Combined Radar and Communications with Phase-Modulated Frequency Permutations

TL;DR

This work analyzes a joint radar-communications system built on a classical stepped-frequency radar waveform, augmented with phase modulation and frequency-permutation signaling. By deriving the ambiguity function and Fisher information matrix, the authors show that phase modulation has negligible impact on radar local accuracy, while random frequency permutations maintain favorable AF features on average. They introduce a Lehmer-code-based mapping to efficiently generate a large set of phase- and permutation- modulated waveforms and develop an ML receiver that uses a Hungarian-algorithm-based implementation to reduce detection complexity; they also provide union-bound, nearest-neighbor, and Rayleigh fading bounds for the block error rate. The results demonstrate that higher data rates can be achieved without compromising radar performance, supported by extensive numerical validations in AWGN and fading channels, highlighting the practical viability of high-rate JCR systems.

Abstract

This paper focuses on the combined radar and communications problem and conducts a thorough analytical investigation on the effect of phase and frequency change on the communication and sensing functionality. First, we consider the classical stepped frequency radar waveform and modulate data using M-ary phase shift keying (MPSK). Two important analytical tools in radar waveform design, namely the ambiguity function (AF) and the Fisher information matrix (FIM) are derived, based on which, we make the important conclusion that MPSK modulation has a negligible effect on radar local accuracy. Next, we extend the analysis to incorporate frequency permutations and propose a new signalling scheme in which the mapping between incoming data and waveforms is performed based on an efficient combinatorial transform called the Lehmer code. We also provide an efficient communications receiver based on the Hungarian algorithm. From the communications perspective, we consider the optimal maximum likelihood (ML) detector and derive the union bound and nearest neighbour approximation on the block error probability. From the radar sensing perspective, we discuss the broader structure of the waveform based on the AF derivation and quantify the radar local accuracy based on the FIM. Extensive numerical examples are provided to illustrate the accuracy of our results.

Figures (13)

• Figure 1: The JCR system model.
• Figure 2: The (a) three-dimensional surface plot and (b) contour plot of the AF of a linear stepped frequency waveform for $L = 8$. The first frequency $f_0 = 0$ Hz and the step $\Delta f = 1/T$Hz.
• Figure 3: The (a) three-dimensional surface plot and (b) contour plot of the AF of a linear stepped frequency and PSK based waveform for $L = 8$ and $M = 4$. The first frequency $f_0 = 0$ Hz and the step $\Delta f = 1/T$Hz. The phase sequence is $[3\pi/4,3\pi/4,3\pi/4,\pi/2,3\pi/4,\pi/2,\pi/4,\pi/2]$.
• Figure 4: The (a) zero-Doppler cuts and (b) zero-delay cuts of the AFs in Fig. \ref{['fig:AF_step']} and Fig. \ref{['fig:AF_step_psk']}.
• Figure 5: The (a) approximated CRLBs in \ref{['eq:crlb_st_step']} and \ref{['eq:crlb_stept']} on delay estimation errors and (b) approximated CRLBs in \ref{['eq:crlb_sw_step']} and \ref{['eq:crlb_stepw']} on Doppler shift estimation errors for stepped frequency based waveforms with and without phase modulation..