ELEC212 Chapter 5 Transform Analysis of LTI Systems

ELEC212 Chapter 5 Transform Analysis of LTI Systems - The...

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1 The Hong Kong University of Science and Technology Department of Electronic and Computer Engineering Prof. Matthew R. McKay ELEC 212 Chapter 5: Transform Analysis of Linear Time-Invariant Systems 2 09-10 Fall ELEC 212 Transform Analysis of LTI Systems The Frequency Response of LTI Systems ± For an LTI system, the DTFT of the system input and output are related by ± Can also express in polar form is referred to as gain of system –i s r e f e r r e d t o phase response of system 3 09-10 Fall ELEC 212 Transform Analysis of LTI Systems Phase Distortion and Delay ± A convenient measure of linearity of phase is group delay . ± The deviation of the group delay from a constant indicates the degree of non- linearity of phase. ± Consider a pure delayed system with a constant delay n d Constant group delay 4 09-10 Fall ELEC 212 Transform Analysis of LTI Systems Example 5.1 Effects of Attenuation and Group Delay
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5 09-10 Fall ELEC 212 Transform Analysis of LTI Systems Design Problem ± In the previous example, we demonstrated a filter characteristic: – It filtered out the undesired frequency signals – The remaining signals however (ie. signals in the passband) suffer from different delays due to the non-linear phase response ± In this lecture, we will discuss the design techniques on maintaining the same delay on the remaining signals (ie. linear phase in the passband) ± We will also discuss how to design stable and causal inverse systems ± First, we will see how to easily derive a system (transfer) function from a LCCDE, and vice-versa … (revision?) 6 09-10 Fall ELEC 212 Transform Analysis of LTI Systems System Functions for System Characteristics ± Recall linear constant coefficient difference equation of LTI system: ± Applying z-transform on both sides, we have ± The system transfer function H ( z ) is defined as ± The impulse response h [ n ] can be evaluated by inverse Z -transform of H ( z ), with an appropriate ROC: 7 09-10 Fall ELEC 212 Transform Analysis of LTI Systems Example 5.2 Second-order system ± Suppose that the system function of a LTI system is given by: ± Find the difference equation. 8 09-10 Fall ELEC 212 Transform Analysis of LTI Systems Inverse System ± Consider system transfer function with zeros at z = c k and poles at z = d k , in addition to possible zeros and/or poles at z = 0 and z = . Then the transfer function of inverse system is i.e. the poles of H i ( z ) are the zeros of H ( z ) and vice versa. ± Note: The ROC of H i ( z ) and H ( z ) must overlap.
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9 09-10 Fall ELEC 212 Transform Analysis of LTI Systems Example 5.5: Inverse System Function ± Suppose that ± The inverse system function is (ROC extends outwards from z = 0.9) Inverse function is not unique 10 09-10 Fall ELEC 212 Transform Analysis of LTI Systems Impulse Response for Rational System Functions ± Recall that any rational function with only first-order poles, M N , can be written as ± If the system is assumed to be causal , it follows that ± If the impulse response has infinite length , the system is called an infinite impulse response (IIR) system ±
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This note was uploaded on 03/27/2011 for the course ELEC 212 taught by Professor Prof.matthewmckay during the Spring '11 term at HKUST.

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ELEC212 Chapter 5 Transform Analysis of LTI Systems - The...

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