Damping factor estimation of damped complex sinusoidal signals using a maximum likelihood approach

This paper presents an approximate Maximum Likelihood (ML) estimation framework for estimating the damping factor in exponentially decaying complex sinusoidal signals. We derive a closed-form approximate ML estimator, explicitly noting its validity for small damping factors. Its efficiency is assessed through the derivation of the Cramer-Rao Lower Bound (CRLB), which serves as the theoretical benchmark for estimator performance. Analytical proof demonstrates that the proposed approximate ML estimator achieves the CRLB under small-damping conditions, confirming its statistical efficiency and optimality. The performance of the estimator is evaluated by analyzing its behavior with respect to critical system parameters such as signal-to-noise ratio (SNR), sample size, signal amplitude, and damping intensity. Our analysis reveals several fundamental relationships: estimation accuracy improves with increasing SNR and a larger sample size, confirming that the approximate ML estimator effectively utilizes additional information. Furthermore, the empirical variance of the approximate ML estimator converges to the CRLB, validating its efficiency. These findings yield practical guidelines for system designers. We establish that weakly damped signals can achieve satisfactory accuracy with moderate resources, whereas strongly damped scenarios necessitate substantially enhanced signal conditions. The proposed framework provides optimal estimation performance while also achieving significantly reduced computational complexity. This combination makes it particularly suitable for applications requiring rapid damping parameter extraction, such as structural health monitoring, radar systems, and vibration analysis.