A FURTHER SUBSPACE METHOD OPTIMIZATION FOR TRACKING REAL VALUED SINUSOIDS IN NOISE

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1 Électonique et tansmission de l infomation A FURHER SUBSPACE MEHOD OPIMIZAION FOR RACKING REAL VALUED SINUSOIDS IN NOISE ŞEFAN SLAVNICU, SILVIU CIOCHINĂ Key wods: Subspace tacking, Fequency estimation, Real-valued data, R-ESPRI, PASd. A futhe optimisation of adaptive subspace-based fequency estimation fo multiple eal valued sine waves is consideed in this pape. Authos poposed in [10] a novel combination of stong adaptive covaiance matix update and optimised block pocessing fo fequency values etieval. he new optimised appoach combines the pojection appoximation subspace tacking technique based on deflation PASd (fo the signal subspace update) with the ESPRI-like fequency estimation of eal-valued sinusoids (fo fequency values etieval). Consequently, a new adaptive subspacetacking algoithm fo fequency estimation is poposed. he method poposed bings a significant eduction in aithmetical complexity at the same level of accuacy. 1. INRODUCION Adaptive subspace tacking fo detemining sine wave time-vaying fequencies is a eseach field still unde study. Not only old techniques have been optimized [6, 7], but also new algoithms have been developed in ode to impove the accuacy of the methods o to decease the computational buden [8]. Howeve, all these methods pesent a majo dawback: they assume that the data ae complex-valued and this implies additional computational effot. All supe-esolution subspace block methods (MUSIC, ESPRI etc.) ae based on a complex-valued signal model, as they have initially been designed fo aay pocessing [4]. Only ecently Mahata and Södestöm developed an ESPRI-like method to estimate the eal-valued sinusoidal fequencies [1, 9]. his new noniteative method, called R-ESPRI by the authos, is based on a eal-valued signal model and bings a spectacula eduction in the numbe of opeations equied to compute the fequency estimates. It is then natual to think about an adaptive method able to take advantage of this much lowe complexity. In the pesent pape we have made a futhe step to Politehnica Univesity of Buchaest, Electonics and elecommunications Depatment, Splaiul Independentei, 313, Buchaest, Romania, slavnicu@ieee.og, silviu@comm.pub.o Rev. Roum. Sci. echn. Électotechn. et Éneg., 5,, p , Bucaest, 007

2 10 Ştefan Slavnicu, Silviu Ciochină the wok pesented in [10] and adapted the well-known pojection appoximation subspace-tacking algoithm based on deflation (PASd []) to the eal-valued signal model. Simila to the method pesented in [10] fo PAS algoithm, the subspace tacking-type PASd is modified fo applying R-ESPRI fo eal sinusoids etieval. We have used the stabilized vesion of PASd deived by the authos in [3] in ode to eliminate some cases of instability specific to least squaes methods due to popagation of ound-off eos [5]. We will compae the pefomances and the complexity of the newly deived algoithm with the stabilized PASd vesion based on the complex-valued data model.. SIGNAL MODEL he signal model is pesented in [10]. We will biefly eview it hee fo pesent pape consistency. he input signal consists in a numbe of sinusoidal signals embedded in white Gaussian noise: x( t) = sk sin( tωk + φk ) + k = 1 n( t), (1) whee s k is the amplitude, ω k is the angula fequency of the of the k th sinusoid and n(t) epesents the coupting additive zeo-mean white noise. he phases { k } k = 1 φ ae andom vaiables unifomly distibuted in the [ π, π] inteval. he compact subspace epesentation dedicated fo eal valued sinusoids diffes fom the classical complex-valued signal model [1]. We have to obtain an altenative snapshot vecto so that its noise-fee pat lies in a subspace of dimension. o that aim, we will intoduce the following input vectos: x = K + () c () t [ x () t x ( t n 1)] whee the snapshot vecto dimension x = K (3) b () t [ x ( t 1) x ( t n )] 1 x = { xc + xb } (4) n >.

