A UNIFIED FRAMEWORK FOR MULTICHANNEL FAST QRD-LS ADAPTIVE FILTERS BASED ON BACKWARD PREDICTION ERRORS

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1 A UNIFIED FRAMEWORK FOR MULTICHANNEL FAST QRD-LS ADAPTIVE FILTERS BASED ON BACKWARD PREDICTION ERRORS César A Medina S,José A Apolinário Jr y, and Marcio G Siqueira IME Department o Electrical Engineering Praça General Tibúrcio, 8 Urca Rio de Janeiro, RJ 9- Brazil csmedina@epqimeebbr, apolin@ieeeorg and mgs@ieeeorg ABSTRACT Fast QR decomposition algorithms based on backward prediction errors are well known or their good numerical behavior and their low complexity when compared to similar algorithms with orward error update Their application to multiple channel input signals generates more complex equations although the basic matrix expressions are similar This paper presents a uniied ramework or a amily o multichannel ast QRD-LS algorithms This amily comprises our algorithms two basic algorithms with two dierent versions each These algorithms are detailed in this work 1 INTRODUCTION Digital processing o multichannel signals using adaptive ilters has recently ound a variety o new applications [1] such as color image processing, multi-spectral remote sensing imagery, biomedicine, channel equalization, stereophonic echo cancellation, multidimensional signal processing, Volterra-type nonlinear system identiication, and speech enhancement This increased number o applications has spawned a renewed interest in eicient multichannel algorithms One class o algorithms, known as multichannel ast QR decomposition least-squares adaptive algorithms based on backward error updating, has become an attractive option because o ast convergence properties and reduced computational complexity In the case o one single channel, a uniied ormulation or Fast QRD-LS algorithms is available in [] In this paper, the basic algorithm proposed in [] is studied and a new ormulation is developed or the amily o multichannel ast QRD algorithm based on backward errors This paper is organized as ollows Is Section we present the common matrix equations or the two Multichannel Fast QRD-LS algorithms Section presents the named MC FQR PRI B based em a priori backward errors updating Section addresses the MC FQR POS B(a posteriori backward errors updating) In Section we present two dierent versions or each algorithm and, among them, two algorithms that hadn t been proposed beore Section shows computer simulations with one o the proposed algorithms Conclusions are summarized in Section This work was partially supported by CAPES, Brazil y Supported by the FAPERJ rom Brazil (grant no E-/11/1)

2 THE MULTICHANNEL ALGORITHM: COMMON EQUATIONS In this work N is deined as the number o ilter coeicients and M is the number o input channels The objective unction to be minimized according to the least-squares (LS) algorithm is deined as ο LS (k) = where e(k) = e(k) = kx i= ki e (i) =e T (k)e(k) (1) e(k) 1= T e(k 1) k= e() is an error vector and may be represented as ollows d(k) 1= d(k 1) k= d() k= x T N () x T N (k) 1= x T N (k 1) W T N (k) = d(k) X N (k)w N (k) () and x T N (k) = x T k x T k1 x T kn +1 () where x T k =[x 1(k) x (k) x M (k)] is the input vector at time instant k I we have U N (k) as the Cholesky actor o X N (k), obtained through Givens rotation matrix Q N (k), then e q (k) =Q N (k)e(k) = dq1 (k) = d q (k) eq1 (k) e q (k) U N (K) W N (k) () Assuming the use o orward prediction, we deine the matrix X N +1 (k) as ollows X N +1 (k) = d (k) X N (k 1) T (M1)x(MN+M ) () where d (k) =[x k 1= x k1 k= x ] T is the (k +1) M orward reerence signal U N +1 (k) can be obtained by applying Givens rotations to () as ollows U N +1 (k) Q(k 1) (k)q (k) X I N +1 (k) MxM (k)q (k) Q(k 1) I MxM d (k) (k)q (k) (k) e q1 (k) d q U N (k 1) 1= x T T (M1)x(MN+M ) d q (k) U N (k 1) E (k) T X N (k 1) T (M1)x(MN+M ) ()

