Question

It is suggested that if you have an FFT subroutine for computing a length-N DFT, the inverse DFT of an N-point sequence X[k] can be implemented using this subroutine as follows: 1. Swap the real and imaginary parts of each DFT coefficient X[k]. 2. Apply the FFT routine to this input sequence. 3. Swap the real and imaginary parts of the output sequence. 4. Scale the resulting sequence by 1/N to obtain the sequence x[n], corresponding to the inverse DFT of X[k]. Determine whether this procedure works as claimed. If it doesn't, propose a simple modification that will make it work

Answer 1

MATLAB code is given below with explanation.

clc;
close all;
clear all;

% let the sequence be
x = [1 1 1 1 1 ];
N = length(x);
% apply fft routine to get X[k]
X = fft(x);

% now swap the imaginary and real parts of X[k]
X1_real = imag(X);
X1_imag = real(X);

% now define swapped X[k]
X1 = X1_real + 1j* X1_imag;

% apply FFT routine to X1[k]
x1 = fft(X1);

% scale the sequence x1
x1 = x1/N

Result:

x1 =

Columns 1 through 5

0 + 1.0000i 0 + 1.0000i 0 + 1.0000i 0 + 1.0000i 0 + 1.0000i

It is observed that the signal x and x1 are not quite the same though their magnitudes are same. It needs to be a little modification to get the sequence x1 same as x by using the same routine fft.

modification:

instead of swapping the real and imaginary parts, parts of X[k], take conjugate of X[k] and apply fft routine to the same.

modified code:

% now conjugate of X[k]
X1 = conj(X);

% apply FFT routine to X1[k]
x1 = fft(X1);

% scale the sequence x1
x1 = x1/N

result:

x1 =

1 1 1 1 1