Adaptive Comb Filtering for Harmonic Signal Enhancement

Transcription

1 1124 IEEE TRANSACTIONS ON ACOUSl'ICS. SPEECH, AND SIGNAL PROCESSIKG, VOI.. ASSP-34, NO. 5, OCTOBER 1986 Adaptive Comb Filterig for Harmoic Sigal Ehacemet Abstract-A ew algorithm is preseted for adaptive comb filterig ad parametric spectral estimatio of harmoic sigals with additive white oise. The algorithm is composed of two cascaded parts. The first estimates the fudametal frequecy ad ehaces the harmoic compoet i the iput, ad the secod estimates the harmoic amplitudes ad phases. Performace aalysis provides ew results for the asymptotic Cramer-Rao boud (CRB) o the parameters of harmoic sigals with additive white oise. Results of simulatios idicate that the variaces of the estimates are of the same order of magitude as the CRB for sufficietly large data sets, ad illustrate the performace i ehacig oisy artificial periodic sigals. I. INTRODUCTION IGNALS that cosist of a sum of sie waves whose S frequecies are itegral multiples of the lowest frequecy (so-called fudametal) are said to be harmoic. May physical sigals are approximately harmoic. Examples iclude voiced speech ad other biological sigals, musical waveforms, helicopter ad boat soud waves, ad outputs of oliear systems excited by a siusoidal iput. To filter oise corrupted harmoic sigals whose parameters are ukow ad possibly time varyig, it is desirable to apply adaptive filterig. Most existig adaptive filters (for istace, i [l] ad [a]) do ot accout for the special structure of the harmoic spectrum, thus, their performace is ot likely to be optimal for such sigals. I Sectio I1 of this paper, we develop a ew adaptive algorithm, specially desiged to ehace harmoic sigals measured with additive white oise. It ca also be used as a adaptive otch filter for elimiatig harmoic iterferece from a measuremet broad-bad process. The parameters of the harmoic sigal, such as the fudametal frequecy, ad the harmoic amplitudes ad phases, are assumed ukow ad are estimated by the algorithm. The proposed algorithm is of recursive predictio error (RPE) or recursive maximum likelihood (RML) type (see [3] ad [4]) ad uses several ostadard features to improve its performace. Mauscript received February 5, 1985; reviscd March 29, The work of A. Nehorai was supported i part by the Natioal Sciece Foudatio uder Grat DCI A. Nehorai was with Systems Cotrol Techology, Ic., Palo Alto, CA He is ow with the Departmet of Electrical Egieerig, Yale Uiversity, New Have, CT B. Porat is with the Departmet of Electrical Egieerig, Techio- Israel Istitute of Techology, Haifa 32000, Israel. IEEE Log Number IEEE The filter appearig i the algorithm cosists of a cascade of secod-order ifiite impulse respose (IIR) sectios, ad has a comb-type frequecy respose. These secod-order sectios are parametrized by a sigle variable: the estimated fudametal frequecy of the harmoic sigal. This yields a estimatio algorithm for the fudametal frequecy which is more efficiet ad accurate tha others (for example, [l] ad [2]) which view the siusoidal frequecies as idepedet. I additio, simi- larly to [SI, special costraits are imposed o the filter coefficiets, givig arbitrarily arrow-bad-pass filters for each harmoic. This improves the performace i compariso to other comb filters of fiite impulse respose (FIR) type (for example, [6] ad [7]), which typically require a large umber of coefficiets to obtai arrow passbads. The harmoic amplitudes ad phases are estimated separately, coditioed o the estimated fudametal fre- quecy. The algorithm is computatioally efficiet; the umber of operatios it requires per time sample is proportioal to the squared umber of the filtered harmoics. Sectio I11 is devoted to performace aalysis of the algorithm. We first derive the Cramer-Rao boud (CRB) for estimatig the parameters of harmoic sigal embedded i white oise, The results of Mote Carlo simulatios are preseted, idicatig that the variaces of parameter estimates are of the same order of magitude as the CRB for a sufficietly large umber of data, but the algorithm is ot fully efficiet i geeral. The algorithm is also tested agaist periodic sigals with a ifiite umber of harmoics. The results demostrate the applicabil- ity of the algorithm to the filterig of artificial periodic waveforms, such as square waves, saw-tooth, triagular waves, ad others. Sectio 1V summarizes the paper. 11. THE ADAPTIVE COMB FILTER This sectio derives the proposed adaptive comb filter (ACF) for harmoic sigal ehacemet ad spectral estimatio. The subsectios below cosider the followig subjects: I) the special model ad parameterizatio of the harmoic sigal with additive white oise, ad a geeral descriptio of the algorithm; 2) the error regressio ad gradiet; 3) the recursive algorithm for estimatig the fudametal frequecy ad ehacig the harmoic sigal; ad 4) the recursive algorithm for estimatig the amplitudes ad phases of the harmoic compoets.

