Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part54

Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part54

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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 10 21 Copyright © Orchard Publications The Fast Fourier Transform (FFT) Now, we will explain how the unnatural order occurs and how it can be re ordered. Consider the discrete time sequence ; its DFT is found from (10.55) We assume that is a power of and thus, it is divisible by . Then, we can decompose the sequence into two subsequences, which contains the even components, and which contains the odd components. In other words, we choose these as and for Each of these subsequences has a length of and thus, their DFTs are, respectively, (10.56) and (10.57) where (10.58) For an point DFT, . Expanding (10.55) for we obtain (10.59) Expanding (10.56) for and recalling that , we obtain fn [] Fm W N mn n0 = N1 = N2 2 f even n f odd n f n f2 n = f n n 1 + = 1 2 1 ,,, = F m f n W = 1 n W = 1 == F m f n W = 1 + W = 1 W e j 2 π ---------- e j 2 π N ------   2 W N 2 = 8N 8 = n1 2 3 7 ,,, , = W N = 7 = f0 f1 W N m W N 2m f3 W N 3m ++ + = + f 4 W N 4m f5 W N 5m f6 W N 6m f7 W N 7m +++ 1 2 a n d 3 = W N 0 1 =
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Chapter 10 The DFT and the FFT Algorithm 10 22 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications (10.60) Expanding also (10.57) for and using , we obtain (10.61) The vector is the same in (10.59), (10.60) and (10.61), and . Then, Multiplying both sides of (10.61) by , we obtain (10.62) and from (10.59), (10.60) and (10.62), we observe that (10.63) for . Continuing the process, we decompose into and . These are sequences of length . Denoting their DFTs as and , and using the relation (10.64) for , we obtain (10.65) and (10.66) F even m [] f2 n W N 2mn n0 = 3 = f0 W N 0 W N 2m f4 W N 4m f6 W N 6m +++ = W N W N W N = 1 2 and 3 ,,, = W N 0 1 = F odd m n 1 + W N = 3 f1 W N 0 f3 W N f5 W N f7 W N == W N W N W N = W N N8 = W N W 8 e j 2 π N ------ e j 2 π 8 = W N m W N m F m W N m W N 3m W N 5m W N 7m = Fm F m W N m F m + = n1 2 3 7 ,,, , = and f 6 {} , , N4 2 = F even1 m F even2 m W e j 2 π ---------- e j 2 π N   4 W N 4 = 1 , = F m W N + = F m W N + =
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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 10 23 Copyright © Orchard Publications The Fast Fourier Transform (FFT) The sequences of (10.65) and (10.66) cannot be decomposed further. They justify the statement
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This note was uploaded on 11/20/2009 for the course EE EE 102 taught by Professor Bar during the Fall '09 term at UCLA.

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Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part54

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