Effects of bias and characteristic phase on the crosscorrelation of in-sequences

A.Z.Ti rkel C.F.Osborne T.E. Hall

Indexing terms: M-sequences, Cross correlation

Abstract: The paper examines the origin of high crosscorrelation between m-sequences of the same length by exhaustive examination of all pairs of sequences for y1 < 18. It is shown that, for composite lengths, there exist forced agreements between terms of the sequences. This leads to long-range order or correlation bias for particular decimations relating the sequences and the relative sequence phases. Decimations leading to biased correlations are predictable. For such bias, one of the sidelobes occurs for characteristic phase alignment. The small, but finite, discrepancy between the predicted and observed sidelobe maxima is analysed and is found to be due to the partitioning of sequence terms at characteristic phase. Sequences of nonprime length are found to exhibit considerable diversity in their cyclotomic set structure and closed-form expressions for the crosscorrelation at characteristic-phase alignment. By contrast, these are particularly simple for Mersenne-prime lengths, where it is found that suppression of peak sidelobes by at least 3dB is achieved by avoiding characteristic-phase alignment. The impact of the theory developed in the paper on CDMA is that sequences with biased correlation, and hence high sidelobes, are predictable and therefore avoidable.

1 Introduction

M-sequences have been studied extensively as the nearest approximation to random sequences. M- sequences and related codes have been used in communications, including spread-spectrum code- division multiple-access (CDMA) communication, error-correction coding and cryptography. Schroeder [l] reviews other disciplines which use these codes. The

0 IEE, 1997 IEE Proceedings online no. 19911319 Paper first received 2nd September 1996 and in revised form 15th April 1997 A.Z. Tirkel is with Scientific Technology Pty. Ltd., 3/9 Bamato Grove, Armadale 3143, Australia C.F. Osbome is with the Department of Physics, Monash University, Clayton 3168, Australia T.E. Hall is with the Department of Mathematics, Monash University, Clayton 3168, Australia

authors applied m-sequences to image watermarking. Tirkel et al. [2] and van Schyndel et al. [3] discuss the feasibility of encoding and the results of decoding of multiple m-sequence-based watermarks on one image, with the sequences being chosen for their constrained crosscorrelation. M-sequences have been extended to nonbinary alphabets, Green [4] and two- and higher- dimensional form, MacWilliams and Sloane [5] and Green [6]. Tirkel and Osborne [7] applied both techniques to produce a two-dimensional colour watermark based on GF(8).

In this paper the discussion is restricted to the impact of high crosscorrelation sidelobes to CDMA. CDMA requires codes with highly peaked autocorrelation and minimum crosscorrelation. This paper considers even periodic correlations. In general, aperiodic and partial correlations are difficult to estimate. In special cases, the even and odd crosscorrelations can be constrained locally, as described by Grob et al. [13] in the construc- tion of quasisynchronous CDMA. Apart from noise and interference, the ratio of the peak value of the autocorrelation to the modulus of the highest sidelobe determines the probability of detection and false alarm during synchronisation. In this respect, m-sequences are optimal, because of their two-valued autocorrelation, and the absence of sidelobe peaks. High crosscorrela- tions btween m-sequences were first noted in a table of Gold and Kopitzke, presented by Sarwate and Pursley [SI. Previous analysis of m-sequences has been con- cerned with crosscorrelation spectra predictable by the trace of elements in GF(27 onto the base field, e.g. Niho [9], Helleseth [lo], Simon [ll]. Golomb [14], showed that, in general, the crosscorrelation has the form (-1 + 44. The research by the present authors shows that it is of the form -1 + 2ki (with k > 1). Some sequence pairs (preferred pairs) exhibit low crosscorre- lation peaks. The largest set of these was named by Sarwate and Pursley [8] as the ‘maximal connected set’. Tirkel et al. (121 showed the size of such sets is too small for most applications.

