G Selecting the best family of binary PRN sequences for GNSS Galileo E1 channel, supported by the Electre III method

1

Abstract—In this article three families of binary PRN sequences

are evaluated as potential candidates for primary codes to be applied in GNSS Galileo E1 channel. Ordering of variants is supported by the ELECTRE III multiple criteria optimization method. The sequences issued by European Space Agency (ESA) are compared with the Weil-based sequences and the Sidelnikov-based sequences. Criteria reflect the essence of the problem, hence they are based on correlation properties of the sequences.

Index Terms—multicriteria optimization, ELECTRE III,

Galileo, Weil sequences, pseudo-random noise, satellite navigation

I. INTRODUCTION

ALOBAL Navigaton Satelite System (GNSS) Galileo is planned to be launched by ESA (European Space Agency) till the end of 2015. By that time, any improvements, which minimize the tracking error can be considered.

Galileo system, similarly to Global Positioning System (GPS) exploits Direct Sequence Spread Spectrum (DS-SS) in CDMA technology. Hence, signals originating from different satellites are transmitted in the same frequency band but they are distinguished by the receiver situated on the Earth through different PRN (Pseudo-Random Noise) sequences. Their autocorrelation function (ACF) should resemble Dirac pulse and the cross-correlation between different sequences should be close to zero [12] – [15]. Each chip is coded as –1 or 1, so the PRN coding sequences are binary. Moreover, there should be at least 66 different binary sequences which fulfill these requirements to provide reliable communications from 33 satellites. Research performed by the Author of this article confirm the steady impact of binary PRN sequences on the tracking error when the CNR (Carrier-to-Noise Ratio) is close to the sensitivity threshold of the receiving terminal [6].

In this article a decision problem of selecting the best family of spreading codes for Galileo E1 channel is solved with the Electre III multiple criteria optimization method. This decision-making technique helps to rank the allowed solutions and to choose the best variant [4].

In last two decades many attempts have been made to improve sensitivity of the GNSS systems. Turunen showed in [11] the impact of convolutional coding on signal acquisition

sensitivity. Replacing incoherent detection with a coherent two-stage way of reception resulted in a steady increase in the sensitivity threshold and better power utilization in subchannels E1B and E1C [13] Rushanen in [12] proposed to implement the modified Weil sequences to GPS L1C channel. A detailed mathematical description of these sequences was outlined in [10]. Two years later Rushanen compared the methods of generating and optimization of spreading codes for Galileo E1 and GPS L1 channels [14]. Consequently, Wallner et al. made three proposals of optimizing pseudo-random secondary codes for Galileo E1 channel [1]: modified interplex modulation, majority voted codes and the Sidelnikov/Lempel/Cohn/ Eastman sequences.

In Galileo E1 channel the signal is modulated with Composite Binary Offset Carrier (CBOC) modulation and the spreading codes are tiered codes such that repetitions of

primary codes modulate the consecutive chips of the secondary code. The length of the primary code is equal to

4092 [9]. In [6] it was shown that the family of binary Sidelnikov-based sequences generated according to the procedure described in [7] yields significantly lower tracking error for CNR=-3dB than the original sequences recommended by ESA in [9]. A detailed inspection confirmed the outstanding properties of this family of sequences.

However, the problem of selecting the family of binary, pseudorandom primary codes for Galileo E1 channel has remained unsolved. In this article three sequence families are considered as potential candidates to be launched in GNSS Galileo system. A comparison between these alternatives is performed with aid of multiple criteria optimization method ELECTRE III, which helps to rank the sequence families. Hence, we can fairly compare different solutions of the problem by using a formal method of evaluation .

In Section II the Electre III method is briefly recollected. This description is enriched with figures and formulae.

In Section III the Electre method is applied to solve the problem under consideration. Mathematical criteria and the preference thresholds are formulated.

In Section IV the results are presented in tables and the final ranking is performed.

The last Section summarizes the article. The conclusions about further research are drawn.

