## Convex Research Topics of Daniel P. Palomar

## Financial Engineering and Econometrics

Signal processing and financial engineering are seemingly different areas that share strong connections underneath. Both areas rely on the statistical analysis and modeling of systems and signals, either from the financial markets or from communication channels. In both cases, accurate characterization is essential to predict the behavior of practical algorithms and optimize their performance. The exploration of these connections reveals ways to capitalize on existing mathematical tools and methodologies developed and widely applied in the context of signal processing applications. As a matter of fact, the techniques underlying optimal strategies for reliable communications over wireless links prove to be very useful in approaching open issues and recurring problems in quantitative finance.

- Yiyong Feng and Daniel P. Palomar,
*A Signal Processing Perspective on Financial Engineering*, Foundations and Trends® in Signal Processing, Now Publishers, vol. 9, no. 1-2, 2016. [pdf] - Konstantinos Benidis, Ying Sun, Prabhu Babu, and Daniel P. Palomar, “Orthogonal Sparse PCA and Covariance Estimation via Procrustes Reformulation,”
*IEEE Trans. on Signal Processing*, vol. 64, no. 23, pp. 6211-6226, Dec. 2016. [R code] [Matlab code] - Ying Sun, Prabhu Babu, and Daniel P. Palomar, “Robust Estimation of Structured Covariance Matrix for Heavy-Tailed Elliptical Distributions,”
*IEEE Trans. on Signal Processing*, vol. 64, no. 14, pp. 3576-3590, July 2016. - Yiyong Feng and Daniel P. Palomar, “SCRIP: Successive Convex Optimization Methods for Risk Parity Portfolio Design,”
*IEEE Trans. on Signal Processing*, vol. 63, no. 19, pp. 5285-5300, Oct. 2015. - Yiyong Feng, Daniel P. Palomar, and Francisco Rubio, “Robust Optimization of Order Execution,”
*IEEE Trans. on Signal Processing*, vol. 63, no. 4, pp. 907-920, Feb. 2015. - Ying Sun, Prabhu Babu, and Daniel P. Palomar, “Regularized Robust Estimation of Mean and Covariance Matrix Under Heavy-Tailed Distributions,”
*IEEE Trans. on Signal Processing*, vol. 63, no. 12, pp. 3096-3109, June 2015. [Matlab code] - Ying Sun, Prabhu Babu, and Daniel P. Palomar, “Regularized Tyler’s Scatter Estimator: Existence, Uniqueness, and Algorithms,”
*IEEE Trans. on Signal Processing*, vol. 62, no. 19, pp. 5143-5156, Oct. 2014. - Mengyi Zhang, Francisco Rubio, Daniel P. Palomar, and Xavier Mestre, “Finite-Sample Linear Filter Optimization in Wireless Communications and Financial Systems,”
*IEEE Trans. on Signal Processing*, vol. 61, no. 20, pp. 5014-5025, Oct. 2013. - Yang Yang, Francisco Rubio, Gesualdo Scutari, and Daniel P. Palomar, “Multi-Portfolio Optimization: A Potential Game Approach,”
*IEEE Trans. on Signal Processing*, vol. 61, no. 22, pp. 5590-5602, Nov. 2013. - Mengyi Zhang, Francisco Rubio, and Daniel P. Palomar, “Improved Calibration of High-Dimensional Precision Matrices,”
*IEEE Trans. on Signal Processing*, vol. 61, no. 6, pp. 1509-1519, March 2013. - Francisco Rubio, Xavier Mestre, and Daniel P. Palomar, “Performance Analysis and Optimal Selection of Large Minimum-Variance Portfolios under Estimation Risk,”
*IEEE Journal on Selected Topics in Signal Processing: Special Issue on Signal Processing Methods in Finance and Electronic Trading*, vol. 6, no. 4, pp. 337-350, Aug. 2012.

## Optimization and Signal Processing in Big Data

Traditional convex optimization methods and signal processing techniques can be employed in big data problems characterized by large amounts of data in high-dimensional spaces. However, big data brings additional difficulties that need to be taken care of such as high-computational cost that require simple and efficient methods, distributed implementations, faulty data in the form of outliers, etc.

