# Data and Pilot Combining for Composite GNSS Signal Acquisition

Authors: Daniele Borio; Letizia Lo Presti
Journal: International Journal of Navigation and Observation
 (2008)

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## Abstract

With the advent of new global navigation satellite systems (GNSS), such as the European Galileo, the
Chinese Compass and the modernized GPS, the presence of new modulations allows the use of special
techniques specifically tailored to acquire and track the new signals. Of particular interest are the new
composite GNSS signals that will consist of two different components, the data and pilot channels. Two
strategies for the joint acquisition of the data and pilot components are compared. The first technique,
noncoherent combining, is from the literature and it is used as a comparison term, whereas the analysis
of the second one, coherent combining with sign recovery, represents the innovative contribution of this
paper. Although the analysis is developed with respect to the Galileo E1 Open Service (OS) modulation,
the obtained results are general and can be applied to other GNSS signals.


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## Body

## 1. Introduction

With the
advent of the new global navigation satellite system (GNSS), such as the
European Galileo and the Chinese Compass, new signals and new modulations have
been introduced in order to guarantee the coexistence and interoperability with
existent systems, like the American GPS, and to fully exploit the technologies
currently available.An example of those new signals is the coherent
adaptive subcarrier modulation (CASM) that will be used for the transmission of
the Galileo signal on the E1 frequency. CASM combines three different signals
in a constant amplitude modulation that allows the use of efficient class C
amplifiers [1, 2].The three signals combined in the CASM are denoted as
theA, B, and C channels. The first one is used for public-regulated service
(PRS), whereas the latter two will provide the open service (OS). The B and C
channels have two different roles: the first one, denoted data channel, will
carry the navigation message whereas the second one, denoted pilot channel,
will be used for determining the pseudoranges between the satellites and the
receiver. The B and C
signals will be transmitted at the same time, at the same
frequency, and they will be separated only by different codes [3]. The A signal is not
completely defined by the galileo interface control document [1] but will probably be
separated in frequency as for the A signal emitted by Giove-A [4], the first experimental
satellite of the Galileo constellation.The presence of new modulations allows one to adopt
special techniques specifically tailored to acquire and track new signals.One solution, when acquiring composite GNSS signals
such as the Galileo E1 OS modulation, consists in ignoring the data channel and
processing only the pilot signal. In this way only half of the useful power is
employed and the GNSS receiver could not be able to acquire and track signals
that would be easily processed if all the useful power were used. Pilot and
data channel combining allows recovering all the available power improving the
acquisition performance and providing more reliable signal detection. For these reasons the design of signal combing techniques is critical for the efficient acquisition of the new GNSS signals.In this paper, Galileo E1 OS signals are considered
and in particular two acquisition strategies for
the efficient combining of data and pilot channels are analyzed. In the first
strategy, called noncoherent combining, the received signal is correlated
separately with the pilot and data local replicas. The correlation outputs are
then squared and summed. This strategy essentially exploits the principle of
noncoherent integration [5–7] employed to extend the integration period over the bit
duration. Its use for data and pilot combining is reported by [8], however its performance in
terms of false alarm and detection probabilities has been only marginally
investigated.The second strategy consists in multiplying the
received signal by two new signals, obtained by summing and subtracting the
data and pilot local replicas, respectively. Then, the maximum of the two
correlations is adopted as decision variable. This strategy can be seen as an
extension of the(b–c) technique proposed by [9, 10] in which only the
correlation with the difference between data and pilot codes was considered.
This second strategy implements the Maximum Likelihood estimator for the code
delay, the Doppler frequency, and the relative phase between data and pilot
channels, and, to the best of our knowledge, has never been applied to the E1
OS signals. This strategy allows the coherent combining of data and pilot
channels and it will be denoted as coherent channel combining with sign
recovery. The proposed method is the adaptation of the acquisition strategy
proposed in [11] to
the Galileo E1 OS signal. In particular [11] considered the case of data and pilot components
transmitted with a ±90 degree phase difference. Moreover, [11] only proposed the method
without providing any analytical characterization of its performance.Both strategies have been analyzed in terms of false
alarm and detection probabilities and closed-form expressions for the
probabilities of coherent channel combining with sign recovery have been
derived. To the best of our knowledge these expressions have never been derived
before and represent one of the innovative contributions of this paper.
Moreover, these formulas are general and can be easily adapted to the case of
other modulations, for example, for the GPS L5 case that has been analyzed only
by simulations [11].In the analysis the coherent integration time is
limited to a single code period. In this way the acquisition block has not to
deal with the problem of bit transitions that can occur every4 milliseconds. Moreover, an integration time of 4 milliseconds should be sufficient to acquire
GNSS signals in high to moderate C/N0 conditions. The problem of bit transitions can
be overcome by using noncoherent integrations. The integration time can be also
increased by considering the pilot channel alone and exploiting the structure
imposed by its secondary code [11]. These issues are however out of the scope of this
paper.Monte Carlo techniques have been used for supporting
the theoretical analysis: simulations and analytical expressions agree well
proving the effectiveness of the developed theory.From the analysis it emerges that the coherent
combining always outperforms noncoherent combining and thus it should be
adopted for the joint acquisition of data and pilot channels.This work is organized as follows: Section2
introduces the CASM and provides a model for Galileo E1 OS signals. In Section
3 the noncoherent and coherent combining algorithms are described and the
expressions for false alarm and detection probabilities are derived. In Section
4 the derived formulas are validated by Monte Carlo simulations. Finally, the
conclusions are presented in Section 5.


