of the autoorrelation estimator. Although the bias of the simple PSD estimator asymptotially approahes zero as the data 23 % %

36 samples inreases, the variane Var { (f) n P 2 (f) (3.8) remains essentially onstant - the square of the desired PSD. Figure 3.3 shows the true PSD of a random signal onsisting of a weak and strong sinusoid embedded in WGN. L ",x (f) strong signal I0',,, 0--, "4 10" weak signal 10- noise I0" 0 "'. I I I f 0 0.! Figure 3.3. The Ideal PSD of a Strong and a Weak Signal Embedded in WGN (19:328) Note that the frequeny axis represents a frational (or normalized) frequeny - the atual frequeny divided by the sampling frequeny f = l/t. Figure 3.4 depits the results *' of the simple PSD estimator for several data reord lengths. As expeted, the variane appears to remain essentially onstant for different values of N. Many researhers agree that the simple PSD estimator generally provides unreliable results, partiularly if there is no a priori knowledge of the true PSD (1:448). 24 L',

37 . - X (f) P (f) :-A xa ~x " IC I l, 0~ }(" N = 128 N = ' 10 ",0,i.,.J..,..'I : L lo" -,, t 4i IO II I*/ II 0 - IC 0 Oi C ''A Px(f),oA - P f.o'~ I0' 0 0, '" N = 512 N 1024,0"" J 1IC" -% I0"I ' 0,o- 10 -o, - I , Figure 3.4. The Simple PSD of Weak and Strong Signal Example for Various Values of N (19:328) 25, *... -,,.,. - -, ,....,. -.,.....,.. -

38 A method to redue the variane of the simple PSD estimator is funtionally desribed in Figure 3.5. The biased autoorrelation estimator is omputed from a given data reord, windowed, and then transformed. ACST R (k) n)p COMPUTATION O[W--INDOW TRASFORMERJT Figure 3.5. A Funtional Desription of the BT Method This method is mathematially expressed as nm O BT(f)= z w(k)r (k)exp(-j2fk) (3.9).1 k=-m where M <- N-I and w(k) is alled the lag window with the following properties 0 < w(k) < w(0) = 1 w(k) = w(-k) (3.10) S w(k) = 0 for Ikj > M * The subsript BT is used to designate this as the Blakman and Tukey PSD estimator - Blakman and Tukey pioneered the development of this estimator. The BT PSD estimator an 4 also be represented as.6 -%-.

39 P BT) DTFT lk)r (k I/ 2T I J W(f-Z)Px(Z)dZ (3.11) -1/2T where W(f), the spetral window, is the disrete-time Fourier transform of the lag window. Thus, the BT PSD estimator is ommonly onsidered a smoothed (or filtered) version of the simple PSD estimator. All is not gain, however, when using the BT PSD estimator. The BT PSD estimator allows for a redution of variane at the expense of inreasing the bias, or equivalently a redution of sidelobes at the expense of deteriorating the frequeny resolution. Kay (82-84) shows that the mean and variane of the BT PSD estimator are, respetively, 1/2T EBT(f) J W(f- )P(;)d (3.12) B-1/2T and Var [p ( ~ n! 1 P 2 (f) w2(k (3.) BT f ) N x " k=-m If a small bias is desired, then M is hosen large suh that the spetral window ats as a dira delta funtion. On the other hand, for small variane M is hosen small as indiated by Eq (3.13). Clearly, a ompromise of aeptable bias and variane has to be determined. In most ases, this 27 ~~~~ 4 % %ft% f4 4 / f d 4 f..

40 "'4 ompromise depends on the partiular appliation of the estimator. Figure 3.6 shows the BT PSD estimator of the spetrum given in Figure 3.4, assuming a Blakman lag window. A omparison of Figures 3.5 and 3.6 indiates that although the weak sinusoid is not deteted, the BT PSID estimator is muh smoother (less variane). Therefore, the BT PSD estimator is onsidered more reliable (19:334). The seletion of the lag window is onsidered an "art" in onventional PSD estimation. The lag window is not to be 4.. P" BT (f) P BT M ) 2 BT N N = d048,,4-513 M = q Blakman window I 0- ' Blakman wdow 1: ' i00-j. io-* - 10'-4 10"1 -I 10-. ei 10- " a Figure 3.6. The BT PSD of the Weak and Strong Signal Example (19:334). onfused with the data window. The data window is unavoidable as implied by Eq (3.5). That is, the autoorrelation estimator is omputed based on a finite 28 'p :,,-.., :.: :.-. :... :,:.:,:.- ',....:.:.:.... -,.. -., --:

41 ,..2.'. number of data samples (the data reord). The Blakman-Tukey approah as defined by Eq (3.9) assumes that the data window is retangular. The lag window, however, is optional and is used primarily to redue the variane (11:63). Unfortunately, there is no algorithm that preisely defines the approah for seleting the most *appropriate lag window. For this reason, many disrete-time lag windows have been proposed for use in PSD estimation. An empirial approah of window seletion is usually done, espeially if there is no a priori knowledge of the random proess. It is beyond the sope of this study to onsider the detailed harateristis of the many proposed lag windows (see any of the above mentioned referenes). However, it suffies to say that the spetral resolution of the BT PSD estimator depends largely on the main lobe of the Nspetral window. Also, the sidelobes of the spetral window greatly influenes the variane of the BT PSD estimator. The implementation of the BT PSD estimator on digital omputers requires P BT(f i)= M k=-m w(k)r x(k)exp(-j2rzfik) = DFT (k)rx(k) (3.14) where f.= i/kt for 0! i : K-1. I The disrete Fourier transform (DFT) differs from the DTFT in that both the time and frequeny representation of the proess are disrete. 29

42 Generally, a fast Fourier transform (FFT) algorithm is used to ompute Eq (3.14) (see Appendix A). The value of K is arbitrary, but usually K >> M, so that sharp details in the PSD estimate will not be missed (21:152). If K >> M, then the argument of Eq (3.14) is zero padded from M+l to K. Zero padding dereases the spaing between spetral line omponents in the DFT and does not really improve the resolution between two losely spaed spetral omponents of the signal (see Figure 3.7). The spetral resolution is improved, partiularly for short data reords, by onsidering modern tehniques of spetral estimation. These tehniques extend the ACS by extrapolation (or predition) rather than appending zeroes. The following setion onsiders the modern tehnique of autoregressive (AR) spetral estimation. Modern Spetral Estimation - Via the AR/Burg Method The autoregressive (AR) spetral estimation tehnique is a parametri method whih attempts to model the data reord with an all-pole rational transfer funtion. Ulryh and Bishop (4:185) argue that many disrete-time random proesses enountered in pratie may be desribed by 30 "4..

