1. A Novel Approach to the Performance
Evaluation of an Arctangent Discriminator for
Phase Locked Loop and application to the carrier
tracking of the Ionospheric Scintillation
Paolo Crosta, Thales Alenia Space Italia (Milan)
BIOGRAPHY
Paolo Crosta graduated in Telecommunications
Engineering at the University of Pisa, Italy in 2002.
In 2004 he joined Laben (now Thales Alenia Space
Italia) where he works in the industrial unit ground
as a SW Engineer, acquiring competences in the
digital signal processing of GNSS receivers for
space and ground applications. He has been
involved in the design of the SW architecture of the
ESA Non-PRS Galileo Receiver (Galileo Receiver
Chain project, part of the Ground Monitoring
Segment GMS). He has also worked on the design
of the Galileo receivers developed in the frame of
the European projects, GARDA and GIRASOLE
and for the development of a Fully SW radio
receiver in the frame of SWAN.
1 INTRODUCTION
The upcoming GNSS systems, Galileo and
modernized GPS, solve the problems related to the
presence of data bits on the GPS signals by
introducing a pilot channel in quadrature with the
data channel and without any navigation data
modulation. Considering the effect on the phase
locked loop (PLL), this new channel allows the PLL
to use discriminators insensitive to phase jumps.
Among the possible choices, the extended
arctangent discriminator (atan2) shows the widest
linear tracking range and its performances are
hereafter studied. The main objective of this paper is
to describe a novel approach to the performance
evaluation of the atan2 discriminator based on the
derivation of a closed-form expression of the
probability distribution at the output of the
discriminator. In terms of thermal noise analysis,
this result allows a computation of the standard
deviation of the discriminator output as function of
the CN0 without the need of any Monte Carlo
simulation. The advantage of this approach is that it
provides an analytical way to compute the error
variance and it differs from the traditional approach
of considering similarity of atan with other
discriminators, assumption which is valid only
under particular conditions. The same approach is
also used to derive a closed form expression for the
slope of the discriminator S-curve at different signal
to noise ratios. As a practical example of application
of the above theory, the effects of a ionospheric
amplitude and phase scintillation on the tracking
error variance at the PLL output are analyzed. This
analysis has been developed in the framework of the
NonPRS Galileo Receiver Chain (GRCN) project
using the complete Galileo receiver realized for the
Galileo Mission Segment (GMS). The preliminary
testing activity on the receiver has been carried out
using a Galileo SIS Spirent simulator (GNSS 7800)
with an high-rate control of the phase and amplitude
of the signal generated at radio frequency. The time-
series of phase and amplitude variations due to
scintillation have been computed with different
values of ionospheric amplitude (S4) using the
GISM (Global Ionospheric Scintillation Model)
simulator.
2 ANALYTICAL DERIVATION OF THE
PDF OF THE EXTENDED ARC-
TANGENT DISCRIMINATOR
The extended arctangent discriminator (atan2) is the
maximum likelihood estimator for the phase
estimation and it is defined as:
( )IQ,arctan2=ϕ (1)

2. where I and Q are the In-Phase and In-Quadrature
correlation values as shown in the block diagram
below describing the signal processing architecture
of a typical GNSS receiver.
Figure 1: Block Diagram of the receiver PLL
Due to the non-linearity of the arctangent operator, a
theoretical expression of the phase error variance is
not used in literature and in general, it is assumed
that the equation applicable to the dot-product
discriminator also suits the arctangent discriminator
well [1].
The variance of the dot-product discriminator is
commonly known and it is the following:






