Universality and Realistic Extensions to the Semi-Analytic ... · Universality and Realistic...

RADIOENGINEERING, VOL. 21, NO. 2, JUNE 2012 647

Universality and Realistic Extensions to the Semi-AnalyticSimulation Principle in GNSS Signal Processing

Ondrej JAKUBOV, Petr KACMARIK, Pavel KOVAR, Frantisek VEJRAZKA

Dept. of Radio Engineering, Czech Technical University in Prague, Faculty of Electrical Engineering,Technicka 2, 166 27 Prague 6, Czech republic

[email protected], [email protected]

Abstract. Semi-analytic simulation principle in GNSS sig-nal processing bypasses the bit-true operations at high sam-pling frequency. Instead, signals at the output branches ofthe integrate&dump blocks are successfully modeled, thusmaking extensive Monte Carlo simulations feasible. Meth-ods for simulations of code and carrier tracking loops withBPSK, BOC signals have been introduced in the literature.Matlab toolboxes were designed and published. In this pa-per, we further extend the applicability of the approach.Firstly, we describe any GNSS signal as a special instance oflinear multi-dimensional modulation. Thereby, we state uni-versal framework for classification of differently modulatedsignals. Using such description, we derive the semi-analyticmodels generally. Secondly, we extend the model for realis-tic scenarios including delay in the feed back, slowly fadingmultipath effects, finite bandwidth, phase noise, and a com-bination of these. Finally, a discussion on connection of thissemi-analytic model and position-velocity-time estimator isdelivered, as well as comparison of theoretical and simu-lated characteristics, produced by a prototype simulator de-veloped at CTU in Prague.

KeywordsAltBOC, BPSK, BOC, GNSS, post-correlation model-ing, semi-analytic simulation, tracking loops.

1. IntroductionIn global navigation satellite systems (GNSS), Monte

Carlo simulations became dominant in assessment of al-gorithm performance. Often, closed-form solutions appearintractable, and explicitly derived formulas need experi-ments with short implementation time or idealistic condi-tions. Straightforward generation of high-rate samples andconsecutive processing of them is a widely used approachwhich offers adequate representation of crucial physical phe-nomena of the communication channel [1], [2].

However, increasing the sampling frequency or algo-rithm complexity may turn such simulations into a tedious

task. This would be the case especially for wideband sys-tems, including all GNSSs. Fortunately, in a large numberof systems the most computationally demanding operationscan be bypassed with an analytically derived closed-formand statistical description of their outputs, taking the inputsas arguments to their description functions. The rest of theoperations are then left untouched. This forms the definitionof the semi-analytic simulation principle [2].

The problem of semi-analytic modeling in GNSS wasfirst addressed in [3], [4]. The high-rate correlation betweenthe received signal and its replica was avoided by express-ing the output as a function of the spreading code autocor-relation function (ACF) and a noise term with known sta-tistical description. The idea turned into a simulator in theform of Matlab toolbox [5], [6], named SATLSim. Theauthors therein developed models for binary-phase-shift-keying (BPSK) and binary-offset-carrier (BOC) modulatedsignals, processed by code and carrier tracking loops. Eval-uation of the tracking jitter, tracking threshold, and meantime to lose lock was discussed.

1.1 Contribution of the PaperIn this paper, we establish a universal framework for

description of GNSS modulations. We will see that anyof these signals, including the most challenging alternate-binary-offset-carrier (AltBOC), can be expressed as a specialcase of linear multi-dimensional modulation. Such descrip-tion would not be further limited if different signals in theinphase and quadrature component were present, includingfor example composite BOC (CBOC) modulation.

Using such description, we derive the semi-analyticmodels for the tracking loops. At the expanse generality,the expressions will have the form of vectors and matrices.

In order to adopt this model, which we call as basicsemi-analytic model, to realistic scenarios, we extend it toaccount for delay in the feed back, slowly fading multipath,finite bandwidth, phase noise, and combination of these. Themodel is called as extended semi-analytic model.

Not necessarily, these models must be employed onlyin the investigation of code and carrier tracking loops. Dis-

648 O. JAKUBOV, P. KACMARIK, P. KOVAR, F. VEJRAZKA, UNIVERSALITY AND REALISTIC EXTENSIONS TO . . .

cussion on the connection of these models with multipathmitigation techniques [7] – [11] and position-velocity-time(PVT) estimation [12], [7] is provided. Investigation of vari-ous advanced algorithms such as vector tracking [13] – [18],direct positioning [19], [20] turns out to be possible usingthe semi-analytic approach, as well.

Finally, a Matlab-based toolbox using object-orientedprogramming with its graphical user interface (GUI) hasbeen developed at CTU in Prague. It provides simulationcharacteristics of feed-back systems for civil GPS, Galileo,and GLONASS signals. It is freely downloadable at [21].

