DFT Matrix

The following example reinforces the discussion of the DFT matrix in §6.10. 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