Performance Analysis of Space-Time Block Codes over Generalized- Fading MIMO Channels Michail Matthaiou ∗ , Nestor D. Chatzidiamantis † , Himal A. Suraweera ‡ , and George K. Karagiannidis † ∗ Department of Signals and Systems, Chalmers University of Technology, Gothenburg, Sweden, email: michail.matthaiou@chalmers.se † Department of Electrical & Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece, emails: {nestoras, geokarag}@auth.gr ‡ Engineering Product Development, Singapore University of Technology and Design, Singapore, email: himalsuraweera@sutd.edu.sg Abstract—This paper elaborates on the performance of orthog- onal space-time block codes (STBCs) for multiple-input multiple- output (MIMO) systems operating in generalized-K fading condi- tions. The considered fading model is generic since it intimately encompasses both small-scale fading (modeled via the Nakagami- m distribution) and large-scale fading (modeled via the gamma distribution). In the following, new exact analytical expressions are derived for the Shannon capacity along with asymptotic expressions in the high and low Signal-to-Noise ratio (SNR) regimes. We also present exact tractable expressions followed by first-order expansions for the marginal outage probability and symbol error rate (SER); further, we quantify the performance of STBCs in terms of diversity order and coding gain. The derived analytical expressions are validated via a set of Monte-Carlo simulations which also demonstrate the implications of the model parameters on the overall performance. I. I NTRODUCTION Orthogonal STBCs represent a simple and reliable transmis- sion scheme which can afford the same diversity order as the classical maximal-ratio combining. Since their original estab- lishment by Alamouti in [1] and the generalization by Tarokh et al. in [2], they have been extensively used in the design and performance analysis of MIMO communication systems. The main advantage of STBCs is that maximum-likelihood (ML) decoding can be performed with linear processing at the receiver, while the MIMO channel can be transformed into an equivalent scalar Gaussian channel with a response equal to the Frobenius norm of the channel matrix [3], [4]. Conceptually, the majority of related studies documented in the literature adopts the common assumption of Rayleigh or Ricean fading statistics (see for instance [5], [6] and references therein among others). Some theoretical investigations and measurement campaigns [7], [8] though, have demonstrated that the Nakagami- distribution [9] yields a better fit with real-time data for various measured channels and, more im- portantly, encompasses both Rayleigh/Ricean distributions as special cases. This reveals that assessing the performance of STBC over MIMO Nakagami- channels is a highly interesting topic. Apart from small-scale fading, each MIMO link is most likely to experience path-loss and shadowing (large-scale fading) with the latter manifestation being rather critical when assessing MIMO performance since it can significantly dimin- ish the benefits of MIMO technology. For this reason, it is of paramount importance to consider the effects of shadowing into the analysis of STBC MIMO systems. As such, we note the work in [10] which modeled the shadowing via the log- normal distribution and provided an integral representation of the outage capacity. In order to circumvent this, the author approximated the outage capacity numerically via Gauss- Hermite polynomials with this technique, however, being time- consuming (especially at low SNRs) and not amenable to further manipulations. Over the past years, lognormal shad- owing has been successfully approximated by the analytically friendlier Gamma shadowing with the resulting fading model being usually referred to as the generalized -distribution [11], [12], which will serve as our reference model henceforth. In this paper, we provide a systematic statistical characteri- zation of STBCs over generalized- fading MIMO channels. The main paper contributions can be summarized as follows: ∙ After presenting exact analytical expressions for the prob- ability density function (PDF) and cumulative density function (CDF) of the instantaneous STBC SNR, we derive a new, analytical formula for the exact Shannon capacity. In addition, we consider the asymptotically high and low-SNR regimes for which insightful expressions are also presented; in the latter case, the notions of minimum energy per bit to reliably convey any positive rate and wideband slope are introduced. ∙ In the second part of our analysis, we focus on the SER and outage probability measures for which exact ana- lytical expressions along with first-order expansions are deduced. Moreover, we introduce a common parametriza- tion in the area of wireless communications to quantify the STBC performance in terms of diversity order and coding (or array) gain. Notation: We use upper and lower case boldface to denote matrices and vectors. The expectation is given by ℰ [∙], while (∙) † represents the Hermitian transpose and tr(∙), ∥∙∥ yield the trace and Frobenius norm of a matrix, respectively. The real part is expressed as Re(∙). II. MIMO STBC SYSTEM MODEL AND STATISTICS OF THE I NSTANTANEOUS SNR We consider a typical point-to-point MIMO system with   receive antennas and   transmit antennas. For the case of composite fading (both small and large-scale fading), we can model the   ×   channel matrix as Z = HΞ 1/2 . The entries of the small-scale channel matrix H ∈ ℂ  × are 2011 IEEE Swedish Communication Technologies Workshop (Swe-CTW) 978-1-4577-1878-6/11/$26.00 ©2011 IEEE 68