Generation of Binary Sequences of Length 10230 Bits Having Better Odd and Even Correlation with Large Linear Complexity for Use in Global Navigation Satellites Systems (GNSS) Applications

Generation of Binary Sequences of Length 10230 Bits Having Better Odd and Even Correlation with Large Linear Complexity for Use in Global Navigation Satellites Systems (GNSS) Applications

Dileep DharmappaMahalinga V. Mandi Ramesh Siddaiah 

Department of Electronics and Communication Engineering (ECE), Sri Siddhartha Academy of Higher Education (SSAHE), Agalakote, Tumakuru 572107, Karnataka, India

Navigation Systems Area, ISRO Telemetry Tracking and Command Network (ISTRAC), ISRO, Bengaluru 560058, Karnataka, India

Department of Electronics and Communication Engineering (ECE), Dr. Ambedkar Institute of Technology, Near Jnana Bharathi, Bengaluru 560056, Karnataka, India

Corresponding Author Email: 
dileep@istrac.gov.in
Page: 
94-102
|
DOI: 
https://doi.org/10.18280/mmep.070112
Received: 
2 September 2018
|
Revised: 
24 December 2019
|
Accepted: 
2 January 2020
|
Available online: 
31 March 2020
| Citation

© 2020 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).

OPEN ACCESS

Abstract: 

Binary sequences used in Global Navigation Satellites Systems (GNSS) are found to be easily decoded due to less linear complexity. The correlation properties are extremely important while selecting the sequences for GNSS Applications. Linear Feedback Shift Register (LFSR) based Sequences are being used in GNSS Systems such as Global Positioning System (GPS) L2 CM (L2 frequency band Civil Moderate) Signal. Due to the short cycling of LFSR these sequences result in poor correlation properties. In this work the properties of binary sequences used in the state of the art GPS L2CM Navigation signal is explored. The odd and even correlation properties of GPS L2CM sequences are presented in detail. These sequences are analyzed for their Linear Complexity property. A method is proposed for generation of 10230 bit length binary sequences. A new set of binary sequences with set size of 47 sequences are generated using chaotic real sequences. The generated sequences are analyzed for odd correlation, even correlation and linear complexity properties. The proposed binary sequences are found to have better correlation and excellent linear complexity properties as compared to GPS L2 CM Sequences, which make them suitable for use in GNSS Applications.

Keywords: 

global positioning system (GPS), global navigation satellites systems (GNSS), chaotic logistic map, auto correlation, cross correlation, linear complexity (LC)

1. Introduction

Binary sequences play vital role in Global Navigation Satellites Systems (GNSS) Systems. GNSS Systems such as Global Positioning System (GPS) by US, European GNSS (GALILEO), Russians Global Navigation Satellite System (GLONASS), Indian Regional Navigation Satellite System (IRNSS) by India, Quasi Zenith Satellite System (QZSS) by Japan and Chinese BeiDou Navigation Satellite System (BEIDOU/COMPASS) use binary sequences for various types of services provided by each GNSS system. User receiver at ground identifies a unique service from a GNSS System using these binary sequences.

GPS system is widely used for location services across the world. Towards enhancing the capabilities of GPS system, a new GPS signal namely GPS L2 band Civil Moderate (GPS L2 CM) signal having binary sequences of length 10230 bits was added in L2 frequency. This signal provides one order better accuracy than the existing GPS C/A (Coarse Acquisition) signal having binary sequences of length 1023 bits.

Binary spreading sequences provide secure and reliable transmission for GNSS Systems. Binary sequences derived from Linear Feedback Shift Register (LFSR) are the commonly used Pseudo Random Binary Sequences in spread spectrum applications [1]. For GNSS Systems, the property of spreading sequence length plays an important role. It is always designed such that the spreading sequence length should be an integer multiple of chip rate. However binary sequences generated using conventional methods such as Gold Codes, which are based on LFSR are restricted to code length (2N-1) for ‘N’ stages. The GPS L2 CM sequences are generated using N = 27 stage LFSR. The length of the binary sequence of 27 stage LFSR is 134217727 bits. However to meet the sequence length constraints these binary sequences are short cycled/reset to 10230 bits in state-of-the art GPS Navigation System. The process of short cycling results in poor correlation properties of the truncated sequences [1]. The desirable properties to be considered in the design of the binary sequences for GNSS Systems are described in the literature [1-5]. However in the existing literature, the detailed analysis of the properties namely odd correlation and linear complexity of the new civilian signal GPS L2 CM sequences are not done.

