(see (7.40)) level of average power of MAI and multipath interference at the correlator output.

7.5 Examples of minimax signature ensembles

The signature ensembles considered in the previous section may be regarded as adequate only in situations where mutual time shifts of users’ signals are entirely controllable by the system and may be kept within the predicted range. If this is not the case, asynchronous CDMA based on shifted replicas of the same sequence risks collisions: the signal of one user may acquire delay, making it indistinguishable from the signal of some other user. This may be the reason for employing minimax signature ensembles, i.e. those whose correlation peaks achieve or approach bounds (7.45) or (7.46). Since the correlation peak of a minimax ensemble is maximized over the whole period, its small value (achievable at the cost of long enough length L) secures the proximity of ensemble correlation properties to the perfect ones (7.43), guaranteeing pseudorandomness of signatures.

A survey of all known minimax ensembles would take a lot of space, so we will confine ourselves to a brief discussion of those that either enjoy wider practical application or seem more indicative among others. Readers interested in learning more about them may consult [9,67,70].

7.5.1 Frequency-offset binary m-sequences

Take a binary f 1g m-sequence fa1, ig of period L ¼ 2n 1 and use it as a signature for the first user. The rest of the K 1 signatures are generated by a symbol-wise multiplication of fa1, ig with discrete harmonics of frequencies (k 1)/L, k ¼ 2, 3, . . . , K:

236 Spread Spectrum and CDMA

which is the (k l)th component of the DFT energy spectrum of the sequence fa1, ig. Since the energy spectrum of fa1, ig is the DFT of its periodic ACF, and the ACF equals1 everywhere except at the zero point, where it is equal to L, we have:

The last sum differs from zero and equals L only for k ¼ l, so that collecting all the results together and passing over to normalized correlations gives:

It is seen now that the squared correlation peak of the ensemble (7.48):

2max ¼ LLþ2 1 L1

i.e. practically coincides with the Welch bound (7.45). Thus, the ensemble under scrutiny is a minimax one, realizing the optimal asynchronous CDMA mode.

The description of the above ensemble one may find, e.g. in [71], yet earlier and independently it was used in the global satellite-based navigation system GLONASS (see Section 11.2). One of the advantages of this signature set versus other polyphase ones is the possibility of generating signatures by a simple offset of carrier frequency. Indeed, incrementing the carrier frequency f0 by (k 1)/LD is equivalent to a linear phase progression between adjacent chips equalling 2 (k 1)/L, which is exactly what is prescribed by the rule (7.48).

Despite many other minimax polyphase ensembles being known, the binary f 1g ones are traditionally considered more attractive from a hardware point of view, and the rest of this section is dedicated to some important examples of binary signature sets.

7.5.2 Gold sets

The following properties of binary f 1g m-sequences may serve to explaining the set construction found by Gold:

1.If a binary f 1g m-sequence fuig of period L ¼ 2n 1 is decimated with the decimation index d, where d is co-prime to L, the resulting sequence fvig is again a binary m-sequence of the same period. To decimate means to pick out every dth symbol of fuig and write symbols thus obtained one by one, so that vi ¼ udi. We call the sequence fvig produced this way a decimation of fuig.

Let us estimate the correlation peak of the Gold ensemble, beginning by calculating correlations of the first L sequences:

It is seen that since the case m ¼ 0 modL and k ¼ l corresponds to the mainlobe of the kth ACF, the situation should be analysed where these equalities are not fulfilled simultaneously. But then either both uiui m and vi kvi l m are just some other shifts of the initial sequences fuig, fvig, or only one of those products is a sequence consisting of only ones. In the first case we have the CCF of the initial m-sequences fuig, fvig taking on only the three values indicated by (7.50) or (7.51), while in the second we have the non-normalized ACF sidelobe of one of the sequences fuig, fvig, i.e. 1.

Consider now the CCF of fak, ig, k ¼ 1, 2, . . . , L and fal, ig, l ¼ L þ 1:

L 1

X

Rp;klðmÞ ¼ uiui mvi k i¼0

If m ¼ 0 modL, uiui m ¼ 1 and the CCF is simply a constant component of fvig, i.e. 1. Otherwise uiui m ¼ ui s for some s and we have the CCF of initial m-sequences obeying the restrictions (7.50) or (7.51). The same is true for the CCF of fak, ig, k ¼ 1, 2, . . . , L and fal, ig, l ¼ L þ 2.

Finally, the CCF of faLþ1, ig and faLþ2, ig is directly the CCF of the initial m-sequences, while their autocorrelation functions, like those of m-sequences, have non-normalized ACF sidelobes equalling 1. Collecting all of the results together, we see that the correlation peak (7.44) of the Gold set is determined by the maximal in modulus value of the original CCF (7.50) or (7.51). After normalizing it to the length L we come to the estimation:

L

with the last approximation corresponding to large length L 1. As is seen, for any odd memory n Gold signature ensembles asymptotically (L 1) attain the Sidelnikov lower bound (7.46), while for the case of even n not divisible by four their loss in max against this bound is about 3 dB.3

3 When n ¼ 0 mod4 a Gold ensemble also exists with the same correlation peak as in the case n ¼ 2 mod4, but with number of sequences smaller by one [67,70].