Adaptive System Identification **new** 2003/07/09 13:08:05.119 GMT-5 2004/02/19 09:59:45.521 US/Central Douglas L. Jones dl-jones@uiuc.edu Douglas L. Jones dl-jones@uiuc.edu Kyle Clarkson kclarks@rice.edu 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.