# 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 ϖ is applied to an LTI system h ( n ), then y ( n ) = x ( n )* h ( n ) = e jn ϖ H ( e j ϖ ) where H ( e j ϖ ) = ∑ ∞ ∞- = n h ( n ) e-jn ϖ . Facts :- The output is the same as the input multiplied by the function H ( e j ϖ ). Sinusoids are eigenfunctions of linear systems.- The function H ( e j ϖ ) depends only on the frequency variable ϖ . It is called frequency response .- The function H ( e j ϖ ) is 2 π-periodic in the frequency variable ϖ . This is obvious since e j ( ϖ +2 π ) = e j ϖ . 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 ϖ ) = ∑ ∞ ∞- = n x ( n ) e-j ϖ n ( DTFT ) provided that the sequence sums absolutely: ∑ ∞ ∞- = n | x ( n ) | < . ∞ • Note that X ( e j ϖ ) 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 ϖ ) - take T = 2 π . Therefore we can find an expression for the Fourier Series coefficients. Definition : The Inverse Discrete-Time Fourier Transform of the function X ( e j ϖ ) is given by x ( n ) = π 2 1 ∫- π π X ( e j ϖ ) e j ϖ n d ϖ ( 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: π 2 1 ∫- π π X ( e j ϖ ) e j ϖ n d ϖ = π 2 1 ∫- π π [ ∑ ∞ ∞- = m x ( m ) e-j ϖ m ] e j ϖ n d ϖ = π 2 1 ∑ ∞ ∞- = m x ( m ) ∫- π π e-j ϖ ( m-n ) d ϖ = = else ; ; 2 n m π ↓ = 2 πδ ( m-n ) = π π 2 2 ∑ ∞ ∞- = m x ( m ) δ ( 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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