Cross-Correlation of Binary m and GMW Arrays

Sequences and arrays with good auto and cross-correlation are useful in communications and digital , .. 'atermarking respectively. Two-valued autocorrelation is relatively easy to achieve. However, sets of pSfudonoise sequences with good cross-correlation arc small and difficult to find. Here, the cross-correlation for particular shifts of m-sequences and GM\V sequences is analysed by looking at these sequences as two-dimensional arrays. The particular shifts are natural in the context of the array decomposition. "square" arrays and arrays ·where the column length is 3 yield new results for m-sequences. The array interpretation also detel'mines the conditions under which the cross-correlation of GIV1\V sequences is identical to the cross-correlation of the m-sequences used in the construction of the GI\1_1. In ILl) the shift sequence) describing the order of cyclic shifts is derived. All m-sequences of the same length can be derived from the decimations of a ''prototype'' msequence. Some decimations result in the array representations of the two m-sequences having the same m-sequence colunms. but different shift sequences. Cross-correlation betv\.'een such sequences is obtained from matching entries in the shift sequence. tn 1l.2 \\'e describe the constructioll of GMW nrrays by USing the shift sequences from the msequence decomposition, by substituting the msequence colunms in the m-array by other sequences with the same autocorrelation and balance property as m-sequences. Cross-correlation of such arrays is identical to that of their parent m-arrays. In Section nI, we examine two palticular decompositions: the· "square" array and the array with column length 3. For the square array cross-correlations for purely veltical nnd purely horizontal shifts for a range of decimations arc found by using the shift sequence. This includes the location of previously unexplained crosscorrelafion peaks. Decomposition of m-sequences into arrays with columns of length 3 allows aU m-arrays to be described by the shift sequence. For many decimations, it is possible to deduce cross-correlation peaks for purely vertical shifts. The results in the two 0-7803-7627-71021$ 1 7.00(C)2002IEEE cases are npplied to GMW arrays in Section IV. II INTRODliCTION In th1s paper, we investigate m and GMW sequences of length 211_1 where n is composite. Such sequences c;:m be represented as arrays \vhose colunms are cyclic shifts of a single shorter balanced sequence, with ideal autoconelatioll or null columns.' Cross-correlation between arrays with the same types of columns becomes a summation of auto correlations of sequence columns, aUlocorrelations of null columns and crossc01Teiations between the two. The first is 2 valued, whilst the latter are both single valued. Cross-. correlation between the arrays becomes a simple addition of column correlations. 11.1 M-Sequence Decomposition Consider an m-sequence s of length 2H_1 where n is composite (n=kmj. where sj=d , ;=0,/ ..... 2"-2 and where (.t is any primitive element of GF(2'1). A binary sequence can be obtained by a trace mapping to the base field GF(2) given by lSi = Tr,"(a') . The sequence can be mapped to subfields GF(2"') or GF(2') by "'s, =Tr"~(a') or's, =Tr;(a ). The m-sequence c~a1§91be \\Titten as a two dimensional --columns of length 2"'_1 or array of T= 2 m -1 2" -1 T=-,-colunms of length 2k_1. In either case, the 2 -1 columns are short m~sequences (of length 2"'-1 or i' -1) or null columns.