1.2 NotationAll vectors in the text are column vectors, denoted with

bold emphasize, as matrices are. The operators Av [.], E [.]denote time average and expectation, respectively. We usesymbol R(.) (.) to represent correlation functions in continu-ous time, symbol R(.) [.] for correlation functions in discretetime. Symbol S(.) (.) denotes power spectral density (PSD).Symbol F [.] is the Fourier transform operator. From thecontext, it will always be clear what exactly each operatorrefers to. Symbol δ(.) denotes Dirac delta function, symbolδ[.] the unit impulse. Symbol [.]i, j denotes an element withindices i, j. Symbol ∑n denotes the summation over index nwith unspecified lower and upper bounds, extending over aninfinitive range. We use a notation TrigT (τ) for the followingtriangular function

TrigT (τ) ={

1− |τ|T , |τ|< T,0, else

(1)

and notation RectT (τ) for the following rectangular function

RectT (τ) ={

1, 0≤ τ < T,0, else. (2)

2. GNSS Signals as LinearMulti-Dimensional ModulationsIn this section, we define the linear multi-dimensional

modulation (LMDM) and represent BPSK(γ), BOC(β,γ),AltBOC(15,10) modulations in that manner. We also show,how signals with inphase and quadrature components repre-senting different signals can be expressed as a kind of suchmodulation. The TMBOC modulation of GPS L2C signal isnot discussed separately, but neglecting the property of thelong period of the L2 CL-code, our model can still be ap-plied. This simplification would make no difference sincethe LMDM representation is developed for the simulationwhere code acquisition is a priori assumed to be done.

2.1 Linear Multi-Dimensional ModulationThe modulated signal s(t) at time t is assumed to be

an additive composition of vector modulation impulses h(t)

multiplied by vector of channel symbols qTn generated at dis-

crete time n, corresponding to n multiples of the channelsymbol period Ts in continuous time, [22]

s(t) = ∑n

qTn h(t−nTs) . (3)

We denote the dimension of the modulation impulse vectorand the channel symbol vector as Nh = dimh(t) = dimqn.The channel symbols qn are a function of the data symbolsdn and the inner states of the discrete part of the modulator.We suppose that the channel symbols are equiprobable, inde-pendent and identically distributed (EP-IID), E

[qn+mqT

n]=

δ [m]I, and that the diagonal elements of the correlation ma-trix RRR h(τ) of the modulation impulse

RRR h(τ) =Z

∞

−∞

h(t + τ)hH(t)dt (4)

have unit energy ∀i ∈ {1, . . .Nh} :∣∣∣[RRR h(0)]i, i

∣∣∣= 1.

2.2 BPSK(γ)According to [7], the BPSK(γ) modulated signal can

be defined as a BPSK direct-sequence-spread-spectrum (DS-SS) signal with chip length Tc = T0/γ where T0 = 1/1023 ms,γ ∈ N

sBPSK(γ)(t) = ∑n

qnh(t−nTs) . (5)

In (5), it holds that qn ∈ {−1, 1}, Nh = 1, and the modu-lation impulse contains the whole number of code periodsNs = Ts/(NcTc) ∈ N per a channel symbol. Symbol Nc de-notes the number of chips in a code period. The modulationimpulse is depicted in Fig. 1 (left). The autocorrelation func-tion of the modulation impulse can be approximated as a tri-angular function in Fig. 1 (right), Rh,BPSK(γ) (τ)≈ TrigTc

(τ),for details see [7].

2.3 BOC(β,γ)The BOC(β,γ) modulated signal is a BPSK(γ) modu-

lated signal multiplied by an alternating periodic signal [23],named subcarrier,

∑i

(−1)i RectTr (t− iTr) (6)

with period 2Tr such that β = T0/(2Tr). We define Nras the number of multiplying rectangles on one chip, thenNr = 2β/γ. The leading edge of the subcarrier is always co-incident with the leading edge of a chip. An illustrative ex-ample of BOC(1,1) modulated signal (Galileo E1b or E1c)is given in Fig. 2. It holds that qn ∈ {−1, 1}, Nh = 1, andthe modulation impulse equals BPSK(γ) modulation impulse(Fig. 1 left) multiplied by the alternating signal in (6). Theautocorrelation function of the modulation impulse Rh(τ)can be derived using similar approach as in [23]

Rh(τ) =1Nr

Nr−1

∑i=0

Nr−1

∑i′=0

(−1)i−i′ TrigTr

(τ−(i− i′

)Tr). (7)


t

h(t)

0 Tc 2Tc 3Tc Ts = NsNcTc

1√Ts

− 1√Ts τ

Rh,BPSK(γ)(τ)

0 Tc−Tc

1

Fig. 1. Modulation impulse h(t) of BPSK(γ) modulation (left), approximated ACF of the modulation impulse Rh(τ) (right).

t

∑i(−1)i

RectTr (t− iTr)

0 Tr 2Tr 3Tr 4Tr 5Tr 6Tr

1

−1

t

sBPSK(1)(t)

0 Tc 2Tc 3Tc

1√Ts

− 1√Ts

t

sBOC(1,1)(t)

0 Tc 2Tc 3Tc

1√Ts

− 1√Ts

×

=

Fig. 2. BOC(1,1) modulation as a BPSK(1) modulation multi-plied by an alternating periodic rectangular signal. NewGalileo ICD [24] slightly modifies the pulse, not con-straining the approach used here.

