Table Of ContentA Complete IF Soft- (cid:127) calculationofthenavigationsolution;and,
(cid:127) SummaryofourMATLABandCtoolkits
ware GPS Receiver: Thispaperwillpresentexamplesandresultswehavegen-
erated using sampled GPS signals collected with a high
speeddatarecorder
A Tutorial about the
Details INTRODUCTION
TheIFSoftwareGPSReceiverhasbeendevelopedaspart
ofa USAFContract entitled “Novel Wide-Band Receiver
Solutionfor‘Near-Far’CommunicationProblem”. TheIF
Kent Krumvieda, DFC; Premal Madhani, CCAR; Chad Software GPS Receiver, hereafter referred to as Baseline
Cloman,DFC; EricOlson,DFC;Dr.JohnThomas,DFC; Receiver,servedboth asaBaselinetocomparetheresults
Dr.PeninaAxelrad,CCAR;Dr.WolfgangKober,DFC. of the Novel Near-Far Resistant (NFR) receiver, and to
provide an architecture in which to host the NFR compo-
nents.
BIOGRAPHY
Mr. Krumvieda has an M.S. in Civil and Environmental DigitalData
Engineering, a B.S. in Chemical Engineering from the
University of Colorado in Boulder, and over 15 years
ChannelA ChannelB Channeln
industrial experience. He has extensive experience
Acquisition Acquisition Acquisition
involving programming for all of Data Fusion Corpora-
tion’s sensor management and tracking efforts. Recently
Tracking Tracking Tracking
...
hewasthechiefsoftwarearchitectforthedevelopmentof
Data Fusion Corporation’s GPS Near Far Resistant
BitSync BitSync BitSync
ReceiverandOrbitalTrackingToolbox.
FrameSync FrameSync FrameSync
ABSTRACT
Research and development continues to extend the capa- CalculateNavigationSolution
bilities and to increase the robustness of Global Naviga-
tion Satellite Systems (GNSSs) receivers. Software PVT
receivers are quite valuable in evaluating potential FIGURE1.BaselineReceiverArchitecture
improvements because of their flexibility. In addition,
The Baseline Receiver Architecture is designed to handle
software receivers afford batch data processing options
‘n’channelsinparallel. Onceaminimumoffourchannels
that are not available in hardware implementations. In
havefinishedFrameSynchronizationandhaveread some
developing such a receiver, we have found that many
data,thenavigationsolutionmaybecomputed. TheBase-
existing texts and papers on software receivers omit
line receiver requires that the input data be at a user
important implementation details. Here we discuss our
definedIntermediateFrequency(IF)andconvertedtodig-
complete Intermediate Frequency (IF) software GPS
italrepresentationatsomeuserdefinedsamplingrate.
receiverandrevealdetailsofthelessonslearnedduringits
construction. Specific topics addressed in this paper In the absence ofany data supplied by theUSAFinitially
include: we implemented a World Simulator and Signal Generator
• linearandnon-linearadaptiveanalog-to-digitalcon- to supply our receiver with data. The World Simulator
version; was of sufficient fidelity to create realistic signal, i.e.,
(cid:127) acquisitionusingcoherentandnon-coherentintegra- spherical earth, circular satellite orbits, multiple coordi-
tion;
natesystems,Doppler,thermalnoise,etc. Afterthesignal
(cid:127) searchtechniques(e.g.,Vernier,MofN,andTong)
hadbeenrealisticallysimulatedthereceiverhadtoconvert
performedinboththetimedomainandthefrequency
the signal from RFto IFand discretely sample the signal.
domain;
(cid:127) limitationsoftheDFTapproachincludingleakage; To realistically model this we implemented a “Real-
(cid:127) zeropaddingandadditionalDFTenhancements; World”LinearAnalogtoDigitalConverter.
(cid:127) trackingutilizingFLLs,PLLsandDLLs;
(cid:127) bitsynchronizationandnavigationdatarecovery;
(cid:127) framesynchronizationandparitydecoding;
1
ANALOG TODIGITAL CONVERSION Clock jitterreferstothetimebetweenaninputclockedge
(ADC) that defines the sampling period and the actual switching
instantinternaltotheADC.
Data Fusion Corporation implemented and we shall dis-
cuss the Idealized Linear ADC, Linear Real-World ADC, Thelockedhistogramtestcanbeusedtoestimateaperture
aswellasthe Non-LinearADC whichhasniceproperties error in flash ADC. To perform this test the same fre-
inhighCWinterference. quency isused forboththe sampling clock and input sine
wave. Afull-scaleorgreateranaloginputissynchronized
tothesamplingclocksothatthesamepointontheinputis
IDEALIZED LINEAR ADC
repetitively sampled. The effects of aperture error will
becomemoreevidentasthesignalslewrateincreases. In
The theoretical idealized transfer function for a ADC is a
order to estimate the ADC aperture error, the standard
straight line with an infinite numberofsteps. Apractical
deviationofthehistogramistypicallyused. Thisisequiv-
idealized transfer function is a uniform staircase with a
alenttotheRMSapertureerror.
finitenumberofsteps.