## 2. Signal and System Model

The signal at
the input of a Galileo receiver, in one-path additive Gaussian noise
environment, can be written as(1)rRF(t)=∑i=1L2PR,iyi(t)+ηRF(t)that is the sum of L useful signals, emitted by L different satellites and with power PR,i,
and of a noise term ηRF(t).
Each signal yi(t),
in the Galileo E1 band, is given by [1, 2]:
(2)yi(t)=23[eB,i(t−τ0,i)−eC,i(t−τ0,i)]cos(2π(fRF+fd,i)t+θi)−13[2eA,i(t−τ0,i)+eA,i(t−τ0,i)eB,i(t−τ0,i)eC,i(t−τ0,i)]×sin(2π(fRF+fd,i)t+θi),where
(i)
eA,i(t), eB,i(t) and eC,i(t) are the three useful signals emitted on the E1
frequencies and corresponding to the A, B, and C channels, respectively, and eA,i(t) is a restricted access signal for PRS whereas eB,i(t) and eC,i(t) are the data and pilot signals for OS;(ii)
τ0,i, fd,i,
and θi are the delay, the Doppler frequency, and the
phase introduced by the transmission channel;(iii)
fRF=1575.42 MHz is the E1 central frequency.eA,i(t), eB,i(t),
and eC,i(t) are real binary sequences that assume value in
the set {−1,1}.
These three signals are combined according to the CASM in order to obtain a
constant envelope signal yi(t).
This result is achieved by introducing the term proportional to the three
useful signals in (2).The input signal (1) is recovered by the receiver
antenna, downconverted, and filtered by the receiver front end. A low-IF
receiver is assumed. In mass-market receivers the bandwidth of the front-end
filter is generally limited to a few MHz and thus the components depending on
the PRS signal in (2) can be considered eliminated by the filtering process.
This is due to the fact that a BOC(15,2.5) modulation [2, 4] is employed for the PRS
signal. This modulation splits the PRS signal power far from the E1 central
frequency where the power spectral densities (PSD) of the OS signals are
concentrated. In this way, the received signal, before the analog-to-digital
(AD) conversion, is given by
(3)rIF(t)=∑i=1L2PR,iy¯i(t)+ηIF(t)=∑i=1L2PR,i23[eB,i(t−τ0,i)−eC,i(t−τ0,i)]×cos(2π(fIF+fd,i)t+θi)+ηIF(t),where fIF is the receiver intermediate frequency and ηIF(t) is the downconverted and filtered noise
component. In (3) the effect of the front-end filtering on the OS components is
considered negligible.Finally, (3) is sampled and AD is converted, obtaining,
by neglecting the quantization impact, the following signal
model:(4)rIF(nTs)=∑i=1L2PR,iy¯i(nTs)+ηIF(nTs)=∑i=1L2PR,i23[eB,i(nTs−τ0,i)−eC,i(nTs−τ0,i)]×cos(2π(fIF+fd,i)nTs+θi)+ηIF(nTs).In the following, the notation x[n]=x(nTs) will indicate a discrete-time sequence x[n],
obtained by sampling a continuous-time signal x(t) with a sampling frequency fs=1/Ts.
For this reason (4) can be rewritten as(5)rIF[n]=∑i=1LCi(eB,i[n−τ0,iTs]−eC,i[n−τ0,iTs])×cos(2πFD,0in+θi)+ηIF[n],where Ci=(4/9)PR,i and FD,0i=(fIF+fd,i)Ts. Ci represents the total power of the i th received signal since the
term(6)(eB,i[n−τ0,iTs]−eC,i[n−τ0,iTs])cos(2πFD,0in+θi)has unitary power. The spectral
characteristics of ηIF[n] depend on the type of filtering along with the
sampling and decimation strategy adopted in the front end. A convenient choice
is to sample the IF signal with a sampling frequency fs=2BIF,
where BIF is the one-sided front-end bandwidth. In this
case, it is easily shown that the noise variance becomes(7)σIF2=E{ηIF2(t)}=E{ηIF2(nTs)}=N0fs2=N0BIF,where N0/2 is the Power Spectral Density of the IF noise.
The autocorrelation function(8)RIF[m]=E{ηIF[n]ηIF[n+m]}=σIF2δ[m]implies that the discrete-time
random process ηIF[n] is a classical independent and identically
distributed (iid) wide-sense stationary (WSS) random process, or a white
sequence. δ[m] is the Kronecker delta. From now on, this
signal model is adopted and the system performance is expressed in terms of
Carrier-to-Noise Ratio Ci/N0.Both data and pilot signals in (5) can be represented
as the product of three terms(9)eB/C,i[n]=cB/C,i[n]sB[n]dB/C,i[n],where cB/C,i[n] is a spreading code of length Nc=4092, sB[n] is the subcarrier signal that in the Galileo
OS signal is the BOC(1,1), and dB/C,i[n] is the navigation message for the data signal
and the secondary code in the pilot case. dB/C,i[n] is constant over one code period. Hereinafter
the product between the code and the subcarrier will be denoted
as(10)e¯B/C,i[n]=cB/C,i[n]sB[n].As a result of code
orthogonality, the different Galileo codes are analyzed separately by the
receiver, and thus hereinafter the case of a single satellite is considered and
the index i is dropped. Thus the resulting signal
is(11)rIF[n]=C(eB[n−τ0Ts]−eC[n−τ0Ts])cos(2πFD,0n+θ)+ηIF[n].


