carrier loop architectures for tracking weak gps...

Carrier Loop Architectures forTracking Weak GPS Signals

ALIREZA RAZAVI

DEMOZ GEBRE-EGZIABHER, Member, IEEEUniversity of Minnesota

DENNIS M. AKOS, Member, IEEEUniversity of Colorado

The performance of various carrier recovery loop architectures(phase lock loop (PLL), Doppler-aided PLL, frequency lock loop(FLL), and Doppler-aided FLL) in tracking weak GPS signals areanalyzed and experimentally validated. The effects of phase orfrequency detector design, oscillator quality, coherent averagingtime, and external Doppler aiding information on delaying lossof lock are quantified. It is shown that for PLLs the metricof total phase jitter is a reliable metric for assessing low C=Nperformance of the tracking loop provided the loop bandwidthis not too small (»> 5 Hz). For loop bandwidths that are not toosmall, total phase jitter accurately predicts carrier-to-noise ratio(C=N) at which loss of lock occurs. This predicted C=N is veryclose to the C=N predicted by bit error rate (BER). However,unlike BER, total phase jitter can be computed in real-time andan estimator for it is developed and experimentally validated.Total phase jitter is not a replacement for BER, since at lowbandwidths it is less accurate than BER in that the receiver loseslock at a higher C=N than predicted by the estimator. Similarly,for FLLs operating at small loop bandwidths, it is found thatnormalized total frequency jitter is not a reliable metric forassessing loss of lock in weak signal or low C=N conditions. Atsmall loop bandwidths, while total frequency jitter may indicatethat a loop is still tracking, the Doppler estimates provided by theFLL will be biased.

Manuscript received June 21, 2006; revised December 7, 2006,released for publication February 5, 2007.

IEEE Log No. T-AES/44/2/926553.

Refereeing of this contribution was handled by G. Lachapelle.

Authors’ addresses: A. Razavi, Dept. of Electrical and ComputerEngineering, University of Minnesota, Twin Cities Campus,Minneapolis, MN 55455; D. Gebre-Egziabher, Dept. of AerospaceEngineering and Mechanics, University of Minnesota, Twin CitiesCampus, 107 Akelman Hall, 110 Union St. SE, Minneapolis,MN 55455, E-mail: ([email protected]); D. M. Akos, Dept. ofAerospace Engineering Sciences, University of Colorado, Boulder,CO 80309.

0018-9251/08/$25.00 c° 2008 IEEE

I. INTRODUCTION

The Global Positioning System (GPS) is asatellite-based utility which provides users withaccurate navigation and timing services worldwide.Because of its accuracy, ubiquity, and low cost of userequipment, it has become the navigation and timingsystem of choice for many users. This is evidenced bythe numerous and varied applications of GPS rangingfrom aircraft navigation to performance monitoring ofathletes [1]. Today, modestly priced differential GPSsystems can provide centimeter level of accuracy.Thus, the outstanding challenge for GPS receiverdesigners and users is not increasing accuracy butenhancing robustness. As such, the threat to GPSrobustness presented by wideband RF interference(RFI) or jamming has attracted a significant amountof current attention [2—5].Wideband RFI reduces the effective GPS signal

to noise ratio by elevating the noise floor. Thus,it can be viewed as an attenuation or weakeningof the received GPS signal. Various approaches toenhance the ability of GPS receivers to track weak,attenuated, or corrupted signals have been proposed.In broad terms these approaches consist of advancedantenna design and signal processing techniques suchas beam steering, null forming, and adjusting RFfront-end gains [4], adjusting or tuning conventionalreceiver parameter such as code and carrier trackingloop bandwidths [6, 7], and multi-sensor fusiontechniques such as tight integration of GPS withinertial navigation systems or ultratight integrationusing vector delay lock loops (VDLL) [8—13]. Thispaper focuses on the latter two of these approaches inthat it presents a detailed parametric study assessingcarrier tracking loop design features, their effect onthe ability of a GPS receiver to track weak signals andhow this ability can be enhanced by using informationfrom other sensors to aid the tracking loops. Stateddifferently, this paper focuses on techniques thatcan be used on the postcorrelation signal in a GPSreceiver to enhance RFI performance.The contributions of this paper are three fold.

First, it evaluates the performance of various trackingloop architectures and quantifies the performanceenhancement they provide in tracking weak GPSsignals. It also assesses the relative importance ofvarious tracking loop design features and componentson the ability of a GPS receiver to track weak signals.Second, it develops error models in support of ananalytical framework for evaluating the performanceof conventional GPS receiver architectures in weaksignal environments. The error models and analysisare experimentally validated. Third, it develops anestimator for total phase jitter and shows the validityof phase jitter as a metric to assess the performance ofa GPS receiver in weak signal environments.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 44, NO. 2 APRIL 2008 697

We consider these contributions to be importantand relevant to current GPS research because of thefollowing reasons: First, they answer the followingquestion: “What kind of performance can be expectedfrom a well-designed conventional GPS receiver whentracking weak signals?” Second, the analytical errormodels and experimental validation methodologydeveloped in this paper provide a unified frameworkfor analyzing and experimentally validating theperformance of carrier tracking loop designs. Withthe recent interest in GPS software radios, indoornavigation and ultratight, multi-sensor fusion withGPS, it is judged that this analysis and experimentalvalidation methodology will be a useful tool forperforming quick end-to-end evaluation of not onlyconventional but advanced receiver designs.

A. Prior Art and Original Contributions

The analytical framework used here is notentirely new and to a large extent is a recasting ofthe methodology developed in the seminal works onGPS carrier tracking loop performance presented in[6] and [7]. In [2], the robustness of GPS code andcarrier tracking loops to RFI are analyzed using themethods developed in [6] and [7]. In [3] and [14] thework in [2] is extended to analyze the performance ofnontraditional carrier tracking loops. In particular, theperformance of Doppler-aided GPS carrier trackingloops is analyzed. The analysis is accomplished byextending the approach of [2] to include realisticmodels for satellite and receiver clock phase noise aswell as developing a method for assessing the effectof errors on the externally derived Doppler estimateused to aid carrier tracking loops.In this work, the analysis methodology of [3] and

[14] are extended to include the following carriertracking loop architectures: phase lock loop (PLL),Doppler-aided PLL, frequency lock loop (FLL),and Doppler-aided FLL. In addition, the effectsof phase discriminator (PD) design and coherentaveraging time on the ability of a receiver to trackweak signals are analyzed and quantified. This paperalso experimentally validates the analysis results of[2], [3], [6], [7], [14] and presents a methodology forperforming these experimental validations.