Selecting the best family of binary PRN

sequences for GNSS Galileo E1 channel,

supported by the Electre III method

Michał PILC, Pozna

Ĕ University of Technology, Poland

G

2 II. THE ELECTRE III METHOD

A. Multiple criteria problems

Multiple criteria (MC) optimization methods are widely applied in many branches of human activity like finance, transportation, agriculture but also telecommunications [4]

According to the target of the decision making process MC decision problems can be classified as [5]:

- problems of choice, where one variant is chosen from a given set of allowed solutions;

- problems of ordering, where the set of variant is ordered; - classifications problems where the variants are assigned to

predefined classes.

The first group of problems is usually formulated as MC mathematical programming tasks, i.e. multiple criteria optimization. Selection of the most promising family of PRN codes for GNSS Galileo system is the classification problem.

Multiple criteria decision making (MCDM) methods are divided by most of professionals into three distinct classes [5]:

a) based on multi-attribute rating technique, which do not consider incomparability of variants e.g. AHP; b) based on outranking relation, which take into account

incomparability of variants, e.g. ELECTRE; c) interactive methods based on the attempt and error

approach in each iteration.

The ELECTRE methods are discussed below.

B. The ELECTRE III method

The ELECTRE methods were introduced by B. Roy. In 1978 Bernhard Roy introduced an MCDM method of ordering the finite set of variants evaluated with a collection of criteria [3]. The ELECTRE methods were later improved and upgraded. [4, Chapter 4].

Mathematical functions used in this MCDM technique, show the degree of dominance of one variant over the other. Its computational algorithm consists of three stages [5]:

1) construction of the matrix of grades and definition of the preference model;

2) creation of the outranking relation; 3) exploitation of the outranking relation.

Let X={ ,x x1 2,!,xN} be a set of variants and

1 2

{ , , , L}

C= c c !c be a set of criterions. Each alternative ,

n

x where 1 n≤ ≤ , is evaluated with a real-valued function N

( )

l n

g x representing the performance of the variant xnwith

respect to the criterion cl, where 1 l≤ ≤ [3]. For every L

criterion c a decision-maker’s preference model is expressed l

with the aid of four relations [4, 5]:

- an indifference relation between variants x andi x , . j i.e. xiIlx ; j

- a weak preference relation of variant xi over x , j xiQlx ; j

- a strong preference relation of variant x over i x , j xiPlx ; j

- a relation of incomparability of variants, xiJlx . j

The weak preference area expresses the decision-maker’s hesitation between indifference and preference [2].

Any ordered pair of variants can be classified to one and the only one relation from the above-mentioned. This operation is performed with help of the following thresholds [2]:

- an indifference threshold, i.e. ql

(

g xl( n)

)

= ; ql

- a preference threshold, i.e. pl

(

g xl( n)

)

= ; pl

- a veto threshold i.e. vl

(

g xl( n)

)

= ; vl

xiQlxj xiIlxj xjQlxi xjPlxi xjJlxi gl

( )

xj −ql gl

( )

xj +ql gl

( )

xj +vl

( )

j

l x l

g −p gl

( )

xj gl

( )

xj +pl g xl( )i Fig 1. Thresholds and relations between variants in ELECTRE III These thresholds classify the pairs of variants with the afore-mentioned relations as it is illustrated in Fig. 1.

Furthermore, for every individual criterion we define a weight expressing its credibility and a direction of preference (maximize, minimize).

In the second stage an outranking degree

(

x xi, j

)

S describing the credibility of the hypothesis that variant xi outranks x is evaluated for every pair of j

alternatives. Its values are placed between 0 and 1. They are based on concordance and discordance indices [2, 5].

For every ordered pair of variants the concordance index is calculated as [2]: ( ) ( ) 1 1 , , L i j l l i j l x x x x C w c w = =

¦

⋅ , (1)

where w is a weight assigned to the l-th criterion (l c ) and l

1 L l l w w = =

¦

. (2)

Similarly we define a discordance index d

(

x xi, j

)

for any

ordered pair of criteria

(

x xi, j

)

.