## 3. OS Signals Acquisition

In traditional
GPS receivers the delay and the Doppler frequencies of received signals are
estimated by using the correlation with local signal replicas opportunely
delayed and modulated. In the acquisition stage, the code delay and Doppler
frequency are estimated as the ones that make the correlation with the local
replica pass a fixed threshold. In the Galileo E1 case however this strategy
cannot be directly employed since two different signals are present. Moreover,
due to the navigation message on the data channel and to the secondary code on
the pilot channel, these two components can be either summed or subtracted.In this Section, we analyze two possible strategies
for signal acquisition for Galileo OS signals. The performance of the two
algorithms is evaluated in terms of false alarm and detection probabilities
that are the probabilities that the decision variable passes a fixed threshold
under two different hypotheses:(H0)
the signal is present and correctly aligned with the local replica;(H1)
the signal is absent or not correctly aligned with local replica.
The plot of the detection probability
versus the false alarm probability is called receiver operating characteristic
(ROC) and it completely defines the system performance [12]. The ROCs are usually
employed for comparative analysis, as effective metric for characterizing an
acquisition system [5, 6]. For these reasons they are adopted in this work and
used as basis for the analysis of the two acquisition algorithms considered in
the paper.


### 3.1. Pilot and Data Noncoherent Combining

The conceptual scheme for the acquisition of
Galileo E1 signals, with the noncoherent combination of data and pilot
channels, is depicted in Figure1. The received signal is multiplied by two
orthogonal sinusoids at the frequency FD,
leading to the in-phase (I) and quadrature (Q) components. These signals are
then split in two different branches where the correlations with the data and
pilot codes are evaluated. The cross-correlations from the two branches are
noncoherently combined leading to the decision variable(12)Snc(FD,τ)=|RB(FD,τ)|2+|RC(FD,τ)|2,where(13)RB(FD,τ)=1N∑n=0N−1rIF[n]e¯B[n−τTs]exp{j2πFDn},RC(FD,τ)=1N∑n=0N−1rIF[n]e¯C[n−τTs]exp{j2πFDn}are the cross-correlations of
the received signal with the data and pilot replicas delayed by τ and modulated by FD.
In (13) N denotes the number of samples used to
integrate the received signal over one-code period. More specifically, N is given by ⌊Tc⋅fs⌋, where Tc=4 milliseconds is the duration of one code
period, fs is the sampling frequency, and ⌊⋅⌋ denotes the floor operator. Since the cross-correlations RB(FD,τ) and RC(FD,τ) have been obtained from the input signal rIF[n],
they also depends on τ0, FD,0,
and θ,
the parameters that characterize rIF[n].
The dependence on these parameters has not been explicitly reported in (13) for
the ease of notation.Figure 1
Conceptual
scheme of the Galileo OS signals acquisition with noncoherent combining.The multiplication by the complex exponential in (13)
is implemented in Figure 1 by the multiplication with the two orthogonal
sinusoids.In AppendixA it is shown that RB(FD,τ) and RC(FD,τ) are two complex Gaussian random variables
that, due to the orthogonality properties of Galileo codes [3], are approximatively independent. Thus |RB(FD,τ)|2 and |RC(FD,τ)|2 are two χ2 random variables with 2 degrees of freedom. From this consideration
and (12), Snc(FD,τ) is a χ2 random variable with 4 degrees of freedom.In order the determine the ROC, the two following
probabilities have to be evaluated:(14)Pfanc(β)=P(Snc(FD,τ)>β|H1),Pdnc(β)=P(Snc(FD,τ)>β|H0).When the signal is not present
or not correctly aligned it is possible to assume [5, 6] that both RB(FD,τ) and RC(FD,τ) are zero mean. Thus Snc(FD,τ) is a central χ2 random variable, whose distribution is
completely characterized by the variance of RB(FD,τ) and of RC(FD,τ).
From (13) we have(15)Var{RB(FD,τ)}=Var{RC(FD,τ)}=Var{1N∑i=0N−1rIF[n]e¯B[n−τTs]exp{j2πFDn}}=1N2∑i=0N−1Var{rIF[n]}=1N2∑i=0N−1Var{ηIF[n]}=N0fs2N=N0BIFN.Equation (15) directly derives
from the noise model reported in Section 2. In this case it is assumed that
the input noise is a white sequence. In a real GNSS receivers the front end can
introduce some correlation among the different noise samples. This correlation
causes a correlation loss [13] that would affect both noise and signal components.
This effect could be included in the analysis but it would introduce no further
insight and thus, for the sake of clarity, it is not considered in this
context.Since bothRB(FD,τ) and RC(FD,τ) are complex random variables with iid real and
imaginary parts, the variance (15) is equally divided between the two
components and thus we can define(16)σn2=N0fs4N=N0BIF2N,σn2 is the variance of the real and of the
imaginary parts of RB(FD,τ) and RC(FD,τ).
Given these premises and exploiting the fact that Snc(FD,τ) is a central χ2 random variable it is possible to derive the
false alarm probability [14](17)Pfanc(β)=exp{−β2σn2}(1+β2σn2).When the signal is present and
correctly aligned Snc(FD,τ) is a noncentral χ2 random variable with noncentrality
parameter(18)λ=|E{RB(FD,τ)}|2+|E{RC(FD,τ)}|2=2|E{RB(FD,τ)}|2that assumes the following
expression [13, 15]:(19)λ=C2sin2(πNΔF)(πNΔF)2K2(Δτ)≈C2,
where
(i)
ΔF=FD,0−FD is the difference between the Doppler
frequencies of the received signal and of the local replica;(ii)
Δτ=(τ0−τ)/Ts is the difference between the delays of the
received signal and of the local replica, normalized with respect to the
sampling interval;(iii)
K(⋅) is the correlation between the incoming code,
filtered by the frontend, and the code generated at the receiver.
When the
Doppler frequency and the delay of the local replica match the ones of the
received signal, the loss (sin2(πNΔF)/(πNΔF)2)K2(Δτ) can be assumed negligible and λ≈C/2.From these considerations it is possible to evaluate
the detection probability [14, 16](20)Pdnc(β)=Q2(λσn2,βσn2)≈Q2(2CNN0fs,4βNN0fs),where Q2(⋅,⋅) is the Generalized Marcum Q function of order
2 [16, 17], defined as(21)QK(a,b)=1aK−1∫b+∞xKexp{−x2+a22}IK−1(ax)dxwhere IK−1(⋅) is the modified Bessel function of first kind
and order K−1 [18].