+=
TNCTNC 02
1
1
2
12
0
δφσ
(2)
where T is the pre-detection integration time and
CN0 is the signal to noise ratio. The validity of this
equation has been proven for practical purposes by
statistically computing the discriminator output
variance under Gaussian noise using Monte Carlo
simulations.
The main objective of this paper is to find an
alternative way to compute the phase error variance
of the arctangent discriminator as function of the
CN0 without the need of any Monte Carlo
simulation.
The first step is to start with the computation of the
cumulative distribution function, cdf, at the output
of the atan discriminator. The cdf is a function F(x)
of a random variable, X, defined for a number x by:
( ) ∫
−∞
=≤=
x
dssfxXPxF
,0
)()(
(3)
where f(s) is the probability density function or pdf.
Assuming that the PLL is truly tracking and the
frequency errors and the code delay errors are small,
the inputs of the discriminator (I&Q prompt
correlators) can be represented by two simple
independent Gaussian variables: the In-Quadrature
(Q) correlation with zero mean and variance σ2
and
the In-phase correlation with mean A and the same
variance. The values of A and σ2
are both linked to
the signal to noise ratio by the following expression:
TNCA 02=σ (4)
In all the following equations, without lacking of
generality, the thermal noise variance (σ2
) is set to a
unitary value in order to consider A as a normalized
signal power.
Given a phase error φ0 between –π/2 e +π/2 radians,
the geometric representation of the I and Q
correlations is shown in the following figure.
Figure 2: I&Q correlations in the complex plane
A phase angle equal to φ0 can be obtained with a
couple of I and Q equal to (I0, Q0), where I0>0 due
to the assumed range value for φ0 and
( )000 tan ϕIQ = (5)
Note that I0 is not uniquely determined but it can be
any value greater than 0.
Computing now the probability of having a phase
angle φ greater than φ0 conditioned to assuming an I0
value on the In-phase correlation, the following
equation is obtained:
{ } ( ){ }
{ }2
)(tan 00
0
0000
πϕ
ϕϕϕ
>+
+∫ =>=>
+∞
P
dIIfIIIQPP
(6)
where
( ) 2
0
2
0
2
1
)( AI
eIf −−
=
π
(7)
and

3. ( ){ }
( )
( ) )2tan(2121
2
1
tan
00
tan
0
2
000
00 2
ϕ
π
ϕ
ϕ
Ierf
dzeIIIQP
I
z
−=
∫==> −
(8)
with erf(.) representing the “error function”
encountered in integrating the normal distribution
and equal to
dtezerf
z
t
∫= −
0
22
)(
π
(9)
In order to obtain the equation for the probability
density function at the output of the atan2
discriminator (fA) is sufficient to apply the following
relationship:
( ) { } 000 ϕϕϕϕ ddPf A >−= (10)
The derivative operator deletes completely the
second term in the addition of (6) because
{ }2πϕ >P does not depend on φ0. The derivation of
the first term, instead, yields to the expression:
( ) zdzeef
Azz
A
2
)(
0
2
)(tan
0
20
2
0
22
)(cos2
1
−
−
∞−
−
∫=
ϕ
ϕπ
ϕ
(11)
This equation can be simplified and the integral
solved, just by noting that the exponential terms can
be written as a single term with the following
exponent:
)(sin
2)(cos2
))(cos(
0
2
2
0
2
2
0
2
ϕ
ϕ
ϕ AAz
−
−
−
(12)
and then obtaining:
( ) ∫=
∞
−
−
−
0
)(cos2
))(cos(
0
2
)(sin
2
0
0
2
2
0
2
0
2
2
)(cos2
zdze
e
f
AzA
A
ϕ
ϕϕ
ϕπ
ϕ
(13)
Finally, for the last step just a variable substitution
in the integral argument is needed to change the
previous integral in a simple definite integral of a
negative exponential function. This yields to the
final expression of the probability density function
at the output of the extended arctangent
discriminator:
( ) ( )
π
ϕ
ϕ
π
ϕ
ϕ
2
)
2
cos
(1
cos
22
2
0
2
sin
00
2
0
22
A
A
A
eA
erf
e
A
f
−
−
+





+
=
(14)
This equation can be extended to the complete range
of the phase errors ([-π, +π]) by applying the same
method used for the angles in [–π/2, +π/2] to the
angles in [-π, -π/2] and in [+π/2, +π].
The pdf described by (14) can be approximated by a
Gaussian pdf only at high CN0. When the signal to
noise ratio increases, the value of A increases and
φ0 tends to assume values close to zero with an high
likelihood: under this assumptions the equation (14)
becomes
( ) 2
0
2
0
2
2
ϕ
π
ϕ
A
A e
A
f
−
=
(15)
that is a Gaussian distribution with zero mean and
variance
TNCA 0
2
2
2
11
==ϕσ
(16)
that is the same of the equation (2), excluding the
squaring loss term, i.e. the second term in the
bracket of the equation (2).
Figure 3 shows the histogram of the results
obtained by bit-true simulations for the output of the
atan2 discriminator with an integration time (T) of
100 ms and a CN0 of 25 dBHz.
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
0
0.5
1
1.5
2
2.5
3
theta [rad]
Probability
Simulation Value
Theor. Formula
Figure 3: histogram of atan2 discriminator
output with CN0 25 dBHz and T 100 ms

