Adaptive Line Enhancement and Spectrum Analysis D. W. TUFTS PROCEEDINGS LETTERS 169

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1 PROCEEDNGS LETTERS 69 is obtained. The analogy of (3) is that the matix S is not stochastic but is an aay of known complex numbes. Diffeentiation with espect to vaiables multiplied by elements of this matix is completely valid. Mogea s () is valid. Howeve, in this equation, S is not a vaiable, so that a - n [det (S)] = (5 adi and my (7) follows. Mogea s statement that his () is obtained if and only if the elements of the eigenspectum ae distinct is a mystey to me. t may be that he means that the cicumstance that the extemum is obtained when all di ae equal negates the applicability of patial diffeentiation. f this is what is meant, this is clealy eoneous. As an example, the function P Q = (zi - ) i= attains the minimum value of zeo fo each zi =. The coect solution is easily found by computing 3Qla.q; i =,,. P and setting each patial equal to zeo. The last majo point of Mogea s communication is the lack of igo in pogessing fom (7) to (8). The validity of the diagonalizability of A by a similaity tansfomation is based upon ecognizing that fo this poblem thee is a unique ML estimato of the covaiance matix. Note that when A is diagonalizable, the unique matix with equal eigenvalues is a scala multiple of the identity matix. So that ifa is diagonalizable it must equal N times the identity matix as in (9). KA is not diagonalizable, thee is not a unique matix with the eigenvalues of N. Fo this situation thee is not a unique ML estimato. REFERENCW [ S. D. Mogea, Comments on An Altenate Deivation of the Maximum Likelihood Estimato of a Covaiance Matix, Poc. EEE, vol. 6, p. 9, Nov. 76. [ R. Nitzbeg, An Altenate Deivation of the Maximum Likelihood Estimato of a Covaiance Matix, hoc. EEE, vol. 63, pp. 63-6, Nov [3] H. Came, Mathematical Methods of Statistics. Pinceton, NJ: Pinceton Univesity Pess, 96, p. 98. C. R. Rao. Linea Statistical nfeence and its ADDlicatwP. New L... Yok:Wiley, 965, p. 89. [5] D. A. S. Fase, Statistics, An ntoduction. New Yok: Wiley, 958, p.. Adaptive Line Enhancement and Spectum Analysis D. W. TUFTS Abmct-The notim that adaptive filtes may estimate?ad tack the fequency of a ph.&-modul.ted sin& m noise bette tha a spectnnn analyze is examined. t using both theoetical and expeimental esults, that spectnun analysis pefoms bette than adaptive filitteling. n dditioa, tbe method of spec- analysis CL be impoved fo ach sppkationa by tbe uae of fxequencyope pocessing, o by the pwdon of a patidly cobeeat tefeence wavefom by maximum posteio pobability estimatica of the phm hctim. Adaptive line enhancement has been poposed as an impoved method fo a) estimating and tacking the fequency of a phasemodulated, siuinusoidal signal in noise and b) deteecting the pesence of Manuscipt eceived Febuay 3, 976; evisedeapil6,9!6. The autho is with the Depatment of Electical Engueenng, Univesity of Rhode sland, Kingston, R 88. such a signal in noise. Widow et al. and Giffiths ague that adaptive filteing may have bette pefomance than spectum analysis in such applications [, [. n this lette ague that, on the contay, spectum analysis appeas to haveadvantages in pefomance which can be demonstated both theoetically and expeimentally. Conside the case in which the fequency of the sinusoid is almost constant ove the delay ange of the adaptive filte. This is the condition given by Giffiths fo effective opeation of the adaptive filte [, p. ). n Giffiths example E [, p. 9 in which the fequency is vaying sinusoidally with a peiod of samples, the delay ange of the adaptive filte was taken to be L = samples. We assume that no a pioi knowledge is available about the fom of fequency vaiation. The ange within which the fequency lies-which could be