GNSS Solutions: What is the carrier phase measurement? How is it generated in GNSS receivers? Simply put, the carrier phase

Transcription

1 GNSS Solution: Carrier phae and it meaurement for GNSS GNSS Solution i a regular column featuring quetion and anwer about technical apect of GNSS. Reader are invited to end their quetion to the columnit, Dr. Mark Petovello, Department of Geomatic Engineering, Univerity of Calgary, who will find expert to anwer them. Hi addre can be found with hi biography at the concluion of the column. What i the carrier phae meaurement? How i it generated in GNSS receiver? Simply put, the carrier phae meaurement i a meaure of the range between a atellite and receiver expreed in unit of cycle of the carrier frequency. Thi meaurement can be made with very high preciion (of the order of millimeter), but the whole number of cycle between atellite and receiver i not meaurable. A good analogy to thi i to imagine a meauring tape extending from the atellite to the receiver that ha numbered marker every one millimeter. Unfortunately, however, the numbering cheme return to zero with every wavelength (approximately 20 centimeter for GPS L1). Thi allow u to meaure the range very preciely, but with an ambiguity in the number of whole carrier cycle. A number of ubtletie mut be conidered when generating thee meaurement in a receiver. To fully appreciate thee require a more detailed look at what we mean preciely when dicuing the carrier phae. All current GNSS atellite tranmit radio frequency (RF) ignal in the L-band. Thee ignal conit of, at the very leat, an RF carrier modulated by a peudorandom noie (PRN) code. When dicuing the phae of a ignal it i important to realize that phae i fundamentally a property of inuoid. Every inuoid ha an amplitude and a phae and can be written in complex notation a A exp j(ωt + θ), where A i the amplitude, ω i the radial frequency, and θ i the phae in radian at t = 0. When the RF carrier i modulated by a PRN code, the reulting ignal i no longer a pure inuoid but can, by Fourier Theorem, be expreed a a linear combination of inuoid. The concept of the phae of a combination of inuoid i le clear than that for a ingle inuoid (e.g., how do we define the phae of the um to two inuoid?). For GNSS ignal proceing we define the ignal phae to be the phae of the carrier ignal. In other word, thi i the phae of the pure inuoid that would reult if the PRN code and any other modulation are wiped off. In the following dicuion, when we mention the ignal phae, thi i what we hall be referring to. Conider a inuoidal ignal tranmitted from atellite at time t tx and received at receiver r at time t rx. Thi ignal can be repreented a where i the received amplitude and i the received phae in cycle. For a plane wave propagating in free pace, the phae of the received RF ignal i given by where f RF i the frequency of the tranmitted inuoid in Hertz (e.g., for GPS L1 f RF MHz), ϕ (0) i the initial phae at the tranmitter, λ = c/f RF i the wavelength, and r(t rx ) = (t rx t tx )c i the range from atellite to receiver. The key point to note i that the received carrier phae give information regarding the range between atellite and receiver. In GNSS application we would like to extract thi information for navigation purpoe. Ideally, we want to obtain a meaure of the range expreed in unit of cycle. Rearranging the previou equation give u: 18 InideGNSS july/augu t

