Matched Transforms for Compact Signal Classification: A Unified Technical Record of Three Studies — Spectral (Fourier), Wavelet, and a Pooled Matched-Transform Dictionary

This document consolidates three companion studies into one narrative. A fixed transform, a compaction step that keeps a few principal coordinates, a magnitude/phase feature map and a linear readout form a strong, cheap classifier — when the transform matches the signal's structure. Part I develops this for the Fourier transform and shows that the most effective single improvement is selecting frequencies by discriminative power (ANOVA F-score) rather than energy; that, with parameters counted in full (each complex weight as two reals), the model is competitive with a spiking network at a fraction of the budget; and that the advantage is conditional on spectral structure. Part II repeats the programme with wavelets, adding the key observation that localized wavelet magnitude recovers temporal order a Fourier magnitude hides in phase, while a fully invariant scattering discards it, and that an orthogonal DWT fails on narrow-band global structure. Part III pools the transforms into one dictionary and lets discriminative selection route itself — a reconstruction of the Local Discriminant Basis — and stress-tests it across six source types (1-D localized and global, two image types, text, video) against a deep network. A final component removes the last manual choice — the classifier head — by making it self-organizing: linear by default, growing gated non-linear blocks only where the data demands them, at negligible convergence cost. Throughout, the framing is candid: these are reconstructions and cross-modality validations of classical ideas, with negative results reported as prominently as positive ones.