ConversionCode Digital σ⋅V
Rangeof Digital Out IdealStraightLine Apertureerror = Ta = S----l--e---w----r-L--a-S--t-B-e--
Analog
Input Output 0101
Values Code where Slewrate = V ⋅2π⋅f
AMP in
4.5-5.5 0101 0100
3.5-4.5 0100 step Staticerrorsarethoseerrorsthataffecttheaccuracyofthe
2.5-3.5 0011 0011
1.5-2.5 0010 converter when it is converting static (D.C.) signals, and
0.5-1.5 0001 0010 can be completely described by just four terms. These
0-0.5 0000
stepwidth(1LSB)
terms are offset error, gain error, integral nonlinearity
0001
Analog (INL) and differential nonlinearity (DNL). Each of these
0000 In termsareexpressedinLSBunits.
0 1 2 3 4 5
FIGURE2.IdealTransferFunction(ADC)
Offset Error is the difference between the nominal and
Where 1 LSB is (one least significant bit), the inherent actualoffsetpoints. Theoffsetpointisthemid-stepvalue
quantization error is ±1/2 LSB. To ensure that our whenthedigitaloutputiszero. Thiserroraffectsallcodes
receiverwasrobusttonon-idealsituationsa“Real-World” by the same amount and can usually be compensated for
LinearADCwasimplemented. byatrimmingprocess.
Gain Error is the difference between the nominal and
LinearReal-WorldADC actual gain points on the transfer function after the offset
error has been corrected to zero. The gain point is the
The simulation model for the real world ADC imple-
mid-step value when the digital output is full scale. This
mented expands on the ideal quantizer by incorporating:
error represents a difference in the slop of the actual and
Number of Bits, Max Value, Minimum Value, Offset
ideal transfer functions and as such corresponds to the
Error, Gain Error, Differential Nonlinearity (DNL) Error,
same percentage error in each step. This error can also
Integral Nonlinearity (INL) Error, Aperture Errors, and
usuallybeadjustedtozerobytrimming.
DynamicErrors.
Differential Nonlinearity (DNL) Error is the difference
ADCs provide an intrinsic sampling function of the
betweenanactualstepwidthandtheidealvalueof1LSB.
dynamic analog input signal. Dynamic errors which are
If the DNL exceeds 1 LSB, there is a possibility that the
determined from the sample and hold (S/H) specification
convertercanbecomenonmonotonic. Thismeansthatthe
are usually defined in terms of two parameters: aperture
magnitude of the output gets small for an increase in the
errorandclockjitter.
magnitudeoftheinput. InanADCthereisalsoapossibil-
Aperture error refers to the variation in the length (aper-
itythattherecanbemissingcodes,i.e.,oneormoreofthe
ture) of the sampling period. Direct measurement of this
possible2nbinarycodesareneveroutput. Tomeasuredif-
errorsource isvery difficult and aspecification isusually
ferential linearity, a delayed sweep and timebase multi-
arrivedatthroughaprocessofestimation. Asaguideline
plier of the oscilloscope can be used to scan individual
tohowmuchapertureerroristolerable,
segments of the sawtooth error waveform. By initially
1 calibratingthex-axistotheidealwidthof1LSB,thevari-
dt = ---------------------------
MAX π⋅f⋅2N+1 ation in width of the sawtooth segments can ten be mea-
suredtofindtheDNL. Otherteststhatcanbeusedarethe
Thisequationisderivedfromafull-scalesinewavewhich
Servo-LoopCodeTransitionMeasurement,DigitalAnaly-
representsaworst-casecondition.
sis,[8].
2
Integral Nonlinearity (INL) Error is the deviation of the Considering asinewaveinput F(f)ofamplitude Aso that
values on the actual transferfunction from a straight line. F(t) = Asinωt which has a mean square value of F2(t),
Thisstraightlinecanbeeitherabeststraightlinewhichis
2(cid:1)π
drawnsoastominimizethesedeviationsoritcanbealine
where F2(t) = --1---- A2sin2(ωt)dt
drawnbetweentheendpointsofthetransferfunctiononce 2π
thegain andoffseterrorshavebeen nullified. TheServo- 0
LoopCodeTransition MeasurementandDigital Analysis, whichisthesignalpower. ThereforetheSNRisgivenby
canbeusedtodetermineDNLandINL,[8].
2 2
A q
SNR(dB) = 10Log ------⁄------
TheEffectiveNumberofBits(ENOB),expressedindeci- 2 12
bels,takesintoaccountseveraldynamicspecificationsand
Spur Free Dynamic Range (SFDR) is the dynamic range
isdeterminedbytheresolutionoftheADC.
(fromfullscaleoftheADC)downtothelevelofthehigh-
SINAD–1.76 est tonesproduced bytheADC whetherthey are harmon-
ENOB = ----------------------------------
6.02 ics, intermods or unrelated spurious tones. Essentially,
this is an indication of how far it is possible to go below
whereSINADistheratioofsignaltonoiseplusdistortion.
the full-scale input signal without hitting noise or distor-
This includes SNR along with the total harmonic distor-
tion. It can be defined as the ratio of the full-scale input
tion(THD)oftheADC. THDistheratioofthesumofthe
signal to the highest harmonic or spurious noise compo-
harmonicdistortionamplitudestotheoriginalinputampli-
nent amplitude. This is usually a different number than
tude. THDiscausedbytheADCnonlinearities.
