Mean Square Projection Error Gradient-based Variable Forgetting Factor FAPI

Transcription

1 3rd Inernaional Conference on Advances in Elecrical and Elecronics Engineering (ICAEE'4) Feb. -, 4 Singapore Mean Square Projecion Error Gradien-based Variable Forgeing Facor FAPI Young-Kwang Seo, Jong-Woo Shin, Wongi Seo, and young-nam Kim refined mehods including OPAST (Orhonormal PAST) [4], NIC (Novel Informaion Crierion) [5], and NP3 (Naural Power mehod 3) [6] have been presened. The final sae of he developmen along his baseline has been recenly marked by an adven of he so-called fas approximaed power ieraion (FAPI) [7]. The FAPI shows excellen performance in global convergence propery, racking accuracy, and orhonormaliy error even when he subspace of inpu daa suddenly changes wih ime. owever, since he FAPI involves a consan forgeing facor (FF) when esimaing a covariance marix as used in he convenional recursive leas squares (RLS) algorihm, i could no cope well wih coninuously ime-varying environmens. The convergence rae of he FAPI is slow when he FF is close o, whereas he misadjusmen is large when he FF is small. Therefore, he FAPI canno simulaneously achieve boh he fas convergence rae and he small misadjusmen due o he consan FF. In order o enhance he convergence performance of he FAPI mehod, his paper applies he gradien-based variable forgeing facor (GVFF) RLS algorihm o he FAPI mehod [8]. The GVFF RLS decreases he FF value when here is a large increase in mean square projecion error (MSPE) and forces he FF oward he ceiling when here is negaive gradien of MSPE. In addiion, in order o improve he racking performance in coninuously ime-varying environmens, we modify he FF conrol equaion in he manner ha he FF has rapidly decreased for a posiive gradien and slowly increased for negaive gradien. This paper is organized as follows. In Secion II, he subspace racking mehods, he PAST, and he FAPI, are briefly described. Secion III inroduces he GVFF RLS algorihm and hen gives a proposed subspace racking mehod called he GVFF FAPI. Secion IV presens some simulaion resuls o demonsrae he performance of GVFF FAPI. Finally, he conclusions of his paper are summarized in Secion V. Absrac This paper proposes a fas subspace racking mehod named gradien-based variable forgeing facor fas approximaed power ieraion (GVFF FAPI). Since he convenional FAPI uses a consan forgeing facor for esimaing covariance marix of source signals, i has a difficuly in being applied o non-saionary environmens such as coninuously ime varying subspace. To cope wih he drawback of he convenional FAPI mehod, we modified he forgeing facor conrol equaion of he GVFF recursive leas squares algorihm and hen applied o he FAPI mehod. Simulaion resuls show ha he proposed GVFF FAPI algorihm is superior o he convenional FAPI in erms of subspace error and roo mean square error (RMSE) of racked direcion of arrival. Keywords FAPI, GVFF RLS, subspace racking, PAST I. INTRODUCTION S UBSPACE racking mehods are an imporan pre-processing sep o reduce he compuaional complexiy in he field of subspace-based adapive sysems []. Insead of updaing whole eigen-srucure, subspace-racking mehods work only wih he signal or noise subspace. This makes he subspace-racking mehods more efficien han convenional subspace esimaion mehods such as eigenvalue decomposiion (EVD) or singular value decomposiion (SVD) []. A review of hisorical advances in he subspace racking mehods can be found in []. In early 99s, i urned ou ha he signal subspace racking problem can be solved wih a dominan complexiy of only O(nr) compuaions per each updae. These kinds of mehods are called by a fas subspace racking mehod and a projecion approximaion subspace racker (PAST) is he mos popular one [3]. Alhough he PAST ouperforms oher mehods in erms of racking accuracy and global convergence propery, i sill suffers from he lack of orhonormaliy in esimaed subspace columns. In order o reduce he orhonormaliy error of