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Chapter_12 - CHAPTER 12 12.1 2 2 2 1 J fb m = E f m-1 n E b...

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244 CHAPTER 12 12.1 Differentiating with respect to a complex variable: 12.2 (a) Suppose we write Then substituting these relations into Burg’s formula: , J fb m , 1 2 -- E f m -1 n ( ) 2 [ ] E b m -1 n -1 ( ) 2 [ ] + ( ) 1 κ m 2 + ( ) = κ m E f m -1 n ( ) b m -1 * n -1 ( ) [ ] κ m * E b m -1 n -1 ( ) f m -1 * n ( ) [ ] + + J fb m , ∂κ m ----------------- κ m E f m -1 n ( ) 2 [ ] E b m -1 n -1 ( ) 2 [ ] + ( ) = E b m -1 n -1 ( ) f m -1 * n ( ) [ ] E b m -1 n -1 ( ) f m -1 * n ( ) [ ] + + κ m E f m -1 n ( ) 2 [ ] E b m -1 n -1 ( ) 2 [ ] + ( ) = 2 E b m -1 n -1 ( ) f m -1 * n ( ) [ ] + f m i ( ) f m -1 i ( ) κ ˆ m n ( ) b m -1 i -1 ( ) + = b m i ( ) b m -1 i ( ) κ ˆ m n ( ) f m -1 i ( ) + = κ ˆ m n ( ) 2 b m -1 i -1 ( ) f m -1 * i ( ) i =1 n f m -1 i ( ) 2 b m -1 i -1 ( ) 2 + i =1 n ------------------------------------------------------------------ = 2 b m -1 i -1 ( ) f m -1 * i ( ) 2 b m -1 n -1 ( ) f m -1 * n ( ) + i =1 n E m -1 n -1 ( ) f m -1 n ( ) 2 b m -1 n -1 ( ) 2 + + ----------------------------------------------------------------------------------------------------------- =
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245 Cross-multiplying and proceeding in a manner similar to that described after Eqs. (12.12) and (12.13) in the text, we finally get (b) The algorithm so formulated is impractical because to compute the updated forward prediction error f m ( n ) and backward prediction error b m ( n ) we need to know the updated . This is not possible to do because the correction term for requires knowledge of f m ( n ) and b m ( n ). 12.3 For the transversal filter of Fig. 12.6 in the text we have: tap-weight vector = k M ( n ) tap-input vector = u M ( i ), i = 1,2,..., n desired response, d ( i ) = The a posteriori estimation error equals The deterministic cross-correlation vector equals We also note that Hence, the sum of weighted error squares equals κ ˆ m n ( ) κ ˆ m n -1 ( ) f m -1 * n ( ) b m n ( ) b m -1 n -1 ( ) f m * n ( ) + E m -1 n -1 ( ) --------------------------------------------------------------------------------- = κ ˆ m n ( ) κ ˆ m n ( ) 1 i n = 0 i n -1, 1 , = e i ( ) d i ( ) k M H n ( ) u M i ( ), i 1 2 n , , , = = φ n ( ) φ n ( ) λ n -i u M d * i ( ) i =1 n = u M n ( ) = E d n ( ) λ n - i d i ( ) 2 i =1 n 1 = =
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246 We note that the inner product is a real-valued scalar. Hence, 12.4 We start with where is the tap-input vector, is the past value of the deterministic correlation matrix, and is its present value. Hence, where I is the identity matrix. Hence, taking the determinants of both sides: (1) But, We may therefore rewrite Eq. (1) as E min n ( ) E d n ( ) φ H n ( ) w ˆ n ( ) = 1 u M H n ( ) k m n ( ) = u M H n ( ) k m n ( ) E min n ( ) 1 k m H n ( ) u m n ( ) = γ M n ( ) = Φ m n ( ) λΦ m n -1 ( ) u m n ( ) u m H n ( ) + = u M n ( ) Φ m n -1 ( ) Φ m n ( ) λΦ m n -1 ( ) Φ m n ( ) u m n ( ) u m H n ( ) = Φ m n ( ) I u m n ( ) u m H n ( )Φ m 1 n ( ) [ ] = λ det Φ m n -1 ( ) [ ] det Φ m n ( ) [ ] det I u m n ( ) u m H n ( )Φ m 1 n ( ) [ ] = det I u m n ( ) u m H n ( )Φ m 1 n ( ) [ ] det I u m H n ( )Φ m 1 n ( ) u m n ( ) = 1 u m H n ( )Φ m 1 n ( ) u m n ( ) = γ m n ( ) =
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247 Hence, we may express the conversion factor as 12.5 (a) The ( m +1)-by-( m +1) correlation matrix may be expressed in the form (1) Define the inverse of this matrix as (2) Hence, from Eqs. (1) and (2): From this relation we deduce the following four equations: (3) (4) λ det Φ m n -1 ( ) [ ] det Φ m n ( ) [ m n ( ) = γ m n ( ) γ m n ( ) λ det Φ m n -1 ( ) [ ] det Φ m n ( ) [ ] ---------------------------------- = Φ m +1 Φ m +1 n ( ) U n ( ) φ 1 H n ( ) φ 1 n ( ) Φ m n -1 ( ) = Φ m +1 1 n ( ) α 1 β 1 H β 1 Γ 1 = I m +1 Φ m +1 n ( )Φ m +1 1 n ( ) = U n ( ) φ 1 H n ( ) φ 1 n ( ) Φ m n -1 ( ) α 1 β 1 H β 1 Γ 1 = U n ( )α 1 φ 1 H n ( )β 1 + U n ( )β 1 H φ 1 H n ( )Γ 1 + φ 1 n ( )α 1 Φ m n -1 ( 1 + φ 1 n ( )β 1 H n ( ) Φ m n -1 ( 1 + = U n ( )α 1 φ 1 H n ( )β 1 + 1 = U n ( )β 1 H φ 1 H n ( )Γ 1 + 0 =
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248 (5) (6) Eliminate β 1 between Eqs. (3) and (5): Hence, (7) which is real-valued. Correspondingly, (8) From Eq. (6): (9) Check: Substitute Eqs. (8) and (9) into the left hand side of Eq. (4):
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