Improved PARAFAC based Blind MIMO System Estimation

Transcription

1 Improved PARAFAC based Blind MIMO System Estimation Yuanning Yu, Athina P. Petropulu Department of Electrical and Computer Engineering Drexel University, Philadelphia, PA, 19104, USA This work has been supported by ONR under grant N CSPL Drexel University Asilomar 05 p. 1/1

2 Blind MIMO System Identification Memoryless (scalar) System Identification Independent component analysis (ICA) Convolutive System Identification Time domain methods need the channel length information Cumulant-based identification [Y. Inouye, K.Hirano, 1997] HOS based reverse filter [J.K. Tugnait, 1997] Frequency domain methods do not need system length information, and can use the results for memoryless MIMO problem. However, we need to resolve frequency-dependent permutation, scaling and phase ambiguities. Second-order Spectra [I. Bradaric, A.P. Petropulu, et 2003] Joint Diagonalization [B. Chen, A.P. Petropulu 2001] CSPL Drexel University Asilomar 05 p. 2/1

3 Blind MIMO System Identification Second-order statistics (SOS) methods can deal short data length and involve low complexity. However, they require channel diversity and only apply to non-white inputs. Subspace type appraoches [K.Abed-Meraim, J.F. Cardoso et.al. 1997] Second-order spectra based methods [I.Bradaric, A.P. Petropulu et.al. 2003] High-order statistics (HOS) methods do not need channel diversity, and also can deal with white inputs (must be Non-Gaussian) Joint Diagonalization [B.Chen, A.P.Petropulu, 2001] High order based inverse filter [J.K.Tugnait 2000] CSPL Drexel University Asilomar 05 p. 3/1

4 PARALLEL FACTORIZATION PARAFAC, a low rank decomposition of three- or higher-way arrays. A 3-way tensor X with dimensions J I K, F -component: x i,j,k = F a i,f b j,f c k,f f=1 expresses the three-way array X as a sum of F rank-one three-way factors, each one of which is the outer product of three factors. In a compact form, X can be expressed in terms of its slices: X i = BD i [A]C T where A is a I F matrix with entries a i,f ; B is a J F matrix with entries b j,f ; C is a K F matrix with entries c k,f. CSPL Drexel University Asilomar 05 p. 4/1

5 PARALLEL FACTORIZATION Definition 1: Consider an I J matrix A. If rank(a) = r then A contains r linearly independent columns. If every l J columns of A are linearly independent, but this does not hold for every l + 1 columns, then A has k rank k A = l. Theorem 1: Let X be a three-way tensor. X can be decomposed into A,B and C uniquely up to permutation and scaling ambiguities if k A + k B + k C 2F + 2 Under Theorem 1, the tensor can be decomposed into: Â = APΛ 1 ˆB = BPΛ2 Ĉ = CPΛ 3 Λ 2 Λ 1 Λ 3 = I CSPL Drexel University Asilomar 05 p. 5/1

6 Problem Formulation Let us consider a N i -input N o -output LTI system: x(k) = L 1 l=0 h(l)s(k l) + n(k) Let H(k) (N o N i ) be the N-point DFT of h(n) H(k) = L 1 n=0 h(n)e j 2π N kn, k = 0,..., N 1 where N > L Our goal is to estimate H(k) based on the system output up to some inherent ambiguities: Ĥ(k) = H(k)PΛe j 2π N km The input can be recovered within a scalar ambiguity and a circular shift. Also, the inputs would be recovered in some unknown order. CSPL Drexel University Asilomar 05 p. 6/1

7 Review of the SPD method Here are the assumptions made in the SPD method [Y.Yu, A.P.Petropulu, 2005]: A1) Each s i (.) is a zero mean, non-symmetrically distributed, i.i.d., stationary process with nonzero skewness. A2) The matrix H(k) is invertible for all k = 0,..., N 1. A3) The k-rank of H(k) satisfies: 3k H 2N i + 2 for every k. A4) n i (.), i = 1,..., N o are zero mean Gaussian stationary random processes with variance σ 2 n, mutually independent and independent of the inputs. Assumption (A2)is satisfied if the channel taps in matrix h(n) are independent, where h(n) is full-rank with probability one. A full rank matrix is also full k-rank. In that case, condition (A3) is equivalent to 3min(N i, N o ) 2N i + 2. For N i N o this is satisfied for N i 2. CSPL Drexel University Asilomar 05 p. 7/1

