A generalized cross-ambiguity function for geolocation

We derive a generalization of the cross-ambiguity function (CAF) based on time-scaling, which we apply to geolocation. We demonstrate by simulations that this scale cross-ambiguity function (SCAF) is much more accurate than CAF for geolocating wide-band transmitters. Under the assumption that the transmitted signal is a sinusoid having slowly varying complex modulation, the conventional CAF process models a received signal as doppler shift of the carrier frequency, composed with a time delay. For a more general class of transmitted signals, including wide-band signals, the CAF model is inadequate. For transmitted signals which are not dominated by a single carrier, it is more accurate to model the received signals as a time-scaled versions of the transmitted signal, composed with a time delay. The resulting SCAF model accounts for doppler shifts of all frequencies present in the emitted signal, and agrees with the CAF model when the transmitted signal has a single dominant carrier. For wide-band signals, SCAF is more accurate than CAF for computing estimates of the time-difference of arrival (TDOA) of a transmitted signal collected at two receivers, and the difference in radial velocities of the two receivers. The accuracy of these two difference estimates is critical, since TDOA and velocity difference separately determine spatial curves whose intersection determines the location of the emitter. SCAF may also be applied to the problem of estimating range and radial velocity of radar targets, for wide-band transmitted signals.