Module2_ZTransform

Module2_ZTransform - Module 2 L R Chen The Z-transform and its Applications Z Module 2 The Z-transform and its applications Z 1 Content Definition

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Module 2 The Z The Z-transform and its Applications transform and its Applications 0 Chen 9/9/201 L R Module 2 The Z The Z-transform and its applications transform and its applications 1 Content • Definition of bilateral Z transform • Definition of bilateral Z-transform • Region of convergence (ROC) • Pole location and temporal behaviour of DT causal signals • Properties of the Z-transform • The inverse Z transform • The inverse Z-transform • Unilateral Z-transform; applications Bibliography • “The Z-Transform” (Chapter 10) in Signals and Systems , 2 nd Edition by A V O h i A S W i l lk dS H N b Module 2 The Z The Z-transform and its applications transform and its applications 2 A. V. Oppenheim, A. S. Willsky, and S. H. Nawab.
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Learning Objectives • Understand the notion of poles and zeros • Understand the notion of poles and •Beab leto compute the Z-transform and specify the ROC for different DT signals • Understand the relation between the ROC and the time- domain properties of a DT signal 0 •Understand pole location and the impact on the temporal behaviour of causal, DT signals Chen 9/9/201 •Beable to compute the inverse Z-transform solve difference equations and analyze DT-LTI systems in the Z-domain L R Module 2 The Z The Z-transform and its applications transform and its applications 3 systems in the Z domain Motivation for the Z-transform • plays same role in analysis of DT signals and LTI systems as the Laplace transform does in CT signals and LTI systems •provides means of characterizing an LTI system and its response to signals by its pole-zero locations Module 2 The Z The Z-transform and its applications transform and its applications 4
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Definition of the direct (bilateral, 2-sided) Z-transform • the Z-transform of the DT signal x [ n ] is defined by the following power series: −∞ = n n z n x z X ] [ ) ( • since the Z-transform comprises an infinite series, it exists only for the values of z for which the series converges 0 region of convergence (ROC) : set of all values of z for which X ( z ) is finite Chen 9/9/201 • whenever we cite a Z-transform, we also need to specify its ROC L R Module 2 The Z The Z-transform and its applications transform and its applications 5 Definition of the direct (bilateral, 2-sided) Z-transform • if we write z = r exp[j θ ] where r = | z | and θ = z , then = jn n e r n x z X θ ] [ ) ( now, −∞ = n = jn n e r n x z X ] [ ) ( −∞ = jn n n e r n x ] [ −∞ = = n n r n x ] [ thus in the ROC of X ( z ), | X ( z )| < ∞⇒ need to ensure that x [ n ] r -n is absolutely summable the values of r which guarantee this −∞ = n Module 2 The Z The Z-transform and its applications transform and its applications 6 the values of condition form the ROC
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Definition of the direct (bilateral, 2-sided) Z-transform •note + = = 1 ] [ ] [ ] [ ) ( n n n r n x r n x r n x z X = −∞ = −∞ = + = 0 ] [ ] [ n n n n n n x r n x = = 0 1 n n r converges converges 0 converges for r < converges for r > r 2 ( r
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This note was uploaded on 07/09/2011 for the course ECSE 304 taught by Professor Chenandbacsy during the Spring '11 term at McGill.

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Module2_ZTransform - Module 2 L R Chen The Z-transform and its Applications Z Module 2 The Z-transform and its applications Z 1 Content Definition

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