A comparison of two methods for joint time-scale and TDOA estimation for geolocation of electromagnetic transmitters

For wide-band transmission, geolocation modeling using the wide-band cross-ambiguity function (WBCAF) is preferable to conventional CAF modeling, which assumes that the transmitted signal is essentially a sinusoid. We compare the accuracy of two super-resolution techniques for joint estimation of the time-scale (TS) and TDOA parameters in the WBCAF geolocation model. Assuming a complex-valued signal representation, both techniques exploit the fact that the maximum value of the magnitude of the WBCAF is attained when the WBCAF is real-valued. The first technique enhances a known joint estimation method based on sinc interpolation and 2-D Newton root-finding by (1) extending the original algorithm to handle complex-valued signals, and (2) reformulating the original algorithm to estimate the difference in radial velocities of the receivers (DV) rather than time scale, which avoids machine precision problems encountered with the original method. The second technique makes a rough estimate of TDOA on the sampling lattice by peak-picking the real part of the cross-correlation function of the received signals. Then, by interpolating the phase of the WBCAF, it obtains a root of the phase in the vicinity of this correlation peak, which provides a highly accurate TDOA estimate. TDOA estimates found in this way are differentiated in time to obtain DV estimates. We evaluate both super-resolution techniques applied to simulated received electromagnetic signals which are linear combinations of complex sinusoids having randomly generated amplitudes, phases, TS, and TDOA. Over a wide SNR range, TDOA estimates found with the enhanced sinc/Newton technique are at least an order of magnitude more accurate than those found with conventional CAF, and the phase interpolated TDOA estimates are 3-4 times more accurate than those found with the enhanced sinc/Newton technique. In the 0-10 dB SNR range, TS estimates found with the enhanced sinc/Newton technique are a little more accurate than those found with phase interpolation. Moreover, the TS estimate errors observed with both super-resolution techniques are too small for a CAF-type grid search to realize in comparable time.