ece306-10 - Z Transform Z Transform and Its Application to...

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1 Z Transform °± ²³´µ¶·´¸´¹º³» ¼ ³½¸¾¿ ´¸¶³ ¶ÀÁ¹ÃÄ»¾½¿¼Àº´À½½¿ ´Àº ŽĶ¿¾Ã½À¾ ¶³ ƹ³µ ƹùÀ¶ °±° ²³´ µ¶³ °±° ²³´µ¶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.
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2 °±° ²³´µ¶³ 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 °±° ²³´µ¶³ ¸ ¹ 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: 0 z ( 0) k z k > ( 0) k z k - > becomes unbounded for z = ∞ 0 z =
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3 °±° ²³´µ¶³ º Z Transform and Its Application to the Analysis of LTI Systems °»¼½¾¿ÀÁ 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 ´ µ 2 3 4 1 1 1 1 ( ) 1, , , , ,... 2 2 2 2 x n · ¸ ¹ ¹ ± ² ± ² ± ² = º » ³ ´ ³ ´ ³ ´ µ ¶ µ ¶ µ ¹ ¹ ¼ ½ 2 3 4 1 2 3 4 1 1 1 1 ( ) 1 ... 2 2 2 2 X z z z z z - - - - ± ² ± ² ± ² = + + + + + ³ ´ ³ ´ ³ ´ µ µ µ 1 0 0 1 1 ( ) 2 2 n n n n n X z z z - - = = ± ² ± ² = = ³ ´ ³ ´ µ µ ° ° 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 °±° ²³´µ¶³ ´
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