MODULE-6 - MODULE 6 DISCRETE-TIME FOURIER TRANSFORM...

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Unformatted text preview: MODULE 6 DISCRETE-TIME FOURIER TRANSFORM Frequency Response Discrete-Time Fourier Transform (DTFT) Sinusoidal Response Properties and Examples of DTFT Page 6.1 INDEX Discrete-Time Fourier Transform Frequency Response Definition of the Discrete-Time Fourier Transform Proof of Inversion of DTFT Comments on DTFT Sinusoidal Response of Linear System Interpretation of DTFT Example - Low-Pass Filter Symmetry of DTFT Properties of DTFT Examples of Discrete-Time Fourier Transforms MAIN INDEX Page 6.2 6. DISCRETE-TIME FOURIER TRANSFORM index READ : Sections 2.7 - 2.9, Chapter 5 of Oppenheim & Schafer. Work as many related problems as possible. Frequency Response If the complex exponential x ( n ) = e jn is applied to an LTI system h ( n ), then y ( n ) = x ( n )* h ( n ) = e jn H ( e j ) where H ( e j ) = - = n h ( n ) e-jn . Facts :- The output is the same as the input multiplied by the function H ( e j ). Sinusoids are eigenfunctions of linear systems.- The function H ( e j ) depends only on the frequency variable . It is called frequency response .- The function H ( e j ) is 2 -periodic in the frequency variable . This is obvious since e j ( +2 ) = e j . So, it has a Fourier Series representation . Page 6.3 Discrete-Time Fourier Transform index Definition : The Discrete-Time Fourier Transform of the sequence x ( n ) is given by X ( e j ) = - = n x ( n ) e-j n ( DTFT ) provided that the sequence sums absolutely: - = n | x ( n ) | < . Note that X ( e j ) is NOT the same as the DFT , which is another transform that we will study later. Absolute summability is sufficient for the DTFT to exist. So is square summability. However, there is no known necessary and sufficient condition. The DTFT is in fact the Fourier Series representation of the periodic function X ( e j ) - take T = 2 . Therefore we can find an expression for the Fourier Series coefficients. Definition : The Inverse Discrete-Time Fourier Transform of the function X ( e j ) is given by x ( n ) = 2 1 - X ( e j ) e j n d ( IDTFT ) Page 6.4 Proof of Inversion of DTFT index The DTFT is of course uniquely invertible . This is easily shown by plugging the DTFT expression into the IDTFT expression (or vice-versa). The IDTFT is: 2 1 - X ( e j ) e j n d = 2 1 - [ - = m x ( m ) e-j m ] e j n d = 2 1 - = m x ( m ) - e-j ( m-n ) d = = else ; ; 2 n m = 2 ( m-n ) = 2 2 - = m x ( m ) ( m-n ) = x ( n ) The proof hinges on the...
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MODULE-6 - MODULE 6 DISCRETE-TIME FOURIER TRANSFORM...

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