2 NEHORAI AND PORAT: ADAPTIVE COMB FILTERING 1125 x(t) WE Wo't) Y(t) j ACF 3 RLS Fig. 1. Block diagram algorithm. proposed of the k= 1 A. The Model Let x(t) be the harmoic sigal whose parameters are summetry the of estimated. to Due Thus, be X(t) = c ck Si (kw,t + 4 k) (1) k= 1 w, is the fudametal frequecy, ad ck ad c$~ are the amplitude ad phase of the kth harmoic compoet of x (t), respectively. The umber is the assumed umber of harmoics i x (t). I cases the actual sigal cosists of a ifiite umber of harmoics, we trucate the ifiite sum at harmoics is chose so that the eergy i the remaiig harmoics is sufficietly small. The remaiig harmoics will be cosidered as part of the, oise, cf. (2) below. The white oise corrupted measuremet at time t is assumed to be Y ( 0 = + u (t) = C ck si (kw,t + &) + u(t) (2) k=l u (t) is a zero-mea white oise with variace u2, We assume that the parameter vector A(q-') is the 2th-order polyomial i the uit delay operator q-', whose zeros are o the uit circle at the sie wave frequecies. We should stress that sice the ulls of A(q-') are at the sie wave frequecies, it does ot follow from (4) that y (t) = u (t). Also ote that A(q-') has 2 poles at the origi which are ot importat for the preset discussio as they are cacelled out i (4). Sice i special our case, the zeros of A(q-') itegral at are multiples of the fudametal frequecy, this polyomial ca be-writte as A(q-') = (1 + akq-' + q-2) (5a) (IIk = -2 COS ko,. A(q- ') A(q-') = 1 + qq-' aqw alq-2+l + q-? (5b) (6) The whiteig filter of y (t) is required to be stable, ad its output has to be u (t) whe excited by y (t). By ispectio, we fid from (4) that the whiteig filter ca be approximated by This fuctio satisfies the stability requiremets whe p < 1. It is characterized by 2 zeros o the uit circle at {e i jko, 1 I k 4 ) ad 2 poles havig the same phase agles at the zeros ad their radii are all p, that is, the poles are at (pe'jkoo, 1 5 k I ]. The parameter p is e = c1 -. c, 4'. - 4~~ (3) chose by the user; typical values are From (4),(3, ad (7), we observe that the error sigal is ukow. A maximum likelihood estimatio of 0 would require a oliear algorithm of dimesio To 4 = wq-9 Y (t) (8) simplify this situatio, we shall divide our algorithm ito approximates the oise u (t) whe p is sufficietly close to two cascaded parts, as is illustrated i Fig. 1. As show oe. i the figure, the first part of the algorithm is the recursive Note that H(q-') ca be used as a otch filter for elimpredictio error adaptive comb filter, which estimates the iatig harmoic disturbace added to a desired broadfudametal frequecy w, ad ehaces the harmoic bad process. The reverse operatio of extractig the harcompoet x(t) of y(t). Based o these results, the ammoic sigal plitudes { Ck} ad phases {&} x (t) from the oisy measuremet y (t) is obare estimated (after paramtaied by eter trasformatio, see later) usig a liear recursive least squares (RLS) algorithm. Let us discuss the whiteig filter of y(t). For this we recall that for sie waves (ot ecessarily harmoics) with additive white oise, y (t) ca be show to satisfy the K is the zero frequecy gai of H(q-'). relatioship [SI Oe of the mai advatages of our algorithm ca be observed already at this poit by otig that the whiteig 4 - l ) Y(t) = A(q--]) u(t) (4) filter H(q-') is modeled by a sigle parameter, amely, the fudametal frequecy of x@). This leads to a computatioally efficiet algorithm for estimatig w,, as is discussed i the sequel. B. fie Error Regressio ad Gradiet The proposed algorithm uses the model (7), (8) ad adjusts the estimate of w, so as to miimize the cost fuctio

3 1126 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-34, NO. 5, OCTOBER 1986 N v, = c E2(t) r=1 N is the umber of data samples. To compute the error ~ (t) we eed to fid its regressio expressio. Let a = [al - * a,] T (1 1) ad let H be the Hakel matrix superscript T deotes traspose. From (7) ad (8) 1 oe obtais the differece equatio E(t) = y (t) + y (t - 2) - p2"c(t - 212) - (p'(t>a (12) H = [ ad CP(~) + ['~l(t) * * ~(t)l' (13a) ~..:-::::~ j p]' *-. Pm- 1 (20) Assume that { Xi, 1 5 i 5 m} are all distict. The The gradiet of ~(t) with respect to u, ca be foud as follows. Let i =. The derivatives -ae(t)/aai ca be show to be give (see also [5]) _- sew - Pi (0 aai A(P~-*) A = 'PFi(t), by (15) i.e., -&(t)/aai are filtered versios of {pi(t)}. To compute the derivatives aai/au, i (14), we shall use the relatioship We ow apply the results of lemma 1 to our case. Due to the symmetry of the polyomial A(q-'), we oly eed to cosider the derivatives of the first coefficiets. Usig (20) ad (21), we get I;;; 1 -- = -[OiHIV (22) -. I.- ah 2 [0 i 7?] is a X 2 matrix whose leftmost colums are zero ad { Xi, 1 5 i 5 2) are the zeros of the polyomial zz~(z-'), amely, h2k- = eikuo = e-jkmo, 1 5 k 5. (17) The expressio i the right-had side of (16) ca be computed usig the followig lemma prove i Appedix A. Lemma I: Let P(z) be a polyomial of order m i z, viz., P(2) = zm + plzm-' + * pm = (Z - X,)(Z - X,) * * (Z - Xm). (18) Let V be the Vademode matrix The matrix 7 i (22) is similar to (19), but here m = 2 ad Xi are give i (17). Now let The