### 3.2. Pilot and Data Coherent Combining with Sign Recovery

By considering
(11) and by collecting the navigation message of the data channel, it is
possible to rewrite the received signal rIF[n] as(22)rIF[n]=CdB[n−τ0Ts]×(e¯B[τ0Ts]−dC[n−(τ0/Ts)]dB[n−(τ0/Ts)]e¯C[n−τ0Ts])cos(2πFD,0n+θ)+ηIF[n].Expressed in this form, (22)
indicates that the navigation message dB[n] is spread by the equivalent pseudorandom sequence(23)eeq[n]=e¯B[n]−dC[n]dB[n]e¯C[n].Since the navigation message dB[n] and the secondary code dC[n] are constant over one code period and since
they can assume only two values, −1 and 1,
only two equivalent spreading sequences are possible(24)eeq[n]={e¯B[n]−e¯C[n],e¯B[n]+e¯C[n].In the coherent combining scheme, the received signal is correlated with
both equivalent codes: the equivalent code that maximizes the cross-correlation
is likely the correct one. Based on this principle the decision variable is
given by(25)Sml(FD,τ)=max{|R+(FD,τ)|2,|R−(FD,τ)|2},where(26)R+(FD,τ)=1N∑n=0N−1rIF[n]{e¯B[n−τTs]+e¯C[n−τTs]}exp(j2πFDn),R−(FD,τ)=1N∑n=0N−1rIF[n]{e¯B[n−τTs]−e¯C[n−τTs]}exp(j2πFDn).This kind of algorithm is based
on the Maximum Likelihood estimator for the code delay, Doppler frequency, and
relative sign between data and pilot channels, since, as shown in Appendix B,
the maximization of the correlation function, with respect to the delay τ,
the frequency FD,
and the ratio dC[n]/dB[n],
corresponds to the maximization of the likelihood function evaluated for the
received signal rIF[n].The acquisition scheme with coherent combining is
reported in Figure2: the received signal is correlated with the two equivalent
codes and the decision variable is obtained by choosing the maximum of the two
cross-correlations. It has to be noted that the variables R+(FD,τ) and R−(FD,τ) can be obtained by opportunely combining the
cross-correlations with the data and the pilot codes(27)R+(FD,τ)=1N∑n=0N−1rIF[n]e¯B[n−τTs]exp(j2πFDn)+1N∑n=0N−1rIF[n]e¯C[n−τTs]exp(j2πFDn)=RB(FD,τ)+RC(FD,τ),(28)R−(FD,τ)=1N∑n=0N−1rIF[n]e¯B[n−τTs]exp(j2πFDn)−1N∑n=0N−1rIF[n]e¯C[n−τTs]exp(j2πFDn)=RB(FD,τ)−RC(FD,τ).In this way the coherent
combining algorithm can be implemented with a computational load similar to the
one required by the noncoherent combining strategy.Figure 2
Conceptual scheme of the Galileo
OS signals acquisition with coherent combining.From (28) and (27) it clearly emerges that R+(FD,τ) and R−(FD,τ) are linear combinations of RB(FD,τ) and RC(FD,τ).
Thus, since RB(FD,τ) and RC(FD,τ) are complex Gaussian random variables, R+(FD,τ) and R−(FD,τ) are also complex
and Gaussian.The ROC for the coherent combining strategy can be
easily obtained by determining the false alarm and detection probability of the
variables|R+(FD,τ)|2 and |R−(FD,τ)|2.