the whole Nyquist fequency ange-is assumed to be known. Unde these cicumstances Vitebi has deived the maximum-likelihood estimate of fequency within eachof the intevals ovewhich it is almost constant [3, ch. lo]. in the pesence of additive Gaussian noise. Let us assume that each such inteval has duation L = samples, as in Giffiths example. The maximum-likelihood estimate is obtained by spectum analysis (SA), which could be ealized using the fast Fouie tansfom algoithm (FFT). Howeve, the size of the aay to be tansfomed would not be samples, but moe likely 56 o 5 samples. That is, the data aay must be heavily padded with zeos to attain the de+d fequency esolution. The fequency esolution is defined in this note to be the accuacy of the measuement of the instantaneous fequency (i.e., the deivative of the phase function) of a phase-modulated sinusoid. Fo a med nonzeo window length ove which the instantaneous fequency is almost constant, the fequency esolution of a zeepadded discete Fouie tansfom OFT) depends on ) the signal-tenoise atio (SNR) of the input data, ) the esidual mismatch between the signal and the closest constant-fequency sinusoid, and 3) the numbe of fequency bins which ae fomed (and, hence, the amount of zeo padding). f thee is no noise with the signal, the fequency esolution monotonically impoves, as moe unifomly spaced fequency bins ae added ove the same fequency ange, until the esidual mismatch of the signal, if any, limits the accuacy of the fequency measuement. n pactical applications the attainable fequency esolution natually depends on the type and pecision of a pioi knowledge of the vaiation of signal fequency o phase, and how that a pioi knowledge is used in the measuement of fequency. Hee, following Giffiths [, p. 9, we assume only that the fequency is almost constant ove the window of nonzeo data. The spectum analysis method can be impoved by testing a small numbe of fequency-slope bins associated with each cente-fequency bin. This enables a fequency-coheent efeence to be usedove a longe span of data, because the fequency ate is being matched as well as the fequency. Hee this is called sloped spectum analysis (SSA). A sinusoidally vaying fequency was simulated with additive noise in ode to tack the fequency by the SA method, based on maximumlikelihood estimation of fequency and by the SSA method. The paametes wee chosen to match those chosen by Giffiths in his example E (, p. 9. Good tacking of fequency (to an accuacy of about.5 Hz) was obtained down to an SNR of db. Examples of these expeimental esults ae pesented in Tables,, and below.thesamesignalandnoisevalues ae used in each of the thee cases, but the noise is scaled to obtain the desied SNR. Only thee values of fequency slope wee used fo the SSA method, namely, zeo and t.37 Hz/s. These values cove the angeof fequencyslope values of the signat The fequency estimates appea to be moe accuate than those obtained by adaptive fdteing. t is difficult to compae the complexity of the SA method with that of adaptive filteing. Although the fome was ealized by a zee padded FFT algoithm to obtain the esults of Tables,, and, this is not the most efficient ealization. Widow et ul. suggest that it may be possible to detect low-level sinusoids in noise moe effectively by use of adaptive line enhancement [l, pp The spectum analyze with which they compae adaptive fdteing is not allowed to coheently integate ove the full set of input samples ove which the fequency is constant. This would impove its pefomance. f a phase-modulated sinusoid is to be detected, then esults of detection theoy can be applied [3, ch. 8, [, ch.. The modeled phase function, any a pioi infomation, and the input data should then be used to estimate the phase function [S, [6] and fom a patiauy coheent efeence wavefom [3, ch. 8.

