# Design of Short Synchronization Codes for Use in Future  GNSS System

Authors: Surendran K. Shanmugam; Cécile Mongrédien; John Nielsen; Gérard Lachapelle
Journal: International Journal of Navigation and Observation
 (2008)

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## Abstract

The prolific growth in civilian GNSS market initiated the modernization of GPS and the
GLONASS systems in addition to the potential deployment of Galileo and Compass GNSS system.
The modernization efforts include numerous signal structure innovations to ensure better
performances over legacy GNSS system. The adoption of secondary short synchronization codes
is one among these innovations that play an important role in spectral separation, bit synchronization,
and narrowband interference protection. In this paper, we present a short synchronization
code design based on the optimization of judiciously selected performance criteria. The new
synchronization codes were obtained for lengths up to 30 bits through exhaustive search and
are characterized by optimal periodic correlation. More importantly, the presence of better
synchronization codes over standardized GPS and Galileo codes corroborates the benefits and
the need for short synchronization code design.


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## Body

## 1. Introduction

The legacy global positioning system (GPS) has
performed well beyond initial expectations in the past but faces stern
impediments in the view point of new civilian GPS applications. Several
initiatives were launched during the last decade to satisfy the demands of
these new civilian applications. Consequently, these efforts led to the birth
of second-generation global navigation satellite systems (GNSSs). These efforts
include the modernization of legacy GPS and the restoration of Russian global
navigation satellite system (GLONASS). The Galileo system, a major European
initiative, is well positioned to benefit from the three decades of GPS and
GLONASS experience [1]. More recently, the GNSS community has witnessed yet
another highpoint with the launch of first medium earth orbit (MEO) satellite
of Chinese Compass GNSS system [2].A major milestone in the modernization initiative is
the inclusion of new civilian signals that will provide the benefits of
frequency diversity besides accuracy and availability improvements [3–5]. These new civilian signals
include numerous structural innovations that will provide the foremost benefit
to the civilian GNSS community. The modernized signals encompass key
innovations such as data-less channel, improved navigation data message format,
secondary spreading code structure, and new modulations schemes [6]. More specifically, both GPS and Galileo systems utilize
secondary short synchronization codes to accomplish
(i)
data symbol synchronization,(ii)
spectral separation,(iii)
narrowband interference protection.
For instance,
the use of short 10-bit and 20-bit Neuman-Hofman (NH) codes, in GPS L5 signals,
readily alleviates the issue of data symbol synchronization. Besides, the
different code period of NH10 and NH20 codes in the data and pilot channels readily
provides the necessary spectral separation. The secondary synchronization code
further enhances the correlation suppression performance of the primary
pseudorandom noise (PRN) code. Finally, it spreads the spectral lines
of primary PRN I5/Q5 codes thereby reducing the effect of narrowband
interference by another 13 dB [4]. The Galileo system also
utilizes short secondary synchronization codes of
various lengths to facilitate the aforementioned tasks [7]. Table 1 lists the secondary
code assignments and their lengths in GPS and Galileo systems.Table 1
Secondary code assignment in GPS and Galileo systems.
GPSGalileoSignal typeCode nameCode lengthSignal typecode nameCode lengthL5-DataNH1010E5a-DataCS2020L5-PilotNH2020E5a-PilotCS1001−50100L1C-PilotOC18001−2101800E1cCS2525E5b-DataCS44E5b-PilotCS10051−1001100E6cCS1001−50100The secondary synchronization codes are predominantly
memory codes except for the L1C, wherein the overlay codes were obtained
through truncated m-sequences (1–63) and gold sequences (64–210)
[8]. There exists a trade-off
between memory codes and codes that are obtained from linear feedback shift
register (LFSR) implementation. While the LFSR-based codes are appealing in the
view point of hardware implementation, they only exist for specific lengths.
The use of truncation technique can alleviate this issue at the expense of
inferior correlation properties. On the other hand, memory codes can be
obtained for any specific lengths with optimal correlation characteristics.
However, exhaustive search of optimal synchronization code becomes more
difficult with increasing code lengths.A limitation arising due to the usage of short
synchronization codes is the degradation in correlation suppression especially
in the presence of frequency errors. For instance, the vulnerability of NH20
code acquisition in the presence of Doppler uncertainties is discussed in
[9]. The isolation of
the main correlation peak to that of secondary peaks can degrade from the
nominal 14 dB to 4.8 dB level under worst case Doppler scenarios [10]. Under these conditions,
the NH code acquisition of weak GPS L5 signals becomes more difficult in the
presence of other strong GPS L5 signals. The existence of better
synchronization codes over standardized NH20 code was later reported in
[10], which is based
on the 20-bit synchronization code originally proposed in [11]. Under specific Doppler
conditions, the new 20-bit code (known as the Merten’s code) showed an
improvement of around 2 dB over the standardized NH20 code in terms of
correlation suppression [10]. However, the performance improvement achieved by the
Merten’s code corresponds to a specific Doppler scenario and thus does not
reflect the actual performance improvement under Doppler uncertainty.
Interestingly, the importance of spreading code selection for the Galileo GNSS
system and the corresponding measures was identified in [12]. Besides, it is also
desirable to develop optimal synchronization codes that offer better resistance
to residual Doppler errors. In this paper, we introduce relative performance
measures such as peak-to-side lobe ratio (PSLR) and integrated side lobe ratio
(ISLR) related to the design of periodic binary codes that are utilized in GNSS
system. More importantly, new optimal secondary synchronization codes were
obtained using these performance measures through exhaustive search for lengths
up to 30 bits. The merits of the proposed synchronization codes are also
compared with standardized codes using the same performance measures. Besides,
the association of the optimal synchronization codes with the systematic codes
such as Golay complementary codes is also established. Numerical simulations were
used to demonstrate the superior acquisition performance of the proposed short
synchronization codes over standardized codes under Doppler uncertainties in
terms of PSLR measure.The remainder of this paper is organized as follows.
In Section2, the advantage of optimal synchronization codes is further
established in the view point of GPS L5 NH code acquisition. More specifically,
we show the inadequacy of NH20 code in comparison to Merten’s 20-bit code under
different Doppler conditions. The relevant performance measures pertaining to
optimal binary periodic synchronization code are introduced in Section 3. The
binary-code search strategy and the various code construction methods are
detailed in Section 4. Besides, the merits of new synchronization codes are
compared with the standardized codes. Acquisition performance analysis is then
carried out in Section 5. The final concluding remarks are made in Section 6.


