Measurement of the differential Doppler shift

Doppler shifts between narrow-band signals observed at one or more pairs of receivers and originated from a remote source of radiation are useful for estimating source location and track. This paper deals with an instrumentationally attractive approach of estimating differential Doppler shifts by making center frequency measurements at each receiver output and subtracting them in a pairwise fashion. For low in-band signal-to-noise ratio conditions, center frequency measurements at different receiver outputs are weakly correlated. In this mode of operation, therefore, the mean-square error in the differential Doppler estimate equals the pairwise sum of the mean-square errors in each of the center frequency measurement. For high signal-to-noise ratio conditions, the various center frequency estimates are strongly correlated. In this mode of operation, the accuracy of the resulting differential Doppler estimate improves with the first power of the signal-to-noise ratio, even though the accuracy of each center frequency estimate approaches an absolute bound independent of the noise spectrum. The optimal (minimum mean-square error) differential Doppler shift estimate is obtained by simultaneous processing of the receiver outputs jointly. Comparison between the former (indirect) and the latter (direct) estimation techniques yields some interesting insights: the accuracy of the direct estimation procedure is proportional to T^-3where T is the observation period. It is basically a coherent procedure. Center frequency measurement and the differential Doppler shift estimate derived from it are basically incoherent procedures and obey the well known T^-1dependence. Under high signal-to-noise ratio conditions, the estimation error in both methods decreases with the first power of the signal-to-noise ratio. Under low signal-to-noise ratio conditions, the accuracy of the suboptimal indirect method is inferior to the optimal direct method by a factor proportional to the inverse first power of the individual signal-to-noise ratio.