2 7 PROCEEDNGS OF THE EEE, JANUARY 977 TABLE F( F(SSA) F(S.) Compaison of input fequency, FO, fequency estimate using sloped spectum analysis, F(SSA), and fequency estimate using spectum analysis F(SA). The signal samples ae those specified by Giffiths [, p. 9 The numbe of fequency bins ove a Hz band is 5. (the bin sepaation is.95 Hz). The fequency slope fo the SSA method is i.37 Hzls. SNR is 7 db. TABLE F( F(SSA) F(SN Compaison of values of input fequency and fequency estimates as in Table, except SNR = db. TABLE F( n) F(SA) F(SSA) Compaison of values of input fequency and fequency estimates as in Table, except SNR = 5 db. B. Widow, et al., Adaptive noise cancelling: Pinciples and applications, Poc. EEE,vol. 63, pp , Dec L. J. Giffith. Ranid measuement of digital instantaneous fequency, EEE lhm. Acous., Speech, -Signal Po#&g, A VOl. Vitebi. ASSP-3, Pin&les pp. 7-, of Coheent Ap Communication. New Yok: McGaw-Hill, Selin, Detection Theoy. Pinceton, NJ: Pinceton Univ. Rem, t 6 A<. D. W. Tufts and J. T. Fancis, An iteative pocedue fo phase estimation based on maximum posteio pobability estimation, to appea in EEE Tmns. nfom. Theoy. - Maximum posteio pobability demodulation of angle mohtion signals, unpublished. Reply by L. J. Giffiths Tufts discussion coectly points out that conventional spectum analysis cau be used to accuately detemine the instantaneous b- quency of a sinusoid which is obseved in the pesence of additive white noise. The advantage of using adaptive filtes fo estimating and tacking instantaneous fequencies is not that such filtes pefom bette than optimal detectos fo a specific poblem. They do not Rathe, the advantage lies in the fact that adaptive estimatos ae extemely obust. That is, they pefom emakably wenove a wide aage of input signal paametes and statistics with no o pioi knowledge egadiug the pecise natue of these paametes. n addition, the pefomance of the adaptive mte has been shown to be elatively insensitive to the two available fjlte paametes-filte length L and the nomalized adaptive popotionality constant Q. nceasing o -asing eithe of these paameten by a facto of two causes little change in adaptive pefomance. Examples of this behavio ae pesented in [. Of cou~se, once the pecise statistical desciption of an input wavefom is available, optimal filtes can be deived and wil undoubtedly outpefom adaptive ates. A note of caution fo this appoach, howeve, must be injected. The pefomance of optimal pocessos is not always obust. f a mistake is made in chaacteizing the input and the wong optimal pocesso is employed, the esults can often be damatically in eo. As an example, if the specmun analysis method (SA) demi in the above coespondence is applied to an input containing Manuscipt eceived May,976. L. J. Gnffiths is with the Depatment of Electical Engineeing, Univesity of Coloado, Boulde, CO 83.

3 PROCEEDNGS LETTERS 7 two closely spaced sinusoids, and a fou-sample analysis inteval is used, the spectum obtained afte adding zeos to achieve the desied fequency esolution will not contain peaks at the appopiate instantaneous fequency points. This is tue even fo the case ofinfimiteinput signal-tenoise atio. Thus seaching fo maxima in the output could easily lead to fequency estimation eos geate than 5 pecent fo the tw-input sinusoid case. n fact, a deivation of the multiple input optimal estimato shows that it diffes significantly fom the diect SA method used by Tufts. n contast, the same fom of the adaptive fequency estimato woks well fo eithe singlea sevealinput fequency-modulated sinusoids, as shown in [. Moe ecent esults have also shown that the adaptive method povides good estimates fo the case of bust-type inputs m which the sinusoids ae pesent only duing shot andom time intevals. n summay, agee completely with Tufts obsevation that one should always apply all available u pioi infomation when designing detection and estimation signal pocessing filtes. Cae must be taken, howeve, to ensue that the pefomance of these filtes does not apidly deteioate when small enos exist between the actual and assumed input statistics. Fo those cases which Seem to occu all too fequently in pactice-that is, when