5.7 DIRECT-SEQUENCE MULTIPLE-ACCESS DESIGNS
5.7.5 A Transform-Domain Design Philosophy
The linear designs of the previous section have excellent periodic correla- tion properties, but short linear spans. A transform domain design approach will now be described which results in a NLFFL design producing balanced binary sequences with optimal correlation properties and longer linear span [32], [42].
Consider two functions,r(x) and s(x), mapping the elements of GF(2d) into
⫹1 and ⫺1. These functions can be viewed as representing NLFFL functions Pmaxⱖ2L>2⫹1.
7 12L⫺2#2L>2⫹121>2⫽2L>2⫺1 Pmaxⱖ aN21J⫺12
NJ⫺1 b1>2⫺ 12L⫺2L>2⫺1⫹r21>2 r⫽2L>2⫹1,
mar1z2
operating on a Galois m-sequence generator with characteristic polynomial ma(z), whose d-stage register contents represent an(see (5.127)).
The resulting binary (⫹1 and ⫺1) sequences, {r(an)} and {s(an)}, have peri- odic cross-correlation at shift tgiven by
(5.292) wherey⫽at.
The inner product of r(yx) and s(x) over xHGF(2d) in (5.292) can be stud- ied in a transform domain using any inner product-preserving transform.Two closely related transforms will be considered for this purpose, the first being thetrace transformwhich is defined as
(5.293) for all lHGF(2d). The following properties of the trace transform can be verified:
(a) Inversion theorem:
(5.294) (b) Multiplicative shifting theorem: For y⫽0,
(5.295) (c) Parseval’s relation:
(5.296) (d) Additive shifting theorem:
(5.297) (5.298) Another equivalent inner-product-preserving transform is defined after a one-to-one linear mapping of GF(2d) onto the space of d-tuples with elements in GF(2). Specifically, let b1, . . . , bd be an arbitrary basis for GF(2d), and let g1, . . . ,gdbe another basis for GF(2d) with the property that (5.299) Tr1bigj2⫽ e1, i⫽j
0, i⫽j.
vd
sˆ1l2⫽rˆ1l⫹y2 for all l iff s1x2 ⫽r1x2 1⫺12Tr1xy2 for all x. s1x2 ⫽r1x⫹y2 for all x iff sˆ1l2 ⫽rˆ1l2 1⫺12Tr1ly2 for all l,
x苸GFa12d2r1x2s1x2⫽l苸GFa12d2rˆ1l2sˆ1l2. s1x2 ⫽r1yx2 for all x iff sˆ1l2 ⫽rˆ1y⫺1l2 for all l.
r1x2⫽ 1
2d>2l苸aGF12d2rˆ1l2 1⫺12Tr1lx2. rˆ1l2 ⫽ 1
2d>2x苸aGF12d2r1x2 1⫺12Tr1lx2
⫽ ⫺r102s102⫹x苸GF12a d2r1yx2s1x2. Rrs1t2⫽ a
2d⫺1 n⫽1
r1an⫹t2s1an2 r11n2,p,rd1n2
Two bases satisfying (5.299) are called complementary.It can be verified [42]
that every basis for GF(2d) has a complementary basis. Let xandlbe arbi- trary elements of GF(2d), represented in complementary bases by
(5.300) Coordinate values can be determined by applying (5.299) and trace linear- ity to (5.301) to yield
(5.301) for all i. These same relations allow Tr(lx) to be interpreted as an inner prod- uct of vectors of coefficients from the expansions (5.300).
(5.302) The mapping from xHGF(2d) to , can be used to define
(5.303) for all xin GF(2d), substitution of (5.302) and (5.303) into the trace trans- form definition (5.293) gives
(5.304) The right side of (5.304) defines the Fourier transform of the function R(x). The inversion theorem, Parseval’s relation, and additive shifting theo- rem for the Fourier transform are easily derived from the corresponding results for trace transforms, in view of (5.304). Both transforms will be used in the design approach to follow.
