Cross correlation of two hyperbolic‐frequency‐modulated (HFM) waveforms

Abstract

In a multi‐user radar/sonar environment, or in the case of a single user but with different pulses transmitted, it is important to know the cross correlation of the waveforms as a function of time delay. The cross‐correlation property of two differently swept LFM (linear‐frequency‐modulated) waveforms is straightforward from the view point of using the stationary‐phase approximation. There, as one waveform's frequency‐time curve slides against the other's, the intersecting angle remains constant over an interaction interval in time delay corresponding to the common frequency interval of the two waveforms. The picture changes vastly when the two LFM waveforms are replaced by two HFM (hyperbolic‐frequency‐modulated) ones. In this paper, the cross correlation over such an interval is established. Using examples, the analytical approximation is shown to compare favorably with numerical simulation results. The difference between the analytic and numerical results is further characterized using the unity volume constraint for narrow‐band cross‐ambiguity function which is lesser known than the auto‐ambiguity counterpart.