2 where i the true integer number of wavelength (cycle) between the atellite and the receiver. So, if the receiver can etimate the received RF phae, then three unknown remain that mut be etimated in order to determine the range. Thee are: 1. The receiver time t rx 2. The initial atellite phae offet ϕ (0) 3. The integer number of cycle between atellite and receiver. In mot carrier phae proceing application, the effect of the firt two are removed by differencing of obervation; differencing between atellite remove error in the receiver time component, while differencing between receiver remove error in the atellite phae offet. So, how doe the receiver generate an etimate of the carrier phae obervation? The hort anwer i by integrating the Doppler frequency (hence, why the carrier phae meaurement i often called the accumulated Doppler range), but thi doe not addre the original quetion. So, intead conider the implified overview of the carrier phae from tranmiion to intermediate frequency, a given in Figure 1. The incoming ignal i firt paed through a low noie amplifier (LNA); then it i multiplied by a locally generated inuoid and filtered, a proce known a mixing. The reulting ignal i a replica of the RF ignal, but hifted down to a lower, intermediate frequency (IF). The mixer ignal i a inuoid whoe phae i given by While only a ingle phae i indicated in Figure 1, the ignal at the antenna i the um of the ignal from all atellite in view. At thi point in the receiver the IF ignal i paed to a number of phae locked loop (PLL) operating in parallel. Figure 2 how a implified linear model of the PLL. Each PLL track the phae of the IF ignal for a given atellite. ϕ r, RF (t rx ) LNA Local Ocillator Once the PLL i locked to the incoming ignal, (mod 1). Comparing Equation (3) and (5), it ϕ r, Mix (t rx ) Freq. Synth ϕ (t rx ) ϕ r, IF (t rx ) FIGURE 1 Signal phae from tranmiion through downconverion to intermediate frequency. At each point the ignal i aumed to be a inuoid, the phae of which i indicated in text in the figure. can be een that only the term f IF t rx i known by the receiver, and hence the carrier phae meaurement, denoted, where f IF i the intermediate frequency. The phae of the ignal at the mixer output, the IF ignal phae, i then given by july/augu t 2010 InideGNSS 19

3 GNSS SOLUTIONS ϕ r, IF δϕ r, IF Filter ϕ r, NCO NCO f r, NCO FIGURE 2 Linear model of a phae locked loop (PLL). A local replica inuoid with phae i generated by a numerically controlled ocillator (NCO). The difference between thi local replica and the IF phae i filtered to generate a command ignal for the NCO, which drive the error ignal to zero. can be generated a follow Note that, rather than being a meaure of the true range, the carrier phae meaurement alo include term due to phae offet in the receiver and the atellite. A uch, thi meaurement can be conidered to be an ambiguou meaure of the peudorange, expreed in cycle of the carrier. A number of problem arie with generating the carrier phae meaurement in thi way, however. Firt, note that we can conider the NCO phae to conit of two component, one due to the intermediate frequency and another that i jut due to the range between atellite and receiver. Conider the imple cae of an IF of 0 Hz. In thi cae the NCO phae conit only of the range component, and the carrier phae meaurement can be obtained directly a the negative of the NCO phae. Unfortunately the integer number of cycle between atellite and receiver i not obervable in the PLL. To generate ueable phae meaurement, the receiver phae obervation mut maintain a contant integer number of cycle offet from the true carrier phae. To do thi, both integer and fractional component of the NCO phae mut be tored. To ee thi more clearly, conider the ituation where phae tracking of the ignal from atellite begin at time t, at which point the phae etimate i initialized to zero (i.e., both fractional and integer component are zero): At ome later time t, phae lock i achieved, and the fractional phae of the local replica ignal i a good etimate of the fractional phae of the down-converted atellite ignal. However, the integer component of the phae i till offet from the true integer component by ome arbitrary unknown number of cycle. Thi offet i referred to a the carrier phae ambiguity. Let denote the integer component of the NCO phae, then the phae obervation i given by 20 InideGNSS july/augu t