4. 5000
10000
15000
20000
30
210
60
240
90
270
120
300
150
330
180 0
Figure 4: angle histogram of atan2 discriminator
output with CN0 25 dBHz and T 100 ms
The Figure 4 is relative to the same simulation but
the representation is different: the same phase angles
are shown in an angle histogram, which is just a
polar plot depicting the distribution of values
grouped according to their numeric range.
In case of low CN0 values, the Gaussian
approximation is not still valid. Due to the extended
arctangent discriminator’s output inherent
boundaries, at very low CN0 values the distribution
becomes non-Gaussian and it tends to a uniform
distribution in the interval [-π, +π] as A tends to 0.
With the same integration time of 100 ms, but
lowering the CN0 at 5 dBHz, this effect is already
visible, as shown in the Figure 5 and Figure 6. In
particular the angle histogram in the second figure
shows immediately a widening of the angle
histogram that is tending to the circular shape of the
uniform distribution.
-4 -3 -2 -1 0 1 2 3 4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
theta [rad]
Probability
Simulation Value
Theor. Formula
Gaussian pdf
Figure 5: histogram of atan2 discriminator
output with CN0 5 dBHz and T 100 ms
500
1000
1500
2000
2500
30
210
60
240
90
270
120
300
150
330
180 0
Figure 6: angle histogram of atan2 discriminator
output with CN0 5 dBHz and T 100 ms
3 DEPENDENCY OF THE S-CURVE
DISCRIMINATOR SLOPE ON THE CN0
As described in [2], the CN0 value in
correspondence of which the distribution becomes
non-Gaussian is also the value when the arctangent
discriminator’s slope at the origin starts to decrease.
This behaviour is better explained in the following
figure.
-4 -3 -2 -1 0 1 2 3 4
-3
-2
-1
0
1
2
3
Input phase error [rad]
Discriminatoroutput[rad]
Atan2 Discriminator S-curve T 100 ms
25
20
15
10
5
Figure 7: S-curve slope in the origin at different
CN0
In particular, in the Figure 7, the mean value of the
atan2 discriminator output is shown as function of
the input phase error for different CN0 values
ranging from 25 up to 5 dBHz with an integration
time of 100 ms. The curve whose slope in the origin
considerably diverges from the unitary value is the
one at 5 dBHz i.e. the CN0 at which it has been
previously proved that the distribution cannot be
approximated with the Gaussian one.
The effect of the reduced slope can be considerable
in the carrier phase tracking loop: it corresponds to a
change in the open-loop gain that leads to a change

5. of the poles in the closed loop transfer function and
it can cause also a possible instability, depending on
the order of the loop.
In order to detail more the relationship between S-
curve slope and CN0, it can be taken the definition
of the derivative of the discriminator S-curve in the
origin as
( ) ( )[ ]
ε
ε
ε
0
lim
0
Φ−Φ
→
E (17)
where E(.) is the mean value linear operator and Ф
the output of the atan2 discriminator.
This equation can be further simplified by using the
definition of the mean value as first moment of the
distribution at the output of the discriminator and
the symmetry properties of the pdf, leading to
( ) ϕεϕϕ
ε
π
πε
dfASlope A +⋅−= ∫
−→
1
lim)(
0
(18)
Finally, by substituting fA with the equation (14),
applying the rule of the integration by parts and the
normalization property of the pdf, the equation (18)
can be rewritten as follows:
( )ππ AfASlope 21)( −= (19)