High crosscorrelation sidelobes are accentuated by the near-far problem, where the interferer is closer to the receiver. Adaptive interference cancellation [ 171 offers some amelioration, but requires a reference channel and compensation for temporal variations in the multipath delay spectrum. Avoidance of high sidelobes by control of decimation and phase is appropriate within a closed system, such as a microcell. Skaug [15] attempted to classify anomalously high crosscorrelation sidelobes. Tirkel [ 161 performed an

211 IEE Proc.-Commun., Vol. 144, No. 4, August 1997

exhaustive study of m-sequence crosscorrelations for n < 18 and found that the peak sidelobes presented in [8] were incomplete, owing to the statistical (Fourier) technique used. His complete table, together with Skaug's heuristic (and partly accurate) classification, allowed him to demonstrate that, for many m- sequences, these anomalously high sidelobes were due to forced agreements between sequence terms, due to a recurrence relation between them, determined by factors of 2" - 1. In this paper, a generalisation of that theory is presented, with more accurate predictions. The key feature is that of characteristic phase.

2 Characteristic phase

A binary m-sequence can be transformed into another m-sequence by the process of proper decimation. In the special case of a decimation by 2, the original sequence is reproduced, except for a phase offset. When this offset is 0, the sequence is defined to be in its characteristic phase. Gold [18] provides a construction which determines the starting vector (shift-register seed) for characteristic phase, based on derivatives of the generating polynomial.

Arazi [19] derives an expression for the phase shift between any two vectors of sequences related by a decimation d. Therefore, the starting vectors for two m- sequences which have their characteristic phases aligned can be determined analytically. This condition has a profound impact on the values which the crosscorrelation can assume. The sequences need not start in their characteristic phase, as long as they share a common offset from their respective starting vectors. Hershey [20], notes that, although numerous publications have been devoted to the characteristic phase, the subject is not yet closed.

3 Forced correlations

In [16] it was demonstrated that, for two m-sequences of length 2n - 1, {a,} and {am}, where d is a proper decimation, the terms apL and a+, will be identical if

It was shown that the above congruence can be satis- fied if p is a factor of 2" - 1. The solutions for d then become

pz dpz modulo (an - 1) (1)

The forced agreements occur when the characteristic phases of the sequences are aligned and for all integer multiples of (2n - l)/p of phase displacement from alignment. This results in crosscorrelation peaks of approximately (2n - l)/p whenever p is small. For n even, the lowest value of p is 3 and these biased corre- lations are highly recognisable. For these and selected values of composite 2" ~ 1, these predictions agree with observed data, as shown in Table 1.

For all solutions to eqn. 2, the correlations are high and close to the predicted values. For some values of n, the peak at characteristic-phase alignment is greater than the phase-shifted peaks, which are all equal. There is no explanation for the unusually high peak crosscorrelations for sequences of Mersenne-prime length, which are free from biased behaviour (no solutions to eqn. 1). For some values of n, namely 9 and 11, higher peaks are observed than those

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Table 1: Forced correlations

n 2"-1 Factors NT 8m,n emax 8b ezax

3* 7 - 2 5 -5 - 5

4 15 3*5 2 7 -9 # 9

5 31 - 6 9 + l l - 11

6 63 32*7 6 15 +23 +21 23 7" 127 - 18 17 -41 - 41 8 255 3*5*17 16 31 +95 +85 65 9 511 7*73 48 33 -113 +73 113

10 1023 3*11*31 60 63 +383 +341 161 11 2047 23*89 176 65 +287 +89 287 12 4095 32*5*7*13 144 127 +I407 ,1365 511

13 8191 - 630 129 +703 - 703 14 16383 3*43*127 756 255 +5631 +5461 897

15 32767 7*31*151 1800 257 +4799 +4681 2495 16 65535 3*5*17*257 2048 511 +22015 +21845 4703

17* 131071 - 7710 513 +5951 - 5951 * signifies a Mersenne prime NT= number of m-sequences of length 2"- 1 e,,, = minimum crosscorrelation value emax= maximum crosscorrelation value 8, = number of agreements forced by solutions of the Dio- phantine equation

= maximum crosscorrelation after removal of decima- tions leading to significant bias

predictable by finite-algebra effects. These peaks are solitary and occur for decimations which b relation to solutions of eqn. 2. These anomalies can be explained by the effects of characteristic-phase alignment.

