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

Huiguang Zhang

TL;DR

The paper tackles wideband spectral monitoring under extreme sub-Nyquist sampling, where classical compressive sensing struggles with basis mismatch and amplitude/phase accuracy. It proposes SNGEM, which deterministically estimates all signal parameters by synchronously acquiring $x[n]$ and its time derivative, enabling reliable generalized eigenvalue solutions even at compression ratios $>10\times$. A key contribution is the closed-form Cramér-Rao bound for the dual-channel amplitude-ratio parameter $R = A/B$ and the frequency $f$, revealing only a $3\mathrm{\,dB}$ penalty compared with full-rate sampling. Monte-Carlo experiments show SNGEM closely approaches the derived CRB across SNRs and significantly outperforms OMP, establishing it as a statistically near-optimal deterministic method for sub-Nyquist parameter spectrum analysis with strong robustness and calibration advantages.

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

Cramér-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

The paper tackles wideband spectral monitoring under extreme sub-Nyquist sampling, where classical compressive sensing struggles with basis mismatch and amplitude/phase accuracy. It proposes SNGEM, which deterministically estimates all signal parameters by synchronously acquiring and its time derivative, enabling reliable generalized eigenvalue solutions even at compression ratios . A key contribution is the closed-form Cramér-Rao bound for the dual-channel amplitude-ratio parameter and the frequency , revealing only a penalty compared with full-rate sampling. Monte-Carlo experiments show SNGEM closely approaches the derived CRB across SNRs and significantly outperforms OMP, establishing it as a statistically near-optimal deterministic method for sub-Nyquist parameter spectrum analysis with strong robustness and calibration advantages.

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
Paper Structure (5 sections, 6 equations, 1 figure)

This paper contains 5 sections, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Signal Model and Dual-Channel Observation
  3. Closed-Form Cramér-Rao Bound for Amplitude-Ratio Frequency Estimation
  4. Monte-Carlo Validation and Comparison with Compressive Sensing
  5. Conclusion

Figures (1)

  • Figure 1: Relative frequency RMSE vs. SNR for $f=100$ MHz, compression ratio 15× ($f_s=133$ MS/s). SNGEM tightly follows the derived dual-channel CRB (solid black), while OMP exhibits irreducible floor $>10^{-4}$ due to grid bias and aliasing noise.

Theorems & Definitions (1)

  • Remark 1