Simultaneous estimation of time-of-arrival, pole, and amplitude parameters for transient exponential signals

The simultaneous estimation of time-of-arrival, damping, frequency, and amplitude parameters of sums of exponentials in white Gaussian noise is cast in terms of a constrained optimization problem. The constraints derive from smoothness conditions at the signal onset (i.e. the signal starts at zero and one of its derivatives may also be zero at the signal onset). Without such constraints, time-of-arrival estimates are shown to be, at best, limited to sampling times, and in the case of the data window beginning after the signal onset, all time-of-arrival estimates are equally valid. Applying the initial condition constraints, an error norm is derived which varies continuously with the time-of-arrival. For this constrained error norm, maximum likelihood estimator performance is demonstrated through simulation of the response of two-pole highpass and bandpass systems to a stepped sinusoid