This document analyzes and simulates pseudo ranging noise codes for geo-stationary satellites, including their Doppler effect. It examines the auto-correlation function and cross-correlation function properties of Gold codes, Kasami codes, and their truncation effects. The simulations were performed using MATLAB and System View. The results show that truncating the codes degrades their correlation properties. Doppler frequency shift on carrier frequencies and codes for geo-stationary satellites was also analyzed and found to be negligible.

1. ACEEE Int. J. on Communications, Vol. 03, No. 02, Nov 2012
Analysis and Simulation of Pseudo Ranging Noise codes for
Geo-Stationary Satellites and its Doppler Effect
P.N.Ravichandran, Sunil Kulkarni, H.S.Vasudevamurthy, M.Vanitha
Digital Systems Group
ISRO Satellite Centre , Bangalore, India
pnravi@isac.gov.in, kulkarni@isac.gov.in, hmurthy@isac.gov.in, vani@isac.gov.in
Abstract — The Geo-Stationary Navigation Satellite System these codes are readily available and can be used with minor
will provides basically two types of services 1) Standard modifications or software change. The generation of these
Positioning Service (SPS) and 2) Restricted Service (RS). codes in on-board will be easy and they provide the required
Both of these services are provided at two frequencies of L ACF and CCF among themselves. The requirements of ACF
and S-Band. The code sequences used in SPS and RS are and CCF properties of the above said codes are simulated
Pseudo Ranging Noise (PRN) codes. In SPS downlink, it is and results are presented. The simulation of Kasami code is
planned to use Gold Codes for navigation data transmission. carried out for comparison purpose. The simulation of Doppler
The RS navigation down link has signals with pilot component effect was carried out for gold code. The effect of Doppler
and data component. The pilot component uses primary code frequency shift on both carrier frequencies and codes used
and secondary code to get final code known as tiered code. in Geo-Stationary Satellites are very much negligible. Doppler
The primary code is truncated Gold code. The secondary frequency shift varies with the distance of satellite from
code is PRN sequence code. The data component of RS service ground. The paper is organized as follows; section II deals
uses truncated PRN sequence code. This paper presents the with PN sequence properties, section III deals with Gold code
performance analysis and simulation results of auto sequence, simulation results and truncation effect. Section
correlation function (ACF) and Cross correlation function IV deals with Kasami code sequence, simulation result and
(CCF) properties for Gold code, Kasami codes and it’s truncation effect. Section V deal with Comparison of Gold
truncation effect. Apart from ACF and CCF, Doppler code and Kasami code with Welch bound. Section VI deals
frequency shift on L & S-band carrier frequencies and with analysis of Doppler frequency shift on carrier frequencies
Doppler frequency shift on L & S band Codes are carried and Doppler frequency shift on Gold code. Section VII deals
out. The simulations of ACF & CCF on codes and Doppler Conclusion at the end.
effects were analyzed using Matlab and System View design
tool and results are compared with Welch bound. The II. PN SEQUENCE PROPERTIES
simulated test results are well within the theoretical limits.
PRN codes are PN Sequence codes, which are random like
Keywords — Gold Code, Kasami code, ACF, CCF and Doppler sequences with symbols ±1 having following properties.
frequency shift. Balance Property: Good balance requires that in each period
of the sequence, the number of one’s differs from the number
I. INTRODUCTION of binary zero’s by at most one digit [1].
The Geo-Stationary Navigation Satellite System Run Property: The appearance of the alternate digit in a
constellation consists of seven operational satellites. Each sequence starts a new run. The length of the run is the number
satellite generates a navigation message in binary notation of digits in the run. Among the run’s of ones and zeros in
based upon data periodically uploaded from ground station each period, it is desirable that about one half the runs of
and modulo-2 sum of this message and a 1.023 MHz PRN each type are of length 1, about one fourth of length 2, one
code sequence is used for SPS and a 2.046MHz PRN code eighth are of length 3, and so on. A ‘run’ is a sub-sequence of
sequence is used for RS [2,6]. For SPS signal generation the 1’s or 0’s.
satellite modulates the resulting bits stream on to L-band
Correlation Property: if a period of the sequence is compared
and S-band carriers using BPSK modulation technique to
term by term with any cyclic shift of itself, it is best if the
create a spread spectrum ranging signal, which it then
number of agreement differs from the number of
broadcasts to the user community. In case of RS signal the
disagreements by not more than one count. The PRN codes
satellite modulates the resulting bit steam on to L-band and
used for spread spectrum require certain mathematical
S-band carriers using Binary Offset carrier (BOC) modulation
properties. They are 1) maximal length sequence 2) Auto
technique to create a spread spectrum signal. Each of the
correlation function and 3) Cross correlation function..
