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Vol. 4, Issue 1, Jan-June 2013, pp- 33-43 DOI: 10.5958/j.1945-919X.4.1a.003

Study on Cyclic Cross-Correlation Behaviour of Maximal Length Pseudo-Random Binary Sequences A. Ahmad*, S. Al-Busaidi, M.J. Al-Mushrafi and J.A. Jervase

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Department of Electrical and Computer Engineering, College of Engineering, Sultan Qaboos University, P.O. Box 33, Postal Code 123, Muscat, Sultanate of Oman *Corresponding author: Email: [email protected]

Abstract: Cyclic cross-correlation function between image pairs of maximal length pseudo-random binary sequences is studied. It is found that the peak of the cross-correlation remains constant for every possible image pairs of the sequences of the same length. However, this unique property is not valid for all the other possible pairs. Keywords: Cyclic cross-correlation, Pseudo-random recursive binary sequence, PRRBS, PN code, m-sequences, LFSR

1. INTRODUCTION Due to their unique properties, maximal length pseudo-random recursive binary sequences (PRRBSs) (often called m-sequences or pseudo-random number (PN) codes), generated by different structures of autonomous linear feedback shift registers (LFSRs), are unexcelled for general use in communications and ranging. The careful investigation of the used code is an essential process before its application, even if that code is one of the relatively safe m-sequences [1-4]. PRRBSs have an advantageous feature from the computational viewpoint, and they tend to have useful structural properties. Due to only these structural properties, recursive binary sequences have enormous applications; for example, direct sequence spread spectrum, PN generation, built-in selftest, encryption-decryption and error detection [5-18]. The study of the cyclic correlation function of two PN codes is of great importance. The situation where large numbers of transmitters, using different codes, share a frequency band necessitates the careful selection of PN codes to avoid interference between the users. The effect of a high degree of correlation between an undesired code received and receiver reference is an increase in the receiver’s false alarm rate and, under extreme circumstances, false recognition of synch by the receiver. An even worse possible effect of poorly chosen codes with high cross-correlation is that a jammer might transmit a code from the set being used. This in turn could cause every receiver within range of the jammer to be affected by partially correlating with their reference codes and thereby causing false synchronisation. In some applications, the cross-correlation properties of PN codes

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are as important as the auto-correlation properties. For example, in code division multiple access systems, where each user is assigned a unique sequence, high values for the cross-correlation of these sequences are undesirable [19-28]. This paper presents the results of a study of the behaviour of cyclic cross-correlation function of maximal length PRRBSs. The results reveal that the peak of the cyclic cross-correlation function of each pair of m-sequences of the same length generated by a corresponding primitive and its reciprocal polynomial of order n (n ≥ 5), respectively, remains constant. However, this function between any pair of m-sequences of the same length generated by two corresponding primitive polynomials (which are not reciprocal to each other) may or may not have the larger peaks.

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2. MATHEMATICAL MODEL FOR CROSS-CORRELATION PROCESS Basically, in principle, cross-correlation is a statistical property of a signal, which is related to averages and probabilities. In other words, correlation is a mean measuring the similarities between signals. It is to be learned here that the convolution and cross-correlation processes are closely related to each other. Typically, convolution and cross-correlation both use shifting, multiplying and integration operations on time waveforms. The mathematical formula for correlation of two signals can be given as shown in Equation (1). (1) where x(t) and y(t) are the signals, the variable ‘t1’ is the time shift applied to y(t) and the c(t) is cross-correlated signal. For discrete signals, Equation (1) can be written as follows: (2) Equation (2) describes the cross-correlation of two binary sequences, namely, a and b, having the sequence length of n. For different length of sequences, the value of n of Equation (2) can be evaluated using Equation (3). n = LCM (n1,n2)

(3)

where n1 and n2 are the lengths of sequences a and b, respectively. Thus, cross-correlation process can be achieved by the procedure given below. • Shift b(i) by j. • Multiply a(i) and b(i+j) together. • Now, find the average value of the resulting multiplication, and this is the integration process described in Equation (1). Example 1 Let a = (0, 1, 0, 1, 0, 1, 1) and b = (1, 1, 0, 1, 0, 0, 1) be the binary sequences where n is 7. Then, cross-correlation ‘c’ can be computed and verified as given below. The cross-correlation can also be visualised in Figure 1. c = [0.00, 1.00, 0.00, 1.00, 1.00, 1.00, 3.00, 1.00, 2.00, 2.00, 1.00, 2.00, 1.00]

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(see Figure 4). Table 3 describes this property for a PRRBS periodicity 2n-1 with the assumption that the clock pulse of LFSR has time period T.

