The Least mean squares filter solution converges to the Wiener filter solution, assuming that the unknown system is LTI and the noise is stationary. Both filters can be used to identify the impulse response of an unknown system, knowing only the original input signal and the output of the unknown system. By relaxing the error criterion to reduce current sample error instead of minimizing the total error over all of n, the LMS algorithm can be derived from the Wiener filter.
Derivation of the Wiener filter for system identification Given a known input signal s [ n ] {\displaystyle s[n]} , the output of an unknown LTI system x [ n ] {\displaystyle x[n]} can be expressed as:
x [ n ] = ∑ k = 0 N − 1 h k s [ n − k ] + w [ n ] {\displaystyle x[n]=\sum _{k=0}^{N-1}h_{k}s[n-k]+w[n]}
where h k {\displaystyle h_{k}} is an unknown filter tap coefficients and w [ n ] {\displaystyle w[n]} is noise. The model system x ^ [ n ] {\displaystyle {\hat {x}}[n]} , using a Wiener filter solution with an order N, can be expressed as:
x ^ [ n ] = ∑ k = 0 N − 1 h ^ k s [ n − k ] {\displaystyle {\hat {x}}[n]=\sum _{k=0}^{N-1}{\hat {h}}_{k}s[n-k]}
where h ^ k {\displaystyle {\hat {h}}_{k}} are the filter tap coefficients to be determined. The error between the model and the unknown system can be expressed as:
e [ n ] = x [ n ] − x ^ [ n ] {\displaystyle e[n]=x[n]-{\hat {x}}[n]}
The total squared error E {\displaystyle E} can be expressed as:
E = ∑ n = − ∞ ∞ e [ n ] 2 {\displaystyle E=\sum _{n=-\infty }^{\infty }e[n]^{2}}
E = ∑ n = − ∞ ∞ ( x [ n ] − x ^ [ n ] ) 2 {\displaystyle E=\sum _{n=-\infty }^{\infty }\left(x[n]-{\hat {x}}[n]\right)^{2}}
E = ∑ n = − ∞ ∞ ( x [ n ] 2 − 2 x [ n ] x ^ [ n ] + x ^ [ n ] 2 ) {\displaystyle E=\sum _{n=-\infty }^{\infty }\left(x[n]^{2}-2x[n]{\hat {x}}[n]+{\hat {x}}[n]^{2}\right)}
Use the Minimum mean-square error criterion over all of n {\displaystyle n} by setting its gradient to zero:
∇ E = 0 {\displaystyle \nabla E=0}
which is
∂ E ∂ h ^ i = 0 {\displaystyle {\frac {\partial E}{\partial {\hat {h}}_{i}}}=0} for all i = 0 , 1 , 2 , . . . , N − 1 {\displaystyle i=0,1,2,...,N-1}
… excerpt ends here. Continue reading the full article.
