Bounds on the Number of Signals with Restricted Cross Correlation
Harrison E. Rowe · IEEE Transactions on Communications · 1982
The author seeks the maximum number of users possible for multiuser, spread-spectrum radio systems, as the signal set is varied, under the following ideal conditions: 1) no fading; 2) synchronous system; 3) orthogonal signal set for each user; 4) matched-filter individual receivers; 5) no additive noise. Welch (1974) has given an upper bound on the number of vectors (or equivalently signals)wpossible in ad-dimensional space, given the maximum cross-correlation magnitudeCmaxbetween any vector pair. However, the error rate for the systems under study may depend on a suitably defined root-mean-square cross correlationCrms, rather than on the maximum cross correlationCmaxof a vector set. Welch's results are extended tod-dimensional vector sets of sizewdivided into alphabets of sizeA, the vectors of each alphabet being strictly orthogonal, and the maximum size of such a vector set is given as function ofCrms. It is demonstrated that as Welch's limit is approached, all cross correlations must approachCmax, and given precise limits on the way in which this must occur. Similarly, close to the present bounds (in terms ofCrms) all rms cross correlations must be close toCrms. Particular examples are given of signal sets close to the present bounds.