Finishing with polyphase codes, note once more that they are not as attractive technologically as binary antipodal ones. Do codes exist which do not yield to the binary ones in implementation simplicity, but—unlike them—possess perfect periodic ACF? Answering this question is the subject of the next section.

6.11.3 Ternary sequences

Consider sequences whose elements ai may assume in addition to binary values 1 also zero value. In other words, the alphabet is now ternary f 1, 0, 1g, which technically means combining BPSK with pauses, i.e. time intervals where chips are missing. Clearly, an extension of the binary alphabet f 1g into the ternary one f 1, 0, 1g does not seriously complicate either the generation or processing circuitry, but, as is shown below, opens the way to obtaining sequences with perfect periodic correlation properties. Remember that one of the main reasons for an interest in spread spectrum in time measuring and resolution is a desire to achieve high performance operating at low peak power, i.e. with signal energy spanning a large time interval. Quite a natural measure of efficiency of energy time-spreading is the peak-factor (see Section 2.7.1), i.e. the ratio between peak and time-averaged power. For any PSK, in particular binary, sequence signal energy is uniformly spread over the period so that peak and average powers are the same and ¼ 1. Inserting Np pauses on the sequence period N, as occurs with the ternary alphabet, will make the energy spreading less uniform and increase the peakfactor as ¼ N/(N Np). Hence, our interest is to design ternary sequences possessing not only perfect periodic ACF, but also small number of zeros Np on the period, i.e. peak-factor not much higher than one. Without this constraint the problem would prove degenerated and have a trivial solution: code with only one non-zero symbol on the period N, corresponding to a single chip repeated with period ND, certainly has perfect periodic ACF.

A number of rules to generate ternary sequences with the properties just stated are now known. The most powerful of them produces sequences with lengths and values of peak-factor obeying the following equations:

where q ¼ pw is a natural power of a prime p and n is odd. Sequences of this sort exist for any combinations of q, n within stipulated limitations, and therefore, a peak-factor as close to 1 as is wished is achievable by just taking q large enough.

The constructions of the ternary sequences meeting (6.32) are based on some fine features of Galois fields. The simplest of them and at the same time covering the majority of lengths indicated by (6.32) corresponds to the case of odd p (q ¼ pw, p > 2) [40,41]. In order to present the idea in the most transparent form, let us give a detailed description of the algorithm only for the case q ¼ p, i.e. w ¼ 1. The easiest way to do it involves p-ary m-sequences.

Let di, i ¼ . . . , 1, 0, 1, . . ., be a p-ary m-sequence, where p is an odd prime. Each of its elements belongs to a prime field GF(p). Let us transform this sequence into a ternary one, mapping its zero elements into a real zero and non-zero elements into their binary

Figure 6.19 Generator of a ternary sequence (6.33)

characters. After this let us change the signs of all elements at the odd positions. The algorithm thus described is formally given by the equation:

where i ¼ . . . , 1, 0, 1, . . . . Figure 6.19 shows the structure implementing this rule and including the m-sequence generator unit, which maps m-sequence elements into characters or zeros, and the multiplier, providing alternation of polarities.

To compute the peak-factor of a ternary sequence (6.33) it is enough to recollect that the period of m-sequence is L ¼ pn 1, and the balance property asserts that on this period there are L0 ¼ pn 1 1 zero symbols. All of them but no others produce zeros in the ternary sequence; therefore, on the periodical segment of L elements of a ternary sequence exactly L0 elements are zeros and the peak-factor:

which coincides with what (6.32) tells us at q ¼ p. To prove that a sequence (6.33) has the period obeying (6.32) and perfect periodic ACF, one more pseudorandom property of m-sequences is exploited, proof of which can be found in [42]. To formulate it denote:

and consider all pairs (di, di m) of elements of p-ary m-sequence separated by m positions if i runs over one period (i ¼ 0, 1, . . . , L 1). Then (pair property) if m is not a multiple of h (m 6¼lh, l being integer), among pairs (di, di m) the pair (0, 0) occurs pn 2 1 times and any other pair (x, y) of fixed x, y 2 GF(p) occurs pn 2 times. Otherwise, if m ¼ lh, in the pairs (di, di m) the second element is strictly determined by the first: di m ¼ ldi, where is, as usual, a primitive element of the field GF(p).

With understanding that a ‘genuine’ (i.e. unknown so far) period N of a ternary sequence (6.33) is some divisor of the original m-sequence period L, let us calculate nonnormalized periodic ACF of the ternary sequence over the interval L, containing L/N periods:

di m6¼0

where terms in the last sum, for which didi m ¼ 0 resulting in zero contribution, are discarded. Consider first the case when shift m is not a multiple of h(m 6¼lh). Then according to the pair property among all pairs (di, di m) in (6.34), any pair (x, y) of nonzero fixed x, y 2 GF(p) occurs exactly pn 2 times. This allows computing (6.34) in the following way:

due to the character property (6.21). Now turn to the case of shifts divisible by h (m ¼ lh). According to the pair property only pairs (di, di m) ¼ (x, lx), x 2 GF(p) enter the sum in (6.34). But due to the balance property each period of a p-ary m-sequence contains exactly pn 1 fixed non-zero elements of GF(p). Therefore, making use of the multiplicative property of characters (6.20):

odd integers and consequently is odd itself. By that l(h þ 1) is even independently of l and Rp(lh) ¼ pn 1(p 1) NL. As is seen, Rp(lh) is the same for any integer l, while from (6.35) Rp(m) ¼ 0 whenever m 6¼lh. This shows that the Rp(m) as a function of m repeats

sequence of this period use the ternary m-sequence of Example 6.6.2: 1, 0, 0, 2, 0, 2, 1, 2, 2, 1, 0, 2, 2, 2, 0, 0, 1, 0, 1, 2, 1, 1, 2, 0, 1, 1, . . . . There are only two non-zero elements in GF(3), of which only 2 is primitive. It is clear that (1) ¼ 1, (2) ¼ 1 and all non-zero elements

184 Spread Spectrum and CDMA

of the m-sequence should be replaced as 1 ! þ1, 2 ! 1, zeros being mapped onto real zero. This produces the ternary sequence þ1, 0, 0, 1, 0, 1, þ1, 1, 1, þ1, 0, 1, 1, 1, 0, 0, þ 1, 0, þ1, 1, þ1, þ1, 1, 0, þ1, þ1, . . . of period 26. Changing the signs of the elements with odd numbers (starting with zero) gives the final ternary sequence þ1, 0, 0, þ1, 0, þ1, þ1, þ1, 1, 1, 0, þ1, 1, þ1, 0, 0, þ1, 0, þ1, þ1, þ1, 1, 1, 0, þ1, 1, . . . , having period N ¼ 13 and peak-factor ¼ 13/9 1:445. Perfection of its periodic ACF may be verified by a direct computation.

One may eliminate alternation of signs of odd-numbered elements in rule (6.33) and in the generator of Figure 6.19 using instead of m-sequences some special linear sequences of the smaller period. For that coefficients fi in the recurrence (6.13) and feedback of the LFSR generator should belong to an appropriate non-primitive irreducible polynomial of degree n. The theory behind it may be found in [41]. Examples of such polynomials of the third degree allowing removal of alternating signs in (6.33) are given in Table 6.5 for p 31. The last two columns contain non-maximal period L of the linear sequence generated by LFSR and period N of the final ternary sequence. One more advantage of these polynomials is that the negative of at least one of their coefficients is 1, simplifying multiplication down to just connection to the adder.

Example 6.11.6. Form the ternary sequence corresponding to p ¼ 3, n ¼ 3 starting with polynomial x3 þ 2x þ 2 of Table 6.5. The recurrence (6.13) then takes the form di ¼ di 2 þ di 3, generating with initial loading d0 ¼ 1, d1 ¼ d2 ¼ 0 the linear sequence over GF(3) 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 0, 1, 2 of period L ¼ 13. After mapping its non-zero elements onto their characters and zeros onto a real zero, the ternary sequence of period N ¼ 13 is formed identical to that of the previous example.

The extension of the construction above to the case q ¼ pw, p > 2, w > 1 is rather immediate and rule (6.32) preserves its validity. The only difference is that an m-sequence fdig in it is now q-ary, i.e. with elements belonging to an extension (as opposed to prime) finite field GF(q). The arithmetic of extension fields is a bit trickier than just modulo q operations, and we do not want to dwell on those details here. The reader may consult [40,41].

Table 6.5 Non-primitive polynomials over prime fields