In our BOC(1,1) example, Nr = 2, Tr = T0/2, hence

Rh,BOC(1,1)(τ) =12(2TrigTr

(τ)−TrigTr(τ−Tr)

−TrigTr(τ+Tr)

).

The autocorrelation function Rh,BOC(1,1)(τ) is depicted inFig. 3.

τ

Rh,BOC(1, 1)(τ)

−Tr

0

Tr 2Tr−2Tr

1

−12

Fig. 3. ACF of BOC(1,1) modulation impulse.

2.4 The IQ Extension of BPSK(γ), BOC(β,γ)In some cases, modulated signals are transmitted in the

inphase and quadrature components at the same time. Pro-vided the channel symbols are generated with the same pe-riod Ts, the modulated signal can be expressed as

s(t) = ∑n

qn,IhI(t−nTs)+ j∑n

qn,QhQ(t−nTs)

= ∑n

qTn h(t−nTs)

where Nh = 2, qn = [qn,I jqn,Q]T , h(t) = [hI(t)hQ(t)]T . Thecorrelation matrix follows

RRR h (τ)≈(

RhI (τ) 00 RhQ (τ)

). (8)

Frequently, the quadrature component has the form of pilotsignal and no data are present, then qn,Q = 1. An examplemay be the Galileo E1b, E1c signals.

2.5 AltBOC(15,10)The AltBOC(15,10) modulation, explicitly defined

in [24], is a nonlinear modulation

sAltBOC(15,10)(t) = ∑n

h(qn, t−nTs) (9)

where qn = [qn,1 qn,2]T , qn,1, qn,2 ∈ {−1, 1}. Symbol

h(qn, t) denotes the modulation function essentially nonzeroon 〈0, Ts) . Here, we assume Ts to be 1 ms = duration of


one code period. Nevertheless, this approach will enable usto easily represent the AltBOC(15,10) signal as an LMDM.We define the following couple of channel symbol vector q′nand modulation impulse vector h′(t) such that

q′n =

[1000]T for qn = q{1}n ,

[0100]T for qn = q{2}n ,

[0010]T for qn = q{3}n ,

[0001]T for qn = q{4}n ,

(10)

h′(t) =

h(q{1}n , t)h(q{2}n , t)h(q{3}n , t)h(q{4}n , t)

(11)

where q{1}n = [−1 −1]T , q{2}n = [−11]T , q{3}n = [1 −1]T ,q{4}n = [11]T . The AltBOC(15,10) can then be rewritten asan LMDM without the EP-IID property of channel symbols

sAltBOC(15,10)(t) = ∑n

q′Tn h′(t−nTs). (12)

The correlation matrix RRR h′(τ) of the modulation impulse isthen, using results from [25],

RRR h′(τ) =

Rh,00 (τ) Rh,01 (τ) Rh,10 (τ) Rh,11 (τ)Rh,01 (τ) Rh,00 (τ) Rh,11 (τ) Rh,10 (τ)Rh,10 (τ) Rh,11 (τ) Rh,00 (τ) Rh,01 (τ)Rh,11 (τ) Rh,10 (τ) Rh,01 (τ) Rh,00 (τ)

(13)

where correlation functions Rh,00, Rh,01, Rh,10, Rh,11 are de-picted in Fig. 4.

3. Basic Model: Semi-AnalyticApproachIn this section, we derive the basic model of the post-

correlator signals for LMDM. We suppose the linear addi-tive white Gaussian noise (AWGN) channel with slowly timevarying parameters (STVP) [26], [27]. This model is herenamed as basic.

3.1 Channel ModelLinear AWGN channel models the received signal x(t)

as the transmitted signal with data symbols d = [. . .dn . . .]T

shifted in time by delay τ0 > 0, multiplied by a carrier withtime varying phase ϕ(t) = 2π fst + ϕ0 where ϕ0 ∈ R is thecarrier phase offset and fs ∈ R is the frequency shift. Suchsignal is then attenuated in the channel by α0 > 0 and is em-bedded in complex WGN w(t)∈C with double-sided powerspectral density 2N0, N0 > 0

x(t) = α0 exp(jϕ(t))s(t− τ0, d)+w(t). (14)

The parameters τ0, ϕ0, fs, α0, N0 are said to be slowly timevarying with respect to Ts. We also make the assumption thatfsTs < π. This condition holds after a successful acquisition.

3.2 Representation of Correlator OutputSignals

Let ZZZ (∆τs) denote the vector of correlator output sig-nals at time nTs in the following form

ZZZ (∆τs) =Z

∞

−∞

x(t)exp(−jϕ(t, fs)

)h∗ (t−nTs− τ0 +∆τs)dt

(15)

where ϕ(t) is the estimate of the carrier phase at time t de-pending on carrier frequency estimate fs. Symbol τ0 is theestimate of the signal delay, and ∆τs is the time shift of thereplicated modulation impulse. The situation is depicted inFig. 5.

