System Identification and Adaptive Filtering in the Short-Time Fourier Transform Domain

Transcription

1 System Identification and Adaptive Filtering in the Short-Time Fourier Transform Domain Electrical Engineering Department Technion - Israel Institute of Technology Supervised by: Prof. Israel Cohen

2 Outline Introduction 1 Introduction Fullband identification Subband identification 2 Problem Formulation MSE analysis Discussion 3 Offline identification Adaptive identification 4

3 Fullband identification Subband identification Example: Acoustic echo cancellation DUET speakerphone, Poenix Audio Technologies.

4 Fullband identification Fullband identification Subband identification Identification of linear systems is a fundamental problem in many practical applications, including acoustic echo cancellation and beamforming. Let h(n) denote an impulse response of a linear system, whose input x(n) and output y(n) are related by y(n) = h(n) x(n) + ξ(n) d(n) + ξ(n), where d(n) is the desired signal and ξ(n) is an additive noise signal.

5 Fullband identification (cont.) Fullband identification Subband identification System identification problem: To derive an estimator ĥ(n) for the system impulse response, given x(n) and y(n). An adaptive algorithm is generally required for the identification process. Time-domain adaptive algorithms often suffer from slow convergence rate and high computational complexity due to long system impulse response.

6 Subband Identification Fullband identification Subband identification Alternatively, subband techniques are used, in which the desired signals are filtered into subbands, then decimated and processed in distinct subbands. The computational complexity is reduced and the convergence rate is improved due to the downsampling operation.

7 Cross-band filters Introduction Fullband identification Subband identification In the time-frequency domain we have y p,k = d p,k + ξ p,k. p - time index k - frequency index To perfectly represent h(n) in the STFT domain cross-band filters between the subbands are generally required: d p,k = N 1 M 1 k =0 p =0 x p p,k h p,k,k, where h p,k,k is the cross-band filter from frequency-band k to frequency-band k.

8 Cross-band filters (cont.) Fullband identification Subband identification Cross-band filters illustration:

9 Cross-band filters (cont.) Fullband identification Subband identification The cross-band filter h p,k,k depends on both h(n) and the STFT analysis/ synthesis parameters: where h p,k,k = { h(n) φ k,k (n) } n=pl φ k,k (n) m ψ(m)e j 2π N mk ψ(n + m)e j 2π N (n+m)k and ψ(n) and ψ(n) are the analysis and synthesis windows, respectively. For fixed k and k, the filter h p,k,k is noncasual in general, with N L 1 noncasual coefficients.

10 Cross-band filters (cont.) Fullband identification Subband identification Practically, relatively few cross-band filters need to be considered. For instance, A mesh plot of the cross-band filters h p,1,k for a synthetic impulse response. L denotes the decimation factor. For subband system identification: An estimator ĥ p,k,k for the cross-band filters is required.

11 Problem Formulation Problem Formulation MSE analysis Discussion Consider an offline system identification in the STFT domain. Let y k = [ y 0,k y 1,k y P 1,k ] T denote a time-trajectory of y pk at frequency-bin k. Let ĥp,k,k be an estimate of the cross-band filter h p,k,k, and let ˆd p,k be the resulting estimate of d p,k using only 2K + 1 cross-band filters around the frequency-band k: ˆd p,k = k+k M 1 k =k K p =0 ĥ p,k,k modnx p p,k modn

12 Problem Formulation (cont.) Problem Formulation MSE analysis Discussion Cross-band filters illustration:

13 Problem Formulation (cont.) Problem Formulation MSE analysis Discussion Let ˆ h be the 2K + 1 estimated filters at frequency-band k. Then, the estimated desired signal can be written in a vector form as ˆd k = kˆ h k, where k is the concatenation of the corresponding input Toeplitz matrices. LS optimization problem: LS solution: ˆ h k = arg min h k yk k h k 2 ( ˆ h k = H ) 1 k k H k y k

14 MSE analysis Introduction Problem Formulation MSE analysis Discussion The (normalized) MSE is defined by { dk } 2 E ˆd k ǫ k (K) = E { d k 2} Assumptions: x p,k and ξ p,k are zero-mean white Gaussian complex signals with variance σ 2 x and σ 2 ξ. x p,k and ξ p,k are statistically independent.

15 MSE analysis (cont.) Problem Formulation MSE analysis Discussion Let η = σ 2 x/σ 2 ξ denote the SNR. MMSE in the k-th frequency-band: ǫ k (K) = α k(k) η + β k (K) where β k (K) 1 [Avargel & Cohen, IEEE trans. Audio, Speech, Language Process.] M α k (K) 2 (2K + 1) P h k M (2K + 1) P 1 h k 2 2K m=0 hk,(k K+m)modN 2.