SNR, see above. SFDR is always specified with signals
A curve fitting algorithm is often used to measure the
present. In some cases the SFDR is much betterthan the
ENOB or resolution of an ADC when digitizing a sine-
SNR. Toseethetonesyoumusteitherzoomonthemwith
waveinput. Thebest-fitsine-wavecalculationmustdeter-
a decimating filter and look over the whole input fre-
mineall fourparameterthat describea sine wave: ampli-
quency range oftheADC or usea biggerand biggerFFT
tude,phase,frequency,andoffset. Thefittedsinewaveis
to push the noise down so that the harmonics, intermods
usedasareferencefromwhichtocalculatetheerrorinthe
an spurious tones can be seen. Just put in a filtered (to
actualdata.
cleanup thegeneratorharmonics)single tonefromagen-
The error introduced by representing a continuous signal erator. TakeabigFFTandobservethehighestharmonic.
asa discretefunction ofa finite numberofstatesiscalled
Non-LinearADC
2
quantization noise, and can be represented by N . The
The design ofan ADC can have significant impact on the
2
total mean square error[11], N , over the whole conver-
performance of a GPS receiver in the presence of CW
sion area oftheADCisthesum ofeach quantization lev- interference. Amoroso [10]proposeda scheme ofthresh-
els mean square multiplied by its associated probability. old adaptation and post quantization weighting that per-
Theerroratthejth stepis Ej = (Vj–VI). WhereVjisthe forms remarkably well making decisions on a relatively
Voltageat the midpoint ofthe jth step andV istheactual smallpercentageofthedemodulatedchips. Formostsys-
I
temsit isdesirabletoquantizeascoarselyasperformance
Voltage. Themeansquareerrorovertheentirestepis
willpermit. ForaonebitperchipADCinthepresenceof
Ej2 = -q-1---⋅ q(cid:1)⁄2Ej2dE = 1-q--2-2-- (1) GdeagursasdiaatnionnoidsueethtoeCSWNRinitserdfeegrreandceedis1m.96ucdhBgr[e1a0t]era.nd the
1
–q⁄2 Amoroso’s 2 bit ADC technique reduces the degradation
Assuming equal steps, the total error, or Mean square in Gaussian interference to less than 0.6 dB, and the per-
2 formanceinCWinterferenceisexcellent. Thisisdoneby
2 q
quantizationnoseis N = ------.
exploitingthenon-GaussiannatureoftheCWinterference
12
in setting the thresholds for the non-linear transfer func-
Signal toNoise Ratio(SNR) refers to the noise level pro-
tion. Statistical control with feedback is used to set the
duced by the ADC down from full scale. SNR can be
thresholds to ensure thresholds are exceeded with the
expressed with a signal present(one toneusually)orwith
required statistical frequency. For example, the sign
no signal present. If the SNR changes as a function of
threshold requires 50% above and 50% below the thresh-
input frequency then there is usually a jitter problem
old,andabout15%ofthechipsfallabovetheuppermag-
reflected in the aperture uncertainty specification, see
nitude threshold and 15% fall belowthe lowermagnitude
above. If the SNR remains flat as the input is exercised
threshold. Actually, the magnitude thresholds are depen-
overtheentireNyquistrange then there usuallyisn’tajit-
dant on the SNR, i.e., the higher the CW interference the
terproblem.
higherthemagnitudethreshold mustbeto achieveperfect
3
chip decisions. Care must be taken to ensure that a sub- allowsforafullsearchofthecodesequence. OncetheC/
stantialnumberofchipssurvivethethresholdingoperation A signal is tracked, interpretation of the HOW within the
to allow for correlation. The transfer function appears in navigationmessagepermitsastraightforwardtransitionto
below: P(Y)CodetrackingforPPSreceivers. Thekeyaspectsof
acquisitionarethedefinitionofthesearchspace,thecrite-
R ria set for signal detection, and the performance of the
algorithmsundervarioussignallevels.
1
T0–∆ TheSearchSpace
-1 T0+∆
The search space must cover the full range ofuncertainty
in the code and Doppler offset. Because the C/A code is
-R
fairly short, typically the rangespace will includeall pos-
FIGURE3.TransferFunctiononNon-LinearADC siblecodeoffsetvalues. Theresolutionofthecodesearch
isusually½chipincrementsbutoftenthesamplingfreqis
WhereRistheweighteddigitaloutput,and T0 isthesign used to specify the resolution. The range of the Doppler
threshold. Samples falling below the magnitude thresh- dimension is governed by the vehicle and GPS satellite
oldsaregivenunityweightinthecorrelatorwhilesamples dynamicsandthestabilityofthereceiveroscillator. Fora
exceedingthemagnitudethresholdsaregivenweightR. It terrestrialusersystem thisistypically in therange of5to
isthesettingofRmuchgreaterthanunitythatenablesthe 10 kHz. The frequency resolution is determined by the
rejection of CW interference. Assigning the samples coherent integration time (or dwell time). The relation is
whichfallbelowthemagnitudethreshold+/-unityensure 2
D = ------ [6]whereDisafrequencybinwidthinHzandT
good performance in Gaussian noise, where discarding 3T
them (assigning them to zero) leads to poor performance isthepredetectionintegrationtimein seconds. Thedwell
inGaussiannoise. time isbasedon theC/N0ratiooftheGPStransmittersto
beacquired.
