**Appendix 5. A2 Spatial Channel Capacity**

**6.5 Capacity Limits of Spatially Distributed MIMO Systems**

### Systems

Up to this point, we have developed a narrowband spatially distributed MIMO system between arbitrarily shaped spatial clusters. The proposed system takes a power constrained function as its input and has a function as its output. The output is then corrupted by additive white Gaussian noise. Now, we can proceed to use this formalism to evaluate the instantaneous capacity and outage capacity probability of the system. Before proceeding to the main results presented in this chapter, we first determine the number of independent channels available between the spatial clusters.

### 6.5.1

### Favourable Propagation Characteristics

Independent channels between transmit and receive cluster is one of the most important characteristics of the favourable propagation in spatially distributed MIMO systems subject to Rayleigh fading. Although there are infinitely many connections between the clusters, finite number of independent practical channels are available to carry information between transmit and receive clusters.

5_{Rayleigh fading model represents excellent approximation of practical scenarios as it de-}

6.5 Capacity Limits of Spatially Distributed MIMO Systems 143

In conventional MIMO systems the rank of the channel matrix determines the maximum number of independent parallel channels between the transmitter array and the receiver array. In the literature [258], the idealistic model assumes that the channel matrix has independent elements corresponding to sufficiently spaced antennas. However, when realistic antenna configurations and scattering environ- ments are considered, the elements of the channel matrix become correlated, and the number of possible independent parallel channels is reduced.

In corollary, in case of spatially distributed MIMO communications, the rank

of the scattering channel matrix H (6.22) would determine the maximum number

of independent parallel channels between the transmit and the receive clusters. Therefore, number of independent parallel channels

K =rank(H)≤min{N, M} (6.25)

with the richness factor

ks = rank(H)

min{N, M} (6.26)

where ks ∈(0,1] characterizes the scattering environment, such that

1. ks = 1 corresponds to a uncorrelated high rank channel model where the

elements of the channel matrix H are independent. Thus, characterizes a

rich scattering environment.

2. ks <1 corresponds to a correlated low rank channel model. Therefore, the

scattering environment is not rich enough to support the full set of indepen- dent modal gains.

### 6.5.2

### Capacity Limits

We are now in position to derive exact closed-form expressions for the instanta- neous capacity and outage probability of spatially distributed MIMO systems for mode-to-mode communications. We start with evaluating the instantaneous ca- pacity of spatially distributed MIMO systems for narrowband transmissions.

Instantaneous Capacity

The instantaneous capacity of the spatially distributed MIMO system can simply be characterized in terms of the mutual information between the transmit and receive clusters.

Hence, based on the characteristics of system model (6.21), we can define the
instantaneous capacity as [11]
Cins = I(s,y|H)
= log|IM +
1
σ2HQH
†_{|}
= log|IN +
1
σ2QH
†
H| (6.27)

where IM and σ2 are defined in (6.23) and Q is defined in (6.20). Further, IN is

theN×N identity matrix. Note that (6.27) follows from the determinant identity

|I+AB|=|I +BA|.

However, the singular value decomposition (SVD) theorem or eigen channel based evaluation of conventional MIMO instantaneous capacity is well established in literature [90,210,259–264]. Utilizing (6.27), the following theorem provides the SVD theorem or eigen channel based instantaneous capacity of spatially distributed MIMO systems.

Theorem 6.1 For a given constraint on the total transmit power PT, the instan-

taneous capacity of any spatially distributed MIMO system for mode-to-mode com- munication over Rayleigh fading channels is given by

Cins = K X k=1 log(1 + 1 σ2Pkλk). (6.28)

whereK = min{N, M}represents the number of independent channels withN and

M denoting the number of transmit and receive modes, σ2 _{is the identical additive}
white Gaussian noise (AWGN) power per receive mode, Pk is the power of the
kth _{channel such that} PK

k=1Pk ≤ PT and λk is the kth eigenvalue of the channel

6.5 Capacity Limits of Spatially Distributed MIMO Systems 145

Proof of the theorem is given in Appendix 1.

Next, we proceed to quantify the outage capacity probability of narrowband spatially distributed MIMO systems.

Outage Capacity Probability

By definition, the outage capacity probability, denoted by Pout is the probability

of not achieving a threshold capacity, outage capacity.

Thus, the outage probability for spectral efficiency R is defined as

Pout = Pr{I(s,y|H)< R} (6.29)

whereI(s,y|H) =Cins is defined in Theorem 1. Now, based on this definition we

provide the following theorem to determine the outage probability of narrowband spatially distributed MIMO systems.

Theorem 6.2 Considering narrowband signal transmissions between arbitrary shaped spatial clusters in terms of orthonormal basis functions encoded in wave modes to describe the outgoing/ incoming signals in transmit/ receive cluster, outage prob- ability of any spatially distributed MIMO system subject to Rayleigh fading is

Pout =
K
Y
k=1
1−e−2Rζk
, 2R _{≥}_{0} _{(6.30)}

where R is the desired spectral efficiency and

ζk =

1

1 +λkPk/σ2

(6.31)

is the rate parameter of Rayleigh distribution. In addition, K, σ2_{,} _{P}

k and λk are

defined in Theorem 1.

A comprehensive proof of the theorem is given in Appendix 2.

In the following section, we elucidate some physical insights of the derived theorems.

### 6.5.3

### Discussion

Observe that Theorem1 and Theorem2represent closed-form expressions for the

instantaneous capacity and the outage probability of narrowband spatially dis- tributed MIMO systems. These expressions indicate that the available mutual information in the spatially distributed MIMO system, as well as the distributed MIMO outage probability, is limited by the number of independent channels avail- able between the transmit and the receive clusters. Further, since the effective

number of independent channels K = min{N, M}, the maximum outcomes from

(6.28) and (6.30) are achieved when equal number of transmit and receive modes are excited.

In this context, for any arbitrary antenna configuration (number of antennas and their location), [24] developed a modal decomposition method that maps the transmitted signals to the transmit modes and the receive modes to received sig- nals. The modal decomposition method showed that the transmit and the receive modes are generated from the spatially constrained transmit and receive antenna arrays, respectively, and the number of antennas is restricted by the number of

effective modes. For instance, if nT and nR denote number of transmit and re-

ceive antennas, respectively, then nT = 2N+ 1,nR= 2M + 1 for circular clusters

and nT = (N + 1)

2

,nR= (M+ 1)

2

for spherical clusters. Furthermore, according

to [24], if we consider transmit and receive clusters of radiusrT andrR, respectively,

then for circular/ spherical clusters, N =deπrT/λe and M =deπrR/λe where λ

represents the wavelength corresponding to the operating frequency. Therefore, the effective number of modes excited in the transmit and the receive clusters is limited by the size of the clusters.

Based on the above discussion, it is evident that the narrowband spatially distributed MIMO systems provide best performances with each spatial cluster containing equal number of randomly distributed antennas. Populating clusters with different number of antennas simply adds redundancy to the system. How- ever, since the effective number of modes is dependent on size of the cluster, the best performance requires equal sized clusters.