The security of the GNSS Systems depends on the linear complexity property of the binary spreading sequences. It is observed from the literature that due to the low linear complexity property binary sequences used in BeiDou System were decoded by Gao et al. [6]. Also similar instance of decoding Galileo In Orbit Validation Element-A (GIOVE-A) codes, was done by Psiaki et al. [7]. Design of binary sequences with high linear complexity is being attempted by many researchers [8-11].

Binary sequences used for GNSS Applications are expected to perform well in real world environment such as multipath, interference etc. Whenever there is a flip in a bit in the navigation sequence or the bit in the secondary code sequence is observed within the integration period of the binary spreading sequence, it is difficult to acquire the signal and this depends on the odd correlation properties of the spreading sequence [12-14].

In this work a method is proposed to derive binary sequences with good even correlation, odd correlation and linear complexity properties. Proposed binary sequences are of length 10230 bits and are derived from chaotic real valued sequences.

The paper is organized as follows. Correlation and linear complexity properties are defined in the Section 2. Section 3 provides correlation and linear complexity properties of GPS L2 CM sequences in detail. In Section 4, proposed method of generating chaotic binary sequence is discussed. The correlation and linear complexity properties of the proposed binary sequences are provided in Section 5. Section 6 concludes the work.

2. Correlation and Linear Complexity Properties

In this section the correlation and linear complexity properties are defined.

2.1 Correlation property

Considering two binary sequences x and y of length N bits the aperiodic correlation is defined [15] and is given by,

$C_{x, y}(l)=\left\{\begin{array}{ll}\sum_{j=0}^{N-1-l} x_{j} y_{j+l}^{*}, & 0 \leq l \leq N-1 \\ \sum_{j=0}^{N-1+l} x_{j-l} y_{j}^{*}, & 1-N \leq l \leq 0 \\ 0, & |l| \geq N\end{array}\right.$   (1)

Here l denotes the shift of one sequence say y relative to the other sequence x.

Even Cross Correlation (even CCR) property between two sequences x and y is provided [15] by,

$\theta_{x, y}(l)=C_{x, y}(l)+C_{x, y}(l-N)$    (2)

Even Auto Correlation (even ACR) property for a single sequence say, sequence x or sequence y is defined [15] as,

$\theta_{x, x}(l)=C_{x, x}(l)+C_{x, x}(l-N)$    (3)

Odd Cross Correlation (odd CCR) property between two sequences x and y is provided [15] by,

$\hat{\theta}_{x, y}(l)=C_{x, y}(l)-C_{x, y}(l-N)$    (4)

Odd Auto Correlation (odd ACR) property for a single sequence say, sequence x or sequence y is defined [15] as

$\widehat{\theta}_{x, x}(l)=C_{x, x}(l)-C_{x, x}(l-N)$    (5)

The importance of properties of even correlation is well understood in the literature for GNSS systems. Any improvement in the correlation is an added advantage for GNSS system.

Especially the odd correlation plays an important role in GNSS systems when there is a flip in the bit of the binary sequence, due to the presence of multipath or interference and that leads to difficulty in acquiring and tracking the satellite signals.

2.2 Linear complexity property

Linear complexity [16] for a binary sequence represents the length of shortest LFSR required to generate the same binary sequence. Berlekamp – Massey algorithm [16] is commonly used for computation of Linear complexity.

Linear complexity property plays vital role in advanced applications of GNSS systems such as security of driverless cars, Unmanned Arial Vehicles (UAV), Internet of Things (IoT) etc. 