4 where i the integer component of the term in parenthee on the firt line of Equation (8), and i known a the integer ambiguity. Provided that continuou phae tracking i maintained, thi term hould be contant. So, for the cae of zero IF, the only conideration for generating ueable phae meaurement i to enure that the integer ambiguity remain contant, that i, if the range increae by one cycle, the integer component of the NCO phae,, alo increment by one cycle. Thi can be accomplihed by imply integrating the NCO frequency and incrementing the number of integer cycle. If the IF i non-zero, there i an extra conideration to be made. Now the NCO phae conit of two component: 1. The IF phae:. Thi component of the NCO phae hould ideally match the IF phae term at the mixer output, f IF t rx. 2. The negative of the phae meaurement: The total NCO phae i then given by: The integer component of the IF phae i irrelevant and can be dicarded. Again, conider that phae tracking of the ignal from atellite commence at time t ; o, the phae meaurement component of the NCO phae i again initialized to zero. Care mut be taken a to how the IF phae component i initialized. Recall that the PLL drive the difference between the incoming IF phae and the total NCO phae to zero; thu, once phae lock i achieved the total NCO phae will be given by Comparing Equation (8) and (10), two extra contant term appear in the phae meaurement in Equation (10). Thee term (third and fourth) are a function both of the time at which phae tracking begin, t, and the initial value of the IF component of the NCO phae,. Thi phae bia term i therefore different for each atellite, which could caue ignificant problem in proceing thee meaurement. However, a receiver deigner i free july/augu t 2010 InideGNSS 21

5 GNSS SOLUTIONS (a) Double Diff. Phae Micloure. (cycle) (b) Double Diff. Phae Micloure. (cycle) Time Since () Time Since () FIGURE 3 Double difference phae micloure for a zero-baeline tet: a) IF phae initialized to 0, b) global IF phae accumulator ued. to chooe the initial value of the NCO phae and thu can enure that thi bia i the ame for all atellite. One imple way to do thi i to mix the IF ignal down to zero, thereby reverting to the zero IF cae. An alternative i to tore a globally acceible IF phae accumulator in the receiver. Thi can be arbitrarily initialized when the receiver i turned on, and it phae i updated continuouly during receiver operation. Whenever a PLL tart to track a ignal, Mark Petovello i an Aociate Profeor in the Department of Geomatic Engineering at the Univerity of Calgary. He ha been actively involved in many apect of poitioning and navigation ince 1997 including GNSS algorithm development, inertial navigation, enor integration, and oftware development. mark.petovello@ucalgary.ca the IF component of it NCO phae i initialized to that of the global IF phae accumulator. The initial value of the IF component for atellite i therefore given by where t 0 i the time at which the receiver i turned on. Inerting thi into Equation (10), we ee that the extra phae bia term i now given by t 0 f IF, which i the ame for all atellite tracked. Thi term can therefore be aborbed into the initial mixer phae term in Equation (10). To demontrate the foregoing explanation, raw IF data wa collected in a zero-baeline configuration with a non-zero value for f IF. The carrier phae meaurement were initialized in two way: a) the IF phae wa initialized to zero, and b) the IF phae wa initialized uing the global IF phae accumulator. The double difference micloure for each cae are plotted in Figure 3. The atellite dependent phae biae are clearly viible in cae a), hown in the figure upper panel, while there are no uch biae in cae b), the lower panel. In ummary, the carrier phae meaurement i a highly precie meaure of the peudorange between atellite and receiver, the generation of ueable carrier phae meaurement in a receiver require a phae locked loop, and the receiver deigner mut take care to enure that 1) the integer ambiguity term i contant, and 2) the initial phae i choen o a to enure a common phae bia in all tracking channel. Cillian O Dricoll Dr. Cillian O Dricoll i a enior reearch engineer at the Univerity of Calgary. Hi reearch interet are in all area of GNSS ignal proceing. Correction On page 22 of the June iue GNSS Solution column, the final term at the end of the firt paragraph i miing a parenthei. It hould be: Sinc(x) = in(πx)/(πx). On page 24, the lat entence in the fourth paragraph hould read: In thi region the approximation i quite good and degrade only in the point where the value given by (3) are of the ame order of the term of the integrated double frequency. Alo, in the eventh paragraph of the ame page, the term τ > 1 ha an inverted ymbol. The complete entence hould read: Moreover, we can oberve that for mall value of τ (that i, τ < 1 chip duration) Equation (3) alo give a good approximation of the AF, a hown in Figure 4. Finally, the title for column editor Mark Petovello hould have been Aociate Profeor, following hi promotion in April. 22 InideGNSS july/augu t