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ACEEE Int. J. on Communications, Vol. 03, No. 03, 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 © 2012 ACEEE 17 DOI: 01.IJCOM.3.3. 1036 ACEEE Int. J. on Communications, Vol. 03, No. 03, 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. © 2012 ACEEE 18 DOI: 01.IJCOM.3.3. 1036 ACEEE Int. J. on Communications, Vol. 03, No. 03, 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. © 2012 ACEEE 19 DOI: 01.IJCOM.3.3.1036 ACEEE Int. J. on Communications, Vol. 03, No. 03, 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 © 2012 ACEEE 20 DOI: 01.IJCOM.3.3. 1036 ACEEE Int. J. on Communications, Vol. 03, No. 03, 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, © 2012 ACEEE 21 DOI: 01.IJCOM.3.3. 1036 ACEEE Int. J. on Communications, Vol. 03, No. 03, 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 © 2012 ACEEE 22 DOI: 01.IJCOM.3.3. 1036 ACEEE Int. J. on Communications, Vol. 03, No. 03, 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 © 2012 ACEEE 23 DOI: 01.IJCOM.3.3.1036