In fact it can be easily shown that the false alarm and the detection probabilities
of the decision variable Sml(FD,τ) are given by(29)P(Sml(FD,τ)>β)=P(max{|R+(FD,τ)|2,|R−(FD,τ)|2}>β)=1−P(max{|R+(FD,τ)|2,|R−(FD,τ)|2}<β)=1−P(|R+(FD,τ)|2<β,|R−(FD,τ)|2<β)=1−P(|R+(FD,τ)|2<β)P(|R−(FD,τ)|2<β).The last line in (29) has been
obtained by exploiting the independence between |R+(FD,τ)|2 and |R−(FD,τ)|2 that derives from the independence of R+(FD,τ) and R−(FD,τ).
In fact we have(30)E{R+(FD,τ)[R−(FD,τ)]*}=E{[RB(FD,τ)+RC(FD,τ)][RB(FD,τ)−RC(FD,τ)]*}=E[|RB(FD,τ)|2−|RC(FD,τ)|2]=0.Equation (30) is zero since |RB(FD,τ)|2 and |RC(FD,τ)|2 are equally distributed and the difference of
their mean cancels out. From (30) R+(FD,τ) and R−(FD,τ) are
uncorrelated and, since they are both Gaussian random variables, independent.
For these reasons |R+(FD,τ)|2 and |R−(FD,τ)|2 are also independent.SinceR+(FD,τ) and R−(FD,τ) are Gaussian random variables, |R+(FD,τ)|2 and |R−(FD,τ)|2 are χ2 distributed with two degrees of freedom. When
the signal is absent, or the local replicas are not aligned with the received
signal, then |R+(FD,τ)|2 and |R−(FD,τ)|2 are both central χ2 random variables [14] and the false alarm
probability of Sml(FD,τ) is(31)Pfaml(β)=1−[1−exp{−β4σn2}]2.Equation (31) has been obtained
by substituting the cumulative density function (cdf) of central χ2 random variables [14] into (29). It can be noted
that the exponential in (31) depends on 4σn2 instead of 2σn2 as for (17). This is due to the fact that the
equivalent code (24) has twice the power of the single pilot and data codes.When the Galileo signal is present and correctly
aligned with the local replica,|R+(FD,τ)|2 and |R−(FD,τ)|2 are no more central χ2 random variables, and the noncentrality
parameters λ+ and λ− have to be determined. λ+ and λ− can be obtained by determining the mean of R+(FD,τ) and R−(FD,τ) under the hypothesis that the local equivalent
code matches or not the navigation bit. In particular we have(32)E[R+(FD,τ)]=E{RB(FD,τ)+RC(FD,τ)}={2E{RB(FD,τ)}=Csin(πNΔF)(πNΔF)K(Δτ)≈CifdC[n]dB[n]=1,0ifdC[n]dB[n]=−1and similarly(33)E[R−(FD,τ)]=E{RB(FD,τ)−RC(FD,τ)}={2E{RB(FD,τ)}=Csin(πNΔF)(πNΔF)K(Δτ)≈CifdC[n]dB[n]=−1,0ifdC[n]dB[n]=1.From these considerations it
emerges that the decision variable Sml(FD,τ),
under the hypothesis of presence of signal and correct alignment, is given by
the maximum between a central χ2 and a noncentral χ2 random variables with two degrees of freedom.
The noncentrality parameter of the noncentral χ2 random variable is given by(34)λ=Csin2(πNΔF)(πNΔF)2K2(Δτ)≈C.Given these premises it is
finally possible to express the detection probability(35)Pdml(β)=1−[1−exp{−β4σn2}][1−Q1(λ2σn2,β2σn2)]≈1−[1−exp{−β4σn2}][1−Q1(2CNN0fs,2βNN0fs)],where Q1(⋅,⋅) is the Marcum Q function of order 1.