he PAST, a number of II. OVERVIEW OF SUBSPACE TRACKING METODS Young-Kwang Seo is wih he School of Elecronic, Elecrical, and Compuer Engineering, Pusan Naional Universiy, Busan, Korea (corresponding auhor o provide phone: ; iloverey@pusan.ac.kr). Jong-Woo Shin is wih he School of Elecronic, Elecrical, and Compuer Engineering, Pusan Naional Universiy, Busan, Korea ( sjjoly@pusan.ac.kr). Wongi Seo is wih Nexwill Co,,Ld., Daejeon, Korea ( nexwill@nexwill.com). young-nam Kim is wih he Deparmen of Elecronics Engineering, Pusan Naional Universiy, Busan, Korea ( hnkim@pusan.ac.kr). A. Projecion Approximaion Subspace Tracker (PAST)[3] The goal of he PAST algorihm is o recursively esimae he principal subspace which is spanned by he eigenvecors associaed wih r dominan eigenvalues of a ime-recursively updaed covariance marix C ( ) of dimension N N as follows: 96

2 3rd Inernaional Conference on Advances in Elecrical and Elecronics Engineering (ICAEE'4) Feb. -, 4 Singapore C ( ) C ( ) x( ) x ( ), () where x() is a daa snapsho vecor of dimension n > r and is a posiive exponenial forgeing facor close o one. An esimaed subspace marix W() is obained as solving he minimizaion problem of exponenially weighed cos funcion defined by u J( W( )) x( u) W( ) W ( ) x ( u). () u All sample vecors available in he ime inerval u are involved in esimaing he signal subspace a he ime insan. The key issue of he PAST is o approximae W ( ) x ( u) in (). In oher words, an inpu daa vecor x ( u) is projeced ono he column space of W () by he expression of y( u) W ( u ) x ( u). This resuls in a modified cos funcion defined by u J '( W( )) x( u) W( ) y ( u). (3) u The compuaion of W () consiss of a daa compression sep (4) and an orhonormalizaion sep (5) of a compressed marix a each updae as C ( ) C ( ) W ( ), (4) W( ) R( ) C ( ), (5) where C () is an n r correlaion marix of an n dimensional daa vecor x() and a r dimensional compressed daa vecor y() as y( ) W( ) x ( ). (6) The exponenially weighed leas squares problem in (3) is well sudied in field of adapive signal processing. The modified cos funcion J (W()) is minimized if R() is equal o C yy () as follows: W C C, (7) ( ) ( ) yy ( ) C ( ) C ( ) x( ) y ( ), (8) C ( ) C ( ) y( ) y ( ). (9) yy yy An efficien way o compue he inverse of C yy () is o apply he marix inversion lemma. Since his process is oally same wih he well-known RLS algorihm, he PAST jus uses i for updaing W() wihou furher derivaion. The PAST mehod requires 3nr +O(r ) compuaions a each updae [3]. Noe ha he PAST mehod is derived by minimizing he For,, modified cos funcion in J (W()) in (3). Therefore, he columns of esimaed W() have lack of he orhonormaliy which depends on he signal-o-noise raio (SNR) and forgeing facor [3]. B. Fas Approximaed Power Ieraion (FAPI)[7] An orhonormal subspace basis is required for some subspace-based esimaion algorihms such as MUSIC [9]. The FAPI mehod applies new projecion approximaion insead of convenional projecion approximaion of he PAST mehod for orhonormal subspace basis as W( ) W ( ) ( ), () where () is a r r orhonormal marix. A family of power ieraion mehods, including he PAST and NP3 mehod, is based on convenional projecion approximaion by W( ) W ( ). These mehods have a consrain ha R() mus be posiive definie marix. ence, hese subspace rackers could no guaranee o converge if R() deviaes from posiive definie marix consrain. On he oher hand, he FAPI is no affeced by his consrain because he FAPI relies on he less resricive approximaion as (). The bes approximaion of W() is obained by solving he following minimizaion problem arg min W( ) W ( ) ( ) F, () Θ() TABLE I FAPI METOD [7] Ir Iniializaion : W(), P() I ( nr) r y( ) W ( ) x( ) h( ) P( ) x( ) g( ) h( ) / [ y ( ) h( )] ( ) ( x ( ) x( ) y ( ) y( )) and he soluion of () is obained by / ( ) I ( ) g ( ) g( ) ( ) p ( ) ( ) ( ) ( ) ( ) ( ) I g ( ) g( ) ( ) p y '( ) y( ) ( ) g( ) ( ) h '( ) P ( ) y'( ) ε( ) P( ) g( ) g( ) h ' ( ) g( ) ( ) ( ) P( ) P( ) g( ) h ' ( ) ε( ) g ( ) e'( ) x( ) W ( ) y( ) W( ) W( ) e'( ) g ( ) End r 97