4 NEHORAI AND PORAT: ADAPTIVE COMB FILTERING 1127 = 2 k si kio, k= 1 si [i( + l)wo] - - (,+ 1) cos i(2 l+ l)iw, si2 (?) si (2) Combiig (22)-(24), I= we ow have I 1 or C. The Adaptive Algorithm for 2(t) ad Go@) With the results above, we ca ow apply the geeral RML or RPE approach to our case. The resultig recursios of the algorithm for ehacig the harmoic sigal ad estimatig the fudametal frequecy are summarized below. Note that i each recursio, the latest available estimates of a, ad A(q-') replace their value i the expressios of the previous subsectio. The explaatios of the algorithm ad its special features are discussed below. 1) The RPE Adaptive Comb Filter: Desig Variables:, X(l), X,, y(l), r(l), p(l), pot P(W). Iitializatio: G,(O), pi(0) = pfi(0) = 0, i = 1, * -,, y(-i) = 0, i = 1, * -, 2. Nomial Values: = umber of bad-pass filters X(l) = , X, Ey2(t)/100 = 0.98, y(1) = 1, r(1) z p(1) = 0.8, po = 0.98, p ( ~ = ) (see also Mai Loop: commets below). a, = 1. Thus, $(t) ca be computed by first evaluatig the sequece {xi} i (25), the computig { aai/ aoo) by (27), ad fially, G0(t) = Go(t - v = [V, * vy - 1

5 1128 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-34, NO. 5, OCTOBER 1986 fi(t) = si [i( + 1) &,(t)] 2 si2 (T) i&,(t) si (y) i- I Vi(t + 1) = c dl@) ai-&) I=O p(t + 1) =' [p& + 1) * p,(t + l)lt VFQ + 1) = [PFl(f + 1) * PFO + 1>IT V(t + 1) = [V,(t + 1) * * * V,(t + l)lt $(t + 1) = pgt + 1) V(t + 1) X(t + 1) = X,X(t) + (1 - X,) y(t + 1) = y(t)/[y(t) + + 1)1 r(t + 1) = r(t) + y(t + 1)[g2(t + 1) - r(t)] (30s) P(t + 1) = POP@) + (1 - Po) P(W). (300 We ow discuss some of the special features which were used to improve the performace of the algorithm. Some of these features were used also i the algorithm of [SI, ad therefore, they are discussed oly briefly. 2) Time-Varyig Pole Moduli: The recursio (30q) for the pole moduli yields a time-varyig p(t) istead of a costat value of the pole moduli. Note that p(t) grows expoetially from p(1) to p(m) with a time costat To = 1/( 1 - p,). Thus, at the begiig of the data processig, the otches of the estimated whiteig filter are wide ad they become arrower as time goes o. This improves the sesitivity of the algorithm to the presece of harmoics ad icreases its covergece rate. To see this, ote that whe the iitial coditios of the algorithm are poor, ad if the otches are too arrow, the iput har- moics are likely to appear at the flat part of the estimated whiteig filter trasfer fuctio. I such cases, the gradiet of the algorithm vaishes ad the algorithm may ot coverge to the desired trasfer fuctio. The omial iitial value p(1) = 0.8 was chose as the smallest value of pole modulus which still has sigificat effect o the otch badwidths. The value of p(m) is chose as a tradeoff betwee accurate asymptotic performace (p(m) as close as possible to oe) ad robustess as well as trackig capability of time-varyig frequecies (smaller values of p(00)). 3) Updatig the Gai ad Trackig Time-Varyig Parameters: The recursio (30r) for the gai y(t) ad the related recursios (30q) ad (30s) are adapted from [4] for statioary sigals. Equatio (30q) of updatig X(t) iflueces the cost fuctio at the trasiet phase ad improves the covergece rate of the algorithm. The above omial values of X( 1) ad X, were foud to yield the best results i our simulatio experimets. For SNR = 0 db, it was useful to take X(1) = 0.45 ad for SNR 2 4 db X(1) = With the above recursios ad for statioary sigals, the resultig step size y(t)/r (t) asymptotically approximates the iverse secod derivative of the cost fuctio evaluated at &,(t - 1). For this reaso, the algorithm ca be viewed as a recursive Gauss-Newto type. To allow trackig of time-varyig parameters, it is commo to choose y(t) = yo, yo is a small positive umber. This correspods to a expoetially weighted cost fuctio characterized by time costat lly, (see [4]). 4) Stability Moitorig: The aalysis of [3] ad [4] shows that for RPE algorithms, oe must check at each time update whether the curret parameter estimates correspod to a stable filter. Otherwise, the parameter estimates must be projected back ito the stable model set. However, i our case, the estimated whiteig filter H(q-') is stable by costructio, ad therefore, stability moitorig is ot eeded. 5) Estimatig the Number of Harmoics: For practical cases the umber of harmoics is ukow, we ote that ca be estimated as follows. Apply the algorithm give above ad the cascaded algorithm of Sectio 11-D-1 with a overestimated umber of harmoics based o prior iformatio o. The ca be estimated o lie usig the umber of harmoics yieldig sigificat amplitudes, i.e., larger tha some predetermied threshold. The use of a uderestimated usually adds a distortio to the filtered sigal. The faster the spectrum evelope of x(t) decreases, the smaller the distortio will be. The use of a overestimated usually adds some oise to the output. 6) A Modified Versio: The algorithm above directly estimates the fudametal frequecy of x (t). Aother versio of the algorithm first estimates the coefficiet a1 [see (5b)l ad the the fudametal frequecy ca be evaluated usig the relatioship a, = cos-' (-cu1/2). This modified versio was foud to be more robust tha the origial algorithm, especially whe most of the eergy is cocetrated i the first harmoic of x(t). (See the examples bf the ext sectio.) This result ca be explaied heuristically by the fact that i such cases, the error gradiet teds to be more liear with respect to al tha with respkct to w, (cosider, for istace, the special case of a sigle sie wave). The modified algorithm is obtaied by