43 .4o FRACTION OF SAMPUNG FREQUENCY FRACTION OF SAMPLING FREQUENCY (a) (b) 0. 0.I E(d) 1.1

44 -. x(n) = - a(k)x(n-k) + Y(n) (3.15) k=1 where a (1),a p(2),...,a p(p) are the P-th AR oeffiients, and ro(n) is the sample of a zero-mean WGN random proess 2 with variane a. The above differene equation is referred p to as an AR-proess beause x(n) is a linear regression on itself with Y?(n) representing the error. The variable P represents the order of the AR-proess and plays a very signifiant role in determining the spetral density. More is said about the signifiane of P shortly. Taking the z-transform on both sides of Eq (3.15) and ombining terms yields the system funtion 1 H(z) = 1 (3.16) 1 + ap kz k k=l Thus, the AR-proess is viewed as being generated by applying a zero-mean WGN proess to an all-pole digital filter (see Figure 3.8). The system funtion is assumed to be both stable and ausal (i.e., all poles lie inside the unit irle). This ondition is neessary to insure that x(n) is WSS (17:179), an assumption made throughout this study. The evaluation of H(z) along the unit irle, z = ~1 exp(j2nf), yields the all-pole transfer funtion H(f) - p (3.17) 1 + Yap(k)exp(-j2nfk) 32

45 Ti(n) x(n) a P(P) a p(1) Figure 3.8. An All-Pole Filter for Generating and AR Proesss (12:250) Therefore, the PSD of the random proess, that implied by the linear disrete model, is ik=1 '.' 2 AR ( f ) = P 2 (3.18) 1 + Za p (k)exp(-j2nfk) for -1/2T -S f -5 1/2T. This indiates that the problem of PSD estimation, via the AR method, is atually a problem of parameter estimation. The AR PSD is defined one the parameters {ap(1), a (2),..., a (P), o 2 and P are determined. A myriad of methods are available for estimating these parameters. The interested reader is referred to Kay (17: ), Marple (21: ), Akaike (3: ), and Parzen (25: ). This study only examines the Burg method of estimating the parameters {a (1), a(2),..., a (P), a2}. Also, the proedure proposed by Akaike for estimating the model order P is briefly -a 33 I

46 i '-'. disussed. However, before proeeding to disuss these two methods, the basis for the high frequeny resolution attributed to the AR PSD method needs to be made lear. DuBroff (31: ) shows that an equivalent representation of the AR PSD is given by P AR(f) X A x (k)exp(-j2nfk) (3.19) where 4 4 R (k) for 0 : 5 k S P n, p -.4 SE(k) a(t)r (k-t) for k > P (3.20) Thus, assuming that the first P+1 true autoorrelation lags are known, via Eq (3.3). Eq (3.20) implies that autoorrelation lags may be reursively extended indefinitely. Although, omputing the AR PSD with Eq (3.19) is not very pratial. The usefulness of this equivalent form is to readily illustrate the basis for the high frequeny resolution. That is, in ontrast to the BT PSD *estimator no windowing and no zero padding ours. Hene, 4% the AR PSD does not possess the sidelobe phenomena of the BT PSD estimator and generally has muh better frequeny 4 resolution. The problem of estimating the parameters {a p (1), a p(2),..., a (P), a2) is now onsidered. ap 2,.. a (., Burg is redited d Io " ',.?'' '" : '' ";.." ";- '' -."" *- : - > " - ',,.". "'4.., ', "- "; ". -' "" -",' - ".'"."- "'

47 for developing the maximum entropy spetral density. Considering how best to estimate the extended autoorrelation lags, Burg argued that the time series (or data reord) haraterized by the known and extrapolatated ACS should have maximum entropy. This implies that the time series is the most random. The rationale for this hoie is that it provides for the flattest and the most minimum bias spetral estimate (17:199). Ulryh and Bishop (4: ) and a host of other researhers show that the maximum entropy spetral density is idential to the AR PSD for 4linear Gaussian random proesses. Hene, the terms AR PSD and Burg PSD are used synonymously in this study.. Burg also suggested that in estimating the parameters f{a (1) a (2),..., a (P), o2}, the Levinson-Durbin reursive algorithm be used. A detailed disussion of this algorithm is found in Kay (17: ). In summary, the Levinson-Durbin reursive algorithm provides estimates of the AR parameters by the following proedure N-1 ". ~ x N (3.21) N=0 For K = 1,2,...,P a k 1 l(i) + Fkak-l (k-i), i=l,...,k-i ak~i) (3.22) k k, i=k and * k ,. "-I ' ' ' ".,, " ''.,, u,, _ " '_ " ' < " " - " - "

48 The mathematial form of Eq (3.23) is analogous to the transmission of power through a terminated two-port devie. Hene, Fk is known in the literature as the refletion oeffiient (12:254). The estimate AR parameters {ap(1), a (2),...,ap(p), ap2} are known one the refletion oeffiient is determined. Burg proposed to estimate the refletion oeffiient by minimizing the sum of the forward and bakward predition error powers. The forward and bakward predition errors are defined, respetively, as p Fk (n) = x(n) + Z ap(k)x(n-k) (3.24a) k=l and P Bk (n) = x(n-p) + I ap(k)x(n-p+k) (3.24b) k=1 These two equations represent predition-error filters operating in the forward and bakward diretion, respetively. The Burg method of determining F k insures that if operating in the forward diretion, then the predition of x(n) given {x(n-1), x(n-2),...,x(n-p)} is the "best" possible predition (i.e.,the mean square error or forward predition error power is minimized). Similarly, if operating in the bakward diretion, then the predition of x(n-p) given {x(n-p+l), x(n-p+2),...,x(n-l), x(n)} is the "best" possible predition (i.e., the bakward predition error power is minimized). Haykin (12: ) shows that 36 * '0:. ~:~~ - &:: 2 -~ ;:-:2:.

