Performance Limits of Spatially Distributed MIMO Systems

In document Information Theoretic Limits for Wireless Information Transfer Between Finite Spatial Regions (Page 56-58)

2.5 Introducing Spatial Constraints into MIMO Systems

2.5.3 Performance Limits of Spatially Distributed MIMO Systems

Systems

The physical limits of the number of degrees of freedom and of the information capacity of spatially distributed narrowband MIMO systems have been considered by a number of authors [40, 41, 45, 47, 48, 191–202]. These limits are universal, in the sense that hold in any propagation conditions and do not depend on specific assumptions on the channel gains, correlations structure, etc. The main result of all of these works is that the number of degrees of freedom and information capacity are physically limited by the size of the spatial region.

Moreover, the performance limits between two spatial volumes containing the antenna arrays was studied by a large number of research works. Earlier works [22,203,204], presented an exact, complete method for defining, evaluating and op- timizing the best-connected orthogonal communication channels, modes or degrees of freedom for scalar waves between two volumes of arbitrary shape and position. In another approach, [23] addressed the capacity limits of compact MIMO arrays, constrained in space but not in the number of antenna elements and compared with traditional unconstrained arrays. In addition, [125, 205] presented a model for MIMO outdoor wireless fading channels and predicted excellent performance

2.5 Introducing Spatial Constraints into MIMO Systems 33

for reasonable value of scattering radius of the transmitter and the receiver, al- most regardless of how large the antenna spacing is. Furthermore, [62] numer- ically studied the electromagnetic degrees of freedom of a noise limited system in two dimensions with random multiple scattering and showed that the average degrees of freedom is strongly dependent on the sizes of the transmit volume, the receive volume and the scattering region, whereas, [28, 60] proposed numerical techniques for estimating channel capacity for MIMO systems where the system is modeled using a transmit volume, a receive volume and a set of reflective scatter- ers. In [25, 26, 149, 206–209] an analytical model was developed for generating the channel gains between arbitrary arrays of transmitter and receiver antennas for general scattering environments and showed that there exists a maximum achiev- able capacity for communication between spatial regions of space, which depends on the size of the regions and the statistics of the scattering environment. More- over, [29, 58, 210] studied the information capacity of Gaussian channels between two volumes under a constraint in the average radiated power.

On top of that the works of [39–44] stated the capacity scaling of narrowband spatially distributed MIMO systems in line of sight propagation environment. Ac- cording to [39], capacity can be significantly improved when users form distributed MIMO arrays and for no given limitation on transmit power, the capacity can scale almost linearly with number of antennas. In another approach, the capacity result in [40] was derived under the assumption that the phases of the channel gains are functions of the locations of the antennas. Further, the capacity scalings pro- posed in [41, 42] suffer from spatial limitations when the clusters are separated by long distances, whereas, [43, 44] studied the performance of distributed MIMO transmissions between far away clusters of antennas.

In addition, the fundamental limits of spatially constrained broadband MIMO systems were studied by [49, 50, 195, 211–214]. The definition of total degrees of freedom of broadband multiple-antenna channels provided in [211,214] agreed with the “rule of thumb” of multiplying the degrees of freedom available in space and frequency. Whereas, [49, 50, 195] showed that for spherically restricted broadband wireless transmissions the parameters space, time and frequency are linked together and the degrees of freedom, as started in [49,50], is proportional to the surface area of the spatial region, the square of the center frequency and bandwidth and the

degrees of freedom of the broadband signal itself. Further, [212,213] indicated that due to the mutual coupling of space and time spectra, it is not possible to exploit time and space diversity gains simultaneously and the ultimate physical capacity limit along both space and time dimensions corresponds either to the maximum achievable information rate of an arbitrarily large MIMO narrowband system, or to the one of a single-input single-output (SISO) ultra-wideband (UWB) system.

2.6

A Simple Framework to Analyze Space-Time

Signals

The classical wave equation is one of the fundamental equations in signal process- ing. It governs how space-time signals radiated from a source distribution generate a wavefield. Thus, to study the fundamental performance limits of spatio-temporal wireless systems we require a solid knowledge in the physics of wave propagation. In this section, we study the solution to the wave equation in detail and exploit it to develop modal analysis techniques which provide insight into the structure of the space-time signals. In order to do so, we start with reviewing the adiabatic equation, which describes the wavefield at any arbitrary point within a homoge-

nous medium, often referred to as the Helmholtz wave equation. We then derive

solutions to the Helmholtz equation following [215, 216], which uses orthogonal basis functions to parameterize a source free wavefield. In the subsequent chap- ters, we use these solutions explicitly to develop modal analysis techniques to solve the general performance limit problems of spatio-temporal wireless communication systems.

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