∗R∞−∞(·)dt

x(t)

e−jϕ(t,fs)h∗ (t− nTs − τ0 + ∆τs)

nTs Z(∆τs)

Fig. 5. Correlator output signals and their calculation. Boldlines represent vectors.

Substituting (3) into (14) and the result into (15), wesee that ZZZ (∆τs) can be decomposed into the useful compo-nent XXX (∆τs) and the noise component NNN (∆τs)

ZZZ (∆τs) = XXX (∆τs)+NNN (∆τs) (16)

where

XXX (∆τs) = α0 exp(j(ϕ− ϕ)) (17)·sinc

((fs− fs

)Ts)

RRR Th (τ0− τ0−∆τS)qn

where ϕ = 2π fsnTs +ϕ0 and ϕ = ϕ(nTs). The noise compo-nent NNN (∆τs) can be modeled as zero mean white Gaussiannoise vector uncorrelated over time n, but correlated overelements with the following covariance matrix CNNN (∆τs) =2N0RRR h (0). However, ZZZ (∆τs) signals for various valuesof ∆τs are evaluated in practice and the vectors NNN (∆τs)then become correlated over each other. This would be thecase for early/late components ZZZE = ZZZ (−∆τ), ZZZL = ZZZ (∆τ),0 < ∆τ < Tc, where 2∆τ is sometimes referred as correlatorspacing. If NNN concatenates the early/late noise componentsNNN =

[NNN T

E NNN TL]T where NNN E =NNN (−∆τ) , NNN L =NNN (∆τ), the

vector NNN is also a zero mean WGN vector with the followingcovariance matrix

CNNN = 2N0

(RRR h (0) RRR h (2∆τ)

RRR h (−2∆τ) RRR h (0)

). (18)

The concatenated noise component NNN can be generated as

NNN = Aηηη (19)


τ

< [Rh,00(τ)]= [Rh,00(τ)]

τ

< [Rh,01(τ)]= [Rh,01(τ)]

τ

< [Rh,10(τ)]= [Rh,10(τ)]

τ

< [Rh,11(τ)]= [Rh,11(τ)]

0−Tc Tc

1

0−Tc Tc

1

0−Tc Tc

1

0−Tc Tc

1

Fig. 4. Correlation functions Rh,00, Rh,01, Rh,10, Rh,11 of the correlation matrix RRR h′ (τ) for LMDM representation of AltBOC(15,10) modulation.

where ηηη ∼ N (0, I) and CNNN = AAH . Matrix A can be ob-tained using Choleski decomposition. Vector ηηη can be eas-ily generated in Matlab or any similiar software. The groupdelay introduced by the integration in (15) is discussed inSection 4.

Strictly speaking, the convenience of post-correlatormodeling is due to the fact that the output signals ZZZ (∆τs) be-come sufficient statistics for the maximum likelihood (ML)estimation of τ0, ϕ0, fs, α0, N0, d, under the above men-tioned conditions [26], [27], [28]. Hence, the semi-analyticmodels of the tracking loops in [5], [6] may have been de-veloped. In this paper, we do not embody the post-correlatormodel into a particular structure, instead, we develop a gen-eral model which can be adopted to an arbitrary system ful-filling the assumptions.

We inherently supposed that the integration time waschosen as channel symbol period Ts. If shorter integrationtimes are desired in the simulation, the modulation impulsevector h(t) may account for smaller number of code periodsor only one. The channel symbols then lose the stationar-ity and EP-IID property. We will see that this fact does notprevent us from using the developed models, except for thebandwidth constriction which is out of scope of this paper.

3.3 SummaryIn order to sum up this section, we recall that a corre-

lator output signal expressed as an LMDM in linear AWGNchannel with STVP can be modeled as depicted in Fig. 6. Inthat figure, output signals for ∆τs = ∆τ1, . . . ,∆τK are gener-ated ∀k ∈ {1, . . . ,K} : 0 < ∆τk < Tc

ZZZ =

ZZZ (∆τ1)...

ZZZ (∆τK)

,

XXX =

XXX (∆τ1)...

XXX (∆τK)

, NNN =

NNN (∆τ1)...

NNN (∆τK)

where the correlation matrix of the concatenated noise com-ponent is

CNNN = 2N0

RRR h (0) · · · RRR h (∆τK−∆τ1)...

. . ....

RRR h (∆τ1−∆τK) · · · RRR h (0)

.

4. Extended Model: RealisticPhenomenaIn this section, we extend our model to a more realistic

situation. Firstly, we show how feed-back delay may be em-bodied in the model. Secondly, we consider slowly fadingmultipath channel with AWGN. We will see that the tapped-delay-line model (TDLM) of the received signal will, thanksto the linearity of correlation, simply result in an additivecomposition of the separate delay-line components. Thirdly,we discover that bandwidth constriction of the received sig-


++

-RT

h (ϑ+ ∆τ1)qn

RTh (ϑ+ ∆τK)qn

.

.

....

.

