Joint Fractional Delay and Doppler Frequency Estimator Under Spectrum Wrapping Phenomenon for LEO-ICAN AFDM Signals

With the rapid development of low earth orbit (LEO) satellites, the design of integrated communication and navigation (ICAN) signals has attracted increasing attention, especially in the field of vehicle-to-everything (V2X). As a new-generation waveform, Affine Frequency Division Multiplexing (AFDM) features high robustness against Doppler effects, a simple modulation structure, and low pilot overhead, making it a promising candidate for high-dynamic LEO satellite scenarios. However, LEO-ICAN AFDM signals face challenges in fractional delay and Doppler frequency estimation. Existing studies that ignore its inherent spectrum wrapping phenomenon may lead to deviations of varying degrees in model construction. This paper conducts an in-depth derivation of AFDM’s input-output relationship under fractional cases, reveals the envelope characteristics of its equivalent channel, and proposes a joint estimation algorithm based on peak-to-sidelobe power ratio (PSPR) detection and early-late gate (ELG) to estimate fractional Doppler frequency and delay. Simulations show that the algorithm has low complexity, low guard interval overhead, and high precision compared with traditional methods.

💡 Research Summary

The paper addresses a critical problem for low‑earth‑orbit (LEO) satellite‑ground integrated communication and navigation (ICAN) systems: accurate estimation of fractional delay and fractional Doppler frequency in Affine Frequency Division Multiplexing (AFDM) waveforms. AFDM, which relies on the Discrete Affine Fourier Transform (DAFT), offers superior Doppler resilience compared with conventional OFDM, but its inherent “spectrum wrapping” – a discontinuity that occurs when the signal’s frequency components fold back onto the baseband – has been largely ignored in prior work. This omission leads to mismatches between theoretical channel models and the actual doubly‑selective channel, degrading estimation accuracy, especially in the high‑mobility LEO scenario where both delay and Doppler are rarely integer multiples of the sampling intervals.

The authors first derive a rigorous input‑output relationship for AFDM that explicitly incorporates the wrapping effect. Starting from the continuous‑time signal with a chirp periodic prefix (CPP), they model the doubly‑selective channel, sample the received waveform, and apply the DAFT demodulation. The resulting received symbol vector y can be written as y = H_eff x + w, where the effective channel matrix H_eff contains a key term F(m,m′). This term is a superposition of piecewise Fourier bases indexed by the chirp segment number q. Under the practical assumption that the maximum normalized delay l_max is much smaller than the number of subcarriers N, the magnitude |F| can be approximated by the product of two distinct patterns: (i) a periodic sinc‑sampling sequence Υ(m,m′) that is sensitive to the fractional Doppler κ, and (ii) a global sinc envelope Θ(m,m′) that depends jointly on the fractional delay ι and κ. Theorem 1 formalizes this approximation and provides a compact expression that reveals how the spectrum wrapping creates coherent structures in the DAFT domain.

Building on this insight, the paper proposes a two‑stage joint estimator.

1. Fractional Doppler Estimation and Compensation (Closed‑Loop) – An embedded single‑pilot (SPA) is used to obtain the integer parts of delay and Doppler. The remaining fractional Doppler κ is searched by maximizing a Peak‑to‑Sidelobe Power Ratio (PSPR) computed from the compensated received signal **˜y