Decentralized Wireless Networks: Spread Spectrum Communications Revisited

This paper addresses a decentralized wireless networks of K separate transmitter-receiver pairs. Users treat each other as noise and there is no central controller to assign the resources to the users. Each user randomly spreads the symbols in its Gaussian codewords by the so-called signatures of spreading gain N. Any receiver is aware of the signatures of its affiliated transmitter, however, it is unaware of the signatures of other users. This makes the interference plus noise at each receiver be mixed Gaussian, and hence, there is no closed expression for the achievable rates of users. Invoking conditional entropy power inequality and a key upper bound on the differential entropy of a mixed Gaussian random vector, we develop a lower bound on the achievable rates of users. This lower bound has the same signal-to-noise ratio (SNR) scaling as that of the exact achievable rate. It is shown that the sum multiplexing gain (SMG) in the network can be made arbitrarily close to (K/N) for any finite values of K and N where K ≤ N. The effect of matched filtering is studied in the particular case where the signatures are constructed over a binary alphabet. It is established that the SMG of the network is larger than (1/2e) regardless of the value of K as long as N = 2 and the signatures are generated according to a proper nonuniform distribution. This paper is concluded by a section on signature design in the finite SNR regime. The main observation is that for any two different methods A and B of designing the signatures, if method A results in a larger achievable rate per user for sufficiently large SNR values, then construction B is likely to yield larger achievable rates for sufficiently small values of SNR. This behavior is attributed to the interplay between two critical factors, namely, the multiplexing gain per user and what we refer to as the interference entropy factor.