.

.

264 X (∆τ1)...

X (∆τK)

375

qn

qn

N = A · η

×

×

.

.

. +

τ0

τ0

ϑ

α0ej(ϕ−ϕ)

X

X (∆τ1)

X (∆τK)

N

Z

N0

Fig. 6. Basic model of the correlator output signals generated as LMDM in linear AWGN channel with STVP. Bold lines represent vectors.

nal will result in bandwidth constriction of the correlationmatrix. The modified correlation matrix would then be pos-sible to store to memory and reload its values when gener-ation of the output signals. Fourthly, we will discuss howphase noise with 1/ f 2 phase noise characteristics may beadded to the true parameters. Finally, a combination of thesephenomena is turned into a complex scheme.

4.1 MultipathLet us suppose that the received signal is composed of

L + 1 line-of-sight components with amplitude αl > 0, car-rier phase offset ϕl ∈ R, and code delay τl > 0

x(t) =L

∑l=0

αl exp(j(2π fst +ϕl))s(t− τl , d)+w(t). (20)

The output signals of the correlators ZZZ (∆τs) can be ex-pressed by first substituting (3) to (20), the result then into(15). We get that

ZZZ (∆τs) =L

∑l=0

XXX l (∆τs)+NNN (∆τs)

with lth useful signal component

XXX l (∆τs) = αl exp(j(φl− ϕ0))sinc((

fs− fs)

Ts)

·RRR Th (τl− τ0 +∆τs)qn.

We use notation φl for the actual carrier phase of the lth com-ponent at time nTs, then it holds that φl = 2π fsnTs +ϕl .

4.2 Finite BandwidthIn this subsection, we will derive the power spectral

density of a GNSS signal as an LMDM. We consider thatchannel symbols are EP-IID. This assumption does not holdfor our IQ representation where one of the channel symbolis constant. We will discuss this case separately.

The power spectral density (PSD) Ss( f ) of a modulatedsignal s(t) equals, according to the Wiener-Khinchin theo-rem,

Ss( f ) = F [Rs (τ)]

whereRs (τ) = AvE [s(t + τ)s∗(t)] .

For all linear multidimensional modulations with stationarychannel symbols, it holds that [22]

Ss ( f ) =1Ts

HH( f )SSS q( f Ts)H( f ) (21)

where

H( f ) = F [h(t)]

=[F [h1(t)] F [h2(t)] . . .F

[hNh(t)

]]T,

SSS q(F) =∞

∑m=−∞

RRR q[m]e−j2πFm,

RRR q[m] = E[qn+mqH

n].

With EP-IID and zero mean channel symbols, we have

RRR q[m] = δ[m]diag(

E[|q0,1|2

], . . . , E

[∣∣q0,Nh

∣∣2])for all our modulations E

[∣∣q0,k∣∣2] = c, c > 0 for all k ∈

{1, . . . ,Nh}, thus

RRR q[m] = cIδ[m],SSS q(F) = cI

and finally

Ss ( f ) =cTs

HH( f )H( f ).

Let SFBW( f ) be the PSD of signal s(t) constrained with anideal low-pass filter of BW/2 band-stop frequency

SFBW( f ) ={

S( f ), | f |< BW/2,000, else.

The constrained PSD of the signal

Ss,FBW ( f ) =cTs

HHFBW( f )HFBW( f )

will constrain the Fourier transform of the modulation im-pulse vector

HFBW( f ) ={

H( f ), | f |< BW/2,000, else.

The truncation of the spectrum of the modulation impulsescan then be accomplished in Matlab by the following steps:


1. Calculate the Fourier transform of the modulation im-pulse correlation matrix

ρρρh( f ) = F [RRR h (τ)] ,

2. truncate it in frequency domain

ρρρh,FBW( f ) ={

ρρρh( f ), | f |< BW/2,0, else

3. and calculate inverse Fourier transform

RRR h,FBW (τ) = F −1 [ρρρh,FBW( f )] .

Considering the IQ extension of BPSK(γ) or BOC(β,γ) mod-ulations, the channel symbol vector would not be generallyEP-IID, since qn,Q = 1. It results in the fact that SSS q( f Ts) 6= cIin (21). Nevertheless, we bypass the problem by suppos-ing that bandwidth constriction will not influence the flowof transmitted channel symbols, but the modulation impulsevector. It substantiates us to use the procedures 1-3 to con-strain the received signal.

The transformations are here represented in continu-ous time domain. However, computationally more attrac-tive would be to accomplish the constriction in discrete timedomain using the fast Fourier transform (FFT), and store thesamples of the constrained correlation matrix in memory. Onload, an interpolation may be employed to better approxi-mate the continuous function.