No one magnitudethreshold setting is optimal forall sys-
temerrorrates. Theadaptive2bitADCmustbeevaluated Theselectionofapaththroughthesearchspaceisafunc-
for each system under consideration, taking into account tionofthevehicledynamicsandrequirementsforacquisi-
processing gain, threshold bit error rate, and relative tion speed and reliability. Typically the Doppler is set to
importance of the performance in Gaussian noise versus theexpectedvalueandthecodeissearchedoverallpossi-
CWinterference. TheSoftwareIFGPSBaselinereceiver bledelays. Ifthisfails,thesearchiscontinuedinthenext
allowstheusertospecifyRand ∆. Thedefaultsettingsof Doppler bin, with the sequence of bins alternating above
3 and 1σ respectively, were found to give good resultsin and below the starting Doppler. Additional consider-
themostcases. ations,suchasselectingaforwardsearchthroughthecode
to avoid false acquisition of a multipath signal are also
included in designing the search process. FIGURE 4.,
ACQUISITION
depictstheacquisitionuncertaintyregion.
This section describes the GPS signal acquisition process
for both conventional and software receiver architecture.
Acquisition is a coarse synchronization process giving 1cell 1/2chip Signallocationversus
searchthreshold
estimates ofthe PRN code offset and the carrier Doppler.
This information is then used to initialize the tracking Obinnedoppler Vt2 Signalmigrationdirection
loops. Vt1
GPS signal acquisition is a two dimensional search pro-
cess in which a replica code and carrier are aligned with Startofsearch
the received signal. The correct alignmentisidentified by (Expectedvalue Searchdirection
ofDoppler)
measurement of the output power of the correlators. In
other words, when both the code and carrier Doppler
matchtheincidentsignal,thesignalisdespreadandacar-
riersignalisrecovered. Theresultofthetwodimensional
search isan estimateofthecodeoffset to withinonehalf-
chipandtheDopplertowithinhalftheDopplersearchbin 1023chips
FIGURE4.Acquisitionuncertaintyregion[6]
size (several hundred Hz). In a conventional receiver,
acquisition is performed using a carrier signal and C/A
code replica. The short length of the C/A code (1 ms)
4
DetectionCriteria threshold is set very high to avoid false detections, then
there is a high probability that weak signals will not be
Theacquisition isbased ona measurement ofthecorrela-
detected. InGPSthisisthetypicalsituation sothatasin-
toroutput. ThecorrelatorsprovideameasureofthetotalI
gletrialdetectionisnoteffective.
and Q signal voltages over the coherent integration time.
The total amplitude is then given by the envelope FIGURE 6. shows the acquisition system block diagram
consisting ofnoncoherent correlator, PRN code generator
I2+Q2. When the replica and reference signals are
andsynchronizationcontrolscheme.
aligned,theamplitudeoftherecoveredsignalisatamaxi-
mum. Duringthedwelltimeineachcell,theIn-phaseand
Quadrature components I and Q respectively are formed
Average I2
breycestirviepdpisniggnoaflf.tThheereefnevreenlocpeecwodheicahndistthheecmareraiesrufrreoomftthhee Isnigpnuatl 90oXphase &Dump (.)2 + H1
X shifter Σ Env>Vt
amplitude of the incoming signal is computed and com- +
pared with a threshold.Inthepresence ofnoise, one must X &AvDeruamgpe (.)2 H2
setathresholdbaseduponanacceptableprobabilitythata Q2
Threshold
noisy measurement that does not contain the signal will
appear to match the replica. Specifically the amplitude
PRNCode
CarrierNCO
threshold issetby [6]: V = σ –2lnP where P isthe generator
t n fa fa
single trial probability of false alarm, σ is the 1-sigma Sync.Control
n scheme
noise amplitude. σn is frequently obtained by calculat- FIGURE6.Acquisitionscheme
ing usingareferencePRNwhichisknowntobeabsent.
Data Fusion Corporation’s GPS software receiver per-
Forthechosenthreshold V ,anycellenvelopethatisator forms a two-dimensional C/A-code search to achieve the
t
above the threshold isdetected as the presence ofthe sig- initial acquisition ofthe GPSsignal. The search involves
nal. Any cell envelope that is below the threshold is replicating all 1,023 C/A-code phase states in the range
detectedasnoise.Thedetectionofthesignalisastatistical dimensionand numerousDopplerstates. Iftherange and
process because each cell either contains noise with the Doppler uncertainty are known, then the search pattern
signalpresentornoisewiththesignalabsent. should cover the 3-sigma values of the uncertainty. Data
Fusion Corporation’s GPS software receiver assumes
EachcasehasitsownprobabilitydensityfunctionasFIG-
Dopplerand rangeuncertainty isunknown. Currently the
URE5.
codephaseissearchedinfractionalchipincrementscorre-
spondingtothesamplingfrequencyandtheDopplerstates
correspondtothereceiver’sdynamics. Forexampleasta-
tionary receiver with, f =1.25 MHz, f =5MHz, and
IF s
T=1msthere wouldbe5000 rangebinscorresponding ~1/
No
Signal 5-chip increments 21 Doppler bins corresponding to
666.67 Hz increments, where f is the intermediate fre-
IF
quency, f isthe samplingfrequency and Tisthecoherent
s
dwellorintegrationtime. TheDopplerbinsizeisdefined
as2/(3T).
