4.8 Practical issues in wiener filter implementation

The weiner-filter, $W_{\mathrm{opt}}=R^{(-1)}P$ , is ideal for many applications. But several issues must be addressed to use it in practice.

Tradeoffs

Larger $N$ more accurate estimates of the correlation valuesbetter $(W_{\mathrm{opt}})$ . However, larger $N$ leads to slower adaptation.

Computational considerations

As presented here, an adaptive filter requires computing a matrix inverse at each sample. Actually, since the matrix $R$ is Toeplitz, the linear system of equations can be sovled with $O(M^2)$ computations using Levinson's algorithm, where $M$ is the filter length. However, in many applications this may be too expensive, especiallysince computing the filter output itself requires $O(M)$ computations. There are two main approaches to resolving the computation problem

• Take advantage of the fact that $R^{(k+1)}$ is only slightly changed from $R^k$ to reduce the computation to $O(M)$ ; these algorithms are called Fast Recursive Least Squareds algorithms; all methods proposed so farhave stability problems and are dangerous to use.
• Find a different approach to solving the optimization problem that doesn't require explicit inversion of thecorrelation matrix.