DFT Matrix

The following example reinforces the discussion of the DFT matrix in §6.12. We can simply create the DFT matrix in matlab by taking the DFT of the identity matrix. Then we show that multiplying by the DFT matrix is equivalent to the calling the fft function in matlab:

>> 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       1          1       1          
   1       0 - 1i    -1       0 + 1i
   1      -1          1      -1          
   1       0 + 1i    -1       0 - 1i

>> S4' * S4          % Show that S4' = inverse DFT (times N=4)
ans =
    4    0    0    0
    0    4    0    0
    0    0    4    0
    0    0    0    4

>> x = [1; 2; 3; 4]
x =
     1
     2
     3
     4

>> fft(x)
ans =
  10          
  -2 + 2i
  -2          
  -2 - 2i

>> S4 * x
ans =
  10          
  -2 + 2i
  -2          
  -2 - 2i

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.