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Introduction to Digital Signal and System Analysis Discrete Fourier Transform X [ k] = + W k 8 s= 0 1 1 s= 0 x[2(2s)] W sk 2 x[2(2s) + 1] W + W sk 2 1 k 4 s= 0 + W x[2(2s + 1)] W 1 k 4 s= 0 sk 2 x[2(2s + 1) + 1] W sk 2 W k 8 W sk 2 2π k 2π k = exp − j , W k 4 = exp − j 8 4 2π sk = − 0 1 exp j , W 2 = 1, W2 = −1 2 Using the following graphical illustration in Figure 6.6, we can see the complex multiplications are only 4+6+7=17 in this case, much fewer than that in the direct calculation of the DFT. x[n] 0 0 0 ⊕ ⊕ X[k] 0 ⊕ 1 2 4 ⊕ ⊕ 1 ⊕ 2 4 2 ⊕ ⊕ 2 ⊕ 3 6 6 ⊕ ⊕ 3 ⊕ 4 1 1 ⊕ ⊕ 4 ⊕ 5 3 5 ⊕ ⊕ 5 ⊕ 6 5 3 ⊕ ⊕ 6 ⊕ 7 7 7 ⊕ ⊕ 7 ⊕ Figure 6.6 2 N=4 point DFT are changed to 4 N=2 point DFTs Download free ebooks at bookboon.com 98
Introduction to Digital Signal and System Analysis Discrete Fourier Transform It can be proved that the FFT algorithm has saved huge computing time by reducing from 2 N complex multiplications to N log 2 N , i.e. saved N 8 8 times of computation. For example, if N=8, = = 2. 67 ; if N=1024, log 2 N log2 8 3 1024 1024 = = 102.4 . i.e. saved 102 times of multiplications. The longer the data length N, the more time can log 1024 10 2 be saved relative to the direct calculation of the DFT. Problems Q6.1 What are the features of the DFT coefficients X[k] of an N-sample signal which is a) Real, b) Real and even, c) Real and odd, and d) Complex Q6.2 For the digital sequence a) x[n] = [1 -1 ], b) x[n]= [3 -2], c) x[n]=[1 -1 0 0], d) x[n] = [1 0 0 1], e) x[n]=[1 2 1 3]. Calculate the Discrete Fourier Transform (DFT) . Q6.3 Explain how the Fast Fourier Transform (FFT) algorithm can be faster than direct calculation of the Discrete Fourier Transform (DFT). Q6.4 Answer the following questions: -- With reference to the Fast Fourier Transform (FFT), why is the length, N, normally chosen as an integer power of 2 -- In brief, what is the reason that the FFT algorithm can be faster than direct calculation of the Discrete Fourier Transform (DFT) -- If the length of a sequence is not yet an integer power of 2, how is it possible to take advantage of the FFT algorithm 99 Download free ebooks at bookboon.com
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