5.7 DIRECT-SEQUENCE MULTIPLE-ACCESS DESIGNS
5.7.3 Cross-Correlation of Binary M-Sequences
It is clear from Table 5.6 that for reasonable register lengths, e.g., 20 to 30 stages, there exist large numbers of cyclically distinct m-sequences.Therefore, it is reasonable to determine whether or not several m-sequences can be used
Pmax 7 12N221>2. c
1NPmax2>2H1bj,t,bk, 02 1NPmax2>2 0N2H1bj,t,bk, 02 0,
Pmax 0Pjk1t2 0 ` a
N n1
112bnt1j2 bn1k2` c
an1j2 112bn1j2, bj,m 1bm1j2 1,p,bmN1j2 2.
together as a DS-SSMA signal set. We approach this question by consider- ing the cross-correlation between a roots-of-unity m-sequence {an} and its r-th decimation {arn}, the latter being another m-sequence whenver ris rel- atively prime to the period of {an}.
Let denote a binary m-sequence,abeing a primitive element of GF(2L), and let {an} be the corresponding roots-of-unity m-sequence. The periodic cross-correlation between {an} and its r-th decimation ,
for all n, is given by
(5.246) Sinceais primitive,anscans through the non-zero elements of GF(2L) as n varies in (5.246), and, after compensating for the insertion of the zero ele- ment into the calculation, (5.246) leads to
(5.247) whereyat. Analytical simplifications of (5.247) have been achieved only for certain values of r. A comprehensive survey of these results, including tabulations of computer searches is given in [37].
One particularly good result will be derived from the following lemmas.
LEMMA5.2. Let
(5.248) and assume e divides k. Then r(y) takes on three values: 0, 2(Le)/2, and 2(Le)/2, with 0occurring for 2L2Ledifferent values of y.
Proof. Squaring and simplifying (5.247) gives
(5.249) Equality (1) results from the substitution of zw xand the form of r, and equality (2) employs trace linearity and the fact that and have the same trace since they are conjugates.
Asxvaries over GF(2L), the trace in the inner sum is 0 and 1 equally often (see trace Property T-4) provided that w2kw2Lk is not zero. Hence, the xw2Lk wx2k
122 w苸aGF12L2112Tr1ywwr2x苸aGF12L2112Tr3x1w2kw2Lk24 112 x,bw苸GF12L2112Tr1ywxr1wx21w2kx2k24
3¢r1y2 42 b
x,z苸GF12L2112Tr3y1xz2xrzr4 r2k1, egcd12k,L2,
¢r1y2 ¢ 1Paa¿1t2 x苸aGF12L2112Tr1xyx¿2,
Paa¿1t2 a
2L1 n1
112Tr1ant2Tr1arn2. anœ arn
5anœ6 5Tr22L1an2 6
sum on xis 0 or 2L, and (5.249) reduces to
(5.250) where
(5.251) The intersection of GF(22k) and GF(2L) is GF(2e), where eis the gcd (2k,L).
The order of every win GF(2e) divides 2e1, and, therefore, assuming e dividesk,
(5.252) where c is an integer. Since the traces of wand w2 are identical, (5.250) reduces to
(5.253) by (5.125). Since is zero for 2Levalues of y by trace property T-4,
(5.254) When is non-zero, the outer trace in (5.253) is zero and one equally as often as wscans through GF(2e) and, therefore,
(5.255) 䊏 Lemma 5.2 provides the mathematical background for computing the peri- odic cross-correlation between an m-sequence and its r-th decimation, when ris of the form 2k1. The application of this result to the cross-correlation of m-sequences requires the following lemma to determine when the r-th decimation of an m-sequence is itself an m-sequence.
LEMMA5.3. For all integers m and n,
(5.256) (5.257) Proof. By Euclidean division,
(5.258) and, using binary representations of 2m1 and 2n1 as m- and n-tuples
manr, 0r 6 n,
gcd12m1, 2n121 iff n>gcd1m,n2 is odd.
gcd12m1, 2n122gcd1m,n21
3¢r1y2 420 for 2L2Le values of y. Tr22eL1y12
3¢r1y2 422Le for 2Le values of y. Tr22eL1y12
3¢r1y2 422Lw苸aGF12e2112Tr22e53Tr22eL1y2Tr22eL1124w6
wrw12e2c#ww2, 5w:w苸GF122k2,w苸GF12L26. 5w:w22kw,w苸GF12L26
5w:w2kw2Lk0,w苸GF12L26 3¢r1y2 422Lw苸a112Tr1ywwr2,
of ones, Euclidean division of 2m1 by 2n1 gives
(5.259) for some integer b. Hence, there is a correspondence between terms in Euclid’s algorithm (see Appendix 5A.2) for determining gcd(m,n) and that for finding gcd(2m1, 2n1), which easily yields a proof of (5.256).