3 3rd Inernaional Conference on Advances in Elecrical and Elecronics Engineering (ICAEE'4) Feb. -, 4 Singapore Convenional GVFF ( ) ( ) I r g( ) e ( )e( ) g ( ),. () where Ir is a r r ideniy marix, g() is a gain vecor, and e() is an error vecor. The paricular soluion of () is calculaed by using he marix inversion and relaed specific derivaions can be found in [7]. The oal procedure of he FAPI is presened in Table and he FAPI requires only n(3r+) + O(r) compuaion a each updae [7]..9 Magniude III. GRADIENT-BASED VARIABLE FORGETTING FACTOR FAST SUBSPACE TRACKING (GVFF FAPI) ALGORITM A. GVFF Recursive Leas Squares Subspace racking mehods wih a consan forgeing facor (FF), such as PAST, NP3, and FAPI, are no suiable for racking in coninuously ime-varying subspace environmens because is convergence rae is slow when he FF is close o one, whereas he misadjusmen is large when he FF is small. In order o achieve he saisfacory performance of a subspace racker in coninuously ime-varying environmens, his paper applies he gradien-based variable forgeing facor (GVFF) RLS algorihm o he FAPI mehod. The GVFF RLS algorihm conrols he forgeing facor by using he gradien based mehod which is derived from an improved mean square error (MSE) analysis of he ime variable error weighing RLS (TWRLS) algorihm [],[]. The TWRLS algorihm can be obained from he minimizaion of he leas squares cos funcion parameerized by desired signal d(i ), weigh vecor (a) Proposed GVFF..9 Magniude w( ), and error weighing funcion i ( ) as follows (b) Fig. The variable FF of GVFF FAPI : (a) using conrol equaion (5), J ( ) i ( ) d(i) w ( )x(u). (b) using conrol equaion (6) (3) i The MSE analysis wih error weighing funcion yields he following gradien updaing equaion [8] e ( ) e ( ) h e ( ), The conrol mechanism can be explained as follows. If he derivaive / in (4) is posiive, his enables he conrol equaion o decrease he forgeing facor whenever here is a large increase in e ( ). On he oher hand, h / in (4) is (4) always negaive and his drives he gradien o negaive afer e ( ) is reduced o a cerain level and forces he forgeing where e ( ) / means a gradien of MSE, and h are facor oward he ceiling. coefficiens o unie he mulinomial of derivaion process, e ( ) and are he esimaed MSE and he esimaed B. GVFF FAPI The GVFF RLS algorihm is applied in FAPI mehod o improve he racking performance in coninuously ime-varying environmens. owever, i is designed o yield beer convergence performance in suddenly ime-varying environmens. This paper modifies he FF conrol equaion in order o enhance he convergence performance in coninuously ime-varying environmen as follows: measuremen noise variance, respecively. Relaed specific derivaions and parameers can be found in [8]. Using he gradien e ( ) /, he gradien based conrol for he variable forgeing facor of he TWRLS algorihm is expressed as ( ) ( ) max e ( ). ( ) (5) # 98

4 3rd Inernaional Conference on Advances in Elecrical and Elecronics Engineering (ICAEE'4) Feb. -, 4 Singapore max e ( ) ( ) ( ) # for (6) max e ( ) ( ) ( ) / # Velociy of DOA e ( ) DOA (angle), Velociy (angle / sec) ( ) for e ( ) The conrol mechanism can be explained as follows. The variable sep-size μ/(-β(-)) in (6) enables he conrol o rapidly reduce he forgeing facor o a sufficienly small level whenever here is a small and posiive e ( ) /. On he DOA Pah Velociy of DOA / oher hand, he variable sep-size μ/(-β(-)) in (6) drives he forgeing facor o slowly increase when e ( ) / is negaive. Figure shows wo cases of variable FF of he GVFF FAPI mehod ha he firs, conrolled by he FF conrol equaion (5), is represened in Fig. (a) and anoher, conrolled by he FF equaion (6), is given in Fig. (b). 