Characteristic-phase alignment enforces different forms of sequence agreement. Consider a sequence of length 2'" - 1, which factors of p , q such that p is the smallest factor (q may be prime or composite). Solutions to eqn. 2 yield the approximate number of forced agreements for the p peaks for decimation d. Characteristic-phase alignment leads to further forced agreements if

qz E 2'dqz ( 3 ) The above follows from the property of characteristic phase: decimations by powers of 2 result in an identical sequence. In eqn. 3, modulo 2" - 1 arithmetic is assumed. Any solutions to eqn. 3 with 0 < k < n will lead to q forced agreements between the sequences in characteristic-phase alignment only. Any solutions for k = 0 which are not redundant with the original recur- rence due to factor p (p and q having gcd = 1) would lead to increased bias in unusual phase locations. So far no evidence of that has been found.

The total bias at characteristic phase alignment B, resulting from both factors is

2n - 1 B,, - P + (2nq-1 e) P (4)

The extra multiplier in the second term is to avoid redundant counting of agreements (pq), which are sub- ject to both recurrences. Examples of such degeneracy splitting can be found for y2 = 10, 12, 15. The search for q as a solution to eqn. 3 was performed by the process of exhaustion, although a systematic method was also developed for this purpose. The revised maximum bias predictions are shown in Table 2. Considerable improvement in agreement is observed, as is evidenced

IEE Pvos -Conmu" Vol 144, No 4, AuguSt 1997

by the small disparity between prediction (bias at CP) and Om,,.

Table 2: Revised maximum-bias predictions

n Factors p q Bias Bias at CP Omax 6

8 3,5,'17 3 - +85 +85 +95 +IO 10 3,11,31 3 1 1 +341 +403 +383 -20 12 3',5,7,13 3 91 +I365 +I395 +I407 +I2 14 3,43,127 3 - +5461 +5461 +5631 +I70 15 7,31,151 7 151 +4681 +4867 +4927 +60 16 3,5,17,257 3 - +21 845 +21 845 +22015 +I70

The 6 column represents the partial correlation of the sequences after removal of the terms forced to agree by the recurrence relations. As shown in Section 4, they are predjctable. Smaller values of bias can exist, due to larger factors of 2" ~ 1.

4 Mersenne-prime lengths

The effects of characteristic-phase alignment on sequences of Mersenne-prime length are deduced by partitioning the sequences in their Characteristic phases. Such sequences can be divided into equivalence classes of elements, whose values are equal because of the property of decimation by powers of 2. Such sets are of the form

(U%, a22, a 2 2 2 , . ' . , a2"-'%)

where modulo 2" - 1 is implied (5) There are (2" - 2)/n such distinct sets and the solitary element ao. When two sequences are in characteristic phase alignment, the crosscorrelation is dependent on agreement or disagreement between complete sets, rather than individual sequence elements. Since (2" - 2)/n is an even number,

(6) A restriction on Ocp is imposed by the balance prjtqion. The number of Is must exceed the number of O s by 1. Therefore the equivalence sets of length n must be evenly divided, while a. must be 1. The agreement of all aos results in

This is still subject to the general restriction on 8 = -1 + 4i. Since n is odd for Mersenne-primes, j must be odd to satisfy both criteria:

8,, = f l f 2jn 0 < j < (2% - 2)/2n

e,, = +1 f 23n (7)

OCp = 1 f 2(2k + l )n (8)

Table 3: Values of crosscorrelation possible for charac- teristic-phase alignment

n ~ " - 1 IemaxI I~"maxI 5 31 7 127 41 21 9 51 1 113 79?

1 1 2047 287 159 13 8191 703 351 17 131071 5951 2239

Therefore, there are only a few values of crosscorrelation possible for characteristic-phase alignment and they are high and of either polarity, as

IEE Proc -Commun , Vol. 144, No 4, August 1997

shown in Table 3. 10muxl denotes the highest absolute correlation, while I O*,,,I is the residual maximum after removing all peaks occurring for characteristic-phase alignment. The reason for inclusion of n = 9 and 11 is explained in Section 5. There appears to be no means of predicting which decimation results in the highest sidelobe.

