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Beyond LMS: an overview of other adaptive filter algorithms

Module by: Douglas L. Jones. E-mail the author

RLS algorithms

FIR adaptive filter algorithms with faster convergence. Since the Wiener solution can be obtained on one step by computing W opt =R-1⁢P W opt R P , most RLS algorithms attept to estimate R-1 R and PP and compute W opt W opt from these.

There are a number of O⁢N2 O N 2 algorithms which are stable and converge quickly. A number of O⁢N O N algorithms have been proposed, but these are all unstable except for the lattice filter method. This is described to some extent in the text. The adaptive lattice filter converges quickly and is stable, but reportedly has a very high noise floor.

Many of these approaches can be thought of as attempting to "orthogonalize" RR, or to rotate the data or filter coefficients to a domain where RR is diagonal, then doing LMS in each dimension separately, so that a fast-converging step size can be chosen in all directions.

Frequency-domain methods

Frequency-domain methods implicitly attempt to do this:

Figure 1
Figure 1 (fig1BeyondLMS.png)
If Q⁢R⁢Q-1 Q R Q is a diagonal matrix, this yields a fast algorithm. If QQ is chosen as an FFT matrix, each channel becomes a different frequency bin. Since RR is Toeplitz and not a circulant, the FFT matrix will not exactly diagonalize RR, but in many cases it comes very close and frequency domain methods converge very quickly. However, for some RR they perform no better than LMS. By using an FFT, the transformation QQ becomes inexpensive O⁢N⁢logN O N N . If one only updates on a block-by-block basis (once per NN samples), the frequency domain methods only cost O⁢logN O N computations per sample. which can be important for some applications with large NN. (Say 16,000,000)

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