3 3 Method optimization fo tacking eal valued sinusoids in noise 11 Fom the above definitions we obtain x = A s + n, (5) whee s (t) is an 1 vecto given by whee φ + = φ ( 1/ ) ω fo 1 k. whee k k A is an n k + { ω t + φ } a 1 cos 1 1 { } s = M, (6) + a cos ωt + φ matix given by ω1 ω cos L cos 3ω1 3ω cos L cos A =. (7) M O M 1 1 cos n ω1 L cos n ω he noise snapshot vecto n (t) in this modified model is given by 1 n = { nc + nb }, (8) n K, (9) c () t = [ n () t n ( t + n 1)] n K (10) b () t = [ n ( t 1) n ( t n )]. One can show [1] that A is a full column ank matix. he impotant fact hee is that the noise-fee pat of x (t) lies in an -dimensional subspace that is diffeent fom the complex-valued data model, whee the dimension of the signal subspace is. Futhe on, let us intoduce { () () } = P E s t s t. (11)

4 1 Ştefan Slavnicu, Silviu Ciochină 4 he noise vectos n c (t) and n b (t ) ae andom vectos, mutually independent, with E{ n() t n () t } = ( σ /) I n whee σ is the noise vaiance. We obviously have that { () ()} σ = E t t = + n R x x A P A I. (1) We may then conside the eigenvalue decomposition = + R S Λ S G Σ G, (13) whee Λ is an diagonal matix containing the dominant eigenvalues of R on the diagonal. he n matix S is composed of the coesponding left eigenvectos. In the same pespective, Σ is a diagonal matix containing the emaining n eigenvalues of R. he n ( n ) matix G is composed of the coesponding left eigenvectos. he columns of G ae othogonal to those of S. One can show (see [1]) that whee S = A C, (14) 1 σ C = PAS Λ I n. (15) he columns of S fom an othonomal basis of the column space of A. he idea is to adaptively obtain an estimate of S fom the data via PASd adaptive subspace tacking method, which will then be pocessed to obtain the fequency estimates. 3. ALGORIHMS 3.1. R-ESPRI he R-ESPRI algoithm is an ESPRI-like estimation method of ealvalued sinusoidal fequencies. he algoithm has been poposed and has been pesented in detail in [9] and [1]. We will esume as in [10] the main aspects of this method as it epesents a key facto in developing ou new adaptive method. R-ESPRI elies on the signal model pesented in Chapte.

5 5 Method optimization fo tacking eal valued sinusoids in noise 13 he basic idea is to make use of two ( n ) n oeplitz matices L 0 (1) = L 0, M O O O O M 0 L (16) L 0 () 1 = L 0. M O O O O M 0 L (17) Fom the definition of holds: A matix (see elation (7)) the following identity whee () D is the following diagonal matix: A (1) = A D, (18) { cos( ω ),, cos( ω )} D = diag 1 K n. (19) Let us also intoduce the following matix Φ 1 = C D C. (0) hen, the algoithm may be deived as follows: a) It is fist equied to estimate Ŝ fom the input data. b) his estimate will be used in estimating Φˆ, fom (14) and (18), as = ˆ Sˆ. (1) ˆ ( 1) 1 () ( S ) c) he eigendecomposition of Φˆ will lead us to Φ d) Knowing Dˆ, fequency values easily esult fom: ω Dˆ, following equation (0). = cos 1 { D ˆ ( k, k)} k = 1, K. () k, his finally gives the fequency estimates. he dimension of the signal subspace is educed to fom i.e. the case of taditional ESPRI method fo eal-valued sine waves.