6 NEHORAI AND PORAT: ADAPTIVE COMB FILTERING 1129 multiplyig +(t) by -= dg,(t) 1 d&,(t) 2 si G,(t) 7) Covergece Aalysis: The covergece of RPEtype algorithms has bee aalyzed i [3] ad [4]. The proof that the assumptios of this aalysis hold for our case is 'techical, ad we omit its detail. Amog the results of the cavergece aalysis, we briefly metio that i the model complete case (i.e., whe the model is cosistet with the data) ad white Gaussia oise, the parameter estimates are asymptotically ubiased, ormally distributed, ad achieve the Cramer-Rao boud (i.e., statistically efficiet). Fially, we ote that the iitializatio r(1) = Ey2(t)/ 100 is oly a approximatio implicitly based o recommedatios i [4, p As oted i [4], i practice, it is commo to take r(1) = 1/C C is a large umber. As C becomes' larger, the value of c3,(0) becomes oly margialfy importat. Iitializatio: C(0) = 0, P(0) = (l0o/snr)z2,. Mai Loop: r(t) = [si Got * * - si &t, cos Got - * * cos ij,t] e(t) = a(t) - r*(t) C(t - 1) (354 (35b) (35c) C(t) = $(t - 1) + P(t) c(t) e(t). (354 I the above algorithm, the variable X(t) is updated similarly to the algorithm i (30). The amplitudes ad phages ca'be evaluated by (35f) D. Estimatig the Harmoic Amplitudes ad Phases I some applicatios, it is desired to estimate the amplitudes ad'phases of the harmoic compoets i a$$tio to the fudametal frequecy. The algqrithm below is suggested for this purpose. To derive the algorithm, we shall first assume that w, is kow, ad apply proper trasformatio to, the desired parameters. We have /I y (t) = C ck si (kw,t + &) k= 1 + u (t) = x ( gk COS kw,t + hk Si k0,t) + U(t) (32) k= 1 I the ext sectio, we will show that the expected relative error of 2, is sigificatly smaller tha those of the amplitudes ad phases whe obtaied by a maximum likelihood estimator. This implies that by replacig w, by its estimate &, the accuracy of the amplitude ad phase estimates is ot expected to deteriorate sigificatly. To improve the estimatio accuracy of this procedure, it is also useful to replace the raw data y(t) by the filtered data a(t) [see (35b)l. Here it should oted be that the oise i a(t) is colored, which may cause some bias i the estimates of the amplitudes ad phases. However, sice this oise is much smaller tha i the raw data y (t), the overall accuracy of this procedure is still better tha if y(t) were used. The iitializatio P(0) = (100/SNR) Z2, ZZ is gk = ck si (334 the 2 X 2 idetity matrix, is oly a approximatio based o recommedatios i [4, p I practice it is hk = ck COS &. (33b) commo to take P(0) = C ZZ C is a large umber This ca be writte as (see r41). 2) Amout of Computatios: The overall algorithm for y (t) = [si w,t * - si wot, adaptive estimatio of the harmoic sigal parameters requires a umber of operatios proportioal to 2 per time cos w,t cos w,t] q + u(t) (34a) sample. Most of the computatio load is take by the computatio of A(q-', t) i (~OC), computatio of the gradiet vector V(t + 1) i (301), ad updatig the RLS gai = [SI * * g, hl * (34b) vector P(t) c(t) i (35c, d). To save computatios, the Equatio (34a) implies that if w, were kow, a RLS algorithm (35) of amplitude ad phase estimatio does ot algorithm could be used to estimate the sequeces { gk} have to be implemeted whe oly c3, ad R(t) are ad (hk}. The the desired amplitudes { Ck} ad phases eeded. { &} could be foud by a simple trasformatio from rectagular to polar coordiates. Sice the fudametal fre PERFORMANCE EVALUATION quecy w, is ukow, we ca replace it by its estimate I this sectio we evaluate the performace of the proobtaied from the algorithm of Sectio II-C. This procedure is summarized below. posed algorithm by several Mote Carlo simulatio rus, ad make comparisos to the Cramer-Rao boud (CRB). I) The RLS Algorithm for the Harmoic Amplitude ad The we illustrate.the covergece of the algorithm for Phase Estimatio: sigals with a large umber of harmoics. Fially, we