FIGURE5.Probabilitydensityfunctionforvariouscases.
In the absence of the signal the envelope has a Rayleigh
distribution and in thepresence ofthesignal theenvelope
hasaRiceandistribution;IandQsignalsarenormallydis-
tributedifthenoiseisgaussian.
InFIGURE5.onecanseethatiftheSNRishigh,itiseasy
to set a threshold that provides both low probability of
false alarm, and also has low risk of a missed detection.
Asthe SNR is reduced itis no longerpossible because of
the significant overlap of the signal distributions. If the
5
ableKandtheconfirmationthresholdAmustbeinitialized
based on the operational environment. The operational
environment will determine the relative importance of
acquisitionspeed versusprobability ofdetectionandfalse
alarm. Maximum acquisition speed will be achieved by
ut setting K=1. To achieve a higher probability of detection
p
ut and a lower probability of false alarm, at the expense of
O
speed,Kmaybesetto2orhigher. WhenKreachesAthe
r
o
at signal isdeclared present. Atypical rangeforA is [8-12]
rrel whichcorrespondstohightolowC/N0respectively.
o
C
Assuming the search space has been defined as described
intheprevioussectionexecutionoftheTongAlgorithmis
Doppler(Hz) C o d e S a m ple s rfeolramtievdelfyorsaimgipvleen. seAarccho-rgrreilda-tcioelnleevnevryeloTpseeconI2ds+.QIf2thies
correlation envelope exceeds the threshold V, then the
t
FIGURE7.CorrelationSurface
counterKisincrementedbyone,elseKisdecrementedby
one. IfK equalsA thenthe signal isdeclared present and
Thuswhen theuncertainty islargeorunknown thesearch
thesearchisover. IfKequalszerothenthesignalisdeter-
pattern is correspondingly large and the search time
mined not to be present and the entire process starts over
increases. Various detectors have been developed to per-
in thenextsearch-grid-cell. Finally,toavoidthepossibil-
formthissearch.
itythattheTongwillgetcaughtinasituationwhereitsat-
isfies neither condition, Data Fusion Corporation
SearchmethodsusedforGPSsignalacquisition
modified the algorithm such that only a finite number of
The false alarmrate and the probability of detection from correlation envelopes will be calculated for a particular
singledwelltimedetectors(singletrialdecisions)areusu- search-grid-cell. When this maximum dwell threshold is
ally unsatisfactory for GPS applications. So single dwell exceededthealgorithmgiveuponthatcell,determinesthe
timeschemesareseldomused. signal is not present and the entire process starts over in
Various search detectors exist. Kaplan [6] identifies the thenextsearch-grid-cell.
Variable Dwell Time and the Fixed Dwell Time detectors
as two types which are commonly used in GPS receiver
designs. AvariabledwelldetectormakesaBoolean deci- K=K+1 K=A? Yes DSiegcnlaalre
Present
sionthatasignalispresentbaseduponapre-definedcrite-
ria in an unspecified orvariable amount of time. Afixed
Yes No
dwelltimedetectormakesaBooleandecisionthatasignal
is present in abased upona pre-defined criteriain afixed
I
amount of time. An example of a sequential variable Q Edentveecltoopre Env>Vt Csaomnetinceulelin
dwelltimesearchdetectoriscalledtheTongdetector. ResetK
andmove
tonextCell
No No
TongSearchDetector
Data Fusion Corporation implemented a modified ver- K=K-1 dwellsK>=d0w?eollrMax? Yes DSiegcnlaalreNot
sionoftheTongdetectordescribedinKaplan. "TheTong Present
detector has a reasonable computational burden and is
FIGURE8.TongSearchdetectoralgorithm
excellent for detecting signals with an expected C/N of
0
25 dB-Hz or higher."[6] To increase effectiveness under OneimportantperformancecriteriaofGPSreceiversisthe
heavy jamming and/or interference conditions a hybrid mean time of acquisition. A variable dwell procedure is
versionhadtobedeveloped. Thehybridversionperforms adopted for its most significant improvement in reducing
an exhaustive search over all code offsets and Doppler the expected acquisition time over a single dwell proce-
bins with the maximum normalized result indicating the dure.
detected signal. Figure8 presents the Tong search detec-
FortheTong-searchdetectorthemean numberofcorrela-
toralgorithm.
tionenvelopesthatmustbecalculated(K =1)todismiss
init
To utilizetheTong algorithma fewvariables must be ini-
acellcontainingnoiseis[6]:
tialized. These variables are K and A. The counter vari-
6
1 In this technique a DFT (discrete fourier transform) is
N = -------------------- (2)
1–2P applied to the incoming GPS signal and multiplied by the
fa
conjugate DFT ofthe reference signal.Taking the inverse
Utilizing the fact that most of the time is spent searching
DFToftheproductgivesthecorrelation resultinthetime
cells that contain noise only, the overall speed of acquisi-
domainforallthe1023codephaseoffsets.Thismethodis
tioncanbeestimated[6]:
computationallymoreefficientandfasterthantheconven-
R = ---d---- = d----(--1-----–----2----P----f-a---)- chips/sec. (3) tional time domain technique. Hence this method is used
NT T inthesoftwareGPSreceiver.
whered=chips/celland T=predetectionintegrationtime TheDFTofasampledsignalx(n)isgivenas:
inseconds.