3 Where Q (k) and Q (k) are series o Givens rotations that zeroes the irst k MN rows and perorms a triangularization process o () The resulting null section can be removed as shown below: U N +1 (k) =Q (k) dq (k) U N (k 1) E (k) where Q (k) is the ixed range matrix o Q (k) Based upon the above equation it is possible to obtain [U N +1 (k)] 1 = " U 1 N E 1 (k) 1 (k 1) U # N (k 1)d q(k)e 1 (k) () Q T (k) (8) and deine E (k) as the zero order orward error covariance matrix as, E (k) dq (k) (k) E (k) From the act that and Q (k) = ^Q (k) Q(k 1) = ^Q(k 1) 1 Q (k 1) 1 Q(k ) (9) (1) (11) and knowing that 1 T Q (k 1) = ^Q (k) we could write rom () d q (k) E (k) ^Q(k 1) T I MxM 1 T Q (k 1) = ^Q (k) ^Q(k 1) I MxM = ^Q (k) ^Q(k 1) = ^Q (k) ^Q(k 1) I MxM = ^Q(k 1) T I MxM ^Q(k 1) I MxM x T k 1= d q (k 1) 1= E (k 1) 1 T Q (k 1) x T k 1= d q (k 1) 1= E (k 1) 1 T Q (k 1) 1 T T Q(k ) I MxM x T k 1= e q1 (k 1) 1= d q (k 1) k= x T M1xM x T k 1= x T k1 k= x T M1xM (1) (1)

4 rom () and (1) could be show that d q (k) E (k) = ^Q (k) e T q1 (k) d q (k) 1= E (k 1) (1) rom (1) and (1) and using ixed order matrix Q (k) rom ^Q(k) e T q1 (k +1) x T k+1 d q (k (k) +1) 1= d q (k) Similarly, rom (1) it is possible to obtain T E (k +1) (k +1) e T q1 (k +1) 1= E (k) Finally, the joint process estimation is perormed by the ollowing expressions [1] eq1 (k +1) d(k +1) d q (k +1) (k +1) 1= d q (k) (1) (1) (1) e (k) =e q1 (k)=l(k) =e(k)=l (k) (18) THE MULTICHANNEL FQR PRI B ALGORITHM In the MC FQR PRI B algorithm, the a priori backward prediction error vector is updated as ollows [] a N (k) = 1= U T N (k 1)x N (k) (19) From the deinition above and (8) it is possible to obtain a N +1 (k +1)= 1 U T (k) N (k 1) xk+1 E T (k) E T (k)d T T q (k)u N (k 1) x N (k) = 1 U T (k) N (k 1)x N (k) E T (k)x k+1 E T (k)d T T q (k)u N (k 1)x N (k) (k) an (k) r(k +1) () where r(k +1)= 1= he T = 1= E T (k) = 1= E T i (k 1)x N (k) (k)x k+1 E T (k)d T T q (k)u N h x k+1 U 1 N q(k) (k 1)d T xn (k) (k) xk+1 W T x N (k) = 1= E T (k)[x k+1 d (k)] = 1= E T (k)e (k +1) (1) i

5 inally [] e (k +1)=l(k)e q1 (k +1) The ollowing equation is used to update Q (k +1)[]: 1=l(k +1) (k +1) 1 a(k +1) () () Equation (1) requires a matrix inversion operation which can be numerically unstable In order to avoid stability problems we can use [] (k +1) 1=l(k +1) r(k +1) The resulting equations are summarized in Table 1 () Table 1: The Multichannel FQR PRI B Equations MC FQR PRI B 1 Obtaining d q (k +1) e T q1 (k +1) (k) d q (k +1) Obtaining E (k +1) T E (k +1) (k +1) Obtaining a N (k +1) a N +1 (k +1)=Q (k) an (k) x T k+1 1= d q (k) e T q1 (k +1) 1= E (k) r(k +1) Obtaining Q (k +1) E (k +1) (k dq (k +1) +1) E (k +1) Obtaining Q (k +1) 1=l(k +1) 1 (k +1) a(k +1) Joint Estimation eq1 (k +1) d(k +1) d q (k +1) (k +1) 1= d q (k) Obtaining the a priori error e (k +1)=e q1 (k +1)=l(k +1) THE MULTICHANNEL FQR POS B ALGORITHM To derive equations or this algorithm, we update the a posterior backward prediction errors vector as in [] n+1 (k +1)=U T N +1 (k +1)X N +1(k +1) ()