4.3 Phase NoiseThe carrier phase noise signal can be modeled as zero

mean Gaussian additive component ϕPN to the carrier phase,with defined phase noise characteristics Lϕ( f ) – normalizedsingle-sided power spectral density of ϕPN . The same defi-nition stands for the code phase noise signal τPN , being addi-tive to the code delay. Its influence is not significant relativeto ϕPN . We model the phase noise characteristics Lϕ( f ) withthe following function

Lϕ( f ) ={

σ2, | f |< f0,a2σ2/ f 2, else,

depicted in Fig. 7. Signals ϕPN , τPN can be obtained by gen-erating white Gaussian noise with variance σ2, and passingit through linear time-invariant (LTI) system with frequencyresponse G( f ) such that (a > 0)

|G( f )|={

1, | f |< f0,a/ f , else.

f

Lϕ(f)

Lϕ(f0) = σ2

f0f1

Lϕ(f1) = a2σ2/f21

Fig. 7. Model of the carrier phase noise characteristics Lϕ( f ).

4.4 Feed-Back DelaySince the integration in (15) can be classified as a sig-

nal propagation through a linear time-invariant (LTI) systemwith RectTs (t +nTs) impulse response, group delay Ts/2 isintroduced by that operation. In discrete time with samplingperiod Ts, this can be modeled with the moving average

12

(ZZZn(∆τs)+ZZZn−1(∆τs)) (22)

where index n is used to denote the time in multiples of in-tegration time Ts.

Additionally, we can model an integer delay in thefeed-back, denoted as Dτ, Dϕ for code and carrier phase, re-spectively, simply by adding FIFO memory. Then, we storethe samples and read them back using a circular buffer, inorder to reduce the computational complexity.

4.5 Combined ModelWe can connect the above mentioned models in order

to get the extended model in Fig. 8. Due to linearity of theTDLM, the finite bandwidth would clearly constrain eachcorrelation matrix. Extension of the phase noise model andfeed-back delay model is straightforward.

5. Complex System Model: PVT Esti-mationIn [5], [6], the semi-analytic models were applied to

investigate behavior of the code and carrier tracking loops.However, the approach might be incorporated to simulationsof much more complex systems. As an example, based onthe satellite constellation and true user PVT, input param-eters to the basic model (τ, fs, ϕ, α, N0) can be generated,then they can propagate through the model, and the outputvalues can be further processed. By further processing wemean for instance PVT estimation. Optionally, feed-back be-tween the PVT estimator and the model may be establishedin order to simulate vector tracking or direct positioning al-gorithms.


+

+

-

+

+

-

z−Dτ

+

z−1

+

RT,FBWh (ϑ0 + ∆τ1)qn

RT,FBWh (ϑL + ∆τ1)qn

RT,FBWh (ϑ0 + ∆τK)qn

RT,FBWh (ϑL + ∆τK)qn

qn

qn

qn

qn

N = A · η

×

×

×

×

P

P

264 X (∆τ1)...

X (∆τK)

375 +

z−Dϕ +

z−1

+

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

τ0 ϑ0

τL ϑL

τ0

τPN

12

N0

X 0(∆τ1)

XL(∆τ1)

X 0(∆τK)

XL(∆τK)

α0ej(φ0−ϕ′)

αLej(φL−ϕ′)

X (∆τ1)

X (∆τK)

X Z

N

ϕ′12

ϕPN

ϕ

Fig. 8. Extended model of the correlator output signals generated as LMDM in slowly fading TDLM in AWGN channel with FBW, feed-backdelay, and phase noise. Bold lines represent vectors.

Section 4 also provides hints how the model can be in-corporated in simulations of multipath mitigation techniqueswhich supposes a slowly fading channel. Slight modifica-tions of the model would suffice most of the proposed meth-ods adopting correlation [8] – [11], [13] – [20], [30], [31].

6. Developed SimulatorA simulator of GNSS tracking loops, using the ex-

tended model as an underlying block, has been developedat CTU in Prague. Its name is GNSSTracker. The simula-tor was designed in the Matlab environment with the object-oriented programming approach. It has an encapsulatinggraphical user interface (GUI). A quick tutorial to the simu-lator is available in [29], the simulator itself at [21].

A GNSS signal may be represented as LMDM in classSetup by definition of the correlation matrix RRR h (τ), thechannel symbol vector qn, and some other constants con-cerning the modulated signal. Class CorrOut represents the

extended model itself. The multipath channel is here re-stricted to a 2-path channel.

Classes DLL, PLL represent a delay-locked loop anda phase-locked loop, respectively, and define the propertiesof the discriminator and loop filter. Both DLL, PLL are sub-classes of abstract class FBS (feed-back system). ClassesMotion, Sources, Signals provide the FBSs with the in-put parameters changing over time as requested by classTracking, which is responsible for the loops’ closure. Theinput signals might be step, ramp, frequency ramp, or signalsimulating the mutual motion between the user and the spacevehicle (SV).

Based on the Monte Carlo simulation principle, char-acteristics of the equivalent model, multipath, mean squareerror (MSE), and synchronization failure can be estimated. Itis ensured by classes Eq model, Multipath, MSE, SyncFail,respectively. Transient responses can be visualized, as well,with methods of the class Tracking. Hierarchy of theclasses is depicted in Fig. 9.


