Estimation of differential Doppler shifts

Abstract

If the acoustic signal radiated by a moving source is observed at two or more locations, the received signals exhibit differential Doppler shifts which provide information about source motion. This paper calculates minimum mean-square errors for the estimates of differential Doppler shifts, which can be obtained in a given observation interval. Both Gaussian and sinusoidal signals are considered. The noise is assumed to be Gaussian and independent from sensor to sensor. Dependence of the estimation errors on observation time, signal-to-noise ratio, and size of the receiving array are studied. The estimation of differential Dopplers is found to be uncoupled from the estimation of differential delays and from the estimation of signal parameters, such as center frequency and bandwidth. A comparison is made between two possible procedures of differential Doppler estimation: coherent processing of the signals received at two sensors and subtraction of separate frequency estimates obtained at each sensor. The two are equivalent for sinusoidal signals, but for large TW Gaussian signals the coherent procedure yields substantially smaller errors.