one has absolutely no eliable u pioi knowledge egading the stuctue of the desied signals- know of no bette oveall pocedue than to use a simple adaptive filte based on the LMS aigoithm. L. J. Giffths, Rapid measuement of digital instantaneous fequency, EEE Tam Acowt., Speech., Signal Pocudng, VO. ASSP-3, pp. 7-, AF Reply3 by B. Widow, J. Glove, J. McCool,6 and J. Teichle We thank Tufts fo his comments on the discussion of adaptive line enhancing that appeas in Appendix D of ou pape Adaptive Noise Cancelling: Pinckples and Applications [. t is clea to us that ou suggestion was in eo that given equal input data, the line enhance might outpefom the DFT m detecting a single sinewave in white Gaussian noise. n this classic case, the DFT implements a finitelength appoximation to the matched filte and is theefoe close to optimal. We believe that ow simulations ae coect, but that ou analysis and intepetation of the esults ae subject to question. t would have been moe meaningful to compae DDFT with d G athe than with DALE as we did in Appendix D. On the basis of this compaison, analysis shows that the adaptive line enhance and the DFT ae equally effective as a single-sinewave detecto. Tufts suggests that in esticting the length L of the DFT window to less than the length of the total data block, the DFT was in effect disadvantaged in ow study. This issue is not clea in Appendix D.We should note hee that the adaptive filte length was esticted in the same way, in that the numbe of weights was made equal to the numbe of DFT points, pecluding any compaative advantage o disadvantage. Such length estictions ae useful in the pesence of an input signal with finite bandwidth. Fo best esults, the DFT length and the adaptive-fdte length should both be set to coespond to the ecipocal of the signal bandwidth, so that the fequency esolution of each method is matched to the signal linewidth. We eget the eo in Appendix D and wish to point out that this section is independent of the est of the pape. We did caution the eade that the concepts... have not been in existence long enough to povide an adequate pespective. Since the line enhance is a new and unusual filteing system, it is sometimes a difficult o complex matte to esolve questions egading its behavio with input signals of finite bandwidth, coloed noise, multiple signals in noise, etc. Peoplewhohaveobseved the pefomance of the line enhance ae vey enthusiastic about its potential and ae anxious to undestand its behavio fom a theoetical standpoint. Expeiments ae easy to pefom but ae sometimes difficult to intepet. n the following expeiments, adaptive-fateing and adaptivelineenhancing techniques have been applied to some inteesting pob- Manuscipt eceived June, 976. B. Widow and J. Teichle ae with the Depatment of Electical Engineeing, Stanfod Univesity, Stanfod, CA 935. J. Glove is with the Depatment of Electical Engineeing, Univesity of Houston, Houston, TX. J. McCool is with the Fleet Engineeing Depatment, Naval Undesea Cente, San Diego, CA 93. lems in spectal analysis. The same kinds of poblems can as be solved by conventional methods based on the DFT. The expeiments pesented in Fig. ae simila to the expeiments descii in Appendix D of [ with the following exception: in the expeiments pesented hee, we use the tansfe function magnitude of the lineenhance adaptive filte, athe than magnitude squaed as we did in Appendix D, fo compaison with the powe density spectum of the pimay input, consisting of a sinewave signal in noise. Fo the expeiment depicted in Fig. l(a), the noise was white; fo and (c), the noise was 5-pecent white, SO-pecent coloed. The coloed noise had 5-pecent bandwidth and was geneated by passing white noise though a twoconjugatepole filte. n each case, the SNR was.56. The adaptive filte had 8 weights; the DFT had 8 points. Thesampling fequency was. The total amount of data usedwas samples in each case. The DFT takes in blocks and unifomly weights the samples, wheeas the line enhance woks on a steady-flow basis and exponentially weights the data ove time. n compaing the data consumption of the two techniques, the data used by the line enhance wee defmed as the numbe