B. Paper Organization

The remainder of the paper is organized asfollows. In the next section, the basic operation ofGPS receivers and, in particular, the operation of thePLLs is presented. Following this, sources of trackingerrors, models for these errors and analysis results oftheir effect on receiver’s performance are presented.Then experimental results validating the analysisresults are presented. A similar treatment of FLLs

then follows. A summary and concluding remarksclose the paper.

II. PHASE TRACKING LOOPS IN GPS RECEIVER

Understanding the issues affecting a GPSreceiver’s ability to track weak signals requires athorough understanding of how a GPS receiver works.Thus, in the next two sections we briefly describehow a GPS receiver works and discuss the errorswhich degrade its ability to track weak signals. Thisinterested reader can find a more detailed treatment ofthe subject in [15], [16].In a GPS receiver, signals from the GPS satellites

are converted to baseband after going throughan antenna, a preamplifier, mixers, filter and ananalog-to-digital converter (A/D). The preamplifier,which includes a band pass filter and a low-noiseamplifier, rejects the out-of-band interference andnoise. The preamplifier is followed by mixers whichdown-convert the RF signal to a convenient IF.This is followed by the IF filter to attenuate outundesired signals and the A/D which produces thedigital IF signal. In order to decode the navigationdata transmitted by the GPS satellites, the basebandreceiver tracks the phase and the code of receivedsignal by estimating and removing any Doppler shift.In this work, we focus on the carrier tracking

loops in a GPS baseband receiver. The purpose ofthe carrier tracking loop is to generate an estimateof the phase or frequency and Doppler shift of thereceived GPS RF carrier. It does this by generatinga replica of the IF carrier which is in phase withincoming signal. Carrier tracking loops which providean estimate of phase are called PLLs. Carrier trackingloops which provide an estimate of the signal’sfrequency are called FLLs. Most GPS receivers usePLLs only. In some instances a combination of bothphase and FLLs are used [6]. Each has its advantagesand disadvantages when it comes to tracking weakGPS signals. In what follows we first focus on PLLs.Discussion of FLLs is postponed to later in the paper.The PLL provides an estimate of 't, the true phase

of the incoming GPS signal seen at the receiver’santenna. Because of various errors, the actualobserved phase at the antenna is different from thetrue phase that was broadcast by the satellite. Thisobserved phase is the incoming signal phase and wedenote it as 'i. It is related to the true phase by thefollowing equation:

'i = 't+ ±'sv+ ±'a+ ±'i: (1)

Errors on the incoming signal caused by GPS satelliteclock instabilities are represented by ±'sv. Errors onthe incoming signal caused by propagation delaysin the ionosphere and troposphere are representedby ±'a. Wideband noise on the incoming signal isrepresented by ±'i.

698 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 44, NO. 2 APRIL 2008

Fig. 1. Block diagram of GPS PLL.

The PLL generates an estimate of 'i (and, thus,indirectly an estimate of 't) by internally generatinga replica of the measured signal at the antenna andsynchronizing the phase of this replica with that of themeasured signal. The phase of the internal replica isthe output of the PLL and is denoted as 'o. The phaseof the replica or the observed output from the trackingloop is related to the measured or input phase by

'o = 'i+ ±'rx+ ±'v+ ±'o+ ±'d: (2)

Errors on the output due to receiver clock (oroscillator) errors are represented by ±'rx. Errorsdue to vibration (which induce phase error inthe receiver’s oscillator) are represented by ±'v.Wideband noise on the output is represented by ±'o.Transient errors due to abrupt platform motion arerepresented by ±'d. Ideally, we would like a PLL toprovide an accurate estimate of the true phase. That is,we would like to minimize the difference between 'tand 'o. This difference is called the phase error andit is difficult, if not impossible, to measure it directlybecause we do not have access to 't. Therefore, thePLL tries to minimize the tracking error ±' instead,which is the difference between the received signalphase and the phase of the replica. That is

±'= 'i¡'o: (3)

The PLL synchronizes the replica with the incomingsignal by driving ±' to zero. The baseband signal thePLL works with to do this is given by [16]

r[k] = AC[k]D[k]cosμ2¼fkfs

+ μ0

¶+ n[k] (4)

where A is the amplitude of the signal, C[k] iscoarse-acquisition (C/A) code, D[k] is the 50 bit/snavigation message, fs is the sampling frequency, f isthe frequency of the baseband signal, μ0 is a constantphase shift, and n[k] represents additive noise.Additive noise includes thermal noise, interference,and jamming. In this paper, we consider interferenceand jamming which are wideband in nature.A GPS receiver’s PLL is shown schematically in

Fig. 1. This loop consists of a mixer, a match filter fordespreading the C/A code, a PD, a loop filter, and a

Fig. 2. Linear model of PLL.

numerically controlled oscillator (NCO). Neglectingthe interference from other satellites and assuming aperfect autocorrelation of the C/A code, the in-phase(I) and quadrature (Q) signals before the PD are givenby [16]

I[k] =q2(C=N)TcohD[k]cos±'+ ºI[k]

Q[k] =q2(C=N)TcohD[k]sin±'+ ºQ[k]

