  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.
