Adaptive IIR filters

Module by: Douglas L. Jones

Adaptive IIR filters are attractive for the same reasons that IIR filters are attractive: many fewer coefficients may be needed to achieve the desired performance in some applications. However, it is more difficult to develop stable IIR algorithms, they can converge very slowly, and they are susceptible to local minima. Nonetheless, adaptive IIR algorithms are used in some applications (such as low frequency noise cancellation) in which the need for IIR-type responses is great. In some cases, the exact algorithm used by a company is a tightly guarded trade secret.

Most adaptive IIR algorithms minimize the prediction error, to linearize the estimation problem, as in deterministic or block linear prediction. y k =∑n=1L v n k y k - n +∑n=0L w n k x k - n y k n 1 L v n k y k - n n 0 L w n k x k - n Thus the coefficient vector is W k = v 1 k v 2 k ⋮ v L k w 0 k w 1 k ⋮ w L k W k v 1 k v 2 k ⋮ v L k w 0 k w 1 k ⋮ w L k and the "signal" vector is U k = y k - 1 y k - 2 ⋮ y k - L x k x k - 1 ⋮ x k - L U k y k - 1 y k - 2 ⋮ y k - L x k x k - 1 ⋮ x k - L The error is ε k = d k - y k = d k - W k T U k ε k d k y k d k W k U k An LMS algorithm can be derived using the approximation E ε k 2= ε k 2 ε k 2 ε k 2 or ∇̂k=∂∂ W k ε k 2=2 ε k ∂∂ W k ε k =2 ε k ∂∂ v 1 k ε k ⋮∂∂ ε k w 1 k ⋮=-2 ε k ∂∂ v 1 k y k ⋮∂∂ v L k y k ∂∂ w 0 k y k ⋮∂∂ w L k y k ∇ k W k ε k 2 2 ε k W k ε k 2 ε k v 1 k ε k ⋮ ε k w 1 k ⋮ -2 ε k v 1 k y k ⋮ v L k y k w 0 k y k ⋮ w L k y k Now ∂∂ v i k y k =∂∂ v i k ∑n=1L v n k y k - n +∑n=0L w n k x k - n = y k - n +∑n=1L v n k ∂∂ v i k y k - n +0 v i k y k v i k n L 1 v n k y k - n n L 0 w n k x k - n y k - n n L 1 v n k v i k y k - n 0 ∂∂ w i k y k =∂∂ w i k ∑n=1L v n k y k - n +∑n=0L w n k x k - n =∑n=1L v n k ∂∂ w i k y k - n + x k - n w i k y k w i k n L 1 v n k y k - n n L 0 w n k x k - n n L 1 v n k w i k y k - n x k - n Note that these are difference equations in ∂∂ v i k y k v i k y k , ∂∂ w i k y k w i k y k : call them α i k =∂∂ w i k y k α i k w i k y k , β i k =∂∂ v i k y k β i k v i k y k , then ∇̂k= β 1 k β 2 k … β L k α 0 k … α L k T ∇ k β 1 k β 2 k … β L k α 0 k … α L k , and the IIR LMS algorithm becomes y k = W k T U k y k W k U k α i k = x k - i +∑j=1L v j k α i k - j α i k x k - i j L 1 v j k α i k - j β i k = y k - i +∑j=1L v j k β i k - j β i k y k - i j L 1 v j k β i k - j ∇̂k=-2 ε k β 1 k β 2 k … α 0 k α 1 k … α L k T ∇ k -2 ε k β 1 k β 2 k … α 0 k α 1 k … α L k and finally W k + 1 = W k -U∇̂k W k + 1 W k U ∇ k where the μμ may be different for the different IIR coefficients. Stability and convergence rate depends on these choices, of course. There are a number of variations on this algorithm.

Due to the slow convergence and the difficulties in tweaking the algorithm parameters to ensure stability, IIR algorithms are used only if there is an overriding need for an IIR-type filter.

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This work is licensed by Douglas L. Jones under a Creative Commons Attribution License (CC-BY 1.0).

Last edited by Kyle Clarkson on May 11, 2004 11:39 am GMT-5.