ofsamples inputted duing fou time constants of the adaptive pocess. nspection of the plots of Fig., all dawn to the same linea scale, shows that in each case, using eithe the DFT o the line enhance, the amplitudes of the signal components pesent (the spikes) wee appoximately the same and the backgound noise levels wee simila. n one case, howeve, whee the signal fequency was close to the peak of the coloed-noise component (Fig. l(c)), the noise output of the line enhance was about 3 db highe than that of the DFT powe spectum. The line enhance in each case was implemented with its delay set at 56 samples, which was adequate to decoelate and eliminate the coloed-noise components. Fig. illustates a diffeent kind of expeiment, whee a lageamplitude signal summed with small-amplitude signals in noise causes difficulties in detecting and/o esolving the small siguals. Fig. (a) shows the fomation of input A as the sum of white noise of unit powe with thee sinusoidal signals: the firt of powe equal to 5 at fequency fi = , the second of powe.5 at fequency f =.565, and the thid of powe.5 at fequency f3 =.875. The sampling fequency is. Notice that signal was one thousand times moe poweful than signal, and that they both wee close in fequency. is a block diagam of the system used in this expeiment, its pimay input indicated by A. ts outputs ae B and C, which epesent, espectively, the eo of the adaptive pocess and the adaptive-filte output. An additional output is the weight vecto of the adaptive fdte, compising its impulse esponse, whichcan be Fouie tansfomed to povide a tansfe function as discussed above. Fig. (c) is a linea plot of the DFT powe spectum of input A. The fequencies of the thee signals ae indicated by the aows. Notice that signal is not esolvable; it is buied in the second sidelobe of signal. Even when this spectum is taken though a Hanning window and plotted on a log scale, signal 3 is visible but smalle in amplitude than many of the sidelobes of signal, and signal is undetectable. When the DFT spectum of the eo B is taken instead, a powe spectum is obtained that clealy shows the weak signal at fequency fz. This spectum, plotted on a linea scale in Fig. (d), also shows the weak signal at fequency f3 and the boad-band backgound noise. The stong signal was totally cancelled as a esult of the adaptive pocess. The plots of Figs. (c) and (d) wee nomalized so that full scale coesponds to the lagest amplitude point of each plot. The specta of both A and B wee taken fom 8 data points. Thee was no ensemble aveaging. The 6 -weight adaptive filte cancelled the stong signal within about fwe cycles of fequency fi, i.e., within about 3 sample peiods. The CFT spectum of eo B was then taken so that the amount of data used in foming it was only slightly geate than the quantity of data used in foming the DFT spectum of inputd. The line enhance used as a stong line cancelle appeas to have impoved the capability of the DFT to esolveand detect weak signals when they ae closein fequency to a stong intefeence. Although an equivalent esult could have been obtained by the DFT alone using a wide window, it pobably would equie substantially moe data. The line enhance canbeused as an altenative to the DFT as a detecto and estimato of weak signals in noise. t also povides useful output signals and can function as a self-tuning filte, to tune in some and tune out othe signals automatically. t is a vey pomising methodology fo spectal analysis and is elated to the maximum

4 7 PROCEEDNGS OF THE EEE, JANUARY 977 PONT 8 DFT 8 WEGHT LMS FLTER ENSEMBLE SZE : 56 3,768 DATA SAMPLES USED 3,768 DATA SAMPLES USED O.Oo O.OO n.75 E.5 (b) Y n 8 PONT DFT ENSEMBLE SZE = 56 3,768 DATA SAMPLES 8 WEGHT LMS FLTER 3,768 DATA SAMPLES USED - O.OO w 8 WEGHT LMS FLTER 3,768 DATP SAMPLES USED 5 F. Cl+xl spectal. d & (Dm)-left, compaed with adaptive lin enhanc@g- ight. (a) %we he in white now. 6) be in So-pacent Mte, 5-pecent coloed noie. (c) sing& line in SO-pecent white, 5-pecent coloed noiw. WHTE NOSE TO LNE ENHANCER PRMARY NPUT ADAPTVE FLTER ( WEGHT VECTOR (b) + '3!.5 J i! (e) enhnce the detection and esolution of signals in noie. enhnce. (c) Powe density pectum of mput A. (d) Powe

