The Enigma of CDMA Revisited

In this paper, we explore the mystery of synchronous CDMA as applied to wireless and optical communication systems under very general settings for the user symbols and the signature matrix entries. The channel is modeled with real/complex additive noise of arbitrary distribution. Two problems are addressed. The first problem concerns whether overloaded error free codes exist in the absence of additive noise under these general settings, and if so whether there are any practical optimum decoding algorithms. The second one is about the bounds for the sum channel capacity when user data and signature codes employ any real or complex alphabets (finite or infinite). In response to the first problem, we have developed practical Maximum Likelihood (ML) decoding algorithms for overloaded CDMA systems for a large class of alphabets. In response to the second problem, a general theorem has been developed in which the sum capacity lower bounds with respect to the number of users and spreading gain and Signal-to-Noise Ratio (SNR) can be derived as special cases for a given CDMA system. To show the power and utility of the main theorem, a number of sum capacity bounds for special cases are simulated. An important conclusion of this paper is that the lower and upper bounds of the sum capacity for small/medium size CDMA systems depend on both the input and the signature symbols; this is contrary to the asymptotic results for large scale systems reported in the literature (also confirmed in this paper) where the signature symbols and statistics disappear in the asymptotic sum capacity. Moreover, these questions are investigated for the case when not all users are active. Furthermore, upper and asymptotic bounds are derived and numerically evaluated and compared to other derivations.