4 Fig. Time varying DOA models of a source signal Signal Subspace Error - SNR db Subspace Error (db) - IV. PERFORMANCE EVALUATION To demonsrae he applicabiliy in direcion-of-arrival (DOA) sysems and he racking performances of he proposed mehod, we compared subspace error [6] and roo mean square error (RMSE) for racked DOA of GVFF FAPI wih ha of he convenional FAPI using he consan forgeing facor (FF). A linear uniform array wih n 8 sensors and a ime-varying DOA according source signal described in Fig. were used in his simulaion. The disance beween he adjacen sensor elemens was a half of he wavelengh and he frequency of inciden signal was kz. The sep-size μ of he conrol equaion of he GVFF FAPI mehod is.. SNR was db and a sequence of snapsho x() ( =,, ) was generaed according o he above signal model and used o rack he signal subspace by FAPI and GVFF FAPI. Afer each subspace was updaed, he MUSIC algorihm was applied o compue RMSE of racked DOA from signal subspace esimae. In his simulaion, he proposed mehod using he conrol equaion (5) and he conrol equaion (6) refer o GVFF FAPI and GVFF FAPI, respecively. I is clear, from Fig. 3 ha he subspace error and RMSE of racked DOA abou GVFF FAPI and GVFF FAPI are smaller han FAPI s. In addiion, we can confirm ha GVFF FAPI shows he superior performance o GVFF FAPI in coninuously ime-varying environmen. owever, he performances beween GVFF FAPI and GVFF FAPI was reversed in he converging siuaion afer DOA suddenly changed a = FAPI GVFF-FAPI GVFF-FAPI (a) RMSE of racked DOAs - SNR db FAPI GVFF-FAPI GVFF-FAPI 8 6 RMSE (b) Fig. 3 Performance comparison beween FAPI and GVFF FAPI, : (a) subspace error, (b) RMSE of racked DOA he FAPI in coninuously ime-varying environmen, we modified he FF conrol equaion of he GVFF RLS algorihm and applied he modified equaion o he FAPI mehod. Simulaion resuls show ha he GVFF FAPI using he modified V. CONCLUSION In his paper, we proposed a subspace racker named as he GVFF FAPI. In order o enhance he racking performance of 99

5 3rd Inernaional Conference on Advances in Elecrical and Elecronics Engineering (ICAEE'4) Feb. -, 4 Singapore he FF conrol equaion gives he ousanding performance han he FAPI and he GVFF FAPI using he original FF conrol equaion. The fuure works would be a heoreical analysis of he modified FF conrol equaion and a comparaive sudy wihoher s FF conrol equaion of VFF RLS algorihms. ACKNOWLEDGMENT This research was suppored by he Korea Insiue for Advancemen of Technology (KAIT), Souh Korea, A developmen of a super-precision radio measuremen sysem -4. REFERENCES [] P. Srobach, The fas recursive row-ouseholder subspace racking algorihm, Signal Processing, vol. 89, no., pp , Dec. 9. [] P. Comon, G.. Golub, Tracking a few exreme singular values and vecors in signal processing, Proceedings of he IEEE, vol. 79, no. 8, Aug. 99. [3] B. Yang, Projecion Approximaion Subspace Tracker, IEEE Trans. Signal Processing, vol. 44, no., pp. 95-7, Jan [4] K. Abed-Meraim, A. Chkeif, and Y.ua, Fas orhonormal PAST algorihm, IEEE Trans. Signal Processing. Le., vol. 7, no. 3, pp. 6-6, Mar.. [5] Y. Miao and Y. ua, Fas subspace racking and neural nework learning by a novel informaion crierion, IEEE Trans. Signal Processing, vol. 46, no. 7, pp , July [6] Y. ua, Y. Xiang, T. Chen, K. Abed-Meraim, and Y. Miao, A new look a he power mehod for fas subspace racking, Digial Signal Processing, Oc. 999 [7] R. Badeau, B. David, and G. Richard, Fas Approximaed Power Ieraions Subspace Tracking, IEEE Trans. Signal Processing, vol. 53, no. 9, pp. Aug [8] Shu-ong Leung and C.F. So, Gradien-Based Variable Forgeing Facor RLS Algorihm in Time-Varying Environmens, IEEE Trans. Signal Processing, vol. 53, no. 8, Aug. 5. [9] R.O. Schmid, Muliple emier locaion and signal parameer esimaion," IEEE Trans. AP, vol. 34, no. 3, pp. 76-8, Mar [] D. T. M. Slock, T. Kailah, Fas ransversal filers wih daa sequence weighing, IEEE Trans. Acous., Speech, Signal Processing, vol. 33, no. 3, pp , Mar [] B. Toplis, S. Pasupahy, Tracking Improvemens in fas RLS algorihms using a variable forgeing facor, IEEE rans. Acous., Speech, Signal Processing, vol. 36, no., pp. 6-7, Feb. 988.