A multitude of even more effective binary codes were found, based on converting linear recurrent sequences over finite fields to the binary f 1g alphabet, as well as on Singer codes. All of them possess a similar appealing feature: a simple SLSF structure whose coefficients take on no more than three different values. Without going into the details, which are sophisticated enough, and referring the curious reader to [49, 50], note that among those sequences families are present having asymptotically vanishing SLSF loss: dB ! 0 dB with N ! 1.

6.13 FSK signals with optimal aperiodic ACF

To discuss briefly the issue of designing FSK signals with good correlation properties let us come back to (5.20), recalling that having a low level of Rp(m) is equivalent to minimizing the number of coincident frequencies in a frequency code F0, F1, . . . , FN 1 and its replica shifted by m positions. Certainly, when the number of chips N (i.e. length) does not exceed the number of frequencies M, it is trivially easy to obtain zero level of sidelobes a(m) ¼ 0, m 6¼0 by using a frequency code whose elements Fi are all different. Practically, however, the case N > M is much more interesting, entailing repetitions among elements Fi, i ¼ 0, 1, . . . , N 1 and, thus, at least one coincidence in the shifted replicas of the frequency code, i.e. a, max 1/N.

Since a frequency sequence may be described as an M N array (Section 5.5), minimizing a, max means inventing an array with minimal possible number of coincident labels (dots) in the array itself and its replica, which is shifted horizontally by m positions. One of the topical problems is constructing the so-called radar arrays defined as M N arrays having only one labelled entry in every column and a, max ¼ 1/N, i.e. the number of abovementioned coincidences within one. The desire to find a radar array as long as possible is understandable, given M, because it would mean minimizing a, max under limitations imposed on the frequency resource. Following [51] let us proof a simplest upper bound on the length of a radar array.

Consider a sequence F0, F1, . . . , FN 1 and note that to have no more than one coincidence all differences between numbers of positions carrying the same frequencies should be different. Indeed, let Fi ¼ Fk, Fs ¼ Ft and i k ¼ s t > 0. Then in the original sequence and its copy shifted by m ¼ i k ¼ s t positions at least two coin-

XX

Now count the number of possible differences between numbers of positions carrying identical frequencies. There are i repetitions of some frequency and hence i(i 1) such

differences for this very frequency. Since there are ni frequencies repeated i times, the

P

total number of differences in question is i i(i 1)ni, and because among the differences no repetition is allowed:

X

i

where the right-hand side gives the maximal number of unequal positive differences among the numbers f0, 1, . . . , N 1g. The trinomial i(i 1) þ 3 2i ¼ i2 3i þ 3 has no real roots, and, hence, is positive at any i. Therefore the sum:

X X X X

½iði 1Þ þ 3 2i&ni ¼ iði 1Þni þ 3 ni 2 ini 0

i i i i

which, being combined with (6.47), (6.48), gives N 1 þ 3M 2N 0 or:

In fact this bound is not the tightest one. More accurate bounds are known, e.g. in [52] the asymptotic result is derived:

p

N 20 þ 6 M; M 1; ð6:50Þ 8

lowering the right-hand side of (6.49) by approximately 0.194M.

Absolutely tight, i.e. really achievable, upper bounds on the length N are now known up to M ¼ 16. The table given in [52] allows the maximal length Nmax of a radar array in this range of M to be expressed as

Example 6.13.1. The frequency code 1, 2, 3, 4, 5, 6, 7, 8, 7, 4, 3, 9, 9, 5, 8, 2, 6, 5, 1, 4, 2, 1, 3, 7, where the numbers of frequencies in an alphabet containing M ¼ 9 frequencies or, equivalently, the numbers of dotted rows in every column of the array are given, has maximal possible length N ¼ 24. Its radar array property, i.e. possessing only one frequency coincidence at all non-zero shifts, is verified by a direct test (Problem 6.54).

In addition, a regular rule for constructing radar arrays of length N ¼ 2:5M exists (see details in [51]) whenever M is even and M/2 is a product of primes having remainder one of division by 4, i.e. M ¼ 10, 26, 34, 58, . . . .

A sonar array is a further generalization of a radar array, preserving the ‘no more than one coincidence’ property for arbitrary non-zero combinations of horizontal and vertical shifts [53]. Physically, this requirement reflects the desire to have a small correlation of signal replicas detuned in both time and frequency. Considering an approach to the choice of frequency space for FSK signals (Section 5.5), frequency shifts turning a current frequency into the adjacent one are more typical of sonar