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Table 3: Pulses generated in PRRBS Number of pulses 1 1 For 1≤ x ≤ n -2 ; 2x-1

Pulse width nT (n-1)T (n-x-1)T

Nature of pulse Active high Active low Active high

For 1≤ x ≤ n -2 ; 2x-1

(n-x-1)T

Active low

5. STUDY In this study, the PRRBS are generated through the simulation of a dynamic model of an n-bit LFSR, where an efficient and easy search method is incorporated to compute all possible primitive polynomials of order n and their reciprocals [5, 6]. The procedure adopted to determine the cyclic cross-correlation between all possible PRRBS (of length 2n--1) is summarised in the form of an algorithm and is given below. Algorithm Begin Let n = length of LFSR used to generate the PRRBSs; Compute the total number of possible primitive polynomials of order n, i.e., NPP; For k = 1, 3, 5, …, NPP-1, do Begin Generate primitive polynomials Pk(x); Compute the reciprocal Pk+1(x) of Pk(x); End do; For i=1 to NPP, do Begin Generate PRRBS ‘Si’ corresponding to Pi(x); For j=i+1 to NPP, do Begin Generate PRRBS ‘Sj’ corresponding to Pj(x); For m=1 to 2n--1, do Begin Generate the sequence Sjm {Sjm is the m-bit cyclic shifted version of the sequence Sj}; Compute the cross-correlation between Si and Sjm; End do; Write maximum and minimum values of cross-correlation for each Si and Sjm; End do; End do; End.

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6. RESULTS Using the algorithm given above, cyclic cross-correlation function is studied for the sequences generated by an n-bit LFSR for n ≥ 5. The exclusion of values of n < 5 (i.e., n = 2, 3, 4) is due to the fact that only one primitive polynomial exists for n = 2 and only two for n = 3 or 4. A representative set of results for n = 6 is shown in Table 4. For n = 6, the number of primitive polynomials is 6 and there exists three image pairs. These are represented by P1(x), P2(x), …,P6(x). The image pairs are {P1(x), P2(x)}, {P3(x), P4(x)} and {P5(x), P6(x)}. Table 5 shows the list of all possible primitive polynomials of order 6. Also shown in Table 4 are the corresponding PRRBSs (with 000001 initial loading of LFSR) generated by each of the respective polynomials. It is observed that the cross-correlation between image pairs has constant maximum and minimum values of 38 and 24, respectively. For any other pair, this property does not hold. This property has been verified for each n; 5 ≤ n ≤ 16.

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Table 4: Maximum/minimum values of cyclic cross-correlation between all possible PRRBSs of length 63 (value 1/value 2: value 1 = maximum and value 2 = minimum). P1(x) P1(x) P2(x) P3(x) P4(x) P5(x) P6(x)

_ 38/24 40/24 36/20 40/24 36/20

P2(x) 38/24 _ 36/20 40/24 36/20 40/24

P3(x)

P4(x)

P5(x)

P6(x)

40/24 36/20 _ 38/24

36/20 40/24 38/24

40/24 36/20 36/20 40/24 _ 38/24

36/20 40/24 40/24 36/20 38/24

36/20 40/24

_ 40/24 36/20

_

Table 5: List of primitive polynomials of order 6 and the corresponding PRRBSs generated by the respective polynomials Primitive polynomial P1(x) = 1 + x5 + x6 P2(x) = 1 + x + x6 P3(x) = 1 + x2 + x3 + x5 + x6 P4(x) = 1 + x + x3 + x4 + x6 P5(x) = 1 + x + x4 + x5 + x6 P6(x) = 1 + x + x2 + x5 + x6

Generated sequence (of length 63) 100000100001100010100111101000111001001011011101100110101011111 100000111111010101100110111011010010011100010111100101000110000 100000101111110010101000110011110111010110100110110001001000011 100000111000010010001101100101101011101111001100010101001111110 100000111100100101010011010000100010110111111010111000110011101 100000110111001100011101011111101101000100001011001010100100111

7. CONCLUSION A unique property of the cyclic cross-correlation function between image pairs of PRRBSs has been discovered. It is found that the maximum and minimum values of cross-correlation remain constant for every possible image pairs of the same length. All the other possible pairs of PRRBSs, however, do not possess this property.

