4F7 Adaptive Filters (and Spectrum Estimation) Least Mean Square (LMS) Algorithm Sumeetpal Singh Engineering Department

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1 4F7 Adaptive Filters (and Spectrum Estimation) Least Mean Square (LMS) Algorithm Sumeetpal Singh Engineering Department 1

2 1 Outline The LMS algorithm Overview of LMS issues concerning stepsize bound, convergence, misadjustment Some simulation examples The normalised LMS (NLMS) Further topics 2

3 2 Least Mean Square (LMS) Steepest Descent (SD) was h (n + 1) = h (n) µ J (h (n)) 2 = h (n) + µe {u (n) e (n)} Often J (h (n)) = 2E {u (n)e(n)} is unknown or too difficult to derive Remedy is to use the instantaneous approximation 2u (n)e(n) for J (h (n)) Using this approximation we get the LMS algorithm e (n) = d (n) h T (n)u(n), h (n + 1) = h (n) + µe (n)u(n) 3

4 This is desirable because We do not need knowledge of R and p anymore If statistics are changing over time, it adapts accordingly Complexity: 2M + 1 multiplications and 2M additions per iteration. Not M 2 multiplications like SD Undesirable because we have to choose µ when R not known, subtle convergence analysis 4

5 3 Application: Noise Cancellation Mic 1: Reference signal d(n) = s(n) }{{} signal of interest + v(n) }{{} noise s (n) and v (n) statistically independent Aim: recover signal of interest Method: use another mic, Mic 2, to record noise only, u(n) Obviously u (n) v (n) but u (n) and v (n) are hopefully correlated Now filter recorded noise u (n) to make it more like v(n) Recovered signal is e(n) = d(n) h(n) T u(n) and not y(n) Run Matlab demo on webpage 5

6 We are going to see an example with speech s(n) generated as a mean 0 variance 1 Gaussian random variable Mic 1 s noise was 0.5 sin(n π ) Mic 2 s noise was 10 sin(n π 2 ) Mic 1 and 2 s noise are both sinusoids but with different amplitudes and phase shifts You could increase the phase shift but you will need a larger value for M Run Matlab demo on webpage 6

7 4 LMS convergence in mean Write the reference signal model as d (n) = u T (n)h opt + ε (n) where h opt denotes the optimal vector (Wiener filter) that h (n) should converge to and ε (n) = d(n) u T (n)h opt For this reference signal model, the LMS becomes h (n + 1) = h (n) + µu (n) ( ) u T (n)h opt + ε (n) u T (n)h(n) = h (n) + µu (n)u T (n) ( h opt h (n) ) + µu (n) ε (n) ( ) (h h (n + 1) h opt = I µu (n)u T ) (n) (n) hopt + µu (n) ε (n) This looks like the SD recursion h (n + 1) h opt = (I µr) ( h (n) h opt ) 7

8 Introducing the expectation operator gives E { } h (n + 1) h {( opt ) (h = E I µu (n)u T ) } (n) (n) hopt + µe } {u (n) {{ ε (n)} } ( =0 { }) I µe u (n)u T (n) E { } h (n) h opt (Independence assumption) = (I µr)e { h (n) h opt } Independence assumption states that h (n) h opt is independent of u (n)u T (n) since h (n) function of u (0), u (1),..., u (n 1) and all previous desired signals, only true if {u (n)} is an independent sequence of random vectors but not so otherwise. Assumption is better justified for a block LMS type update scheme where the filter is updated at multiples of some block length L, i.e. when n = kl and not otherwise 8

9 We are back to the SD scenario and so E {h (n)} h opt if 0 < µ < 2 λ max Assuming h (n), being a function of u (0), u (1),..., u (n 1), is independent of u (n) is not true. However, behaviour predicted using this simplifying assumption agrees with experiments and computer simulations The point of the LMS was that we don t have access to R, so how to compute λ max? Solution by a time average: using the fact that for square matrices tr(ab) =tr(ba), and R = QΛQ T, 9

10 M λ k = tr (Λ) k=1 ( ) = tr ΛQ T Q = tr (QΛQ T) = tr (R) { } = ME u 2 (n) and we see that λ max < M k=1 λ k = ME { u 2 (n) } Note that we can estimate E { u 2 (n) } by a simple sample average and the new tighter bound on the stepsize is 2 0 < µ < ME { u 2 (n) } < 2 λ max With a fixed stepsize, {h (n)} n 0 will never settle at h opt, but rather 10

11 oscillate about h opt. Even if h (n) = h opt then h (n + 1) h opt = µu (n) e (n), and because u (n)e(n) is random, h (n + 1) will move away from h opt Assume d(n) = u(n) T h opt + ε(n) where ε(n) is a zero mean independent noise sequence. Best case for E { e 2 (n) }, using e(n) = u T (n)h opt + ε (n) u T (n)h(n), is when h (n) = h opt, so that E { e 2 (n) } = E { ε 2 (n) } (= J min ). However, since h (n) h opt, we would expect E { e 2 (n) } > J min. We call E { e 2 (n) } J min J min the misadjustment 11

12 5 LMS main points Simple to implement Works fine is many applications if filter order and stepsize is chosen properly There is a trade-off effect with the stepsize choice large µ yields better tracking ability in a non-stationary environment but has large misadjustment noise small µ has poorer tracking ability but has smaller misadjustment noise 12

13 6 Adaptive stepsize: Normalised LMS (NLMS) We showed that LMS was stable provided 2 µ < ME { u 2 (n) } What if E { u 2 (n) } varied, which would be true for a non-stationary input signal LMS should be able to adapt its step-size automatically The instantaneous estimate of ME { u 2 (n) } is u T (n)u(n) µ Now replace the LMS stepsize with u T (n)u(n) = µ u(n) 2 where µ is a constant that should < 2, e.g., 0.25 < µ < We make µ smaller because of the poor quality estimate for ME { u 2 (n) } in the denominator This choice of stepsize gives the Normalized Least Mean Squares 13

14 (NLMS) e (n) = d (n) u T (n)h(n) µ h (n + 1) = h (n) + u (n) 2e (n)u(n) where µ is relabelled to µ. NLMS is the LMS algorithm with a datadependent stepsize There is another interpretation of the NLMS. It can be shown h (n + 1) is the solution to the following optimization problem (Example Sheet): h (n + 1) = arg min h h (n) h s.t. d (n) u T (n)h = 0 In light of new data, the parameters of an adaptive system should only be perturbed in a minimal fashion. Another motivation for the NLMS is to reduce the sensitivity of the filter update to large amplitude fluctuations in u (n) But small amplitudes will now adversely effect the NLMS. The solution 14

15 is µ h (n + 1) = h (n) + u (n) 2 + ǫ e (n)u(n) where ǫ is a small constant, e.g Comparing NLMS and LMS Compare the stability of the LMS and NLMS for different values of stepsize. You will see that the NLMS is stable for 0 < µ < 2. You will still need to tune µ to get the desired convergence behaviour (or fluctuations of h (n) once it has stabilized) though. Run the NLMS example on the course website 15

16 8 Nonlinear LMS LMS algorithm uses linear combination of u (n), u (n 1),..., u (n M + 1) (i.e. u (n)) to predict d (n) y (n) = h T (n)u(n) Why not nonlinear, e.g., quadratic y (n) = + M 1 i=0 M 1 i=0 h 1,i (n)u(n i) M 1 j=i h 2,i,j (n)u(n i)u(n j) This case can be written in the usual form y (n) = θ T (n) } {{ } Linear Parameters Ψ (n) } {{ } Function of u(n) and all we have done with the LMS can be applied 16

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