Cram\'er-Rao Bound Analysis and Near-Optimal Performance of the Synchronous Nyquist-Folding Generalized Eigenvalue Method (SNGEM) for Sub-Nyquist Multi-Tone Parameter Estimation

TL;DR

This paper demonstrates that the SNGEM method achieves near-optimal, sub-Nyquist multi-tone parameter estimation performance, closely approaching the theoretical Cramer-Rao bounds even at high compression rates.

Contribution

It derives accurate CRB for amplitude ratio parameters and shows SNGEM's near-optimal performance compared to classical methods under sub-Nyquist sampling.

Findings

SNGEM achieves machine accuracy in noise-free conditions.

SNGEM closely approaches CRB at all SNR levels.

Classical compressive sensing methods exhibit irreducible errors.

Abstract

The synchronous Nyquist folding generalized eigenvalue method (SNGEM) realizes full frequency/amplitude/phase estimation of multitone signals at extreme sub-Nyquist rates by jointly processing the original signals and their time derivatives. In this paper, accurate Cramer-Rao bounds for amplitude ratio parameter R=A/B=1/(2\pif) are derived for two channels with equal SNR. Monte-Carlo simulations confirm that SNGEM achieves machine accuracy in noise-free conditions and closely approaches the derived CRB at all SNR levels, even at 10- 20x compression, whereas classical compressive sensing OMP exhibits irreducible error flattening due to DFT grid bias and aliasing noise. These results establish SNGEM as a statistically nearly optimal deterministic sub-Nyquist parameter spectrum analysis