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Adaptive System Identification

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

goal:

To approximate an unknown system (or the behavior of that system) as closely as possible
Figure 1
Figure 1 (fig1AdaptiveSysID.png)

The optimal solution is R-1P=W 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 d k x k h k i 0 x k - i h i Now

P=( E d k x k - j )=( E i =0 x k - i h i x k - j )=( i =0 h i E x k - i x k - j )=( i =0 r xx ji )=( r xx 0r1rM1|rMrM+1 r1r0| r2r1| r0r1|r2r3 rM1rM2r1r0|r1r2 )h0h1h2 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
(1)
If the adaptive filter HH is a length-MM FIR filter ( hm=hm+1==0 h m h m 1 0 ), this reduces to P=Rh-1 P R h and W opt =R-1P=R-1(Rh)=h W opt R P R R h h FIR adaptive system identification thus converges in the mean to the corresponding MM samples of the impulse response of the unknown system.

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