This paper provides an overview of state-of-the-art radio propagation and channel models for wireless multiple-input multiple-output (MIMO) systems. We distinguish between physical models and analytical models and discuss popular examples from both model types. Physical models focus on the double-directional propagation mechanisms between the location of transmitter and receiver without taking the antenna configuration into account. Analytical models capture physical wave propagation and antenna configuration simultaneously by describing the impulse response (equivalently, the transfer function) between the antenna arrays at both link ends. We also review some MIMO models that are included in current standardization activities for the purpose of reproducible and comparable MIMO system evaluations. Finally, we describe a couple of key features of channels and radio propagation which are not sufficiently included in current MIMO models. I. INTRODUCTION AND OVERVIEW Within roughly ten years, multiple-input multiple-output (MIMO) technology has made its way from purely theoretical performance analyses that promised enormous capacity gains [1], [2] to actual products for the wireless market (e.g., [3], [4], [5]). However, numerous MIMO techniques still have not been sufficiently tested under realistic propagation conditions and hence their integration into real applications can be considered to

In this contribution, models of wireless channels are derived from the maximum entropy principle, for several cases where only limited information about the propagation environment is available. First, analytical models are derived for the cases where certain parameters (channel energy, average energy, spatial correlation matrix) are known deterministically. Frequently, these parameters are unknown (typically because the received energy or the spatial correlation varies with the user position), but still known to represent meaningful system characteristics. In these cases, analytical channel models are derived by assigning entropy-maximizing distributions to these parameters, and marginalizing them out. For the MIMO case with spatial correlation, we show that the distribution of the covariance matrices is conveniently handled through its eigenvalues. The entropy-maximizing distribution of the covariance matrix is shown to be a Wishart distribution. Furthermore, the corresponding probability density function of the channel matrix is shown to be described analytically by a function of the channel Frobenius norm. This technique can provide channel models incorporating the effect of shadow fading and spatial correlation between antennas without the need to assume explicit values for these parameters. The results are compared in terms of mutual information to the classical i.i.d. Gaussian model.

Abstract — The impact of scattering condition and array configuration on performances are inseparable in early analyses of multiple-antenna systems. An array-independent scattering model is introduced where three basic scattering mechanisms are modeled. Performance results become more intrinsic property of the scattering channel itself. For linear arrays of length L in an environment of total angle spread |Ω|, the ergodic capacity is shown to increase linearly with L|Ω | for large arrays. When antenna arrays reduce to practical sizes, the capacity scaling depends on the SNR as well. This implies that the number of antennas used should also depend on the SNR. In terms of outage capacity, the trade-off between spatial multiplexing gain and diversity gain is shown to be very sensitive to the underlying scattering mechanisms. Finally, as |Ω | varies with the propagation range, the trade-off among multiplexing gain, diversity gain, and propagation range is studied. Index Terms — Multiple antennas, multiple-input multipleoutput (MIMO) systems, physical channel modeling, antenna

In this contribution, the double directional model derived within the maximum entropy framework in [1] is studied. For a given power profile, an asymptotic analysis (in the number of antennas) is conducted on the achievable transmission limit using tools of random matrix theory. It is shown in particular that the optimal throughput is achieved when the scatterers have equally distributed powers. For special cases, the mutual information is proven to be asymptotically Gaussian.

In this contribution, the performance of a downlink code division multiple access (CDMA) system with orthogonal spreading and multicell interference is analyzed. A useful framework is provided in order to determine the optimal base station coverage for wireless frequency selective channels with dense networks where each user is equipped with a matched filter. Using asymptotic arguments, explicit expressions of the spectral efficiency are obtained and provide a simple expression of the network spectral efficiency based only on a few meaningful parameters. Contrarily to a common misconception which asserts that to increase spectral efficiency in a CDMA network, one has to increase the number of cells, we show that, depending on the path loss and the fading channel statistics, the code orthogonal gain (due to the synchronization of all the users at the base station) can compensate and even compete in some cases with the drawbacks due to intercell interference. The results are especially realistic and useful for the design of dense networks.

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Abstract — We investigate the problem of establishing the joint probability distribution of the entries of a Multiple-Input Multiple-Output (MIMO) spatially correlated flat-fading channel, when little or no information about the channel properties are available. We show that the entropy of a random positive semidefinite matrix is maximized by the Wishart distribution. We subsequently obtain the Maximum Entropy distribution of the MIMO transfer matrix by establishing its distribution conditioned on the covariance, and by later marginalizing over the covariance matrix. The obtained distribution is isotropic, and is described analytically as a function of the Frobenius norm of the channel matrix. I.

The problem of modelling channels is crucial for the efficient design of wireless...

In [1], a unified framework for constructing MIMO models based on the principle of maximum entropy was provided. Several MIMO directional and double directional models were derived and in each case, the mutual information was shown to have asymptotically a Gaussian behavior. Interestingly, the results were shown to be accurate, through simulations, in the finite regime with an astonishing small number of antennas. In this paper, the models are analyzed and validated based on a wideband outdoor measurement campaign carried out in Oslo during summer 2002. The measurements were performed at a center frequency of 2.1 GHz with a bandwidth of 100 MHz in three different urban scenarios: a regular street grid scenario, an open city place and an indoor cell site.

We apply the replica method to analyze vector precoding, a method to reduce transmit power in antenna array communications. The analysis applies to a very general class of channel matrices. The statistics of the channel matrix enter the transmitted energy per symbol via its R-transform. We find that vector precoding performs much better for complex than for real alphabets. As a byproduct, we find a nonlinear precoding method with polynomial complexity that outperforms NP-hard Tomlinson-Harashima precoding for binary modulation on complex channels if the number of transmit antennas is slightly larger than twice the number of receive antennas. I.