Pseudo Ranging Noise (PRN) codes provides the mechanism
to identify each satellite in the constellation. The PRN codes Maximum length sequences: all maximum length sequence
proposed for SPS & RS systems are Gold code, Truncated are called m-sequence, in order to generate m-sequence, the
Gold and PN sequence code. Since user receiver chipsets for generator polynomial G(x), must be from the class of
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2. ACEEE Int. J. on Communications, Vol. 03, No. 02, Nov 2012
polynomials known as primitive polynomial. This implies in The Welch bound is the theoretical minimum of the maximum
simple terms, that G(x) cannot be factorized into lower-order value of cross-correlation that can be obtained for a given
polynomial. the maximum length sequences(MLS) are code length L within a set of M codes [3]. The cross correlation
pseudorandom binary sequences generated using maximal between any pair of binary sequences of period L=2n-1 in a
linear feedback shift registers. the m-sequence own their set of size M=L+2 is given by Θmaxe” L(sqrt(M-1/LM-1))
name to the fact they can be reproduced by a shift register and Θmaxe” sqrt(L ) for large code size M[4]. The Welch
with n-taps resulting in a maximum length of 2n -1 chips. bound depends directly on the length of the code.
Maximum length sequences are spectrally flat, with the
exception of a zero continuous term. The PN sequences must
III. GOLD CODE SEQUENCE
exhibits good correlation properties. The table-1 shows the
number of m-sequence for selected shift register stage Gold code sequences are constructed by exclusive-or of two
T ABLE.I. M-SEQUENCES
m-sequences of the same length with each other[4]. Thus,
for a Gold sequence of length L = 2n-1, one uses two LFSR,
each of length 2n-1. If the LFSRs are chosen appropriately,
Gold sequences have better cross-correlation properties than
maximum length LFSR sequences. The advantage of Gold
code is in generating larger number of codes size[1]. Gold
and Kasami showed that for certain well-chosen m-sequences,
the cross correlation only takes three possible values, namely
-1, -t(n) or t(n)-2. Two such sequences are called preferred–m
sequences [5].
TABLE.II. GOLD CODE SEQUENCE
Non-Maximal sequences: a sequence generated by a non-
primitive generator polynomial G(x) may have a period of
less than 2n-1 and hence this sequence is not an m-sequence
or non-maximal sequence.
Auto correlation Function(ACF) : the ACF reefer’s to the
degree of correspondence between a sequence and a phase
shifted replica of itself(time shifted). The ACL properties Here t(n) depends solely on the length of the LFSR used. In
are near ideal for code acquisition or synchronization, where fact, for a LFSR with ‘n’ memory elements, Gold code family
perfectly aligned condition of q=0 between the received and size M= 2n+1, n=shift register stages. The code size increases
locally stored sequences has to be detected. The ACF is of with increasing the number of stage of shift register con-
most interest in choosing code sequence that gives the least struction as shown in table-2.
probability of false synchronization. TABLE. III. GOLD CODE FULL LENGTH AND T RUNCATED SEQUENCE
Cross Correlation Function (CCF): When the received signal
with a different PN sequence than that of the receiver is
mixed with the locally generated PN sequence, it must result
in minimum signal strength. This would enable receiver to
receive only the signal matching the PN codes. This property
is known as orthogonality of PN sequence.
Preferred maximum length m-sequences: These sequences
are used to generate Gold and Kasami codes. These
sequences produces 3 valued ACF and CCF for Gold and
Kasamicodes.
Welch bound: Designing codes optimized for any of the
potetial application is practically impossible, using code-
centric metric is more appropriate. This is the reason why the
Welch bound has gained importance in recent years as
suitable metric for evaluating PRN codes.
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3. ACEEE Int. J. on Communications, Vol. 03, No. 02, Nov 2012
The complete family of gold codes for a given generator is The simulation result in figure-1 shows the ACF of full length
obtained by using different initial load conditions in one of sequence 8191. The result shows three valued ACF. They
the shift register polynomial G2(x) in a given pair of are 1, -0.0155048 and +0.0155048. The peak is at time t=0.
polynomials G1(x) and G2(x). In Gold code, the maximum
cross-correlation for large code size family is θmax=”2L for
‘n’-odd and θmax =2"L for ‘n’ even, which means maximum
cross correlation is lower by “2 for n-odd and by 2 for n even
when compared to Welch bound.
