4.1 Adaptive iir filters

Adaptive IIR filters are attractive for the same reasons that IIR filters are attractive: many fewer coefficients may be neededto achieve the desired performance in some applications. However, it is more difficult to develop stable IIRalgorithms, they can converge very slowly, and they are susceptible to local minima. Nonetheless, adaptive IIRalgorithms are used in some applications (such as low frequency noise cancellation) in which the need for IIR-type responses isgreat. 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. $$y_k=\sum_{n=1}^L v_n^k{y}_{k-n}+\sum_{n=0}^L w_n^k{x}_{k-n}$$ Thus the coefficient vector is $$W_k=\left(\begin{array}{c}{v}_1^k\\ v_2^k\\ \\ v_L^k\\ w_0^k\\ w_1^k\\ \\ w_L^k\end{array}\right)$$ and the "signal" vector is $$U_k=\left(\begin{array}{c}{y}_{k-1}\\ y_{k-2}\\ \\ y_{k-L}\\ x_k\\ x_{k-1}\\ \\ x_{k-L}\end{array}\right)$$ The error is $${}_k=d_k-y_k=d_k-W_k^T{U}_k$$ An LMS algorithm can be derived using the approximation $({}_k^2)={}_k^2$ or $$({}_k)=\frac{\partial^1{}_k^2}{\partial W_k}=2{}_k\frac{\partial^1{}_k}{\partial W_k}=2{}_k\left(\begin{array}{c}\frac{\partial^1{}_k}{\partial v_1^k}\\ \\ \frac{\partial^1{w}_1^k}{\partial {}_k}\\ \end{array}\right)=-2{}_k\left(\begin{array}{c}\frac{\partial^1{y}_k}{\partial v_1^k}\\ \\ \frac{\partial^1{y}_k}{\partial v_L^k}\\ \frac{\partial^1{y}_k}{\partial w_0^k}\\ \\ \frac{\partial^1{y}_k}{\partial w_L^k}\end{array}\right)$$ Now $$\frac{\partial^1{y}_k}{\partial v_i^k}=\frac{\partial^1\sum_{n=1}^L v_n^k{y}_{k-n}+\sum_{n=0}^L w_n^k{x}_{k-n}}{\partial v_i^k}=y_{k-n}+\sum_{n=1}^L v_n^k\frac{\partial^1{y}_{k-n}}{\partial v_i^k}+0$$ $$\frac{\partial^1{y}_k}{\partial w_i^k}=\frac{\partial^1\sum_{n=1}^L v_n^k{y}_{k-n}+\sum_{n=0}^L w_n^k{x}_{k-n}}{\partial w_i^k}=\sum_{n=1}^L v_n^k\frac{\partial^1{y}_{k-n}}{\partial w_i^k}+x_{k-n}$$ Note that these are difference equations in $\frac{\partial^1{y}_k}{\partial v_i^k}$ , $\frac{\partial^1{y}_k}{\partial w_i^k}$ : call them ${}_i^k=\frac{\partial^1{y}_k}{\partial w_i^k}$ , ${}_i^k=\frac{\partial^1{y}_k}{\partial v_i^k}$ , then $({}_k)=\begin{pmatrix}{}_1^k & {}_2^k & & {}_L^k & {}_0^k & & {}_L^k\\ \end{pmatrix}^T$ , and the IIR LMS algorithm becomes $$y_k=W_k^T{U}_k$$ $${}_i^k=x_{k-i}+\sum_{j=1}^L v_j^k{}_i^{k-j}$$ $${}_i^k=y_{k-i}+\sum_{j=1}^L v_j^k{}_i^{k-j}$$ $$({}_k)=-2{}_k\begin{pmatrix}{}_1^k & {}_2^k & & {}_0^k & {}_1^k & & {}_L^k\\ \end{pmatrix}^T$$ and finally $$W_{k+1}=W_k-U({}_k)$$ where the $$ may be different for the different IIR coefficients. Stability and convergence rate depends on thesechoices, 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 areused only if there is an overriding need for an IIR-type filter.