Table 3 shows that avoidance of characteristic-phase alignment leads to approximately 3 dB suppression of the highest crosscorrelation sidelobe. This can be arranged by introducing a deliberate phase offset between sequences. In cases where all available sequences must be used, this results in a uniform spacing of n chips between sequences, with respect to alignment. This is easy for a cellular or microcellular forward-channel CDMA, where a single base-station transmitter is involved. The reverse channel can only be controlled where the delay/multipath spread is less than n chips.

5 General effects of characteristic-phase alignment

Consider the partitioning of sequences of nonprime length in their characteristic phase. In general, n does not divide 2" - 2. Since the maximum length of an equivalent set is n, this implies that there must be sets of length less than n. These sets are composed of

where i < n and 2 2 ~ E 2 Mod (2" - 1) (9) The solution to the above Diophantine equation is

cw - i * I

x = k - 2% - 1

with k being an odd integer less than n (10) If 2" - 1 has any factors of the form 2' - 1, short sets

will exist. Some values of n have several factors of this form and some 2' - 1 can be composite themselves. In addition, some solutions for x are degenerate. Degener- acies can arise when: (i) x is a solution to eqn. 10 for il and i2, where il < i2 and 2l1 - 1 is a factor of 2 j 2 - 1; and (ii) among solutions for the same value of i, due to dif- ferent values of k leading to x being within the same equivalence class. Another constraint is that i must divide n. n = 11 is included with the Mersenne-primes because there are no factors of the form 2' - 1 so there are (211 ~ 2)/11 = 186 full-length sets and the 0th element. Other cases of composite 2" - 1 are enumerated in Table 4. In column 2, the number in parentheses denotes the set size, while the boldface multiplier denotes their number. The bal- ance rule constrains short sets as listed in column 3. Crosscorrelations for characteristic-phase alignment are listed in column 4.

6 Conclusions

This paper presents a formalism for the prediction of peak sidelobes in the periodic crosscorrelation between m-sequences of equal length. This permits CDMA-code designers to make simultaneous use of the maximum number of sequences with a lower, self-induced inter- ference level. For prime-length sequences this involves the avoidance of characteristic-phase alignment. For composite lengths, sequences related by predictable

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Table 4

n Structure Constraints Cross-correlation

6 1“ I, 1*2,2*3,9*6 8 I*?, 1*2,3*4,30*8 a. = 0, a85 = a170 = 0 7 + 8 i + 1 6 j , l ~ i ~ - 2 , 1 5 ~ j ~ - 1 5

9 I* I, 2*3,56*9 a. = 0 1 + 6i + 18j, 1 2r i 2 -1,28 B j -28

10 1*1,1*2,6*5,99*10 a. = 0, a341 = aEB2 = 1 1 3 + 1 0 ~ + 2 0 j , 3 s i . - - 3 , 4 9 a j ~ - 5 0

a0 = 0, a21 = a42 = 1 9 + 6; + 1 2 j, 1 i 2 -1,4 B j I -5

12 I* I, 1*2,2*3,3*4,9*6,335* 12 ? ?

14 1*1,1*2,18*7,1161*~4 a. = 0, a5461 = = 1 17 + 14i+ 28j, 9 2 ;e-9, 580 r j r - 5 8 1 15 I* I, 2*3,6*5,2182* 15 a. = 0 1 + 6 i + 1 2 j + 3 0 k , 1 > i > - 1 , 3 > j 2 - 3 , 1091 >kz- l091

16 1*1,1*2,3*4,30*8,4080*16 ao=0,a21845=a43690=0 7 + 8 i + 16j+32k, 1 >\;I, 15>1jl,2040>Ikl

decimations must be avoided in characteristic-phase alignment and other predictable phase shifts. The observed peak sidelobe values are accounted for for n < 18. The significance of forced agreements between and within sequences is examined. The formalism developed can be extended to all values of n and to nonbinary alphabets.

7 Acknowledgments

The authors thank Monash University for the continu- ing appointment of A.Z. Tirkel as Visiting Research Fellow and for making available the computing and other facilities of the Department of Physics. The man- uscript benefited greatly from the inspirational support of E.M. Gray of Griffith University.

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