7 1130 IEEE TRANSACTIONS ACOUSTICS, SPEECH, SlGNAL AND PROCESSING, VOL. ASSP-34, NO. 5, OCTOBER 1986 illustrate the algorithm s performace for artificial sigals with a ifiite umber of harmoics. A. The Asymptotic CRB for Harmoic Sigals Asymptotic formulas for the Cramer-Rao bouds of the parameters i the model (1) are derived i Appedix B. These formulas are valid whe N, the umber of data samples, satisfies N >> l/uo. The mai results of Appedix B are summarized as follows: f r quecy f, = 0.08, ad u(t) is a zero mea uit variace white Gaussia oise. The harmoic amplitudes {ck) were chose to yield both the desired sigal-to-oise ratio give by SNR(dB) = 10 log (XiZl C~I2o2) ad the desired spectrum. To make the examples as close as possible to practical situatios, the spectrum evelope was chose such that its decrease i gai was -6 db per octave, or harmoic powers proportioal to 114. This type phases are ukow ad amplitudes are kow or ot phases are kow ad amplitudes are kow or ot 20 Var (&) 2 - i all cases (36c) N Var (&) I y = C k2ci. (360 k= 1 Remarks: Note the quatity y/2a2 = Et=, k2ci/202 appearig i the CRB of the fudametal frequecy, is ot the SNR, but a ehaced SNR, i which the various harmoics are weighted both by their eergies ad by their relative frequecies. I particular, higher frequecy harmoics have larger ehacemet tha low-frequecy harmoics. It is of iterest to ote that the results (36a) ad (36b) ca also be writte as f 12 I Var (Go) I I SNR w:ffn3 3 SNR w&n3 SNR = X, = Ci/(202) ad the effective badwidth = E;=, k2c$z, =, C:. The factor 1/(SNR w$f) appears also i the CRB o time delay estimatio (see e.g., [IO]). B. Mote Carlo Simulatios The followig examples illustrate the bias ad variace of the estimated parameters, compared to the CRB. The iput data were 5 y(t) = c C, si 2~0.08kt + u(t), (37) k= 1 i.e., the umber of harmoics = 5, fudametal fre- fudametal frequecy is ukow ad amplitudes are kow or ot fudametal frequecy is kow ad amplitudes are kow or ot of evelope is ofte used to model the voiced speech spectrum evelope geerated by the vocal tract. The algorithm was applied, usig a DEC VAX computer with double precisio arithmetic, to estimate the parameters of the sigal (37) with desig variables p(]) = 0.8, po = 0.98, p(m) = 0.96, ad X, = For SNR = 0 db, we used X(l) = 0.45 ad for larger SNR s values X(1) = As was explaied earlier, sice i this case the umber of harmoics is relatively small ( = 5), ad most of the eergy of x(t) is cotaied i its first harmoic, it was was useful to update the parameter &1 rather phases are ukow ad amplitudes are kow or ot phases are kow ad amplitudes are kow or ot tha &. This was foud to improve the robustess of the algorithm. The algorithm was tested uder differet data legths ad sigal-to-oise ratios. Each of the statistics below was computed from 100 idepedet Mote Carlo experimets. The parameter estimates Gl(t) were set to values such thatl(0) was equal to 0.05 at the begiig of each experimet. Table I summarizes the resultig sample statistics of the ormalized fudametal frequecy estimates for the sigals i (37). I the simulatios, we ecoutered some realizatios which gave outlier performace, i.e., the estimates were ot close to the typical behavior. Out of 100

8 NEHORAI AND PORAT: ADAPTIVE COMB FILTERING 1131 TABLE I STATISTICAL RESULTS FOR THE FUNDAMENTAL FREQUENCY ESTIMATE OF THE SIGNAL (37) N SNR Stadard (Samples) (db) Bias X10-6 Deviatio XlO- CRB XlO (7) (3) (6) (3) fo TABLE 11 STATISTICAL RESULTS FOR THE AMPLITUDE ESTIMATES OF THE SIGNAL (37) C1 c2 c3 c4 c5 Stadard N SNR Stadard Stadard (Samples) (db) True Bias Deviatio True Bias Deviatio True Bias Deviatio True Bias Deviatio True Bias Deviatio CRB , TABLE 111 STATISTICAL RESULTS FOR THE PHASE ESTIMATES OF THE SIGNAL (37) 61 $ Stadard Stadard Stadard Stadard Stadard N SNR Bias Deviatio CRB Bias Deviatio CRB Bias Deviatio CRB Bias Deviatio CRB Bias Deviatio CRB (Samples) (db) x ~ O - ~ x10-l x10-l X10-2 x10-l x10-l x ~ O - ~ x1o-i x10- x ~ O - ~ ~10- x10-l x ~ O - ~ x10- x , experimets i each frequecy estimate, the umber of Usig the estimates of the fudametal frequecy, the outliers is idicated i paretheses i the table. These secod part of the algorithm was used to estimate the harcases were elimiated from the statistical computatios. moic amplitudes ad phases for the same sigals above. This is justified by the fact that the Cramer-Rao boud is Table I1 presets the resultig sample statistics for the derived uder small error assumptio. The decisio o amplitude estimates. The results of the table show that the outlier was based o a arbitrary threshold of i amplitude estimates have a egative bias. This bias bethe error of the fudametal frequecy estimates. The comes larger for the higher frequecy harmoics whose umber of cases i which this happeed decreased with power i these examples is relatively small. The egative the data legth ad sigal-to-oise ratio. bias is due to the error i the fudametal frequecy es- Table I presets also the CRB of the fudametal fre- timates, which teds to atteuate the sie wave ampliquecy for the above examples. The ratio of the actual tudes. However, the bias decreases with the data legth stadard deviatio to the CRB appears to decrease as N ad sigal-to-oise ratio. icreases, at least for high SNR s. However, this ratio is Table I11 presets the sample statistics of the phase esot close to oe, so the estimates are ot fully efficiet. timates for the above sigals, ad the CRB computed by We applied sigificace tests to the bias (usig the t the square root of (36d). Similarly to the previous estistatistic) ad foud that w, is ubiased with cofidece mates, we observe that the stadard deviatios of the level 0.95, i all cases except for N = 200 ad SNR = 0 db. The CRB was computed usig (36a). phases decrease with the data legth ad the sigal-tooise ratio, but they do ot approach the CRB i geeral.