(5)

where C=N is the carrier-to-noise ratio, Tcoh is thecoherent averaging time, ±' is the tracking error andit is the difference between the incoming signal phaseand estimated phase by the PLL for kth sample, andºI[k] and ºQ[k] are the normalized, zero-mean, unitvariance additive noise components on the in-phaseand quadrature parts of the signal, respectively. Asstated before, the noise density in C=N includesthe effect of thermal noise as well as widebandinterference and jamming.The PD shown in Fig. 1 estimates the tracking

error ±' from the I and Q signals. In early analogimplementations, the PD used to be an I ¤Q PD.Even though most current digital implementationsuse a tan¡1 PD, a significant amount of the literatureon GPS carrier tracking loop performance analysisassumes an I ¤Q type of discriminator. That is, theanalysis does not reflect the actual receivers in widespread use today. One of the objectives of this paperis to recast tracking loop analysis in terms of a tan¡1

discriminator and show that there is a performancedifference between the two discriminators. This isdiscussed in more detail later in the paper.The output of the PD goes through a loop filter to

generate a signal which drives the NCO. The NCOgenerates a replica signal whose phase is synchronizedto that of the incoming signal. A linear model forthe analog version of the PLL depicted in Fig. 1 isshown in Fig. 2. This linear model is more suitable foranalytical work. Thus, for simplicity, we use an analogmodel at this stage. The closed-loop transfer functionfor the linear model shown in Fig. 2 is given by

H(S) ='o(S)'i(S)

=F(S)G(S)

1+F(S)G(S)(6)

where 'i is the incoming signal phase, 'o is thetracking loop’s output or equivalently the tracking

RAZAVI ET AL.: CARRIER LOOP ARCHITECTURES FOR TRACKING WEAK GPS SIGNALS 699

loop’s estimate of 'i, G(S) is the transfer function ofthe NCO (given by S¡1), and F(S) is the loop filter’stransfer function. In this work, we use a third-ordertracking loop. Thus, the transfer function H(S) isgiven by

H(S) =2!nS

2 +2!2nS+!3n

S3 +2!nS2 +2!2nS+!3n: (7)

It is shown in [20] that this design is an optimum onein the sense that it reduces the effect of dynamics inthe presence of additive white noise.

III. TRACKING LOOP ERRORS

One metric that is normally used to determine ifloss of lock has occurred in a PLL is the total phasejitter ¾', which is defined as [6]

¾' =q¾2'o +¾

2'c+¾'d3· 15 deg (8)

where ¾'o represents the phase jitter on the output ofthe tracking loop due to wideband noise. Phase jitteron the output due to correlated or colored noises isrepresented by ¾'c . Phase jitter due to the trackingloop's inability to quickly respond to abrupt phasechanges resulting from abrupt motion is representedby ¾'d . For phase jitter values greater than 15 deg, theassumption that a PLL can be modeled as the linearfeedback system as shown in Fig. 2 is questionable.Thus, for phase jitter values greater than 15 deg itis assumed that the tracking loop will lose lock. Thevalidity of this assumption is examined later in thepaper. Each of the three terms which contribute tototal phase jitter and their dependence on trackingloop parameters is discussed next.

A. White Noise Phase Jitter (¾2'o )

The ¾2'o term characterizes the noise on thetracking loop’s output caused by input whitenoise and depends on the type of PD used. If thediscriminator is realized as a simple multiplicationof the I and Q signals (±'=Q ¤ I), it can be shownanalytically that ¾2'o is given by [16]

¾2'o =BL

C=N(1¡ 2BLTcoh)μ1+

12TcohC=N

¶

' BLC=N

μ1+

12TcohC=N

¶: (9)

Equation (9) indicates that for this type of PD, ¾2'o isa function of the coherent averaging time Tcoh and theone-sided PLL loop bandwidth BL, where BL is givenby

BL =Z 1

0

kH(f)k2kH(0)k2 df: (10)

The coherent averaging time appears in the secondterm of (9) and this term represents the effectsof squaring loss. Note that weak GPS signals arecharacterized by small C=N values. Thus, to increasethe performance of a conventional GPS receiverin tracking a weak signal, the coherent averagingtime can be increased or the carrier tracking loopbandwidth can be decreased.In digital implementations such as would be

found in most current GPS receivers, a tan¡1 typeof discriminator is used. This discriminator is themaximum likelihood estimator for phase [21] andminimizes the effect of squaring loss. Followingthe same process in [16] and considering the firsttwo terms in a Taylor series expansion for tan¡1, theoutput of this PD is expressed as

±'= tan¡1μQ+ ºQI+ ºI

¶

¼ tan¡1μQ

I

¶+

11+ x2

¯̄̄̄I=Q

μQ+ ºQI+ ºI

¡ QI

¶

= tan¡1μQ

I

¶+

I2

I2 +Q2

μIºQ¡QºII(I+ ºI)

¶

¼ tan¡1μQ

I

¶+I ¤ ºQ¡Q ¤ ºII2 +Q2

: (11)

Therefore, the variance of the phase jitter due to whitenoise becomes

¾2'o =BLC=N

: (12)

From (12) it is apparent that with a tan¡1

discriminator, ¾2'o depends only on BL. Unlike an I ¤Qdiscriminator, increasing the coherent averaging timedoes not improve the performance of the tan¡1 PD.Therefore, with a tan¡1 discriminator, the only way toimprove performance in tracking weak signals is bydecreasing the one-sided loop bandwidth. However, asis shown next, the correlated errors (colored noises)present in the GPS carrier tracking loop place a lowerlimit on the minimum achievable BL.

