# On the Outage Capacity of Orthogonal Space-time Block Codes Over Multi-cluster Scattering MIMO Channels

###### Abstract

Multiple cluster scattering MIMO channel is a useful model for pico-cellular MIMO networks. In this paper, orthogonal space-time block coded transmission over such a channel is considered, where the effective channel equals the product of complex Gaussian matrices. A simple and accurate closed-form approximation to the channel outage capacity has been derived in this setting. The result is valid for an arbitrary number of clusters of scatterers and an arbitrary antenna configuration. Numerical results are provided to study the relative outage performance between the multi-cluster and the Rayleigh-fading MIMO channels for which .

## I Introduction

Deployment of pico-cellular networks in dense teletraffic areas such as train stations, office buildings and airports is becoming increasingly popular due to their abilities to extend coverage areas and increase network capacity. In general, it is difficult to model the pico-cellular channel as it involves a wide range of physical mechanisms. Among these mechanisms, however, a distinctive feature is that the transmitted signal propagates through a sequence of clusters (layers) of scatterers until it reaches the destination. This multi-layered scattering channel is typical in modeling indoor propagation between floors in a building [2007Saunders, Chap. ]. For transceivers equipped with multiple antennas, the effective end-to-end channel becomes a product of the multiple input multiple output (MIMO) channel matrices of each layer. In literature, this multiple cluster scattering MIMO channel was considered in [2002Muller, 2002aMuller], and physical motivation for this channel model can be found in [2002Andersen, Sec. ].

Despite the needs to understand the fundamental limits, such as the channel capacity, of the multiple scattering MIMO channels, results in this direction are quite limited. Closed-form expressions of the ergodic capacity have been derived respectively in [2013AKW] and [2013AIK] for equal and unequal number of scatterers in each cluster. The ergodic capacity scaling law has been established in [2013WZTH]. However, for practical transmission schemes such as Orthogonal Space-time Block Codes (OSTBCs), the corresponding information-theoretic quantities have not been addressed in literature. OSTBCs are particularly attractive open-loop transmit diversity schemes that decouple the MIMO channel into scalar channels. Thus, decoding is reduced from a vector detection problem to a scalar one, which significantly decreases the decoding complexity [1999Tarokh, 2003Larsson]. Moreover, OSTBCs require little computational cost for encoding and achieve full spatial diversity gain [2003Larsson]. The use of OSTBCs facilitates the implementation of outer code, i.e. each of the equivalent scalar channels, cf. (3), can be encoded independently with a powerful outer code such as Turbo code.

We consider OSTBCs coded transmissions over the multiple cluster scattering MIMO channels, and study the corresponding outage capacity. Outage capacity is a relevant performance measure when the transmission of each codeword spans only one or finitely many fading realizations. This is the scenario of a pico-cellular network, where the mobile terminals are moving at walking speed, so that the channel gain, albeit random, varies so slowly that it can be assumed as constant along a coding block [2001Biglieri]. For such a delay-limited system, the average capacity over the ensemble of channel realizations, i.e. the ergodic capacity, can not characterize the achievable transmission rates [2001Biglieri]. In this paper, we propose a simple closed-form approximation to the outage capacity of the multiple cluster MIMO channels based on the derived exact moment expressions. The proposed approximation is valid for arbitrary but finite transceiver sizes and scatterers per cluster. The result is obtained by making use of finite-dimensional singular values distribution for products of complex Gaussian matrices as well as the moment based approximation. Interestingly, the proposed approximation becomes exact as the channel degenerates to a conventional Rayleigh fading channel. Simulations are conducted to show the usefulness of the proposed approximation as well as to compare the outage performance with the conventional MIMO channels. Based on the analytical and numerical results, we gain physical insight into the behavior of the outage capacity of the considered channel model.

The rest of the paper is organized as follows. In Section II we outline the system model studied in this paper, which includes the channel model, the signal model as well as the outage capacity formulation. Section III is devoted to the analysis of the outage capacity of the considered system model. Simulations are presented in Section LABEL:sec:simu to examine the outage performance in various realistic scenarios. In Section LABEL:sec:con we conclude the main findings of this paper. Proofs of all the technical results are provided in the Appendices.

## Ii System Model

### Ii-a Channel Model

Consider a single user MIMO system with transmit and receive antennas. Information transmitted to the receiver goes through successive scattering clusters, each having () scatterers, as shown in Fig. 1. The channels between non-consecutive clusters as well as the direct link between the transmitter and the receiver are ignored. As a result, the effective channel between the transmitter and the receiver equals the product of channel matrices

