Global Positioning System (GPS)

Using messages received from a minimum of four satellites, a GPS receiver is able to determine the satellite positions and time sent.

When four satellites are observed simultaneously, the position (x, y, z) and receiver clock offset can be found from a single observation. The x, y, z component are reference as [x_n, y_n, z_n, t_n] where n is the number of satellite. The time received is mark as t_rn. The GPS receiver can compute the indicated transit time of the messages by formula, t_rn – t_tn. Assume the signal propagated at the speed of light, c, the distance travelled can be determine by multiply the indicated transit time by the speed of light, (t_rn – t_tn)c. The calculation above is done by ignoring the noise or error. In the ideal case of no errors, the GPS receiver will be at an intersection of the surfaces of four spheres.

But when there is error, we cannot use formula above to calculate the exact position. Therefore, we have to consider the clock error or bias. This calculation is based on concept of trilateration.

The equation of the sphere surface is

; n=1,2,3,4

Another useful form of these equations is in term of the pseudo range

; n=1,2,3,4

Where x, y, z = receiver’s coordinates

t_c = time correction for the GPS receiver’s clock

n = satellite’s number

t_tn = signal transmitted time for each satellite

t_rn = signal received time for each satellite

c = speed of light (3 x 10⁸ m/s)

This mean, when there is four satellite used, there will be four equation needed for the calculation.

The accuracy in this technique is about 1-5 meters.

The main problem in code phase pseudo range is that the bits (or cycles) of the pseudo random code are so wide (a cycle width equal to almost a microsecond). Therefore, there might be some error when signals sync up.