(

)

(

)

, , i j l i j l x x c x x d 1 dl

(

x xi, j

)

cl

(

x xi, j

)

0 gl( )xi −vl gl( )xi −pl gl( )xi −ql g xl( )i gl

( )

xj Fig 2. Definition of the concordance index and the discordance index in

MCDM method ELECTRE III

Formulae expressing both coefficients are given in [2] or [5,p. 62]. If the l-th criterion is to be minimized, then the values of these indices are given in Fig. 2. Finally, the degree of

outranking is evaluated as [2]: ( ) (₍ )₎ ( ₍ ) ₎ ( ) ( ) ( , ) , if , , , 1 , , otherwise , 1 i j i j l i j l i j l i j _{l i j} i j i j x x l J x x x x x x C d C x x S d x x x x C x x C ∈ ­ ∀ ≤ °° = ® _⋅ − ° ₋ °¯

∏

, (3)

3

where J

(

x xi, j

)

is the set of criteria for which

(

i, j

)

(

i, j

)

l x x x x

d ≥C . For any ordered pair of variants, the values of S

(

x xi, j

)

are stored in an outranking matrix

( )

(

i, j

)

N N x x S _× = S . (4)

In the last stage of ELECTRE III the variants are ordered based on the degree of outranking S

(

x xi, j

)

. At the beginning

of this stage we define the following value [5, p.64]:

( ) , ,

max

i j i j x x X x x S λ ∈ = . (5)

Moreover, a cut-off threshold s( )λ distillates only these pairs of alternatives

(

x xi, j

)

for which S

(

x xi, j

)

≥ −λ s( )λ .

Then, for any variant x a qualification index k Q x( k) is

calculated as the difference between the number of variants outranked by x and the number of variants outranking k x [2]. k

Finally, the ordering of variants is performed with aid of two classification algorithms: ascending preorder (from the worst to the worst) and descending preorder (from the best to the worst) [5, p.64].

III. PROBLEM STATEMENT

Choice of the best family of pseudo-random binary codes for GNSS Galileo E1 channel can be performed with help of multiple criteria optimization methods. The ELECTRE III MCDM method can be successfully applied to order the possible solutions of the problem under consideration. Let us pay attention to three PRN sequence families which can be regarded as variants or alternatives in the language of MC optimization, meanwhile:

1) the Weil-based sequences created from p=4093 and then modified and optimized as in [10] –  ; 1

2) the family of sequences recommended by ESA called further the original sequences –  ; 2

3) the Sidelnikov-based sequences generated with the algorithm described n [7] –  . 3

Let

{

(1)_, (2)_{, ,} (N)

}

j = x x x

 ! , where j∈

{

1, 2,3

}

be one of the afore-mentioned families of PRN sequences. This set consists of N=66 members such that:

(

)

( ) ( ) ( ) ( ) 1 2 {1,2 , , } , , , n n n n L n N x x x ∈

∀

= x ! " . (6) and xl( )n ∈ −

{ }

1;1 is the l-th term of the n-th sequence in the

family  . For these three families of PRN codes the following criteria can be calculated:

1) maximum imbalance of sequences inside one family

j  , where j∈

{

1, 2,3

}

, i.e.:

( )

( ) 1 1 1

max

L n j l n N l g x ≤ ≤ = =

¦

 ; (7)

in ideal case this value is equal to 0, which means that in every PRN code inside the considered family the number of 1’s is the same as the number of –1’s; 2) average imbalance of sequences inside one family, i.e.

( )

( ) 2 1 1

max

L n j l n N l g x ≤ ≤ = =

¦

 ; (8)

3) maximum (over N sequences) of the highest absolute

out-of-phase values of the autocorrelation function (ACF) of sequence ( )n j ∈ x  , i.e.: ( ) ( ) ( ) 3 1 0 1

max max

L n n j l k l n N k l g x x₊ ≤ ≤ ≠ = ­ ½ = ® ¾ ¯

¦

¿  , (9)

where k+ is taken modulo L and l n∈

{

1, 2,!,N

}

; 4) average (over N sequences) of the highest out-of-phase

values of the ACF of sequences ( )n j ∈ x  , i.e.: ( ) ( ) ( ) 4 0 1 1 1

max

N L n n j l k l k n l g x x N = ≠ = + ­ ½ = ® ¾ ¯ ¿

¦

¦

 ; (10) 5) maximum (over 1 ( )

