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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 • Ztransform is an alternative representation of a discrete signal. • ZTransform is important in the analysis and characterization of LTI systems • ZTransform 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. • Ztransform provides us with a mean of characterizing an LTI system and its response to various signals by its polezero locations. 2 &¡& ¢£¤¥¦£ ¢ Z Transform and Its Application to the Analysis of LTI Systems • The ztransform of a discrete time signal x(n) is defined as The direct ZTransform The direct ztransform ( ) ( ) n n X z x n z ∞ =∞ = & where z is complex variable. j z re θ = The inverse procedure is called inverse ztransform . So relationship can give as ( ) ( ) z x n X z ↔ [ ] ( ) ( ) X z x n Z ≡ We also denote the ztransform as &¡& ¢£¤¥¦£ § ¨ Z Transform and Its Application to the Analysis of LTI Systems becomes unbounded for Since the ztransform 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 Ztransform Sequence In many cases , we can express the sum of the finite or infinite series for ztransform in a closedform 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 ztransform in a closedform 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: Ztransform Find the ztransform &¡& ¢£¤¥¦£ ¤ 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.
 Winter '08
 Aliyazicioglu

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