Computation of the Inverse DCT (IDCT) Using DCT Or IFFT

Question

Is there a way to compute the inverse discrete cosine transform (type-2) by leveraging either a DCT, FFT, or IFFT algorithm? I have seen ways to compute the DCT using FFTs, and I've seen ways to compute IFFT using FFT. I can't quite find a simple example with description.

Answer 1

Have a look at Fast DCT Algorithm (PDF Version).

It has both DCT and Inverse DCT using DFT (FFT).
They show how to do a DCT and Inverse without the reflection trick.

The standard (Less efficient then above) way doing so is:

numElements = 10;

vX = randn(1, numElements);

disp(vX); %<! Diusplay the input signal

vDctRef = dct(vX);

% Forward DCR using FFT
vXX     = [fliplr(vX), vX]; %<! Mirroring
vXDft   = fft(vXX);

vGrid = [0:((2 * numElements) - 1)];

vShiftGrid = exp(-1j * 2 * pi * (numElements - 0.5) * vGrid / (2 * numElements));

vXDft2 = real(vXDft ./ vShiftGrid);

vDct    = vXDft2(1:numElements) / sqrt(2 * numElements);
vDct(1) = vDct(1) / sqrt(2);

disp(vDctRef)
disp(vDct)

% Inverse DCT Using FFT
vDct(1) = vDct(1) * sqrt(2);
vDct    = vDct * sqrt(2 * numElements);

vXDft = [vDct, 0, -fliplr(vDct(2:numElements))] .* vShiftGrid;
vXX = ifft(vXDft);

vX = real(fliplr(vXX(1:numElements)));
disp(vX);

A reference code for the IDCT by the more efficient method (As in reference):

numElements = 10;

% Signal (DCT Coefficients)
vXDct = randn(1, numElements);

% Reference Inverse IDCT
vXRef = idct(vXDct);

% Inverse IDCT Using FFT
vGrid = [0:(numElements - 1)];

vShiftGrid      = exp((1j * pi * vGrid) / (2 * numElements));
vShiftGrid      = vShiftGrid * sqrt(2 * numElements);
vShiftGrid(1)   = vShiftGrid(1) / sqrt(2);


vTmp = vShiftGrid .* vXDct ;
vTmp = real(ifft(vTmp));

vX = zeros(1, numElements);

for ii = 0:((numElements / 2) - 1)
    vX((2 * ii) + 1) = vTmp(ii + 1) ;
    vX((2 * ii) + 2) = vTmp(numElements - ii) ;
end

disp(vXRef);
disp(vX);

Another reference is given by Lecture Notes for the Course MAT-INF2360 - Fourier Theory, Wavelet Analysis and Non Linear Optimization.
Have a look at 4.2 Efficient implementations of the DCT at page 115.

Enjoy...

Remark
When it is written DCT above it refers to DCT Type II.