Bounds on the Sum Capacity of Synchronous Binary CDMA Channels

Abstract

In this paper, we obtain a family of lower bounds for the sum capacity of Code Division Multiple Access (CDMA) channels assuming binary inputs and binary signature codes in the presence of additive noise with an arbitrary distribution. The envelope of this family gives a relatively tight lower bound in terms of the number of users, spreading gain and the noise distribution. The derivation methods for the noiseless and the noisy channels are different but when the noise variance goes to zero, the noisy channel bound approaches the noiseless case. The behavior of the lower bound shows that for small noise power, the number of users can be much more than the spreading gain without any significant loss of information (overloaded CDMA). A conjectured upper bound is also derived under the usual assumption that the users send out equally likely binary bits in the presence of additive noise with an arbitrary distribution. As the noise level increases, and/or, the ratio of the number of users and the spreading gain increases, the conjectured upper bound approaches the lower bound. We have also derived asymptotic limits of our bounds that can be compared to a formula that Tanaka obtained using techniques from statistical physics; his bound is close to that of our conjectured upper bound for large scale systems.

A review on synchronous CDMA systems: optimum overloaded codes, channel capacity, and power control

“…The sum capacity for CDMA as a special case of MAC is also defined in [43]. For the noiseless case, the channel capacity of a system with binary signature matrix A will be equal to Hosseini…”

“…The authors of [43] have tried to find the lower bounds for both noisy and noiseless channels in binary CDMA systems by choosing a random signature matrix and then derive the expected value of the sum capacity of the channel corresponding to this random matrix. In other words, the lower bound is the average sum channel capacity of a typical signature matrix.…”

“…In other words, the lower bound is the average sum channel capacity of a typical signature matrix. According to [41][42][43], the capacity of a channel with random signature matrix will be higher than the expected value with high probability.…”

“…All upper bounds that are derived for noisy and noiseless channels are based upon a conjecture which implies that the input vectors have uniform distribution [43]. In [41], the authors used this conjecture for the special case when the noise has Gaussian distribution.…”

“…In [20,21], the authors have found lower and upper bounds for the channel capacity in the general case without using the statistical physics approach for both noiseless and noisy channels. In [43], the same task was performed for the binary case.…”