Reflection Geometry

A diagram depecting the geometry of a GPS signal reflection from a sphere is shown below.  The signal is transmitted from the satellite, and the receiver detects both the direct signal and the reflected signal.

Where:

• R is the location of the receiver antenna
• S is the location of a GPS satellite
• P is the location of the reflection point.
• C is the center of the sphere
• is the normal vector to the sphere at point P

Application of the Law of Cosines yields:

Where .  Applying the Law of Cosines again gives:

We know from the Least Time Principle that for P to be a reflection point,   must be a minimum.  This is equivalent to  .  By constraining  , the above equations can be solved numerically for  .  Knowing  , the location of point P on the surface of the sphere is known.  Knowing the reflection location, the delay of the signal with respect to the direct signal can be computed by:

Where c is the speed of light (299,792,458 m/s).

PRN Correlation

The presence of a reflected signal alters the shape of the correlation function, and introduces an error in the pseudorange measurement.  Consider the figure below.  Here is shown the correlation of a direct signal and a reflected signal.  The reflected signal is weaker than the direct signal, and offset by a number of chips equal to:

Where N is the CA code length (1023 chips), and TCA is the CA code chipping period (977.5e-9 sec).

As shown in the above figure, the reflected signal adds to the direct signal and produces a distorted correlation function.  A GPS receiver determines the pseudorange to a satellite by using a normalized early minus late envelope discriminator (see the figure below).  The early and late envelopes are equalized (i.e. early minus late equals zero) and the prompt envelope corresponds to the correlation peak and thus to the PRN delay, which yields the pseudorange.

The correlation figure above is for a pure PRN signal, which results in a symmetric delta function, and the prompt envelope does indeed correspond to the correlation peak..  However, since a reflected signal changes the shape of the correlation function, the receiver correlator (by equalizing the early and late envelopes) reports an erroneous PRN delay.  The delay error is defined as the difference between the measured shift and the true shift (due to the distorted correlation function):

The delay error results in a pseudorange error thusly:

OverviewSimulation