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HW_7(483)S09_Solns

HW_7(483)S09_Solns - EE 483 Introduction to Digital Signal...

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1 EE 483 – Introduction to Digital Signal Processing Spring 2009 Homework #7 Solutions 7.1 + + + + + = + + + + = - - - - - - - - 2 1 2 1 2 1 2 1 4 4 2 5 5 2 4 1 2 1 4 2 5 5 . 4 2 5 . 4 ) ( z z z z z z z z z H ( 29 ) ( ) ( 2 1 1 0 z A z A + = , where 1 ) ( 0 = z A and 2 1 2 1 1 4 2 5 5 2 4 ) ( - - - - + + + + = z z z z z A are stable allpass transfer functions. Now H 4 ( e j ϖ ) 2 = 1 4 { A 0 ( e j ϖ ) A 0 * ( e j ϖ ) + A 1 ( e j ϖ ) A 0 * ( e j ϖ ) + A 0 ( e j ϖ ) A 1 * ( e j ϖ ) + A 1 ( e j ϖ ) A 1 * ( e j ϖ ) } . Since ) ( 0 z A and ) ( 1 z A are allpass functions, A 0 ( e j ϖ ) = e j φ 0 ( ϖ ) and A 1 ( e j ϖ ) = e j φ 1 ( ϖ ) . Therefore, H 4 ( e j ϖ ) 2 = 1 4 2 + e j ( φ 0 ( ϖ ) - φ 1 ( ϖ )) + e - j ( φ 0 ( ϖ ) - φ 1 ( ϖ )) { } 1 as maximum values of e j ( φ 0 ( ϖ ) - φ 1 ( ϖ )) and e - j ( φ 0 ( ϖ ) - φ 1 ( ϖ )) are . 1 H 4 ( z ) is stable since ) ( 0 z A and ) ( 1 z A are stable allpass functions. Hence, H 4 ( z ) is BR. 7.2 . Labeling 1 V the output variable of the top multiplier connected to input 1 X we then analyze Figure P7.10(b) and obtain 2 1 1 1 2 1 1 1 ), ( X z V Y X z X k V m - - + = - = , ) 1 ( 2 2 X k X k m m + - = and . ) 1 ( 1 2 1 1 1 2 X k X k z X V Y m m + + = + = - Hence, the transfer matrix of the two-pair is given by τ ( . 1 1 1 1 - + - = - - z k k z k k m m m m Using Eq. (7.128b) we then arrive at the chain matrix Γ .
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