7. Simulation ResultsThe correctness of the model and the functionality

of the simulator have been verified in the following man-ner. Well known characteristics were simulated and com-pared with their theoretical values. Characteristics knownfrom bit-true simulations were compared with the simulatedones in GNSSTracker. To be more specific, discriminatorcharacteristics, tracking jitters, multipath characteristics ofBPSK(1), BOC(1,1), AltBOC(15,10) modulated signals invarious system setups were consulted with [7], [23], [25],[30].

In this paper, we present simulation results of Alt-BOC(15,10) fully optimal processing introduced in [31].The method employs four complex early/late correlators(Nh = 4, dim(ZZZ) = 8) and can be classified as joint-maximum-likelihood estimation of the signal parameters andchannel symbols [26], [27]. Each of the four correlators cal-culates a metric for decision about the pair of actual channelsymbols. The output of the selected branch is then fed to theDLL and PLL discriminators.

In the simulation, we consider the extended model fromFig. 8. The setup of the simulation is in Tab. 1. In Fig. 10, thecharacteristics of discriminator, loop noise variance, multi-path, and mean square error are presented.

Setup

DLL, PLLCorrOut Sources

Motion

MultipathEq_model Tracking

Signals

MSE SyncFail

Fig. 9. Class hierarchy in GNSS tracking toolbox(GNSSTracker). Classes with the shape of roundedrectangle are intended for simulations. The dashed linedenotes an input which is optional.

It is apparent from the figure that the discriminatorcharacteristics of the DLL has multiple stable lock pointswhich is a consequence of the alternating shape of the cor-relation function ℜ [Rh,00 (τ)] in Fig. 4. The narrow peak of

ℜ [Rh,00 (τ)] at τ = 0 and narrow correlator spacing ∆τ resultin a steep zero crossing of the characteristics. The peaks arerounded due to the finite bandwidth. PLL discriminator char-acteristics is linear over 〈−π,π〉 which complies with [7].

Modulation AltBOC(15,10)Bandwidth BW = 80 MHz

Correlators’ Setup Optimal [31]Early/Late Spacing ∆τ = Tc/18DLL Discriminator PowerPLL Discriminator Atan2

DLL Filter 3. order, 10 Hz, Dτ = 1PLL Filter 3. order, 20 Hz, Dϕ = 1

SNR = α20/(2N0) 20 dB

DLL Input Signal Step: 0→ 0.02TcPLL Input Signal Step: 0→ π/3

Carrier Phase Noise f0 = 1 Hz, L( f0) =−100 dBc/Hzf1 = 100 Hz, L( f1) =−120 dBc/Hz

Tab. 1. Simulation setup.

The variance of the DLL equivalent loop noise exhibitsmultiple local minima, whereas the variance of the PLLequivalent loop variance is almost constant over the rangeof the interest. It corresponds to the shape of either discrim-inator characteristics.

The peaks of the alternating DLL multipath character-istics decline from its first maximum similarly as in [30].The next two following subfigures depicting transient re-sponses on the step functions document that both feed-backsystems react relatively promptly. Be the sampling period ofthe simulation 1 ms, the DLL would get into the steady statewithin 0.3 s and the PLL within 0.7 s. The characteristics ofmean square error depending on SNR were obtained as thesteady-state values of the characteristics depicting the meansquare error in time.

8. ConclusionsIn this paper, we established a universal framework for

description of GNSS modulations as special cases of the lin-ear multi-dimensional modulation (LMDM). Namely, we in-troduced how BPSK(γ), BOC(β,γ), AltBOC(15,10) modula-tions can be represented in that manner. We extended theapproach for IQ representation of the modulation, includingCBOC modulation. A discussion on representation of TM-BOC signals was delivered.

Based on the general description of the signals, ba-sic semi-analytic model for simulations with correlation be-tween the received signal and replicas of its modulation im-pulses has been developed. The model can represent an ar-bitrary number of correlations for various time shifts of thereplicated modulation impulses. To represent more realisticscenarios, an extended model accounting for slowly fadingmultipath, finite bandwidth, phase noise, feed-back delay,and combination of these has been developed.


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.06

−0.04

−0.02

0

0.02

0.04

0.06DLL - Discriminator Characteristics

Delay Offset [Chips]

Dis

crim

inat

orO

utp

ut

[Chip

s]

−4 −3 −2 −1 0 1 2 3 4−3

−2

−1

0

1

2

3PLL - Discriminator Characteristics

Carrier Phase Offset [rad]

Dis

crim

inat

orO

utp

ut

[-]

−2 −1.5 −1 −0.5 0 0.5 1 1.5 210−6

10−5

10−4

10−3 DLL - Loop Noise Variance

Code Delay [Chips]

Var

iance

[Chip

s*C

hip

s]

−4 −3 −2 −1 0 1 2 3 410−3

10−2

10−1

100

101 PLL - Loop Noise Variance

Carrier Phase Offset [rad]

Var

iance

[rad

*rad

]

0 0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5DLL - Multipath Error

Delay Offset [Chips]

Err

or[M

eter

s]

