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SolSet6 - EE 562a Homework Solutions 6 November 5 2009 1...

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Unformatted text preview: EE 562a Homework Solutions 6 November 5, 2009 1 1. (a) G z ( z ) = ∞ X t =-∞ g ( t ) z- t =- 1 X t =-∞ y (- t ) z- t = ∞ X t =1 y ( t ) z t =- y (0) + ∞ X t =0 y ( t ) 1 z- t =- y (0) + Y z 1 z The subtraction of y (0) does not affect the convergence of Y z ( 1 z ) , which converges when 1 z > a , or equivalently, when | z | < 1 a . (b) H z ( z ) = ∞ X t =-∞ h ( t ) z- t =- 1 X t =-∞ y (- t- 1) z- t = ∞ X t =0 y ( t ) z t +1 = z ∞ X t =0 y ( t ) 1 z- t = z Y z 1 z Multiplication by z does not affect the convergence of Y z ( 1 z ) , which (as in (a)) converges when | z | < 1 a . (c) Assuming that the z-transform of g c ( t ) has a non-empty region of convergence, G c z ( z ) = ∞ X t =-∞ [ g ( t ) + y ( t )] z- t = G z ( z ) + Y z ( z ) =- y (0) + Y z 1 z + Y z ( z ) . For this to be a two-sided z-transform, z must be in the range a < | z | < 1 a so that both transforms exist in the same region of the z-plane, and hence we must have a < 1 for the two-sided transform G c z ( z ) to exist within a non-empty region of convergence. (d) X z ( z ) = z τ ∞ X t =0 y ( t ) z- t = ∞ X t =0 y ( t ) z- t + τ = ∞ X t =- τ y ( t + τ ) z- t and hence x ( t ) = y ( t + τ ) , t ≥ - τ , otherwise. Certainly | X z ( z ) | = | z | τ | Y z ( z ) | and and therefore the sum in X z ( z ) converges if and only if the sum in Y z ( z ) converges. Therefore z is in the region of convergence of X z ( z ) if and only if | z | > a . 2. Solution: From the definition of the discrete-time Fourier transform, X ( f ) = ∞ X t =-∞ x ( t ) e- i 2 πft . Using this we have Z 1 / 2- 1 / 2 X ( f ) Y * ( f ) df = Z 1 / 2- 1 / 2 ∞ X t =-∞ x ( t ) e- i 2 πft ! Y * ( f ) df 2 EE 562a Homework Solutions 6 November 5, 2009 = ∞ X t =-∞ x ( t ) Z 1 / 2- 1 / 2 e- i 2 πft Y * ( f ) df ! = ∞ X t =-∞ x ( t ) Z 1 / 2- 1 / 2 e i 2 πft Y ( f ) df ! * = ∞ X t =-∞ x ( t ) y * ( t ) 3. Solution: (a) z ( u,t ) = ax ( u,t ) + by ( u,t ) = ⇒ m z ( t ) = m z = am x + bm y , and using centered notation, K z ( t + τ,t ) = K z ( m ) = E { z o ( u,t + τ ) z * o ( u,t ) } = E { [ ax o ( u,t + τ ) + by o ( u,t + τ )][ a * ( x * o ( u,t ) + b * y * o ( u,t )] } = | a | 2 K x ( τ ) + | b | 2 K y ( τ ) + a * b E { x * o ( u,t ) y o ( u,t + τ ) } + ab * E { x o ( u,t + τ ) y * o ( u,t ) } = | a | 2 K x ( τ ) + | b | 2 K y ( τ ) e K z ( t + τ,t ) = e K z ( τ ) = E { ( z ( u,t + τ )- m z )( z ( u,t )- m z ) } = a 2 e K x ( τ ) + b 2 e K y ( τ ) . where the fact that x ( u,t ) and y ( u,t ) are independent for all integers t,t , was used factor the expectations in the cross-covariance and psuedo-cross-covariance derivations. The PSD is found by taking the discrete-time Fourier transform of the correlation function: S z ( f ) = F { R z ( τ ) } = F { K z ( τ ) + | m z | 2 } = F {| a | 2 K x ( τ ) + | b | 2 K y ( τ ) + | am x + bm y | 2 } = F {| a | 2 ( K x ( τ ) + | m x | 2 ) + | b | 2 ( K y ( τ ) + | m y | 2 ) + 2 Re ab * m x m * y } = | a |...
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This note was uploaded on 04/03/2010 for the course EE 562a taught by Professor Toddbrun during the Fall '07 term at USC.

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SolSet6 - EE 562a Homework Solutions 6 November 5 2009 1...

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