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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-1⁢P=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 ⁢0r⁢1……r⁢M−1|r⁢Mr⁢M+1… r⁢1r⁢0⋱⋱⋮|⋮⋮… r⁢2r⁢1⋱⋱⋮|⋮⋮… ⋮⋮…r⁢0r⁢1|r⁢2r⁢3… r⁢M−1r⁢M−2…r⁢1r⁢0|r⁢1r⁢2… )⁢h⁢0h⁢1h⁢2⋮ 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 ( h⁢m=h⁢m+1=…=0 h m h m 1 … 0 ), this reduces to P=R⁢h-1 P R h and W opt =R-1⁢P=R-1⁢(R⁢h)=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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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

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