B. Colored Noise Phase Jitter (¾2'c )

The colored noise (or correlated errors) in GPSphase tracking loops originate from satellite clockphase jitter, the receiver clock phase jitter, vibration,and propagation delay due to the ionosphere. Thecontribution of the colored noises to the total phasejitter can be determined by

¾2'c =Z 1

0kHe(f)k2Gc(f)df (13)

where the error transfer function, He(f), is given by1¡H(f). The magnitude of the error transfer function


kHe(f)k2 for the third-order tracking loop given in (7)is equal to

kHe(f)k2 =!6

!6 +!6n=

f6

f6 +μ1:22¼

¶6B6L

: (14)

Gc(f) in (13) is the one-sided power spectral density(PSD) of colored noise sources such as receiver clockerror (Grx(f)), satellite clock error (Gsv(f)), andthe vibration effect (Gv(f)). The total colored noisevariance ¾2'c is the combination of the various errorsources such as satellite clock noise (¾2'sv ), receiverclock noise (¾2'rx) and others that are discussed next.1) Receiver Clock Phase Jitter (¾2'rx): The phase

jitter due to the receiver clock depends on the qualityof the clock. The overall PSD of the clock phase errorcan be expressed as [22]

Grx(f) =0X

k=¡4hkf

k, fl · f · fh (15)

where fl and fh are the inverses of the longestobservation period and sampling period, respectively.Grx(f) is considered zero outside of this frequencyrange. A given clock’s coefficients hi can bedetermined experimentally. We use the experimentallyderived coefficients from [23]. These coefficientsare summarized in Table I for a typical temperaturecompensated crystal oscillator (TCXO) and anovenized crystal oscillator (OCXO). These arecommonly used oscillators in GPS receivers. Usingthe procedure outlined in [22], a closed-formexpression for phase jitter due to clock error in athird-order PLL can be approximated as

¾2'rx =0:9048(2¼)3h¡4

B3L+0:8706(2¼)2h¡3

B2L

+1:2566(2¼)h¡2

BL(16)

where the values of the last two coefficients, h¡1 andh0, are neglected because their effect on phase jitter isminimal. For a typical TCXO and OCXO, the phasejitter values as a function of BL computed using thismodel are shown in Fig. 3.2) Satellite Clock Phase Jitter (¾2'sv ): The

contribution of the satellite clock error to the totalphase jitter is described in [25]. This error descriptionspecifies the error at a single operating point only.That is, the error at a 10 Hz bandwidth is specifiedto be 0.1 rad (5.7 deg). This information is insufficientto develop a PSD for satellite clock jitter. Therefore,we follow the assumption in [2] which considers thenominal satellite clock error PSD to be identical to theTCXO PSD in shape and passes through 0.1 at 10 Hz.However, as shown in [3] and [14] this error model isvery conservative and overestimates the satellite clockphase error. A more realistic model can be obtained

Fig. 3. Contributions to total phase jitter from various phaseerror sources.

TABLE INumerical Values for Coefficients in OCXO and TCXO Clock

Error Models

Coefficient OCXO TCXO

h0 5:50£ 10¡8 5:00£ 10¡8h¡1 5:00£ 10¡5 6:19£ 10¡5h¡2 6:50£ 10¡4 9:60£ 10¡4h¡3 9:00£ 10¡7 6:00£ 10¡3h¡4 1:00£ 10¡7 6:00£ 10¡4

by dividing the above noted PSD (which we call thenominal model) by a factor of one hundred [3]. Thesatellite clock phase jitter as a function of BL is shownin Fig. 3.3) Vibration-Induced Phase Jitter (¾2'v ): Another

colored noise source in GPS application is due to theeffect of vibration on the receiver oscillator. As shownin [26] an oscillator which is subjected to mechanicalvibrations, exhibits an output phase error with a PSDdescribed by

Gv(f) = (jkgjfo)2Gmv(f)f2

(17)

where kg represents the oscillator’s sensitivity tomechanical vibration or g-sensitivity and it is givenin units of parts per g. fo is the oscillator’s centerfrequency and Gmv(f) is the single-sided PSD ofthe mechanical vibration. A typical value for theg-sensitivity is approximately 10¡9 parts/g [14]. PSDof the mechanical vibration is application dependent.For example, in aviation applications, we would usethe PSD for instrument panel installation in a jettransport such as the one given in [27] and plottedin Fig. 4. The phase jitter resulting from this PSD isshown in Fig. 3.4) Phase Jitter due to Atmospheric Effects (¾2'a):

For completeness, in addition to the above notedcolored errors, we should consider the errors due tosignal propagation delays and distortion caused by thetroposphere and ionosphere layers. The troposphere


Fig. 4. Mechanical vibration PSD for jet powered airplane(from [27]).

effect can be ignored in many applications [14] butthe ionosphere errors cannot. The ionosphere error hasa temporal character which depends on the time ofday, the season, and the solar cycle. This means thatthe error due the ionosphere cannot be representedas a stationary random process and, thus, cannot bemodeled using a PSD. Instead, it is more convenientto consider an upper bound on this error. A modelwhich employs this argument is given in [28] anddetermines the upper limit of frequency shift forL1 due to the ionosphere. This frequency shift isspecified to be 0.085 Hz for a terrestrial GPS user.Thus, given this frequency shift, the phase jitter dueto the ionosphere can be handled the same way wehandle the effect of dynamic stress as discussed in thenext section.

C. Dynamic Stress Effects (¾2±'d )

Dynamic stress refers to the errors in trackingphase that are the result of abrupt platform motion. Itis a transient error of the tracking loop in response todiscontinuous input signals. For example, any suddenchange in the user-to-satellite velocity, acceleration, orjerk will produce dynamic stress. For a third-orderPLL, the peak of dynamic stress error (in units ofradian) due to user-to-satellite velocity and jerk areexpressed as [16]

¾'d =2¼¢fmax4BL

(18)

¾'d =2¼(5:67)jmax

¸B3L(19)

where ¢fmax represents the maximum frequencychange due to the velocity change, jmax is themaximum jerk in g/s, and ¸ is the wavelength of theGPS signal. For the analysis in this paper we assumethat jmax is equal to 0.25 g/s which is the maximumexpected jerk in certain aviation application [29].

In applications such as personal navigation jmax canbe much higher. It should be noted that (18) canalso be used to determine the phase jitter due to theionospheric delay when ¢fmax is chosen to be equal to0.085 Hz.Fig. 3 shows the contributions to the total phase

jitter of the various errors discussed above. Thisfigure shows that phase jitter due to dynamic stressis the dominant colored noise source which places alimit on the low achievable loop bandwidth in GPSapplications. In the absence of dynamic stress errors,the dominant colored error source is the phase noisefrom the TCXO. Not surprisingly, in the absence ofall other errors, a tracking loop using an OCXO hasbetter performance than one using a TCXO. Thisunderscores the importance of oscillator quality onGPS receiver performance.