Next section provides the correlation property and linear complexity property of 10230 bit GPS L2 CM binary sequences.

3. Correlation and Linear Complexity Properties of GPS L2 CM Sequences

GPS L2 CM sequences of length 10230 bits are generated using the method provided in the GPS Interface Control Document (ICD) [17] and the correlation and linear complexity properties are analyzed.

3.1 Generation of GPS L2 CM sequences

The GPS L2 CM code is a new civilian signal which has improved navigation accuracy which repeats for every 20 ms and are LFSR based codes reset after 10230 bits having chip rate of is 511.5 Kbps.

These codes are being transmitted from block II series replenishment satellites (IIR-M), block II series follow on satellites (IIF) and also subsequent blocks of GPS Satellites. Block diagram of the GPS L2 CM signal generator is shown in Figure 1. The L2 CM sequences are obtained by short cycling the LFSR for every count of 10230 chips by resetting with a specified initial state. The polynomial used to generate the L2 CM signal is 1112225171 (octal) of degree 27. The 27 bit LFSR provides sequences of length (227-1) which is equal to 134217727 bits out of which 134207497 bits are discarded to get the required 10230 bit sequences. A set of 37 binary sequences of length 10230 bits are generated from the method given in Figure 1 by selecting different initial values for LFSR and all the 37 sequences are short cycled to 10230 bits. The initial values to generate the 37 sequences are given in Table 1.

LFSR based sequences are most commonly used binary sequences for GNSS. However for GNSS Signals, it is desirable to have binary sequences period to be multiple of on-board fundamental frequency (10.23 MHz) i.e. multiple of 1023. Considering 10th multiple of 1023 bits which is equal to 10230 bits and is the required binary sequence length of GPS L2 CM sequences. However it is not possible to generate 10230 bit period, binary sequences using LFSR. Hence the sequences from LFSR are truncated/short cycled to 10230 bits. All the 37 GPS L2 CM sequences of length 10230 bits are generated using MATLAB.

Polynomial:- 1+X3+X4+X5+X6+X9+X11+X13+X16+X19+X21+X27

Figure 1. GPS L2 CM Binary Sequence generator

Table 1. Initial values of 27 stage LFSR for 37 GPS L2 CM sequences

SV ID

GPS PRN No

Initial Shift register Stage(Octal)

End Shift register Stage(Octal)

1

1

742417664

552566002

2

2

756014035

034445034

3

3

002747144

723443711

4

4

066265724

511222013

5

5

601403471

463055213

6

6

703232733

667044524

7

7

124510070

652322653

8

8

617316361

505703344

9

9

047541621

520302775

10

10

733031046

244205506

11

11

713512145

236174002

12

12

024437606

654305531

13

13

021264003

435070571

14

14

230655351

630431251

15

15

001314400

234043417

16

16

222021506

535540745

17

17

540264026

043056734

18

18

205521705

731304103

19

19

064022144

412120105

20

20

120161274

365636111

21

21

044023533

143324657

22

22

724744327

110766462

23

23

045743577

602405203

24

24

741201660

177735650

25

25

700274134

630177560

26

26

010247261

653467107

27

27

713433445

406576630

28

28

737324162

221777100

29

29

311627434

773266673

30

30

710452007

100010710

31

31

722462133

431037132

32

32

050172213

624127475

33

*

500653703

154624012

34

*

755077436

275636742

35

*

136717361

644341556

36

*

756675453

514260662

37

*

435506112

133501670

3.2 Even and odd ACR properties of GPS L2 CM sequences

Investigation is carried out for even and odd ACR properties. The results of maximum off peak even ACR and maximum off peak odd ACR values are tabulated in Table 2. The off peak even and odd ACR values are found to be multiple valued because of truncation. From Table 2, it is found that maximum even ACR for the set of 37 sequences is -26.90 dB and the maximum odd ACR is found to be -27.24 dB. The histogram of off peak even ACR and the off peak odd ACR is shown in Figure 2 and Figure 3 respectively. In Figure 2, it is found that only for 5 sequences out of 37 sequences are having off peak even ACR value of -29.2 dB. In Figure 3, it is found that only for 4 sequences out of 37 sequences are having off peak odd ACR value of -28.5 dB.

