>> dft_matrix = [1 1 1 1; 1 j -1 -j; 1 -1 1 -1; 1 -j -1 j];
>> dft_matrix

dft_matrix =

   1.0000 + 0.0000i   1.0000 + 0.0000i   1.0000 + 0.0000i   1.0000 + 0.0000i
   1.0000 + 0.0000i   0.0000 + 1.0000i  -1.0000 + 0.0000i   0.0000 - 1.0000i
   1.0000 + 0.0000i  -1.0000 + 0.0000i   1.0000 + 0.0000i  -1.0000 + 0.0000i
   1.0000 + 0.0000i   0.0000 - 1.0000i  -1.0000 + 0.0000i   0.0000 + 1.0000i

>> signal = [1; 2; 3; 4];
>> signal_dft = dft_matrix*signal

signal_dft =

  10.0000 + 0.0000i
  -2.0000 - 2.0000i
  -2.0000 + 0.0000i
  -2.0000 + 2.0000i

>> signal_dft = (1/2)*signal_dft

signal_dft =

   5.0000 + 0.0000i
  -1.0000 - 1.0000i
  -1.0000 + 0.0000i
  -1.0000 + 1.0000i

>> cir = [0; 0; 1; 0]

cir =

     0
     0
     1
     0

>> cir_idft = ctranspose(dft_matrix)*cir

cir_idft =

     1
    -1
     1
    -1