5 PROCEEDNGS LETTERS 73 entopy technique [], [3]. Since the line enhance is entiely diffeent in stuctue fom the DFT, it may be moe easily implemented in some cases. We hope to publish a compehensive wok on adaptive spectal analysis in the futue, contingent on moe extensive undestanding of the behavio of the line enhance with inputs of the types descibed above. ( B. Widow, et. az, Adaptive noise cancelling: Pinciples and applications, Poe. EEE vol. 63, pp , Dec [] J. P. Beg, Maximum en topy spectal analysis, pesented at the 37th Annu. Meeting, SOC. Exploation Geophysicists,Oklahoma City, OK, 967; [3] L. J. Giffiths, Rapid measuement of instantaneous fequency, EEE Tans. Acoust., Speech, Signal Pocesdng, vol. ASSP-3, pp. 9-, Ap Realization of a Discete Fouie Tansfom (DFT) Module fo ncopoation m F R Rocessos A. POMERLEAU, M. FOURNER, AND H. L. BUJS Absmet-Fo applications equiing and in-place teatment, it is often advmtageoun to ealize Specinlpupoee computes. This pape descibes a discete Fouie fo iacopoation in fast Fouie tmsfom p- The module is highly suitable fo ed input applicztions equiing high+ m- fomatba t awbutes one point to all fnquency channels in one clock cycle. 'bis teatment is not only well suited fo the pesent technology, but appeas to be moe attactiveinviewofecenttendsin digital cicuity.. FORMULATON n the evaluation of a N point discete Fouie tansfom fo highspeed applications with FFT algoithms whee N =, each of the n passes may be computed in a manne such that each input is pocessed sequentially and its contiiution is amibuted to all fequency channels in one clock cycle [. n such a case, the basic equation to evaluate each point tansfom without ecouse to the FFT algoithm is - ki F(j)=-x f(k)exp-ni-, k=o j=o,l,.-.,-l. n this expession, f(k) is the input function and F(j) ae the Fouie tansfom coefficients Fo the geneal case, the evaluation of F(j) equies complex opeations to uniquely detemine the spectum. Fo a seialinput paallel-output cicuit, it would then equie complex multiplies and complex addes. Howeve, this numbe can be geatly educed when is a multiple of, since thee ae only ( - )/ diffeent absolute values of the eal and imaginay pats of the oots of, neglecting and. Futhemoe, when the input function is eal and only the nonedondant tems ae evaluated, an additional eduction in the numbe of mathematical opeations is possiile. The implementation of a module in which the data ae teated in magnitude, while the sign and that of the tigonometic coeffkients ae taken into account in the addes, then equies the following. ) eal addes with thei associate memoies. ) ( - )/ multiplies opeating in paallel on eal numbes. This quantity is equied since thee ae only ( - )/ absolute diffeent values in the oots of, neglecting and, if is a multiple of. 3) A elatively small quantity of multiplexes, since some of the addes have always access to the same weighted values. ) Contol units having thee main functions: a) to diect the weighted data towad the pope accumulato; b) to add o subtact the weighted data; c) to apply the inhiiit function (multiplication by zeo). Tabb shows the numbe of addes, multiplies, and multiplexes fo diffeent values of. Manuscipt eceived Januay, 976; evised Mach 9,976. The authos ae with the Depatment of Electical Engineeing, Lava Univesity, Quebec, Canada. Complex opeation: A complex opeation Q defined by a complex multiplication followed by a complex addition. TABLE NUMBER OF COMPONENTS REOURED FOR -PONT TRANSFORM mu tiplexes mu ti pl ies accumulatos quant-f inputs a- Fig.. Eight-point Fouie tansfom module.. REALZATON Figue shows the ealization of an -point Fouie tansfom. t is a seial-input paallel-output cicuit whee all the bits foming a wod ae popagated in paallel. A. Coding n each module, the input data wods ae epesented by B bits expessed in magnitude and sign. The multiplications ae done in magnitude only and the addes opeate in s complement code. Theefoe, a code convesion cicuit is not equied between the multiplie and the accumulatos. B. Multiplies As shown in Table, it equies only one multiplie fo a adix 8 Fouie tansfom. Hee the data ae epesented by E bits, while the 3