Affine Frequency Division Multiplexing for Compressed Sensing of Time-Varying Channels

TL;DR

This work tackles reliable recovery of doubly sparse time-varying wireless channels (sparse in both delay and Doppler) using Affine Frequency Division Multiplexing (AFDM). By linking delay-Doppler sparsity to the hierarchical sparsity framework, it introduces a sparse recovery approach based on AFDM measurements and HiHTP, complemented by HiRIP-based performance guarantees. The analysis shows that with appropriately chosen AFDM parameters, the recovery can achieve near-optimal performance with reduced pilot overhead and sub-Nyquist sampling for radar sensing. The results indicate a practical advantage of AFDM over OFDM and OTFS in terms of estimation overhead and required sampling rates, enabling efficient sensing and communication in high-mobility, high-frequency scenarios.

Abstract

This paper addresses compressed sensing of linear time-varying (LTV) wireless propagation links under the assumption of double sparsity i.e., sparsity in both the delay and Doppler domains, using Affine Frequency Division Multiplexing (AFDM) measurements. By rigorously linking the double sparsity model to the hierarchical sparsity paradigm, a compressed sensing algorithm with recovery guarantees is proposed for extracting delay-Doppler profiles of LTV channels using AFDM. Through mathematical analysis and numerical results, the superiority of AFDM over other waveforms in terms of channel estimation overhead and minimal sampling rate requirements in sub-Nyquist radar applications is demonstrated.

Figures (5)

• Figure 1: Time-frequency representation of three subcarriers of OFDM and AFDM ($c_1=\frac{P}{2N}$). Each subcarrier is represented with a different colour.
• Figure 2: Examples of channels satisfying (a) Type-1 delay-Doppler sparsity, (b) Type-2 delay-Doppler sparsity, (c) Type-3 delay-Doppler sparsity
• Figure 3: Time-frequency content of one AFDM pilot and its echoes, before and after analog de-chirping and sampling ($\tau_{\max} \triangleq (L-1)\Delta t$, $T_{\rm CPP} \triangleq L_{\rm CPP}\Delta t$)
• Figure 4: MSE and pilot overhead for $N=4096, L=30, Q=7, p_{\rm d}=0.2$, $N_{\rm ofdm,symb}=16$, $N_{\rm otfs}=16$, $M_{\rm otfs}=256$. Overhead: $N_{\rm p,td}N_{\rm p,fd}+(N_{\rm ofdm,symb}-1)(L-1)$ for OFDM, $\min(4Q+1,N_{\rm otfs})\min(2L-1,M_{\rm otfs})$ for OTFS, $N_{\rm p}\left((L-1)P+1\right)+(L-1)P+4Q$ for AFDM.
• Figure 5: Two examples of the set $\mathcal{D}_l$ ($P=1$) (a) for an $l$ resulting in a whole diagonal, (b) for an $l$ resulting in a wrapped diagonal. In each one of the two examples, the grid points forming $\mathcal{D}_l$ are shown surrounded by red rings.

Theorems & Definitions (9)

• Definition 1: Hierarchical sparsity, hierarchical
• Definition 2: HiRIP, hierarchical
• Definition 3: Delay-Doppler double sparsity, afdm_gc
• Lemma 1
• proof
• Theorem 1: HiRIP for AFDM based measurements
• proof
• Corollary 1: Recovery guarantee for AFDM based measurements
• proof