6 14 Ştefan Slavnicu, Silviu Ciochină R-PASD ALGORIHM In this chapte we will deive a novel adaptive method fo estimating the signal subspace Ŝ fom the input data. his method is based on the stabilized Pojection Appoximation Subspace acking based on Deflation (PASd) algoithm poposed in [] and modified fo the eal data model pesented in Section of this pape. We will theefoe efe to this algoithm as R-PASd. Fom the authos knowledge, this method has neve been published in this fom befoe. n Let x R be the input data vecto at time t defined as in elation (5), with H the coelation matix R = E{ x x }. Note hee that epesents the numbe of eal sinusoids, which is half the numbe of complex sinusoids used in taditional PAS o PASd methods fo tacking fequencies of eal sine waves. his is the fist level of eduction in aithmetical complexity. he second level comes fom the use of R-ESPRI technique, a much simple method adapted fo eal sinusoids envionments, instead of ESPRI, taditionally poposed fo complex-valued signal envionments. We ae inteested to ecusively estimate the signal subspace Ŝ, theefoe to compute the signal subspace estimate at the time instant t fom the subspace estimate at t 1 and the new aiving sample vecto x. As in [] we know that if we conside the following cost function (RLS citeion): t t i i= 1 J( W ) = β x ( i) W W x ( i) = [ ] = t R t W R W + + t () t () t () t () t () t W R W W W, (3) whee W (t) is a eal-valued n matix, then we may appoximate that gives us a modified cost function W x ( i) W ( i 1) x ( i) = y ( i), (4) t i= 1 t i J' ( W ) = β x ( i) W y ( i). (5) Following the theoy in [], we can pove that the matix W R n ( < n) minimizing J' ( W ) is a good estimate fo the signal subspace S ˆ of the coelation matix R (t).

7 7 Method optimization fo tacking eal valued sinusoids in noise 15 Equation (4) pesents an appoximation fo the unknown pojection of x (i) onto the column of W (t). his appoach is the key of the so-called PAS (Pojection Appoximation Subspace acking) method. he PASd algoithm ([, 3]) is deived fom the PAS appoach and is based on deflation technique. Within this algoithm, the pincipal components ae estimated sequentially. Fist, PAS algoithm is applied with = 1 fo updating the most dominant eigenvecto. Fom x (t) itself, we extact the pojection of the cuent data vecto to this eigenvecto. In the updated data vecto, the second eigenvecto becomes the most dominant one. So it can be extacted in the same way as explained above. his geneates an iteative method fo extacting all desied eigencomponents sequentially. able 1 biefly pesents the subspace tacking R-PASd algoithm adapted fo sinusoidal fequencies identification. able 1 Stabilized R-PASd algoithm fo Real-Valued Fequency Estimation = numbe of eal sinusoids n = dimension of x [ 1 1 1] d ( 0) = 1443 L ; 4 FOR t = 1,, DO 1 x = xc + xb FOR i = 1 DO { } yi = wi H ( t 1) x( t 1) d i = βd i ( t 1) + yi e = x( t 1) wi ( t 1) yi 1 L 1 w(0) = M O M L 1 * wi = wi ( t 1) + e( t)[ yi / di ] x = x( t 1) w i yi END FOR W (:, i) = wi S ˆ = W f = R ESPRI( Sˆ ) END FOR n

8 16 Ştefan Slavnicu, Silviu Ciochină 8 Hee x (t) epesents the input vecto at time t. At a given time t, the estimated coelation matix W is obtained by setting its columns to be the wi (t) vectos. Following the MALAB notation we may say that: W (:, i) = w. (6) Estimated fequencies vecto f is obtained by applying R-ESPRI method to the othonomal basis W of signal subspace. i 4. SIMULAION RESULS 4.1. EVALUAION OF ARIHMEICAL COMPLEXIY We evaluate the computational buden fo the main loop of each algoithm in ode to bette compae the two methods R-PASd and PASd. Pefomance of subspace tacking-type algoithms depends not only on the numbe of sinusoids, but also on the dimension n of the input vecto x (t). We obtain the following estimations (whee opeation means eal numbes addition o multiplication): PASd : 16n + 8 opeations / iteation RPASd : 8n + n + 4 opeations / iteation. We can clealy see that R-PASd equies less opeations than PASd fo computing the update of the signal subspace estimate W. Futhe gain in computational buden comes fom the use of R-ESPRI instead of ESPRI fo the values of the fequency estimates. A detailed compaison of these two block methods fom computational buden point of view may be found in [9]. Fom extensive simulations, we may state that the oveall computational buden fo R-PASd is about 70% as compaed to PASd fo the same input vecto dimension. We have checked the esults with MALAB flops outine. 4.. ALGORIHMS BEHAVIOR IN SAIONARY ENVIRONMENS We study the statistical popeties of both R-PASd and PASd algoithms in stationay envionments. We ae inteested to see if the eduction in aithmetical complexity affects the algoithm pefomance. We pesent the esults obtained fo the two algoithms when etieving two sinusoids of nomalized fequencies f =.1, f 0., embedded in backgound white noise. 1 0 =

