Background and Motivation

The previous chapters have investigated the dimensionality and representation of a spatial multipath field. Chapter 2 reviewed the approximation of a spatial field using a finite set of spatial modes. Chapter 4 investigated the properties of these modes. Chapter 5 demonstrated that the dimensionality creates a limit for the ability to resolve the direction of arrival of a source.

Wireless communications systems achieve the transmission of information through the ex- citation and detection of electromagnetic fields. Whilst these fields exist continuously over space and time, they are constrained in complexity by virtue of the wave equation described in Chapter 1. A means of characterising the complexity or diversity of possible spatial fields over a finite volume is to consider the dimensionality or degrees of freedom. This was the subject of Chapter 2. Where there is only a single degree of freedom, it will only be possible to achieve one channel or independent path of communication between the sender and re- ceiver. Additional degrees of freedom or dimensionality allow additional independent paths which can be used to achieve a higher power or spectral efficiency in a communications system.

The degrees of freedom of a spatial field is directly related to the size and shape of the region of interest. This has a bearing on the accuracy and amount of information that can

be obtained from the spatial field. In this chapter we are concerned with effect of the region size on an appropriate channel model. The additional impact of the region shape could be incorporated using some of the results from Chapter 4. This chapter presents a framework for modelling the spatial channel incorporating the effective spatial dimensionality.

The development and use of spatial channel models is an important area of research for multiple antenna (MIMO) communication systems. In practice, the capacity that can be achieved will be limited by the extent to which the spatial environment supports parallel independent data paths. Models for the spatial propagation channel are therefore important for the design, development and testing of system designs. A good channel model will be simple and provide a channel simulation that is consistent with measured data. The channel model must capture the important characteristics of the physical channel.

There are two categories of stochastic channel models. Geometric or double directional models [70] describe the statistics of physical multipath component parameters (directions of arrival and departure, delay and amplitude). Analytic models approximate the complete statistics of the antenna transfer parameters [63] and provide a simple means for generating random channel matrices representative of a measured environment.

The geometric channel models require a large number of parameters to describe the general characteristics and distributions of the paths. Parameters include the number of discrete paths, the distributions of path direction, the angular spread of each path and correlations between paths. As we have shown, it is only possible to resolve the directions of a fixed number of paths given the receive region size, therefore this type of geometric model tends to have redundancy in the parameters.

Analytic models provide a simple alternative. However, both the spatial aspects of the chan- nel and the characteristics of the antenna arrays are captured in the model. In this way, any simulation is restricted to the specific arrays used for the measurement.

In this chapter we address the following question:-

Is there an alternative approach to creating a model of the spatial channel without reference to specific directional paths ?

This work is an extended version of a paper that was presented at the Vehicular Technology Conference, May 2006 [125].

6.1 Introduction

6.1.2

Review of MIMO Channel Models

Consider a MIMO system withNT transmit elements andNRreceive elements. The trans- mitted signalssand received signalsyare related by

y=Hs+w (6.1)

whereHis theNR×NT matrix of complex channel coefficients andw is the noise vector at the receiver.

The statistical models considered [30, 66, 67, 270] assume the channel to be well modelled by second order statistics. This is generally true of non line of sight MIMO channels such as those expected in indoor environments. In this case the elements ofHare zero mean [26, 63]. The correlation matrix for the channel coefficients,R_H is obtained,

R_H =En−→H−→HHo, (6.2)

where_·H is the Hermitian operation, and−→_· is the vector operation which stacks the columns of a matrix. ExpectationsE_{·} are taken over all channel matrix realisations. The matrix R_His anNTNR×NTNRcomplex positive definite Hermitian matrix with(NTNR)2degrees of freedom. It is possible to approximateR_H with fewer parameters. Specific examples are the Kronecker model withN2

T +NR2 parameters [30], the virtual channel model withNTNR parameters [66] and the recent Weichselberger model withN2

T +NR2 +NTNR−NT −NR parameters [270].

A review of these models [63] demonstrated that the Weichselberger model provided the best match to measured data. It also has the largest parameter space. Whilst the virtual channel separates the propagation channel from the array geometry, it was shown to overestimate channel diversity and capacity.

In this chapter, we present a MIMO channel model with the following properties: • A simple analytic framework for generating channels.

• Ability to match measured channel data. • The minimum number of internal parameters. • Separability of antenna array and spatial channel.

The proposed model quantifies the relationship between the size of the array and number of internal modelling parameters. Further, modelling accuracy can be adjusted with a single

parameter. It applies to two-dimensional environments, with straight forward extension to three dimensions.

Section 6.2 presents a new model framework to satisfy question posed above. This is fol- lowed by a discussion in Section 6.3 highlighting the advantages of the proposed model. Section 6.4 demonstrates the properties of the proposed model through simulation and ap- plication to real MIMO data sets.

Whilst many MIMO channel models assume separability of the receiver and transmitter cor- relations, this approach has come under scrutiny [68, 69]. Recent work by Lamahewa et al. provides a parametric extension to the Kronecker style model to introduce joint correlations between the angle of departure from the transmitter and the angle of arrival [271–273]. The main contribution of this chapter is the development of a stochastic model that captures the joint distribution of the receiver and transmitter correlations from experimental data.