Returning to the periodic cross-correlation computation of (5.292), we now develop a bound on cross-correlation. Applying Parseval’s relation (5.296) gives
(5.305) Therefore,
(5.306) where the sum over lin (5.306) has been limited to those lfor which the corresponding term is non-zero, i.e.,
(5.307) and
(5.308) Parseval’s relation also implies that
(5.309)
l苸sar
0rˆ1l2 02⫽l苸GF12a d20rˆ1l2 02⫽x苸GF12a d20r1x2 02⫽2d, ysr⫽ 5l:rˆ1y⫺1l2⫽06⫽5ly:l苸sr6.
ss⫽¢ 5l:sˆ1l2 ⫽06 0Rrs1t2 0 ⱕ1⫹l苸ysa
r¨ss
0rˆ1y⫺1l2 0 0sˆ1l2 0, Rrs1t2 ⫽ ⫺r102s102 ⫹l苸aGF12d2rˆ1y⫺1l2sˆ1l2.
R苲1l2 rˆ1l2⫽ 1
2d>2x苸va
d
R1x21⫺12ltx⫽¢ R苲1l2. R1x2⫽¢ r1x2
x苸vd
Tr1lx2⫽xtL.
xi⫽Tr1gix2, li⫽Tr1bil2 x⫽ a
d
i⫽1xibi, l⫽ a
d i⫽1ligi.
which yields a bound on the trace transform, namely
(5.310) Applying the definition on the left side of (5.310) to (5.306) gives a general bound on correlation magnitude.
(5.311) The above development suggests the following procedure for designing a set of distinct NLFFL functions operating on identical m-sequence shift registers, to produce sequences with good periodic auto- and cross-correla- tion properties:
Property P-1.
Assume that the transforms of all functions in are non-zero on the same subset of GF(2d) and are all zero outside . Notice that
(5.312) for all . Hence, a set of balanced sequences will be achieved in this design if and only if 0 is not in .
Property P-2.
Choose the set (with ) so that is as small as possible for all non-integer yin GF(2d). To calculate the average size of the intersection, letai⫽1 for each ifor which , and let ai⫽0 otherwise. Then for y⫽ at, we have
(5.313) and bounding the maximum intersection by the average intersection gives
(5.314) Property P-3.
Design the set of functions so that for each randsin ,
(5.315)
x苸aGF12d2r1x2s1x2 ⫽0⫽l苸aGF12d2rˆ1l2sˆ1l2.
b,r⫽s b
ⱖ 0s0 1 0s0 ⫺12 2d⫺2 .
y苸GF12maxd2 y僆GF122
0ys傽s0 ⱖ 1
2d⫺2 a
2d⫺2
t⫽1 a
2d⫺2 i⫽0ai⫹tai
0ys傽s0 ⫽ a
2d⫺2 i⫽0ai⫹tai, ai苸s
s傽s 0僆s
s
s r苸b
0僆s3rˆ102 ⫽0⫽x苸aGF12d2r1x235r1an2 6 is balanced, s
s
b b
0Rrs1t2 0 ⱕ1⫹ 2BrBs0ysr¨s0. Br⫽¢ max
l 0rˆ1l2 02ⱖ 2d 0sr0 ,
This orthogonality guarantees that 0Rrs(0)0is 1. Since in the transform domain can be viewed as a -tuple, the achievable size of is limited by , i.e.,
(5.316) Property P-4.
Design each function in so that its trace transform has constant magni- tude, namely on by (5.309). If this objective can be achieved and if the bounds (5.314) and (5.316) can be achieved with equality, then by (5.311) it is possible to find orthogonal NLFFL functions such that
(5.317) for all t⫽0 mod 2d⫺1.
The final parameter, namely , can be chosen with the aid of the Welch bound, which states that 0Rrs(t)0must be on the order of 2d/2 when is large. Assuming that (5.317) is valid and tight, the Welch bound (5.241) indi- cates that should be on the order of 2d/2, with larger values of result- ing in designs which cannot meet that bound with equality. The above design approach, with , will be successfully demonstrated in the next section.