8. REFERENCES [1] [2] [3]

Golomb, S.W., (1982). “Shift Register Sequences”. Aegean Park Press, Laguna Hills, CA, Ahmad, A., (2008). “Investigation of Typical Properties of Some LFSR Structures”, Journal of System Science and Engineering, 17(1), pp. 65–69. Ahmad, A. and Al-Maashri, A., (2008). “Investigating Some Special Sequence Length Generated

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[4] [5]

[6] [7]

[8]

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[9]

[10]

[11]

[12]

[13]

[14]

[15] [16]

[17] [18] [19] [20] [21]

A. Ahmad, S. Al-Busaidi, M.J. Al-Mushrafi and J.A. Jervase Through an External Exclusive-NOR Type LFSRs”, International Journal Electrical and Computer Engineering (PERGAMON, Elsevier Science), 34, pp. 270–280. Murtagi, R.N., (1996). “Pseudo-Noise Sequences for Engineers”, IEE Electronic and Communication Engineering Journal, 8(2), pp. 79-87. Ahmad, A., (2010) “A Simulation Experiment on a Built-In Self Test Equipped with Pseudorandom Test Pattern Generator and Multi-Input Shift Register (MISR)”, International Journal of VLSI Design & Communication Systems, 1(4), pp. 1-12. Ahmad, A., (2005) “Testing of Complex Integrated Circuits (ICs) – The Bottlenecks and Solutions”, Asian Journal of Information Technology, 4(9), pp. 816–822. Al-Lawati, A. and Ahmad, A., (2004). “Realization of a Simplified Controllability Computation Procedure – A MATLAB-SIMULINK Based Tool”, Sultan Qaboos University Journal for Scientific Research - Science and Technology, 8, pp. 131–143. Ahmad, A., Al-Lawati, A.M.J. and Al-Naamany, A.M., (2004). “Identification of Test Point Insertion Location via Comprehensive Knowledge of Digital System’s Nodal Controllability Through a Simulated Tool”, Asian Journal of Information Technology (AJIT), 3(3), pp. 142–147. Ahmad, A., (2002). “Investigation of a Constant Behaviour Aliasing Errors in Signature Analysis due to the Use of Different Ordered Test-Patterns in a BIST Testing Techniques”, Journal of Microelectronics and Reliability (PERGAMON, Elsevier Science), 42, pp. 967–974. Al-Naamany, A.M. and Ahmad, A., (2002). “Development of a Strong Stream Ciphering Technique Using Non-Linear Fuzzy Logic Selector”, Mobile and Wireless Communications, Kluwer Academic Publishers, 234, pp. 199 – 206. Ahmad, A., Al-Musharafi, M.J. and Al-Busaidi, S., (2002). “Study and Implementation of Properties of m-Sequences in MATLAB-SIMULINK – A Pass/Fail Test Tool for Designs of Random Generators”, S.Q.U. Journal of Scientific Research – Science and Technology, 7(1), pp. 147–156. Ahmad, A., (2002). “Constant Error Masking Behaviour of an Internal XOR Type Signature Analyzer due to the Changed Polynomial Seeds”, Journal of Computers & Electrical Engineering (PERGAMON, Elsevier Science), 28(6), pp. 577–585. Ahmad, A., Al-Musharafi, M.J., Al-Busaidi, S., (2002). “Design and Study of a Strong Stream CryptoSystem Model for e-Commerce”, International Council for Computer Communication Publishers, Washington DC, USA (The ACM Library), 1, pp. 619–630. Ahmad, A., Al-Musharafi, M.J., Al-Busaidi, S., Al-Naamany, A. and Jervase, J.A., (2001). “An NLFSR Based Sequence Generator for Stream Ciphers”, Proceedings (SETA’01) - An International Conference on Sequences & Their Applications, Norway (Bergen), 13-17 May 2001, pp. 11–13. Ahmad, A., (1997) “Achievement of Higher Testability Goals Through the Modification of Shift Register in LFSR Based Testing”, International Journal of Electronics (UK), 82(3), pp. 249-260. Ahmad, A., Al-Lawati, A.M.J., Jervase, J.A. and Zabalawi, I.H., (1996). “The Study of the Effect of Rotationally Delayed Transmission of Data on Error Masking Behaviour of Different Types of Signature Analysis Schemes”, Journal for Scientific Research - Science and Technology, 1, p. 88. Nanda, N.K., Ahmad, A. and Gaindhar, V.C., (1989). “Shift Register Modification for Multipurpose Use in Combinational Circuit Testing”, International Journal of Electronics (UK), 66(6), pp. 875–878. Ahmad, A. and Nanda, N.K., (1989). “Effectiveness of Multiple Compressions of Multiple Signatures”, International Journal of Electronics (UK), 66(5), pp. 775–787. Ding, C. and Tang, X., (2010). “The Cross-Correlation of Binary Sequences with Optimal Autocorrelation”, IEEE Transactions on Information Theory, 56(4), pp. 1694–1701. Idris, D.M., (2006). “Autocorrelations of Random Binary Sequences”, Combinatorics, Probability and Computing, Cambridge University Press, 15, pp. 663–671. Deng, X., Fan, P. and Suehiro, N., (2000). “Sequences with Zero Correlation Over Gaussian Integers”, Electronics Letters, 36(6), pp. 552–553.