## 2. Need for Improved Synchronization Codes

An issue with
short synchronization codes is limited correlation suppression performance due
to their short code length. For instance, the correlation suppression
performance of NH20 code can be degraded by as much as 8 dB from the nominal
14 dB in the presence of Doppler uncertainty [9]. In [10], the authors reported a degradation of 9.2 dB for NH20
code under specific Doppler scenarios. To further illustrate this, the GPS L5
NH20 code and Galileo E1c CS25 code correlation
outputs for different Doppler bins are plotted in
Figure 1. The acquisition criterion in Figure 1 was obtained following the
analysis reported in [10]. For instance, the residual Doppler during the
acquisition of NH20 and CS25 code was set to 12 Hz; and this residual Doppler
was searched between 0 and 250 Hz in steps of 25 Hz.Superposition of secondary code correlation
outputs for various Doppler offsets. (LHS) GPS L5 NH20 code (RHS) Galileo E1c
CS25 code.
(a)(b)In Figure1, we can readily observe the degradation in
correlation main peak isolation for NH20 from the nominal 14 dB to 4.8 dB as
reported earlier in [10]. On the other hand, the Galileo E1c CS25 code
degraded from the nominal 18.4 dB down to 5.5 dB. The additional 3 dB degradation
in CS25 code acquisition can be attributed to the longer coherent integration
time (i.e., 25 millie seconds rather than 20 millie seconds) and nonzero
out-of-phase correlation in the original CS25 code. Accordingly, the
acquisition of weak GPS L5 signals or Galileo E1c signals can be hindered in
the presence of strong GPS L5 and Galileo E1c signals from other satellites.
While the correlation suppression performance can be improved with longer
length codes, judicial selection of synchronization codes can offer better
correlation suppression for the same code length. For example, in [10], the authors reported a
correlation suppression gain of around 2 dB for Merten’s code over standard NH20
code under specific Doppler scenario. The LHS plot in Figure 2 shows the
superposition of the Merten’s 20-bit synchronization code (M20) correlation
outputs for the same Doppler setting as in Figure 1. The RHS plot shows the
correlation suppression performance for the standardized NH20 and the M20 code
for various residual Doppler’s. The Doppler was searched between 0 to 250 Hz in
steps of 25 Hz.(LHS) Superposition of secondary code correlation outputs
for various Doppler offsets for M20 code (RHS) PSLR performance as a function of
residual Doppler.
(a)(b)The RHS plot in Figure2 readily shows the 2 dB
improvement accomplished by the M20 code over the standardized NH20 code for
the residual Doppler of 12 Hz. In other words, the M20 code can tolerate
another 10 Hz of residual Doppler for the same PSLR of 4.8 dB achieved by the
NH20 code. The M20 code resulted in an average performance improvement of
around 1.7 dB over the NH20 code for the range of residual Doppler’s. The
performance improvement in M20 code can readily be accredited to its better
correlation characteristic. For instance, the periodic correlation of the
different synchronization codes of length 20 (see Table 2) is summarized
below(1)RNH10={10,−2,2,−2,−2,2,−2,−2,2,−2},RNH20={20,0,0,0,0,0,−4,0,4,0,−4,0,4,0,−4,0,0,0,0,0},RCS20={20,0,0,0,0,0,4,0,−4,0,−4,0,−4,0,4,0,0,0,0,0},RM20={20,0,0,0,0,−4,0,−4,0,0,0,0,0,−4,0,−4,0,0,0,0}.The periodic correlation output
of the M20 code, RM20, has lesser number of out-of-phase correlation when
compared to both NH20 and CS20 codes. Accordingly, one can expect its code
acquisition performance to be superior even in the presence of residual
Doppler. It is worth emphasizing here that the NH10 and NH20 codes were not
obtained from exhaustive search, whereas the M20 code was obtained through