8 SPD: Channel Estimation Under assumptions (A1) and (A4), the third-order cross-cumulants of the system outputs c 3 lij (τ, ρ) equals: Cum[x l (k), x i (k+τ), x j (k+ρ)] = The cross-bispectrum equals: C 3 lij(k 1, k 2 ) = N i p=1 N i p=1 L 1 γs 3 p m=0 h lp (m)h ip(m+τ)h jp (m+ρ) γ 3 s p H lp ( k 1 k 2 )H ip( k 1 )H jp (k 2 ), k 1, k 2 = 0,..., N 1 The l-th slice of the tensor C 3 (k 1, k 2 ) (N o N o N o ) equals: C 3 l (k 1, k 2 ) = H ( k 1 )Γ 3 D l [H( k 1 k 2 )]H T (k 2 ) where Γ 3 = Diag{γ 3 s 1,..., γ 3 s Ni }. CSPL Drexel University Asilomar 05 p. 8/1

9 SPD: Channel Estimation First we apply the PARAFAC decomposition to the tensor C 3 ( m + rδ, δ). Under assumption (A3) and via Theorem 1, the tensor can be decomposed into: Â 0 = H(m δ)pλ1, ˆB 0 = H (m)γ 3 PΛ 2, Ĉ 0 = H(δ)PΛ3 For r = [1, 2,...2N 1], define: Â l (r) = (Â (r 1)) 1 C 3 l ( m + rδ, δ)(ĉ T 0 ) 1, l = 1,..., N o Â(r) = [diag(â 1 (r)),..., diag(â No (r))] T Â(0) = Â 0 It can be shown that: Â(r) = H(m rδ δ)pk ((r))2 e j(φ 1+rΦ 2 ) where Φ 1,Φ 2 are diagonal matrices, K 1,K 0 are diagonal matrices with positive elements, and ((.)) 2 denotes modulo 2 operation. CSPL Drexel University Asilomar 05 p. 9/1

10 SPD: Solve the Phase Ambiguity Let us consider N to be even, and δ to be co-prime to N, then H(m rδ δ) can be obtained within trivial ambiguities as: Ĥ(m rδ δ) = Â(r)[Â 1 (N + i)â(i)] r/n, r = 0,..., N 1 = H(m rδ δ)pk ((r)) 2 e j(φ 1+ 2π N kr) Let us take m δ = Rδ, where R is some integer. We get: Ĥ( rδ) = H( δr)pk ((r+r)) 2 e j(φ 1 + 2π N kr) Applying an N/2-point IDFT on the even samples of Ĥ( ((r)) Nδ): ĥ(n) = h δ ((n + k)) N/2 PK ((R)) 2 e jφ 1 which is an upsampled by δ version of h(n) circularly shifted by k. By downsampling ĥ(n) by δ we can get a circularly shifted version of h(n). CSPL Drexel University Asilomar 05 p. 10/1

11 Proposed ISPD method Drawback of the SPD method: In each iteration, uses only one slice of output tensor in order to recover one row of the channel response matrix, not robust. Require the MIMO system has N o N i, limit the applications. Here we propose an approach (Improved SPD) that fully exploits the information in the output tensor, and can recover the entire channel response matrix. The proposed method not only achieves lower error values but also becomes applicable to MIMO systems with more inputs than outputs. CSPL Drexel University Asilomar 05 p. 11/1

12 ISPD method, continued Let us stack the matrices C 3 i ( m + rδ, δ) for i = 1,..., N o to form a tall matrix U A (r). It holds: U A (r) = (B r C r )A T r, (N 2 o N o ) where is the Khatri-Rao(column-wise Kronecker) Product. Instead of assumption (A2), we have (B2): The Khatri-Rao product (H(k 1 ) H(k 2 )) has left inverse for all k 1, k 2 = 0,..., N 1. Under (B2), we can solve A r as: A r = ((B r C r ) 1 U A (r)) T where (B r C r ) 1 is the left pseudo inverse of (B r C r ). CSPL Drexel University Asilomar 05 p. 12/1

