The first-order Doppler effect has, in general, been used in electronic tracking, but for the greater accuracies now desired in velocity measurements, this approach is no longer satisfactory. A fundamental study of the Doppler effect is presented in this paper. Five steps are developed which enable the exact derivation of the Doppler equations for any system. Six different configurations of transmitter, receiver, and vehicle are investigated, and the results are applied to a number of Doppler systems. It is determined that, for velocity inaccuracies <1.0 fps, the second-order relativistic or classical equations must be used. The receipt of a zero Doppler shift has also been investigated, and it does not necessarily imply zero line-ofsight velocity. Nomenclature a = proportional frequency shift factor produced by a beacon 6 = constant frequency shift factor produced by a beacon c = velocity of light / = frequency h = factor relating the constant frequency shift to the proportional shift produced by a beacon N = wave normal of a plane wave t = time parameter v = velocity magnitude V = velocity relative to the earth w = phase velocity X = position vector Subscripts i = receiver-fixed reference system when subscript 0 refers to transmitter d = Doppler received by a beacon r = receiver reference system t = transmitter reference system v = vehicle 0 = fixed reference system (receiver or transmitter as the case may be)