exhaustive search [11]. The very existence of the NH20, M20, and CS20
corroborates the presence of multiple solutions for
the code design problem. Besides, the search for periodic code is expected to
yield multiple solutions due to the existence of equivalence classes [13]. Hence, it is necessary to
obtain the binary codes that satisfy the optimal correlation characteristics
and select the best possible code judiciously using relevant performance
measures.Table 2
Optimal binary synchronization code search result.
Code lengthNumber of codesPSLR (dB)ISLR (dB)Code lengthNumber of codesPSLR (dB)ISLR (dB)48 (1)∞∞186,047 (168)19.12.4510 (1)143.21975 (2)22.610647 (8)9.50.9205,079 (45)143.1728 (2)16.94.1211,259 (30)16.94.2832 (2)622215,839 (360)20.82.99108 (8)9.51.72391 (2)27.31210360 (16)141.4241,535 (32)15.691144 (4)20.86.1257,000 (260)18.44.31296 (4)9.54.52631,615 (608)22.33.413104 (4)22.37.127775 (144)19.14.9141,791 (128)16.91.92823,743 (424)16.94.11559 (4)23.58293,247 (56)19.74.616255 (16)122.73035,039 (584)23.53.9172,175 (64)15.12.3


## 3. Optimal Synchronization Code—Figure of Merits

Better
synchronization code can be obtained by optimizing the corresponding
correlation characteristics of the individual codes. As we are dealing with binary codes of short period, the
optimization of correlation characteristics can be achieved in an exhaustive
fashion. It is however, necessary to derive performance measure or measures
that readily embody the correlation characteristics of a binary code [12]. The two important performance
measures pertaining to optimal synchronization codes are the peak-to-side lobe
ratio (PSLR) [14] and
the integrated side lobe ratio (ISLR) [15]. Besides, the synchronization
codes are also expected to be balanced for desirable spectral
characteristics. To define PSLR and ISLR, we first express the periodic
auto-correlation of the binary code of length N (i.e., x=[x0,x1,…,xN−1]), at shift i, as(2)R(i)=∑k=0N−1x(k)x(k−imodN),i=0,1,2,…,N−1,where x(k)∈{+1,−1} and mod is the modulo
operation. The PSLR for the binary code x(k) with the
periodic auto-correlation, R(i), is given by(3)PSLR(x)=R(i=0)2max|R(i≠0)|2,i=0,1,2,…,N−1.Maximizing the PSLR measure
minimizes the out-of-phase correlation that eventually aids in reducing false
acquisition. On the other side, ISLR measures the ratio of auto-correlation
main lobe (or peak) energy to its side lobe energy [15]. The ISLR of a binary code
is defined as(4)ISLR(x)=N22∑i=1N−1|R(i)|2,i=0,1,2,…,N−1.Maximizing the ISLR measure
readily limits the effect of out-of-phase correlation from all shifts. It will
be emphasized here that the maximization of ISLR often maximizes the PSLR
measure. Finally, the balanced property of a binary code is related to the mean
value of the code and is given by(5)μ(x)=1N∑k=0N−1x(k).For binary code sets design, as
in the case of OC1800 in GPS and CS100 in Galileo, it is also desirable to
minimize the mutual interference experienced by the individual codes from other
codes. Minimizing the magnitude of cross-correlation readily limits the effect
of mutual interference between any two codes. The mean square correlation (MSC) measure embodies this mutual correlation and can be utilized during
multiobjective synchronization code optimization. For any two codes xp(k) and xq(k) of length N pertaining to
the code set comprising of M unique codes,
the mutual correlation or the MSC is given by(6)MSC(p,q)=2∑i=0N−1|Rp,q(i)|2,p≠q,where Rp,q(i) is the periodic
cross-correlation between the codes xp(k) and xq(k), and is given
by(7)Rp,q(i)=∑k=0N−1xp(k)xq(k−imodN),i=0,1,2,…,N−1.The aforementioned mean square
correlation is closely related to the well-known total squared correlation
measure utilized in CDMA spread code optimization [16].


