The math of ambiguity: what is the acquisition ambiguity function and how is it expressed mathematically?

One of the core issue in charge of a GNSS receiver is the acquisition of the satellites in view signals. Such an operation is based on the evaluation of a 2-D correlation function, called Ambiguity Function (AF), which allows both the satellite detection and the estimation of its signal parameters (i.e., code offset and Doppler frequency). The detection process, based on AF evaluation, involves picking a PRN code and then searching all possible combinations of local code offset and Doppler shift fd. In order to take the decision on the presence or absence of the searched satellite, the absolute value of the resulting AF is then compared with a predefined threshold.Different are the techniques to evaluate the Ambiguity Function (i.e., serial search, FFTs). Hereafter, a mathematical dissertation covers different aspects of the AF formula in absence of noise.