Table 2. Off peak maximum even ACR and odd ACR for 37 GPS L2 CM sequences

GPS L2 CM Sequence Number

Off Peak Maximum even ACR Value

Off Peak Maximum odd ACR Value

1

-28.833

-27.408

2

-28.928

-28.602

3

-28.928

-28.070

4

-28.833

-27.569

5

-28.466

-29.120

6

-30.040

-27.774

7

-29.120

-28.787

8

-29.217

-29.267

9

-28.648

-28.694

10

-28.740

-28.975

11

-29.416

-29.071

12

-28.833

-29.120

13

-28.556

-28.511

14

-28.288

-28.833

15

-28.027

-29.366

16

-26.905

-28.928

17

-28.113

-29.169

18

-27.858

-27.942

19

-29.416

-29.416

20

-28.648

-28.648

21

-28.288

-27.368

22

-28.376

-28.556

23

-28.556

-28.740

24

-28.928

-29.071

25

-28.288

-27.691

26

-28.833

-28.556

27

-28.648

-27.250

28

-28.376

-28.787

29

-28.833

-28.511

30

-27.448

-28.511

31

-27.289

-29.267

32

-29.316

-27.488

33

-27.858

-28.694

34

-27.368

-28.975

35

-29.217

-28.511

36

-27.528

-28.156

37

-29.316

-28.070

Figure 2. Histogram of maximum off peak even ACR for 37 GPS L2 CM sequences

Figure 3. Histogram of maximum off peak odd ACR for 37 GPS L2 CM sequences

3.3 Even and odd CCR properties of GPS L2 CM sequences

Investigation is also carried out for even and odd CCR properties. The histogram of results of all 666 combinations of pairwise maximum even CCR values and pairwise maximum odd even CCR values are plotted in Figure 4 and Figure 5 respectively. From Figure 4 and Figure 5, the even and odd CCR values are found to be multiple valued because of truncation. Also in Figure 4, it is found that the maximum pairwise even CCR value is -25.39 dB and the occurrences of pairwise even CCR values are found to be populated near -28.5 dB. In Figure 5, it is found that the maximum pairwise odd CCR value is -26.32 dB and the occurrences of pairwise odd CCR values are found to be populated near -28.5 dB.

From the correlation results following observations are made

  • The odd correlation and even correlation functions of the GPS L2 CM sequences are multiple valued due to truncation of LFSR sequences.
  • The GPS L2 CM Binary sequences are found to have better odd correlation performance. The odd ACR and odd CCR values are found to be better than the even ACR and even CCR values by 0.34 dB and 0.93 dB respectively. It is to be noted that the GPS L1 C/A PRN code designed during the first generation of GPS is found to have better even correlation properties when compared to odd correlation properties [18].

Figure 4. Histogram of pairwise maximum even CCR of all 666 combinations for 37 GPS L2 CM sequences

Figure 5. Histogram of pairwise maximum odd CCR of all 666 combinations for 37 GPS L2 CM sequences

3.4 Linear Complexity properties of GPS L2 CM sequences

Linear complexity (LC) values are computed for all 37 sequences using Berlekamp – Massey algorithm [16] and it is found to be 27. In the next section a method is proposed to generate binary sequences having better properties than GPS L2 CM sequences.

4. Proposed Method of Generating Chaotic Binary Sequence

Proposed method [18] for deriving binary sequences from chaotic sequences is shown in Figure 6.

Due to its simplicity Logistic map is the commonly used chaotic map [19-21]. This work considered chaotic sequences generated by Logistic Map defined by the equation [21]

$S_{k+1}=B S_{k}\left(1-S_{k}\right)$    (6)

Sk is real valued chaotic output with bound 0<Sk<1, B is bifurcation factor. The bifurcation factor is in the range 3.57< B < 4 and will always ensure that the output from this system is chaotic in nature.

