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Efficient Estimation of Sum-Parameters for Multi-Component Complex Exponential Signals with Theoretical Cramer-Rao Bound Analysis

Huiguang Zhang

TL;DR

This work tackles parameter estimation for sums of complex exponentials by introducing a low-dimensional, permutation-invariant set of sum-parameters ($\Sigma, \Omega, \Phi$). It derives exact closed-form Cramér-Rao bounds for these parameters under deterministic and stochastic models, revealing that the frequency sum-parameter benefits from total signal power without explicit weighting. The Efficient Global Estimation Method (EGEM) is proposed, offering $O(N\log N)$ complexity and asymptotic efficiency across a wide range of conditions, and simulations show EGEM approaching the theoretical bounds and outperforming Zoom-Interpolated FFT and Root-MUSIC, even with limited data. The approach has strong implications for large-scale, real-time multi-component signal analysis and motivates future extensions to nonuniform sampling, damped models, and hybrid data-driven techniques.

Abstract

This paper addresses the challenging problem of parameter estimation for multicomponent complex exponential signals, commonly known as sums of cisoids. Traditional approaches that estimate individual component parameters face significant difficulties when the number of components is large, including permutation ambiguity, computational complexity from high-dimensional Fisher information matrix inversion, and model order selection issues. We introduce a novel framework based on low-dimensional sum-parameters that capture essential global characteristics of the signal ensemble. These parameters include the sum of amplitudes, the power-weighted frequency, and the phase-related sum. These quantities possess clear physical interpretations representing total signal strength, power-weighted average frequency, and composite phase information, while completely avoiding permutation ambiguities. We derive exact closed-form Cramer-Rao bounds for these sum-parameters under both deterministic and stochastic signal models. Our analysis reveals that the frequency sumparameter achieves statistical efficiency comparable to single-component estimators while automatically benefiting from power pooling across all signal components. The proposed Efficient Global Estimation Method (EGEM) demonstrates asymptotic efficiency across a wide range of signal-to-noise ratios, significantly outperforming established techniques such as Zoom-Interpolated FFT and Root-MUSIC in both long- and short-sample regimes. Extensive numerical simulations involving 2000 Monte-Carlo trials confirm that EGEM closely approaches the theoretical performance bounds even with relatively small sample sizes of 250 observations.

Efficient Estimation of Sum-Parameters for Multi-Component Complex Exponential Signals with Theoretical Cramer-Rao Bound Analysis

TL;DR

This work tackles parameter estimation for sums of complex exponentials by introducing a low-dimensional, permutation-invariant set of sum-parameters (). It derives exact closed-form Cramér-Rao bounds for these parameters under deterministic and stochastic models, revealing that the frequency sum-parameter benefits from total signal power without explicit weighting. The Efficient Global Estimation Method (EGEM) is proposed, offering complexity and asymptotic efficiency across a wide range of conditions, and simulations show EGEM approaching the theoretical bounds and outperforming Zoom-Interpolated FFT and Root-MUSIC, even with limited data. The approach has strong implications for large-scale, real-time multi-component signal analysis and motivates future extensions to nonuniform sampling, damped models, and hybrid data-driven techniques.

Abstract

This paper addresses the challenging problem of parameter estimation for multicomponent complex exponential signals, commonly known as sums of cisoids. Traditional approaches that estimate individual component parameters face significant difficulties when the number of components is large, including permutation ambiguity, computational complexity from high-dimensional Fisher information matrix inversion, and model order selection issues. We introduce a novel framework based on low-dimensional sum-parameters that capture essential global characteristics of the signal ensemble. These parameters include the sum of amplitudes, the power-weighted frequency, and the phase-related sum. These quantities possess clear physical interpretations representing total signal strength, power-weighted average frequency, and composite phase information, while completely avoiding permutation ambiguities. We derive exact closed-form Cramer-Rao bounds for these sum-parameters under both deterministic and stochastic signal models. Our analysis reveals that the frequency sumparameter achieves statistical efficiency comparable to single-component estimators while automatically benefiting from power pooling across all signal components. The proposed Efficient Global Estimation Method (EGEM) demonstrates asymptotic efficiency across a wide range of signal-to-noise ratios, significantly outperforming established techniques such as Zoom-Interpolated FFT and Root-MUSIC in both long- and short-sample regimes. Extensive numerical simulations involving 2000 Monte-Carlo trials confirm that EGEM closely approaches the theoretical performance bounds even with relatively small sample sizes of 250 observations.

Paper Structure

This paper contains 11 sections, 7 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Signal Model and Sum-Parameter Definition
  3. Cramér-Rao Bound for Sum-Parameters
  4. Fisher Information Matrix Derivation
  5. Closed-Form Cramér-Rao Bounds
  6. Efficient Global Estimation Method (EGEM)
  7. Performance Evaluation
  8. Simulation Setup
  9. Results and Analysis
  10. Conclusion
  11. Detailed Derivation of the Fisher Information Matrix for Sum-Parameters

Figures (2)

  • Figure 1: Frequency estimation normalized RMSE versus SNR for $N=2000$ samples. The proposed EGEM algorithm tightly tracks the derived Cramér-Rao bound across the complete SNR range, demonstrating asymptotic efficiency.
  • Figure 2: Performance comparison under short-sample conditions ($N=125, 250, 500$). EGEM maintains near-CRB performance even with $N=250$ samples, while Zoom-IpFFT exhibits significant error floor and Root-MUSIC performance degrades substantially.