Generalized Ambiguity Functions for Ultra Wide Band Random Waveforms

The generalized ambiguity function of a random ultra wide waveform can be defined from the cross-correlation between the waveform and a time-scaled and delayed version of it, where the time scaling is related to the relative velocity of the target and the delay to the range. As a result of the randomness, the ambiguity function fluctuates between different realizations. A full characteerization includes its statistical distribution but often second orders staatistics are sufficient. The ambiguity function is then described as a sum of its average and a noise component defining a noise floor. The average value includes in the integrand a time-scaled version of the autocorrelation function of the random waveform, while the variance of the ambiguity function defines the noise floor of the side lobes. Particularly useful in characterizing the ambiguity function of a random waveform is the second moment statistics describing the sum of the squared aveerage and the variance. Formulations are developed analytically for Gaussian waveforms, phase-modulation by continuous Gaussian noise and randomized step frequency radar.