ZT_sol - not have any poles outside the unit circle a The...

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EE3210 Supplementary Exercise: z-Transform (Solution) 1. (a) , and the ROC contains all z except 0. The Fourier transform exists because the ROC includes the unit circle. 5 ) ( = z z X (b) . 3 1 z , 3 1 9 ) 3 ( ) 3 / 1 ( ) 3 / 1 ( ] [ ) ( 2 2 2 2 < + = = = = = = = −∞ = −∞ = z z z z z z n x z X m m m m m n n n n n The Fourier transform does not exist because the ROC does not include the unit circle. 2. The z-transform of x [ n ] is: 3 1 , 3 1 1 ) 2 1 ( 3 1 1 ) 2 1 ( ) 3 1 ( ) 2 1 ( ) 3 1 ( ) 2 1 ( ) 3 1 ( ) 2 1 ( ) 3 1 ( ) 2 1 ( ) 4 cos( ) 3 1 ( ) ( 4 / 4 / 0 4 / 0 4 / 0 4 / 0 4 / 0 < + = + = + = = = = −∞ = −∞ = −∞ = z z e z e z e z e z e z e z n z X j j n n n j n n n n j n n n n j n n n n j n n n n π 4 / 3 1 j e z = 4 / 3 1 j e z = The poles are at and . 3. For a system to be both causal and stable, the corresponding z-transform must
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Unformatted text preview: not have any poles outside the unit circle. a) The given z-transform has a pole at infinity. Therefore, it is not causal. b) The poles of this z-transform are at z = 1/4 and -3/4. Therefore, it is causal. 4. a) y [ n ] = x [ n-1] b) Y ( z ) = X ( z ) / z . Hence H ( z ) = 1/ z . The ROC is all z except zero. c) It is stable because the ROC of its transfer function includes the unit circle. d) Inverse system: H ( z ) = z ....
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