## 4. Optimum Code Search Results

For short code
length, the synchronization code optimization can be accomplished through
exhaustive search of binary codes with optimal correlation characteristics. The
developed exhaustive search technique utilized fast Fourier transform
(FFT)-based block processing and matrix manipulations to speed up the search
process. Both PSLR and ISLR were utilized for the objective maximization.
Optimal synchronization codes for lengths up to 30 were obtained through
exhaustive search. Interestingly, the search process yielded large number of
codes that were optimal based on the aforementioned performance measures. Table2 lists the number of codes alongside the unique solutions within braces, the
PSLR and ISLR values, respectively.The large number of codes arise from existence of the
equivalence classes due to the shift invariance property of the periodic codes
[13]. For example, the
code x(k), its negated version, its time reversed, or its
shifted version will be characterized by similar PSLR and ISLR measures. To
obtain unique solutions, the search technique discarded codes if their maximum
cross-correlation is equal to the code length. Accordingly, any two codes xp(k) and xq(k) satisfy the
following cross-correlation constraint are
considered unique:(8)max|Rp,q(i)|<N,i=0,1,2,…,N−1.Besides, the codes are
time-reversed and hence were tested for (8). While the balance property (i.e., μ(x)) was not
included during the code selection, its significance will be emphasized during
the acquisition performance analysis. In Table 2, the binary codes whose
lengths are similar to the standardized codes are highlighted in red. In
[17], the authors
theoretically established the optimal periodic correlation of a balanced binary code as(9)R(i)={0or−4Nmod4=0,2or−2Nmod4=2,i≠0.The periodic correlation of
optimal binary code for both odd and even lengths was further established in
[18], and is expressed
below(10)R(i)={0or4or−4Nmod4=0,1or−3Nmod4=1,2or−2Nmod4=2,−1or3Nmod4=3,i≠0.From (1) and (9), we see that
both NH10 and M20 possess optimal periodic correlation. Besides, the Galileo
CS25 code was also optimal as it satisfied the periodic correlation expressed
in (10). On the other hand, both NH20 and CS20 are not optimal in the view
point of (9), but can be considered optimal in terms of PSLR measure. The
inferior periodic correlation of NH20 does not come as a surprise as the
original NH codes were not obtained by exhaustive search [19]. It should be noted here
that all the secondary codes utilized in GPS and Galileo system are not
balanced (i.e., sum of individual code phases is not equal to zero) and thus
(9) cannot be applied in a strict sense, but indicates the conditions for
optimality. Numerical analysis later confirmed the fact that even unbalanced
binary code is characterized by periodic correlation as predicted in (9).All the binary codes obtained through exhaustive
search indeed satisfied the periodic correlation as expressed in (10) and
thereby asserting the optimality of the developed binary codes. The optimal
10-bit and 25-bit code obtained through exhaustive search resulted in similar
PSLR and ISLR performance measures to that of NH10 and CS25 codes in accordance
to (10). On the other hand, the 20-bit code obtained via exhaustive search
resulted in better ISLR performance even as the PSLR performance was the same.
Moreover, the new 20-bit code had similar correlation characteristics as that
of M20 code introduced earlier. In Table 2, we can also observe that odd-length
codes generally yielded better PSLR and ISLR performance. More specifically, the binary
codes for lengths
N=5,7,11,13,15 showed similar
PSLR and better ISLR, even when compared to twice their code lengths (i.e., N=10,14,22,26,30). The high
PSLR and ISLR values observed for code lengths
N=5,7,11,13,15,23 can readily be
attributed to their ideal correlation characteristics as expressed in (10).
However, it is recognized that the choice of secondary code length in GNSS
system can be influenced by other parameters besides correlation
characteristics alone.Further analysis of the optimal binary code of length
20 revealed the existence of close association of optimal binary codes to that
of the well-known Golay complementary pairs [20]. The Golay complementary pairs have been extensively
utilized in a number of applications ranging from radar signal processing
[21] and communication
[22] to multislit
spectrometry [20]. Two
binary codes xa(k) and xb(k)
are said to be
Golay complementary pair, if they satisfy the following
constraint:(11)RG(i)=Ra(i)+Rb(i)={2N,i=0,0,i≠0,where Ra(i) and Rb(i)
are the periodic
correlation of xa(k) and xb(k), respectively. RG(i) is the periodic
correlation function of the Golay complementary pair. Besides, the individual
codes in a Golay complementary pair are referred as Golay codes. The periodic
correlation in (11) immediately asserts the advantage of Golay complementary
codes in the view point of code design. For example, the NH10 code and the first-half of the NH20