## 3.1. Pilot and Data Noncoherent Combining

The conceptual scheme for the acquisition of
Galileo E1 signals, with the noncoherent combination of data and pilot
channels, is depicted in Figure1. The received signal is multiplied by two
orthogonal sinusoids at the frequency FD,
leading to the in-phase (I) and quadrature (Q) components. These signals are
then split in two different branches where the correlations with the data and
pilot codes are evaluated. The cross-correlations from the two branches are
noncoherently combined leading to the decision variable(12)Snc(FD,τ)=|RB(FD,τ)|2+|RC(FD,τ)|2,where(13)RB(FD,τ)=1N∑n=0N−1rIF[n]e¯B[n−τTs]exp{j2πFDn},RC(FD,τ)=1N∑n=0N−1rIF[n]e¯C[n−τTs]exp{j2πFDn}are the cross-correlations of
the received signal with the data and pilot replicas delayed by τ and modulated by FD.
In (13) N denotes the number of samples used to
integrate the received signal over one-code period. More specifically, N is given by ⌊Tc⋅fs⌋, where Tc=4 milliseconds is the duration of one code
period, fs is the sampling frequency, and ⌊⋅⌋ denotes the floor operator. Since the cross-correlations RB(FD,τ) and RC(FD,τ) have been obtained from the input signal rIF[n],
they also depends on τ0, FD,0,
and θ,
the parameters that characterize rIF[n].
The dependence on these parameters has not been explicitly reported in (13) for
the ease of notation.Figure 1
Conceptual
scheme of the Galileo OS signals acquisition with noncoherent combining.The multiplication by the complex exponential in (13)
is implemented in Figure 1 by the multiplication with the two orthogonal
sinusoids.In AppendixA it is shown that RB(FD,τ) and RC(FD,τ) are two complex Gaussian random variables
that, due to the orthogonality properties of Galileo codes [3], are approximatively independent. Thus |RB(FD,τ)|2 and |RC(FD,τ)|2 are two χ2 random variables with 2 degrees of freedom. From this consideration
and (12), Snc(FD,τ) is a χ2 random variable with 4 degrees of freedom.In order the determine the ROC, the two following
probabilities have to be evaluated:(14)Pfanc(β)=P(Snc(FD,τ)>β|H1),Pdnc(β)=P(Snc(FD,τ)>β|H0).When the signal is not present
or not correctly aligned it is possible to assume [5, 6] that both RB(FD,τ) and RC(FD,τ) are zero mean. Thus Snc(FD,τ) is a central χ2 random variable, whose distribution is
completely characterized by the variance of RB(FD,τ) and of RC(FD,τ).
From (13) we have(15)Var{RB(FD,τ)}=Var{RC(FD,τ)}=Var{1N∑i=0N−1rIF[n]e¯B[n−τTs]exp{j2πFDn}}=1N2∑i=0N−1Var{rIF[n]}=1N2∑i=0N−1Var{ηIF[n]}=N0fs2N=N0BIFN.Equation (15) directly derives
from the noise model reported in Section 2. In this case it is assumed that
the input noise is a white sequence. In a real GNSS receivers the front end can
introduce some correlation among the different noise samples. This correlation
causes a correlation loss [13] that would affect both noise and signal components.
This effect could be included in the analysis but it would introduce no further
insight and thus, for the sake of clarity, it is not considered in this
context.Since bothRB(FD,τ) and RC(FD,τ) are complex random variables with iid real and
imaginary parts, the variance (15) is equally divided between the two
components and thus we can define(16)σn2=N0fs4N=N0BIF2N,σn2 is the variance of the real and of the
imaginary parts of RB(FD,τ) and RC(FD,τ).
Given these premises and exploiting the fact that Snc(FD,τ) is a central χ2 random variable it is possible to derive the
false alarm probability [14](17)Pfanc(β)=exp{−β2σn2}(1+β2σn2).When the signal is present and
correctly aligned Snc(FD,τ) is a noncentral χ2 random variable with noncentrality
parameter(18)λ=|E{RB(FD,τ)}|2+|E{RC(FD,τ)}|2=2|E{RB(FD,τ)}|2that assumes the following
expression [13, 15]:(19)λ=C2sin2(πNΔF)(πNΔF)2K2(Δτ)≈C2,
where
(i)
ΔF=FD,0−FD is the difference between the Doppler
frequencies of the received signal and of the local replica;(ii)
Δτ=(τ0−τ)/Ts is the difference between the delays of the
received signal and of the local replica, normalized with respect to the
sampling interval;(iii)
K(⋅) is the correlation between the incoming code,
filtered by the frontend, and the code generated at the receiver.
When the
Doppler frequency and the delay of the local replica match the ones of the
received signal, the loss (sin2(πNΔF)/(πNΔF)2)K2(Δτ) can be assumed negligible and λ≈C/2.From these considerations it is possible to evaluate
the detection probability [14, 16](20)Pdnc(β)=Q2(λσn2,βσn2)≈Q2(2CNN0fs,4βNN0fs),where Q2(⋅,⋅) is the Generalized Marcum Q function of order
2 [16, 17], defined as(21)QK(a,b)=1aK−1∫b+∞xKexp{−x2+a22}IK−1(ax)dxwhere IK−1(⋅) is the modified Bessel function of first kind
and order K−1 [18].