2N N− pairs of sequences) of the 1

highest absolute cross-correlation values between pairs of sequences ( )m_, ( )n j ∈ x x  , i.e.: ( ) { } ( ) ( ) 5 1 0 ,1, , 1 1

max max

L m n j l k l k m n N L l x x g + ∈ ≤ < ≤ − = ­ ½ = ® ¾ ¯

¦

¿  ! ; (11) 6) average (over 1 ( )

2N N− pairs of sequences) of the 1

highest absolute cross-correlation values between pairs of sequences _x( )m_,x( )n _{∈ , i.e.:} ( ) { } 1 ( ) ( ) 6 0 ,1, , 1 1 2 1 2 ( 1)

max

L N n m n j l k l k L l n m x x g N N − + ∈ − = = = ­ ½ = ₋ ® ¾ ¯

¦

¿

¦ ¦

 ! ; (12) 7) the highest RMS tracking error (for all PRN code pairs

among one family) observed for SNR=33 dB evaluated by Galileo TUT simulator [8] in simulations lasting 4s:

( )

{ }

7 j RM S tra_err [ ]

g  = m ; (13)

IV. SIMULATION RESULTS

The values of the afore-mentioned functions (7-12) were determined with Matlab scripts.

The RMS tracking error of Galileo receiver was evaluated with Simulink-based Galileo TUT simulator documented in [8]. The simulation results are stored in Table I.

In the next stage the q, p, v threshold values and the weights are assigned to every criterion. These thresholds were chosen arbitrarily based on the prior knowledge of the essence of the considered problem. They are stored in Table II.

TABLE I

CRITERIA VERSUS ALTERNATIVES IN CASE OF THE PROBLEM UNDER CONSIDERATION

Alternatives

Criteria Weil ( ) ESA 1 ( ) Sidelnikov 2 ( ) 3 ( ) 1 j g  0 0 2 ( ) 2 j g  0 0 0.2 ( ) 3 j g  182 220 130 ( ) 4 j g  160 193 126 ( ) 5 j g  202 244 130 ( ) 6 j g  185 211 128 ( ) 7 j g  58 716 36

4

Next, the concordance credibility degree and the discordance credibility degree are evaluated for every criterion and for every pair of sequence families. These parameters are stored in Table III.

Finally, we evaluate the degree of outranking for all pairs of variants based on formula (3). An example of calculations is given for pairs

(

 ₂, ₃

)

, that is

( ) ( ) 7 2 3 2 3 2 3 2 3 1 ( , ) 1 0.84 0.2 0.04 , , 1 ( , ) 1 0.2 d S C C − − = = ⋅ = − −         ;(14)

Similarly, S

(

 ₂, ₁

)

=0.12. The rest of values of the outranking relation is equal to the respective concordance credibility degree.

Fig. 3 Final ranking of the considered PRN sequence families

In the last stage of the ELECTRE III method we evaluate 1

λ= (5) and presume ( ) 0.1sλ = .

Hence, three pairs are considered in the ranking phase, i.e.

(

  1, 2

)

(

  , 3, 1

)

(

  . 3, 2

)

For all families of the PRN sequences the qualification index is computed as follows: Q

( )

1 =0, Q

( )

2 = −1,

( )

3 2

Q  = . The ranking in ascending preorder and in descending preorder is the same.

V. CONCLUSION AND FURTHER RESEARCH The final ranking presents itself accordingly:

1st _place: 3

 – Sidelnikov-based sequences; 2nd place: 1 – modified Weil-based sequences; 3rd _place:

2

 – Weil-based sequences.

However, a few remarks have to be made. First, the criteria and the weights assigned to them should be evaluated very carefully. A slight change in parameters fixed arbitrarily beforehand may change the decision, which should be avoided.

ACKNOWLEDGMENT

TUT Simulink Galileo E1 simulator was used to receive the results, which is here acknowledged.

REFERENCES

[1] S. Wallner et al. “Revised PRN Code Structures for Galileo E1 OS,”

ION GNSS 21st _{International Technical Meeting of the Satellite}

Division, 2008.

