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We developed an algorithm which allows obtaining a Gram matrix having a particular pattern for crosscorrelations of the signature sequences. When it is used for real numbers, unit tight frame signature sets are obtained, as they are required by 2G or 3G CDMA systems. When this algorithm is used for complex numbers, complex valued signature sets with constant chip magnitude are obtained. The results are derived in the context of synchronous CDMA systems and numerical results are provided to illustrate the mathematical formalism.

Resorting to some facts from analytic number theory and the following simple but important property of complex exponentials, we propose a new propagation delay estimation scheme for asynchronous CDMA systems over Gaussian channels: a circular shift in a complex exponential vector looks like multiplying it by a unit-magnitude complex number with shift-appropriate phase angle. In spite of the most existing schemes, the proposed method is a deterministic approach and therefore its performance does not depend on the near-far ratio and sampled data used for statistics estimation. It will be shown that over noise-free Gaussian channels, the propagation delay can be deterministically estimated by transmitting only two training symbols. We will dramatically improve the performance in noisy environments by modifying the method through consuming several more symbols for training.