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**Unformatted text preview: **Chapter 15 THE DISCRETE 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] The magnitude and phase of the DTFT X ( e j ) of a sequence x ( n ) are continuous functions of the angular frequency . In this way, we have a transform that associates continuous functions with a sequence x ( n ). The idea of the Discrete Fourier Transform ( DFT) is to associate with x ( n ) a sequence in the transform domain as well (rather than a continuous function). It turns out that the DFT is a very powerful tool that is widely used in the fields of signal processing and communications. One of the primary reasons for its success is the existence of efficient (fast) algorithms for its evaluation. We discuss the DFT and some of its properties in this chapter. Motivation To motivate the DFT, we start by recalling that the DTFT X ( e j ) is a periodic function of with period 2 . Assume now that we divide the frequency axis into subintervals of width 2 /N radians/sample each, for some integer N . In particular, the interval [0 , 2 ] is divided into N such subintervals. We then evaluate the DTFT X ( e j ) at multiples of 2 N and define the sampled sequence: X ( k ) = X ( e j ) fl fl = 2 k N , k = ...,- 1 , , 1 ,... That is, the complex numbers { X ( k ) } are obtained by sampling the DTFT every 2 N radians/sample. Since X ( e j ) is periodic, we obtain that the sequence X ( k ) will also be periodic with period N . Thus it is sufficient to consider only one period of X ( k ), say X ( k ) = X ( e j ) fl fl = 2 k N , k = 0 , 1 ,...,N- 1 . The question now is what is the significance of this new sequence { X ( k ) } and how does it relate to the original sequence x ( n )? 156 157 1. Relation to the time-domain . We answer the latter question first. The N values of X ( k ) over one period ( k = 0 ,...,N- 1) can be related to x ( n ) as follows: X ( k ) = &quot; X n =- x ( n ) e- jn # = 2 k N , k = 0 , 1 ,...,N- 1 , = X n =- x ( n ) e- j 2 k N n , k = 0 , 1 ,...,N- 1 , = [ ... + x (- N ) + x (0) + x ( N ) + x (2 N ) + ... ] + [ ... + x (- N + 1) + x (1) + x ( N + 1) + x (2 N + 1) + ... ] e- j 2 N + [ ... + x (- N + 2) + x (2) + x ( N + 2) + x (2 N + 2) + ... ] e- j 4 N + ... + [ ... + x (- 1) + x ( N- 1) + x (2 N- 1) + x (3 N- 1) + ... ] e- j 2( N- 1) N Now introduce a periodic sequence (of period N ) that is defined by x p ( n ) = X l =- x ( n + lN ) ....

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