A Design Framework that Unifies 6G Modulation Schemes for Double Selectivity

There is significant recent interest in designing new modulation schemes for doubly-selective channels with large delay and Doppler spreads, where legacy modulation schemes based on time-frequency signal representations underperform. Multiple modulation schemes, e.g., in the delay-Doppler, chirp, time-sequency, and other domains, have been proposed in the literature for this purpose, with varying implementation details. In this letter, we establish that all previously proposed modulation schemes for doubly-selective signaling are instances of a single family of complex Hadamard-modulated pulse trains. When the delay and Doppler spread of the doubly-selective channel is limited to a certain support, all modulation schemes in this waveform family offer equivalent, full diversity achieving performance with no symbol fading and low channel estimation overhead. The existence of this waveform family also enables flexible, multi-waveform co-existence – allowing a common transceiver architecture to generate multiple waveforms in the family, that may each be flexibly allocated to different users and services.

💡 Research Summary

The paper addresses a critical challenge for future 6G wireless systems: reliable communication over doubly‑selective channels that exhibit large delay and Doppler spreads. Conventional modulation schemes such as OFDM, which are based on a time‑frequency (TF) representation, suffer severe performance degradation in such environments because of symbol fading and high channel‑estimation overhead. Recent literature has proposed a variety of alternative schemes—Zak‑OTFS, ODDM, AFDM, OTSM, among others—each operating in a different domain (delay‑Doppler, chirp, time‑sequency, etc.). However, these proposals have been presented in isolation, with disparate implementation details and without a unifying theoretical perspective.

The authors demonstrate that all these apparently distinct modulation formats are special instances of a single family of waveforms, denoted Ψ, which they call complex Hadamard‑modulated pulse trains. A Ψ waveform is constructed from a periodic pulse train of length (M N) (where (M) and (N) are the numbers of delay and Doppler bins, respectively). The construction uses two degrees of freedom: an arbitrary unitary matrix (U) of size (M N \times M N) and a set of (N \times N) complex Hadamard matrices (H(i)) indexed by the delay‑bin index (i). Each basis element takes the form

\