ch4_dft - MIT OpenCourseWare http:/ocw.mit.edu HST.582J /...

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MIT OpenCourseWare http://ocw.mit.edu HST.582J / 6.555J / 16.456J Biomedical Signal and Image Processing Spring 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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Harvard-MIT Division of Health Sciences and Technology HST.582J: Biomedical Signal and Image Processing, Spring 2007 Course Director: Dr. Julie Greenberg HST582J/6.555J/16.456J Biomedical Signal and Image Processing Spring 2007 Chapter 4 - THE DISCRETE FOURIER TRANSFORM ± c Bertrand Delgutte and Julie Greenberg, 1999 Introduction The Fourier representation of signals that we studied in Chapter 3 is important for understand- ing how filters work and what a spectrum is, but it is not a practical tool because the DTFT is a continuous function of frequency and therefore its computation would in general require an infinite number of operations. The purpose of this chapter is to introduce another representation of discrete-time signals, the discrete Fourier transform (DFT), which is closely related to the discrete-time Fourier transform, and can be implemented either in digital hardware or in soft- ware. The DFT is of great importance as an efficient method for computing the discrete-time convolution of two signals, as a tool for filter design, and for measuring spectra of discrete-time signals. While computing the DFT of a signal is generally easy (requiring no more than the execution of a simple program) the interpretation of these computations can be difficult because the DFT only provides a complete representation of finite-duration signals. 4.1 Definition of the discrete Fourier transform 4.1.1 Sampling the Fourier transform It is not in general possible to compute the discrete-time Fourier transform of a signal because this would require an infinite number of operations. However, it is always possible to compute a finite number of frequency samples of the DTFT in the hope that, if the spacing between samples is sufficiently small, this will provide a good representation of the spectrum. Simple results are obtained by sampling in frequency at regular intervals. We therefore define the N -point discrete Fourier transform X [ k ] of a signal x [ n ] as samples of its transform X ( f ) taken at intervals of 1 /N : X [ k ] = ± X ( k/N )= ± x [ n ] e j 2 πkn/N for 0 k N 1( 4 . 1) n = −∞ Because X ( f ) is periodic with period 1, X [ k ] is periodic with period N , which justifies only considering the values of X [ k ] over the interval [0 ,N 1]. 4.1.2 Condition for signal reconstruction from the DFT An important question is whether the DFT provides a complete representation of the signal, that is, if the signal can be reconstituted from its DFT. From what we know about sampling, Cite as: Julie Greenberg, and Bertrand Delgutte. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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we expect that this will only be possible under certain conditions. Specifically, we have seen in Chapter 1 that, if we take N samples
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This note was uploaded on 11/07/2011 for the course AERO 16.422 taught by Professor Juliegreenberg during the Spring '07 term at MIT.

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ch4_dft - MIT OpenCourseWare http:/ocw.mit.edu HST.582J /...

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