9 9 Method optimization fo tacking eal valued sinusoids in noise 17 We calculate the bias and vaiance of the estimated fequencies fo vaious signal lengths N and fo vaious signal-to-noise atios. In each case, we un 100 independent simulations. Each time we compute the Came-Rao bound (CRB) to veify the accuacy of the estimates. ables and 3 pesent the statistical pefomances R-PASd and PASd algoithms, espectively. We see that R-PASd oveall pefoms about the same as PASd at a lowe aithmetical complexity. We also see that both algoithms convege in less than 100 iteations. able Statistical esults fo R-PASd (n = + 5) N SNR (db) Bias f 1 Va f 1 Bias f Va f CRB ` 0.11 able 3 Statistical esults fo PASd (n = + 5) N SNR (db) Bias f 1 Va f 1 Bias f Va f CRB

10 18 Ştefan Slavnicu, Silviu Ciochină CONCLUSIONS In the pesent pape we have moved fowad to the wok pesented in [10] and adapted the well-known pojection appoximation subspace-tacking algoithm based on deflation (PASd [, 3]) to the eal-valued signal model. hus, we deive anothe novel subspace method based on pojection appoximation, optimized fo tacking eal sinusoids in noise. We name this method R-PASd. he new algoithm uses the eal data model. We compae its pefomances to a stabilized vesion of complex-valued PASd algoithm. We conclude that R-PASd has about the same pefomances as PASd in stationay envionments, but at lowe complexity. It seems that we can futhe mitigate the majo dawback in the use of subspace tacking-type algoithms, thei high aithmetical complexity. his pape follows-up the field of optimizing othe adaptive subspace tackng methods like [6] o [7] fo estimating fequencies of eal valued sinusoids in noise. his is also the pespective fo ou futue studies. Received on 16 July 006 REFERENCES 1. K. Mahata,. Södestöm, ESPRI-like Estimation of Real-Valued Sinusoidal Fequencies, IEEE ans. on Signal Pocessing, 5, 5, pp , B. Yang, Pojection Appoximation Subspace acking, IEEE ans. on Signal Pocessing, 43, 1, pp , S. Ciochină, S. Slavnicu, Compaison between wo RLS Methods fo Adaptive Sinusoids Retieval: Notch Filteing and Subspace acking, Buletinul Ştiinţific al Univ. Politehnica din imişoaa, ans. on Electonics and Communications, 47(61), 1, P. Stoica, R.L. Moses, Intoduction to Spectal Analysis, Pentice Hall, S. Ljung, L. Ljung, Eo Popagation Popeties of Recusive Least-squaes Adaptation Algoithms, Automatica, 1,, pp , K. Abed-Meaim, A. Chkeif, Y. Hua, Fast Othonomal PAS Algoithm, IEEE Signal Poc. Lettes, 7, 3, pp. 60 6, S. Attalah, K. Abed-Meaim, Fast Algoithms fo Subspace acking, IEEE ans. on Signal Pocessing, 8, 7, pp , R. Badeau, G. Richad, B. David, Sliding Window Adaptive SVD Algoithms, IEEE ans. on Signal Pocessing, 5, 1, pp. 1 10, K. Mahata,. Södestöm, Subspace Estimation of Real-Valued Sinusoidal Fequencies, Dept. Infom. echnol., Uppsala Univ., ech. Rep., Uppsala, Sweden, Jan S. Slavnicu, S. Ciochina, Subspace Method Optimized fo acking Real-Valued Sinusoids in Noise, Poc. Signals, Cicuits and Systems, 005 (ISSCS 005, Vol., July 005, pp