AR - R xx [0] R xx [1] R xx [2] ...R xx [ p ] R xx [1] R xx...

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On Autoregressive Processes Autoregressive (AR) random signals are signals derived from white noise with an all-pole power spectral density of the form: P xx ( z ) = σ 2 v A ( z ) A * ( 1 z * ) , where v [ n ] is the innovations process that drives the linear system with the monic minimum phase system function H ( z ) = 1 A ( z ) . In the time-domain these signals satisfy the difference equation: x [ n ] + p X k =1 a p [ k ] x [ n - k ] + v [ n ] . The corresponding recursion for the autocorrelation sequence is given by: R xx [ k ] + p X r =1 a p [ r ] R xx [ k - r ] = σ 2 v δ [ k ] . For k > 0, the right hand side reduces to zero and therefore: R xx [ k ] + p X r =1 a p [ r ] R xx [ k - r ] = 0 . (1) For k = 0, the expression yields the result: σ 2 v = R xx [0] + p X r =1 a p [ r ] R xx [ r ] . (2) These two equations can be combined into a matrix framework of the form:
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Unformatted text preview: R xx [0] R xx [1] R xx [2] ...R xx [ p ] R xx [1] R xx [0] R xx [1] ...R xx [ p-1] . . . . . . . . . . . . R xx [ p ] R xx [ p-1] R xx [ p-2] ...R xx [0] 1 a p [1] a p [2] . . . a p [ p ] = 2 v . . . . (3) In a compact matrix form we can rewrite this system as: R xx a = 2 v 1 , This equation system forms the basis for the LevinsonDurbin recursion and will also be the basis for the development of adaptive lattice structures later in the course....
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