Example 7.5.1. Construct Gold sequences of length L ¼ 23 1 ¼ 7. An ensemble of that small length is impractical but useful for elucidating the idea. Let us start with the binary f0, 1g m-sequence first met in Example 6.6.1: fui0g ¼ f1, 0, 0, 1, 0, 1, 1g. The decimation index d ¼ 3

sequence fa1, i g ¼ fþ g. Shifting fvi0g to the right by one position and adding modulo 2 to fui0g gives the sequence f1, 1, 1, 0, 0, 0, 1g, which after transition to symbols f 1g gives the second Gold sequence f þ þ þ g. Six more Gold sequences are obtained by further shifts of fvi0g, adding modulo 2 to fui0g and changing symbols into f 1g. Along with fui0g and fvi0g transformed into f 1g sequences, we obtain K ¼ 23 þ 1 ¼ 9 sequences altogether. Checking the value of the correlation peak in this simplest case makes little sense, since with L ¼ 7 no non-normalized periodic correlation, but the ACF mainlobe, may exceed 5, predicted by (7.53). Building Gold ensembles of greater lengths and checking their optimality is the subject of Problem 7.40.

Gold ensembles enjoy great popularity in modern CDMA systems. Suffice it to say that they are employed in the space-based global navigation system GPS for multiplexing satellite signals, and in the 3G mobile radio UMTS standard for scrambling CDMA codes, etc.

7.5.3 Kasami sets and their extensions

The idea of constructing Kasami sets is very close to that described above for the Gold

fvig ¼ fudig of the period being a factor of L. It may be shown that if fuig is initialized

Then L1 Kasami signatures of length L are formed as symbol-wise products of the initial m-sequence fuig with L1 different cyclic replicas of fvig, and one more signature is a ‘long’ sequence itself:

ak;i ¼ uivi k; k ¼ 1; 2; . . . ; L1

aL1þ1;i ¼ ui

ð7:55Þ

p

where i ¼ . . . , 1, 0, 1, . . .. There are K ¼ L1 þ 1 ¼ 2h ¼ L þ 1 such signatures of period L in total. Of course, again, multiplication of f 1g sequences fuig, fvig may be

special sequence based on the so-called bent function. The details of this are tricky enough and will not be discussed here, but the important thing is that any bent sequence has normalized CCF with any of the first L1 Kasami sequences (7.55) not exceeding by its modulus the correlation peak of both the Kasami and bent sequence ensembles.

unique in the sense that among all known binary ensembles with correlation peak

7.5.4 Kamaletdinov ensembles

More binary minimax ensembles exist [9,67]; however, some of them differ from the described ones only in a fine structure of sequences but not in length L, size K and correlation peak max. Against this background the ensembles discovered by Kamaletdinov [76] are of a particular interest, covering the range of lengths differing from those of Gold and Kasami sets.

In order to make the idea easier to understand, we describe a somewhat narrowed version of Kamaletdinov sets, although with no loss as to the length range or parameters achievable. To outline the first Kamaletdinov scheme let us take prime odd p > 3 of the

will produce the same final result). Let us treat a position number i of the sequence symbol as an element of GF(p), i.e. being reduced modulo p, and form p þ 1 p-ary sequences dk, i over GF(p) (i.e. with elements from this field) as follows:

i ¼ . . . , 1, 0, 1, . . .. One may see that every sequence in (7.57) is formed as a sum of sequences of co-prime periods p and p 1 ( p 1 ¼ 0 ¼ 1), and therefore has the period L ¼ p(p 1). Now perform a mapping of sequences (7.57) onto the binary alphabet f 1g using the extended binary character:

The binary set thus generated has parameters:

Length L may be made large enough only by a proper choice of p (p 1), in which case

(p þ 3)2/L ¼ (p þ 3)2/(p2 p) 1 and 2max ! 1/L, showing after comparing with (7.45) at least asymptotical optimality of the ensemble in the correlation peak.

Example 7.5.3. Let p ¼ 7. Direct verification confirms that ¼ 3 is a primitive element in GF(7). Then the sequences f i g and f i g are both of period p 1 ¼ 6: f. . . , 1, 3, 2, 6, 4, 5, 1, 3, . . .g

K ¼ p þ 1 ¼ 8 7-ary sequences of period L ¼ p(p 1) ¼ 42. For instance, the first of them is

fþ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þ þþg. It is rather tiresome to compute their ACF and CCF ‘by hand’, and Problem 7.42 provides Matlab software to support it.

The second Kamaletdinov construction exploits the p-ary (p ¼ 4h þ 3 ¼ 3mod4) linear sequence fcki g obtained by decimation with the index d ¼ p 1 of p 1 shifts fdiþkg, k ¼ 1, 2, . . . , K ¼ p 1 of the p-ary m-sequence fdig having memory n ¼ 2, i.e. length p2 1. Since d divides p2 1 the sequence fcki g ¼ fddiþkg has the period (p2 1)/(p 1) ¼ p þ 1. Now let us build p 1 sequences over GF(p):

and map them onto the binary alphabet f 1g according to (7.58). This generates the ensemble with parameters:

Again, for the case of long lengths (p 1) the ratio (p þ 1)2/L ¼ (p þ 1)2/(p2 þ p) ! 1 and 2max ! 1/L, demonstrating at least asymptotical optimality of the ensemble.

Example 7.5.4. This time there is no exclusion for p ¼ 3 and the p-ary m-sequence fdi g of memory n ¼ 2 and length p2 1 ¼ 8 may be formed by the primitive polynomial over GF(3) of the second degree f (x) ¼ x2 þ x þ 2, or equivalently, by a recurrent equation di ¼ 2di 1 þ di 2.

4: f. . . , 1, 1, 2, 2, 1, 1, 2, 2, . . .g and f. . . , 0, 2, 0, 1, 0, 2, 0, 1, . . .g. After symbol-wise addition with