Bounds on the Sum Capacity of Synchronous Binary CDMA Channels

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.

[1]  Sergio Verdú,et al.  The capacity region of the symbol-asynchronous Gaussian multiple-access channel , 1989, IEEE Trans. Inf. Theory.

[2]  Sergio Verdú,et al.  Randomly spread CDMA: asymptotics via statistical physics , 2005, IEEE Transactions on Information Theory.

[3]  J. K. Skwirzynski New concepts in multi-user communication , 1981 .

[4]  Erik Ordentlich Maximizing the entropy of a sum of independent bounded random variables , 2006, IEEE Transactions on Information Theory.

[5]  Saieed Akbari,et al.  Errorless codes for over-loaded synchronous CDMA systems and evaluation of channel capacity bounds , 2008, 2008 IEEE International Symposium on Information Theory.

[6]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[7]  Sergio Verdu,et al.  Multiuser Detection , 1998 .

[8]  Sergio Verdú,et al.  Optimum asymptotic multiuser efficiency of randomly spread CDMA , 2000, IEEE Trans. Inf. Theory.

[9]  Peter Harremoës,et al.  Binomial and Poisson distributions as maximum entropy distributions , 2001, IEEE Trans. Inf. Theory.

[10]  Srinivasa R. S. Varadhan,et al.  Asymptotic probabilities and differential equations , 1966 .

[11]  Muriel Médard,et al.  The effect upon channel capacity in wireless communications of perfect and imperfect knowledge of the channel , 2000, IEEE Trans. Inf. Theory.

[12]  Tommy Guess,et al.  Effects of Spreading and Training on Capacity in Overloaded CDMA , 2008, IEEE Transactions on Communications.

[13]  Thomas M. Cover,et al.  Elements of information theory (2. ed.) , 2006 .

[14]  Rudolf Ahlswede,et al.  Multi-way communication channels , 1973 .

[15]  Toshiyuki Tanaka,et al.  A statistical-mechanics approach to large-system analysis of CDMA multiuser detectors , 2002, IEEE Trans. Inf. Theory.

[16]  Babak Hassibi,et al.  How much training is needed in multiple-antenna wireless links? , 2003, IEEE Trans. Inf. Theory.

[17]  E. J. Weldon,et al.  Coding for T-user multiple-access channels , 1979, IEEE Trans. Inf. Theory.

[18]  Shlomo Shamai,et al.  Spectral Efficiency of CDMA with Random Spreading , 1999, IEEE Trans. Inf. Theory.

[19]  Nicolas Macris,et al.  On the concentration of the capacity for a code division multiple access system , 2007, 2007 IEEE International Symposium on Information Theory.

[20]  Nicolas Macris,et al.  Tight Bounds on the Capacity of Binary Input Random CDMA Systems , 2008, IEEE Transactions on Information Theory.

[21]  Venkat Anantharam,et al.  Optimal sequences and sum capacity of synchronous CDMA systems , 1999, IEEE Trans. Inf. Theory.

[22]  Andrea Montanari,et al.  Analysis of Belief Propagation for Non-Linear Problems: The Example of CDMA (or: How to Prove Tanaka's Formula) , 2006, 2006 IEEE Information Theory Workshop - ITW '06 Punta del Este.

[23]  Saieed Akbari,et al.  A Class of Errorless Codes for Overloaded Synchronous Wireless and Optical CDMA Systems , 2009, IEEE Transactions on Information Theory.

[24]  Aaron D. Wyner,et al.  Recent results in the Shannon theory , 1974, IEEE Trans. Inf. Theory.

[25]  Toshiyuki Tanaka Replica Analysis of Performance Loss Due to Separation of Detection and Decoding in CDMA Channels , 2006, 2006 IEEE International Symposium on Information Theory.

[26]  T. Cover Some Advances in Broadcast Channels , 1975 .

[27]  Jacob Wolfowitz,et al.  Multiple Access Channels , 1978 .

[28]  James L. Massey,et al.  Optimum sequence multisets for synchronous code-division multiple-access channels , 1994, IEEE Trans. Inf. Theory.

[29]  R. Ahlswede The Capacity Region of a Channel with Two Senders and Two Receivers , 1974 .