GOLD CODE SIMULATION RESULTS
TABLE. IV. INITIAL CONDITION OF DIFFERENT POLYNOMIAL
The Gold code simulation is carried out using System
View and Matlab. The Gold code sequence for primary codes
of RS were generated using the generator polynomials G1(x) Figure. 2. CCF of Full Length Sequence
and G2(x) with 13 stage shift register. The modulo-2 addition The simulation result in figure-2 shows the CCF of full length
of output sequences of G1(x) and G2(x) gives the primary sequence 8191.There result shows three valued CCF. They
code. The initial conditions used in different polynomials are 15.5048223*10E-3,-0.1220852154*10E-3&15.748992*10E-
identification as shown in Table 4. The Gold code (Primary 3 respectively.
codes) sequences have special ACF and CCF properties as
compared to normal m-sequences [5]. The both ACF and CCF
values are 3 valued spectrums and low CCF between different
codes belonging to same Gold family . The Table-3 gives the
simulation results of both full length(8191 bits long) and
truncated ( 8184 bits long) Gold sequences for different
Identification. The full length (8191) of Gold sequence have a
value of ACF = -36 dB and CCF=-36 dB. When the same
length of code is truncated to last 7-bit we get 8184, we
obtain ACF=-29.5 dB, CCF=-28 dB. The truncated Gold
sequence (8184) is poorer by 6.5 dB in ACF and 8 dB in CCF
values as compared to full length Gold code as shown in
table-3. the polynomial G1(x) is checked with remaining
polynomials and results are presented in table-3.
Figure. 3. ACF of Truncated Length
Figure-3 shows the simulation result of ACF for truncated
Figure. 1. ACF of Full length sequence
sequence 8184, we can see that there is no 3-valued ACF.
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4. ACEEE Int. J. on Communications, Vol. 03, No. 02, Nov 2012
IV. KASAMI CODE SEQUENCE
Kasami codes are the binary sequences sets having very
low cross-correlation [4]. There are two different sets of
kasami sequences, kasami sequences of ‘small set’ and
sequences of ‘large set’. A small set sequences has code size
M=2n/2 with code length L=2n-1 as shown in table-6. The
kasami code takes auto-correlation and cross correlation
values of {-1,-t(n),t(n)-2} where t(n)=2n/2+1. The Kasami
sequences are asymptotically optimal in the sense of
achieving lower Welch bound. The large set of kasami
sequences of period 2n-1, for n-even, contains both the Gold
sequences and the small set of kasami sequences as subsets.
The code size is M=23n/2 if n=0(mod 4) and M=23n/2 + 2n/2 for n=
2(mod 4), n=shift register stages [5]. All the values of ACF
Figure-4: CCF of Truncated Length Sequence
and CCF are five valued functions.The table-6 shows the
Figure-4 shows the simulation result of CCF for truncated different code size of kasami small set. The code size increase
sequence 8184 from full length 8191. The resultant value is with increase in shift register stages. compared to Gold codes,
multi-valued CCF. Even 1 bit truncation causes the ACF and Kasami provides lesser code size. Code size will be the limiting
CCF loses its properties. factor for selecting Kasami small set. Hardware implementation
Table-5 shows the simulation results of Gold sequence for 13 of Kasami code is difficult than Gold code sequence
and 15 stages. The simulation is carried out in steps of integer TABLE. VI. KASAMI (SMALL SET)
multiples of 1023 and found the truncation effect. We can see
that more the truncation of bits from full length sequence,
poorer is the CCF values.
