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Digital Signal Processing

Lecture# 8

Chapter 5

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Transform Analysis of LTI System

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Overview

In Chapter 2 we devolved Fourier transform of DTS&S (Discrete time signal & System).

In Chapter 3 we extend the representation to to Z-Transform.

In both (Ch 2& 3) Emphasis was on the transform & their properties, with brief preview of their use in analysis of LTI system.

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Overview

In this chapter our main focus is on detailed representation and analysis of LTI Systems using Fourier and Z –Transforms.

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Overview

As devolved in Ch 2 ,an LTI system can be characterized in the time domain by its impulse response h [n], with output y [n] due to input x [n] is given by Convolution Sum.

Then Fourier Transform Provides an equal complete characterization of LTI system

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Overview

In Ch 3 we devolved Z-Transform as Generalization of Fourier Transform and we showed that Y (z) (the z-transform of output of LTI system) is related to X (z) (the

z-transform of input of LTI system) and

H (z) (the z-transform of system impulse response) by

Y (z)= H (z) X (z)

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Overview

We will See in this Chapter that both Frequency and Z- transform are extremely useful in analysis and representation of LTI Systems, because we can readily infer many properties of system from these two transforms.

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Frequency response of LTI System

Convolution Sum

is referred to magnitude response or gain. is called phase response or phase shift

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Linear Phase Means Delay

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Ideal LPF Frequency Response

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Group Delay

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Group Delay

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Example: Input Signal

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Frequency Response of Filter

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Output Signal

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Problem with IIR System

Variable Group Delay Distort the output

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Rational System Function

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Inverse System

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Frequency Response of Rational System Function

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Any Question?????

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Real Even (or Odd) Signals If a signal is even in addition to being real,

then its DTFT is also real and even. This follows immediately from the Hermitian symmetry of real signals, and the fact that the DTFT of any even signal is real:

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Example

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Continued…

This is true since cosine is even, sine is odd, even times even is even, even times odd is odd, and the sum over all samples of an odd signal is zero. I.e.,