49 when the forward and bakward predition error powers are minimized, the resultant errors, Eq (3.24), take on the form of WGN proesses. If this happens, then Eq (3.24) is equivalent to the AR proess desribed by Eq (3.15). The sum of the forward and bakward predition error powers is given by Pk = E{ IFk(n) 2 + IBk(n) 12 } (3.25) The refletion oeffiient F k is obtained by minimizing the above expression. Before doing so, it is onvenient to make the following substitution for the forward and bakward error F k(n) = Fk-l(n) + rfkbk-l(n-1) (3.26a) to and Bk(n) = Bk-l(n-1) + FkFk-l(n) (3.26b) These two relations provide a reursive method for determining the errors and are obtained by simply 3ubstituting Eq (3.22) into Eq (3.24). After the substitution of Eq (3.26) into (3.25), pk is differentiated with respet to -k and set equal to zero. Applying some simple algebrai manipulations, it is easy to show that F k is given by N-1 Fk-l(n)Bk-l(n- 1 ) F =-2 N-1 (3.27) k _ (JFk-(n) )2 + Bk (n-l) 2) 0,,*.,,... n 37

50 "4 where the expetations are approximated by spatial averages. Kay (17: ) shows Eq (3.27) guaranties that F k :5 1. Hene, the Burg method for determining the refletion oeffiient provides poles whih are on or inside the unit irle, resulting in a stable or marginally stable filter. In an attempt to try and put the above method into some perspetive, the omputations are to proeed as follows: 1. The initial onditions are: ~2 - N Rx N 1 2 (3.28a) and F 0 (n) = B 0 (n) = x(n), n = 0,1,2,...,N-1 (3.28b) 2. Compute the refletion oeffiient, Eq (3.27), for k =1 3. Compute the variane, Eq (3.23), for K = 1 4. Compute the AR oeffiients, Eq (3.22), for k = 1 5. Compute the predition error updates, Eq (3.26), for K = 1 6. Inrement K by one, go bak to step 2, and repeat. 7. Stop omputation after the desired order P is reahed. To illustrate these steps onsider the simple first-order AR proess given by x(n) = -a (1)x(n-l) + )?(n) (3.29) Multiplying both sides of this equation by x(n-k) and taking the expeted values for k=0 to k=l yields 38 I

51 ~ """R(0) R (1) 1 ""," x 12 (330 Rx(1) Rx(0) al(1) 0 This matrix notation is ommonly referred to as the set of first-order Yule-Walker equations (4:190). The unknown 2 parameters are obviously {a1(1), F 1 }. The proedure outlined in the above seven steps yields the following relations N-I F = -2 -_- n x(n)x(n-l) (3.31) n-i (Ix(n) 2 + Ix(n1) 12] 2= (l-f l2); O (3.32) Na () = F (3.33) F (n) = x(n) + F, x(n-1) (3.34) and B (n) = x(n-1) + F 1 x(n) (3.35) Thus, given x(n) for n = 0 to n = N-i, the estimate parameters i(1), I 2 are easily obtainable. Note that for higher orders of P, steps two through six are repeated until the desired AR parameter estimates a(1), ap, a (P), F are obtained. These estimates are substituted into Eq (3.18) in order to obtain the AR (or Burg) PSD estimator, namely 39 im:

52 i2 ""PAR P ( f ) P 2 (3.36) 1 + z a p (k)exp(-j2nzfk) k=1 Unfortunately, the reursive algorithm outlined above does -not provide any onstraint on the order P. That is, the AR '. parameter estimates may be reursively determined for any order P. However, if the value of P is too low, then the AR PSD estimator results in a smooth spetral estimate with poor frequeny resolution. On the other hand, if the value of P is too high, then the estimator results in a spetral estimate with spurious peaks. This phenomenon is illustrated in the next hapter. A number of researhers, suh as Akaike (3: ), Parzen (25: ), and Kashyap (32: ), have proposed different shemes to determine the "orret" model order P. The word orret is in quotes, beause model order.* determination is generally a non-trivial problem and in most - ases the shemes suggested serve only as guidelines (22: *" 2-11). Ulryh and Bishop (4: ) empirially found that the proedure suggested by Akaike provides the best estimate of P for AR linear Gaussian random proesses. This proedure simply requires that P be seleted suh that the - final predition error (FPE) given by 0.22" 40 :...

53 N + P 2 N P jj.p7 FPE N+ P N 2 37 I -P )11.5- PJO' is minimized. The idea here is to selet some maximum model order L S N - 1. Then suessively ompute the FPE for integer values of P = 1 to P = L. The value of P resulting in the minimum FPE is the "orret" model order. This approah is desirable if there is no a priori knowledge of the random proesses. For this study, however, model orders are determined empirially sine there is suh a "wealth" of knowledge about the simulated random proesses (see Chapter IV). Observe that the denominator of Eq (3.36) is simply the squared magnitude of the DFT of the sequene (1),...,a (P),where a p (0) = 1. Thus, one the AR parameter estimates {ai(1), ap(2),..., ap(p), Cp are reursively determined as disussed above, the AR PSD estimator, Eq (3.36), is determined by using a FFT algorithm (see Appendix A). The statistis of the AR PSD estimator are generally impossible to obtain (17:211). However, Makhoul (20: ) shows that for large data reords (i.e. N approahes infinity) the mean and variane of the AR PSD estimator are, respetively SPAR } PAR(f) (3.38a) 41

54 * -and Var PAR(f)} P(f) (3.38b) Although these results may be of very little pratial importane. That is, they may not provide good approximations for pratial problems. Nonetheless, it is instrutive to note the variane dependene on the model order P. Observe that the model order seletion represents for AR spetral estimation the lassial trade-off between resolution and variane. Comparison of the BT and Burg Methods A omparison of the BT and Burg methods seems the logial progression for the final development of this hapter. Also, the disussion in this setion leads rather niely to the analysis presented in the next hapter. Several areas are worth onsidering when omparing the BT and Burg methods. The most prominent are: omputational omplexity, resolution, and variane. The Burg method is generally more omputationally burdensome. This is evident by onsidering Eq (3.36). As eluded to earlier the denominator in Eq (3.36) an be effiiently evaluated by an FFT routine. The additional omputation results from having to reursively determine the appropriate AR parameter estimates before invoking the FFT routine; however, if resolution is the only onern. Cooper and Kaveh (7:320) show that the model order required by the Burg method is 42

