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

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.