[2] H. F. Li, J. J. Wang, “An improved ranking method for ELECTRE III,” in International Conference on Wireless Communications, Networking

and Mobile Computing 2007, WiCom 2007, pp. 6659–6662.

[3] B. Roy, “The outranking approach and the foundations of the ELECTRE method,” Theory and Decision, vol. 31, pp. 49-73, 1991. [4] J. Figueira (ed.) “Multiple criteria Decision Analysis,” Chapter 4:

Electre methods, 2005 Springer Science + Business Media, Inc.,

Boston, 2005.

[5] J. ĩak „Wielokryterialne wspomaganie decyzji w transporcie drogowym (Multiple Criteria Decision Support in Road Transport),” Postdoctoral Dissertation, PoznaĔ University of Technology, Poland, 2005.

[6] M. Pilc “Metody generowania ciągów pseudolosowych dla kanału E1 systemu nawigacji satelitarnej Galileo (Methods of generating PRN sequences for channel E1 in GNSS Galileo),” Krajowa Konferencja Radiokomunikacji Ruchomej, Radiofonii i Telewizji (National Conference on Mobile Communications, Radio and Television), PoznaĔ, 2011

[7] N. Y. Yu, G. Gong “New Construction of M-ary Sequence With Low Correlation From the Structure of Sidelnikov Sequences,” University of Waterloo, Canada, 2010

[8] S. Lohan et.al “TUT Simulink Galileo E1 signal simulator,” technical

documentation, Tampere University of Technology, Finland, 2009 [9] Galileo Open Service Signal in Space, Interface Control Document, OS

SIS ICD Draft 1, European Space Agency (ESA). 2008.

[10] J. Rushanen (the MITRE Corporation), “Weil sequences: A family of Binary Sequences With Good Correlation Properties,” International

Symposium on Information Theory, Seatle (USA), 2006

[11] S. Turunen (NOKIA corporation), “Can code redundancy Be Used to Improve the GNSS Receiver Acquisition Sensitivity?,” in Position

Location and Navigation Symposium, pp. 1094-1099, IEEE/ION, 2006.

[12] J. Rushanen (the MITRE Corporation), “The Spreading and Overlay Codes for the L1C,” Navigation, vol. 54, No. 1, pp. 43-51, 2007. [13] K. Sun “Composite GNSS Signal Acquisition in Presence of Data Sign

Transition,” 2010 International Conference on Indoor Positioning and

Indoor Navigation, Zürich, Switzerland, 2010.

[14] S. Wallner, J. Avila-Rodriguez, G. W. Hein, J. Rushanen “Galileo E1 OS and GPS L1C Pseudorandom Noise Codes – Requirements, Generation, Optimization and Comparison,” ION GNSS 20th

International Technical Meeting of the Satellite Division, 2008.

TABLE II

THRESHOLD VALUES, WEIGHTS AND DIRECTIONS OFPREFERENCE FOR THE INSPECTED CRITERIA

Criteria Units ql pl vl wl dp ( ) 1 j g  - 4 50 1500 10 min ( ) 2 j g  - 2 25 750 10 min ( ) 3 j g  - 10 60 1000 20 min ( ) 4 j g  - 5 30 500 15 min ( ) 5 j g  - 20 120 1500 5 min ( ) 6 j g  - 10 60 750 10 min ( ) 7 j g  [m] 5 45 800 30 min TABLE III

CONCORDANCE DEGREE FOR THE OBSERVED PAIRS OF PRN SEQUENCE FAMILIES

Ordered pair of variants

Criteria (  1, 2) (  1, 3) (  2, 1) (  2, 3) (  3, 1) (  3, 2) ( ) 1 j g  1 1 1 1 1 1 ( ) 2 j g  1 1 1 1 1 1 ( ) 3 j g  1 0.16 0.44 0 1 1 ( ) 4 j g  1 0 0 0 1 1 ( ) 5 j g  1 0.48 0.78 0.06 1 1 ( ) 6 j g  1 0.03 0.68 0 1 1 ( ) 7 j g  1 0.58 0 0 1 1 ( i, j) C   1 0.43 0.39 0.20 1 1 Sidelnikov ESA Weil-based