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Receiving an output signal that is a weighted sum of Doppler and delay shifted versions of an input signal arises in various engineering systems including radar, sonar, and communications. It is often desired to estimate the Doppler and delay parameters from noisy measurements of the output signal. An estimation algorithm is developed herein for the Doppler frequencies, delays, and amplitudes, for a specific choice of the input. The algorithm is based on a computationally efficient search-free frequency estimation technique for the sum of complex exponentials. Asymptotic performance bounds are developed for the estimated parameters. Simulation results show that the variance of the estimation error is close to the bound above a threshold signal-to-noise ratio, indicating that the estimation algorithm is statistically efficient.