9 1132 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-34, NO. 5, OCTOBER 1986 L o Normalized Frequecy Normalized Frequecy.0851 I,060 1 t ou- ' :O Tlme (samples) i,(tj , I.-.o Normalized Frequecy I 1 Applicatio of sigificace tests cofirmed that the phase estimates were ubiased with cofidece level 0.95 for all cases, except for N = 200 ad SNR = 8 db, 3, was biased. C. Covergece Examples The followig examples demostrate the covergece of the algorithm by two simulatio rus for sigals with a differet umber of harmoics. "I v..o.i Normalized Frequecy Fig. 2. Estimatio results for harmoic sigal with additive white oise. = 5, SNR = 0 db. (a) Power spectral desity of the oise-free harmoic sigal. (bj Power spectral desity of the measuremet. (cj The fudametal frequecy estimate. (dj Trasfer fuctio of the coverget comb filter at f = (e) Power spectral desity of the algorithm's output. Fig. 2(a)-(e) presets the results for the sigal with five harmoics ad fudametal frequecy f, = 6 ad SNR = 0 db. The harmoic amplitudes decrease by 6 db/octave. Fig. 2(a) ad (b) depicts the power spectral desity of the oise-free sigal x(t) ad the oisy measuremet y(t), respectively. The algorithm was applied with iitial coditiofo(0) = 0.05 ad desig variables as described above i the Mote Carlo simulatios, except that p(00) was set to This larger value was chose to achieve

10 NEHORAI AND PORAT: ADAPTIVE COMB FILTERING 1133 higher reductio of oise. Fig. 2(c) shows the fudametal frequecy estimate of the algorithm versus time. We ote that for larger sigal-to-oise ratios, the covergece was usually faster ad smoother. Fig. 2(d) illustrates the magitude of the correspodig comb filter trasfer fuctio after covergece give by G(ej., t) = 1 - t2(eiw, t)/~(t) (38) at t = 2000, ad t) is the estimated whiteig filter of the measuremet, ad K(t) its zero frequecy gai. At t = 2000, the bias i the fudametal frequecy was 8.00 X Fig. 2(e) depicts the power spectral desity of the filtered output R(t) after covergece. This figure was draw by applyig FFT to the last 1024 data poits of R(t) from t = 977 to t = The ext example illustrates the ability of the algorithm to filter oisy sigals with a relatively large umber of harmoics. For this purpose, the harmoic sigal was with 20 harmoics, fudametal frequecy & ad SNR = 0 db. Similarly to the above example, the harmoic amplitudes were adjusled to yield a spectrum evelope of -6 db/octave. The algorithm was ru with the same desig variables as before ad iitial coditio A,(O) = Sice, i this example, the umber of harmoics was relatively large, the algorithm that updates ;,(t) was used. The results are preseted i Fig. 3. The bias i the fudametal frequecy estimate was X at t = The result of the last example is iterestig as it demostrates that the algorithm is applicable for sigals with a large umber of harmoics ad large orders of A(q- ) ad A(pq- ). This is remarkable i compariso to the behavior of most existig system idetificatio algorithms whose usual largest possible order is betwee 5-10 for these types of SNR coditios ad ukow iput. The reaso for this good behavior of the algorithm is our use of a sigle parameter model i the algorithm for estimatig the fudametal frequecy. The power spectral desities of Figs. 2(b) ad 3(b) may lead oe to thik that FFT ca be used to estimatef, easily, as they clearly show the harmoic distributio of the sigal. However, this is oly because&, was chose to be a submultiple of the umber of FFT poits (X, = &) i this simulatio. I practice, usuallyf, does ot satisfy this coditio ad the the leakage betwee the FFT bis makes it prohibitive to estimate L), especially i low SNR s. Moreover, the FFT is a batch method, which caot be applied adaptively i cotrast to the proposed algorithm. D. Ehacemet of Periodic Sigals As metioed i Sectio 11, the proposed adaptive comb filter is also useful for ehacemet of periodic sigals with a ifiite umber of harmoics by trucatig the low-eergy high-frequecy harmoics. This is illustrated by the followig example. This example is of a triagular wave whose period is 12 samples ad additive white oise with SNR = 0 db. The modified algorithm that updates &,(t) was applied with five passbads, desig variables p(1) = 0.8, po = 0.98, p (a) = 0.995, X(l) = 0.45, X, = 0.98, ad &,(O) = -27r cos The above relatively large value of p(-) was chose to achieve high reductio of oise. Fig. 4(a)- (c), respectively, shows the origial triagular wave sigal, the oisy measuremet, ad the algorithm filtered output after covergece, from t = 1900 to t = At t = 2000, the fudametal frequecy bias was 1.75 X We have also.applied the algorithm to filter a square wave sigal plus white oise. The resultig filtered sigal was slightly worse tha i the triagular wave. This is explaied by the fact that the spectrum evelope of a square wave decreases by 6 db/octave, while that of a triagular wave decreases by 12 db/octave. Thus, the trucatio of higher harmoics causes larger distortio i the square wave tha i the triagular wave. We omit the results due to space limitatio. Sigals such as the above square wave ad triagular wave which have so-called symmetric half-wave (i.e., x(t) = -x(t - T/2) T is the period) are characterized by zero odd idexed harmoics. Whe it is a priori kow that the harmoic compoet of the iput is of this type, it is desirable to apply a special comb filter with oly eve idex passbads. The algorithm of Sectio I1 ca be easily modified for such applicatios. IV. CONCLUSION I this paper, we have preseted a ew adaptive algorithm for harmoic sigal ehacemet ad parametric spectral estimatio. Its computatioal efficiecy advatage stems from the separatio of the solutio ito two cascaded parts, as is illustrated i Fig. 1. The first part ehaces the harmoic sigals ad estimates its fudametal frequecy. The secod part estimates the harmoic amplitudes ad phases. I this way, the oliear part of the algorithm ivolves oly oe parameter-the sigal fudametal frequecy. This eables the algorithm to workwith sigificatly larger order ARMA polyomials tha i geeral system idetificatio schemes. Aother improvemet compared to geeral RPE algorithms is the stability of the filter. Thus, stability moitorig, which is usually ecessary for geeral RPE algorithms, is ot eeded i our scheme. Simulatio results idicate that, for sufficietly large data sets, the variaces of the estimated parameters are geerally of the same order of magitude as the Cramer-Rao boud, but the algorithm is ot fully statistically efficiet i geeral. Other variats of the algorithm preseted i this paper ca be derived. For example, istead of usig costat badwidth passbads, oe ca use so-called costat-q passbads, i.e., let the badwidth of each passbad be proportioal to its cetral frequecy. Aother possibility is to iclude oly certai harmoics, say, the odd oes, based o a priori iformatio about the sigal. The ubiquity of harmoic sigals both i ature ad i artificial eviromets suggests a wide variety of potetial applicatios of the proposed algorithm. These iclude ehacemet of oisy biological sigals, such as voiced

