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Unformatted text preview: Module 3 Time and Frequency Analysis of Time and Frequency Analysis of DT Signals and Systems DT Signals and Systems 10 DT Signals and Systems DT Signals and Systems R Chen 9/16/20 L Module 3 Module 3 : Time and Frequency Analysis of DT Signals and Systems : Time and Frequency Analysis of DT Signals and Systems 1 Content • Analysis of DT LTI systems in the Z domain; stability • Analysis of DTLTI systems in the Zdomain; stability • Stability of 2 nd order DT systems • Digital filters Bibliography 010 Bibliography • “The Ztransform” (Section 10.7) of Signals and Systems , 2 nd edition by A V Oppenheim A S Willsky and S H Nawab R Chen 9/16/20 edition by A. V. Oppenheim, A. S. Willsky, and S. H. Nawab. • “Time and Frequency Characterization of Signals and Systems” (Sections 6.0, 6.2, 6.3,6.6.2) of Signals and Systems , L Module 3 Module 3 : Time and Frequency Analysis of DT Signals and Systems : Time and Frequency Analysis of DT Signals and Systems 2 2 nd edition by A. V. Oppenheim, A. S. Willsky, and S. H. Nawab. Learning Objectives • Be able to relate causality and stability to the ROC and • Be able to relate causality and stability to the ROC and the location of the system poles • Understand the stability of 2 nd order DT systems based on pole location • Know the properties of ideal filters and the different types of filters 10 filters • Understand the concept of minimumphase R Chen 9/16/20 • Be able to design various but simple digital filters L Module 3 Module 3 : Time and Frequency Analysis of DT Signals and Systems : Time and Frequency Analysis of DT Signals and Systems 3 Analysis of DTLTI systems in the Zdomain • transfer functions of systems described by linear difference equations M N consider taking Ztransforms and applying the shifting property, ∑ ∑ = = − + − − = k k k k k n x b k n y a n y 1 ] [ ] [ ] [ ) ( ) ( z H z b z Y M k k k = = ∑ = − 010 1 ) ( 1 z a z X N k k k + ∑ = − R Chen 9/16/20 system transfer function is RATIONAL ! L Module 3 Module 3 : Time and Frequency Analysis of DT Signals and Systems : Time and Frequency Analysis of DT Signals and Systems 4 Response of DTLTI system having rational H(z) consider ) ( ) ( ) ( z A z B z H = ) ( z N and let , i.e. input signal has a rational Ztransform • system is initially relaxed so that ) ( ) ( z Q z X = y [1] = y [2] = … = y [ N ] = 0 then ) ( ) ( z N z B z X z H z Y 10 then  assume that the system has simple poles p 1 , …, p N and that the ) ( ) ( ) ( ) ( ) ( z Q z A z X z H z Y = = R Chen 9/16/20 1 N input signal has simple poles q 1 , …, q N where p k ≠ q m for all k and m assume there is no polezero cancellation L Module 3 Module 3 : Time and Frequency Analysis of DT Signals and Systems : Time and Frequency Analysis of DT Signals and Systems 5 assume there is no polezero cancellation Response of DTLTI system having rational H(z) using partial fraction expansion, we have ∑ ∑ + = L k N k Q A z Y ) ( thus...
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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.
 Spring '11
 ChenandBacsy

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