Group-Theoretic Wideband Radar Waveform Design

TL;DR

This paper introduces a novel group-theoretic approach to designing wideband radar waveforms by leveraging affine group representations, tight frames, and difference sets to optimize the ambiguity function.

Contribution

It develops a new algebraic framework for wideband radar waveform design using affine groups and tight frames, extending narrow-band analysis to wideband signals.

Findings

Designed waveforms exhibit lower sidelobes in the WAF.

Constructed a codebook based on difference sets achieving the Welch bound.

Connected the tight frame design with wavelet transform properties.

Abstract

We investigate the theory of affine groups in the context of designing radar waveforms that obey the desired wideband ambiguity function (WAF). The WAF is obtained by correlating the signal with its time-dilated, Doppler-shifted, and delayed replicas. We consider the WAF definition as a coefficient function of the unitary representation of the group $a\cdot x + b$. This is essentially an algebraic problem applied to the radar waveform design. Prior works on this subject largely analyzed narrow-band ambiguity functions. Here, we show that when the underlying wideband signal of interest is a pulse or pulse train, a tight frame can be built to design that waveform. Specifically, we design the radar signals by minimizing the ratio of bounding constants of the frame in order to obtain lower sidelobes in the WAF. This minimization is performed by building a codebook based on difference sets in order to achieve the Welch bound. We show that the tight frame so obtained is connected with the wavelet transform that defines the WAF.