(1) |

where the dimensions of the -th channel are . Each channel is assumed to be an i.i.d. Rayleigh fading MIMO channel, i.e. the entries of follow the standard complex Gaussian distribution and are independent of each other. The assumption of the i.i.d. Rayleigh channel requires the so-called richly scattered physical environment, where there exist a large number of statistically independent reflected paths with random amplitudes [2005Tse, Chap. ]. Thus, there needs to exist rich scattering environments creating and . Between these two environments, all scattering happens through the scatterers in cluster , which can be thought as keyholes. Examples of such channel model include the channel between floors in a building, where inside each floor there is an i.i.d. scattering environment, but between the floors there is restricted propagation through the scatterers [2007Saunders]. As the number of scatterers of all clusters goes to infinity with the antenna size kept fixed, it is expected that the channel (1) reduces to a conventional Rayleigh fading channel. This intuitively clear fact will be proven in Section LABEL:sec:rela. Note that the model (1) also describes the multi-hop amplify-and-forward MIMO relay channels when assuming noiseless relays [2011Fawaz]. Obviously, for the channel (1) becomes the conventional MIMO channel. We notice that the channel (1) is also referred to as the ‘Rayleigh product MIMO channel’ in literature [2008Jin].

### Ii-B Signal Model

We consider linear space-time coded transmissions over the multiple cluster scattering MIMO channels (1). We assume quasi-static flat fading channels. Namely, the channel remains constant for at least the transmission of an entire frame (say symbols), and may vary from frame to frame. The resulting signal model within one frame reads

(2) |

where the matrix denotes the received signals. The entries of the noise matrix are i.i.d. and follow the standard complex Gaussian distribution. In line with the convention [2002Muller, 2013AKW, 2013AIK], the effective channel is normalized by so that the total energy of the normalized channel^{1}^{1}1 denotes the matrix trace operation, and denotes the conjugate-transpose. , cf. (LABEL:eq:m1), will not grow with . In (2), the matrix denotes the linear OSTBC mappings of transmitted symbols in such a way that is proportional to an identity matrix. Since the encoding matrix spans symbol times to encode symbols, the code rate equals , which is also referred to as the delay-optimality of the code. It is shown in [1999Tarokh] that full rate OSTBCs exist for any number of transmit antennas using any real constellation such as PAM. For any complex constellation such as PSK/QAM, half rate OSTBCs exist for any number of transmit antennas, while full rate OSTBC only exists for two transmit antennas, a.k.a. the Alamouti scheme. For specific cases of two, three, and four transmit antennas, rate , , OSTBCs for complex constellations are given in [1999Tarokh]. Without loss of generality, we assume complex constellations in the following discussions.

Due to the orthogonality property of OSTBCs, i.e. , the MIMO channel is decoupled into independent scalar complex AWGN channels after decoding. The resulting equivalent SISO signal model reads [2003Larsson, Th. 7.3]

(3) |

where denotes the transmitted symbol and the noise follows a complex Gaussian distribution with mean zero and variance . We denote by the total transmit power per symbol time, which equals the transmit SNR. Here, denotes the Frobenius norm, and we define . With the above notations, the effective SNR of the equivalent signal model (3) at the output of the STBC decoder equals

(4) |

### Ii-C Outage Capacity

Since the OSTBCs decouple the MIMO channel into independent SISO channels, the problem reduces to the study of the corresponding scalar channels. In particular, the capacity of the multi-cluster scattering MIMO channels (in nats/s/Hz) equals times the capacity of the SISO system (3), divided by the number of time instants used for the transmission:

(5) |

As we are interested in the delay-limited system, where each codeword sees one channel realization, the fundamental limit of such a system is best explained in the capacity versus outage formalism. Namely, for a given rate the outage probability, i.e. the Cumulative Distribution Function (CDF) of , is obtained as

(6) |

where denotes the CDF of . The resulting outage capacity for a given outage probability equals

(7) |

where denotes the inverse function of . The outage capacity can be understood as the capacity guaranteed for of the transmissions. The benefit of using the above performance metrics is manifested by the fact that the outage probability is directly related to the Packet Error Rate (PER) when codewords span only one fading block. Namely, assuming that the transmitted codeword (packet) is decoded successfully if the transmission rate is less than the capacity for the given channel realization and declaring a decoding error otherwise, then the outage probability equals the PER. The outage probability is achievable [2006Prasad] in the sense that for any , there exists a code of sufficiently large block length for which the PER is upper-bounded by . Thus, outage capacity provides useful insights on the performance of a delay-limited coded system.

## Iii Outage Capacity Analysis

### Iii-a Exact Moments of

It is seen from (6) that analyzing the outage capacity requires the distribution of the random variable . Since the maximal rank of the channel matrix is

(8) |

the Hermitian matrix has nonzero eigenvalues, which we denote by . It is shown in [2013AIK], cf. (LABEL:eq:ker) and the subsequent discussions, that the joint density of nonzero eigenvalues of is invariant under any permutation of the matrix dimensions . Thus, without loss of generality we set and denote

(9) |

We can now write the random variable of interest as

(10) |

where the support of is .

Although the exact distribution of seems difficult to obtain, simple yet accurate approximations can be constructed based on the moments of . In Propositions and Corollaries we present closed-form expressions of the integer moments of . Before showing these results, we need the following lemma.

###### Lemma 1.

The joint density of the ordered nonzero eigenvalues of , , reads {},{}{νn,…,ν2,ν1+j-1},{}