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