Detection of Gold Codes Using Higher-Order Statistics

Mamdouh Gouda Misr University for Science & Technology

Cairo, Egypt

Adel El-Hennawy Ain Shams University

Cairo, Egypt

Ahmed Ezzat Mohamed Ain Shams University

Cairo, Egypt

Abstract— In this paper the higher-order statistics (HOS) specified in terms of triple correlation function (TCF) of Gold code which is commonly used in spread-spectrum systems is studied. A detection method of Gold sequence is presented and the proposed gold code receivers based on the peak feature of the TCF are analysed. These receivers take advantage of the TCF and use it for self-synchronization. We study two types of TCF receivers; one-stage receiver which is used with divisible by three code length and two-stage receiver which can be used with any code length. A comparison between the performances of both receivers in Additive White Gaussian Noise (AWGN) channel is performed to prove the immunity of the receivers against different noise levels. Keywords— Gold code; spread spectrum; triple-correlation; HOS; Detection;

I. INTRODUCTION

Direct sequence spread spectrum (DSSS) signal is the typical form of low probability of interception signals, which has been used as secure communication for several decades [1]. Thus the estimation of information sequence and pseudo noise sequence are of great importance in electronic warfare or wireless communication supervision. Gold sequences have favourable cross-correlation properties [2]. If the linear feedback shift register (LFSR) are chosen appropriately, Gold sequences have better cross-correlation properties than maximum length LFSR sequences. Previous work proposed a method based on higher-order statistical (HOS) analysis where triple correlation function (TCF) was used to estimate the primitive polynomial of m-sequences [3].

II. TRIPLE CORRELATION OF M-SEQUENCE

Triple correlation function is useful for distinguishing m-sequences where the shift and add property produces TCF peaks. The locations of TCF peaks are used to find the LFSR function which generates an m-sequence [4].

Suppose � (i) to be an m-sequence of length L = 2n – 1 where n is the number of stages used in LFSR and chip duration �c, let the code chips map as (0, 1) � (-1, 1) so that addition modulo-2 can be replaced by multiplication, its TCF is [5]:

( ) ( )1 21

1, , ( ) ( ) ( )

L

p qi

C C p q i i iL

τ τ ν ν ν=

= = � (1)

In which, ( )p iν and ( )q iν are generated by shifting

( )iν by 1 cpτ τ= and 2 cqτ τ= respectively.

According to the shift and add property for certain shift pairs (p, q), we have ( ) ( ) ( )p qi i iν ν ν= . So for these pairs:

[ ]2

1

1( , ) ( ) 1

L

i

C p q iL

ν=

= =� (2)

For other pairs (p�, q�), ' '( ) ( ) ( )p q si i iν ν ν= , where

( ) ( )s i iν ν≠ , then:

1

1 1( ', ') ( ) ( )

L

si

C p q i iL L

ν ν=

−= =� (3)

So, the shift and add property of m-sequences results in TCF peaks. The location of the peaks differs from a feedback connection to another and so they represent a characteristic feature for each sequence.

III. TRIPLE CORRELATION OF GOLD CODE

A Gold code family can be generated by a preferred pair of m-sequences (u, �) of period L = 2n – 1 and expressed as [6]:

( ) { }2 1, , , , , ,.., LG u u u u T u T u Tν ν ν ν ν ν−

= ⊕ ⊕ ⊕ ⊕ (4)

Where Tk� is the sequence � delayed by k bits and ⊕ indicates modulo-2 addition. For the L Gold sequences, individual bits may be expressed:

( ) mod ,0 1ki i i k Lw u i Lν

+= ⊕ ≤ ≤ − � (5)�

Gold codes have no triple correlation peaks like m-sequence as they don’t obey the shift-and-add property unless the length of the code is divisible by three [7]. The triple correlation function of Gold codes contains sufficient information for the detection of m-sequences that constitutes the Gold code. The TCF of wk is defined:

( )1

0

1,

Lk k k k

i i p i qi

C p q w w wL

−

+ +

=

� �= � �� (6)

Using shift and add rule:

, {0,1,..., 1}

,

p kp

p

k ki i p i i k i p i k p

i i p i k i k p

i r i S p kp

li r kp p

w w u u

u u

u r and S L

w l S r

ν ν

ν ν

ν

+ + + + +

+ + + +

+ +

+

=

=

= ∈ −

= = −

(7)

2011 First International Conference on Informatics and Computational Intelligence

978-0-7695-4618-6/11 $26.00 © 2011 IEEE

DOI 10.1109/ICI.2011.65

361

2011 First International Conference on Informatics and Computational Intelligence

978-0-7695-4618-6/11 $26.00 © 2011 IEEE

DOI 10.1109/ICI.2011.65

361

Substituting (7) in (6):