N(cid:2)–1 (cid:3) (cid:4)
(cid:1) k(cid:2)
Forexample foraPfa of0.1%, T=1 msec, and 1/5th chip/ X(k) = x(n)exp –j2πn---- (4)
N
cell, the code search rate is 160chips/sec. Note that the
n=0
search speed increases as probability of false alarm
MultiplyingtheDFTsoftwosignalsx(n)andy(n)andtak-
decreases. For this example the minimum time to acquire
ing theinverse DFT ofthe product corresponds to convo-
the signal would be when the search starts in the cell
lutioninthetimedomain.Howeversincewecorrelatethe
where the signal is present. In this the time to acquire
incoming GPSsignalwith thereferencesignalinthetime
would be 1/160 = 6.3 ms. The maximum time to acquire
domain, this corresponds to multiplying the conjugate
the signal would be when the signal is present in the last
DFT of x(n) with DFT of y(n), and then taking inverse
cellinthesearchpattern.Ifthefrequencyuncertaintyis+-
DFToftheproduct[13].
5KHz and the frequency bin-width is 500Hz, then the
maximum time to acquire the signal is 2---1----x---1---0---2---3-- = 134.27 FortheNpointDFTofanN-pointsequence, N2 additions
160 and multiplications are required which is the same as the
secs. timedomain.Howeverifthesequencelengthislimitedto
Themean timetoacquireinsecondsforvariousSNR and a power of 2, then using FFT (Fast Fourier transform)
integration times is shown in the table below. P =0.1, N
fa NlogN additions and ----logN multiplications would be
2
TABLE1. TongMeanTimetoAcquire required.Thisresultsinareductionincomputationaltime,
atthecostofaccuracy.
SNRdB T=1ms T=3ms T=5ms
-20 6.32 18.28 28
DFTAcquisitionversusTimeDomainAcquisition
-25 6.59 19.12 31
FIGURE 9. is the difference between the time (evolving)
-30 * 19.72 32.6
acquisitionenvelopeandtheDFTenvelopeovertheentire
-35 * 19.93 34
correlation length. FIGURE10.isthedifferencebetween
K=2, A=10, fs=5 MHz, IF=1.25 MHz, the correct offset the time (circularly convolved) acquisition envelope and
was at the 2500th sample, * indicates No detection. The the Data Fusion Corporation envelope over the entire
threshold was kept constant over each column and the correlationlength.
searchwasdoneinthebininwhichthesignalwaspresent.
AsmentionedearliertoincreaseeffectivenessinlowC/N
0
environments a hybrid version was implemented. This
hybrid version included the concept of maxDwell men-
tioned earlier as well as an exhaustive search over the
entire search grid. To perform thissearch in a reasonable
amount oftime a Discrete FourierTransform version was
developed.
DFTAcquisition
The advantage of the DFT version is that it calculates the
correlationforaentirerangedimension(selectedDoppler)
inasinglestep. ThedisadvantageisthatwhenDoppleris
FIGURE9.DifferenceofTime&FrequencyAcqEnvelope
non-zero the reference signal when convolved produces
someerrors.
7
integration lengths are presented in Table2 on page8 and
Table3onpage8.
TABLE2.NormalizedMeansandStandardDeviations
CI mean std mean std
T-DFT T-DFT T-DFTcs T-DFTcs
1 -1.1896e-4 2.8492e-3 -1.1896e-4 2.8492e-3
2 9.1226e-6 2.6206e-3 9.1226e-6 2.6206e-3
3 -1.2191e-2 2.5486e-2 -4.3753e-5 1.6592e-3
4 -5.2558e-3 1.7034e-2 -1.8417e-5 1.3777e-3
5 1.1784e-3 1.5624e-2 -2.5415e-5 1.5112e-3
6 6.7638e-3 1.9594e-2 -1.4562e-4 1.9978e-3
7 -1.6811e-3 1.0512e-2 2.5115e-5 1.0752e-3
8 -1.8648e-3 9.8580e-3 -1.1505e-5 1.0840e-3
FIGURE10.DifferenceofTime&FrequencyAcqEnvelope
9 -2.3152e-3 1.1959e-2 -6.6621e-5 1.3151e-3
FIGURE10.indicatesthatthemajorityoftheerrorisdue 10 5.4746e-5 8.4567e-4 5.4746e-5 8.4567e-4
tonon-zeroDoppleronthereferencesignal. Thisisdueto
thewaythereferencesignaliscreated. Thereferencesig- TABLE3.AmplitudeIncreaseduetoCoherentIntegration
nalhasconstantDoppler CI Theory Time DFT TimeC DFTcs
x(i) = sin(ωt+πC) (5) 1 1 1 1 1 1
2 1.4142 1.1767 1.1845 1.1845 1.1845
where ω = 2π(f +f ),CistheC/Achipasafunctionof
IF d 3 1.7321 2.1147 1.4798 1.4798 2.1155
time incorporating chip dilation and contraction, and i is
4 2 2.4013 2.0638 2.0638 2.4013
from 1 to the length of y. For the DFT acquisition circu-
5 2.2361 2.1279 2.2785 2.2785 2.1238
larly convolved where in the Time domain acquisition x
evolves in time and therefore has no discontinuities in 6 2.4495 1.5051 1.9285 1.9285 1.4993
phase. Arelativelysimplefixwhichmitigatesmostofthe 7 2.6458 2.8843 2.7043 2.7043 2.892
errorsinvolvesstrippingoffthe (f +f ) beforeperform- 8 2.8284 2.7554 2.6433 2.6433 2.7649
IF d
ingtheDFT. 9 3 2.3827 2.2155 2.2155 2.3796
10 3.1623 3.0214 3.0329 3.0329 3.0329
where the subscript cs refers to performing carrier strip-
pingbeforeperformingtheDFT.
