# The benefit of Eigendecomposition of DCT and DST

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.

• You're putting the cart in front of the horse. "Theoretically, any illustration of {something} is worthwhile": Beg to differ, that's not the case! Same is true for the DFT: the EVD of the DFT is surprisingly complicated and has to the best of my knowledge no direct application in signal processing! Mar 19 at 18:28
• You can of course always come up with an application – but that would feel like a question too broad. Maybe it's more promising to take this from the other side. You're a pure mathematician, so DTTs are already plenty applied math for you; from which side did you come, and how did you end up caring about the DCT, DST, and their spectra? Mar 19 at 18:33
• Oh, you haven't accepted an answer in more than 2 years. Please fix that! Not accepting answers if you got one that answers your question is bad style and hurtful to the effects that drive this community. I won't consider writing an answer in this extreme case! Mar 19 at 18:35
• Many thanks for all your comments, however I could not find any proper clue in your statements.
– ABB
Mar 19 at 19:09
• "the EVD of the DFT ... has ... no direct application in signal processing!" Well, the fact that all the eigenvalues have magnitude 1 means that the DFT is about as well-conditioned numerically as you can possibly get. This saves you a lot of time trying to find something better, or worrying about it in general. Mar 19 at 20:07

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.