Unification of Signal Transform Theory

We unify the discrete Fourier transform (DFT), discrete cosine transform (DCT), Walsh-Hadamard, Haar wavelet, Karhunen-Loève transform (KLT), and several others along with their continuous counterparts (Fourier transform, Fourier series, spherical harmonics, fractional Fourier transform) under one representation-theoretic principle: each is the eigenbasis of every covariance invariant under a specific finite or compact group, with columns constructed from the irreducible matrix elements of the group via the Peter-Weyl theorem.

The unification rests on the Algebraic Diversity (AD) framework, which identifies the matched group of a covariance as the foundational object of second-order signal processing.

The data-dependent KLT emerges as the trivial-matched-group limit; classical transforms emerge as the cyclic, dihedral, elementary Abelian, iterated wreath, and hybrid wreath cases, with composition rules for direct, wreath, and semidirect products.

We also mark the boundary of the construction: the structured points that correspond to no group are the eigenstructures of non-Schurian association schemes, lying just outside the matched-group catalog.

A polynomial-time algorithm, the DAD-CAD relaxation cast as a double-commutator generalized eigenvalue problem, discovers the matched group of any empirical covariance without expert judgment, with noise-aware variants via the commutativity residual $\delta$ and algebraic coloring index $\alpha$.

The fractional Fourier transform is treated as the metaplectic $SO(2)$ case, and a structural principle relates matched group size inversely to transform resolution.

Modern applications (massive-MIMO, graph neural networks, transformer attention, 3D vision, brain connectivity, single-cell genomics, quantum informatics) are sketched with their matched groups.