The benefit of Eigendecomposition of DCT and DST

Question

I am Ph.D in pure mathematics and interested in signal processing.

Theoretically, any illustration of the eigendecomposition of the discrete trigonometric transforms (DTTs) is worthwhile.

Q. What real applications can be followed by having the eigendecomposition of the discrete sine and cosine transforms? Any reference is appreciated.

Answer 1

Okay, here is a question:

We know that if we define the DFT as:

$$\begin{align} X[k] &= \mathcal{DFT} \Big\{ x[n] \Big\} \\ &\triangleq \frac{1}{\sqrt{N}} \sum\limits_{n=0}^{N-1} x[n] \, e^{-j2\pi nk/N} \end{align} $$

and

$$ y[n] \triangleq X[n] $$

(note the substitution of $n$ in for $k$.) then

$$ Y[k] = \mathcal{DFT} \Big\{ y[n] \Big\} $$

then, if the DFT is defined as above:

$$ Y[n] = x[-n] $$

where periodicity is implied: $x[n+N]=x[n]$ for all $n$.

Now, if we require even symmetry

$$x[n] = x[-n] \quad \forall n \in \mathbb{Z}$$

then the DFT, $X[k]$, will also have even symmetry and two repeated operations of the DFT will result in the original input. So then define

$$ y[n] \triangleq x[n] + X[n] $$

And we will know that

$$\begin{align} Y[k] &= \mathcal{DFT} \Big\{ y[n] \Big\} \\ &= y[k] \\ \end{align}$$

So, from that, all sorts of eigenfunctions can be defined. Then we gotta make sure that they form a basis. But it wouldn't have to be pure sinusoids.