oversaturated autonomously, providing n þ nov signatures, so that all signatures from different subspaces remain orthogonal. The reason for so doing is to split an overall multiuser algorithm into N/n parallel ones, each operating in the n-dimensional subspace independently of the others. With a moderate n these partial algorithms are simple, making the whole receiver structure technologically feasible. The total number of users achievable in such a system is:

Nn ðn þ novÞ ¼ N 1 þ nnov

The problem of optimizing an ensemble of this sort is in a sense non-trivial, and just adding nov supplementary signatures to the n primary orthogonal ones does not solve it. We refer the curious reader to [58,59] for details. Another alternative is the design of signature ensembles allowing implementation of multiuser algorithms in various computational-effective iterative forms [60,61].

7.2.3 Welch-bound sequences

Let us get down to another scenario, where a priori tough limitation on the receiver complexity makes acceptable only the simplest, i.e. single-user or conventional, recep-

^

tion algorithm. In this case a decision bk on the current data symbol bk of the kth user is defined only by correlation (7.8), as though no interference except AWGN is present at the receiver input. With no loss of generality, we may admit that a current symbol is received at the interval (0,T] and put delay k and phase k in (7.8) equal to zero:

ZT

zk ¼ Y_ ðtÞS_kðtÞdt ð7:25Þ

0

When all signatures are perfectly synchronized and their number K does not exceed N, orthogonal signatures are again the best choice, since they make a single-user algorithm identical to a multiuser (ML) one. Certainly, no MAI arises in this case so ignoring all the signals of the other users does not undermine the receiver optimality. In contrast to this, the case of an oversaturated (K > N) system is of separate interest, because all signatures then cannot be orthogonal and MAI is unavoidable. Returning to (7.11), let us present the observed complex envelope as:

K

Y_ ðtÞ ¼ S_ðt; b0Þ þ N_ ðtÞ ¼ X b0lS_lðtÞ þ N_ ðtÞ

l¼1

where N_ (t) is the noise complex envelope and designation b0 ¼ (b01, b02, . . . , b0K ) symbolizes again (as in (4.8)) the genuine (i.e. unknown at the receiver) data pattern transmitted by K users to distinguish it from the one b ¼ (b1, b2, . . . , bK ) hypothesized in the course of the decision. After substituting this into (7.25) we obtain:

vectors, which we are interested to have as small as possible. There are K(K 1) such vector pairs entering TSC, so that average squared correlation 2 per pair is:

2 ¼ TSC K KðK 1Þ

giving, together with (7.30), the lower bound on this parameter:

From the way of obtaining (7.30), we may deduce how to come to the Welch boundensemble. Of course, only a non-trivial case of oversaturation should be discussed, since ways of generating orthogonal sequences have been considered previously. First of all, equality in (7.29) is a sufficient (and, of course, necessary) condition of equality in (7.30), or, considering the equation preceding (7.29), sequences for which:

then (7.32) means nothing but orthogonality of the rows of A. Therefore, to build up an oversaturated (K > N) ensemble of Welch-bound sequences, one should just construct an N K matrix A with orthogonal rows. Since the dimension of rows of such a matrix is greater than their number, there is no principal prohibition on its existence. Then the desired sequences are simply columns of A.

We may now estimate the floor (i.e. noise-neglected) SIR for an oversaturated Welchbound ensemble. The total MAI power PI may be found from (7.27) and (7.28) as PI ¼ 4E2(TSC K). Since this quantity is MAI summed over K single-user receivers,

¼

an average output MAI power per receiver will be PIk PI /K. The useful (i.e. caused by the kth signature) effect at the kth receiver output expressed by the first term of (7.26) has power 4E2 (PSK modulation assumed), so that the floor power SIR with respect to an average MAI power according to (7.30) is: