Analysis of the ambiguity function for an FM signal derived from the Lorenz chaotic flow

In prior work, we showed that any one of the state variables of the Lorenz chaotic flow can be used effectively as the instantaneous frequency of an FM signal. We further investigated a method to improve chaotic-wideband FM signals for high resolution radar applications by introducing a compression factor to the Lorenz flow equations and by varying two control parameters, namely ρ and β, to substantially increase the bandwidth of the signal. In this paper, we obtain an empirical quadratic relationship between these two control parameters that yields a high Lyapunov exponent which allows the Lorenz flow to quickly diverge from its initial state. This, in turn, results in an FM signal with an agile center frequency that is also chaotic. A time-frequency analysis of the FM signal shows that variable time-bandwidth products of the order of 105 and wide bandwidths of approximately 10 GHz are achievable over short segments of the signal. Next, we compute the average ambiguity function for a large number of short segments of the signal with positive range-Doppler coupling. The resulting ambiguity surface is shaped as a set of mountain ridges that align with multiple range-Doppler coupling lines with low self-noise surrounding the peak response. Similar results are achieved for segments of the signal with negative range-Doppler coupling. The characteristics of the ambiguity surface are directly attributed to the frequency agility of the FM signal which could be potentially used to counteract electronic counter measures aimed at traditional chirp radars.