A parametric form for the received “instantaneous frequency” of a pure tone is derived for the case of relative straight line motion between source and receiver. Ferguson and Quinn [J. Acoust. Soc. Am. 96, 821-827 (1994)] have recently described an algorithm which, given a sequence of high-accuracy frequency estimators, produces estimators of the closest distance between the source and receiver, the relative speed, the rest frequency, and the time at closest approach. In this paper, an improved algorithm which uses amplitude estimators as well is introduced, and the statistical behavior of the estimators derived. The technique is automatic and will work even if only a section of the Doppler track is available.