−+−=
−
2
1
2
1)( 2
2
A
erfAeASlope
A
π
(20)
( )[ ]TCNerfTCNeSlope TCN
00 11 0 −+−= −
π (21)
The previous formula perfectly matches with the
results obtained in [2]: in particular, the results
obtained by Monte Carlo simulations and the
analytical expression of the equation (21) are
completely in line with the plot shown in the figure
4.4 of [2], as depicted in the following figure.
15 20 25 30 35 40
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
CN0
[dBHz]
DiscriminatorSlope
Atan2 Slope with T 4 ms
Figure 8: Atan2 Slope as function of the CN0
In the previous plot, the range of the CN0 values
have been changed to [15, 40] dBHz, assuming a 4
ms integration time in order to have a common term
of comparison with [2].
The benefits of computing an analytical expression
for this slope’s variation seem to be very promising.
For example, if the digital coefficients of the loops
are designed using any sort of stability criteria, this
equation can be useful to translate the stability
margin of the open loop gain into a minimum CN0
threshold. Or even, it could be used to dynamically
equalize the discriminator slope’s variation by
estimating the CN0 value and computing the
expected change in the slope. These aspects will be
gone into thoroughly in future papers.
4 NUMERICAL INTEGRATION FOR THE
VARIANCE AT DISCRIMINATOR
OUTPUT
Concerning the computation of the variance at the
discriminator output, it can be evaluated by just
applying the definition of variance to the analytical
expression of the probability distribution function
defined in the equation (14).
Unluckily the derivation of a closed-form
expression for the variance is a very tough road to
hold, because the following integral seems to be
unsolvable.
ϕ
ϕ
ϕϕ
π
π
σ
π
π
ϕ
ϕ
d
A
erfe
A
e
A
A












+
⋅+=
∫
−
−
−
2
cos
1cos
223
2
sin
2
2
2
2
22
2 (22)
With A tending to 0, the previous equation tends to
π2
/3 that is the variance of a random variable
uniformly distributed in the range [-π, +π]. Whilst
with A tending to +∞, the asymptotic behaviour is
the one obtained with the Gaussian approximation
as previously described in the equation (16).
Knowing the closed-form expression for the
probability density function at the output of the
atan2 discriminator is still possible to compute the
variance by numerically integrating the equation
(22). The advantage of this approach is that it
provides a relatively fast way to compute the error
variance without Monte Carlo simulations and it
differs from the traditional approach of considering
similarity of the arctangent discriminator with other
discriminators, assumption that is valid only under
particular conditions. This discriminator variance
for typical tracking loops is directly proportional to
the phase tracking error variance that can be written
as follows (see [1]):

6. ( ) [ ]222
, 5.012 radBTBT discT σσϕ −=
(23)
where B is the PLL bandwidth and T the integration
time.
In case of coherent discriminator or dot-product
discriminator the previous equation leads
respectively to:
( ) [ ]2
0
2
,
5.01
rad
N
C
BTB
T
−
=ϕσ (24)
for the coherent one and to:
( ) [ ]2
00
2
,
2
1
1
5.01
rad
N
CT
N
C
BTB
T










+
−
=ϕσ
(25)
for the dot-product with the squaring loss term.
The variance of the atan2 discriminator has been
computed by numerically integrating the expected
value of the squared phase in the domain [-π, +π].
The results are shown in the following figure.
Figure 9: Comparison of Discriminator output
variances with T 100 ms
An upper bound limit for this discriminator variance
is the variance of the dot-product discriminator that
differs from the coherent discriminator only for the
squaring loss term. This squaring loss term does
lead to a very huge effect in the phase error that
makes the variance diverge for low CN0s. This
difference could be completely negligible specially
with a long integration period like 100 ms because it
is visible only at very low CN0s where the PLL is
supposed to lose the lock. As a practical example of
application where this effect is no more negligible is
the computation of the carrier tracking thermal noise
jitter variance in presence of strong amplitude
scintillations as described hereafter.
5 ATAN2 DISCRIMINATOR ROBUSTNESS
TO PHASE AND AMPLITUDE
SCINTILLATION
The effects of the Ionospheric Scintillation
occurring when the radiowave traverses drifting
ionospheric irregularities are essentially fading and
phase fluctuations of the received signal that can
vary widely with frequency, magnetic and solar
activity, time of day, season and latitude ([3]).
The signal is modulated by the passage through the
irregularities so that the level instantaneously both
increases and decreases. The amplitude, phase and
angle of arrival of the signal will change during
periods of scintillation. The intensity of the
amplitude scintillation is characterized by the
variance in received power with the S4 index
commonly used for intensity scintillation and
defined as the square root of the variance of
received power divided by the mean value of the
received power. Attempts made to model the
observed amplitude probability density function
have led to a model distribution function based upon
the use of a Nakagami-m distribution with m = S4
-2
.
At the same time, the phase scintillation is modelled
in terms of a power spectral density. In a log-log
scale, the phase error spectral density has a linear
trend whose parameters are the slope (p) and the
spectral strength (T). This is confirmed by the
analytical model of phase error spectrum, which is
described in [4] as:
( )
( ) 2/22
0
p
ff
T
fS P
+
=ϕ
(26)
where T is the spectral strength corresponding to the
power at 1 Hz, f is the frequency and f0 is a cut-off
frequency used to prevent divergence in PSD
integration at low frequencies.
In particular, the effect of the ionospheric
scintillation on the carrier tracking loops is twofold:
• the amplitude fluctuations will model the
trend of the thermal noise jitter in the time
introducing a dependency of the thermal
jitter from the S4 parameter
• the phase scintillations will be filtered by
the closed loop transfer function of the PLL
introducing another jitter completely
characterized by the parameters T and p
used in the equation (26).
Considering also the contribution of the local
oscillator, the tracking error variance at the output of
the PLL can be expressed as the sum of the
following three contributions.