In the proposed method the real valued chaotic sequences are converted to binary sequences governed by the following equation [18]

$Y_{k}=\left\lfloor\left(S_{k}\right) * E \right\rfloor \bmod (G)$    (7)

Using Eq. (6), iteratively real valued infinite sequence {Sk}, k = 0,1,2, ..., is generated. Each real valued sequence Sk is multiplied by a large integer E and modulo operation is performed over another integer G by selecting G and E as G < E. As shown in Figure 6, the integer part Qk is considered and fraction is rejected. Resulting integer Yk is mapped to binary directly to obtain output sequence Pk.

The binary sequences generated using the proposed method are investigated for correlation and linear complexity properties in the next section.

Figure 6. Proposed method for generation of binary sequences using chaotic function for GNSS applications

5. Correlation and Linear Complexity Properties of the Proposed Binary Sequences

A set of 100 binary sequences of length 10230 bits were generated from the proposed method by selecting the initial values randomly and setting integer G=4, B = 3.99 and E=32767. MATLAB simulations are performed to obtain 47 binary sequences from the proposed method and the initial values of all the 47 binary sequences are listed in Table 3.

Table 3. Initial values for generation of proposed binary sequences

Binary Seqeunce Number

Initial Value

1

0.656560157622049

2

0.379428069773115

3

0.956473170430473

4

0.837323305625539

5

0.186769582443968

6

0.917937385063156

7

0.449003245887149

8

0.317813959862539

9

0.210370839512595

10

0.421562177994972

11

0.115709453804408

12

0.313397650853574

13

0.452906953664362

14

0.707796016680767

15

0.354081569750622

16

0.500947441246977

17

0.353750830722854

18

0.563401008883089

19

0.656157553061328

20

0.326484624595631

21

0.682479882613947

22

0.248820030659104

23

0.677767695635484

24

0.365907196587940

25

0.479347941435340

26

0.576065641236740

27

0.216510493002707

28

0.929057543334895

29

0.832411734145950

30

0.651115097100317

31

0.682275215512988

32

0.435351966932894

33

0.148961992745456

34

0.305386520123207

35

0.682400839938918

36

0.685798293358962

37

0.547081326731316

38

0.903628961195317

39

0.784678357417467

40

0.790692597585520

41

0.974153022080358

42

0.725750333188450

43

0.686225815491298

44

0.735791042001489

45

0.892298117955924

46

0.656560157622049

47

0.379428069773115

5.1 Even and odd ACR properties of proposed sequences

Investigation is carried out for even and odd ACR properties. The results of maximum off peak even ACR and maximum off peak odd ACR values are tabulated in Table 4. From Table 4, it is found that maximum even ACR for the set of 47 sequences is -29.02 dB and the maximum odd ACR is found to be -27.28 dB. The histogram of off peak even ACR and the off peak odd ACR is shown in Figure 7 and Figure 8 respectively. From the Figure 7, it is found that 32 sequences out of 47 sequences are having off peak maximum even ACR in the range -29 dB to -29.2 dB. From the Figure 8, it is found that more than 9 sequences out of 47 sequences are having off peak odd ACR value better than -28.5 dB.

Table 4. Off peak maximum even ACR value and off peak maximum odd ACR value of the proposed binary seqeunces