IV. PERFORMANCE ANALYSIS OF GPS CARRIERLOOPS IN WEAK SIGNAL ENVIRONMENTS

Using the error models described above and totalphase jitter as a performance metric, the ability of aPLL to track weak GPS signals can be evaluated. Asnoted in the discussion associated with Fig. 3 above,for a stationary or slow moving receiver the effect ofdynamic stress may be negligible. On the other hand,for a receiver on a moving platform one of the largesterror sources and, thus, a barrier to reducing BL, isdynamic stress error. Therefore, for a mobile receiverremoving dynamic stress errors would be a logicalfirst step towards decreasing BL and enhancing theability of a GPS receiver to track weak signals.This above noted dynamic stress error is a function

of the platform velocity, acceleration, and jerk.Platform velocity and acceleration can be estimatedor sensed using other complementary sensors suchas Doppler radars, inertial measurement units, orautomotive wheel speed sensors. Thus, given anestimate of the platform to satellite relative velocity,dynamic stress can be estimated a priori and removedor compensated for in a feed-forward fashion asshown in Fig. 2. We call this scheme of mitigatingdynamic stress errors “Doppler aiding.”All sensors have errors and, thus, the estimate

of velocity or acceleration used to remove thedynamic stress errors will have some inaccuracies.For example, if we only consider dynamic stressarising from velocity changes, the sensor inaccuracieswill lead to a residual dynamic stress error orDoppler estimation error of which variance can becharacterized as follows [3]:

¾2¢f =eEf±VRX±VTRXgeT

¸2: (20)

Equation (20) is an expression for the variance of¢f, the error in the estimated Doppler shift. e is theline-of-sight vector from the user platform to the


Fig. 5. Phase jitter versus C=N for unaided PLL (i.e., no externalDoppler information) with tan¡1 PD.

Fig. 6. Phase jitter versus C=N for Doppler-aided PLL withtan¡1 PD.

satellite, ¸ is the wavelength of the GPS carrier, and±VRX is the vector error in the user velocity whichcan be computed using one of the aforementionedsensors and satellite ephemeris information. As shownin [3] it is reasonable to expect Doppler estimationaccuracies of about 0.1 Hz (1¡¾) using modestlypriced complementary sensors. Using this value as ¢fin (18) provides an estimate of the residual dynamicstress error after compensation using an externalsensor has been performed.Figs. 5 and 6 show the total phase jitter of a

conventional and a Doppler-aided PLL as a functionof C=N, respectively. In both figures the horizontalsolid black line is the 15 deg threshold for lossof lock. As shown in these figures, the minimumtrackable C=N in an unaided PLL is 23 dB/Hz whilea Doppler-aided loop can track a GPS signal with20 dB/Hz C=N. Thus Doppler aiding provides a3 dB/Hz margin against interference and jamming.Using an OCXO oscillator instead of TCXO improvesthis margin by another 1 dB/Hz (from 23 dB/Hz to19 dB/Hz). This margin is obtained using optimumPD (tan¡1) and when using this discriminator it cannotbe improved by increasing the coherent averagingtime Tcoh.

A. Experimental Validation of Performance Analysis

As noted in the Introduction, one of objectives ofthis paper is to experimentally validate the analyticalmethodology developed earlier in the paper. We alsoare interested in knowing whether phase jitter asdefined in [6], [7] and (8) is an accurate metric forassessing the weak signal tracking performance ofa GPS receiver. If one has access to the true valueof the incoming phase then the total phased jittercan be computed by comparing the output of thecarrier tracking loop with the known input phase.In actual applications, one does not have access tothe true signal phase and, thus, phase jitter has to beestimated. In what follows, we develop an algorithmfor estimating total phase jitter for a stationary or slowmoving user. This requires only digital IF samplescollected from the receiver. In the case where the GPSreceiver is moving, the algorithm developed is stillapplicable provided information about the receiver’smotion is available from another external sensor suchas an inertial navigation system (INS).1) Estimator for Total Phase Jitter: Referring

back to (8), for a static receiver we note that the totalphase jitter is a function of the error in the estimatedphase due to white noise ¾2'o and the power of thecorrelated noise after the phase detector ¾2'c . Forsimplicity, we call ¾2'o white jitter and ¾

2'ccorrelated

jitter. In practice the total phase jitter cannot bemeasured directly in a PLL. The reason for this canbe seen by examining the block diagram of the typicalPLL given in Fig. 2. In this figure we see that neitherof the total phase jitter components (¾2'o and ¾

2'c) can

be measured separately as the signal at any stageof the tracking loop is corrupted by both white andcorrelated noises. Thus, the challenge in designingan estimator for the total phase jitter is extracting ¾2'oand ¾2'c from the quantities that can be measured.One of the quantities that can be easily measured isthe tracking error. In an ideal or error-free scenario,the tracking error should converge to zero when thetracking loop is locked. Thus, any deviation in thetracking error indicates the presence of noises and thisfact can be used to derive an estimator for phase jitterin the following manner.Let ¾2±' be the tracking error power measured at

the output of the phase detector. Ignoring the effectof dynamic stress for a stationary receiver, ¾2±' can beapproximated by

¾2±' = ¾2'c+¾2w (21)

where the effect of white noise on ¾2±' is representedby ¾2w. Since ¾

2±' is a measurable quantity, our

objective is to use this quantity to estimate white jitterand correlated jitter. Equation (21) shows that if ¾2w isknown, ¾2'c or correlated jitter is easily estimated by

¾2'c = ¾2±'¡¾2w: (22)


To estimate ¾2'o or white jitter we note that ¾2w is the

sum of the power of input white noise after the phasedetector ¾2'i and the output white noise on the PLL’sestimate of phase ¾2'o . Thus, we can write ¾

2w as

¾2w = ¾2'i+¾2'o : (23)

In addition, ¾2'o is related to the power of inputwhite noise after the phase detector by the followingrelation:

¾2'o = 2BLTcoh¾2'i: (24)