7. 2
,
2
,
2
,
2
oscTS ϕϕϕϕ σσσσ ε
++= (27)
The three components indicates respectively the
scintillation error, the thermal noise error and
oscillator phase noise contributions.
In literature, it is shown [4] that the thermal noise in
presence of ionospheric scintillation is given by the
following expressions (closed form valid only for
S4 <0.707)
( )
( )2
40
2
402
,
1
212
1
1
SCN
SCNT
Bn
T
−








−
+
=ϕσ
(28)
where T is the pre-detection integration time
(assumed here T = 100 ms), CN0 is the signal to
noise ratio, B is the loop bandwidth. Setting S4 equal
to zero, the previous equation is very similar to the
equation (2) that does not take into account the real
atan2 discriminator variance previously computed.
The limitation on the maximum S4 in the Thermal
Noise Equation (28) is not corresponding to any
physic reason but it is mainly due to the method
used for its computation and the impact of the
discriminator output on this method.
With an high value of S4 (S4>0.707), indeed, the
probability of having amplitude fades of tenths of
dBs becomes appreciable as shown by the
Nakagami-m distribution depicted in the following
figure with S4 = 0.9.
-25 -20 -15 -10 -5 0 5 10
0
0.2
0.4
0.6
0.8
1
Amplitude fluctuation [dB]
Nakagami distribution with S4 = 0.9
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.2
0.4
0.6
0.8
1
Amplitude fluctuation linear
Nakagami distribution with S4 = 0.9
Figure 10: Amplitude Fluctuations distribution
(S4 0.9)
With such losses, the values of CN0 temporarily
seen at the receiver correlators can easily reach the
limit below which it has been proved that the
similarity of the arctangent discriminator with other
discriminators is no longer applicable.
In order to compute the variance also in case of high
S4 values and the atan2 discriminator is necessary to
apply the same method used to derive the equation
(28) at the variance of the atan2 discriminator
computed by numeric integration. This method is
based on conditional probabilities and it consists in
using the Nakagami distribution to weigh the
thermal noise variance whose values depend on the
signal amplitude. So the effect of the scintillation on
the carrier tracking thermal noise jitter variance is
obtained assuming:
( ) ( ) [ ]2
0
22
, raddAApA sssT ∫
∞
= ϕϕ σσ
(29)
where As is the amplitude of the scintillation with
the following Nakagami distribution p(As):
( )
( )
0,
2
2
12
≥
ΩΓ
= Ω
−−
s
mA
m
m
s
m
Ae
m
Am
Asp
s (30)
with [ ]2
sAE=Ω and 2
41 Sm = .
The results of equation (29), obtained by numerical
integration, are shown in the following figures: the
values are continuous and monotonically increasing
as S4 increases. If compared with the plot related to
the equation (28), they perfectly match for each CN0
in case of low S4 while as S4 is close to 0.707 the
equation (28) tends to greater values or even to
negative ones for S4>0.707.
10
15
20
25
30
0.2
0.4
0.6
0.8
1
-10
0
10
20
30
CN0 dBHzS4
rad2
10
15
20
25
30
0.2
0.4
0.6
0.8
1
0
0.5
1
1.5
2
2.5
3
CN0 dBHzS4
rad2
Figure 11: Tracking error variance at the PLL
output for different S4 and CN0: approximated
equation (up), numeric integration (down)

