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  • richb's DSP

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    By: Richard BaraniukAs a part of collection:"Adaptive Filters"

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Adaptive System Identification

Module by: Douglas L. Jones

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 j-i= r xx 0r1……rM-1|rMrM+1…r1r0⋱⋱⋮|⋮⋮…r2r1⋱⋱⋮|⋮⋮…⋮⋮…r0r1|r2r3…rM-1rM-2…r1r0|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-1Rh=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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