I am wondering how and if it is possible to define a Fourier transform or Wavelet transform on DNA sequences which are basically arrays with the values $\{T,C,G,A\}$ in them.
I have found a paper which uses a "Voss 4D binary indicator representation":
Yin, C., Chen, Y. and Yau, S.S.T., 2014. A measure of DNA sequence similarity by Fourier Transform with applications on hierarchical clustering. Journal of theoretical biology, 359, pp.18-28.
Their code is supposed to be available online FFTDNA4D.m, but the link is broken. The general idea is to define an indicator mapping per nucleotide type ($\alpha \in \{T,C,G,A\}$) for the DNA sequence $s(0), s(1), \dots, s(N-1)$
$$ \begin{equation} u_{\alpha}(n)=\begin{cases} 1, & \text{if $s(n)=\alpha$}\\ 0, & \text{otherwise} \end{cases} \end{equation} $$
Then the DFT is given by:
$$ U_x(k) = \sum_{n=0}^{N-1} u_x(n) e^{-2\pi i \frac{kn}{N}} $$
the power spectrum is then given by:
$$ \Phi(k) = \sum_{x\in \{ A, T, G, C\}} |U_x(k)|^2 \quad k = 0,1, \dots, N-1 $$
Is there a better way to do it? This method masks out the differences between the different types of DNA bases.