Estimation of the Frequency and Decay Factor of a Decaying Exponential in Noise

In this paper, we examine the estimation of the parameters of a decaying complex exponential in noise. The strategy adopted consists of a computationally simple two stage scheme where an interpolation stage refines the coarse estimate obtained from an initial maximum bin search. The interpolators of Quinn, and of Aboutanios and Mulgrew, developed for undamped exponentials are extended to the damped case. In the process, we show that Quinn's estimator can be viewed as a linearized version of Bertocco's algorithm. Theoretical analysis demonstrates that the resulting estimators exhibit similar behavior to the undamped case, leading us to propose two alternative hybrid implementations that yield a significant improvement in the estimation performance. Unlike the undamped case, however, we show that there exists a finite number of samples for which the estimation performance is best, and which we determine in terms of the damping factor. This enables us to adjust the actual number of samples should it deviate significantly from the optimum. Extensive simulation results are presented to support the theoretical findings.