(5.80) where ris a primitive p-th root of unity in the field of complex numbers, e.g.,
(5.81) (In the special case when p 2, then r 1 and bnHGF(2) {0, 1} is mapped into anH{1,1}.) Consequently, additional modulo pof two ele- ments bm and bnfrom GF(p) in the domain of the mapping (5.80) is iso- morphic to complex multiplication of two elements in the range of the mapping, i.e.,
(5.82) with addition of integers in the exponent being both real and modulo psince ris the p-th root of unity.
An m-sequence {bn} of elements GF(p), satisfying the linear recursion (5.83) bn a
L i1
qibni, amanrbmbn, rexp1i2p>p2.
anpbn, HminqL11qJ2. bqJ1, aqLJ1,
NqL1, 0a0 qJ,
0a0 b
Statistical Properties of M-Sequenceshttp://jntu.blog.com 291
can be mapped by (5.80) into a sequence {an} of complex p-th roots of unity satisfying the multiplicative recursion
(5.84) Note that {an} generally does not satisfy a linear recursion of degree Lover the complex numbers. However, the resulting complex sequence {an} often retains the name m-sequence.
Since the m-sequence {an} consists of complex numbers, it is possible to evaluate its periodic autocorrelation function Paa(t), defined by
(5.85) where ( )*denotes conjugation. The nearly ideal periodic correlation func- tion of m-sequences is described in the following result.
Property R-5.
Let {an} be a complex m-sequence of period N, composed of p-th roots of unity. The periodic correlation Paa(t) of {an} has the form
(5.86) Proof. The case t0 mod Nis obvious. Note that NpL1,Lbeing the linear span of the corresponding m-sequence {bn} over GF(p), and by R-2 the number of occurrences of any specified non-zero symbol in one period of {bn} is pL1. Hence, for t0 mod N,
[by (5.80)]
[by R-3]
[by R-2]
(5.87) When the sequence period Nis large compared to the processing gain, as it is in many systems, the full period correlation Paa(t) loses some of its value as a design parameter. Correlation calculations in this case typically are car- ried out over blocks of Ksymbols, where Kmay be larger than the linear span Land much smaller than the period N. A more appropriate statistic for study in this case is the partial-period correlationdefined as
(5.88) Paa1K, n, t2 a
K1
anjtan*j 1.
1rL1a
p1 j0
rj a
N n1rbnt¿1t2 Paa1t2 a
N n1
rbntbn
Paa1t2 e N, t0 mod N 1, t0 mod N.
Paa1t2 a
N n1
antan*, an q
L i1
anqii
.
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which computes the cross-correlation between two blocks of Ksymbols from {an}, one block located tsymbols from the other.
Unlike full-period correlation calculations, partial-period correlation val- ues depend on the initial location nin the sequence where the correlation computation begins. Hence, an explicit description of the function Paa(K,n, t) must include values over the range 0 n,Nof n. Often the size of Npre- cludes direct calculation of all these values, and computable time averages therefore are substituted to give a statistical description of the partial- period correlation function. Denoting the time-average operation by 89, the first and second time-average moments of Paa(K,n,t) are given by
(5.89) (5.90) where the parameter over which the average is being computed is n.
Property R-6.
Let {an} be a complex m-sequence of period N, composed of p-th roots of unity. The time-averaged first and sceond moments of the partial-period cor- relation function of {an} are given by
(5.91) and
(5.92) respectively, for KN.
Proof. The first moment calculation is straightforward:
(5.93) and (5.91) follows directly from Property R-5.
The derivation of the second moment uses the additional fact that Property R-3 for an m-sequence over GF(p) translates via (5.80) into the property
(5.94) antan*ant¿1t2, for all n,
1 Na
K1 k0
Paa1t2, Paa1K, n, t2 1
Na
N n1a
K1 k0
anktan*k
0Paa1K, n, t2 02 •Ka1K1
N b, t0 mod N
K2, t0 mod N
Paa1K, n, t2 eK>N, t0 mod N K, t0 mod N 0Paa1K, n, t2 02 1
Na
N
n10Paa1K, n, t2 02, Paa1K, n, t2 1
Na
N n1
Paa1K, n, t2
Statistical Properties of M-Sequenceshttp://jntu.blog.com 293
when t0 mod N. Hence,
(5.95) Noting that there are Kterms for which jk, and K(K1) terms for which j k, the final result follows immediately by applying Property R-5 to
(5.95).
As a check, note that the results of Property R-6 reduce to those of the full period case R-5 when KN.
For comparison, consider a periodic sequence {xn} composed of Ninde- pendent, identically distributed (i.i.d.) random variables, uniformly distrib- uted over the elements of GF(p),p prime. Furthermore, let {zn} be the corresponding complex sequence determined by the usual mapping (5.80) to p-th roots of unity. Clearly the elements of {zn} are i.i.d. random variables, uniformly distributed on the p-th roots of unity. Both the full-period and par- tial-period time-average autocorrelation functions of {zn} are random vari- ables whose ensemble-average moments can be evaluated, using the independence assumption and the fact that
(5.96) where Edenotes the ensemble average operator. The first moment of the partial period correlation of {zn} is easily shown to be
(5.97) Evaluation of the second moment of Pzz(K,n,t) uses the fourth moment
(5.98) to yield
(5.99) eK2, t0 mod N
K, t0 mod N.
