GEOG 862
GPS and GNSS for Geospatial Professionals

The Integer Ambiguity

PrintPrint
Diagram showing one way ranging between a GPS Satellite and a GPS receiver
One-Way Ranging
Source: GPS for Land Surveyors

The solution of the integer ambiguity, the number of whole cycles on the path from satellite to receiver, would be more difficult if it was not preceded by pseudoranges, or code phase measurements in most receivers. This allows the centering of the subsequent double-difference solution. In other words, a pseudorange solution provides an initial estimate of the candidates for the integer ambiguity within a smaller range than would otherwise be the case, and, as more measurements become available, it can reduce them even further.

After the code-phase measurements narrows the field, there are several methods used to solve the integer ambiguity. In the geometric method, the carrier phase data from multiple epochs are processed, and the constantly changing satellite geometry is used to find an estimate of the actual position of the receiver. This approach is also used to show the error in the estimate by calculating how its results hold up as the geometry of the constellation changes. This strategy requires a significant amount of satellite motion to succeed, and, therefore, takes time to converge on a solution. In the filtering approach, independent measurements are averaged to find the estimate with the lowest noise level. Then, there is the method that searches through the range of possible integer combinations, and finds the one with the lowest residual. Both the search and filtering methods are heuristic. They depend on trial and error. These methods cannot assess the correctness of a particular solution, but they can provide the probability, given certain conditions, that the answer is within given limits.

In the end, most GPS processing software uses some combination of all three ideas. All of these methods narrow the field of the integer ambiguity solution by beginning at an initial position estimate provided by code-phase measurements.

We mentioned earlier that the pseudorange is the beginning gambit with a GPS receiver in most cases to determine the approximate position of the receiver using the code phase. After the code phase measurement narrows the field of positioned, there are several methods to solve the integer ambiguity. The N, the number of full wave cycles between the receiver and the satellite at lock on.

There is the geometric method. The carrier phase data from multiple epochs are processed and the constantly changing satellite geometry is used to find an estimate of the actual position of the receiver. It works pretty well, but requires satellite motion and takes time to converge. It is not the fractional initial phase in a GPS receiver's determination that takes time; it is the solution of the N or the integer ambiguity. Another approach to solving the integer ambiguity is filtering. Independent measurements are averaged to find the estimated position with the lowest noise level. A third uses a search through the range of possible integer ambiguity combinations from which it calculates the one with the lowest residuals. The search method and the filtering method depend on heuristic calculations. In other words, cut and try, or trial and error. These approaches can't assess the correctness of the particular answer, but they can calculate the probability that the answer is correct, within a set of limits. Most GPS receivers use a combination of methods. Nearly all narrow the field by beginning with an initial position established by the code phase measurements. They then use one or more of methods in combination to come up with the most probable integer value for the solution of the integer ambiguity, the key to carrier phase observations.

Signal Squaring

There is a method that does not use the codes carried by the satellite’s signal. It is called codeless tracking, or signal squaring. It was first used in the earliest civilian GPS receivers, supplanting proposals for a TRANSIT-like Doppler solution. It makes no use of pseudoranging and relies exclusively on the carrier phase observable. Like other methods, it also depends on the creation of an intermediate or beat frequency. But with signal squaring, the beat frequency is created by multiplying the incoming carrier by itself. The result has double the frequency and half the wavelength of the original. It is squared.

There are some drawbacks to the method. For example, in the process of squaring the carrier, it is stripped of all its codes. The chips of the P code, the C/A code, and the Navigation message normally modulated onto the carrier by 180° phase shifts are eliminated entirely. As discussed earlier, the signals broadcast by the satellites have phase shifts called code states that change from +1 to –1 and vice versa, but squaring the carrier converts them all to exactly 1. The result is that the codes themselves are wiped out. Therefore, this method must acquire information such as almanac data and clock corrections from other sources. Other drawbacks of squaring the carrier include the deterioration of the signal-to-noise ratio, because when the carrier is squared, the background noise is squared, too. And cycle slips occur at twice the original carrier frequency.

But signal squaring has its up-side as well. It reduces susceptibility to multipath. It has no dependence on PRN codes and is not hindered by the encryption of the P code. The technique works as well on L2 as it does on L1 or L5, and that facilitates ionospheric delay correction. Therefore, signal squaring can provide high accuracy over long baselines.

So, there is a cursory look at some of the different techniques used to process the signal in the RF section. Now let's look at the microprocessor of the receiver.