Lecture 7

# Lecture 7 - Lecture 7 The z-Transform and LTI Systems...

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1 Lecture 7 – The z-Transform and LTI Systems ECSE304 Signals and Systems II ECSE 304 Signals and Systems II Lecture 7: The z-Transform and Linear Time Invariant Systems Reading: O and W, Section 10.7 and 10.8 Richard Rose McGill University Dept. of Electrical and Computer Engineering 2 Lecture 7 – The z-Transform and LTI Systems ECSE304 Signals and Systems II • Analysis of DLTI Systems using z-Transforms – Causality – Stability • LTI difference systems and rational transfer functions • Relating System Behavior to Transfer Functions • Block Diagram Representations Outline

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3 Lecture 7 – The z-Transform and LTI Systems ECSE304 Signals and Systems II • Match the impulse responses with the correct pole-zero plots • See “Lecture 7 Example Problems” for Solutions a ___ b ___ c ___ d ___ e ___ a) b) c) d) e) Review: Inverse z-Transforms (j) (k) (l) (m) (n) Time sequence indices used in online notes 4 Lecture 7 – The z-Transform and LTI Systems ECSE304 Signals and Systems II • For a DLTI system with impulse response , the convolution property of z-Transforms gives: is the system function or transfer function • Properties of DLTI systems causality and stability • … are associated with the characteristics of Poles , zeroes , and ROC Analysis of DLTI Systems Using z-Transforms [] x n y n hn () Hz X z Yz HzXz ROC R R xh () () , = ⊃∩ H z H z
5 Lecture 7 – The z-Transform and LTI Systems ECSE304 Signals and Systems II • Establish necessary and sufficient conditions on to guarantee the causality of a DLTI system • For a system to be causal, must be right-sided and for • Therefore: – The ROC of is the exterior of a circle in the z plane – The ROC of must include Why does causality imply that ? For , as , for This is not true for A DLTI system is causal iff the ROC of its system function is the exterior of a circle including infinity Causality of DLTI Systems () H z hn [] = 0 n < 0 H z H z z =∞ z →∞ 0 n n H zh n z = = ∑ 0 n z 0 n > 0 n < h zR O C =∞⊂ 6 Lecture 7 – The z-Transform and LTI Systems ECSE304 Signals and Systems II Region of Convergence for the z-Transform Sequence Type Region of Convergence Right Sided Sequence Left Sided Sequence Two Sided Sequence

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7 Lecture 7 – The z-Transform and LTI Systems ECSE304 Signals and Systems II A DLTI System represented by rational transfer function, is causal iff: a) The ROC is the exterior of a circle with radius equal to the magnitude of the outermost pole, and b) With expressed as the ratio of polynomials in z, the order of the numerator is less than or equal to the order of the denominator Causality of DLTI Systems () H z 8 Lecture 7 – The z-Transform and LTI Systems ECSE304 Signals and Systems II Example:
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## This note was uploaded on 08/17/2011 for the course ECSE 304 taught by Professor Chenandbacsy during the Spring '11 term at McGill.

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Lecture 7 - Lecture 7 The z-Transform and LTI Systems...

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