Lecture 4 1. Block-based LMS 3. Block-based LMS 4. Standard LMS 2. bw(n + 1) = bw(n) + µu(n)e (n), bw(k + 1) = bw(k) + µ u(kl + i)e (kl + i),

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1 Standard LMS 2 Lecture 4 1 Lecture 4 contains descriptions of Block-based LMS (7.1) (edition 3: 1.1) u(n) u(n) y(n) = ŵ T (n)u(n) Datavektor 1 1 y(n) M 1 ŵ(n) M 1 ŵ(n + 1) µ Frequency Domain LMS (FDAF, ) (edition 3: ) J(n) = u(n)e(n) e(n) Σ + d(n) Block-based LMS 3 Block-based LMS 4 Instead of updating the filter vector for every sample as for the standard LMS, bw(n + 1) = bw(n) + µu(n)e (n), the filter vector is updated once every lth sampel, L 1 X bw(k + 1) = bw(k) + µ u(kl + i)e (kl + i), i= where the sample index n and block index k are related as The gradient estimation can in this case be written as L 1 X Φ(k) = u(kl + i)e (kl + i), i= which is referred to as a time averaged gradient vector. Instead of updating the filtervector with a gradient vector at every sample, the filter vector is updated for every Lth sample with the sum (which is a weighted average) of the gradient vectors for the last L samples. n = kl + i.

2 Block LMS 5 { }} { 1 M u(n) u(k) = ŵ (k)u(k) = [ y(km + M 1),..., y(km)] T Datamatris M L Convergence properties for the Block-LMS 6 ŵ(k) M 1 ŵ(k + 1) µ Both the LMS and the Block-LMS minimizes J(n) = E{ e(n) 2 }. Both the LMS and the Block-LMS converges towards the Wiener solution. The Block-LMS uses a better estimate of the gradient. This will, however, not result is a faster convergence. J(k) = u (n)e T (n) = [e(km + M 1),..., e(km)] Σ + d(k) = [d(km + M 1),..., d(km)] Convergence properties for the Block-LMS 7 Convergence properties for the Block-LMS 8 The convergence criteria for the Block-LMS is < µ < 2 Lλmax. This upper limit for µ makes the Block-LMS converge slower than the LMS, especially for large eigenvalue spread. If a block length of L is chosen to increase the calculations speed, the Block-LMS may become slower in converence speed because of the stricter limit of µ. The natural choice of the block length (L) is the filter length (M). If L > M the gradient estimation is based on more information than the filter If L < M the entire filter is not used because not enough samples are included in the block. Misadjustment is the same as for the LMS.

3 Summary of the Block-LMS 9 Two strategies for reduction of the complexity 1 1. y(kl + i) = bw T (k)u(kl + i), L st = bw T (k)u(k). 2. e(kl + i) = d(kl + i) y(kl + i) = d(k). 3. bw(k + 1) = bw(k) + µ P L 1 i= u(kl + i)e(kl + i) bw(k + 1) = bw(k) + µu(k)e T (k). 1. Adaptive IIR filters can generate long impulse responses for few weights stability problems 2. Frequency Domain Adaptive Filters (FDAF). This strategy is based on the Block-LMS but the heavy calculations are made in the frequency domain. These methods can also be used to improve the convergence properties pof the LMS algorithm. In applications with long filters, e.g. echo cancellation, the complexity becomes very high. The equations that take time are the filtering in (1) and the cross correlation in (3). FDAF 11 Problem 12 In order to increase the calculation speed of the LMS algorithm, the filtering (convolution) and the gradient estimation (crosscorrelation) can be done in the frequency domain instead of the time domain. Strategy: 1. FFT of input and error signals 2. Convolution and crosscorrelation corresponds to multiplication in the frequency domain 3. IFFT Multiplication in the frequency domain corresponds to circular convolution, butr in order to maintain the properties of the LMS, linear convolution must be used. Filtering must be done with linear convolution. Gradient estimation should be done with linear convolution. Then the method is called Fast LMS. If linear convolution is not used here the method is called Unconstrained FDAF. Two advantages of the FDAF: 1. Faster calculation 2. Independent coefficients

4 Fast LMS (FDAF med gradientvillkor) 13 [u(k 1), u(k)] u(n) Datavektor FFT U(k) 1 2M diag Y(k) IFFT Spara sista blocket Properties of the Fast LMS 14 U H (k) = diag(fft([u R (k), u R (k 1)])) Ŵ(k) Ŵ(k + 1) α FFT φ(k) Lägg till nollblock sist φ(k) Spara första blocket IFFT ( ) D(k) Lägg till FFT nollblock först Σ E(k) + Fast LMS is based on the Block-LMS and converges similarly. To calculate and update the filter in the frequency domain opens new possibilities. In the LMS, each filter weight represents a mix of the different eigenmodes. In the Fast LMS, each filter weight is directly connected to as certain eigenmode (frequency range). The filter weights of the FastLMS is therefore updated independently of each other. d(k) = [d(km),..., d(km + M 1)] T Properties of the Fast LMS, cont. 15 Fast LMS, Update equations 16 The convergence speed for the Fast LMS can be optimized for each mode separately. The convergence speed for the i:th mode depends on µλ i. A measure of λ i is the average power in the frequency bin of the i:th mode, P i = U i 2. If µ i = α P i, all modes will converge equally fast (WSS). If the imput signal is not WSS, P i must be estimated recursively P i (k) = γp i (k 1) + (1 γ) U i (k) 2 The stepsize parameter µ is here substituted by a diagonal 2M 2M matrix µ = αd(k), where D(k) = diag[p 1, P 1 1,..., P 1 2M 1 ]. U(k) = diag(fftˆu((k 1)M)...u(kM 1), u(km)...u((k+1)m 1) T ) = Last M elements of IFFT[U(k)c W(k)] d(k) = ˆd(kM) d(km + 1)... d((k+1)m 1) T = d(k)» E(k) = FFT P(k) = γp(k 1) + (1 γ)u H (k)u(k) D(k) = P 1 h (k) = diag P 1 (k) P 1 1 (k)... i P 1 2M 1 (k) φ(k) = First M elements of IFFT[D(k)U H (k)e(k)]» cw(k + 1) = W(k) c φ(k) + αfft

5 FDAF utan gradientvillkor 18 Unconstrained FDAF 17 u(n) Datavektor 1 2M [u(k 1), u(k)] FFT diag U(k) Y(k) IFFT Spara sista blocket Three out of the five FFT/IFFT operations in the Fast LMS is used to perform the filtering with linear convolution (required). Ŵ(k + 1) Ŵ(k) Two out of the five FFT/IFFT operation in the Fast LMS is used to perform the gradient estimation with linear convolution. This is called the time-domain constraint. α FDAF can be used without satisfying this constraint, with gradient estimation based on circular convolution. Then the update equeation becomes cw(k + 1) = c W(k) + µu H (k)e(k) U H (k) = diag(fft([u R (k), u R (k 1)])) ( ) Lägg till FFT nollblock först Σ E(k) + d(k) = [d(km),..., d(km + M 1)] T Properties of the Unconstrained FDAF 19 Suggested reading 2 Does not converge towards the Wiener solution. Larger misadjustment. Poorer gradient estimation results in twice as many iterations in order to reach the same misadjustment as the Fast LMS. Faster calculations. Haykin chap. 7 ( for extra depth) edition 3: 1 ( for extra depth) Exercises: Haykin exercise 7.2, edition 3: 1.2 Computer exercise theme: Apply FastLMS to echo cancellation of speech signal.