Equation (24) is derived from the definition ofBL where the power of input white noise can beapproximated by 2Tcoh times the input noise powerdensity. Therefore:

¾2w =1+2BLTcoh2BLTcoh

¾2'o : (25)

Referring to Fig. 3, we note that for large values of BLthe output error of the tracking loop is primarily dueto white noise. Thus, by setting BL to a large enoughvalue BLmax (in our implementation using a signal witha nominal C=N and a Tcoh = 1 ms this was 20 Hz) wecan assume that ¾2±'(BLmax )¼ ¾2w(BLmax ). Therefore, forany other BL, ¾

2'ocan be approximated as

¾2'o =2BLTcoh

(1+2BLmaxTcoh)¾2±'(BLmax ): (26)

Equation (26) enables us to estimate ¾2'o , white jitter,for different loop bandwidths. Then ¾2'c , correlatedjitter, is estimated by (22) and (25). The methodologydiscussed above to estimate ¾2'o and ¾

2'cand, thus, the

total phase jitter from the phase error power can besummarized by the following process.

1) First compute the input white noise power ¾2'ias follows:

a) Set the tracking loop bandwidth to a largevalue such as 20 Hz (BLmax ).b) Wait for the PLL to reach to steady state.c) Collect samples of ±' (PD output) and

compute ¾2±'(BLmax ).d) Using ¾2±' compute ¾

2'i= (¾2±'(BLmax ))=(1+

2BLmaxTcoh).This value of ¾2'i is constant for all BL as long as theinput C=N is not changed.2) For any bandwidth of interest, BL, compute

¾2'0 = 2BLTcoh¾2'i.

3) Collect samples of the discriminator outputafter the loop reaches steady state at this new BL. Usethese samples to compute ¾2±'.4) Compute ¾2'c = ¾

2±'¡ (¾2'i +¾2'o).

5) Compute the total phase jitter by ¾' =q¾2'o +¾

2'c.

Using this methodology, we can now analyzeand experimentally validate the effect of various

Fig. 7. Tracking error (±') versus loop bandwidth for PLL withtan¡1 PD and TCXO (experimental data with signal C=N equal to

45 dB/Hz).

Fig. 8. Pulse jitter versus loop bandwidth for PLL with tan¡1 PD(experimental data with signal C=N equal to 45 dB/Hz).

parameters on the ability of a GPS receiver to trackweak signals. We can also validate whether phasejitter is a good metric for tracking loop performance.2) Effect of Tcoh and Discriminator Design: First,

we evaluate the effect of coherent averaging time onthe total phase jitter when a tan¡1 PD is used. To dothis, we measure the tracking error power ¾2±' as afunction of BL for three different coherent averagingtimes from a receiver using an OCXO. Fig. 7 showsa plot of ¾2±' for a 1, 5, and 20 ms coherent averagingtime. Note that in this experiment the IF samples froma stationary GPS receiver are used and the C=N ofthe GPS signal is fixed and close to its nominal (orclear sky) value. In what follows, the total phase jitteris estimated from the tracking error power using themethodology developed above.The total phase jitter for the three different

coherent averaging times which are estimated by theabove method are plotted in Fig. 8. This figure alsoincludes a calculated phase jitter estimate based on theanalytical model described earlier. It should be notedthat the loop bandwidth cannot be greater than 1=2Tcohbecause of Nyquist rate. For example, when Tcoh isequal to 20 ms, a BL of more than 25 Hz does


Fig. 9. Pulse jitter versus C=N for PLL using tan¡1 PD, OCXOand high loop bandwidth (experimental data for Satellite 10).

Fig. 10. Pulse jitter versus C=N for PLL using tan¡1 PD, OCXOand low loop bandwidth (experimental data for Satellite 10).

not mean that information with a frequency content of25 Hz can be tracked. This is why in Fig. 8 the plotfor Tcoh = 20 ms has large phase jitter values for BLvalues greater than 10 Hz.While Fig. 7 shows that increasing the averaging

time decreases the amount of tracking error, Fig. 8shows that the phase jitter for different Tcoh in thelocked bandwidth remains unchanged. As shownin Fig. 8, the locked bandwidth is approximatelybetween 1 and 10 Hz. Fig. 8 confirms our analysisresult that increasing coherent averaging time doesnot lower the C=N of the GPS signal which can betracked if a tan¡1 discriminator is used. The differencebetween the analytical phase jitter plotted in Fig. 8and the measured phase jitter is because the genericclock error model used to derive analytical result isdifferent from the actual error model of the clock inthe receiver from which the IF samples are collected.3) Effect of C=N on Phase Jitter: Next we

validate the analysis results showing the relationbetween phase jitter and C=N. To do this we changethe C=N of the original IF data used in Figs. 7 and8 by adding artificial (or numerically generated)wideband noise. The new C=N is easily determinedby looking at the new noise floor. Then for a givenC=N value, we compute the phase jitter for varioustracking loop bandwidth values.

Fig. 11. Pulse jitter versus C=N for PLL using tan¡1 PD, OCXOand high loop bandwidth (experimental data for Satellite 18).

Fig. 12. Pulse jitter versus C=N for PLL using tan¡1 PD, OCXOand low loop bandwidth (experimental data for Satellite 18).

Results of this experiment are plotted in Figs. 9and 10. These figures compare the analytical estimateand experimentally computed value of the total phasejitter for various loop bandwidth in a receiver using anOCXO tracking signals from GPS satellite (SV) 10.Fig. 9 shows the results for higher loop bandwidthsand Fig. 10 shows the results for the lower loopbandwidths. The solid lines with solid circles areexperimental data where the phase jitter is computedfrom the output of a static receiver. As can be seen inFig. 9 the agreement between the experimental andanalytical estimate of phase jitter is good. In Fig. 10,however, we see that as the bandwidth of the trackingloop becomes smaller, the difference between theexperimental and analytical results becomes larger.This divergence is explained by the fact that the linearmodel assumed for the phase lock loop starts breakingdown at the low loop bandwidths. This is explained inmore detail when we look at bit error rate (BER) next.Before that discussion, however, as an additional datapoint we present the results from a similar analysison the signals from SV18. These results are shownin Figs. 11 and 12. It can be seen that the results aremore or less the same as those presented earlier forSV10.


