Indoor Positioning with GNSS-Like Local Signal Transmitters - Global Navigation Satellite Systems: Signal,Theory and Applications - page 315

arriving signal. Let us consider that the frequencies of all the reflected signals are identical 15 .
Then, the physical phenomenon that occurs is simply an addition, in amplitude, phase and
delay, between all the reflected signals (and the first one, which should be the direct path 16 ).
The problem is now to be able to get rid of all contributions except the first one (which
should be the direct path under our assumption). Such a time discrimination is somehow
equivalent to the synchronisation problem and requires theoretically a radio bandwidth
proportional to the time discrimination interval wanted: in our case, where nanoseconds are
sought, this bandwidth is too large and other approaches must be found.

Let us now come back to the specific problem of multipath in GNSS, and to GPS for
illustration. The way time separation is obtained, from the transmitter to the receiver, is
based on the famous auto correlation function (ACF) of the codes. A typical such function is
given in figure 7.

In case of multipath, we are interested by the main lobe of the ACF. Let us consider only one
reflected path (in addition to the direct path) for simplicity of explanations, knowing that
this is clearly not a real situation. If the reflected path is delayed by more than one and an
half chip, the ACF of the incident signal (which is composed of the direct and reflected
paths) with the receiver generated replica has the shape given in figure 8. Remember that a
GPS chip length is given by 1/1023 milliseconds, hence 977.5 nanoseconds, which in turn
corresponds to 293 meters. Thus, figure 8 is characteristic of a reflected path delayed by
more than 440 meters. The receiver will be able without any problem to find the direct path
considering (this is the assumption that is classically made) that the first peak of the ACF is
the value being sought.

ACF

1

ACF

Early-Late
discriminator

0

-1

Time

Fig. 7. Typical autocorrelation function of a GPS code
15 This could not be true in case of reflection on moving objects, such as cars for example. But in our
indoor case, we are going to consider this hypothesis as correct.
16 The direct path could not be present and then the first received signal would also be a reflected path.
This situation is one that we are not going to deal with in this chapter.