Fluctuations of the Shannon capacity in a Raleigh model of wireless communication

Using the fact that the Shannon capacity C of a Raleigh model of wireless channels is a linear statistic of the channel matrix, we calculate its variance var[C]. We find that the expected value C of the Shannon capacity is typical in the model considered, that is the coefficient of variation var[C]/C is small. The efficiency of a wireless channel is determined by its Shannon capacity, C = log 2 (1 + ρ |H| 2) where ρ is the signal-to-noise ratio and H is the transfer characteristic of the channel [1]. The Shannon capacity describes the rate of information transfer (in bits per second, bps). A crucial question in the design of multiple-anntenna arrays is: how does the channel capacity increase with the number of channels? Consider an array of n T transmitters and n R receivers as shown schematically in Fig. 1. The scattering medium is characterised by a n T × n R channel matrix H with complex matrix elements H kl determining the amplitude of the lth receiving antenna arriving from transmitter k. In realistic situations, the scattering medium changes as a function of time, and so does the capacity. Typically it fluctuates randomly; it was therefore suggested [2, 1] to calculate an average capacity as an ensemble average over random matrices H. In [2, 1] an idealised model (called Raleigh model in the following) was considered: H kl were taken to be uncorrelated random variables with zero mean and unit variance. This corresponds to a regular array of antennae, spaced λ/2 apart (λ is the wave length). In this case, the capacity is given by C(H) = log 2 det 1 + ρ n T HH † (1) and its average was calculated in [2]. Remarkably, the average capacity was found to scale linearly with the number of transmitters (receivers) for a large number of antennae. This observation has attracted considerable attention, and the average Shannon capacity in related, but more general models (incorporating correlations between the matrix elements H kl) has been studied in great detail [3, 4, 5, 6]. It was found that correlations between the matrix elements H kl reduce the Shannon capacity somewhat, but by increasing the number of antennae, the Shannon capacity can be increased significantly. Empirical studies have indeed shown substantial efficiency gains for such antenna arrays [7]. An important question is however: how typical is the expected value of …