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A#24 - (c Find the transform of z W n w(d Now compute and...

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Assignment #24 ECSE-2410 Signals & Systems - Spring 2009 Due Tue 04/28/09 1(5). Find the z -transform of the repeating sequence, [ ] { } ,... 0 , 1 , 0 , 1 , 0 , 1 = n x , for . Sketch pole-zero plot 0 n for ( ) z X 2(15). Find the closed form analytic expressions for the inverse z -transforms of the following. Use tables and properties . Assume for 0 ] [ = n x 0 < n . (a)(5) 1 3 1 1 1 ) = z z z ( X (b)(5) () 2 1 3 1 1 1 ) ( = z z z X [] ( ) dz z dX z n nx property, use (c)(5) 2 2 1 1 2 1 1 1 ) ( + = z z z X (use partial fraction expansion) 3(20). Find the inverse transform of () + + = 2 1 1 16 1 4 1 1 3 2 1 8 4 1 ) ( z z z z X . 4(20). This example verifies the convolution property [ ] [ ] { } ( )( ) z Y z X n y n x Z = . Consider signals and . () 2 1 1 + + = z z z X () 3 2 1 + = z z z z Y (a) Find inverse transforms, and [] n x [ ] n y . (b) Determine the convolution, [] [ ] [ ] n y n x n w =
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Unformatted text preview: (c) Find , the transform of . ( ) z W [ ] n w (d) Now compute and show it agrees with (c). ( ) ( ) ( ) z Y z X z W ⋅ = 5(10). Given [ ] [ ] n u n y n step ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = 4 1 1 , use z-transforms to find the impulse response, . [ ] n h 6(20). Find the step response of a system with ( ) 2 1 1 1 25 6 1 17 3 2 ) ( − − − − + − + = z z z z z H . 7(10). A first-order, recursive, digital lowpass filter is described by ] 1 [ 2 1 ] [ ] 1 [ 3 1 ] [ − + = − − n x n x n y n y . (a)(5) Find the impulse response, , using z-transforms. ] [ t h (b)(5) Plot the pole-zero diagram of H ( z ) in the z-plane. Is the system stable?...
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