Fig. 13. BER versus C=N for PLL with tan¡1 PD and OCXO(experimental data for Satellite 10).

Fig. 14. BER versus C=N for PLL with tan¡1 PD and OCXO(experimental data for Satellite 18).

Also note that there is a difference betweenanalytical results presented in Fig. 6 and the analyticalresults in Figs. 9—12 due to the absence of vibrationand dynamic stress in the experimental data.4) Assessment of Phase Jitter as Loss-of-Lock

Metric: The above results give us confidence thatour mathematical models and analysis methodologyis correct. Thus, we can now determine whethertotal phase jitter is a good metric for assessingtracking loop performance. To do this the resultspresented in Figs. 9—12 are compared with theexperimental BER for navigation bits. Fig. 13 showsa plot of the theoretical BER assuming additivewhite Gaussian noise (AWGN) channels, perfect C/Acode wipe-off, and coherence between the incomingsignal and receiver generated replica. Wheneverthe experimentally determined BER exceeds thistheoretical value, we know that tracking loop haseither lost lock or is out of the linear region whereour analytical methodology cannot predict or capture.Considering the data for SV10, for the higher loopbandwidths Figs. 13 and 9 show that the C=N valuewhere this occurs is about the same as the C=N valuewhere the total phase jitter exceeds the 15± threshold.For the lower bandwidths, however, the phase jitteris still less than 15± at the point at where the BERplot suggests a loss of lock. This suggests that the

total phase jitter is a reasonable metric for assessingwhen loss lock has occurred in PLL providedthe tracking loop bandwidth is not less thanapproximately 5 Hz.A similar evaluation using data collected from a

receiver using an OCXO is shown in Fig. 14. Figs. 13and 14 confirm that total phase jitter is a reasonablemetric for assessing loss of lock in a PLL whenthe tracking loop bandwidths are not too small. Forexample, considering the case when using an OCXO,both Figs. 11 and 14 are in agreement in that theyshow that a PLL with 5 Hz loop bandwidth can trackthe signal with C=N down to 20 dB/Hz.

V. FREQUENCY LOCK LOOP

As shown in the last section, a static receiverwith a carrier tracking loop architecture based ona PLL and an OCXO cannot track signals whenC=N is less than 17 dB/Hz. A mobile receiver withno Doppler-aiding cannot track a GPS signal whenthe C=N is less than approximately 23 dB/Hz. As aresult, it would be useful to consider other trackingloop architectures such as FLLs and compare theirperformance to that of PLLs. As we will show, theFLL-based carrier tracking loop is less sensitive tocolored noise than one based on PLLs. Thus, theFLL minimum achievable noise bandwidth will besmaller when dynamic stress effect is mitigated byDoppler-aiding. There are some drawbacks, however,to using an FLL architecture. First, it cannot providethe precise carrier phase solution available from aPLL-based architecture. Second, FLL architecturerequires a noncoherent code tracking loop to maintaincode synchronization. Finally, recovery of navigationbits is complicated, if not impossible, when FLL isused.

A. Frequency Lock Loop Loss of Lock

For FLLs, we can define a metric equivalent to thetotal phase jitter to evaluate loss of lock. This metricis the total frequency jitter and as shown in [7] isa reasonable metric for assessing loss of lock. Thismetric is defined as [6]

¾f =q¾2fo +¾

2±fc+¾fd3· 112Tcoh

Hz (27)

where ¾f represents the total frequency jitter, ¾forepresents the frequency jitter related to widebandnoise, ¾fc represents the effects of colored noises, and¾fd represents the effect of platform dynamics.The frequency jitter due to white noise ¾2fo is given

by [6]

¾2fo =μ

12¼Tcoh

¶2 4FBfC=N

μ1+

1TcohC=N

¶(28)


where F is equal to 1 or 2 for high and low C=N,respectively. This relation assumes that an (I1 ¤ Q2¡I2 ¤Q1)=(t2¡ t1) type frequency detector is beingused where Ii and Qi are the in-phase and quadraturecomponents of the signal at time ti. Bf is theone-sided loop noise bandwidth of the FLL. The lastterm, (1+ (1=(TcohC=N))), is due to the squaring losswhich can be avoided by using a maximum likelihoodestimator of frequency given by (tan¡1(I2,Q2)¡tan¡1(I1,Q1))=(t2¡ t1). Thus, consistent with whatwas done in a previous section, we use the maximumlikelihood estimator of frequency for our analysis.The contributions of colored noises to the total

frequency jitter can be computed using error modelsintroduced in the last section and the followingrelation:

¾2fc =Z 1

0kHe(f)k2Gcf(f)df: (29)

In this equation, Gcf(f) is the one-sided PSD of thecolored frequency noise. This PSD is approximated byusing the phase error PSD, Gc, as follows:

Gcf(f) = f2Gc(f): (30)

In this work, we used a second-order FLL withdamping factor equal to 1=

p2. It was judged that this

would provide a fair comparison with the third-orderPLL discussed earlier. With this FLL the error transferfunction He(f) is

kHe(f)k2 =!4

!4 +!4n=

f4

f4 +μ

10:53 ¤ 2¼

¶4B4f

:

(31)

The ¾fd term in (27) is the contribution of dynamicstress, which, for a second-order FLL is given by [6]

¾fd =(0:53)2 ¤ 9:8 ¤ jmax

¸B2f(32)

where ¸ represents the wavelength of the GPS signal.The contributions to frequency jitter normalized by

coherent averaging time from the various error sourcesare presented in Fig. 15. In this figure, the coherentaveraging time is chosen to be 1 ms. In Fig. 15, thehorizontal solid black line shows the 1¡¾ thresholdfor frequency jitter normalized by coherent averagingtime.The frequency jitter due to satellite clock and

receiver clock appear constant and less than theloss-of-lock threshold for the bandwidths plotted.They can cross the threshold at lower loop bandwidthswhen the effect of random walk in frequency (h4)becomes dominant. This figure makes clear that thedominant frequency jitter sources are white noise anddynamic stress. Since the error due to dynamic stressis not a function of Tcoh (see (32)), (27) shows thatincreasing the coherent averaging time degrades the

Fig. 15. Contributions to total frequency jitter from various errorsources.

