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HW6 (S 2007) - n-1-1 6 y n-2 x n when x n = δ n-1 3 δ...

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ELCT 321 Digital Signal Processing Homework Assignment # 6 (Chapter 7: Z-Transform) by Dr. Yong-June Shin Assigned: March 29, 2006 Due: April 3, 2006 1. (25 Points) Determine z-transforms of following discrete signals de- scribed in time domain: (a) x 1 [ n ] = ( - 0 . 5) n · u [ n ] (b) x 2 [ n ] = 3 · (0 . 5) n - 1 · u [ n - 2] (c) x 3 [ n ] = ( - 1) n · u [ n ] (d) x 4 [ n ] = n · (0 . 5) n · u [ n ] (Hint: Differentiation) (e) x 5 [ n ] = sin( ω 0 n ) · u [ n ] (Hint: Euler’s equation) 2. (25 Points) Determine inverse z-transforms of following transfer func- tion described in z- domain: (a) H 1 ( z ) = 1 - 0 . 8 z - 1 1+0 . 9 z - 1 (b) H 2 ( z ) = z - 2 1 - 0 . 8 z - 1 (c) H 3 ( z ) = 1 - z - 1 1 - 1 6 z - 1 - 1 6 z - 2 (d) H 4 ( z ) = 1+2 z - 1 + z - 2 1 - 3 2 z - 1 + 1 2 z - 2 (e) H 5 ( z ) = e - z (Hint: Taylor series expansion) 3. (40 Points) Solve following difference equations by use of z-transform:
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Unformatted text preview: [ n-1]-1 6 y [ n-2] + x [ n ] , when x [ n ] = δ [ n ]-1 3 δ [ n-1] (b) y [ n ] = 2 . 5 y [ n-1]-y [ n-2] + x [ n ]-5 x [ n-1] + 6 x [ n-2] , when x [ n ] = δ [ n ] (c) y [ n ] = 1 2 y [ n-1]-1 4 y [ n-2] + x [ n ] , when x [ n ] = δ [ n-1] (d) y [ n ] =-y [ n-2] + x [ n ] , when x [ n ] = 5 δ [ n-2] 1 4. (10 Points) Consider the causal system defined by the pole-zero pat-tern provided below. Figure 1 Pole-zero pattern (a) Determine the systems function ( H ( z )). (b) Determine the stability of the systems with reason(s). 2...
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