Full length articleReceive antenna selection in diversely polarized MIMO transmissions with convex optimization
Introduction
Multiple-Input Multiple-Output (MIMO) systems have received increased attention because they significantly improve wireless link performance through capacity and diversity gains [1]. A major limiting factor in the deployment of MIMO systems is the cost of multiple analog chains (such as low noise amplifiers, mixers and analog-to-digital converters) at the receiver end. Antenna selection at the transmitter/receiver is a powerful technique that reduces the number of analog chains required, yet preserving the diversity benefits obtained from the full MIMO system. With antenna selection, a limited number of transmit/receive chains are dynamically multiplexed between several transmit/receive antennas. MIMO antenna selection techniques have thus been extensively studied, and there are several antenna selection criteria. For full-diversity space–time codes, a subset of available antennas can be selected to maximize the channel norm [2]. For spatial-multiplexing systems, antennas can be selected to minimize the error ratios [3]. A useful tutorial paper on antenna selection can be found in [4]. Various selection algorithms applied to MIMO OFDM systems can be found in [5]. An exhaustive search based on maximum output SNR is proposed in [6], when the system uses linear receivers. Since an exhaustive search is computationally expensive for large MIMO systems, several sub-optimal algorithms with lower complexity are derived at the expense of performance. A selection algorithm based on accurate approximation of the conditional error probability of quasi-static MIMO systems is derived in [7]. In [8], the authors formulate the receive antenna selection problem as a combinatorial optimization problem and relax it to a convex optimization problem. They employ an interior point algorithm, i.e., a barrier method, to solve a relaxed convex problem. However, they treat only the case of capacity maximization. An alternative approach to receive antenna selection for capacity maximization that offers near optimal performance at a complexity significantly lower than the schemes in [9] but marginally greater than the schemes in [10] is described in [11]. In [12], [13] a new approach to antenna selection is proposed, based on the minimization of the union bound, which is the sum of the all Pairwise Error Probabilities (PEPs). Our approach is based on formulating the selection problem as a combinatorial optimization problem and relaxing it to obtain a problem with a concave objective function and convex constraints. We follow the lines of [8], [11], and extend it to systems with both spatial and angular correlation, so-called True Polarization Diversity (TPD) [14], [15], [16] arrays. We optimize the performance of systems with such arrays of antennas which are both spatially separated and also inclined at a certain angle. A model for combined spatial and angular correlation functions is also given in [17], but we adhere to the work from Valenzuela [14], [15], [16]. We apply a simple norm based antenna selection method to a polarization diverse array. Application of receive antenna selection on a polarized array can be found in [18], [19]. We proceed to apply optimization now to the system of arrays with Dual-Polarized (DP) and Triple-Polarized (TP) antenna structures. Application of receive antenna selection on a polarized array can be found in [18], [19]. We first model the DP and TP systems with respect to many channel characteristics, e.g., -factor, channel correlations and cross-polarization discrimination (XPD). A good investigation on the modeling of DP MIMO channels is in [20]. In [21] the author models TP systems and presents the performance in terms of outage probabilities. We then compare the results with the Spatially Separated–Single-Polarized (SS–SP) systems with the same channel characteristics. We extend our DP and TP systems to Spatially-Separated Dual-Polarized (SS–DP) and Triple-Polarized (SS–TP) systems. These systems are a combination of both spatial and polarization domains. The remainder of this paper is organized as follows: Section 2 details the generic model and the structure of polarization diverse arrays. We also describe the correlation models for both spatial and angular arrays in this section. We proceed forward to describe the channel model for three dimensional antenna structures in 3. In Section 4, the performance metric to optimize with receive antenna selection is outlined with details. A description of the performance as an optimization problem is presented in 5 for 2D and 3D structures. Important simulation results for polarization diverse systems and triple-polarized systems and comparison with the performance of Uniform Linear Arrays (ULA) are discussed in Section 6. We conclude our work in Section 7.
Section snippets
Channel model for 2D antenna structures
We consider a MIMO system with transmit and receive antennas. The channel is assumed to have frequency-flat Rayleigh fading with additive white Gaussian noise (AWGN) at the receiver. The received signal can thus be represented as where vector represents the th sample of the signals collected at the receive antennas, sampled at symbol rate. The vector is the th sample of the signal transmitted from the
Channel model for 3D antenna structures
The channel is modeled as a Ricean fading channel, i.e., the channel matrix can be composed of a fixed (possibly line-of-sight) part and a random (fast fading) part according to where is the Ricean -factor, is a deterministic matrix and is a random matrix. The random matrix consists of complex Gaussian entries which are independent from one channel realization to the next. In other words, if then models a pure Rayleigh fading channel and if then it models a
Receive antenna selection in MIMO
The earliest works on antenna selection have been in the context of Single-Input Multiple-Output (SIMO) systems. For example, selection diversity, where the receiver only selects the strongest antenna signal has long been used in SIMO systems [28]. Receive antenna selection in MIMO systems offer more degrees of freedom than in SIMO systems. We focus here on receive antenna selection for capacity maximization. The capacity of the MIMO system described in Section 2 is given by the well known
Optimization algorithm for antenna selection
We formulate the problem of receive antenna selection as a constrained convex optimization problem [29] that can be solved efficiently using numerical methods such as interior-point algorithms [30]. Similar to [11], the is defined such that, By definition, if , and 0 otherwise. Now, consider an diagonal matrix that has as its diagonal entries. Thus, the MIMO channel capacity with antenna selection can be re-written
Results
In this section, we evaluate the performance of the proposed antenna selection algorithm via Monte Carlo simulations [31]. We solve the optimization algorithm using the MATLAB based tool for convex optimization called CVX [31]. We use ergodic capacity as a metric for performance evaluation, which is obtained by averaging over results obtained from 1000 independent realizations of the channel matrix . For each realization, the entries of the channel matrix are uncorrelated ZMCSCG random
Conclusions
In this work we investigated a polarization diverse antenna array with compact 2D aperture, to optimize the performance in terms of ergodic capacity. We used the relaxation of a binary integer constraint to have a convex optimization algorithm and solved it using disciplined convex programming. The optimization algorithm not only finds the best antennas for selection but also finds the optimum orientation angles for antenna elements within an array. We also compared the results with an array
Acknowledgments
This work has been funded by the Christian Doppler Laboratory for Wireless Technologies for Sustainable Mobility, KATHREIN-Werke KG, and A1 Telekom Austria AG. The financial support by the Federal Ministry of Economy, Family and Youth and the National Foundation for Research, Technology and Development is gratefully acknowledged. The authors would like to thank Christoph F. Mecklenbräuker for his valuable comments and fruitful discussions. This work has also been funded by the Higher Education
Aamir Habib was born in Rawalpindi, Pakistan. He did his Masters in Mobile & Satellite Communications from the University of Surrey, UK and another from the Center for Advanced Studies in Engineering, Islamabad in Computer Engineering. He has been working at the Institute of Space Technology, Islamabad since June 2000.
Currently he is working in MIMO Communications for his Doctoral thesis sponsored by the Higher Education Commission, Pakistan in collaboration with the Austrian Exchange service
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Aamir Habib was born in Rawalpindi, Pakistan. He did his Masters in Mobile & Satellite Communications from the University of Surrey, UK and another from the Center for Advanced Studies in Engineering, Islamabad in Computer Engineering. He has been working at the Institute of Space Technology, Islamabad since June 2000.
Currently he is working in MIMO Communications for his Doctoral thesis sponsored by the Higher Education Commission, Pakistan in collaboration with the Austrian Exchange service (OeAD) Austria at the Institute of Telecommunications, Vienna University of Technology.