Scale cross-ambiguity and target resolution

We derive the correct formulation of the Doppler problem in which the received signal is scaled in time by a scale factor dependent on the relative radial velocity of the transmitter and the receiver, normalized by the propagation speed in the transmission medium. For sine waves, the conventional Doppler model in which the received signal is translated in frequency is equivalent to the scale model. However, for signals with non zero bandwidth, the difference can be quite significant, as we demonstrate. We further demonstrate simple methods, based on the discrete Fourier transform (DFT) and phase differentiation of the scale CAF, by which delay and scale may be accurately estimated.