On the Doppler Bias of Hyperbolic Frequency Modulation Matched Filter Time of Arrival Estimates

The purpose of this paper is to introduce an extended matched filter for hyperbolic frequency modulation (HFM) waveforms in active sonar systems, along with an exact closed-form solution for the Doppler bias in time of arrival estimates when using this filter. Conventional HFM matched filtering employs a single undilated replica of the transmitted signal and estimates the time of an echo's arrival using the peak squared magnitude of the processed data. It is well known that when the echo is dilated (the term dilate is used in a generic sense to mean either stretching or compressing) due to the Doppler effect, such an estimate of the time of arrival is biased by an amount depending on the range rate v, however, a closed-form solution for this Doppler bias does not exist. Using the extended HFM matched filter in place of a conventional one allows for an exact closed-form solution for the Doppler bias. The closed-form solution for the Doppler bias of time of arrival estimates using the extended HFM matched filter is presented. This solution applies to both broadband and narrowband HFM signals, and suggests that a previously published approximation of the bias for a conventional HFM matched filter is valid only for narrowband HFM signals.