55 -- generally muh less than the number of lags required by the BT method. Under this ondition, the omputation time of the Burg method is omparable to that of the BT method. As disussed previously, the frequeny resolution of the BT method is determined largely by the lag window (or spetral window). A simple exerise in Fourier transforming readily shows that the frequeny resolution in inversely proportional to M, where M 5 N - 1 (see Eq (3.9)). Marple (21:664) shows that the frequeny resolution of the Burg A method for sinusoidal proesses is approximately given by Af AR 1.03 (3.39) * P[SNR(P+I)] 0.31 Thus, the frequeny resolution, via the Burg method, f) * dereases with dereasing SNR but also inreases with inreasing model order number P. The frequeny resolution of the BT method is totally independent of input SNR. More is said about frequeny resolution for both methods in the next hapter. A omparison of Eqs (3.13) and (3.38b) shows that for a * given data reord length N the model order number P is analogous to the lag window length M. That is, the variane of both methods tends to inrease as their respetive * parameter inreases. The following hapter provides a qualitative omparison of the two methods for several data reords. V 43 %

56 IV. Analysis Introdution The purpose of this hapter is to graphially examine the two spetral analysis tehniques presented in the previous hapter. The two tehniques are applied to several known data reords and omparisons made. Also, a "smart routine" is introdued that attempts to automatially d.termine peaks orresponding to atual frequenries of given PSD plots. A major "limitation" of the reeiver system desribed in Chapter I is that of pulse duration. That is, the pulse reeived by the reeiver is relatively short in duration, resulting in only a few data samples available for proessing. The data reord resulting from the reeived pulse is, therefore, onsidered a short data reord. As mentioned in Chapter III, omputing the PSD of short data reords may provide less than desirable results. The Avionis Lab suggested that the number 64 is a good average representative number of data samples available for eah reeived pulse. Hene, eah data reord onsidered in this hapter onsists of only 64 data samples. Reall, that the reeived pulse may be omposed of a number of sinusoids. That is, two or more hostile emitters may be operating simultaneously. As stated in Chapter I, eah emitter is O, assumed to transmit at a single operating frequeny. reeived pulse, and therefore the data reord are omposed The 44.5

57 of sinusoids orresponding to the operating frequeny of eah emitter. For example, if two emitters are operating simultaneously at different frequenies, then the data reord onsists of two equivalent baseband sinusoids orresponding to the two emitter frequenies. The data reords evaluated are intended to represent "real world" type hostile emitter signals. A psuedo random WGN proess is added to eah data reord and represents the noise orruption aused by the medium of propagation. Again, the BT and Burg methods of spetral estimation are applied to eah data reord and omparisons made. Of the available lag windows provided by the ISPX software pakage, the hamming window yields the best ompromise between ol frequeny resolution and leakage. This window is used in determining the BT PSD estimator for all ases onsidered with M = N/2 (or M = 32), see Eq (3.9). As mentioned earlier, determining the model order number P of an AR proess is a non-trivial task, and therefore will be determined empirially. However, data reords that are exlusively sinusoidal (i.e., no embedded noise) require a model order number P = 2m for m number of sinusoids (17:213). This riterion will serve as a basis for empirially determining the orret model order P. X C-<- 45

58 *Problem One The first problem onsidered is a data reord onsisting of a single sinusoid embedded in WGN. This input data reord is given by x(n) = sin[2n(0.2)n] + g(n) (4.1) where g(n) is a psuedo random WGN proess. Figures 4.1 through 4.3 show this input proess for signal-to-noise ratios (SNR's) of 10, 15, and 20 db, respetively. Figure 4.4 shows the BT PSD estimator of Eq (4.1) for eah speified SNR. Observe that the frequeny resolution defined by the main lobe entered about the frational frequeny 0.2 is independent of SNR. This point is further illustrated by superimposing these three results as indiated in Figure 4.5. Figure 4.6 depits the result of overlapping the Burg PSD estimator for eah speified SNR. The model order seleted is two. Observe that as the SNR dereases the respetive spetral peaks are broadened and displaed from the true position (indiated by the vertial dashed line). This illustration learly shows that the frequeny resolution of the Burg PSD estimator is diretly proportional to the input SNR. Furthermore, it is noted that only the spetral peak orresponding to the 20 db SNR orretly resolves the true frational frequeny of 0.2. The broadening and displaement of the other two spetral "*-* peaks are due to the inreased noise (or lower SNR). A quantitative relationship that explains this phenomenon is a 46 --

59 -. 47

60 I z 00.0 Time Figure 4.2. WGN, SNR =15 Sixty-Four Samples of a Single Sinusoid in db 48 I t0

61 10 OC 44

62 [-.r- -.J J 10 4j '*- 050 Frequeny -"4 '. ifigure 4.4a. BT Estimator of the Single Sinusoid in WGN, 4-" SNR = 10 db,. 50 S.,%,-, -.-.,- ".',.,.. ".-,,., ,.-.,.-, : , - _

63 4-' -' E 10- CU Frequeny Figure 4.4b. BT Estimator of the Single Sinusoid in WGN, SNR 1 5 db.5 *

64 CI -- 4 Frequeny Figure 4.4. SNR 20 db BT Estimator of the Single Sinusoid in WGN, 52

65 ,. IA 4 *P - I 101 an 2 d Frequeny , OK e -'"-Figure 4.5. Superposition of the BT Estimator for SNR's [ S% 10, 15, and 20 db - 53

66 'to I' 0 -" -I o"1 0T 20 2 d.i'snr = 20 db SNR = 15 db..--- N = i0 db~ _.~.,-..., -... U-\.,.,,,. o- 0,.. Frequeny 4 Figure 4.6. Superposition of the Burg Estimator for SNR's 10, 15, and 20 db with P = 2 54

