Alternative “Bases” for Gradient Based Optimization of Parameterized FM Radar Waveforms

Even for a fixed time-bandwidth product there are infinite possible spectrally-shaped random FM (RFM) waveforms one could generate due to their being phase-continuous. Moreover, certain RFM classes rely on an imposed basis-like structure scaled by underlying parameters that can be optimized (e.g. gradient descent and greedy search have been demonstrated). Because these structures must include oversampling with respect to 3-dB bandwidth to account for sufficient spectral roll-off (necessary to be physically realizable in hardware), they are not true bases (i.e. not square). Therefore, any individual structure cannot represent all possible waveforms, with the waveforms generated by a given structure tending to possess similar attributes. Unless of course we consider over-coded polyphaser-coded FM (PCFM), which increases the number of elements in the parameter vector, while maintaining the relationship between waveform samples and the time-bandwidth product. Which presents the potential for a true bases, if there is a constraint either explicit or implicit that will constrain the spectrum. Here we examine waveforms possessing different attributes, as well as the potential for a true basis which may inform their selection for given radar applications.