The ambiguity function of random binary-phase-coded waveforms

The cross-correlation technique for signal detection and parameter estimation is widely used in communication and radar systems. For popular binary waveforms, the properties of their ambiguity functions have been studied for years. However, the results are generally restricted to specific codes which are either deterministic or pseudorandom. From the uncooperative reception viewpoint, without a priori knowledge of the emitted signal in many cases, the cross-correlating signals may be considered truly random. Here, in a statistical sense, the ambiguity function of a truly random binary-phase-coded waveform is analytically derived in which the normally used deterministic cross-correlation process is replaced by its ensemble average. Various Doppler responses are presented and discussed. The results are compared with those obtained by transmitting an aperiodic maximum-length pseudo-random sequence. It is shown that the ambiguity function of the latter case is closely represented by the emsemble-average response of the truly random binary signal of equal length.