67 formidable task. In fat, very little is known quantitatively about noisy AR-proesses (4: ). *- " However, it is qualitatively instrutive to onsider. speifi ases. As will be shown in the following examples, the model order number P an sometimes be inreased to mitigate the effets of noise for a given SNR. A word of aution, however, is in order here. As disussed in the previous hapter if the model order number P is too large, spurious peaks in the spetral estimate will appear. This is simply explained by onsidering Eq (3.36), repeated here -for onveniene PA (f) = p (4.2) PAR ( )p 2 1 a p (k)exp(-j2nfk) k=1 Note that if P is too large, then extra poles are produed. These extra poles, ommonly referred to as noise poles, have a tendeny to situate themselves too lose to the unit irle in the z-plane, resulting in unwanted (or spurious) peaks. Problem Two This problem and the remaining two problems onsider Sdata reords with a minimum SNR of 15 db. The Avionis Lab indiated that 15 db is typially the lowest SNR of interest.. for their appliations. The problem addressed here is a data reord onsisting 0 55

68 WW.- p of two unequal amplitude sinusoids embedded in WGN. This data reord is given by x(n, = 0.2sin[2n(0.2)n] + sin[2n(0.25)n1 + q(n) (4.3) Figure 4.7 shows this input proess for a SNR of 15 db relative to the weaker sinusoid. The BT PSD estimator is presented in Figure 4.8. Notie that the main lobe of the weaker signal is not present. Obviously, the BT PSD estimator is not apable of distinguishing both sinusoids. This is a ommon dilemma for all onventional shemes of spetral estimation, partiularly for data reords onsisting of weak and strong sinusoids. This is explained by onsidering Eq (3.9), repeated here for onveniene * ~ M P BT (f)= z w(k)rx(k)exp(-j2rfk) (4.4) k=-m The BT PSD estimator involves a linear operation on a weighted ACS derived from a given data reord. Thus, the spetrum of the sum of two sinusoids (unorrelated) is simply the sum of their respetive spetra. The amplitude of the spetra is diretly proportional to the power in the sinusoids. Therefore, the amplitude of the spetra orresponding to the strong sinusoid is greater than that orresponding to the weaker sinusoid. In fat, so muh so that the main lobe amplitude of the weaker sinusoid is buried (or masked) by the sidelobe amplitudes of the strong sinusoid. It is important, however, to point out that the 56

69 I W ok TIX F T -,.-. 7 Time Figure 4.7. SNR =15 db Sixty-Four Samples of Two Sinusoids in WGN, 57

70 .- 1 Im -.- -i. J4 Frequeny Figure 4.8. i SNR :5 db BT Estimator of Two Single Sinusoid in WGN,

71 inability of the estimator to detet both sinusoids is not due to poor frequeny resolution. That is, the two sinusoids are spaed suffiiently in frequeny to be resolved by this estimator. Again, to reiterate the problem is that the main lobe of the weaker signal is aneled by the sidelobes of the strong signal. If the problem onsidered two equal amplitude sinusoids, then both sinusoids would have been resolved. Reall, that the resolution of the BT PSD estimator is govern largely by the lag window hosen. Of ourse, a lag window with a faster roll-off rate ould have been hosen possibly to retrieve the weaker sinusoid. But as stated earlier, of the available lag windows provided by ISPX, the hamming window *u). provides the best ompromised between frequeny resolution and leakage. Problem four onsiders an example in whih sinusoids are so losely spaed in frequeny, that it is impossible to resolve them using the BT method regardless of power. The Burg PSD estimator is shown in Figure 4.9. As stated earlier, the model order P an be inreased in order to mitigate the effets of noise for a given SNR. Observe that for a model order number of four, the appropriate order for no noise, the results of the Burg PSD estimator are no better than the results of the BT PSD estimator. Only the strong signal is deteted. For model order numbers of six and ten, S" * the Burg PSD estimator is apable of deteting both the strong.. " ' 59. S ~ - - -

72 v~im -4> * A Frqun Fi u e 4 9. B r s i a o f w ig e S n s i n WN SN 5d wt w7w ~60 e3

73 Frqun Fiue49. Br siao f w igesnsl nwn EN 5d wt '1

74 4, A I 410 Frequeny Figure 49. Burg Estimator of Two Single Sinusoid in WGN, SNR =15 db with P

75 and weaker sinusoids. Note the resolution of the estimator is slightly improved for the model order number * -ten. However, a spurious peak is just starting to appear. As explained in the previous setion, spurious peaks are the result of noise poles. This phenomenon an also be explained from a statistial point of view by onsidering Eq (3.38b), repeated here for onveniene, Var par(f)) =-- N PAR If) (4.5) It is observed that as model order P inreases for a given data reord length N, the variane inreases. Thus, P an * be onsidered analogous to the length of the lag window M for the BT PSD estimator, see Eqs (3.12) and (3.13). Problem Three A data reord onsisting of one or more delayed sinusoids is another interesting and often ommon ourrene. For simpliity, a modified version of the previous example is onsidered. is delayed by sixteen time units. That is, the strong signal Thus, the data reord is O given by " x(n) = 0.2sin(2n(0.2)n] + sin[2n(0.25)(n-16)] + g(n) (4.6) This proess is shown in Figure Again, the BT PSD estimator is unable to distinguish both sinusoids as illustrated in Figure A omparison of Figures 4.8 and indiates that the shift appears to have no apparent effet on the frequeny resolution. This should not be too A,

76 1.0 WJ V 01 - Ti me Figure Sixty-Four Samples Of Two Sinusoids in WGN, SNR =15 db -V 64 0

77 ..". C 10 'V " C- II ':" Frequeny.-?Figure BT Estimator of Two Single Sinusoid in WGN, SNR =1 5 db 4% I 65 k N

78 surprising onsidering that the BT PSD estimator is a FFT based operator. The results of the Burg PSD estimator are shown in Figure Comparing these results with those of Figure 4.9 indiate that the time shift auses the required model order number to inrease. A model order number P = 20 is required to resolve both sinusoids. In the previous example, the two sinusoidal peaks first appeared at P = 6. The reason for this inreased model order number is due the overall lower SNR. Problem Four The final problem onsiders a data reord onsisting of four losely spaed sinusoids. This data reord is given by x(n) = 0.2sin[2n(0.2)nJ + 0.7sin[2n(0.22)n] + sin(2n(0.24)] + 0.5sin(2n(0.26)] + g(n) (4.7) Figure 4.13 illustrates this proess. The BT PSD is given in Figure Observe that the sinusoids are too losely spaed to be resolved by this estimator. The main lobe shown is a result of the destrutive and onstrutive interferene of the four sinusoids and in itself does not represent any one of the sinusoids. the Burg PSD estimator for P = 24. Figure 4.15 presents Note that all four sinusoids are resolved with no inauraies ause by the spurious peaks. That is, the highest spurious peak is about 24dB below the lowest desirable peak. This gives rise to 66 A["~.LMV~J.tA...2.k.A.t x...a. t >.V ~ t.