16 MSE analysis (cont.) Problem Formulation MSE analysis Discussion The resulting MMSE satisfies ǫ k (K + 1) > ǫ k (K), for η 0 (low SNR) ǫ k (K + 1) ǫ k (K), for η (high SNR) Let η k (K + 1 K) denote the SNR-intersection point of the curves ǫ k (K) and ǫ k (K + 1). η k (K K 1) η k (K + 1 K) η k (K + 1 K) 1 P (P is the length of x p,k in frequency-band k)

17 MSE analysis (cont.) Problem Formulation MSE analysis Discussion The resulting MMSE satisfies ǫ k (K + 1) > ǫ k (K), for η 0 (low SNR) ǫ k (K + 1) ǫ k (K), for η (high SNR) Let η k (K + 1 K) denote the SNR-intersection point of the curves ǫ k (K) and ǫ k (K + 1). η k (K K 1) η k (K + 1 K) ε k (K) ε k (K+1) η k (K + 1 K) 1 P MSE (P is the length of x p,k in frequency-band k) η k (K+1 K) SNR

18 Discussion Introduction Problem Formulation MSE analysis Discussion Increasing the number of cross-band filters not necessarily implies a lower steady-state MSE in subbands. As the SNR increases or as more data becomes available, additional cross-band filters can be estimated and a lower MMSE can be achieved. The input data length is restricted to enable tracking capability during time variations in the impulse response. Therefore, during fast time variations in the system, less cross-band filters are useful.

19 Experimental results Offline identification Adaptive identification Setup Sampling rate is 16 khz. An office impulse response with T 60 = 300 ms. A 16 ms length Hamming synthesis window with 50% overlap.

20 Experimental results (cont.) Offline identification Adaptive identification White Gaussian signals (k = 1): K = 0 K = 1 K = 2 K = 3 K = K = 0 K = 1 K = 2 K = 3 K = ε k (K) [db] 10 0 ε k (K) [db] 10 0 η 1 (1 0) η 1 (1 0) 10 η 1 (2 1) 10 η 1 (2 1) 20 η 1 (3 2) 20 η 1 (3 2) η 1 (4 3) η 1 (4 3) SNR [db] SNR [db] (a) P = 200 (b) P = 1000

21 Experimental results (cont.) Offline identification Adaptive identification Acoustic echo cancellation application: x(n) is a speech signal and the local disturbance ξ(n) consists of a zero-mean white Gaussian local noise. Performances are evaluated using the echo-return loss enhancement (ERLE): E { d 2 (n) } ERLE(K) = 10log { ( E d(n) ˆd ) } 2 K (n) where ˆd K (n) is the inverse STFT of the estimated echo signal using 2K + 1 cross-band filters.

22 Experimental results (cont.) Offline identification Adaptive identification Acoustic echo cancellation application: K = 0 K = 1 K = 2 K = 3 K = 4 Fullband K = 0 K = 1 K = 2 K = 3 K = 4 Fullband ERLE(K) [db] ERLE(K) [db] SNR [db] SNR [db] (a) P = 190 (b) P = 322

23 Experimental results (cont.) Offline identification Adaptive identification Adaptive identification: Let ĥ k (p) denote the 2K + 1 adaptive cross-band filters at frequency-band k. The LMS update formula: ĥ k (p + 1) = ĥ k (p) + µe p,k x k (p) 0 < µ < [tr(r k )] 1 where e p,k = y p,k ˆd p,k is the error signal, µ is the step-size and x k (p) is the input signal to the corresponding adaptive filters.

24 Experimental results (cont.) Offline identification Adaptive identification White signals: k = 0, µ = 0.3µ max, SNR 20 db

25 Experimental results (cont.) Offline identification Adaptive identification White signals: k = 0, µ = 0.3µ max, SNR 20 db K = 0 K = 1 K = 2 K = 3 30 ε p,k (K) [db] Iteration Number (p)

26 Introduction We have investigated the influence of cross-band filters on the performance of system identification in the STFT domain. The number of useful cross-band filters is shown to be dependent on the power and length of the input signal. An innovative and efficient adaptive algorithm with variable cross-band filters may be constructed to improve both convergence rate and steady-state MSE. The cross-band filtering research may be exploited for improving RTF identification and multichannel processing algorithms.

27 Thank you...