Mathematical Analysis of the Peak Sidelobe Level of the Ambiguity Function for Random Phase Codes

By using uniformly distributed phases for each bit, one can generate a waveform with an almost ideal aperiodic ambiguity function (thumbtack). Moreover,a set of such codes will potentially exhibit low cross-correlation, making them a family of codes suitable for multiple and diverse applications, ranging from 1) MIMO radars, 2) addressing the joint radar-communication problems, 3) mitigating the interference issues of coexistence of multiple radar systems operating in the same region, and 4) increasing the robustness of the system to jamming. This article presents the full analysis of the probability density function of the peak sidelobe level of such codes, as well as a full treatment of the ambiguity function to show mathematically what could be obtained by generating random phase codes. A simple and effective greedy algorithm for generating multiple codes with improved performance is devised increasing the practicality of this waveform family, and numerical results are provided to support the theoretical ones.