MofN
M ofNisa fixed dwell time search detector.TheM ofN
search detector takes N envelopes and compares them to
the threshold, ifM ormore of them exceed the threshold,
then the signal is declared present. The P =0.1, M=7,
fa
TABLE4. MofNMeanTimetoAcquire
SNRdB T=1ms T=3ms T=5ms
-20 24.98 75 125
FIGURE11.DifferenceofTime&FrequencyAcqEnvelope -25 * 74.87 124.95
-30 * 75.06 125.1
Examining FIGURE 11. the errors, as defined as the dif-
ference from the Time domain acquisition, have been -35 * * 125.05
reducedby100,000times. Resultsfromdifferentcoherent N=10, f =5 MHz, IF=1.25 MHz, the correct offset was at
s
the 2500th sample, * indicates No detection. The thresh-
old was kept constant over each column and the search
wasdoneinthebininwhichthesignalwaspresent.
8
Comparing Table1 on page7 with Table4 on page8 you Letusassumeaninputsignalx(t)withnonoise:
canseethatTonggenerallyacquiresfasterthanMofN.
x(t) = D(t)⋅sin(ω t) (6)
c
Vernier whereD(t)isthemodulateddatasignal,tistime, ω isfre-
Vernier search detector as it has been implemented by quency and thesubscripts c, r, ereferto carrier, reference
Data Fusion Corporation is simply a time domain Tong anderrorrespectively.
detectorwithalimited2Dsearchspace. Thesearchspace We also assume that ω =ω -ω is fairly small. x(t) is
e r c
is typically +/- 2 chips and +/- 2 DFT Doppler bins at a
divided into 2 channels, in-phase (I) and quadrature (Q).
finerresolution. Vernierisusedforpolishingtheresultof
The I-channel is multiplied by the reference signal,
the DFT Tong to achieve better estimates of Doppler and
sin(ω t+θ),andlow-passfiltered,whichgives
code offset. Vernier can resolve Doppler down to 83.3/T r
Hzandcodeoffsetdowntoaboutatenthofachip. These
1
I(t) = ---cos(ω t+θ)⋅D(t) (7)
polishedresultsarethenpassedtotheTrackingloops. 2 e
where θ is the phase difference between the carrier and
TRACKING the reference. The Q-channel is multiplied by the refer-
encesignalplusa90°phaseoffset, cos(ω t+θ),andlow-
Theacquisition loop givesa coarse estimate ofthecarrier r
DopplerandPRNcodeoffsetoftheincomingsignal.Con- passfiltered,whichgives
trolisthenhandedovertothetrackingloops,thefunction
ofwhich are to track the variations in the carrierDoppler Q(t) = –12---sin(ωet+θ)⋅D(t). (8)
and code offset due to line ofsight dynamics between the
These two output values are combined to form the com-
satelliteandthereceiver.
plex number x (t) = I(t)+jQ(t). The phase angle of
The unknown parametersofinterest in the incoming GPS IQ
signal are the carrier Doppler and carrier phase and the xIQ(t) isanindicatorofthephase errorbetween theinput
PRN code Doppler and code phase. These parameters are and reference carriers. Thus we can use x (t) in the
IQ
functions of time because of the relative motion between
phase detector portion of a tracking loop to provide the
the satelliteandtheuserreceiver.Henceit isimportantto phaseerrorθ (t). Atypicaldiscriminatorlookslike
e
tracktheseparametersandgetaverygoodestimateofthe
(cid:3) (cid:4)
same. Another important function ofthe tracking loops is (cid:1)Q(t)(cid:2)
θ (t) = k ⋅atan ----------- (9)
todemodulatetheNavigationdatafromtheincomingGPS e p I(t)
signal. The PRNcode phase can be used to determine the
where k is the discriminator gain. (This is explored fur-
pseudorange whereas the carrier phase is used to accu- p
therinthetrackingloopsectionsbelow.)
ratelydeterminethesmallchangesinthepseudorange.