Proposed Binary Sequence Number

Off Peak Maximum even ACR Value

Off Peak Maximum odd ACR Value

1

-29.217

-27.650

2

-29.023

-29.120

3

-29.416

-29.169

4

-29.120

-27.733

5

-29.723

-29.120

6

-29.120

-28.421

7

-29.416

-28.648

8

-29.517

-28.740

9

-29.023

-27.984

10

-29.416

-28.648

11

-29.023

-28.556

12

-29.619

-27.691

13

-29.217

-28.833

14

-29.120

-28.421

15

-29.120

-29.023

16

-29.723

-29.416

17

-29.120

-29.619

18

-29.023

-28.648

19

-29.217

-27.528

20

-29.316

-28.200

21

-29.217

-28.975

22

-29.023

-29.023

23

-29.120

-28.648

24

-29.316

-29.416

25

-29.517

-28.027

26

-29.120

-28.833

27

-29.316

-28.376

28

-29.217

-29.466

29

-29.316

-28.027

30

-29.217

-28.466

31

-29.023

-27.488

32

-29.120

-28.376

33

-29.120

-29.120

34

-29.619

-28.556

35

-29.217

-28.602

36

-29.023

-28.288

37

-29.217

-28.511

38

-29.416

-29.120

39

-29.120

-28.200

40

-29.023

-29.120

41

-29.023

-27.528

42

-29.217

-28.511

43

-29.023

-27.984

44

-29.316

-28.556

45

-29.933

-27.569

46

-29.120

-28.156

47

-29.217

-28.511

5.2 Even and odd CCR properties of proposed sequences

Investigation is carried out for even and odd CCR properties. The histogram of results of all 1081 combinations of pairwise maximum even CCR values and pairwise maximum odd even CCR values are plotted in Figure 9 and Figure 10 respectively. From the Figure 9, it is found that the maximum pairwise even CCR value is -26.01 dB and the occurrences of pairwise even CCR values are found to be populated near -28.5 dB. From Figure 9, it is found that the maximum pairwise odd CCR value is -26.32 dB and the occurrences of pairwise odd CCR values are found to be populated near -28.5 dB.

Figure 7. Histogram of maximum off peak even ACR for 47 proposed binary sequences

Figure 8. Histogram of maximum off peak odd ACR for 47 proposed binary sequences

The histogram results of all 1081 combinations of pairwise maximum even CCR and pairwise maximum odd CCR for a set of proposed 47 binary sequences are as shown in Figure 9 and Figure 10 respectively. From the histogram it is observed that for a set of 47 binary sequences of length 10230 bits, the maximum pairwise even cross correlation is -26.01dB and maximum pairwise odd cross correlation is found to be -26.32 dB.

Table 5 provides comparison of the correlation properties of the proposed sequences and GPS L2 CM sequences.

Table 5. Comparison of Correlation properties of proposed sequences and GPS L2 CM sequences

Sl No

Property

GPS L2 CM sequences

Proposed Sequences

1

Even ACR

-29.02 dB

-26.90 dB

2

Odd ACR

-27.48 dB

-27.24 dB

3

Even CCR

-26.21 dB

-25.39 dB

4

Odd CCR

-26.32 dB

-26.32 dB

From the Table 5 following observations are made

  • Proposed binary sequences outperform the GPS L2 CM sequences in maximum off peak even ACR by 2.12 dB.
  • Proposed set of binary sequences are found to have better even CCR by 0.82 dB as compared to the GPS L2 CM sequences.  
  • Proposed set of binary sequences are found to be having better maximum off peak odd ACR by 0.24 dB.
  • Odd CCR property is same for both the binary sequences.

Figure 9. Histogram of pairwise maximum even CCR of all 1081 combinations for 47 proposed binary sequences

Figure 10. Histogram of pairwise maximum odd CCR of all 1081 combinations for 47 proposed binary sequences

5.3 Linear Complexity properties of proposed sequences

The Linear complexity (LC) property profile of the generated 47 binary sequences of length 10230 bits are computed using Berleykamp- Massey Algorithm [16]. In Table 6, the values of LC for all the 47 proposed binary sequences along with their initial values are tabulated. It is found from Table 4 that the linear complexity for 10230 bits binary sequences proposed is 5115 ± 3. Security of the proposed sequences is excellent as compared to GPS L2 CM sequences.

The Linear complexity profile for proposed binary sequences is as shown in Figure 11. The dotted line represents the expected linear complexity value of 5115 for ideal random sequences of length 10230 bits. The continuous line represents the Linear Complexity of the Proposed 47 binary sequences. From the Figure 11 it is observed that the proposed binary sequences possess the near ideal linear complexity property.

