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3015_Discussion_13_Solutions

3015_Discussion_13_Solutions - University of Minnesota Dept...

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University of Minnesota Dept. of Electrical and Computer Engineering EE3015 – Signals and Systems Discussion Session #13: z -Transform 1. Determine the z -Transform and the region of convergence of the following signal: | 1 | 3 1 ] [ = n n x First find the z -Transform of | | 3 1 ] [ n n y = and use the shifting property to obtain the z - Transform of ] [ n x . Signal ] [ n y can be written as: ] 1 [ 3 ] [ 3 1 ] 1 [ 3 1 ] [ 3 1 ] [ + = + = n u n u n u n u n y n n n n This is the sum of a right-sided and left-sided signal. The z -Transform is the sum of the individual transforms: 3 : , 3 1 1 ] 1 [ 3 3 1 : , 3 1 1 1 ] [ 3 1 1 1 < ⎯→ > ⎯→ z ROC z n u z ROC z n u z n z n Thus, 3 3 1 : , 3 1 1 3 1 1 1 ) ( 1 1 < < = z ROC z z z Y Using properties: Re{ z } Im{ z } 3 1/3
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3 3 1 , 3 1 3 1 1 ) ( ) ( ] 1 [ ] [ 3 3 1 ), ( ] [ 1 1 1 1 1 < < = = ⎯→ = < < ⎯→ z z z z z z Y z z X n y n x z z Y n y z z The term 1 z that appears after shifting does not affect the ROC. 2. A discrete-time causal system is described by the difference equation: ] 1 [ 4 5 ] [ 2 ] 2 [ 8 1 ] 1 [ 4 1 ] [ + = + n x n x n y n y n y a) Find the transfer function ) ( z H and its region of convergence. Is this system stable?
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