11 1134 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-34, NO. 5, OCTOBER 1986 r o @ Normalized Frequecy Normalized Frequecy (a) (b).5 ::::I,,,, o Time (samples) f,(t) Frequecy Normalized V..o Fig. 3. Estimatio results Normalized Frequecy (e) for harmoic sigal with additive white oise. = 20: SNR = 0 db. speech ad heart wave forms. Aother importat appli- oise, e.g., to estimate a radar s modulated pulse repeticatio is trackig of propeller-egie vehicles, for i- tive frequecy by a passive radar. Further research has to stace, submaries ad helicopters. The alterative use, be carried out to ivestigate the performace i such apas a adaptive harmoic otch filter, is useful for har- plicatios. moic oise cacellatio. This may be used, for example, to reduce the oise i helicopter cockpit commuicatio. Our aalysis has provided ew results for the CRB of the parameter estimates of harmoic sigals i additive Other applicatios are whe there is a eed to estimate white oise. Oe iterestig result is that the power of artificial harmoic sigal parameters i the presece of each harmoic appearig i the CRB of the fudametal

12 NEHORAI AND PORAT: ADAPTIVE COMB FILTERING 1135 tively low. Clearly this is due to the fact that the highfrequecy harmoics yield more iformatio o the fudametal frequecy. The CRB results of this paper are geeral ad potefially useful i evaluatig the performace of other harmoic spectral estimatio algorithms. A iterestig topic of further research is the evaluatio of the optimal values of p,, p(l), p(m), X(l), ad X,. This subject is curretly beig ivestigated I Time (samples) (a) APPENDIX A PROOF OF LEMMA 1 Itroduce the polyomials k Pk(z) = pizk-i i=o OIkSWZ (All po = 1. Note that P,(z) = P(z) ad that Hece, m- I ' I Time (samples) 3. I 2. i m-1 = zpm-l(z) - X? - c p ixy i=l = ZP, - l(2) + pm = P(z). (A3) Dividig both sides by (z - X,) we have Time (samples) (c) Fig. 4. Ehacemet results for triagular wave with additive white oise. = 5, SNR = 0 db. (a) The oise-free sigal. (b) The oisy measuremet. (c) The filtered output. This ca also be writte (A51 frequecy estimate is multiplied by its squared relative frequecy. Thus, higher frequecy harmoics have ehaced SNR i the CRB. This mathematical result gives a ew explaatio to a kow physical pheomeo by which high-frequecy, harmoics i speech are importat to its itelligibility although their eergy is usually rela- I/ ad H were defied i (19) ad (20), respectively. Next, ote from (18) that Hece, substitutig z = X1, X2, + *, X, we get

