CROSS-CORRELATION OF M-SEQUENCES - Some Unusual Coincidences Anatol.Z.TIRKEL(Senior Member IEEE) Scientific Technology, 21 Walstab St. East Brighton 3187, Australia Visiting Fellow, Department of Physics, Monash University, Clayton 3168, Australia.
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ABSTRACT M-sequences have been studied extensively as the nearest approximation to random sequences. In general, the problem of computing cross-correlations between m-sequences by analytical techniques has been declared untractable. In this research, cross-correlations for all sequences of lengths up to 218-1 have been examined numerically, in the hope of finding some predictable patterns. One such pattern which emerged from this numerical analysis was the existence of cross-correlation peaks well in excess of those predictable by statistical techniques [1].
This
paper demonstrates that these “anomalous” peaks are due finite algebra effects. These results suggest the existence of some algebraically computable correlations, apart from those already known from Galois Field theory [2]. The technique developed here can be used to determine the pairs of sequences with high cross-correlation peaks, the approximate value of these peaks and the relative phasing of the sequences. Elimination of such pairs of sequences results in a dramatic reduction in the peak cross-correlation among the set of remaining sequences. These have been named "Constrained Connected Sets". When used in conjunction with sequences whose crosscorrelations are predictable by Galois Field analysis [2], this technique may prove useful in the design of code families to meet specific cross-correlation requirements. 1. INTRODUCTION M-sequences have found numerous applications in modern communication systems, including spread spectrum Code Division Multiple Access (CDMA). These applications require large sets of codes with
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highly peaked autocorrelation and minimum cross-correlation.
The salient feature of the auto
correlation is the ratio of the peak value to the modulus of the highest sidelobe. Apart from noise or interference, this is the key parameter which determines the probability of detection and false alarm during synchronization. In this respect, m-sequences are optimal, because of their two-valued auto correlation, and the absence of sidelobe peaks. The critical properties governing the cross correlation performance are its RMS and peak values. The mean and RMS values are constrained owing to first and second moment identities as described by Niho [2] and are independent of the choice of sequences. However, the peak values depend on the sequences chosen and their respective phases. To reduce interference, it is desirable to constrain these peak values to a minimum. All m-sequences of the same length can be derived from each other by a process of proper decimation. Hence, in order to achieve the above result, it is only necessary to deduce the value(s) of the decimation to achieve the desired property. In this context, only periodic cross-correlations are considered. 2. BACKGROUND The m-sequences of length 2n-1 form an extension field of the Galois Field GF(2). Each sequence is related to all other sequences of the same length by a proper decimation. When describing the crosscorrelation between two sequences, it is sufficient to specify the decimation relating the sequences. As mentioned in the abstract, the general problem of computing cross correlations algebraically is intractable. Therefore, in general, these correlations are computed numerically. However, has been well known that certain pairs of sequences have predictable and optimal cross correlation peaks. These are called preferred pairs, and for sequences of length 2n-1 they exhibit peak cross correlation values of 2[(n+2)/2]+1 for n even and 2[(n+1)/2] +1 for n odd. The cross correlation for preferred pairs is three valued 1,-1+p,and -1-p, where p=2(n+e)/2 and e = (n,k) and 0< k
