Information Theory and Wireless Channel Modeling

The problem of modelling channels is crucial for the efficient design of wireless systems [1, 2, 3]. The wireless channel suffers from constructive/destructive interference signaling [4, 5]. This yields a randomized channel with certain statistics to be discovered. Recently ([6, 7]), the need to increase spectral efficiency has motivated the use of multiple antennas at both the transmitter and the receiver side. Hence, in the case of i.i.d Gaussian entries of the MIMO link and perfect channel knowledge at the receiver, it has been proved [8] that the ergodic capacity increase is min(nr,nt) bits per second per hertz for every 3dB increase (nr is the number of receiving antennas and nt is the number of transmitting antennas) at high Signal to Noise Ratio (SNR). However, for realistic channel models, results are still unknown and may seriously put into doubt the MIMO hype. As a matter of fact, the actual design of efficient codes is tributary of the channel model available: the transmitter has to know in what environment the transmission occurs in order to provide the codes with the adequate properties: as a typical example, in Rayleigh fading channels, when coding is performed, the Hamming distance (also known as the number of distinct components of the multi-dimensional constellation) plays a central role whereas maximizing the Euclidean distance is the commonly approved design criteria for Gaussian channels (see Giraud and Belfiore [9] or Boutros and Viterbo [10]). As a consequence, channel modelling is the key in better understanding the limits of transmissions in wireless and noisy environments. In particular, questions of the form: ”what is the highest transmission rate on a propagation environment where we only know the mean of each path, the variance of each path and the directions of arrival?” are crucially important. It will justify the use (or not) of MIMO technologies for a given state of knowledge. Let us first introduce the modelling constraints. We assume that the transmission takes place between a mobile transmitter and receiver. The transmitter has nt antennas and the receiver has nr antennas. Moreover, we assume that the input transmitted signal goes through a time variant linear filter channel. Finally, we assume that the interfering noise is additive white Gaussian.