code are Golay complementary pair as shown in Figure 3. Hence, there exists a
possibility of utilizing this underlying structure
to accomplish better acquisition abilities. Unfortunately, the NH10
code and second half of NH20 code are not Golay
complementary pairs.Figure 3
Correlation output of Golay complementary codes (NH10
and first half of NH20).Motivated by this observation, the optimal binary
codes of length 20 obtained via exhaustive search were tested for Golay
complementary pair. Interestingly, many binary
codes of length 20 obtained through exhaustive search (i.e.,S202 in Table 3) satisfied the
Golay complementary condition. For example, the Golay complementary
pairs
G10a and G10b can be constructed from the
even and odd samples of S202(hex value “05D39” and “FA2C6” also give rise to Golay pairs) listed in Table 3, and the corresponding Golay codes are given
by(12)G10a=[−1,1,−1,1,−1,−1,−1,−1,1,1],G10b=[1,−1,−1,1,1,−1,1,1,1,1].More importantly, the individual
Golay codes G10a and G10b were also optimal having
periodic correlation in accordance to (9). Moreover, the Golay codes of length N/2 obtained from
an optimal code of length N were also
optimal. Consequently, the 45 optimal binary codes of length 20 (see Table 2)
were tested for Golay complementary condition. Surprisingly, 75% (32 out of 45
codes) of the 20-bit optimal binary codes satisfied the Golay complementary
condition. A corollary of this conjecture indicates the possibility of
constructing optimal codes of length N from Golay
complementary pairs of length N/2. The construction of binary codes by multiplexing
Golay complementary pairs readily guarantees that every alternate shift will
result in zero correlation due to the complementary correlation output of
individual Golay codes. Interestingly, the aforementioned property of the Golay
codes was utilized for signal acquisition in
ultrasonic operations [23]. To further verify this corollary, we constructed a
binary code from Golay complementary pairs of length 20 (hex values “CD87F” and
“CE5AA”). The resulting binary code of length 40 (hex value “F0F6916EEE”)
demonstrated optimal periodic correlation as predicted by (9). Thus, it is
possible to construct optimal binary codes of larger lengths by utilizing the
aforementioned association between optimal codes and the Golay complementary
codes. Besides, the highly regular structure of binary Golay complementary
codes readily allows for an efficient construction [24].Table 3
Secondary synchronization code—performance measures (μ(x), PSLR, and ISLR are defined in (5), (3), and (4), resp.).
Secondary code performanceStandard codesProposed codesCode identifierCode length|μ(x)|PSLR (dB)ISLR (dB)Code identifierCode length|μ(x)|PSLR (dB)ISLR (dB)CS440.5∞∞S440.5∞∞NH10100.2141.5S10100141.5NH20200.2144S201200144CS20200.2144S202200.1144.9CS25250.218.46.3S203200.2144M440.5∞∞S251250.218.46.3M10100.4141.5S252250.218.46.3M20200.1144.9M25250.218.46.3Motivated by the aforementioned observation, we
constructed synchronization codes of lengthN=100 from optimal
codes of lengths 10, 20, and 25. The specific choice of code length was dictated
by the fact that the desired code length 100 was divisible by 10, 20, and 25.
The final code length of 100 was obtained by manipulating the individual codes
of length 10, 20, and 25 with the augmentation codes of length 10, 5 and 4. Let xp(k) and xs(k) be the primary
and the augmentation code of length Np and Ns. Thus, we have N=NsNp, where N=100, Ns={10,5,4}, and Np={10,20,25} in our case.
The final binary code, x(k), of length N can be obtained
as follows:(13)x(k)=∑m=0Ns−1∑n=0Np−1xs(m)xp(n)g(k−mNNs−nNNp),k=0,1,2,…,N−1,where g(k) is the
rectangular pulse function and is given by(14)g(k+ΔT)={10≤ΔT<Tb,0elsewhere,where Tb is the basic
bit duration over which the xk is constant.
For example, the 100-bit code, x(k) (hex value “C7F526E3FA9371FD49A7015B2”), was obtained from the primary code, xp(k) (hex
value “380AD90”), and the augmentation code, xs(k) (hex value “1”). In Table 2, we saw that there exists 7,000
codes of length 25 with 260 unique solutions but we only need 100 unique codes.
Thus, we utilized the following constraints on the PSLR and ISLR
measures to limit the number of
codes:
(15)PSLR≥21.9dB,ISLR≥3dB.The PSLR and ISLR
thresholds in (15) were duly obtained from the
average PSLR and ISLR measures of the Galileo G100 code set [25]. Finally, the
cross-correlation constraint expressed in (8) was also utilized to obtain
unique solutions. Consequently, a total number of 105 unique codes were
obtained in this fashion which satisfied the aforementioned conditions. The
hexadecimal representations of the individual codes are listed in Table 6. It
is worth noting here that not a single Galileo G100 code as well as the
proposed 100-bit codes satisfied the optimal periodic correlation based on (9).
The following section establishes the merits and limitations of the proposed
binary synchronization codes in comparison to the standardized secondary
synchronization codes.