6 From (8) and the expression above, we obtain N +1 (k +1)=Q (k +1) (k +1) E T E T (k +1) N (k) p(k +1) U T N (k) (k +1) E T (k +1)d T q (k +1)x k+1 E T xk+1 T (k +1)U N (k) x N (k) U T N (k)x N (k) (k +1)d T T q (k +1)U N (k)x N (k) () where p(k +1)= h E T = E T (k +1) (k +1)x k+1 E T (k +1)d T T q (k +1)U N h x k+1 U 1 N (k)d q(k +1) T xn (k) i (k)x N (k) = E T (k +1) xk+1 W T x N (k) = E T (k)[x k+1 d (k)] = E T (k +1)e (k +1) () i Also rom [], we have the ollowing expression, similar to its single dimension counterpart Q (k +1) 1 = l(k +1) N (k +1) (8) The complete algorithm is summarized in Table THE DIFFERENT VERSIONS The dierent versions presented in this section are based on the implementation o a particular matrix equation For both algorithms we have a matrix equation with the ollowing structure (step and in Tables 1 and, respectively) u(k +1)=Q (k +1) v(k) i(k +1) (9) where i(k +1)is a vector o order M, and is updated without using any prior knowledge o any element o u(k +1) For the irst version o each algorithm, we assume that the inverse o Q (k +1)corresponds to Q T (k +1)

7 Table : The Multichannel FQR POS B Equations MC FQR POS B 1 Obtaining d q (k +1) e T q1 (k +1) x T k+1 d q (k (k) +1) 1= d q (k) Obtaining E (k +1) T e T q1 (k +1) E (k +1) (k +1) 1= E (k) Obtaining Q (k +1) E (k +1) (k dq (k +1) +1) E (k +1) Obtaining N (k +1) N +1 (k +1)=Q (k +1) N (k) p(k +1) Obtaining Q (k +1) 1 l(k +1) Q (k +1) = N (k +1) Joint Estimation eq1 (k +1) d(k +1) d q (k +1) (k +1) 1= d q (k) Obtaining the a priori error e (k +1)=e q1 (k +1)=l(k +1) Thereore, it is possible to show that v 1 (k +1) v N (k +1) i(k +1) = Y Y L1 L I j cos Mj+s+1 sin Mj+s+1 I MN+Mjs sin Mj+s+1 cos Mj+s+1 I s u (k +1) u 1 (k +1) u N (k +1) () where L1 is rom j = to MN 1 and L is rom s = to M 1 It is possible to update v(k) based upon u(k +1)i u N +1 (k +1)is previously known For the second version o this algorithm, equation (9) is used directly The resulting operations are shown

8 below u (k +1) u 1 (k +1) u N (k +1) Y Y L L = I j cos Mj+s+1 sin Mj+s+1 I MN+Mjs sin Mj+s+1 cos Mj+s+1 I s v 1 (k +1) v N (k +1) i(k +1) where L is rom j = MN 1 to and L is rom s = M 1 to (1) 1 The Multichannel FQR PRI B: Versions 1 and Another way to derive () is as ollows From () we can get B U N +1 (k) = E (k) C () Obtaining the inverse [U N +1 (k)] 1 = " E (k) 1 CB 1 E (k) 1 B 1 # () From (19) and using () a N +1 (k +1)= 1= = " h E (k) 1 CB 1i T B T E (k) T # 1= E T (k) xk+1 x k+1 x N (k) () It could be see that the last element o a N (k +1)was known beore its updating process This property is used in MC FQR PRI B 1, which is the irst algorithm version introduced in this paper Table shows pseudo code or MC FQR PRI B step using Matlab cl (Obtaining a N (k +1)) The second version o the MC FQR PRI B algorithm is based on the direct computation o a N (k +1)according to the matrix equation This version was introduced in [] and the corresponding implementation is presented in Table The Multichannel FQR POS B: Versions 1 and Deriving () in the same way as (), it could be show that the last element o N (k +1)was known prior to its updating process and was equal to E (k +1) T xk+1 This property generates the irst algorithm version

