176 Spread Spectrum and CDMA

Nevertheless they remain promising from the point of view of a, max, since the lower border (6.4) gives a, max 1:5/257. Applying to them the above procedure results in the sequence with maximal non-normalized aperiodic sidelobe equal to 12, i.e. a, max ¼ 12/257 or 26:6 dB (compare this with the longest binary Barker code, for which a, max ¼ 1/13 or 22:3 dB). The sequence obtained after eliminating the last symbol turns into the code of length N ¼ 256 with the same maximal non-normalized sidelobe and a, max ¼ 12/256 ¼ 3/64, i.e. again approximately 26:6 dB. Its aperiodic ACF is shown in Figure 6.16a. Interestingly, in the 3G mobile UMTS standard the primary synchronization code is a binary sequence of this very length, N ¼ 256, having aperiodic sidelobes up to 1/4 (Figure 6.16b), i.e. much higher as compared to the sequence just found. On the other hand, the choice of a code for a cell search in UMTS was subject to many other requirements, including implementation issues which might have overpowered the criterion of good autocorrelation.

Figure 6.16 Aperiodic ACF of two binary codes of length 256: the code of Example 6.10.2 (a) and primary synchronization code of UMTS (b)

Figure 6.17 presents one more illustration of the optimization of binary codes in the maximal aperiodic ACF sidelobe, showing the dependence of a, max on length N for

presumably the best binary sequences taken from [25–27,34]. The dashed line shows the p p

curve a, max 0:77/ N approximating the dependence a, max ¼ f (N) as a/ N with a fitted by the least-squares method. As is seen, the accuracy of this approximation is rather good, especially for N > 100.

6.11 Sequences with perfect periodic ACF

As has been indicated time and again, numerous applications exist where the periodicity of the signals makes their periodic correlation properties primarily important. In other words, good periodic ACFs are not only a powerful intermediate tool to design good aperiodic sequences but very valuable in themselves. Examples of this sort include

a0, a1, . . . , aN 1, converting symbols þ1 and 1 to 1 þ c and 1 þ c, respectively. The periodic ACF of the sequence thus obtained is found directly:

a0, a1, . . . , aN 1. Equation (6.8) shows that for any minimax sequence meeting (6.12)

PN 1 Rp(m) ¼ N þ (N 1)( 1) ¼ 1 ) a~0 ¼ 1. Since changing the signs of all

m¼0

elements does not affect ACF, we may consider only sequences with a~0 ¼ 1. Again, for any minimax sequence meeting (6.12), the first sum in the right-hand part of (6.28)

This equation in two real unknowns (real and imaginary parts of c) has an infinite number of solutions. Let us find those that are potentially most interesting. If a real alphabet is desired Re(c) ¼ c and jcj2 ¼ c2 so (6.29) is a quadratic equation:

c2 N2 c N1 ¼ 0

with roots c1, 2 ¼ 1 Nþ1. New binary non-antipodal symbols 1 þ c and 1 þ c may now

N

be divided by 1 þ c to retain þ1 as one of the symbols in the new alphabet. After this we come to the rule of converting a binary minimax sequence with periodic ACF (6.12) into one with perfect ACF: elements 1 should be changed to:

þ

Example 6.11.1. The m-sequence or Legendre sequence of length N ¼ 127 is transformed into

The solution above produces an alphabet with two opposite symbols of unequal magnitude, i.e. results in amplitude modulation (Figure 6.18a). Another possible option

where cos F ¼ (N 1)/(N þ 1) (Figure 6.18b).

Figure 6.18 Non-antipodal binary alphabets

Example 6.11.2. For N ¼ 127 cos F ¼ 63/64 and F ¼ arccos (63/64) 10 803000. Changing all negative elements of the binary m-sequence or Legendre sequence of length N ¼ 127 toexp (jF) produces a sequence with perfect periodic ACF.

The alphabet transformation just discussed, which has been proposed and reopened repeatedly [38,39], can hardly be recognized as very effective practically. As is seen and confirmed by examples, it prescribes rather exotic values of code complex amplitudes, the setting and holding of which with adequate precision may appear technologically infeasible.

6.11.2 Polyphase codes

Involving non-binary PSK modulation with M > 2 opens the way to numerous polyphase sequences with perfect periodic ACF. There are various rules for their construction, but more or less all of them originate in two of the most popular algorithms. The first, corresponding to Chu (or quadratic residue) codes, is very straightforward and approximates in a discrete form the law of linear frequency modulation (cf. Section 6.2). The Chu code exists for an arbitrary length N and is generated as:

It is easy to check that ai ¼ aiþN for all i and therefore, N is at least a multiple of the code period. Calculation of periodic ACF will in passing eventually clarify the issue of a

When m ¼ 0 modN the last sum equals N, while the coefficient in front of it turns into 1. For any other m exp (j2 im/N) depends on i, and the sum above is a sum of the roots of unity of some degree, or, equivalently, a geometric series with the common ratio exp (j2 m/N). Summation of the series gives:

The denominator of the last fraction never turns into zero unless m ¼ 0 modN, and therefore Rp(m) ¼ 0 at all shifts but multiples of N. Hence, the Chu code defined by the first row in (6.30) has period N and perfect periodic ACF. The solution for an odd N is carried out similarly (Problem 6.29).

Despite Chu codes making a rather convincing academic example of PSK sequences with perfect periodic ACF, their practical feasibility is pretty doubtful, since the size of the phase alphabet grows linearly with length and distances between adjacent phases becomes very small. Because of that excessive demands arise towards the precision of forming code symbols, the fineness of representing a phase, susceptibility to environmental conditions, etc.

The same shortcomings are characteristic (to a slightly lower extent, though) of the second popular family of polyphase sequences: Frank codes. They also realize stepapproximation of the linear frequency modulation, but much more roughly, and exist only for lengths that are squares of integers N ¼ h2 ¼ 4, 9, 16, 25, 36, 49, . . . . Their generation rule is:

ai ¼ exp j2 i i ; i ¼ . . . ; 1; 0; 1; . . . ð6:31Þ h h

where, as usual, bxc stands for rounding non-negative x towards zero.

Proof of perfection of periodic correlation properties of Frank codes differs from that

above only in minor details and is left for Problem 6.30. As is seen from a comparison of p

(6.31) and (6.30) the phase step of Frank codes is reduced N times, so the alphabet size grows with N markedly slower.

þ1, þ1, þ1, þj, 1, j, þ1, 1, þ1, 1, þ1, j, 1, þj. Perfection of its periodic ACF may be tested by a direct computation.