Logarithmic Frequency Waveforms

This article introduces a new class of Doppler-tolerant and estimation-capable radar waveforms called logarithmic frequency waveforms (LFWs). These waveforms are created by mapping the magnitudes, phases, and spacing of good digital radar codes onto a set of carriers defined over a logarithmically warped frequency axis. This mapping preserves the digital code's desirable autocorrelation mainlobe-to-sidelobe ratio when matched filtering is done over the warped frequency variable. The logarithm converts the multiplicative Doppler shift into an additive translation of the code's autocorrelation; thus, the location of the matched filter output's maximum is used to estimate the waveform's Doppler shift. This applies even if the code is neither inherently Doppler tolerant nor Doppler detection capable. We introduce two major classes of LFWs, logarithmic frequency codes, and logarithmic frequency rulers. We numerically illustrate their ambiguity functions and compare their Doppler estimation performance in additive white Gaussian noise against the linear chirp (linear frequency modulated) outfitted with the extended matched filter and the hyperbolic-frequency-modulated waveform.