( )1

0

1,

p

Lk l k

i r i qi

C p q w wL

−

+ +

=

= � (8)

The TCF is thus the cross-correlation of two Gold sequences from G(u,�). This cross-correlation is 3-valued, so:

( ) 1 0 1

1 0 1

( 1)/2

( 2)/2

, ,

:

( ) / , 1/ , [ ( ) 2] /

1 2( )

1 2

k

n

n

C p q or

Where

t n L L t n L

for odd nt n

for even n

θ θ θ

θ θ θ

−

−

+

+

=

= − = − = −

� += �

+

(9)

(6) may be rewritten:

( )1

0

1,

Lk

i i k i p i k p i q i k qi

C p q u u uL

ν ν ν−

+ + + + + + +

=

= � (10)

Consider the case where (p�, q�) is a TCF peak location for m-sequence u, i.e. ' 'i i p i qu u u

+ += for 0 � I � L-1. As

2' ' 1i i p i q iu u u u

+ += =

( )1

' '0

1', '

Lk

i k i k p i k qi

C p qL

ν ν ν−

+ + + + +

=

= � (11)

If (p�, q�) is a TCF peak for u it can’t be a TCF peak for �. As phase is irrelevant to the TCF:

( )' '

' '

1

0

1

0

1', '

1

p q

p q

Lk

i k i k ri

L

i i ri

C p qL

L

ν ν

ν ν

−

+ + +

=

−

+

=

=

=

�

� (12)

But, this expression for the ACF of � yields just one value as rp�q� cannot be zero:

( )1

', 'kC p qL

−= (13)

A similar reduction occurs when (p�, q�) is a TCF peak for �, and automatically not for u:

'' ''

1

'' ''0

1

'' ''0

1( '', '')

1, 0

1

p q

Lk

i i p i qi

L

i i S p qi

C p q u u uL

u u SL

L

−

+ +

=

−

+

=

=

= ≠

−=

�

� (14)

All gold codes of the same length have the same two peaks (L/3, 2L/3) and (2L/3, L/3) as illustrated in Fig. 1 which shows the TCF for Gold code of length 63 generated by the preferred polynomials X6+X+1 and X6+X5+X2+X+1. It is obvious that there are other peaks due to the cross-correlation between the two generating polynomials of Gold code. The other peaks’ values can be calculated from (9) to be -0.0159, -0.2698 and 0.2381 which can be removed by taking a threshold for better detection. Fig. 2 shows the peaks using a threshold that is 1/3 of the maximum peak. The contour plot shown in Fig. 3 demonstrates the position of the peaks (42, 21) for Gold code of length 63.

Fig. 4 shows the triple correlation peaks for a Gold code of length 1023 generated by the preferred polynomials X10+X3+1 and X10+X8+X3+X2+1 using a threshold that is 1/3 of the maximum peak.

Figure 1. Peaks of triple correlation for Gold code of length 63, without noise effect

Figure 2. Peaks of triple correlation for Gold code of length 63, without noise effect using a threshold of 1/3

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Figure 3. Distribution of the triple correlation for Gold code of length 63,

without noise effect using a threshold of 1/3

Figure 4. Peaks of triple correlation for Gold code of length 1023, without

noise effect using a threshold of 1/3

The peak locations calculated in this section are summarized in Table 1.

TABLE 1. DELAY SHIFTS IN TCF PEAKS OF TWO GOLD CODES

Period (Chips)

Preferred Polynomial [1]

Preferred Polynomial [2]

Delay shifts for peaks in triple

correlation 63 [6 1 0] [6 5 2 1 0] (42,21), (21,42)

1023 [10 3 0] [10 8 3 2 0] (682,341), (341,682)

IV. PROPOSED TC GOLD CODE RECEIVERS

Two kinds of receivers are demonstrated; 1-stage and 2-stage receivers. The first type can be used if the Gold code length is divisible by three. On the other hand, the 2-stage receiver can be used with any code length where it uses the TC peaks of the pair of m-sequences that forms the Gold code.

Fig. 5 shows the block diagram of the 1-Stage triple correlation gold code receiver. This stage consists of two delay elements (p, q), a modulo-2 adder, Integrator and a Decision device (sign detector). This is the same receiver

used for m-sequence codes [8]. The delays are tuned to the triple correlation peaks of the used Gold code (p�, q�) which was calculated in the previous section. The integration period is from 'T L pΔ = − to L where ' 'p q> . The final

decision is taken by the decision device defined as:

( ) 1, ( ) 0 ( ) 1, ( ) 0y t if b t and y t if b t= − < = > � (15)�

Figure 5. Triple correlation Gold code 1-Stage receiver

Fig. 6 shows the 2-Stage triple correlation gold code receiver, where (-p�, -q�) and (-r�, -s�) are the TCF optimum peaks for the m-sequences u and � that constitute the Gold code respectively.

