Periodic Ambiguity Function of CW Signals with Perfect Periodic Autocorrelation

A periodic ambiguity function (PAF) is discussed which describes the response of a correlation receiver to a CW signal modulated by a periodic waveform, when the reference signal in the receiver is constructed from an integral number N, of periods T, of the transmitted signal The PAF is a generalization of the periodic autocorrelation function, to the case of non-zero Doppler shift. We show that the PAF of N periods is obtained by multiplying the PAF of a single period by the universal function sin(Nπvt)/N sin(πvT), where v is the Doppler shift. The PAF is then used to study the performance of the correlation receiver, in the presence of Doppler shift, to phase modulated signals which exhibit perfect periodic autocorrelation when there is no Doppler shift. The PAF of these signals exhibit universal cuts along the delay and Doppler axes. These cuts are functions only of T, N and the number M, of modulation bits in one period.