## 5. Acquisition Performance Analysis

Having obtained
the optimal binary codes of various lengths, we now turn our focus on the
evaluation of the proposed codes in comparison to
the standardized codes utilized in GPS and Galileo system. In this paper, the
structure proposed in Tran and Hegarty [26] was adopted for the secondary code acquisition,
wherein the primary code is assumed to be acquired within half chip duration
alongside residual Doppler. The secondary code is acquired by correlating the
primary code correlation outputs with the locally generated secondary code
samples. The residual Doppler was assumed to be within ±250Hz. During
the secondary code acquisition, the residual Doppler was also searched within ±250Hz in steps
of 25 Hz.The Galileo CS4 code is already established as the
optimal code and will not be dealt during the acquisition performance analysis.
Table3 lists the μ(x), the PLSR, and the ISLR measures of the standardized
Merten’s and the proposed codes of various lengths. While the 20-bit
synchronization codes achieved similar PSLR measure as that of 10-bit codes,
their ISLR performances were much better than that
of 10-bit codes. In Table 3, it can be noticed that there are 3 different sets
of S20 code (S201, S202, and S203) and two sets of S25 code
(S251 and S252). While these different codes
are optimal in terms of correlation characteristics, their correlation
characteristics differed in the presence of the residual Doppler with some
outperforming the other codes. In Table 3, we see that the designed codes were
not only optimal in terms of PSLR and ISLR measures, they were also more
balanced. The advantage of the M20 and S202 over the NH20 and CS20 codes
is readily asserted by the higher ISLR values. Interestingly, the other 20-bit
codes S201 and S203 demonstrated better
acquisition performance in comparison to M20 and S202 codes despite being inferior
in ISLR measure. In the case of CS100 and S100
codes, the autocorrelation and cross-correlation protection
were evaluated using a number of measures. The PSLR
measure based on the auto-correlation was same for both CS100 and S100 codes
despite being suboptimal in the view point of (9). The cross-correlation PSLR
(CPSLR) measure was also obtained for CS100 and S100 codes. The CPSLR measures the
ratio between the auto-correlation main peak of code (R(i)) to the
maximum of the cross-correlation peak (Rp,q(i)) and
it is given by(16)CPSLR=R(i=0)2max|Rp,q(i≠0)|2.Table 4 lists the maximum,
minimum, mean, and the standard deviation of CPSLR, ISLR, and MSC measures for
the Galileo CS100 and the proposed S100 codes. While the standardized CS100 code is attractive in terms of CPSLR, the
proposed S100 codes were appealing in the view point of ISLR. The MSC
performance of both the codes was similar. The distribution of the CPSLR and
ISLR measures of the CS100 and S100 codes is plotted in Figure 4 for better
comparison. In Figure 4, we see that the standard CS100 codes achieved 1 dB
improvement over proposed S100 codes for 50% of the times in terms of CPSLR. On
the other hand, the proposed codes showed an 0.9 dB
improvement over standard CS100 codes for 50% of the times in terms of ISLR.
The CPSLR degradation observed in proposed S100 codes is inherent to its
construction. Alternatively, one can utilize evolutionary techniques for the
multiple-objective code optimization encountered in CS100 code design [27].Table 4
Galileo CS100 and proposed S100 codes performance.
Secondary code performanceCPSLR (dB)ISLR (dB)MSC (dB)CS100S100CS100S100CS100S100Max14.9146.65.943.543.6Min62.93.75.142.642.8Mean11.29.14.95.64343.1Std. Dev1.22.40.60.30.10.2PSLR and ISLR performance of Galileo CS100 and the proposed S100 codes.
(a)(b)In the preceding section, we inferred the existence of
multiple solutions due to the code periodicity and Table2 listed the number of
codes that accomplished the optimal correlation characteristics as predicted by
(10). To further arrange them, the individual codes were utilized for code
acquisition and their corresponding PSLR measure was obtained in the presence
of residual frequency error. For example, the PSLR of the 10-bit and the 20-bit
codes in the presence of 12 Hz residual error is plotted in Figure 5. In the
case of 20-bit synchronization code, the ISLR measure was relaxed to 4 dB so as
to include the remaining synchronization codes. Accordingly, we evaluated the
PSLR performance of all the 20-bit codes (5079 codes as listed in Table 2)
obtained via exhaustive search. Figure 5 readily confirms the existence of
optimal synchronization codes that are better then the standardized codes in
terms of PSLR measure. However, a question may arise on the specific Doppler
setting and whether that could influence the PSLR performance. Further analysis
did confirm this conjecture due to the existence of codes that were superior
for certain Doppler scenarios.PSLR performance in the presence of residual Doppler (LHS) code 10-bit (RHS) 20-bit code.
(a)(b)Thus, the average of the PSLR over a range of Doppler
(namely from 0 Hz to 25 Hz) was utilized as the selection criterion for code
selection. Under the new average PSLR measure, the codes that accomplished
