Is there mathematical relationship between the FFT and DCT transforms

Question

I need to model the relationship between the DCT and DFT transforms (If it exists). I mean real signal $x \to y = \textrm{DCT}(x) \to z = \textrm{DFT}(y)$, so I need to get the relationship between the $x$ and $z$ if possible.

For more details:

Assume I have a real signal $x[m],\ m=1,2,….N$, the $N$-point IDCT transform of the signal $x[m]$ is $X[n]$ which can be written as follows (let's ignore the coefficients for simplicity):

$$X[n] = \sum_{m=1}^N x[m] \cos\left( \frac{(2m+1)n\pi}{2N} \right)$$

I need to get the relationship between the FFT of $X[n]\ $ and the signal $x[m]$, so taking the $2N-$point DFT for the signal $X[n]\ $ (I upsampled the signal to have the DFT of each point resulted from the IDCT), that will give:

$$Y[v] = \sum_{n=1}^{2N} X[n] e^{-\frac{j2\pi mn}{2N}} = \sum_{n=1}^{2N} \left[ \sum_{m=1}^N x[m] \cos\left( \frac{(2m+1)n\pi}{2N} \right) \right] e^{-\frac{j2\pi mv}{2N}}$$

with some mathematical operations, we can get:

$$Y[v] = \sum_{n=1}^{2N} \left( \left[ \sum_{m=1}^N x[m] \cos\left( \frac{(2m+1)n\pi}{2N} \right) \right] \cos\left(\frac{2\pi mv}{2N}\right) - j\left[\sum_{m=1}^N x[m] \cos\left( \frac{(2m+1)n\pi}{2N} \right) \right] \sin\left(\frac{2\pi mv}{2N}\right) \right)$$

I am trying to get the relationship between the $Y[v]$ and $x[m]$ from the relationship above if it exists. How can we get it?

Answer 1

You have some confusion in the indices, maybe the correct expression would be

$$Y[k] = \sum_{n=1}^N \left[ \sum_{m=1}^N x[m] \cos\left( \frac{(2m+1)n\pi}{2N} \right) \right] e^{-\frac{j2\pi k n}{N}}$$

The DCT is the real part of the odd-indices coefficients of double length DFT

Expressing DCT in terms of complex harmonics

$$z[n] = \sum_{m=1}^N x[m] \left( \exp\left( j \frac{(2m+1)n\pi}{2N} \right) + \exp\left( -j \frac{(2m+1)n\pi}{2N} \right) \right)$$

If you take $X(n)$ being the fourier transform of $x$, in the reals instead of the integers, you could write $z[n] = X(m+1/2)+X(-m-1/2)$, $X(n)$ has Hermitian symmetry and the result agrees with the fact that $z[n]$ is real.

I think that the most efficient way to express the relation between $Y[m]$ and $x[m]$, is that $Y = DFT(DCT(x))$, unless you want to upsample your signal.

The denominator $2N$ in the exponential, the summation is from $1$ up to $N$, and the numerators may be even, so cannot simplify the exponent, the way to go is to compute a $2N$-samples, DFT.

If you create an intermediate signal with odd samples $v_{2i+1} = x_i + x_{N - i}$ and even samples $v_{2i}=0$, and compute $V = DFT^{-1}(v)$, then the first $N$ elements of $V$ equals to $z$, and the $DFT(V) = DFT(DFT^{-1}(v)) = v$