79 '% to,.i. >.- 0, 0 0." 0,- 0.: 0, 01" [.! Frequeny 1~ -I 2% "-r- I re qt n SNR = 15 db with P =4 67..t.,;,-z-,,,..., -.-..,,.-.?,.," , -,... '..,., ,,.~ ~C C -..,.,. -.. = CC i, -A I C 4, I.i -: i,- " -..

80 '" Figure 4.12b. Burg Estimator of Two Single Sinusoid in WGN, '. SNR = 15 db with P =

81 -. %.4,.4.* "4-4 0,. -'""Fgue4.2 SN ur stmao o wosige iusldi. ' SNR= 15dB wth P= 2 '*o "' '.-. :?:.0 '2") ' "_- :";", '-' ' s - - -"" ',',.. '".';' -- ''. :':'.'".,; Frequeny -. ", h".i"' :....,." """ ","',.: --- "- "'".

82 V Time Figure SNR =15 db Sixty-Four Samples of Four Sinusoids in WGN, 70

83 ?. I- W % I -: -. Frequeny -71.'..'-.Figure BT Estimator of Four Single Sinusoid in WGN, ' i" SNR = 15 db '-5, I,",-, 0,w, 7 1 O.* l~'ii ' '_:"%, ' '":,'." ' '.',- _,' -,i /. -.,/ / k..j'

84 ,,', "" u,-4 A 27, Frequeny 4- I I "-: Figure Burg Estimator of Four Single Sinusoid in WGN, " SNR = 1.5 db with P =24 H 72

85 developing a threshold deteting sheme for determining peaks orresponding to atual frequenies - the fous of the next setion. Threshold Detetion Routine The above four problems are by no means an exhaustive list of all possible signal types. However, the data olleted ertainly suggest that the Burg method of spetral estimation is far superior to the BT method for short data reords. This, of ourse, assumes that the input SNR is muh greater than unity. Under the assumption that a suffiient input SNR (typially 2 15 db) is available, the method of hoie is obviously the Burg method. Thus, the threshold detetion routine developed is based on applying the Burg method to short data reords. The foregoing analysis assumes a priori knowledge of the input data reords. This assumption is neessary in order to investigate the merits of both spetral estimators. In pratie, however, a priori knowledge is generally not known about the input data reords. In this ase, it is often very diffiult to determine preisely the peaks orresponding to atual frequenies and the peaks resulting from inauraies of the spetral estimator. For example, given the PSD plot of Figure 4.12 without having any knowledge of the input data reord, it might easily be interpreted as resulting from a data reord onsisting of - six sinusoids. Reall, Figure 4.12 is atually Lie result 73

86 .1 of a data reord onsisting of two sinusoids at frational frequenies of 0.2 and The other peaks are spurious peaks resulting from the inauraies or statistis of the Burg estimator. The obvious dilemma is that of differentiating between atual peaks and spurious peaks. -One possible approah might be that of threshold detetion. The problem proposed by the Avionis Lab is to devise a routine that is apable of automatially deteting a maximum of five atual peaks and their orresponding frequenies. -. That is, if a data reord onsists of m sinusoids (unknown to the observer), then is it possible to detet up to five atual peaks and their frequenies. In order to ome up with an "adaptive" threshold value, several test ases are investigated (see Appendix B). The test ases presented in Appendix B represent only a few of the ases atually onsidered. However, they serve to summarize the implied results of the many ases investigated. The test ases inlude data reords, with 64 data samples, onsisting of one to five independent sinusoids. The upper limit five is a software onstraint (see Chapter II). The amplitude (or power) of eah sinusoid is varied. It is observed, that if the amplitudes of the sinusoids vary by two orders of magnitude or more (e.g. a data reord onsisting of three sinusoids with respetive amplitudes of 0.1, 2.0 and 30), then retrieving the lower amplitude sinusoid is virtually impossible (see Figure B.7). 74

87 However, if the amplitudes of the sinusoids vary by no more than one order of magnitude, then "all" sinusoids may possibly be retrieved (see Figures B.l through B.6). The test ases suggest that a suitable adaptive threshold is 28 db below the highest peak in the spetrum. The word adaptive is used rather loosely in that a threshold of 30 db below the highest peak does not adapt for all ases. This method of determining the threshold is purely empirial, resulting from determining an average value below the highest peak for all test ases onsidered. 4The ideal model order number P required to detet five sinusoids is ten. However, as previously pointed out the orruption of noise auses this number to inrease. The (000 test ases suggest that a model order number of P = 24, using the threshold value previously stated, allows the estimator to orretly resolve atual peaks in most ases for data reords onsisting of up to five sinusoids. It is pointed out, however, that a minimum SNR of 15 db is used in eah ase onsidered. Varying this parameter will hange 4 the required value of P Also the software limitation of a maximum of five sinusoids in a data reord ertainly does not represent all possible data reord types. In pratie *it may be neessary to hange the value of P if it is believed that the atual data reord onsist of many sinusoids (i.e., m >> 5). Obviously, there are several * parameters that effet the value of P; thus, it is not N...

