The Math of Ambiguity - Inside GNSS - Global Navigation Satellite Systems Engineering, Policy, and Design

# The Math of Ambiguity

Q: What is the acquisition ambiguity function and how is it expressed mathematically?

A: One of the main tasks of a GNSS receiver is the acquisition of the signals-in-space (SISs) of all the satellites in view. This operation is based on the evaluation of a 2-D correlation function, called the ambiguity function (AF), which allows both the satellite detection and estimation of the received signal parameters, namely the code phase offset (code offset) and Doppler frequency/shift.

Q: What is the acquisition ambiguity function and how is it expressed mathematically?

A: One of the main tasks of a GNSS receiver is the acquisition of the signals-in-space (SISs) of all the satellites in view. This operation is based on the evaluation of a 2-D correlation function, called the ambiguity function (AF), which allows both the satellite detection and estimation of the received signal parameters, namely the code phase offset (code offset) and Doppler frequency/shift.

The AF is evaluated for each PRN code across all possible combinations of local code offset  and Doppler shift. This concept was well described in Michael Braasch’s “GNSS Solutions” contribution in the March-April 2007 issue of Inside GNSS.

In order to decide on the presence or absence of the searched satellite, the maximum absolute value of the resulting AF is then compared with a predefined threshold. In fact, if the PRN code sought is present in the SIS the AF exhibits a well-defined peak.

. . .

At this point a number of questions arise. What is the validity region of this approximation in the plane (∆τ,∆f)? Is it possible to have a closed-form expression valid in the whole plane (∆τ,∆f)? It is possible to obtain a similar formula for the cross-correlation terms?

Why do we ask these questions? The fact is that new scenarios with new applications are appearing every day within the satellite navigation world. Most of these require very demanding receiver performance (for example, the capability of dealing with degraded scenarios, indoor navigation, and so forth) and new block processing techniques.

In some cases we can approach the study of these new scenarios with a variety of theoretical tools, and the results can eventually be validated by simulation or by real-life experiments. In these cases to have a closed-form expression of the AF, together with its quality of approximation, would greatly help the study — and thus solution — of the problem.