Table 6. Initial values for generation of proposed binary sequences and linear complexity property of proposed binary sequences

Binary Seqeunce Number

Initial Value S0

LC value of generated sequence

1

0.656560157622049

5113

2

0.379428069773115

5116

3

0.956473170430473

5116

4

0.837323305625539

5116

5

0.186769582443968

5115

6

0.917937385063156

5116

7

0.449003245887149

5115

8

0.317813959862539

5115

9

0.210370839512595

5116

10

0.421562177994972

5116

11

0.115709453804408

5117

12

0.313397650853574

5115

13

0.452906953664362

5116

14

0.707796016680767

5114

15

0.354081569750622

5116

16

0.500947441246977

5115

17

0.353750830722854

5115

18

0.563401008883089

5116

19

0.656157553061328

5116

20

0.326484624595631

5115

21

0.682479882613947

5115

22

0.248820030659104

5115

23

0.677767695635484

5115

24

0.365907196587940

5115

25

0.479347941435340

5116

26

0.576065641236740

5115

27

0.216510493002707

5114

28

0.929057543334895

5115

29

0.832411734145950

5115

30

0.651115097100317

5115

31

0.682275215512988

5115

32

0.435351966932894

5115

33

0.148961992745456

5115

34

0.305386520123207

5116

35

0.682400839938918

5115

36

0.685798293358962

5115

37

0.547081326731316

5114

38

0.903628961195317

5115

39

0.784678357417467

5113

40

0.790692597585520

5116

41

0.974153022080358

5115

42

0.725750333188450

5114

43

0.686225815491298

5115

44

0.735791042001489

5112

45

0.892298117955924

5116

46

0.656560157622049

5116

47

0.379428069773115

5115

Figure 11. Linear Complexity profile of proposed set of 47 binary sequences of length 10230 bits in comparison of ideal random sequences Linear Complexity (LC) profile

6. Conclusion

In this work, a set of 37 GPS L2 CM binary sequences are generated. The odd correlation, even correlation and the linear complexity properties are analyzed for GPS L2 CM binary sequences. It is observed that the GPS L2 CM Sequences are obtained by truncating very long LFSR sequence and hence the correlation properties are degraded due to truncation. It is also found that the maximum off peak even ACR value is -26.90 dB, maximum of peak odd ACR value is -27.24 dB, maximum pairwise even CCR value is -25.39 dB, maximum pairwise odd CCR value is found to be -26.32 dB and LC value is 27.

A method is also proposed to generate binary sequences of length 10230 bits. A set of 47 binary sequences of length 10230 bits are generated. The generated binary sequences are analyzed for odd correlation, even correlation and linear complexity properties. For the set of 47 sequences of length 10230 bits it is found that maximum off peak even ACR value is -29.02 dB, maximum off peak odd ACR value is -27.24 dB, maximum pairwise even CCR value is -26.02 dB, maximum pairwise odd CCR value is found to be -26.32 dB and the LC is found to be 5115 ±3.

It is found that the proposed sequences as compared to GPS L2 CM binary sequences are better in having maximum off peak even ACR of 2.12 dB greater, pairwise maximum even CCR of 0.82 dB grater, maximum off peak odd ACR by 0.24 dB higher value. The generated sequences are found to have large LC of 5115 ±3 as compared to LC of 27 for GPS L2 CM binary sequences and hence provides excellent inherent security.

Due to the better correlation properties and large linear complexity the proposed sequences are suitable for use in secure GNSS Applications.

Nomenclature

Sk

real valued chaotic output. This is a dimension less quantity

B

bifurcation factor for chaotic logistic map. This is a dimensionless quantity

E

large positive integer, dimensionless quantity

G

positive integer, dimensionless quantity

Yk

integer valued output from the chotic generator. This is a dimension less quantity

Subscripts

k

kth symbol output from the proposed sequence generator

x

binary sequence x

y

binary sequence y

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