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Equivalence of Several 6G Modulation Schemes for Doubly-Selective Channels

Nishant Mehrotra, Sandesh Rao Mattu, Robert Calderbank

TL;DR

This work addresses modulation design for doubly-selective channels with large delay and Doppler spreads by introducing two core properties: non-selectivity, which ensures full delay-Doppler diversity, and predictability, which reduces pilot overhead. It analyzes five schemes—OFDM, AFDM, ODDM, OTSM, and Zak-OTFS—within a unified discrete-time model with frame size $BT$ and channel spreading function $\mathbf{h}[k,l]$, showing that OFDM is selective and non-predictable while AFDM, ODDM, OTSM, and Zak-OTFS are non-selective, predictable, and, under unitary mappings, equivalent. The paper proves a unitary equivalence among the latter four modulations (via pulsone-based constructions and GDAFT relations) and supports the theory with Vehicular-A channel simulations that indicate equal per-carrier energy and similar BER under perfect CSI, contrasted with OFDM’s degraded performance. The results offer a principled basis for choosing and reinterpreting multicarrier schemes in 6G scenarios, highlighting the practical impact of non-selectivity and predictability on spectral efficiency and detection performance. Key contributions include formalizing the non-selectivity/predictability framework, establishing equivalence among AFDM/ODDM/OTSM/Zak-OTFS, and demonstrating the approach’s validity through detailed numerical experiments.

Abstract

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 do not perform well. In this paper, we develop a framework for analyzing such modulations using two characteristics -- non-selectivity and predictability -- which directly relate to the diversity and spectral efficiency that the modulations achieve. We show that modulations in the delay-Doppler, chirp and time-sequency domains are non-selective, predictable and equivalent to one another, whereas time-frequency modulations are selective and non-predictable.

Equivalence of Several 6G Modulation Schemes for Doubly-Selective Channels

TL;DR

This work addresses modulation design for doubly-selective channels with large delay and Doppler spreads by introducing two core properties: non-selectivity, which ensures full delay-Doppler diversity, and predictability, which reduces pilot overhead. It analyzes five schemes—OFDM, AFDM, ODDM, OTSM, and Zak-OTFS—within a unified discrete-time model with frame size and channel spreading function , showing that OFDM is selective and non-predictable while AFDM, ODDM, OTSM, and Zak-OTFS are non-selective, predictable, and, under unitary mappings, equivalent. The paper proves a unitary equivalence among the latter four modulations (via pulsone-based constructions and GDAFT relations) and supports the theory with Vehicular-A channel simulations that indicate equal per-carrier energy and similar BER under perfect CSI, contrasted with OFDM’s degraded performance. The results offer a principled basis for choosing and reinterpreting multicarrier schemes in 6G scenarios, highlighting the practical impact of non-selectivity and predictability on spectral efficiency and detection performance. Key contributions include formalizing the non-selectivity/predictability framework, establishing equivalence among AFDM/ODDM/OTSM/Zak-OTFS, and demonstrating the approach’s validity through detailed numerical experiments.

Abstract

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 do not perform well. In this paper, we develop a framework for analyzing such modulations using two characteristics -- non-selectivity and predictability -- which directly relate to the diversity and spectral efficiency that the modulations achieve. We show that modulations in the delay-Doppler, chirp and time-sequency domains are non-selective, predictable and equivalent to one another, whereas time-frequency modulations are selective and non-predictable.

Paper Structure

This paper contains 23 sections, 2 theorems, 27 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Doubly-Selective Channels
  3. Modulation Schemes for Doubly-Selective Channels
  4. OFDM
  5. AFDM
  6. ODDM
  7. OTSM
  8. Zak-OTFS
  9. Desirable Properties of Modulations
  10. Non-Selectivity
  11. Predictability
  12. Which Modulations are Non-Selective & Predictable?
  13. OFDM
  14. AFDM
  15. ODDM
  16. ...and 8 more sections

Key Result

Lemma 1

A non-selective modulation with basis $\boldsymbol{\phi}$ satisfies: for all $i,j \in \mathbb{Z}_{MN}$ and $k_1,k_2 \in \mathbb{Z}_{M}$, assuming the channel spreading function $\mathbf{h}[k,l]$ in eq:prelim2 has support $k \in \mathbb{Z}_{M}$, $l \in \mathbb{Z}_{N}$.

Figures (2)

  • Figure 1: (a) AFDM, ODDM, OTSM and Zak-OTFS are non-selective with no variation in per-carrier energy, whereas OFDM is selective with large per-carrier energy variation. (b) AFDM, ODDM, OTSM and Zak-OTFS have equal uncoded data detection performance (assuming perfect channel knowledge), however, OFDM has degraded performance due to its selectivity.
  • Figure 2: Uncoded $4$-QAM data detection performance with channel estimation using a separate pilot frame with pilot SNR equal to data SNR. The non-selective modulations (AFDM, ODDM, OTSM and Zak-OTFS) exhibit equivalent performance.

Theorems & Definitions (4)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Lemma 2