88 possible to ome up with a value suitable for all possible senarios. S The "smart" routine that determines the atual peaks and their orresponding frequenies, inorporating the above riteria, is also presented in Appendix B. For purposes of illustration, onsider the PSD plot of Figure Figure 4.16 is atually the Burg PSD of Problem Three for P = 24 note the many spurious peaks. The results of applying the * and "smart" routine are as follows: f_= f = The reason for the disrepany between the omputed frequenies and the atual frequenies is due to way in * whih the Burg PSD is omputed. That is, the Burg PSD is omputed based on 256 disrete data points over the frequeny range 0 to 0.5. Thus, the omputed frequenies are atually multiples of 0.5/256 (or 1/512) and represent lose approximations. * The following hapter presents onlusions and reommendations of the above analysis. I. 76. i% ",-d".," -". ".,J-. T A -.'''.- 4..'''. "'' "..,''.'' '.. '% '.,,. ' '.r i.. " " ".-., -"......" -.'. '.- ''

89 4..0 '3/ 10- -, 0-28 db 4 - -, S - " ~ "4 Threshold %." Frequeny. 07 '." Figure Burg Estimator of Two Single Sinusoid in WGN,, SNR = 15 db with P = 24

90 -V. '- -. V. Conlusions and Reommendations Conlusions This study has investigated the performane of two popular but distintly different methods of spetral estimation from an EW reeiver point of view. In an EW a,... environment the duration of the reeived pulse is usually relatively short, resulting in only a few data samples available for proessing. Thus, the method used for *.8 spetral estimation has to be apable of providing reasonable results for short data reords. Several "real world" data reords, eah onsisting of 64 data samples, were analyzed using both the BT and Burg methods of spetral estimation. The Burg method was found to yield far superior results in terms of frequeny resolution. However, its performane was determined to be a funtion of the input SNR. It was noted that for low input SNR (i.e., SNR -510 db) the results of the Burg method degraded substantially. Another parameter affeting the resolution apability was V.- the model order number P. Inreasing the value of P has a tendeny of inreasing the frequeny resolution, as well as introduing spurious peaks into the spetrum. Therefore, the value P has to be seleted arefully. The test ases that were analyzed suggest a P = 24 for deteting atual peaks of data reords onsisting of up to five independent sinusoids. A minimum SNR of 15 db was used in eah test oi 78

91 . ase. Evaluation of the test ases suggested that an appropriate sheme for differentiating between atual peaks and spurious peaks was that of threshold detetion. '.4" Reommendations This study is not intended to present an all enompassing approah of spetral estimation for short data reords. The two tehniques presented in this study represent only a some fration of the many methods proposed and urrently being evaluated by several researhers Ifollows: (9:1383). Several appropriate reommendations are as 1. Investigate some of the other methods of spetral estimations. For example, the method proposed by Pisarenko (42: ) whih provides a very aurate disrete spetrum for data reords onsisting f deterministi harmonis in WGN. 2. Inrease the maximum allowable sinusoids in a data reord urrently provided by ISPX. This will allow for *developing a more omprehensive data base; thus, allowing for more onlusive data. 3. Apply the two methods disussed in this study to other types of data. For example, the Burg might have great promise if applied to two dimensional imaging data. 4. Investigate and possibly inorporate some of the shemes that have evoled of the past deade for 79 -.

92 -. determining the orret model order number P into the ISPX 4% software pakage. 08

93 Appendix A: Subroutines Used for Spetral Estimation Several algorithms are used in the analysis setions of this paper. This appendix provides a brief explanation of eah r,..tine with a FORTRAN listing. All FORTRAN listings appear exatly as they appear in the ISPX software pakage. The listings were not generated by the author. They were used merely as a "tool" for analyzing several data reords. Subroutine Blakman-Tukey subroutine blaktukey(x,n,mode,wind,m,nexp,pbt) This program omputes the Blakman-Tukey spetral estimator as given by (3.9). Either the biased or unbiased autoorrelation estimator may be used as well as a lag window. The spetral estimate is evaluated at the frequenies F=-I/2+(I-I)/L for I=,2,...,L. The number of C frequenies is given by L=2**NEXP. C Input Parameters: C x -Complex array of dimension Nxl of data points. n -Number of data points. mode -Set equal to zero for unbiased autoorrelation estimator; otherwise, biased estimator used. wind -Real array of dimension 2M+1 of lag window weights; wind(l),...,wind(m+l),o..,wind(2m+l) orrespond to w[-m],...,w[0],...,w[m]. m -Largest lag desired. * nexp -Power of two whih determines number of frequeny samples desired, L=2**NEXP; must be hosen so that L is >= 2*M+2. Output Parameters: * pbt -Real array of dimension L=2**NEXPxl of samples of the Blakman-Tukey spetral estimate, where pbt(i) orresponds to the spetral estimate at frequeny F=-l/2+(I-l)/L. External Subroutines: * PREFFT,FFT * 81 6

94 Notes: The alling program must dimension the arrary x,wind,pbt. The array w,r,rorr,p must be dimensioned >= the variable dimension shown, or equal to 2**NEXP. Also, the array RCORR should be dimensioned >= 2M+l. C omplex x(l), w(512), r(512), reorr(129) dimension window(1), p(512), pbt(l) pi=4.*atanc 1.) ompute the autoorrelation estimates from the date. ml=m+l all orrelation(n,ml,mode,x,x,rorr) Window the M+1 autoorrelation estimates and insert them into the last M+1 loations of rorr. Then, fill first m points of rorr array with the omplex onjugates of the last m points (shift the autoorrelation sequene to the right by m samples so that FFT may be * used). r (ml) =wind (ml) *rorr (1) do 10 i=1,m r(ml+i )=wind(ml+i)*rorr( i+l) 10 r(ml-i)=wind(ml-i)*onjg(rorr(i+1)) Zero pad the array of windowed autoorrelation samples to obtain an array of dimension equal to L. 2* *nexp i=2*m+2,l=256 N~~Y* 1 = do 2ij 20 r(i)=io.,0.) Compute FFT of the autoorrelation sequene. i nvr s =-1 npad= 1 all prefft(1,npad, invrs,nexp,w) norm= 0 all fft( l,npad,nexp,norm,w,r) the right by m samples. do 30 i=1,1 * f=(i-1.)/1 arg=2.*pi*f*m Compensate for shifting the autoorrelation sequene to 30 p(i)=real(r(i)*exp~mplx(o.,arg)) Transpose halves of FFT output so that first PSD sample is at a frequeny of -1/2. do 40 I=1,1/2 * pbt(i+1/2)=p(i) 40 pbt(i)=p(i+l/2) return end 82.A