superior correlation suppression are listed in Table5. The S10 and S201 codes achieved the overall
best performance in terms of average PSLR taken over a range of Doppler’s. It
should be emphasized here that both these codes were balanced and thus
asserting the significance of the balanced property introduced earlier. Figure 6 shows the PSLR performance of the standard, Merten’s and the proposed 10-bit
and 20-bit synchronization codes during two-dimensional acquisition in the
absence of background noise. The residual Doppler was searched between 0 Hz and
250 Hz in steps of 25 Hz as reported in [10].Table 5
Hexadecimal representation GPS/Galileo and proposed secondary codes
(highlighted colour represents equivalence).
Code identifierCode lengthNumber of hex symbolsNumber of zero paddingHex valueCS4410ENH101032F28NH202050FB2B1CS202050842E9CS252573380AD90M4410DM101032CBCM202050FA2C6M252573E3FA930S4410BS1010323B0, 3C8S201205014B37, 14B37S202205005D39, 6345FS2032050315B0, 640E5S251257321228F8, DFB45C0S2522573AD04C18Table 6
Hexadecimal representation of proposed S100 codes.
Hexadecimal values of S100 codesCode length = 100, no. of hex.
symbols = 25, no. of zero padding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 of residual Doppler on secondary code acquisition (LHS) 10-bit code (RHS) 20-bit code.
(a)(b)The LHS plot in Figure6 readily affirms the
limitation of standard NH10 code and the advantage of utilizing the M10 and the
proposed S10 code. Later it will be shown that the proposed S10 code
correlation can be better than that of M10 code in the presence of frequency
offset. Amongst the 20-bit codes, the Galileo CS20 code had the worst
performance in accordance to result shown in Figure
5. Both the M20 code and the proposed S202 code resulted in same
performance as they belong to the same equivalence class. The S203 code demonstrated similar
performance as that of the NH20 code. Finally, the proposed S201 code showed the best
performance in terms of PSLR under Doppler conditions. The S201 code although suboptimal in
terms of ISLR still performed better owing to its balanced property.The correlation performance degradation in NH20 code
as a function of frequency offset was analyzed in [10]. To further validate this
initial observation and also to compare the correlation suppression performance
of the proposed codes, numerical simulations were carried out. Figure 7 shows
the PSLR performance for both 10-bit and 20-bit synchronization codes as a
function of frequency offset. For the 10-bit code, one can readily notice the advantage of the
proposed S10 code over the M10 and NH10 codes. In the case of 20-bit code, the
standard NH20 and the CS20 codes performed better in comparison to the M20, S201, and S202 codes. On the other hand, the
S203 resulted in the overall best
performance and readily showed a PSLR gain of around 2.5 dB over standard NH20
and CS20 codes. However, the S201 is still attractive as it
yielded the best PSLR performance as shown in Figures 6 and 7. The
aforementioned analysis for a similar setting was carried out for the 25-bit
code, which included the CS25, M25, and the proposed S201 and S202 codes. Note that the M25 and
CS25 codes are essentially similar and are expected
to perform similar. Figure 8 shows the effect of residual Doppler on secondary
code acquisition and the PSLR performance as a function of frequency offset.PSLR performance in the presence of frequency
offset (LHS) 10-bit code (RHS) 20-bit code.
(a)(b)25-bit code performance. (LHS) effect of residual Doppler on secondary code
acquisition (RHS) PSLR performance as a function of frequency offset.
(a)(b)The standard CS25 code and that of M25 code were
exactly same as far as frequency offset is concerned. However, the standard
CS25 resulted in better PSLR performance as shown in LHS plot of Figure8. On
the other hand, both the proposed codes demonstrated superior PSLR performance.
Interestingly, the codes
S201 and S202 were complementary in their
PSLR performance as shown in Figure 8. However, the code S202 can be considered optimal for
not only achieving better PSLR performance (around 2 dB) in the presence of
residual Doppler, it also retained similar PSLR performance to that of standard
CS25 code for a wide range of frequency offsets.Finally, the code acquisition performance of the
standard CS100 and the proposed S100 codes was also evaluated in a similar
manner. The residual Doppler range was reduced to 7.5 Hz so as to reflect the
longer coherent integration utilized in acquiring these codes. Figure9 shows
the average PSLR performance of the standard and the proposed codes. The
standard CS100 code demonstrated better performance in regards to the proposed
S100 codes under both settings. The proposed code despite being characterized
by better ISLR measure was still limited by its construction method from code
of short length. Nevertheless, it readily corroborates the use of alternative
solutions for the multiple code design problem.100-bit code performance. (LHS) effect of residual
Doppler on secondary code acquisition (RHS) PSLR performance as a function of
frequency offset.
(a)(b)