Once we’ve achieved lock, (ω and θ have been mini-
e
I/QDemodulation
mized), the I-channel containsthe navigation data. With
WeuseI/QdemodulationinallofthestandardGPStrack- goodlock,weget
ingloops. Thissimpleprocesshastwofunctions:
1
(cid:127) Demodulatesthenavigationdatafromthecarrier I(t)≅---D(t)+noise (10)
2
(cid:127) Providesanindicatorofthefrequencyandphaseerrors
betweenthereferenceandinputsignals and
ThissectiondiscussesI/Qdemodulationoutsidethescope Q(t)≅noise (11)
of GPS to simplify the later tracking loop sections.
When the input and reference signals are in phase the I-
Figure12 shows a typical time-domain I/Q demodulation
channel contains the data signal at half amplitude and the
circuit. Ourdiscussioninthissectionusesthecontinuous
Q-channelcontainnothingbutnoise.
timedomain,butitappliesequallywelltodigitalprocess-
ing.
Integration&Dump
xi(t) LPF I(t)
I/Q results assume a continuous time domain with x (t)
x(t) scions((ωωrtrt++ θ θ)) xIQ(t) = I(t)+j⋅Q(t) updating instantly in response to changes in the referIeQnce
orinput signals. The digital GPSreceiver,however, inte-
LPF
xq(t) Q(t) gratestheinputdataoveratimeintervalandthendumpsa
FIGURE12.StandardI/QDemodulationDiagram single x value -- which changes the characteristics of
IQ
9
the x output. In this section we examine the effects of GenericTrackingLoop
IQ
integration & dump on the phase angle of x . For sim- The goal of a tracking loop is to produce a replica (refer-
IQ
ence) signal that matches some input x(n). Figure13
plicity’ssake,weassumethereisnonoiseandthenaviga-
showstheblockdiagramforagenerictime-domaindigital
tion data equals a constant value of 1 for the duration of
trackingloop.
theintegration.
Let us assume an initial phase offset θand a constant fre-
quencyerrorω ,asdefinedabove. Without integration&
dump,wehavee x(n) Σ k ε(n) LPF εf(n) y(n)
p
+ (F(z))
-
x (t) (12)
IQNoIntegration x (n)
ref
1
= ---⋅[cos(ω t+θ)–j⋅sin(ω t+θ)]
2 e e
1 –j(ω t+θ)
= ---⋅e e NCO
2 (k )
v
withthephaseangle
FIGURE13.StandardTrackingLoopDiagram
phase(x ) = (–1)(ω t+θ) (13)
IQNoIntegration e The digitalinput signal, x(n), isrun through a discrimina-
Given an integration interval, T >0, a start time t , a tor (dotted box) to produce the error value ε(n). The dis-
CI 1
criminator is represented as a simple subtraction with an
1
stoptime t = t +T ,andascalefactor --------,theintegra- associated gain of k but actual GPS discriminators are
2 1 CI T p
CI
more complex. The output is low-pass filtered to reduce
tionis
noise. Theresultingvalueisusedbythenumerically-con-
(cid:1) (cid:3) (cid:4)
x (t) = ---1----- t2(cid:1)1---e–j(ωet+θ)(cid:2)dt (14) trolled oscillator (NCO), with a gain of kv, to produce a
IQ T 2 new replica signal for the next iteration of the loop. We
CI t1
wanttotheloopoutput,y(n),tobeasclosetozeroaspos-
whichresolvesto
sible – at which point the replica signal will equal the
(cid:3) (cid:4) (cid:3) (cid:4) inputcarrier.
(cid:1)π (cid:2) (cid:1)π (cid:2)
j ---–ωt –θ j ---–ωt –θ
x (t) = --------1---------- e 2 e2 –e 2 e1 (15) Note: The following discussion is for the PLL tracking
IQ 2ω T
e CI loopbutisapplicabletootherloopsaswell.[3]
This final answer is difficult to simplify. Data Fusion Theperformanceanalysisofatrackingloop isquitecom-
Corporation is willing to state without formal proof, that plex,particularlyduringthepull-inprocessorinthepres-
thephaseanglemagnitudeofEquation(15)is ence of noise. In order to simplify the analysis, a linear
(cid:3) (cid:4) model of the feedback tracking loop shown below in
phase(x ) = ω ⋅(cid:1)t +-T---C----I(cid:2) +θ (16) Figure14isoftenused[5].
IQ e 1 2
Discriminator
which is the average phase angle over the integration LoopFilter
range. Note the difference between this and Equation X1(s) + Σ k F(s)
p
(13). -
To show the significance of this difference, let us take a
simple example. Assume θ = 0 and t1 = 0, but ωe≠0. X2(s)
After a time interval T , we perform the integration and k
CI v
dumpandgetthefollowing NCO
phase(xIQNoIntegration) = ωeTCI+θ (17) FIGURE14.LinearmodelofaPLLinLaplace(‘s’)domain.
ω T Thelooptransferfunctionisgivenby:
phase(x ) = -----e------C----I+θ (18)
IQ 2
k k F(s)
The phase angles are quite different, and we should keep H(s) = -s---+----v-k----p-k------F----(--s---)- (19)
thisinmindwhenanalyzingcarriertrackingloops. v p
wherek isthediscriminatorgain, k istheNCOgain,and
p v
F(s)istheloopfiltertransferfunction.
10