First, it is required to determine the optimum delay shifts for each generating polynomial. So, TCF for each polynomial is calculated and the location of peaks is determined as shown in Table 2.

There are other TCF peaks but the chosen ones are the optimum as they satisfy two factors [9]. First factor is choosing the minimum difference between delay pairs and the second factor is to choose the minimum delay pairs.

TABLE 2. OPTIMUM DELAY SHIFTS OF THE PREFERRED M-SEQUENCE PAIRS THAT FORM TWO GOLD CODES

Period (Chips)

Preferred Polynomial

[1]

Optimum delay shifts

Preferred Polynomial

[2]

Optimum delay shifts

63 [6 1 0] (5,6) [6 5 2 1 0] (3,11) 1023 [10 3 0] (7,10) [10 8 3 2 0] (25,49)

At (-p�, -q�), Stage-1 output:

' ' ' ' ' 'i i i p i p i q i q i i p i q i ju u uν ν ν ν ν ν ν− − − − − − +

= = (16)

At (-r�, -s�), Stage-2 output:

' ' 1i j i j r i j sν ν ν+ + − + −

= � (17)�

The integration period is from 'T L pΔ = − to L

where ' ' ' 'p q r s> > > .

In the next section we will compare between the performances of the two types of receivers.

�T

L

Figure 6. Triple correlation Gold code 2-Stage receiver

�T

L

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V. PERFORMANCE OF TC GOLD CODE RECEIVERS IN AWGN

CHANNEL

In this section, several simulations have been completed to testify the performance of the previous two types of receivers. Let the received signal be r (t) = z (t) + n (t) where z (t) is the transmitted signal and n (t) is zero mean AWGN. Calculation of Bit Error Rate (BER) is done for the received data of Gold code by varying Eb/N0 for code length of 63 and 1023. The simulation result for a code length of 63 and Processing Gain (PG) of 63 is shown in Fig. 7. As shown by the analysis, the performance of the 1-stage receiver is better than the 2-stage receiver.

The performance for the Gold code receivers of length

1023 is shown in Fig. 8 and Fig. 9 with PG of 1023 which is much better than Gold code receivers of length 63.

Figure 7. Eb/N0 vs. BER for the receivers of length 63

Figure 8. Eb/N0 vs. BER for the 1-Stage receiver of length 1023

Figure 9. Eb/N0 vs. BER for the 2-Stage receiver of length 1023

VI. CONCLUSIONS

Triple correlation analysis provides a powerful means for detecting Gold codes used in modern cellular spread spectrum communication systems. The TC receivers allow the detection of Gold code in high noise environment. The results show that the 1-stage receiver is used only for divisible by three code length, while the 2-stage receiver is used for any code length and the 1-Stage receiver has better performance than the 2-stage receiver. Also, it was shown that the performance of both receivers improves by increasing the Processing Gain.

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and Sons, New York, 1994. [2] R. Gold, "Maximal Recursive sequences with 3-valued cross

correlation functions", IEEE Trans. Inform. Theory, vol. 14, 1968, pp. 154-156.

[3] E. R. Adams, M. Gouda & P. C. J. Hill, "Detection and characterization of DS/SS signals using higher-order correlation", Proceedings of IEEE 4th ISSSTA, Mainz, Germany, Sep. 1996, pp. 27-31.

[4] X. Zhi-cai, X. Bo, "Identification of Feedback Logic for m-Sequence Based on Competitive Network" WiCom 2007, pp. 1426- 1428.

[5] K. Batty and E. R. Adams, "Detection and blind identification of m-sequence codes using higher-order statistics" Proceedings of the IEEE Signal Processing Workshop, 1999, pp. 16-20.

[6] D.V. Sarwate and M.B. Pursley, "Crosscorrelation properties of pseudorandom and related sequences", Proceedings of the IEEE, Vol. 68, May 1980, pp. 593-619.

[7] E. S. Warner, B. Mulgrew & P.M. Grant, "Triple Correlation Analysis of Binary Sequences for Codeword Detection", IEE Proc.in Sig Proc., vol. 141, Oct. 1994, pp.297-302.

[8] M. Gouda, "Proposed triple correlation receiver for CDMA", ICACT 2010, pp. 909 – 914.

[9] M. Gouda, A. Abdin, E. Essawy "Optimum detection of data modulated m-sequences using triple correlation receiver", Signals, Circuits and Systems, 2008, pp. 1-4.

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