The DFT Matrix

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. The DFT Matrix

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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DFT Matrix

The following example reinforces the discussion of the DFT matrix. We can simply create the DFT matrix in Matlab by taking the FFT of the identity matrix. Then we show that multiplying by the DFT matrix is equivalent to the FFT:

>> eye(4)
ans =
     1     0     0     0
     0     1     0     0
     0     0     1     0
     0     0     0     1


>> S4 = fft(eye(4)) ans = 1.0000 1.0000 1.0000 1.0000
1.0000 0.0000 - 1.0000i -1.0000 0.0000 + 1.0000i 1.0000 -1.0000 1.0000 -1.0000
1.0000 0.0000 + 1.0000i -1.0000 0.0000 - 1.0000i
>> S4’ * S4 % Show that S4’ = inverse DFT (times N=4) ans = 4.0000 0.0000 0 0.0000 0.0000 4.0000 0.0000 0.0000 0 0.0000 4.0000 0.0000 0.0000 0.0000 0.0000 4.0000
>> x = [1; 2; 3; 4] x = 1 2 3 4
>> fft(x) ans = 10.0000
-2.0000 + 2.0000i -2.0000
-2.0000 - 2.0000i >> S4 * x ans = 10.0000
-2.0000 + 2.0000i -2.0000
-2.0000 - 2.0000i

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