13 1136 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-34, NO, 5, OCTOBER 1986 The CRB for 8 is give by the iverse of J. It should be oted that a expressio similar to (B3) was used i [9] for multiple sie waves with idepedet frequecies usig complex sigal formulatio. Let O(x) deote a quatity that is asymptotically liear i x, i.e., such that L o 1 ( 4 the ozero etry at the right-had side appears at its kth row. Hece, VT.. = -diag II (A, - A,>; 1 I k I m (i+k i= 1 Premultiplyig by (VT)- ad makig use obtai the desired expressio (21). (A9) of (A6), we Also let O( 1) deote a bouded quatity. The we have for all kw, such that ko,/27r is ot a iteger ad N >> 1 /w, N E ti si kw,t = O(Ni); i = 0, 1, 2,. * 0-35) t= 1 N 2 ti cos kw,t = O(Ni) i = 0, 1, 2 * -. (B6) t= I Assume further that w, ad i (1) are such that kw,/2 is oiteger for all 1 I k 5 2. (Note that this ad the above assumptio o kw, are always satisfied for a fiite umber of harmoics which are sampled at a rate larger tha the Nyquist rate.) The it ca be show that APPENDIX B THE ASYMPTOTIC CRAMER-RAO BOUND FOR THE PARAMETERS OF HARMONIC SIGNALS IN WHITE NOISE The Cramer-Rao is a geeral lower boud o the error covariace of ubiased estimators. This appedix derives the asymptotic Cramer-Rao boud for the parameter estimates of harmoic sigals i additive white oise. The assumed model is as i (1) parameterized by 8 defied i (2). To derive the CRB for this model, we will first derive the correspodig boud for the alterative model (32) parameterized by 8 = [w, g1 * g,, h,. * - h,lt (B1) {gk} ad {hk} were defied i (33a) ad (33b), respectively. The desired CRB for (1) ad 0 will the be evaluated usig the relatioship betwee the two models. Uder the above model, the joit probability desity of the measuremets { y(1) y(n)} is From (B2) the geeral expressio for the (k, Z)th elemet of the Fisher iformatio matrix is Hece, the Fisher iformatio matrix for the trasformed parameter vector 8 of (Bl) ca be writte as 1 J=-(J J2)

14 NEHORAIANDPORAT:ADAPTIVE COMB FILTERING 1137 I our case, from (33) these derivatives are (B 16) 6kl is the Kroecker delta fuctio. Substitutig (B23) ad (B21) ito (B22), we obtai, after straightforward but legthy computatios, the result (B17a) CRB(@ = 2u2 jl i 0 (B17b) (B17c) yn2 IN The matrix J;' ca be easily computed to yield - 1 Now from (B15) we get J-' = 2u2[J;' - J&.p + J7'J2J;'J*J;1 - * ] (B 19) provided the ifiite sum coverages. From (B16) ad (B18), we have for sufficietly large N Hece, for large N, J;'J2J;' is egligible compared to 5; Similarly, we ca prove for the rest of the terms i (B19); hece, for large eough N, the CRB for 8 is CRB@ = 2~~5;' (B21) J;' was give i (B18). To evaluate ow the CRB for the origial parameter vector 8 i (3), we will use the followig geeral relatioship: T CRB(8) = E] CRB($) [$] the (k, Z)th etry of [a19la8] is aok/a8,. (B22) The Cramer-Rao bouds u = [l, 2 * * IT (B25a) D = diag (l/c:}. (B25b) o the fudametal frequecy w,, the amplitudes (ck}, ad phases (4k}) ca ow be foud from the diagoal etries i (B24) as summarized i (36). The structure of CRB(0) implies that the asymptotic CRB of w; ad (4R) are idepedet of whether the am- plitudes are kow or ukow, ad vice versa. From (B24) we ca also fid that the asymptotic CRB of o, give the phases is Usig a similar otatio we also fid that CRB (4(w0) = 202 D. N (B27)

15 1138 IEEE TRANSACTIONS ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASP-34, NO. 5, OCTOBER 1936 r REFERENCES B. Widrow ad J. R. Glover, Jr. et al., Adaptive oise cacellig: Priciples ad applicatios, Proc. IEEE, vol. 63, pp , Dec B. Freidlader, A recursive maximum likelihood algorithm for ARMA lie ehacemet, IEEE Tras. Acoust., Speech, Sigal Processig, vol. ASSP-30, pp , Aug L. Ljug, Aalysis of a geeral recursive predictio error idetifi- catio algorithm, Autoratica, vol. 17, o. 1, pp , Ja L. Ljug ad T. Soderstrom, Theov ad Practice of Recursive Ide- tljicatio. Cambridge, MA: M.I.T. Press, A. Nehorai, A miimal parameter adaptive otch filter with costraied poles ad zeros, IEEE Tras. Acoust., Speech, Sigal Processig, vol. ASSP-33, pp , Aug J. A. Moorer, The optimum comb method of pitch period aalysis of cotiuous digitized speech, IEEE Tras. Acoust., Speech, Sigal Processig, vol. ASSP-22, pp , Oct J. S. Lim, A. V. Oppeheim, ad L. D. Braida, Evaluatio of a adaptive comb filterig method of ehacig speech degraded by white oise additio, IEEE Tras. Acoust., Speech, Sigal Processig, VO~. ASSP-26, pp , Aug S. M. Kay ad S. L. Marple, Jr., Spectrum aalysis-a modem perspective, Proc. IEEE, vol. 69, pp , Nov D. C. Rife ad R. R. Boorsty, Multiple toe parameter estimatio from discrete-time observatios, Bell Syst. Tech. J., vol. 55, o. 9, pp , NOV C. W. Helstrom, Sfatistical Theory of Sigal Detectio. Elmsford, NY: Pergamo, 1968, pp Boaz Porat (S 79-M 82), for a photograph ad biography, see p. 130 of the February 1986 issue of this TRANSACTIONS.