ece306-10 - 1 Z Transform &¡...

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Unformatted text preview: 1 Z Transform &¡ ¢£¤¥¦§¤¨¤©ª£« ¬ £­¨®¯ ¤¨¦£ ¦°±²©³´«®­¯¬°ª¤°­­¯ ¤°ª µ­´¦¯®³­°® ²¦£ ¶©£¥ ¶©³©°¦ &¡& ¢£¤ ¥¦£ &¡& ¢£¤¥¦£ § Z Transform and Its Application to the Analysis of LTI Systems • Z-transform is an alternative representation of a discrete signal. • Z-Transform is important in the analysis and characterization of LTI systems • Z-Transform play the same role in the analysis of discrete time signal and LTI systems as Laplace transform does in the analysis of continuous time signal and LTI systems. • Z-transform provides us with a mean of characterizing an LTI system and its response to various signals by its pole-zero locations. 2 &¡& ¢£¤¥¦£ ¢ Z Transform and Its Application to the Analysis of LTI Systems • The z-transform of a discrete time signal x(n) is defined as The direct Z-Transform The direct z-transform ( ) ( ) n n X z x n z ∞- =-∞ = & where z is complex variable. j z re θ = The inverse procedure is called inverse z-transform . So relationship can give as ( ) ( ) z x n X z ↔ [ ] ( ) ( ) X z x n Z ≡ We also denote the z-transform as &¡& ¢£¤¥¦£ § ¨ Z Transform and Its Application to the Analysis of LTI Systems becomes unbounded for Since the z-transform is an infinite power series, it exists only for the series convergence. The region of convergence (ROC) of X(z) is the set of all values of z for which X(z) attains a finite value. Example Z-transform Sequence In many cases , we can express the sum of the finite or infinite series for z-transform in a closed-form expression. ( ) {1,2,5,7,0,1} ↑ = x n 1 2 3 5 ( ) 1 2 5 7 X z z z z z---- = + + + + ROC: ≠ z ( 0) k z k > ( 0) k z k- > becomes unbounded for z = ∞ z = 3 &¡& ¢£¤¥¦£ § Z Transform and Its Application to the Analysis of LTI Systems &¨©ª«¬­® In many cases , we can express the sum of the finite or infinite series for z-transform in a closed-form expression. 1 ( ) ( ) 2 n x n u n & ¡ = ¢ £ ¤ ¥ 2 3 4 1 1 1 1 ( ) 1, , , , ,... 2 2 2 2 x n ¦ § ¨ ¨ & ¡ & ¡ & ¡ = © ª ¢ £ ¢ £ ¢ £ ¤ ¥ ¤ ¥ ¤ ¥ ¨ ¨ « ¬ 2 3 4 1 2 3 4 1 1 1 1 ( ) 1 ... 2 2 2 2 X z z z z z---- & ¡ & ¡ & ¡ = + + + + + ¢ £ ¢ £ ¢ £ ¤ ¥ ¤ ¥ ¤ ¥ 1 1 1 ( ) 2 2 n n n n n X z z z ∞ ∞-- = = & ¡ & ¡ = = ¢ £ ¢ £ ¤ ¥ ¤ ¥ ­ ­ 1 1 1 2 z- < 1 2 z > If or 1 1 ( ) 1 1 2 X z z- =- 1 2 z > ROC: Z-transform Find the z-transform &¡& ¢£¤¥¦£ ¤ Z Transform and Its Application to the Analysis of...
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This note was uploaded on 03/11/2008 for the course ECE 306 taught by Professor Aliyazicioglu during the Winter '08 term at Cal Poly Pomona.

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ece306-10 - 1 Z Transform &¡...

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