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[22] Paterson, K.G. and Lothian, P.J.G, (1998). “Bounds on Partial Correlations of Sequences”, IEEE Transactions on Information Theory, 44(3), pp. 1164–1175. [23] Penzhorn, W.T., (1997). “Correlation Attacks on Stream Ciphers Using the Lempel-Ziv Algorithm”, IEEE International Symposium on Information Theory, pp. 179. [24] Penzhorn, W.T., (1996). “Correlation Attacks On Stream Ciphers”, IEEE Africon 4th, 2, pp. 1093-1098. [25] Komo, J.J., (1989). “Crosscorrelation of m-Sequences Over Nonprime Finite Fields” Electronics Letters, 25(4), pp. 281-288. [26] Nguyen, T., (1988). “Coset Correlation of m-Sequences”, IEEE Transactions on Information Theory, 34(1), pp. 141-143. [27] Park, W.J., Jr. and Komo, J.J., (1988). “The Autocorrelation of m-Sequences Over Nonprime Finite Fields”, IEEE Transactions on Aerospace and Electronic Systems, 24(4), pp. 459-461 [28] Sarwate, D.V. and Pursley, M.B., (1980).” Cross-Correlation Properties of Pseudorandom and Related Sequences”, Proceedings of the IEEE, 68(5), pp 593-619. [29] Ahmad, A., (2005). “Development of State Model Theory for External Exclusive NOR Type LFSR Structures”, Enformatika, 10, pp. 125–129. [30] Ahmad, A., Al-Musharafi, M.J. and Al-Busaidi, S., (2001). “A New Algorithmic Procedure to Test m-Sequences Generating Feedback Connections of Stream Cipher’s LFSRs”, Proceedings IEEE conference on electrical and electronic technology (TENCON’01), Singapore, 19–22 August 2001, 1, pp. 366–369. [31] Ahmad, A. and Elabdalla, A.M., (1997). “An Efficient Method to Determine Linear Feedback Connections in Shift Registers that Generate Maximal Length Pseudo-Random Up and Down Binary Sequences”, Computer & Electrical Engineering - An Int’l Journal (USA), 23(1), pp. 33-39. [32] Ahmad, A., (1994). “Critical Role of Polynomial Seeds on the Effectiveness of an LFSR-Based Testing Technique”, International Journal of Electronics (UK), 77(2), pp.127–137. [33] Ahmad, A., Nanda, N.K. and Garg, K., (1990). “Are Primitive Polynomials Always Best in Signature Analysis?”, IEEE design & Test of Computers (USA), 7(4), pp. 36–38. [34] Ahmad, A., Nanda, N.K. and Garg, K., (1988). “A Critical Role of Primitive Polynomials in an LFSR Based Testing Technique”, IEE Electronics Letters (UK), 24(15), pp. 953–956.

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