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set4 - Chapter 2 Identication Part 3 SISO Linear Prediction...

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a39 a38 a36 a37 Chapter 2 – Identification Part 3 – SISO Linear Prediction P. M. Van Dooren Department of Mathematical Engineering Catholic University of Louvain K. A. Gallivan School of Computational Science Florida State University Spring 2006 1
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a39 a38 a36 a37 SISO Linear Prediction So far we have assumed known input: impulse input impulse response output specific input sequence { u i } → output sequence { y i } It is also possible to consider stochastic problems, e.g., input sequence { u i } is white noise This means E ( u i u * k + i ) = 0 when k negationslash = 0 We assume E ( u i u * i ) = 1 2
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a39 a38 a36 a37 Filter Constraints Definition 4.1 An all-pole infinite impulse response (IIR) filter is defined by a transfer function and recurrence with the forms H ( z ) = βz n D ( z ) = β D ( z ) z - n = β 1 + P n 1 α i z - i βu k = y k + n X 1 α i y k - i where D ( z ) is a polynomial of degree n . Such a filter is also called autoregressive. 3
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a39 a38 a36 a37 Remark 4.1 If white noise input is applied to an all-pole IIR filter then the output is noise with some color which is reflected in the crosscorrelation of the output signal samples, i.e., E ( y i y * i + k ) . (The conjugate can be ignore for real scalar signals.) Remark 4.2 Identifying such a system can be done for several reasons and under several assumptions. Designing a colored noise generator with a particular color: In this case one specifies the crosscorrelation of the output color desired and deduces the system coefficients. Designing a whitening filter: After identifying the system it would be run backwards, i.e., the colored noise would become the input and the coefficients used in an FIR manner. The system would orthogonalize the output samples in a stochastic sense given a particular color of input noise. 4
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a39 a38 a36 a37 The Problem Problem 4.1 Given a stationary signal environment, design an all-pole IIR SISO filter such that white noise input, { u i } produces output noise { y i } with a specified cross-correlation. The key tasks are Relate the filter coefficients to the problem constraints. Derive a fast reliable algorithm to compute the coefficients. 5
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a39 a38 a36 a37 The Basic Identities We have E ( u k u k - i ) = δ ik which is 1 if i = k . E ( y k y k - j ) = r j = r - j Use the filter expression to relate the two expected values. For j > 0 we have y k = βu k Σ n 1 α i y k - i y k - j y k = βu k y k - j Σ n 1 α i y k - i y k - j E ( y k - j y k ) = E ( βu k y k - j Σ n 1 α i y k - i y k - j ) r j = 0 Σ n 1 α i r j - i E ( βu k y k - j ) = 0 since y k - j depends on u i i < k which are uncorrelated by assumption. 6
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a39 a38 a36 a37 The Basic Identities For j = 0 we have y k y k = βu k y k Σ n 1 α i y k - i y k E ( y k y k ) = E ( βu k y k Σ n 1 α i y k - i y k ) r 0 = βE ( y k u k ) Σ n 1 α i r i r 0 = β 2 Σ n 1 α i r i Therefore, r 0 + n X 1 α i r i = β 2 r j + n X 1 α i r j - i = 0 7
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a39 a38 a36 a37 Example First write the equations for i = 0 , 1 , 2 , 3 2 6 6 6 6 6 4 r 0 r 1 r 2 r 3 r 1 r 0 r - 1 r - 2 r 2 r 1 r 0 r - 1 r 3 r 2 r 1 r 0 3 7 7 7 7 7 5 2 6 6 6 6 6 4 1 α 1 α 2 α 3 3 7 7 7 7 7 5 = 2 6 6 6 6 6 4 β 2 0 0 0 3 7 7 7 7 7 5
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