Is it possible to define Fourier or wavelet transforms on DNA sequences? 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 &quot;Voss 4D binary indicator representation&quot;:

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 $s(0), s(1), \dots, s(N-1)$
$$
\begin{equation}
  u_{\alpha}(n)=\begin{cases}
    1, &amp; \text{if $s(n)=\alpha$}\\
    0, &amp; \text{otherwise}
  \end{cases}
\end{equation}
$$
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.
First, a code for paper A measure of DNA sequence similarity by Fourier Transform with applications on hierarchical clustering. Journal of theoretical biology is available at GenomeDFT (github).
Then, on the transforms. Discrete Fourier transforms can be defined on other domains than vector spaces. There are Fourier transforms on groups, rings or finite fields. On the latter, it is commonly called a number-theoretic transform (NTT). This possibility entails to making sense of group/ring/field axioms for the nucleotides. For instance, it should be possible to use a finite field of characteristic four. Apparently weird mathematical structures could be interesting when considering (C,G) or (A,T)  patterns, or the redundancy in the codon/amino-acid correspondence.
Some papers mention the Ramanujan-Fourier transform (A Novel Method for Comparative Analysis of DNA Sequences by Ramanujan-Fourier Transform, apparently with the same first author of the paper you mentioned). With less structured transformations, it is also possible to evaluate periodicity, as for instance in Categorical spectral analysis of periodicity in nucleosomal DNA.
There also were some works on wavelets, either continuous or discrete. On the first flavor, by A. Arneodo and colleagues, and more recently in 2019 with
Wavelet-Based Genomic Signal Processing for Centromere Identification and Hypothesis Generation. With discrete wavelet transform, there is for instance Identification of exonic regions in DNA sequences using cross-correlation and noise suppression by discrete wavelet transform, Application of discrete wavelet transform for analysis of genomic sequences of Mycobacterium tuberculosis, Wavelet analysis of DNA sequences.
Finally, on the mappings. There is for instance Digital signal processing methods for biosequence comparison, 1991 or Mapping Equivalence for Symbolic Sequences: Theory and Applications, 2009.
As asked by Marcus, your actual purpose might direct you to an appropriate combination.