Statistical_and_Adaptive_Signal_Processing_Spectra..._----_(Pg_163--203).pdf

Autocorrelation in terms of model parameters if we

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Autocorrelation in terms of model parameters. If we normalize the autocorrelations in (4.2.31) by dividing throughout by r( 0 ) , we obtain the following system of equations Pa = − ρ (4.2.33) where P is the normalized autocorrelation matrix and ρ = [ ρ( 1 ) ρ( 2 ) · · · ρ(P ) ] H (4.2.34) is the vector of normalized autocorrelations. This set of P equations relates the P model coef fi cients with the fi rst P (normalized) autocorrelation values. If the poles of the all-pole fi lter are strictly inside the unit circle, the mapping between the P -dimensional vectors a and ρ is unique . If, in fact, we are given the vector a , then the normalized autocorrelation vector ρ can be computed from a by using the set of equations that can be deduced from (4.2.33) A ρ = − a (4.2.35) where A ij = a i j + a i + j , assuming a m = 0 for m < 0 and m > P (see Problem 4.6). Given the set of coef fi cients in a , ρ can be obtained by solving (4.2.35). We will see that, under the assumption of a stable H(z) , a solution always exists. Furthermore, there exists a simple, recursive solution that is ef fi cient (see Section 7.5). If, in addition to a , we are given d 0 , we can evaluate r( 0 ) with (4.2.20) from ρ computed by (4.2.35).Autocorrelation values r(l) for lags l > P are found by using the recursion in (4.2.18) with r( 0 ), r( 1 ), . . . , r(P ) . EXAMPLE 4.2.3. For the AP(3) model with real coef fi cients we have r( 0 ) r( 1 ) r( 2 ) r( 1 ) r( 0 ) r( 1 ) r( 2 ) r( 1 ) r( 0 ) a 1 a 2 a 3 = − r( 1 ) r( 2 ) r( 3 ) (4.2.36) d 2 0 = r( 0 ) + a 1 r( 1 ) + a 2 r( 2 ) + a 3 r( 3 ) (4.2.37) Therefore, given r( 0 ) , r( 1 ), r( 2 ), r( 3 ) , we can fi nd the parameters of the all-pole model by solving (4.2.36) and then substituting into (4.2.37). Suppose now that instead we are given the model parameters d 0 , a 1 , a 2 , a 3 . If we divide both sides of (4.2.36) by r( 0 ) and solve for the normalized autocorrelations ρ( 1 ), ρ( 2 ), and ρ( 3 ) , we obtain 1 + a 2 a 3 0 a 1 + a 3 1 0 a 2 a 1 1 ρ( 1 ) ρ( 2 ) ρ( 3 ) = − a 1 a 2 a 3 (4.2.38) The value of r( 0 ) is obtained from r( 0 ) = d 2 0 1 + a 1 ρ( 1 ) + a 2 ρ( 2 ) + a 3 ρ( 3 ) (4.2.39) If r( 0 ) = 2 , r( 1 ) = 1 . 6 , r( 2 ) = 1 . 2 , and r( 3 ) = 1, the Toeplitz matrix in (4.2.36) is positive de fi nite because it has positive eigenvalues. Solving the linear system gives a 1 = − 0 . 9063 , a 2 = 0 . 2500 , and a 3 = − 0 . 1563. Substituting these values in (4.2.37), we obtain d 0 = 0 . 8329. Using the last two relations, we can recover the autocorrelation from the model parameters. Correlation matching. All-pole models have the unique distinction that the model parameters are completely speci fi ed by the fi rst P + 1 autocorrelation coef fi cients via a set of linear equations. We can write d 0 a r( 0 ) ρ (4.2.40) Manolakis, Dimitris, et al. Statistical and Adaptive Signal Processing : Spectral Estimation, Signal Modeling, Adaptive Filtering and Array Processing, Artech House, 2005. ProQuest Ebook Central, .
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