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3.1 Adaptive system identification

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To approximate an unknown system (or the behavior of that system) as closely as possible

The optimal solution is R P W

Suppose the unknown system is a causal, linear time-invariant filter: d k x k h k i 0 ∞ x k - i h i Now

P d k x k - j i ∞ 0 x k - i h i x k - j i ∞ 0 h i x k - i x k - j i ∞ 0 h i r xx j i r xx 0 r 1 r M 1 | r M r M 1 r 1 r 0 | r 2 r 1 | r 0 r 1 | r 2 r 3 r M 1 r M 2 r 1 r 0 | r 1 r 2 h 0 h 1 h 2
If the adaptive filter H is a length- M FIR filter ( h m h m 1 0 ), this reduces to P R h and W opt R P R R h h FIR adaptive system identification thus converges in the mean to the corresponding M samples of the impulse response of the unknown system.
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Source:  OpenStax, Adaptive filters. OpenStax CNX. May 12, 2005 Download for free at http://cnx.org/content/col10280/1.1
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