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**Unformatted text preview: **Chapter 12 DISCRETE-TIME FOURIER TRANSFORM Copyright c ° 1996 by Ali H. Sayed. All rights reserved. These notes are distributed only to the students attending the undergraduate DSP course EE113 in the Electrical Engineering Department at UCLA. The notes cannot be reproduced without written consent from the instructor: Prof. A. H. Sayed, Electrical Engineering Department, UCLA, CA 90095, [email protected] From this point onwards in the book we proceed to examine how the transform domain representation of discrete-time signals and systems provides a very useful characterization of their behavior in the so-called frequency domain as opposed to the time domain. The first step in our exposition is to introduce the Discrete-Time Fourier Transform (DTFT) of a sequence and to study some of its properties. The Discrete-Time Fourier Transform (DTFT) Assume X ( z ) is the z- transform of a sequence x ( n ) and that its ROC includes the unit circle | z | = 1. That is, ∞ X n =-∞ | x ( n ) z- n | < ∞ for any | z | = 1 . This condition is equivalent to saying that the sequence x ( n ) is absolutely summable : ∞ X n =-∞ | x ( n ) | < ∞ . For such sequences we can evaluate X ( z ) at any point on the unit circle, i.e., at any point z of the form z = e jω for any ω ∈ [- π,π ], and get X ( e jω ) = ∑ ∞ n =-∞ x ( n ) e- jωn The quantity X ( e jω ) so-defined is referred to as the discrete-time Fourier transform (DTFT) of the sequence x ( n ). The DTFT of an absolutely summable sequence x ( n ) can therefore be computed in one of two ways: 1. Either find the z- transform X ( z ) and replace the variable z by e jω or 116 117 2. Evaluate the series X ( e jω ) = ∞ X n =-∞ x ( n ) e- jωn directly. Examples of DTFTs As a first example, the DTFT of the sequence x ( n ) = { . 5 , 1 , . 5 } is, by definition, X ( e jω ) = 1 + 0 . 5 e jω + 0 . 5 e- jω = 1 + cos ω. Some common DTFTs arise from the following additional examples: 1. Unit-sample sequence . The DTFT of δ ( n ) is X ( e jω ) = 1. Likewise, The DTFT of δ ( n- n o ) is X ( e jω ) = e- jωn o for any integer n o . 2. Exponential sequence. The DTFT of the exponential x ( n ) = α n u ( n ) is, by definition, X ( e jω ) = ∞ X n =0 ( αe- jω ) n . We therefore have the sum of the terms of a geometric series with ratio αe- jω and, consequently, X ( e jω ) = 1 1- αe- jω if | α | < 1 Alternatively, we already know that the z- transform of x ( n ) is z/ ( z- α ) for | z | > | α | . Replacing z by e jω we obtain the above DTFT. However, for this substitution to be valid we need to guarantee that the ROC of X ( z ) includes the unit circle, which in this case requires that we impose the condition | α | < 1....

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