8. The loop parameters used in the previous
simulations are an integration time of 100 ms and a
loop bandwidth of 5 Hz.
This comparison shows the possibility of using this
alternative method to compute the tracking error
jitter in case of amplitude scintillations in order to
extend the validity of this analysis also at high S4
values where the other techniques present in
literature seem to be not valid.
6 GRC-N RECEIVER: PRELIMINARY
SCINTILLATION RESULTS
In order to check the results obtained by the
previously described method, a preliminary testing
campaign has been carried out on the GRCN
receiver, a complete Galileo receiver receiver
embedded into the Galileo Ground Stations of the
Galileo Mission Segment [5]. The same analysis
described in this paper has been developed and used
in the framework of the NonPRS Galileo Receiver
Chain (GRCN) project to characterize the receiver
performances and to satisfy the specifications of the
Galileo Mission Segment (GMS) specially in
scenarios with extreme conditions of ionospheric
scintillation.
The preliminary testing activity on the receiver has
been carried out using a Galileo SIS Spirent
simulator (GNSS 7800) with a real-time control of
the phase and amplitude of the signal generated at
radio frequency. The time-series of phase and
amplitude variations due to scintillation have been
sampled at 100 ms and computed with different
values of ionospheric amplitude (S4) using the
GISM (Global Ionospheric Scintillation Model)
developed at IEEA [6]. The GISM simulator is able
to model second order propagation effects such as
radio wave scintillations using the Multiple Phase
Screen technique (MPS). It consists in a resolution
of the Parabolic Equation (PE) for a medium
divided into successive layers, each of them acting
as a phase screen. The simulator provides the
statistical characteristics of the transmitted signals,
in particular the scintillation index, the fade
durations and the cumulative probability of the
signal. An example of Amplitude and Phase
Scintillations generated by GISM with an S4 equal
to 0.8 are shown in the following figure.
0 5 10 15 20 25 30 35 40 45 50
-30
-20
-10
0
10
Power[dB]
Time s
Scintillation S4
0.8
0 5 10 15 20 25 30 35 40 45 50
-100
0
100
200
Phase[degree]
Time s
Figure 12: Phase and Amplitude Scintillation
Time series generated by GISM simulator with
S4 0.8
In order to distinguish the effects of Amplitude
Scintillation from the ones of Phase scintillation, the
simulator has been programmed to apply Amplitude
Scintillation only at a specific set of visible satellites
and Phase Scintillation only at the other ones.
The following results have been obtained setting the
receiver carrier tracking loop, a third order digital
PLL, with user-reconfigurable parameters, to an
integration time of 100 ms and bandwidth of 5 Hz:
obviously the used discriminator is an extended
arctangent discriminator.
The effects of the different Scintillations (Phase and
Amplitude) are visible just looking at the I and Q
correlations.
In case of only Phase Scintillations, as shown in
Figure 13, the instantaneous power given by the
sum of the squared I and Q is almost constant during
the simulation and the effect of the Scintillation is
more visible in the Q correlation (red curve) than in
the I correlation (green curve).
180 200 220 240 260 280 300 320 340 360
-2
-1
0
1
2
3
4
x 10
5 Iprompt
- red Qprompt
- green
Correlationvalue
Figure 13: IQ correlations in case of Phase
Scintillation S4 0.6 and CN0 40 dBHz