In truncated Gold sequence of length 8184 obtained from full
length sequence 8191 gives better cross correlation value as
compared to a truncated Gold sequence of length 8184
obtained from full length sequence 32767, even though full
length Gold sequence 32767(n=15) is 6dB better than the full
length Gold sequence 8191(n=13)
TABLE-V: EFFECT OF T RUNCATION ON G OLD CODE
T ABLE. VII. EFFECT OF TRUNCATION ON KASAMI CODE
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5. ACEEE Int. J. on Communications, Vol. 03, No. 02, Nov 2012
In table-7 shows that more the truncation of bits from full value in practical. When we simulate kasami and Gold codes
length sequence, poorer is the CCF values. For example for a both the codes provides the CCF of -16.90dB and -11,37 dB
truncated kasami sequence of length 8184 obtained from full respectively. The simulated result shows kasami code reaches
length sequence 16383 gives better CCF value as compared 82.5% then Gold of 46%. The 14 stage shift register provides
to a truncated kasami sequence of length 8184 obtained from the Welch bound of -42.10 dB. This is the maximum attain-
full length sequence 65535, even though full length kasami able value in practical. When we simulate kasami and Gold
sequence 65535(n=16) is 6dB better than the full length kasami codes, both the codes provides the CCF of -42.07dB and -
sequence 16383(n=14). When we compare a truncated Gold 36.08 dB respectively. The simulated result shows kasami
sequence length 8184 obtained from full length sequence code reaches 98% then Gold of 49%. The simulation is re-
8191 & kasami sequence 8184 obtained from full length se- peated for different stages of shift register The increase in
quence 16383, the kasami sequence cross correlation value shift register stages, provides better CCF for both kasami
is 0.3dB better than gold sequence. and Gold codes. When comparing CCF of both Gold code
and Kasami code results, kasami code sequence provides
V. COMPARISSION OF GOLD CODE & KASAMI CODE WITH better CCF.
WELCH BOUND
VI. ANALYSIS OF DOPPLER FREQUENCE SHIFT
TABLE. VIII. CODES SIZE OF G OLD AND KASAMI SEQUENCES
The average radius of the earth is around 6,368 Km[6]. The
radius of Geo-Stationary satellite orbit is approximately
42,164Km. This height is approximately the shortest distance
between a user on the surface of the earth and the satellite.
In a selected length of code sequence, Gold code has
more code size compared to kasami and implementation of
Gold code sequences is simpler than Kasami sequence. For
example 14-stage kasami provide only 128 codes size, where
as same length of Gold code provides larger code size around
16385, even though kasami code provides better CCF than
Gold as per Welch bound. Still Gold sequence is more attractive
due to large code size and easy implementation.
TABLE. IX. COMPARISON OF WELCH B OUND WITH KASAMI & GOLD CODES
Figure. 5. Earth and Elliptical Orbit
In most of the Geo-Stationary receivers are designed to
receive signals from satellite above 5 degrees. Let us assume
that the receiver can receive signal from satellite at the zero
degree point. The shortest distance to the satellite is at zenith
d1=7931Km. The distance from a satellite on the horizon to
the user is
Table-9 shows the results of Gold and kasami with re-
spect to Welch Bound for CCF. The Kasami code provides
better CCF than Gold codes. The Welch bound of 6-stage
shift register is -17.42 dB. This is the maximum attainable,
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6. ACEEE Int. J. on Communications, Vol. 03, No. 02, Nov 2012
Where ‘c’ is speed of light, If the user is on the surface of the to assume the maximum Doppler frequency shift is double
earth, maximum differential delay time from two different of fdr1 & fdr2. These values determines the search
satellites should be within 112(138-26) ms. frequency range in the acquisition program.
All the calculations are based on sidereal day time [6], it is A. The Doppler frequency shift in L & S-band codes:
23 hours, 56 min, 4.09 sec. This is the time for the satellite to rotate
The L and S-bands of SPS and RS uses Gold code,
once around the earth. The angular velocity is
truncated Gold code and PRN code. For L-band, The chip
rate is 1.023MHz and carrier frequency of 1176.45MHz. For S-
band chip rate is 2.046 MHz and carrier frequency of
2492.028MHz respectively. The L-band code chip rate is 1150
As this angle θ, the satellite is at the horizontal position times lower than its carrier frequency, and S-band code chip
referenced to the user. Where ’rs’ is average radius of the rate is 1218 times lower than its carrier frequency. hence the
satellite 42,164 km and ‘Us’ orbital velocity of satellite Doppler frequency shift on the both L and S-band codes are
quite negligible. Doppler frequency shift on both L and S-
bands are as follows
The time difference between an apparent solar day and side
real day is 3 min, 55.91 sec. the satellite will travel
approximately
3.075 m/s * 253.9s - 0.780 km (6)
If the satellite is close to horizon, the corresponding angle If the receiver moves at high speed, these values can be
is doubled.