95 Subroutine Burg subroutine brug(x,n,ip,a,sig2) This program implements the Bu~g method for estimation of the AR parameters (3.21)-(3.27). Input Parameters: x -Complex array of dimension Nxl of data points. n -Number of data points. ip -AR model order desired. C C Output Parameters: a -Complex array of dimension IPxl of AR filter parameter estimates arranged as A(1) to A(IP). sig2 -White noise variane estimate. C Notes: The alling program must dimension the X,A arrays. The arrays EFK,EFK1,EBK,EBK1,AA,RHO must be dimensioned >= (n,n,n,n,ipxip,ip respetively). C omplex x(1),a(1),efk(512),ebk(512),efkl(512), ebkl(512),aa(128,128),: sumn,sumd dimension rho(128) Compute the estimate of the autoorrelation at lag zero (3.21). rho0=0 do 10 i=l,n 10 rhoo=rho0+abs(x(i))**2/n Initialize the forward and bakward predition errors(7.39). do 20 i=2,n efkl(i)=x(i) 20 ebkl(i-l)=x(i-l) Begin reursion. do 80 k=l,ip Compute the refletion oeffiient estimate (3.27). sumn=(0.,0.) sumd=(0.,0 ) do 30 i=k+l,n sumn=sumn+efkl(i)*onjg(ebkl(i-1)) 30 sumd=sumd+abs(efkl(i))**2+abs(ebkl(i-l))**2 aa(k,k)=-2.*sumn/sumd Update the predition error power (7.40). if(k.eg.l)rho(k)=(l.-abs(aa(k,k))**2)*rho0 if(k.gt.l)rho(k)=(l.-abs(aa(k,k))**2*rho(k-1) if(ip.eq.l) go to 90 if(k.eg.l) go to 50 Update the predition error filter oeffiients (3.27). 83 A.f-- "

96 ,. r = -r - : -fl b. r_..,.r r - w, ;- - ;z ' i -,, i " ' " ' do 40 j=l,k-1 40 aa(j,k)=aa(j,k-l)+aa(k,k)*onjg(aa(k-j,k-1)) Update the predition error filter oeffiients (3.24). 50 do 60 i=k+2,n efk(i)=efkl(i)+aa(k,k)*ebkl(i-l) 60 ebk(i-l)=ebkl(i-2)+onjg(aa(k,k)*efkl(i-1) do 70 i=k+2,n efkl(i)=efk(i) 70 ebkl(i-l)=ebk(i-l) 80 ontinue Find final values of the predition error power, whih is the white noise variane estimate, and the predition oeffiients, whih are the AR filter parameter estimates. 90 sig2=rho(ip) do 100 i=l,ip 100 a(i)=aa(i,ip) return end O Subroutine PreFFT subroutine prefft(n,npad,invrs,nexp,w) Th.program sets up the mex exponential table needed to ompute the fast Fourier transform of an array of omplex datasamples using a deimation-in-frequeny algorithm. Pruning is performed if zero padding is requested. The output table ontained in the array is input to the program FFT whih omputes the fast Fourier transform of the data Input Parameters: n -number ofdata samples npad -Set to 1 for no zero padding (N-point transform), 2 for double padding (2N-point transform), 4 for quadruple padding (4N-point transform). invrs -Set to -1 for forward transform, 1 for inverse transform. Output Parameters: nexp -Indiates power of two exponent suh that n=2**nexp. Set to -1 to indiate error ondition if n is not a power of two in whih ase program terminates prematurely. w -Complex array of dimension n*npadxl ontaining exponential table S%

97 Notes: The alling program must dimension the omplex array w greater than or equal to n*npad. omplex w(l),u nexp=1 5 nt=2**nexp if (nt.ge.n) go to 10 nexp=nexp+l go to 5 10 if(nt.eg.n) go to 15 nexp=-1 return 15 nt=n*npad ang=8.*atan(l.)/nt u=mplx(os(ang),invrs*sin(ang)) w( 1)=(1.,0.) do 20 i=2,nt 20 w(i)=w(i-l)*u return end Subroutine FFT *[ subroutine f ft (n, npad, nexp, norm, w, X) Input parameters: n,npad,nexp,w -See parameter list for subroutine "PREFFT" norm -Set to 0 for forward transform, else the sum is divided by n for inverse transform. X -Complex array of dimension nxl of data samples. Output parameters: x -Complex array of dimension n*npadxl of transform values. Notes: C The alling program must dimension arrays x,w. 0' omplex x(l),w(l),t,u if(nexp.eq.-l)return go to (30,20,10,10)npad 10 n2=n*2 nt=n2*2 x(n2+1)=x(1) do 12 K=2,n 12 x(n2+k)=x(k)*w(k) N' '"

98 es' x(n+1)=x(l) nx=n2+l x(n+nx)=x(nx) jj=3 do 14 k=2,n x(n+k)=x(k)*w(jj) nx=2n+k x(n+nx)=x(nx)*w( ii) 14 jj=jj+2 mm =4 go to x(n+1)=x(1) do 22 k=2,n 22 x(n+k)=x~k)*w~k) mm =2 nt =n *2 go to nt=n 35 ll=n do 70 k=1,nexp * nn=11/2 jj=min+1 do 40 i1l,nt,11 kk= i+nn t =x( i )+(kk) x(kk)=x(i)-x(kk) 40 x(i)=t if(nn.eq.1) go to 70 do 60 j=2,nn u=w( ii) do 50 i=j,nt,11 kk=i+nn t=x( i)+x(kk) x (kk ) = Cxi) -xckk) ) u 50 x(i)=t 60 jj=jj+mm l1=nn * nun=mm*2 *70 ontinue nv2=nt/2 nml=nt-1 j=1 -~ do 90 i=1,nml if~i.ge.i) go to 80 t=x( j X(j)=X(i) x(i)=t 80 k=nv2 85 if(k.ge.j) go to 90 J~.. =j -k ~. k=k/2 go to 85

99 -A9061 SECTRAL. ANiALYSIS OF SHO RT DATA RECCRDSMU AIR FORCE 2/2 INST OF TTCN MRIGNT-PATTERSON AFS ON SCNOOI. OF ENGINEERING T E CARTER DEC 87 AFIT/GE/ENG/DT-9 UNCLASSIFIED F/G 17/4 U ENEEEEEIIIII IEEE.