GNSS Solutions • January/February 2015
Why are carrier phase ambiguities integer?Share via: Slashdot Technorati Twitter Facebook It is well known that carrier phase ambiguities are integer values. Intuitively, this is hard to understand with a common counterargument progressing along these lines: even if the receiver measures the instantaneous phase of the incoming signal (thus removing any fractional cycle component at the receiver end), the phase of the signal at the satellite cannot be guaranteed to be zero, so how can the ambiguity be integer? In this article we explain why the carrier phase ambiguities are indeed integer. To keep things simple, we begin by assuming the propagation errors (ionosphere, troposphere, etc.) and clock errors are zero. This is not limiting but greatly simplifies the analysis and interpretation because it only leaves geometric terms (i.e., range and range rate).
Geometric Interpretation of Carrier Phase Recall that carrier phase observations are obtained by integrating the measured Doppler shift of the signal (details in the next section). This is why the term accumulated Doppler range (ADR) is often used to describe the same observation. In the absence of errors, the Doppler shift, f_{Doppler}, of the received signal is Equation (1) (see inset photo, above right) where λ is the carrier phase wavelength and ρ̇ is the geometric range rate between the receiver and satellite (the overdot represents the timederivative of the geometric range, ρ). Integrating the Doppler over time gives the carrier phase observation at time t, ϕ(t), as Equation (2) (see inset photo, above right) where t_{0} is the start of the integration period (usually when the signal is first acquired), is the change in range over the integration period, and ρ(t_{0}) is the initial range and represents the integration constant. Equation (2) also shows that the carrier phase observation is a measure of the change in range over time. Setting t = t_{0} in equation gives Equation (3) because the change in the range over zero time interval is zero. Of course, the initial range is generally unknown (after all, that is what the receiver is trying to measure) and thus can be loosely interpreted as being ambiguous. This is the geometric analogy to the carrier phase ambiguity.
Carrier Phase Generation The Doppler shift of the received signal is given by f_{Doppler} = f_{dc} − f_{IF} (4) where f_{dc} is the frequency of the signal after downconversion, and f_{IF} is the nominal intermediate frequency (IF). The downconverted frequency is given by f_{dc} = f_{Rx} − f_{LO} (5) where f_{Rx} is the received signal frequency, and f_{LO} is the frequency of the receiver’s local oscillator (after being mixed up or down from its fundamental frequency). The IF is given by f_{IF} = f_{SV} − f_{LO} (6) where f_{SV} is the nominal signal frequency (e.g., 1,575.42 MHz for GPS L1). Using a frequency lock loop (FLL), the receiver’s NCO tries to generate a frequency, f_{NCO}, that matches the downconverted frequency as closely as possible. The error (discriminator output), δ f_{NCO}, is passed to the loop filter that computes the feedback to NCO. The carrier phase equivalent of the equation (4) is ϕ(t) = ϕ_{dc}(t) − ϕ_{IF}(t) (7) where the term on the left is the carrier phase observation, ϕ_{dc} is the phase of the signal after downconversion in the frontend, and ϕ_{IF} is the phase of the receiver’s IF signal. For convenience, we assume that f_{IF} = 0 such that ϕ_{IF} is constant. From equation (6), this is equivalent to f_{SV} = f_{LO} (ignoring relativistic effects). In this case, and assuming the receiver’s oscillator is phase synchronized with the satellite (recall our initial assumption of perfect clocks), it follows that ϕ_{IF} = 0. Let us now consider the signal phase after downconversion, which is given by ϕ_{dc}(t) = ϕ_{Rx}(t) − ϕ_{LO}(t) (8) where ϕ_{Rx }is the phase of the signal at the receive antenna (i.e., before downconversion), and ϕ_{LO} is the receiver’s locally generated phase. Analogous to the FLL, a phase lock loop (PLL) tries to drive the NCO phase to the downconverted phase. Accounting for tracking errors, δϕ_{NCO}, we can write ϕ_{NCO} = ϕ_{dc} + δϕ_{NCO} (9) Since the receiver does not know ϕ_{dc}, it instead uses ϕ_{NCO} as its best estimate. In other words, equation (7) can be approximated and then simplified (using equation [9] then [8] and ϕ_{IF} = 0) as follows:
ϕ(t) ≈ ϕ_{NCO}(t) − ϕ_{IF}(t) (10) where ϕ_{SV} is the phase of the satellite (equivalent to the receiver’s phase for the assumptions made). We can break this down further by realizing that the received phase is equal to the phase of the satellite when the signal was transmitted. Knowing that the time of propagation of the signal is T = ρ/c, we can write Equation (11) (see inset photo, above right) Substituting equation (11) into (10) gives Equation (12) (see inset photo, above right) This shows us that the carrier phase observation under ideal conditions equals the true range plus tracking errors. The latter is zero mean with typical noise and multipath contributing approximately one millimeter and two to three centimeters of error, respectively. But where’s the carrier phase ambiguity, you ask? To answer this, we need to recognize that the NCO is really only concerned with matching the phase of the local and received signals within one cycle. More specifically, the carrier phase discriminators in the tracking loops (not shown) cannot distinguish between one cycle and another and thus converge to the nearest cycle. In other words, while the previously described development implicitly assumed ϕNCO = ϕ_{dc}, in reality ϕ_{NCO} = mod(ϕ_{dc},1 cycle) (13) where mod(a,b) is the modulus (remainder) of a / b. Practically, this means the NCO phase is ambiguous by an integer number of cycles and explains why the ambiguity is integer. Also worth noting is that the carrier phase ambiguities are determined when the signal is first acquired. After this time, the change in range/phase is captured by integrating the measured Doppler shift. In other words, with reference to equation (2), the integration constant is determined at t_{0}.
Discussion Ultimately, the ambiguity term will absorb any mean error in phase tracking error. With a PLL, these errors are zeromean and thus are not problematic. For an FLL, the nonzero tracking error would be absorbed. We should also note that the IF phase of the receiver plays a role in the “integerness” of the ambiguities. Earlier, we assumed the receiver phase was synchronized with the satellite’s phase; however, this is not true in general, and any offset will be absorbed by the ambiguity term. This error is effectively random at turnon (due to the random nature of the oscillator’s phase) and thus cannot be easily compensated. This is part of the challenge of ambiguity resolution with precise point positioning (PPP) algorithms. Fortunately, for double difference processing, this effect cancels. Similar to the IF phase, any unaccounted for delays in the receiver hardware (e.g., interchannel delays, etc.) will affect the integerness of the ambiguities. Fortunately, many of these effects can be calibrated with proper techniques. Finally, although the previously described development ignored error sources, including these in the development is relatively straightforward and the same conclusion results. The only difference is that equation (12) would include all of the normal error terms and, of course, the ambiguity! Copyright © 2018 Gibbons Media & Research LLC, all rights reserved. 