## 3.2. Pilot and Data Coherent Combining with Sign Recovery

By considering
(11) and by collecting the navigation message of the data channel, it is
possible to rewrite the received signal rIF[n] as(22)rIF[n]=CdB[n−τ0Ts]×(e¯B[τ0Ts]−dC[n−(τ0/Ts)]dB[n−(τ0/Ts)]e¯C[n−τ0Ts])cos(2πFD,0n+θ)+ηIF[n].Expressed in this form, (22)
indicates that the navigation message dB[n] is spread by the equivalent pseudorandom sequence(23)eeq[n]=e¯B[n]−dC[n]dB[n]e¯C[n].Since the navigation message dB[n] and the secondary code dC[n] are constant over one code period and since
they can assume only two values, −1 and 1,
only two equivalent spreading sequences are possible(24)eeq[n]={e¯B[n]−e¯C[n],e¯B[n]+e¯C[n].In the coherent combining scheme, the received signal is correlated with
both equivalent codes: the equivalent code that maximizes the cross-correlation
is likely the correct one. Based on this principle the decision variable is
given by(25)Sml(FD,τ)=max{|R+(FD,τ)|2,|R−(FD,τ)|2},where(26)R+(FD,τ)=1N∑n=0N−1rIF[n]{e¯B[n−τTs]+e¯C[n−τTs]}exp(j2πFDn),R−(FD,τ)=1N∑n=0N−1rIF[n]{e¯B[n−τTs]−e¯C[n−τTs]}exp(j2πFDn).This kind of algorithm is based
on the Maximum Likelihood estimator for the code delay, Doppler frequency, and
relative sign between data and pilot channels, since, as shown in Appendix B,
the maximization of the correlation function, with respect to the delay τ,
the frequency FD,
and the ratio dC[n]/dB[n],
corresponds to the maximization of the likelihood function evaluated for the
received signal rIF[n].The acquisition scheme with coherent combining is
reported in Figure2: the received signal is correlated with the two equivalent
codes and the decision variable is obtained by choosing the maximum of the two
cross-correlations. It has to be noted that the variables R+(FD,τ) and R−(FD,τ) can be obtained by opportunely combining the
cross-correlations with the data and the pilot codes(27)R+(FD,τ)=1N∑n=0N−1rIF[n]e¯B[n−τTs]exp(j2πFDn)+1N∑n=0N−1rIF[n]e¯C[n−τTs]exp(j2πFDn)=RB(FD,τ)+RC(FD,τ),(28)R−(FD,τ)=1N∑n=0N−1rIF[n]e¯B[n−τTs]exp(j2πFDn)−1N∑n=0N−1rIF[n]e¯C[n−τTs]exp(j2πFDn)=RB(FD,τ)−RC(FD,τ).In this way the coherent
combining algorithm can be implemented with a computational load similar to the
one required by the noncoherent combining strategy.Figure 2
Conceptual scheme of the Galileo
OS signals acquisition with coherent combining.From (28) and (27) it clearly emerges that R+(FD,τ) and R−(FD,τ) are linear combinations of RB(FD,τ) and RC(FD,τ).
Thus, since RB(FD,τ) and RC(FD,τ) are complex Gaussian random variables, R+(FD,τ) and R−(FD,τ) are also complex
and Gaussian.The ROC for the coherent combining strategy can be
easily obtained by determining the false alarm and detection probability of the
variables|R+(FD,τ)|2 and |R−(FD,τ)|2.
In fact it can be easily shown that the false alarm and the detection probabilities
of the decision variable Sml(FD,τ) are given by(29)P(Sml(FD,τ)>β)=P(max{|R+(FD,τ)|2,|R−(FD,τ)|2}>β)=1−P(max{|R+(FD,τ)|2,|R−(FD,τ)|2}<β)=1−P(|R+(FD,τ)|2<β,|R−(FD,τ)|2<β)=1−P(|R+(FD,τ)|2<β)P(|R−(FD,τ)|2<β).The last line in (29) has been
obtained by exploiting the independence between |R+(FD,τ)|2 and |R−(FD,τ)|2 that derives from the independence of R+(FD,τ) and R−(FD,τ).
In fact we have(30)E{R+(FD,τ)[R−(FD,τ)]*}=E{[RB(FD,τ)+RC(FD,τ)][RB(FD,τ)−RC(FD,τ)]*}=E[|RB(FD,τ)|2−|RC(FD,τ)|2]=0.Equation (30) is zero since |RB(FD,τ)|2 and |RC(FD,τ)|2 are equally distributed and the difference of
their mean cancels out. From (30) R+(FD,τ) and R−(FD,τ) are
uncorrelated and, since they are both Gaussian random variables, independent.
For these reasons |R+(FD,τ)|2 and |R−(FD,τ)|2 are also independent.SinceR+(FD,τ) and R−(FD,τ) are Gaussian random variables, |R+(FD,τ)|2 and |R−(FD,τ)|2 are χ2 distributed with two degrees of freedom. When
the signal is absent, or the local replicas are not aligned with the received
signal, then |R+(FD,τ)|2 and |R−(FD,τ)|2 are both central χ2 random variables [14] and the false alarm
probability of Sml(FD,τ) is(31)Pfaml(β)=1−[1−exp{−β4σn2}]2.Equation (31) has been obtained
by substituting the cumulative density function (cdf) of central χ2 random variables [14] into (29). It can be noted
that the exponential in (31) depends on 4σn2 instead of 2σn2 as for (17). This is due to the fact that the
equivalent code (24) has twice the power of the single pilot and data codes.When the Galileo signal is present and correctly
aligned with the local replica,|R+(FD,τ)|2 and |R−(FD,τ)|2 are no more central χ2 random variables, and the noncentrality
parameters λ+ and λ− have to be determined. λ+ and λ− can be obtained by determining the mean of R+(FD,τ) and R−(FD,τ) under the hypothesis that the local equivalent
code matches or not the navigation bit. In particular we have(32)E[R+(FD,τ)]=E{RB(FD,τ)+RC(FD,τ)}={2E{RB(FD,τ)}=Csin(πNΔF)(πNΔF)K(Δτ)≈CifdC[n]dB[n]=1,0ifdC[n]dB[n]=−1and similarly(33)E[R−(FD,τ)]=E{RB(FD,τ)−RC(FD,τ)}={2E{RB(FD,τ)}=Csin(πNΔF)(πNΔF)K(Δτ)≈CifdC[n]dB[n]=−1,0ifdC[n]dB[n]=1.From these considerations it
emerges that the decision variable Sml(FD,τ),
under the hypothesis of presence of signal and correct alignment, is given by
the maximum between a central χ2 and a noncentral χ2 random variables with two degrees of freedom.
The noncentrality parameter of the noncentral χ2 random variable is given by(34)λ=Csin2(πNΔF)(πNΔF)2K2(Δτ)≈C.Given these premises it is
finally possible to express the detection probability(35)Pdml(β)=1−[1−exp{−β4σn2}][1−Q1(λ2σn2,β2σn2)]≈1−[1−exp{−β4σn2}][1−Q1(2CNN0fs,2βNN0fs)],where Q1(⋅,⋅) is the Marcum Q function of order 1.