9. This behavior is mainly caused by the increased
residual phase errors that make a part of the received
signal power move on the Q component with an
angle rotation.
In case of only Amplitude Scintillations, that are the
only ones affecting the thermal noise error
component of the total phase jitter, the correlations
show a completely different behavior as shown in
the following figures for S4 values of 0.8 and 0.3.
1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500
-5
0
5
10
15
20
25
x 10
4 Iprompt
- red Qprompt
- green
Correlationvalue
Figure 14: IQ with S4 0.8 and CN0 30 dBHz
In particular, comparing the Figure 14 with the
Amplitude Time series shown in Figure 12, the
same amplitude fades’ dynamics are visible in the I
correlation values because they are expressions of a
real variation in the received instantaneous power.
1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Carrier Error
rad
Figure 15: output atan2 S4 0.8 CN0 30 dBHz
Moreover, the output of the atan2 discriminator,
depicted in the Figure 15 for a S4 of 0.8, shows an
increased variance in correspondence of the
amplitude fades of the Figure 14.
The same behavior is still visible in the Figure 16
and the Figure 17 with a reduced effect due to the
reduced S4 (0.3).
The modulation of the discriminator variance with
the received signal power is the main basis of the
method used in the equation (29).
0 100 200 300 400 500 600
-5
0
5
10
15
20
x 10
4 Iprompt
- red Qprompt
- green
Correlationvalue
Figure 16: IQ with S4 0.3 and CN0 30 dBHz
0 100 200 300 400 500 600
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Carrier Error
rad
Figure 17: output atan2 S4 0.3 CN0 30 dBHz
To validate the results of the equation (29) a set of
different tests has been carried out, computing the
carrier phase tracking variance for CN0 values
ranging from 30 dBHz up to 40 dBHz with 2 dB
steps and for three S4 values (0.3, 0.6, 0.8)
representing of different Scintillation conditions.
The tests have been carried out only on the E5b
receiver board, just due to time constraints: the
behavior in all the other Galileo carriers will be
verified during the final testing session.

10. 31 32 33 34 35 36 37 38 39 40
2
4
6
8
10
12
14
x 10
-3
CN0 dBHz
rad2
PLL tracking variance with S4
= 0.3, 0.6, 0.8
Figure 18: Carrier Phase Tracking Variance:
Test results (red squared markers) and
theoretical curves (blue) for S4 = [0.3, 0.6, 0.8]
As shown in Figure 18, the test results (red squared
markers) show a good match with the theoretical
ones specially with the greatest value of S4.
7 CONCLUSIONS
A mathematical study of the extended arctangent
discriminator for a PLL carrier tracking loop has
been presented. The first direct result of this
analysis is the computation of a closed-form
equation for the probability density function at the
output of the discriminator. This equation is not just
a mathematical play but it has been applied to
analyse thoroughly many aspects of the arctangent
discriminator.
An immediate application has been the derivation of
a numerical integration method to calculate the
exact variance at the output of the atan2
discriminator.
After that, a closed-form equation for the slope of
the S-curve of the atan2 discriminator has been
computed and it seems very promising specially for
the tracking of signals at low CN0 and it will be the
subject of further publications.
Finally, as a practical example of application, a
method to compute the effect of amplitude phase
scintillations on the tracking phase jitter has been
shown. This analysis shows a good agreement with
literature in case of low or medium ionospheric
amplitude but it allows computing variance values
also in case of low CN0 values and high S4 where
other analyses are limited or not valid. All the
results have been backed by Monte Carlo
simulations and by the preliminary testing activity
on the Galileo GRCN receiver.
REFERENCES
[1] Van Dierendonck, A.J. (1997), GPS Receivers in
Global Positioning System: Theory and Application
Volume I, Progress in Astronautics and Aeronautics
Volume 164, AIAA, pp. 329-408.
[2] Julien, O. (2005), Design of Galileo L1F Receiver
Tracking Loops, PhD Thesis, published as UCGE
Report No. 20227, Department of Geomatics
Engineering, The University of Calgary.
[3] Aarons, J.; Whitney, H.E.; Allen, R.S., Global
morphology of ionospheric scintillations,
Proceedings of the IEEE, Volume 59, Issue 2, Feb.
1971 Page(s): 159 – 172
[4] Conker R., El-Arini M.B., Hegarty C.J., Hsiao,
T.Y., Modelling the Effects of Ionospheric
Scintillation on GPS/SBAS Availability, The MITRE
Corporation, August 2000 MP 00W0000179
MITRE PRODUCT
[5] G. Franzoni, G. Pinelli et al., Galileo receiver chain
- The Non-PRS ground reference receiver, ENC
2008
[6] Béniguel,Y.(2002),Global Ionospheric Propagation
Model (GIM): a propagation model for
scintillations of transmitted signals, Radio Sci.,
May/June 2002.