Signals like ECG and EEG are oscillatory, and analysing them means decomposing the waveform into components you can compute quickly and, ideally, explain to a clinician. Those two demands often pull apart: the fastest decompositions produce components that mean nothing in particular, and the interpretable ones tend to be expensive.

Two frameworks have grown up separately addressing this. Adaptive Fourier Decomposition builds on orthonormal Takenaka-Malmquist expansions, while the Frequency-Modulated Mobius model is a parametric decomposition built on Mobius transformations. Each has its own literature, its own notation and its own users.

This paper establishes a formal connection between them. Unification results are quietly valuable in a field like this: when two methods turn out to be views of one structure, every result proved for one becomes available to the other, and practitioners stop having to pick a camp before they can start work.