Matrix-Monotonic Optimization for MIMO Systems

For MIMO systems, due to the deployment of multiple antennas at both the transmitter and the receiver, the design variables e.g., precoders, equalizers, training sequences, etc. are usually matrices. It is well known that matrix operations are usually more complicated compared to their vector counterparts. In order to overcome the high complexity resulting from matrix variables, in this paper we investigate a class of elegant multi-objective optimization problems, namely matrix-monotonic optimization problems (MMOPs). In our work, various representative MIMO optimization problems are unified into a framework of matrix-monotonic optimization, which includes linear transceiver design, nonlinear transceiver design, training sequence design, radar waveform optimization, the corresponding robust design and so on as its special cases. Then exploiting the framework of matrix-monotonic optimization the optimal structures of the considered matrix variables can be derived first. Based on the optimal structure, the matrix-variate optimization problems can be greatly simplified into the ones with only vector variables. In particular, the dimension of the new vector variable is equal to the minimum number of columns and rows of the original matrix variable. Finally, we also extend our work to some more general cases with multiple matrix variables.Comment: 37 Pages, 5 figures, IEEE Transactions on Signal Processing, Final Versio

Robust Tomlinson-Harashima precoding for non-regenerative multi-antenna relaying systems

Conference Theme: PHY and FundamentalsIn this paper, we consider the robust transceiver design with Tomlinson-Harashima precoding (THP) for multi-hop amplify-and-forward (AF) multiple-input multiple-output (MIMO) relaying systems. THP is adopted at the source to mitigate the spatial inter-symbol interference and then a joint Bayesian robust design of THP at source, linear forwarding matrices at relays and linear equalizer at destination is proposed. Based on the elegant characteristics of multiplicative convexity and matrix-monotone functions, the optimal structure of the nonlinear transceiver is first derived. Based on the derived structure, the optimization problem is greatly simplified and can be efficiently solved. Finally, the performance advantage of the proposed robust design is assessed by simulation results. © 2012 IEEE.published_or_final_versionThe 2012 IEEE Wireless Communications and Networking Conference (WCNC), Paris, France, 1-4 April 2012. In IEEE Wireless Communications and Networking Conference Proceedings, 2012, p. 753-75

Some stochastic inequalities for weighted sums

We compare weighted sums of i.i.d. positive random variables according to the usual stochastic order. The main inequalities are derived using majorization techniques under certain log-concavity assumptions. Specifically, let $Y_i$ be i.i.d. random variables on $\mathbf{R}_+$. Assuming that $\log Y_i$ has a log-concave density, we show that $\sum a_iY_i$ is stochastically smaller than $\sum b_iY_i$, if $(\log a_1,...,\log a_n)$ is majorized by $(\log b_1,...,\log b_n)$. On the other hand, assuming that $Y_i^p$ has a log-concave density for some $p>1$, we show that $\sum a_iY_i$ is stochastically larger than $\sum b_iY_i$, if $(a_1^q,...,a_n^q)$ is majorized by $(b_1^q,...,b_n^q)$, where $p^{-1}+q^{-1}=1$. These unify several stochastic ordering results for specific distributions. In particular, a conjecture of Hitczenko [Sankhy\={a} A 60 (1998) 171--175] on Weibull variables is proved. Potential applications in reliability and wireless communications are mentioned.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ302 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

On the superiority of improper Gaussian signaling in wireless interference MIMO scenarios

©2016 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.Recent results have elucidated the benefits of using improper Gaussian signaling (IGS) as compared to conventional proper Gaussian signaling (PGS) in terms of achievable rate for interference-limited conditions. This paper exploits majorization theory tools to formally quantify the gains of IGS along with widely linear transceivers for MIMO systems in interferencelimited scenarios. The MIMO point-to-point channel with interference (P2P-I) is analyzed, assuming that received interference can be either proper or improper, and we demonstrate that the use of the optimal IGS when received interference is improper strictly outperforms (in terms of achievable rate and mean square error) the use of the optimal PGS when interference is proper. Then, these results are extended to two practical situations. First, the MIMO Z-interference channel (Z-IC) is investigated, where a trade-off arises: with IGS we could increase the achievable rate of the interfered user while gracefully degrading the rate of the non-interfered user. Second, these concepts are applied to a two-tier heterogeneous cellular network (HCN) where macrocells and smallcells coexist and multiple MIMO Z-IC appear.Peer ReviewedPostprint (author's final draft

Maximum mutual information design for amplify-and-forward multi-hop MIMO relaying systems under channel uncertainties

Conference Theme: PHY and FundamentalsIn this paper, we investigate maximum mutual information design for multi-hop amplify-and-forward (AF) multiple-input multiple-out (MIMO) relaying systems with imperfect channel state information, i.e., Gaussian distributed channel estimation errors. The robust design is formulated as a matrix-variate optimization problem. Exploiting the elegant properties of Majorization theory and matrix-variate functions, the optimal structures of the forwarding matrices at the relays and precoding matrix at the source are derived. Based on the derived structures, a water-filling solution is proposed to solve the remaining unknown variables. © 2012 IEEE.published_or_final_versionThe 2012 IEEE Wireless Communications and Networking Conference (WCNC), Paris, France, 1-4 April 2012. In IEEE Wireless Communications and Networking Conference Proceedings, 2012, p. 781-78

The role of asymptotic functions in network optimization and feasibility studies

Solutions to network optimization problems have greatly benefited from developments in nonlinear analysis, and, in particular, from developments in convex optimization. A key concept that has made convex and nonconvex analysis an important tool in science and engineering is the notion of asymptotic function, which is often hidden in many influential studies on nonlinear analysis and related fields. Therefore, we can also expect that asymptotic functions are deeply connected to many results in the wireless domain, even though they are rarely mentioned in the wireless literature. In this study, we show connections of this type. By doing so, we explain many properties of centralized and distributed solutions to wireless resource allocation problems within a unified framework, and we also generalize and unify existing approaches to feasibility analysis of network designs. In particular, we show sufficient and necessary conditions for mappings widely used in wireless communication problems (more precisely, the class of standard interference mappings) to have a fixed point. Furthermore, we derive fundamental bounds on the utility and the energy efficiency that can be achieved by solving a large family of max-min utility optimization problems in wireless networks.Comment: GlobalSIP 2017 (to appear