## 6. Conclusions

The design of
secondary synchronization code for GNSS system is important due to its role
in acquisition and tracking. A limitation arising due to the usage of short
secondary code is the apparent degradation in correlation isolation especially
in the presence of residual frequency errors. This paper introduced the various
performance measures that can be utilized for secondary synchronization code
optimization. Consequently, these performance measures were utilized to obtain
optimal codes of various lengths via exhaustive search. This paper also
established the association between the optimal codes and the systematic codes
such as Golay complementary codes. The proposed secondary synchronization codes
of lengths 10, 20, and 25 obtained in this fashion
readily demonstrated superior correlation isolation performance in the presence
of residual frequency errors. The developed S100 codes although appealing in
terms of ISLR measure demonstrated inferior acquisition performance over
standardized CS100 codes. Truncation of LFSR codes or code design using genetic
algorithms can produce code sets with better correlation characteristics. The
significance of the correlation isolation improvement demonstrated by the new
synchronization codes in terms of probability of false alarm and detection is
currently being investigated. Finally, judicious design of short
synchronization codes can offer optimal correlation suppression and efficient
signal generation.Example 1.
The NH10 Code represented by the hexadecimal
value “F28” is obtained as follows:(17)F⟶1111,2⟶0010,8⟶1000.Hence, “F28”⟶1,1,1,1,0,0,1,0,1,0,0,0. The last two digits
highlighted in red are discarded, and the zero
symbols are mapped in to −1. (i.e., 0→−1).



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*Source: 246703-2008-05-06.xml*