## 4. Simulation Analysis

In order to
validate the theoretical analysis developed in the previous sections, the
Galileo E1 OS has been simulated according to the parameters reported in
[1] and acquired
according to the two methods considered in this work. The simulation parameters
are reported in Table 1. More in detail
a GNSS signal has been simulated according to model (11). The spreading codes
from [1] has been used
to generate the Galileo E1 OS signals, and the acquisition systems depicted in
Figures 1 and 2 have been implemented in order to evaluate the decision
statistics (12) and (25). The false alarm and detection probabilities have been
estimated by verifying if the decision variables, under both (H1) and (H0) hypotheses, passed or not the decision
threshold obtained by inverting (20) and (31). 2⋅105 trials have been used for the estimation
process. Frequency and delay errors have not been simulated and, for the
evaluation of the detection probability, the incoming signal has been
considered perfectly aligned with local replica. These errors were not
considered in the theoretical model and thus they have not been considered in
the simulation scheme as well. The impact of frequency and delay residual
errors has been extensively studied in [15] and can be easily included in the models developed in
previous sections. This topic is however out of the scope of this paper.Table 1
Simulation
parameters.
ParameterValueSampling frequencyfs=4.092 MHzIntermediate frequencyfIF=fs4=1.023 MHzReceiver bandwidthBIF=fs2=2.046 MHzCode rate1.023⋅106 chip/sCode length4092 chipIntegration timeNTs=4 msThe simulations results are reported in Figures3, 4, and 5, where the
cases of C/N0=25,30,
and 35 dB-Hz have been considered. These C/N0 values have been chosen since they represent
marginally strong and weak signal conditions. The probabilities evaluated by
Monte Carlo simulations match the theoretical models developed in previous
sections and the theoretical and simulated ROCs overlap in all the considered
cases. Simulations effectively support the theoretical models proving the
effectiveness of the analysis developed in Section 3.Figure 3
Simulated and theoretical ROCs for noncoherent and
coherent acquisition systems.C/N0=25 dB-Hz.Figure 4
Simulated and theoretical ROCs for noncoherent and coherent acquisition
systems.C/N0=30 dB-Hz.Figure 5
Simulated and theoretical ROCs for noncoherent and coherent acquisition
systems.C/N0=35 dB-Hz.It can be noted, from Figures3, 4, and 5, that the
coherent acquisition algorithm always outperforms the noncoherent combining
strategy. Moreover, it can be observed that the two methods tend to have
similar performance for low C/N0.
This performance degradation of coherent combing with sign recovery is probably
due to the fact that, for low C/N0,
the estimation of the relative phase between data and pilot becomes unreliable
preventing an effective combination of the two channels transmitted for the
Galileo E1 OS.


## 5. Conclusions

In this paper,
two different acquisition strategies for the acquisition of the Galileo E1 OS
signals have been considered and deeply analyzed. The first strategy, the
noncoherent combining, is from the literature whereas the analysis of the
second one, coherent combining with sign recovery, is new and represents the
innovative contribution of this paper. An analytical model for the false alarm
and detection probabilities of both algorithms has been derived and Monte Carlo
simulations have been used for supporting the theoretical analysis. From the
theoretical model and simulations it is shown that the coherent combining
algorithm outperforms the noncoherent combing proving the effectiveness of the
proposed method. Thus, provided that both algorithms require similar
computational loads, the coherent combining acquisition algorithm results in
preferable to noncoherent combining and should be adopted for the joint
acquisition of the E1 OS data and pilot channels.



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*Source: 738183-2008-05-12.xml*