9 Table : The MC FQR PRI B: Version 1 aux = inv(e ) * x(k+1,:) /(lambda ^ ()); tempa = a; a(m*(n-1)+1:l) = aux; or j = 1:N-1 or i = 1:M or s = 1:M a(m*(n-j)-i+1)=(tempa(m*(n-j+1)-i+1)-aux(m-s+1)*sin p(m*(n-j+1)-i+1,s)) /cos p(m*(n-j+1)-i+1,s); tempa(m*(n-j+1)-i+1)=a(m*(n-j)-i+1); aux(m-s+1)=-a(m*(n-j)-i+1)*sin p(m*(n-j+1)-i+1,s)+aux(m-s+1) *cos p(m*(n-j+1)-i+1,s); Table : The MC FQR PRI B: Version or j = 1:N or i = 1:M or s = 1:M temp1 = a (M*j + i); temp = r(m - s + 1); a (M*(j-1)+i)=temp1*cos p(m*(j-1)+i,s)-temp*sin p(m*(j-1)+i,s); a (M*j + i)=a (M*(j-1)+i); r (M-s+1)=temp1*sin p(m*(j-1)+i,s)+temp*cos p(m*(j-1)+i,s); a(m*n + 1:M*N + M) = r; known as the MC FQR POS B 1 Table presents the implementation o its step (Obtaining N (k +1)) The second version o the Multichannel FQR POS B algorithm is based on the direct computation o N (k +1) according to the matrix equation This version was introduced in [] and its implementation is presented in Table SIMULATIONS This section presents simulation results o the Multichannel FQR POS B algorithm version 1 The algorithm was used in a system identiication problem with M = channels, each one with N = 1 coeicients The input signal to each channel was colored noise with eigenvalue spread equal to and observed noise with variance corresponding to db The orgetting actor was set to = :9 Fig 1 presents the MSE (in db) over an ensemble o indepent runs It is worth noting that all other versions present identical learning curves in ininite precision

10 Table : The MC FQR POS B: Version 1 aux = inv(e ) * x(k+1,:) ; temp = ; (M*(N-1)+1:L) = aux; or j = 1:N-1 or i = 1:M or s = 1:M (M*(N-j)-i+1)=(temp(M*(N-j+1)-i+1)-aux(M-s+1)*sin p(m*(n-j+1)-i+1,s)) /cos p(m*(n-j+1)-i+1,s); temp(m*(n-j+1)-i+1) = (M*(N-j)-i+1); aux(m-s+1) = -(M*(N-j)-i+1)*sin p(m*(n-j+1)-i+1,s)+aux(m-s+1) *cos p(m*(n-j+1)-i+1,s); Table : The MC FQR POS B: Version p = inv(e) *eq1 * gamma; or j = 1:N or i = 1:M or s = 1:M temp1 = (M * j + i); temp = p(m - s + 1); (M*(j - 1)+i)=temp1*cos p(m*(j-1)+i,s)-temp*sin p(m*(j-1)+i,s); (M*j + i) = (M*(j - 1) + i); p(m-s+1)=temp1*sin p(m*(j-1)+i,s)+temp*cos p(m*(j-1)+i,s); (M * N + 1:M*N + M) = p; CONCLUSIONS This paper presents a uniied ramework or Multichannel FQR algorithms based on backward error updating Dierent algorithms were derived using the same notation, thereore highlighting their similarities and dierences 1 Alternative and more eicient orms such as the MC FQR PRI B algorithm in [] are currently under investigation 1 Matlab cl codes, are available or downloading via internet at the web page apolin

11 MC FQR_POS_B_1 MSE MIN 1 MSE db k Figure 1: Learning Curve o the MC FQR POS B, Version 1 8 REFERENCES [1] N Kalouptsidis and S Theodoridis, Adaptive Systems Identiication and Signal Processing Algorithms, Englewood Clis, NJ: Prentice Hall, 199 [] J A Apolinário Jr, M G Siqueira, and P S R Diniz, On Fast QR Algorithms Based on Backward Prediction errors: New Results and Comparisons, in Proc First Balkan Con on Signal Process, Commun, Circuits, and Systems, Istanbul, Turkey, June [] A A Rontogiannis and S Theodoridis, Multichannel Fast QRD-LS Adaptive Filtering: New Technique and Algorithms, IEEE Transactions on Signal Processing, vol, No 11, Nov 1998 [] J A Apolinário Jr, New Algorithms o Adaptive Filtering: LMS with Data-Reusing and Fast RLS Based on QR Decomposition, DSc Thesis, COPPE/UFRJ, Rio de Janeiro, RJ, Brazil, 1998 [] M G Bellanger and P A Regalia, The FLS-QR algorithm or adaptive iltering: The case o multichannel signal, Signal Process vol, pp 111, 1991