0 100 200 300 400 500 600 700 8000

0.1

0.2

0.3

0.4

0.5

0.6

0.7DLL - Mean Square Error in Time

Time [Sampling Period]

Err

or[M

eter

s]

0 100 200 300 400 500 600 700 8000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045PLL - Mean Square Error in Time

Time [Samples]

Err

or[M

eter

s]

0 5 10 15 20 25 3010−2

10−1

100

101 DLL - Mean Square Error on SNR

SNR[dB]

Err

or[M

eter

s]

0 5 10 15 20 25 3010−4

10−3

10−2

10−1 PLL - Mean Square Error on SNR

SNR[dB]

Err

or[M

eter

s]

Fig. 10. Simulation results of the setup in Tab. 1. The multipath error is evaluated for the difference of carrier phase offsets ϕ1−ϕ0 = π and ratioof amplitudes α0/α1 = 2. The time axes of the mean square error figures are expressed in samples of the post-correlator processing.

The fact that the models need not be applied only to thetracking loops was discussed. We proposed it for simula-tions of complex systems including PVT estimation, vectortracking, direct positioning, multipath mitigation techniques,or simply elsewhere where the high-rate correlation in theabove stated form appears.

Finally, we introduced a Matlab-based simulator withthe extended model as a basic building block and demon-strated several characteristics of optimal AltBOC(15,10)processing by the code and carrier tracking loops.

AcknowledgementsSafeLOC is a project that is supported by public funds

under Contract No. 20110198 with the Technology Agencyof the Czech Republic in the program to support researchand development ALFA.

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[5] BORIO, D., ANANTHARAMU, P. B., LACHAPELLE, G. Semi-analytic simulations: An extension to unambiguous BOC tracking.In Proceedings of ION ITM 2010: International Technical Meetingof the Institute of Navigation. San Diego (CA, USA), 2010, p. 1 - 14.

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[10] JONES, J., FENTON, P. C., SMITH, B. Theory and per-formance of the pulse aperture correlator (NovAtel technicalreport). 13 pages. [Online] Cited 2011-10-11. Available at:http://webone.novatel.ca/assets/Documents/Papers/PAC.pdf.

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International Technical Meeting of the Satellite Division of The Insti-tute of Navigation (ION GNSS 2005). Long Beach (CA, USA), 2005,p. 1095 - 1102.

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About Authors. . .

Ondrej JAKUBOV was born in Liberec, Czech republic.He received his MSc in electrical engineering from the CTUin Prague in 2010. He is a postgraduate student at the sameuniversity at the Department of Radio Engineering. His re-search interests include GNSS signal processing algorithmsand receiver architectures. His research focuses on advancedpositioning and tracking methods. He is an active memberof the team developing multi-system, multi-frequency, andmulti-antenna GNSS receiver, named the Witch Navigator.

Petr KACMARIK was born in 1978. He received his MScdegree from the CTU in Prague in 2002. During his post-graduate studies (2002 - 2006) he was engaged in the in-vestigation of satellite navigation receiver techniques in hardconditions and wrote his PhD thesis with its main topic onGNSS signal tracking using DLL/PLL with applicability toweek signal environment. He defended the thesis in 2009.Since 2006 he works as an Assistant Professor at the Facultyof Electrical Engineering of the CTU in Prague. His mainprofessional interests are satellite navigation, digital signalprocessing, implementations and simulations of signal pro-cessing algorithms.


Pavel KOVAR was born in Uherske Hradiste in 1970. Hereceived his MSc and PhD degrees from CTU in Prague in1994 and 1998, respectively. Since 1997 to 2000 he workedin Mesit Instruments as a designer of avionics instruments.Since 2000 he is with the Faculty of Electrical Engineeringof the CTU in Prague as an Assistant Professor and since2007 as an Associate Proffesor (Doc.). His interests are re-ceiver architectures, satellite navigation, digital communica-tions and signal processing.

Frantisek VEJRAZKA was born in 1942. He received hisMSc degree from the CTU in Prague in 1965. He servedas an Assistant Professor (1970) and Associate Professor(1981) at the Department of Radio Engineering. He is afull professor of radio navigation, radio communications,and signals and systems theory since 1996. He was ap-

pointed the Head of the Department of Radio Engineering(1994 – 2006), vice-dean of the Faculty (2000 – 2001) andvice-rector of the CTU in Prague (2001 – 2010). His main(professional) interest is in radio satellite navigation wherehe participated on the design of the first Czech GPS receiver(1990) for the Mesit Instruments. He was responsible for thedevelopment of Galileo E5 receiver for Korean ETRI dur-ing 2008 and E1 and E5a receiver in 2009. Prof. Vejrazkahas published 11 textbooks, more than 200 conference pa-pers and many technical reports. He is the former presidentof the Czech Institute of Navigation, Fellow of the Royal In-stitute of Navigation in London, member of the Institute ofNavigation (USA), vice-president of IAIN, vice-chairman ofCGIC/IISC, member of IEEE, member of Editorial Board ofGPS World, Editorial Board of InsideGNSS, etc.

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