It is now clear from the previous discussion that the receiver has to calculate the distance of the GPS antenna (range) w.r.t each of the satellites that it is tracking1.
The GPS satellite signal structure plays a major role in this range estimation. GPS transmissions utilize Direct Sequence Spread Spectrum (DSSS CDMA) modulation technique in which each satellite is associated with a unique code (called the Pseudo-random code or PRN code). The data from each satellite is first encoded using this PRN code and then transmitted using BPSK modulation.
There are two types of codes that a GPS satellite uses – the coarse acquisition or the C/A code and the precise P code. Most of the commercial GPS equipments only utilize the C/A code for position determination (More on the signal structure in future blogs). For the moment let us just note that these codes are a special category of PRN codes called the “Gold codes” which have exceptional auto-correlation and cross-correlation property. These properties mean that –
Hence in order to decode the data from the satellite, the receiver must produce a replica of the PRN code (which acts like a key to the locked data), which not only matches with the satellite Id but also matches the time shifts incurred due to propagation from satellite t user. Let us see this in more detail.
Consider the following case in which the satellite has transmitted the signal at t = t0 s. But due to the propagation delay, it has reached the receiver after a delay of td seconds –
The task of the receiver is thus to generate the replicas of the PRN code for the satellite with all possible shifts and determine that amount of shift for which the incoming PRN code matches the generated code (i.e. has the maximum correlation).
Now, if the time at which the satellite has transmitter the signal is known (t0) and the satellite clock and receiver clock are in perfect synchronization, then the shift required to match the incoming signal would exactly provide us with the propagation delay td. The range or the distance is immediately determined as this ∆t multiplied by the speed of light c (approximately 3 x 108 m/s).
But, the story is not that simple. The satellite clock is a very high precision atomic clock and the clock at the receiver is a low precision clock. Hence the propagation delay thus obtained suffers from what is called “user clock bias”. One good thing about this clock bias is that since all the satellites are perfectly synchronized (or in other words, their errors can be estimated to high degree of accuracy), this user clock bias is a common error to all the satellite signals. This clock bias is the fourth variable in the range equations that needs to be estimated.
The range information thus obtained does not truly provide us with the geometric range of the satellite to the user but contains
Let us assume that the satellite clock bias can be accurately estimated and the atmospheric errors can be neglected (as of now). We can thus construct the range equations from the pseudo-range measurements from (at-least) four satellites as:
R1=√((x1 - xu)2 + (y1 - yu)2) + (z1 - zu)2) + ctu
R2=√((x2 - xu)2 + (y2 - yu)2) + (z2 - zu)2) + ctu
R3=√((x3 - xu)2 + (y3 - yu)2) + (z3 - zu)2) + ctu
R4=√((x4 - xu)2 + (y4 - yu)2) + (z4 - zu)2) + ctu
Where,
The GPS system not only provides us with accurate position information but also accurate velocity and time information as well. Hence the complete GPS solution is called a PVT solution (Position, Velocity and Time). We shall discuss more on how each of them is extracted in future blogs.
The GPS satellite signal structure plays a major role in this range estimation. GPS transmissions utilize Direct Sequence Spread Spectrum (DSSS CDMA) modulation technique in which each satellite is associated with a unique code (called the Pseudo-random code or PRN code). The data from each satellite is first encoded using this PRN code and then transmitted using BPSK modulation.
There are two types of codes that a GPS satellite uses – the coarse acquisition or the C/A code and the precise P code. Most of the commercial GPS equipments only utilize the C/A code for position determination (More on the signal structure in future blogs). For the moment let us just note that these codes are a special category of PRN codes called the “Gold codes” which have exceptional auto-correlation and cross-correlation property. These properties mean that –
- The PRN code of a satellite has very little or no similarity with the shifted version of itself
- The PRN code of any one satellite has very little or no similarity with the PRN codes (even shifted versions) of all other satellites
Hence in order to decode the data from the satellite, the receiver must produce a replica of the PRN code (which acts like a key to the locked data), which not only matches with the satellite Id but also matches the time shifts incurred due to propagation from satellite t user. Let us see this in more detail.
Consider the following case in which the satellite has transmitted the signal at t = t0 s. But due to the propagation delay, it has reached the receiver after a delay of td seconds –
The task of the receiver is thus to generate the replicas of the PRN code for the satellite with all possible shifts and determine that amount of shift for which the incoming PRN code matches the generated code (i.e. has the maximum correlation).
But, the story is not that simple. The satellite clock is a very high precision atomic clock and the clock at the receiver is a low precision clock. Hence the propagation delay thus obtained suffers from what is called “user clock bias”. One good thing about this clock bias is that since all the satellites are perfectly synchronized (or in other words, their errors can be estimated to high degree of accuracy), this user clock bias is a common error to all the satellite signals. This clock bias is the fourth variable in the range equations that needs to be estimated.
The range information thus obtained does not truly provide us with the geometric range of the satellite to the user but contains
- Geometric range from satellite to user
- User clock bias factor
- Offset between the system time and satellite clocks
- Other delays due to atmospheric errors (ionospheric and tropospheric)
Let us assume that the satellite clock bias can be accurately estimated and the atmospheric errors can be neglected (as of now). We can thus construct the range equations from the pseudo-range measurements from (at-least) four satellites as:
R1=√((x1 - xu)2 + (y1 - yu)2) + (z1 - zu)2) + ctu
R2=√((x2 - xu)2 + (y2 - yu)2) + (z2 - zu)2) + ctu
R3=√((x3 - xu)2 + (y3 - yu)2) + (z3 - zu)2) + ctu
R4=√((x4 - xu)2 + (y4 - yu)2) + (z4 - zu)2) + ctu
Where,
- R1 to R4 are the pseudo-ranges from four satellites being tracked
- (x1,y1,z1) correspond to the position of satellite 1 in x, y, z coordinates (and similarly for other satellites)
- (xu,yu,zu) correspond to the position of user in x, y, z coordinates is the range due to user clock bias
The GPS system not only provides us with accurate position information but also accurate velocity and time information as well. Hence the complete GPS solution is called a PVT solution (Position, Velocity and Time). We shall discuss more on how each of them is extracted in future blogs.
