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**Unformatted text preview: **Purdue University: ECE438 - Digital Signal Processing with Applications 1 ECE438 - Laboratory 6: Discrete Fourier Transform and Fast Fourier Transform Algorithms (Week 1) October 6, 2010 1 Introduction This is the first week of a two week laboratory that covers the Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT) methods. The first week will introduce the DFT and associated sampling and windowing effects, while the second week will continue the discussion of the DFT and introduce the FFT. In previous laboratories, we have used the Discrete-Time Fourier Transform (DTFT) extensively for analyzing signals and linear time-invariant systems. (DTFT) X ( e jω ) = ∞ summationdisplay n =-∞ x ( n ) e- jωn (1) (inverse DTFT) x ( n ) = 1 2 π integraldisplay π- π X ( e jω ) e jωn dω. (2) While the DTFT is very useful analytically, it usually cannot be exactly evaluated on a com- puter because equation (1) requires an infinite sum and equation (2) requires the evaluation of an integral. The discrete Fourier transform (DFT) is a sampled version of the DTFT, hence it is better suited for numerical evaluation on computers. (DFT) X N ( k ) = N- 1 summationdisplay n =0 x ( n ) e- j 2 πkn/N (3) (inverse DFT) x ( n ) = 1 N N- 1 summationdisplay k =0 X N ( k ) e j 2 πkn/N (4) Here X N ( k ) is an N point DFT of x ( n ). Note that X N ( k ) is a function of a discrete integer k , where k ranges from 0 to N − 1. Questions or comments concerning this laboratory should be directed to Prof. Charles A. Bouman, School of Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907; (765) 494- 0340; [email protected] Purdue University: ECE438 - Digital Signal Processing with Applications 2 In the following sections, we will study the derivation of the DFT from the DTFT, and several DFT implementations. The fastest and most important implementation is known as the fast Fourier transform (FFT). The FFT algorithm is one of the cornerstones of signal processing. 2 Deriving the DFT from the DTFT 2.1 Truncating the Time-domain Signal The DTFT usually cannot be computed exactly because the sum in equation (1) is infinite. However, the DTFT may be approximately computed by truncating the sum to a finite window. Let w ( n ) be a rectangular window of length N : w ( n ) = braceleftBigg 1 : ≤ n ≤ N − 1 : else . (5) Then we may define a truncated signal to be x tr ( n ) = w ( n ) x ( n ) . (6) The DTFT of x tr ( n ) is given by: X tr ( e jω ) = ∞ summationdisplay n =-∞ x tr ( n ) e- jωn (7) = N- 1 summationdisplay n =0 x ( n ) e- jωn . (8) We would like to compute X ( e jω ), but as with filter design, the truncation window...

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