B. Simulation results of Doppler frequency on ACF
&CCF for Gold code sequence:
If the satellite is close to zenith, the corresponding angle is
Table. X. Gold Code Sequence Length of 8184
The satellite position changes about 0.00107degree to 0.00564
degree per day at the same time with respect to fixed point on
the earth surface,
The maximum Doppler velocity occurs when the satellite is at
horizon position. From the orbital speed, one can calculate
the maximum Doppler velocity Vdm, which is along the
horizontal direction.
This speed is equivalent to a high speed military aircraft. The
Doppler frequency shift caused by a land vehicle is often
very small, even if the motion is directly towards the satellite
to produce the highest Doppler effect.
For the L-band carrier frequency f1=1176.45 MHz. the
maximum Doppler effect is
For the S-band carrier frequency f2=2491.75MHz. the
maximum Doppler effect is
The table-10 shows the simulation results of truncated
For a stationary observer, the maximum Doppler frequency Gold code length of 8191 to 8194 with last 7 bit skipped
shift is around ±1.82Hz & ±3.85Hz for both L and S-bands sequence. The Doppler effect is checked for both ACF &
respectively. If the receiver is used for low-speed vehicle, the CCF with different frequency offsets. The frequency offset
Doppler shift can be considered as ±1.82Hz & ±3.85Hz. If the is reduced from ±2Hz to ±5KHz. When we compare the
receiver is used in a high speed vehicle, it is reasoned equation-12 and 13 with Gold code simulation result, the
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7. ACEEE Int. J. on Communications, Vol. 03, No. 02, Nov 2012
Doppler effect on the code is very much negligible. the rate of change of frequency is very much negligible, the
tracking system is designed with frequency change is lesser
C. Average rate of Change of Doppler frequency: than 1Hz.
The frequency update rate is important parameter for
tracking system. The tracking system is combination of delay VII. CONCLUSION
lock loop (DLL), demodulator, Bit-synchronizer and lock The analysis and simulations are carried out for Gold code,
detector. The required Doppler frequency shift is estimated small set Kasami code . The Gold codes, truncated Gold code
and accordingly DLL parameters are designed in tracking and PRN sequence codes are proposed in both SPS and RS
system and embedded in receiver. The angle  in which systems of Geo-stationary satellite. The observation result
satellite is in horizontal position to user. The maximum shows that for full length Gold sequences are of 3 valued
Doppler velocity occurs when the satellite is in horizontal Auto correlation and cross correlation spectrum. In truncated
position Gold, it was observed that the ACF and CCF are no more 3
valued spectrums. The Gold code sequence provides a larger
number of code sizes with good cross correlation than the PN
sequence. Implementation of Gold code sequence is much
The angle for Doppler frequency change from maximum to simpler. The Doppler frequency effect on carrier frequency
zero is around 1.4191 radian. and Gold code sequence is negligible when it is used in Geo-
stationary satellites. In view of the above and the fact that
Gold code gives fairly good performance in ACF and CCF, the
Gold code is preferred than other codes in Geo-Stationary
The satellite in Geo-stationary location takes 23 hours 56 Satellites.
minutes and 4.09 sec to travel 2π. Hence time taken to cover
1.4191 radian is REFERENCES
[1] R.C.Dixion,”Spread Spectrum with Commercial Application,
During this time Doppler frequency changes from 1.82 Hz to Wiley,1994.
[2] “Report of the ccommittee for identification of Services, Signal
zero in L-band is
structure and CDMA codes “
[3] Stefan Wallner, Jose-Angle Avila-Rodriguez,Guenter W.Hein
“ Galileo E1 OS and GPS L1C Pseudo Random Noise Codes
During time in equation-16, the Doppler frequency changes requiirements, generation,optimization and comparission”.
from 3.857 Hz to zero in S-band is [4] Esmael H.Dinan and Bijan Jabbari “Spreading codes for Direct
sequence CDMA and wideband CDMA celluar networks”IEEE
Communication magzine, Sept-1998
[5] Kimmo Kettunen “Code Selection for CDMA”, Licenticate
course on Signal Processing in Communication “Sept-1997
This is very slow rate of change in frequency. From these [6] James Bao-Yen Tsui “ Fundamentals of Global Positioning
vales, the tracking system will able to track the PRN codes System Receiver” ISBN 0-471-20054-9,2000 John Wiley &
and extracts both data and clock without any problem. Hence Sons,Inc
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