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Models, Methods and Waveforms for Estimation and Prediction of Doubly Sparse Time-Varying Channels

Wissal Benzine, Ali Bemani, Nassar Ksairi, Dirk Slock

TL;DR

The work tackles estimation and prediction for doubly sparse time-varying channels in the delay-Doppler domain. It first establishes an on-grid DS-LTV framework with hierarchical sparsity, showing AFDM offers reduced pilot overhead for sparse channels compared to SCM, OFDM, and OTFS. To capture fractional Doppler shifts, it develops a DPSS-based off-grid BEM representation and an efficient LMMSE estimator, followed by a DPSS-driven channel extrapolation/prediction method with reduced-rank optimality. Numerical results validate the theoretical findings, demonstrating AFDM’s advantages in estimation and prediction under delay-Doppler sparsity and the practical viability of DPSS-based off-grid modeling. Altogether, the paper presents a comprehensive modeling, estimation, and prediction suite for DS-LTV channels with clear implications for high-mobility wireless systems.

Abstract

This paper investigates channel estimation for linear time-varying (LTV) wireless channels under double sparsity, i.e., sparsity in both the delay and Doppler domains. An on-grid approximation is first considered, enabling rigorous hierarchical-sparsity modeling and compressed sensing-based channel estimation. Guaranteed recovery conditions are provided for affine frequency division multiplexing (AFDM), orthogonal frequency division multiplexing (OFDM) and single-carrier modulation (SCM), highlighting the superiority of AFDM in terms of doubly sparse channel estimation. To address arbitrary Doppler shifts, a relaxed version of the on-grid model is introduced by making use of multiple elementary Expansion Models (BEM) each based on Discrete Prolate Spheroidal Sequences (DPSS). Next, theoretical guarantees are provided for the precision of this off-grid model before further extending it to tackle channel prediction by exploiting the inherent DPSS extrapolation capability. Finally, numerical results are provided to both validate the proposed off-grid model for channel estimation and prediction purposes under the double sparsity assumption and to compare the corresponding mean squared error (MSE) and the overhead performance when the different wireless waveforms are used.

Models, Methods and Waveforms for Estimation and Prediction of Doubly Sparse Time-Varying Channels

TL;DR

The work tackles estimation and prediction for doubly sparse time-varying channels in the delay-Doppler domain. It first establishes an on-grid DS-LTV framework with hierarchical sparsity, showing AFDM offers reduced pilot overhead for sparse channels compared to SCM, OFDM, and OTFS. To capture fractional Doppler shifts, it develops a DPSS-based off-grid BEM representation and an efficient LMMSE estimator, followed by a DPSS-driven channel extrapolation/prediction method with reduced-rank optimality. Numerical results validate the theoretical findings, demonstrating AFDM’s advantages in estimation and prediction under delay-Doppler sparsity and the practical viability of DPSS-based off-grid modeling. Altogether, the paper presents a comprehensive modeling, estimation, and prediction suite for DS-LTV channels with clear implications for high-mobility wireless systems.

Abstract

This paper investigates channel estimation for linear time-varying (LTV) wireless channels under double sparsity, i.e., sparsity in both the delay and Doppler domains. An on-grid approximation is first considered, enabling rigorous hierarchical-sparsity modeling and compressed sensing-based channel estimation. Guaranteed recovery conditions are provided for affine frequency division multiplexing (AFDM), orthogonal frequency division multiplexing (OFDM) and single-carrier modulation (SCM), highlighting the superiority of AFDM in terms of doubly sparse channel estimation. To address arbitrary Doppler shifts, a relaxed version of the on-grid model is introduced by making use of multiple elementary Expansion Models (BEM) each based on Discrete Prolate Spheroidal Sequences (DPSS). Next, theoretical guarantees are provided for the precision of this off-grid model before further extending it to tackle channel prediction by exploiting the inherent DPSS extrapolation capability. Finally, numerical results are provided to both validate the proposed off-grid model for channel estimation and prediction purposes under the double sparsity assumption and to compare the corresponding mean squared error (MSE) and the overhead performance when the different wireless waveforms are used.

Paper Structure

This paper contains 18 sections, 8 theorems, 74 equations, 16 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model: Linear time-varying (LTV) channels
  3. Estimation of doubly sparse LTV channels with the on-grid approximation
  4. First approximation: on-grid doubly sparse LTV channels
  5. Relation to hierarchical sparsity hierarchical
  6. Theoretical analysis of sparse recovery of DS-LTV channels with different waveforms
  7. Estimation and extrapolation of doubly sparse LTV channels with off-grid Doppler shifts
  8. Off-grid doubly sparse linear time-varying channels
  9. Background: DPSS Basis Expansion Model
  10. Off-grid modeling using multiple shifted elementary BEMs
  11. DS-LTV off-grid LMMSE channel estimation
  12. Application to channel extrapolation and prediction
  13. Numerical results
  14. Conclusions
  15. Proof of Theorem \ref{['theo:HiRIP_SCM_OFDM']}
  16. ...and 3 more sections

Key Result

Lemma 1

With probability $1-e^{-\Omega\left(\min\left((2Q+1)p_{\rm D},Lp_{\rm d}\right)\right)}$, $\boldsymbol{\alpha}$ is $\left(s_{\rm d},s_{\rm D}\right)$-sparse under Assumption assum:technical.

Figures (16)

  • Figure 1: Examples of channels satisfying (a) Type-1, (b) Type-2, (c) Type-3 delay-Doppler sparsity
  • Figure 2: SCM frame composed of data samples and $N_{\rm p}$ pilot symbols, each of the latter surrounded by $2L-1$ guard samples.
  • Figure 3: A frame of $N_{\rm ofdm,symb}$$N_{\rm fft}$-long OFDM symbols, $N_{\rm p,t}$ of which having $N_{\rm p,f}$ pilot subcarriers
  • Figure 4: An AFDM symbol with $N_{\rm p}$ pilot symbols and their guard samples.
  • Figure 5: An OTFS symbol composed in the Zak domain of data samples (red), a pilot sample (blue) and guard samples (light blue and red)
  • ...and 11 more figures

Theorems & Definitions (26)

  • Definition 1: On-grid Delay-Doppler double sparsity, afdm_gc
  • Definition 2: Hierarchical sparsity
  • Definition 3: HiRIP, hierarchical
  • Lemma 1
  • proof
  • Theorem 1: HiRIP for SCM and OFDM based measurements
  • proof
  • Theorem 2: HiRIP for AFDM based measurements
  • proof
  • Corollary 1: Recovery guarantee for compressive sensing of DS-LTV channels
  • ...and 16 more