Fig. 16. Analytical estimates of total frequency jitter versus C=Nfor unaided FLL (optimum [tan¡1(I2,Q2)¡ tan¡1(I1,Q1)]=t2 ¡ t1

frequency detector).

performance (relative to loss of lock) of an FLL for agiven C=N and ¾fd .1) Performance Analysis of an Unaided and

Doppler Aided Frequency Lock Loop: When thevarious errors shown in Fig. 15 are superimposed,the result is the normalized total frequency jittershown in Fig. 16. Fig. 16 shows that an FLL cantrack a signal with C=N as low as 15 dB/Hz. If weconsider the case where the FLL is Doppler aided, theeffect of the error in the externally estimated Dopplershift can be approximated as a step in frequency.With this approximation and using the normalizedfrequency jitter as the performance metric, Fig. 17shows that we might conclude that a signal witha C=N less than 10 dB/Hz can be tracked whenthe tracking loop bandwidth is reduced to 0.1 Hz.However, the normalized frequency jitter may not bethe best measure of FLL performance in low C=Ncondition. The reason for this is the fact that at thesmall tracking loop bandwidths required for trackinglow C=N signals, the Doppler estimate generatedby an FLL is biased. This can be clearly seen inFig. 18 which shows experimental results from anFLL in a stationary GPS receiver. Since the receiveris stationary, the change in Doppler will be due tosatellite motion. The black dashed line in Fig. 18


Fig. 17. Estimates of total frequency jitter versus C=N forDoppler-aided FLL (optimum [tan¡1(I2,Q2)¡ tan¡1(I1,Q1)]=t2¡ t1

frequency detector).

shows this Doppler change due to satellite motionas a function of time estimated by a PLL with a10 Hz bandwidth. The remaining other lines show theDoppler estimate from an FLL with bandwidth of 10,1, and 0.1 Hz. While the FLL provides an accurateDoppler estimate at bandwidths of 10 and 1 Hz, theDoppler estimate is biased when bandwidth is 0.1 Hz.From Fig. 18 we see that this bias is approximatelyequal to 1.5 Hz which corresponds to a 30 cm/spseudo-range rate bias on L1. Thus, while the FLL ata 0.1 Hz bandwidth does not lose lock as measured bythe total frequency jitter metric, the loop is not veryuseful because the estimated Doppler frequency iserroneous.

VI. SUMMARY AND CONCLUSIONS

This paper evaluated the performance of variousGPS carrier tracking loops architecture. The effect

Fig. 18. Time history of Doppler shift on GPS L1 RF carrier due to satellite motion (experimental data). Note biased estimate whenBL = 0:1 Hz.

of phase or frequency detector design, oscillatorquality, coherent averaging time, and external Doppleraiding information on delaying loss of lock werequantified. It was shown that for PLLs the metric oftotal phase jitter is a reliable metric for assessing lowC=N performance of the tracking loop. It accuratelypredicts when loss of lock occurs and yields resultssimilar to BER provided the tracking loop bandwidthis not too small. Unlike BER, however, total phasejitter can be computed in real-time and an estimatorfor it was developed and validated. Similarly, forFLLs it is found that normalized total frequencyjitter is a reliable metric for assessing tracking loopperformance in weak signal or low C=N conditionsonly when the tracking loop bandwidths are not toosmall.It was shown that with Doppler aiding and

tracking loop bandwidth, tuning a 4 dB/Hz marginagainst RFI can be achieved. The margins are largerand on the order of 8 dB/Hz when bandwidthadjustments are used with an FLL. However, thelarger margins come at a cost of lower qualityDoppler estimates and loss of the ability to decode theGPS navigation message. With PLLs using a tan¡1

phase detector, the margin does not improve withincreasing the coherent averaging time. However,increasing the coherent averaging time (Tcoh) degradesthe performance of FLLs with regard to loss of lock.The results in the paper assumed that the residual

dynamic stress are small. That is, they are eitherremoved by an external estimate of Doppler orthe user’s motion is such that the jerk values aresmall. This is a valid assumption when the user is anairplane but may not be valid for ground vehicle orpersonnel navigation applications. Thus, depending onthe application, much of the margin gained by


bandwidth adjustments may be lost due to largedynamic stresses. Thus, each application must beevaluated before making the assessment as to whetherbandwidth adjustment is warranted. With the adventof software radios, however, the added overheadthat comes with bandwidth adjustment is small and,therefore, incorporating this feature into all receiversmay be worthwhile.

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Alireza Razavi received his B.Sc. degree in electrical engineering in 1998 fromSharif University, Iran. He received his M.S. degree in electrical engineering in2005 from the University of Minnesota, Twin Cities Campus, Minneapolis, wherehe is a Ph.D. student in the Department of Electrical and Computer Engineering.His M.S. thesis research focused on techniques for enhancing the weak signal

tracking performance of GPS receivers. His current research focus is on theapplication of optimization theory to sensor network.Mr. Razavi is a student member of the Institute of Navigation.

Demoz Gebre-Egziabher (M’99) received his Ph.D. in aeronautics andastronautics from Stanford University, Stanford, CA.His is an Assistant Professor of Aerospace Engineering and Mechanics at

the University of Minnesota, Twin Cities Campus, Minneapolis. His research isin the areas of navigation, guidance, and control with a particular emphasis onapplication of estimation theory to avionics sensor fusion and system integrationissues.

Dennis M. Akos (S’87–M’97–MS’07) received his Ph.D. degree in electricalengineering from the Avionics Engineering Center, Ohio University, Athens.He is an assistant professor in the Aerospace Engineering Science Department

of the University of Colorado, Boulder. His research interests include globalnavigation satellite systems, software defined radio, applied/digital processing,and radio frequency design.

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