E5 0Pzz1K, n, t2 026 a
K1
j0 a
K1 k0
E5znjtzn*jzn*ktznk6 E5znjtznj* znkt* znk6 •
1, t0 mod N
1, kj and t0 mod N 0, kj and t0 mod N E5Pzz1K, n, t2 6 eK, t0 mod N
0, t0 mod N E5zn60,
1 Na
K1 j0 a
K1 k0
Paa1jk2. 1
Na
N n1 a
K1 j0 a
K1 k0
anjt¿1t2an*kt¿1t2
0Paa1K, n, t2 02 1 Na
N n1a
K1 j0
anjtan*ja a
K1 k0
anktan*kb*
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Comparisons of (5.97) and (5.99) for a random sequence with (5.91) and (5.92) for an m-seuence both indicate that when KV N, then the time- averaged mean and correlation values of an m-sequence are very close to the corresponding ensemble averages for a randomly chosen sequence. This fact and the balance properties of m-sequences are used to justify their approximation by a random sequence of bits in later analyses of SS system performance.
Both the full-period and partial-period correlation computations have a particularly simple characterization when {bn} is a sequence of elements from GF(2) and {an} is a sequence of 1’s and 1’s. In this case when t 0 mod N,
(5.100) where the weight wt(x) of a vector xdenotes the number of non-zero ele- ments in x. Hence,Paa(K,n,t) is a simple affine transformation of the weight of a K-tuple from {bn}, beginning at element index nt(t). This relation, a direct result of the shift-and-add property, simplifies the tabulation and analysis of partial-period correlation statistics.
While the results of the last three sections appear to present a relatively complete theory and are to a great extent available in [12], the study of par- tial-period correlation has remained a topic of research interest for many years. Bartee and Wood [13] used an exhaustive search to find the binary m- sequence, of each possible period up to 2141, which possessed the largest value for the minimum K-tuple weight,D.
(5.101) Equivalently by (5.101), Bartee and Wood found the m-sequences {an} which possess the maximum value of max0nNmax0 t NPaa(K,n,t), for fixed lin- ear span Land correlation length K.
Table 5.8 displays characteristic polynomials of m-sequences which pos- sess the largest value of Damong all m-sequences of linear span L. These polynomials are specified in octal, i.e., the coefficients of the polynomial are the bits in the binary representation of the octal number. The binary repre- sentation of the octal number is determined simply by converting each octal symbol to its equivalent three-bit binary representation. For example,
(5.102) represents z13z11z10z9z8z21. Leading zeros resulting from the octal to binary conversion may be ignored.
27405 1octal2 010 111 100 000 101 1binary2 D min
1nNwt11bn,p, bnK12 2
Kwt11bnt¿1t2,p, bnt¿1t2K12 2 a
K1
j0112bnjt¿1t2 Paa1K, n, t2 a
K1
j0112bnjtbnj
Statistical Properties of M-Sequenceshttp://jntu.blog.com 295
296 Pseudonoise Generators
Table 5.8 Characteristic polynomials (in octal) of m-sequences with linear span L,having largest value D(in parentheses) of minimum K-tuple weight.The optimum polynomial is not unique.(Abstracted from [13].) LK102030405060708090100200 27777777 (6)(13)(20)(26)(33)(40)(46)(53)(60)(66)(133) 313131313131313 (5)(11)(16)(22)(28)(33)(40)(45)(51)(56)(113) 423232323232323 (4)(9)(16)(20)(25)(32)(36)(41)(48)(52)(105) 575454575454575 (3)(8)(15)(19)(24)(30)(34)(39)(45)(50)(101) 6155103103147147147155 (2)(7)(12)(17)(22)(29)(33)(38)(42)(47)(99) 7367367313211345345211 (2)(7)(12)(17)(21)(26)(31)(36)(41)(46)(96) 8703703747747703453703 (2)(6)(10)(15)(20)(25)(30)(34)(39)(44)(96) 91131171517151773177314231773 (1)(5)(10)(14)(19)(24)(29)(34)(38)(43)(91) 102033350735073507246135253525 (5)(9)(14)(18)(22)(27)(32)(37)(42)(89) 114173746176557655710740555607 (4)(8)(13)(17)(22)(27)(31)(36)(40)(89) 1217147142711060517147170251272717025 (4)(8)(12)(17)(21)(25)(30)(35)(39)(86) 1334035211032740531231232313733532467 (4)(7)(11)(16)(20)(24)(29)(33)(38)(85) 1441657730716557564457641676133365277 (11)(15)(19)(23)(28)(32)(37)(83) p1ch5b 8/17/01 10:25 AM Page 296
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Lindholm [14] derived expressions for the first five time-average moments of the partial period correlation values of m-sequences (including those of Property R-6). He noted that only the first two moments were independent of the characteristic polynomial of an m-sequence with specified period.
Cooper and Lord [15], following a study of Mattson and Turyn [16], noted that for nin the vicinity of a run of L1 zeros in an m-sequence, Paa(K,n, t) had lower correlation values, as tvaried around zero, for m-sequences whose characteristic polynomials contained more non-zero coefficients.
Wainberg and Wolf [17] compared the moments of K-tuple weight distrib- utions for m-sequences over GF(2) to the corresponding moments for a purely random sequence, and, extending Lindholm’s work, carried this out through the sixth moment with K100 for several sequences.
Fredriccson [18] carried this idea closer to the realm of coding theory by noting that the set of N-tuples consisting of one period of an m-sequence and all its cyclic shifts, and the all-zeros N-tuple, together form a linear code which is the dual of a single-error-correcting Hamming code. Since the set {(bn, . . . ,bnK1):n1, 2, . . . ,N} of K-tuples along with the all-zeros K-tuple form a punctured version of this m-sequence code, the set’s dual is a short- ened Hamming code. The weight distribution of the shortened Hamming code can be related in turn to the moments of the weight distribution of K- tuples from {bn} by the MacWilliams-Pless identities [19]. Bekir [20] extended this approach by applying moment techniques [21], [22] to generate upper and lower bounds on the distribution function of partial-period correlation values.