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Module2_ZTransform

Module2_ZTransform - Module 2 L R Chen The Z-transform and...

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Module 2 The Z The Z-transform and its Applications transform and its Applications 0 R 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 0 • The inverse Z-transform • Unilateral Z-transform; applications Bibli h R Chen 9/9/201 Bibliography • “The Z-Transform” (Chapter 10) in Signals and Systems , 2 nd Edition by A V O h i A S Will k d S H N b L R 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 • Be able to 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 R Chen 9/9/201 • Be able to compute the inverse Z-transform • Be able to 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 id f h t i i LTI t d it t • provides means of characterizing an LTI system and its response to signals by its pole-zero locations 0 R Chen 9/9/201 L R 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 R 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 ] [ ) ( θ −∞ = jn n n e r n x ] [ θ 0 −∞ = = n n r n x ] [ R Chen 9/9/201 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 L R Module 2 The Z The Z-transform and its applications transform and its applications 6 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
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