A proof of (5.257) begins by applying (5.256) to give
(5.260) and noting that
(5.261) since these numbers differ by two and are odd. Therefore,
(5.262) 䊏 When the decimation coefficient ris relatively prime to 2L1, Theorem 5.8 indicates that the resulting decimation of an m-sequence is also an m- sequence. Placing Lemma 5.2 in correlation terms and restricting the deci- mation by rto produce an m-sequence, gives the following theorem.
THEOREM5.17. Let{an}be a roots-of-unity m-sequence with period 2L1, and let Paa (t)be the periodic cross-correlation of {an}with , its deci- mation by r. Let r and e be defined by (5.248), with L/gcd(k,L)odd. Then
(5.263) where 0 t2L1.
Final determination of the counts shown in this theorem can be made after showing that tPaa (t)1.
Information concerning other pairs of m-sequences with good periodic cross-correlation is given in Table 5.16. It is assumed in this listing that ris relatively prime to 2L1, and eis the greatest common divisor of Land 2k.
Note that while Welch’s proof of the second listed result is unpublished, it is the m3 case of the third listed result. Each of the entries in Table 5.16 gives rise to peak cross-correlation magnitudes on the order of 2(L1)/2when Lis odd, and 2(L2)/2whenLis even. However, this table does not list all such cases.
Paa¿1t2 •1 for 2L2Le1 values of t,
121Le2>2 for 2Le121Le22>2 values of t,
121Le2>2 for 2Le121Le22>2 values of t
5anœ6 3n>gcd1m,n2 is odd.
3gcd12m,n2 0m gcd12m1, 2n121
gcd12m1, 2m12 1,
gcd112m12 12m12, 2n122gcd12m,n21, 12m12b12n12 12r12, 02r1 6 2n1,
Periodic cross-correlation calculations can be carried out analytically on m-sequences of differing periods, sometimes with good results. This is illus- trated in the following theorem and proof.
THEOREM5.18. Letabe a primitive element of GF(2L),L even, and let (5.264) ar being a primitive element of GF(2L/2). Then the roots-of-unity m- sequences of periods 2L1 and 2L/21, with elements defined by
(5.265) and
(5.266) respectively, have periodic cross-correlation (over 2L 1 sequence ele- ments)values12L/2at all shifts.
Proof. The cross-correlation between {an} and at shift tis given by
(5.267)
wherex an,yat, and the field element 0 has been added to the sum range. The elements of GF(2L) can be represented in the form
(5.268) whereuandyare elements of GF(2L/2) and bis the root of an irreducible quadratic polynomial over GF(2L/2) (see Appendix 5A.5, Theorem 5A.9)
xuby
1x苸aGF12L2112Tr22L1yx2Tr22L>21xr2
Paa¿1t2 a
2L1
n1112Tr22L1ant2Tr22L>21arn2 5anœ6 anœ 112Tr22L>21arn2
an 112Tr22L1an2 r2L>21,
Table 5.16
Maximum absolute value of the cross-correlation between m-sequences with characteristic polynomials ma(z) and ,aa primitive element of GF(2L).
r max0Paa[pr](t)0 Comments
2k1 2(Le)/21 [30]
22k2k1 2(Le)/21 (Welch, unpublished)
(2mk1)/2k1) 2(Le)/21 modd, [37]
2(L2)/21 2(L2)/21 L0 mod 4, [37]
2L11 2(L2)/2 [38]
mar1z2
Then, using trace properties (5.123) and (5.125) on (5.268) gives
(5.269) the latter step using the fact that yandy2have the same trace. The interior sum over yin (5.269) is zero by tace property T-4 unless the coefficient of y is zero, in which case the sum is 2L/2. Furthermore, the coefficient of y in (5.265) is zero for exactly one value of u, namely
(5.270) and it follows that
(5.271) 䊏 Theorems 5.17 and 5.18 will now be applied to the design of large sets of sequences with good periodic correlation properties.