13 ISPD method, continued Let us define the iteration of ISPD method for r = 1,..., N as: Â(r) = ((Â (r 1) Ĉ0) 1 U A (r)) T Â(0) = Â0 It can be shown that (see Appendix): Â(r) = H(m rδ δ)pk ((r))2 e j(φ 1+rΦ 2 ), r = 0,..., N where Φ 1,Φ 2 are diagonal matrices, K 1,K 0 are diagonal matrices with positive elements, and ((.)) 2 denotes modulo 2 operation. Let us consider N to be even and co-prime to δ. The phase ambiguity can be solved in the same manner as in the SPD method. CSPL Drexel University Asilomar 05 p. 13/1

14 Simulation The inputs were taken to be i.i.d. single-sided exponentially distributed; the additive noise processes were white, zero-mean, complex Gaussian with identical variances. The cross-polyspectrum was estimated via the indirect class method, and the sample cross-cumulant estimate was windowed by a Hamming window, and the data length used to obtain the cross-cumulant estimates is denoted by T. Each bandlimited subchannel (i,j=1,2) was generated by: h ij (n) = r 1 c(0.25(n 10), 0.11)+r 2 c(0.25(n 6), 0.11)+r 3 (0.25(n 8), 0.11) where c(n, α) is a raised cosine function with delay m and rolloff α, and r i s are zero-mean Gaussian random variables. The channel length is set to L=6, and the window size used to estimate the cumulant is set to L e > L. CSPL Drexel University Asilomar 05 p. 14/1

15 Simulation The metric used here is the overall NMSE (ONMSE), ONMSE ij = 1 N i N o N o N i i=1 j=1 1 Mc M c l=1 Le 1 k=0 (ĥij(k) h ij (k)) 2 Le 1 k=0 (h ij(k)) 2 A lower bound of the ONMSE was generated by assuming input is known and obtaining the channel estimate by cross-correlating input and output. Each curve is based on 50 Monte-Carlo runs, and we took L e = 10, T = 8000 in all methods. CSPL Drexel University Asilomar 05 p. 15/1

16 Simulation Results SPD T=4k, Le=10 NSPD T=4k, Le=10 SPD T=8k, Le=10 NSPD T=8k, Le=10 SPD T=8k, Le=6 NSPD T=8k, Le=6 lower bound ONMSE SNR ONMSE Comparison with different T and L e CSPL Drexel University Asilomar 05 p. 16/1

17 Simulation Results 1 Cum Prob method of [7] method of [4] SPD ISPD lower bound ONMSE Cumulative distribution of ONMSE over 50 independent channels CSPL Drexel University Asilomar 05 p. 17/1

18 Conclusions We present a more robust iterative scheme under the frequency domain framework for the identification of a multiple-input multiple-output (MIMO) system driven by white, mutually independent unobservable inputs. The proposed SPD method, requires only one PARAFAC decomposition. The frequency response samples are then obtained via a recursive scheme. The proposed SPD method does not require prewhitening, which will increase the global length of the channel response, and can achieves lower ONMSE. The proposed ISPD method, can achieve lower ONMSE value compare with the SPD method, and can be even applied to systems with more inputs than outputs. CSPL Drexel University Asilomar 05 p. 18/1

19 Appendix I A(1) = ((A (0) Ĉ0) 1 U A (1)) T = ((A 0P 0Λ 10 C 0 P 0 Λ 30 ) 1 U A (1)) T = ((B 1 (Γ 3 ) 1 P 0 Λ 10 C 1 P 0 Λ 30 ) 1 U A (1)) T = ((B 1 P 0 (Γ 3 p ) 1 Λ 10 C 1 P 0 Λ 30 ) 1 U A (1)) T = (((B 1 C 1 )P 0 (Γ 3 p) 1 Λ 10Λ 30 ) 1 U A (1)) T = ((P 0 (Γ 3 p) 1 Λ 10Λ 30 ) 1 (B 1 C 1 ) 1 U A (1)) T = (Λ 1 30 (Λ 10) 1 Γ 3 pp T 0 A T 1 ) T = A 1 P 0 Λ 20 Γ 3 p ej(φ 10 Φ 30 ) Similarly, based on C 3 l ( m + rδ, δ) we can get: A(r) = H(m rδ δ)p 0 K r e j(φ 10+r(Φ Γ 3 p Φ 30 )), where K r = Λ 20 Γ 3 p for r odd Λ 10 for r even CSPL Drexel University Asilomar 05 p. 19/1