- Ying Sun, Prabhu Babu, and Daniel P. Palomar, “Majorization-Minimization Algorithms in Signal Processing, Communications, and Machine Learning,”
*IEEE Trans. on Signal Processing*, vol. 65, no. 3, pp. 794-816, Feb. 2017. - Konstantinos Benidis, Ying Sun, Prabhu Babu, and Daniel P. Palomar, “Orthogonal Sparse PCA and Covariance Estimation via Procrustes Reformulation,”
*IEEE Trans. on Signal Processing*, vol. 64, no. 23, pp. 6211-6226, Dec. 2016. [R code] [Matlab code]

- Licheng Zhao, Prabhu Babu, and Daniel P. Palomar, “Efficient Algorithms on Robust Low-Rank Matrix Completion Against Outliers,”
*IEEE Trans. on Signal Processing*, vol. 64, no. 18, pp. 4767- 4780, Sept. 2016. - Yang Yang, Marius Pesavento, Mengyi Zhang, and Daniel P. Palomar, “An Online Parallel Algorithm for Recursive Estimation of Sparse Signals,”
*IEEE Trans. on Signal and Inform. Proc. Over Networks*, vol. 2, no. 3, pp. 290-305, Sept. 2016. - Yang Yang, Gesualdo Scutari, Daniel P. Palomar, and Marius Pesavento, “A Parallel Decomposition Method for Nonconvex Stochastic Multi-Agent Optimization Problems,”
*IEEE Trans. on Signal Processing*, vol. 64, no. 11, pp. 2949-2964, June 2016. - Yiyong Feng and Daniel P. Palomar, “Normalization of Linear Support Vector Machines,”
*IEEE Trans. on Signal Processing*, vol. 63, no. 17, pp. 4673-4688, Sept. 2015. - Ying Sun, Prabhu Babu, and Daniel P. Palomar, “Regularized Robust Estimation of Mean and Covariance Matrix Under Heavy-Tailed Distributions,”
*IEEE Trans. on Signal Processing*, vol. 63, no. 12, pp. 3096-3109, June 2015. - Junxiao Song, Prabhu Babu, and Daniel P. Palomar, “Sparse Generalized Eigenvalue Problem via Smooth Optimization,”
*IEEE Trans. on Signal Processing*, vol. 63, no. 7, pp. 1627-1642, April 2015. [Matlab code] - Ying Sun, Prabhu Babu, and Daniel P. Palomar, “Regularized Tyler’s Scatter Estimator: Existence, Uniqueness, and Algorithms,”
*IEEE Trans. on Signal Processing*, vol. 62, no. 19, pp. 5143-5156, Oct. 2014. - Gesualdo Scutari, Francisco Facchinei, Peiran Song, Daniel P. Palomar, and Jong-Shi Pang, “Decomposition by Partial Linearization: Parallel Optimization of Multi-Agent Systems,”
*IEEE Trans. on Signal Processing*, vol. 62, no. 3, pp. 641-656, Feb. 2014.

- Xiaopeng Fan, Junxiao Song, Daniel P. Palomar, and Oscar C. Au, “Universal Binary Semidefinite Relaxation for ML Signal Detection,”
*IEEE Trans. on Communications*, vol. 61, no. 11, pp. 4565-4576, Nov. 2013. - Daniel P. Palomar and Yonina C. Eldar, Eds.,
*Convex Optimization in Signal Processing and Communications*, Cambridge University Press, 2009.

## Sequence Design and Reconstruction

Carefully designed sequences lie at the heart of virtually any digital system commonly used in our daily lives. Examples include GPS synchronization, multiuser CDMA systems, radar ambiguity function shaping, cryptography for secure transactions, and even the watermarking of digital images or videos. Each particular application requires a careful and delicate design of sequences or codes that enjoy some desirable properties such as low autocorrelation sidelobes for synchronization sequences, small cross-correlations in multiuser systems, and a proper response to Doppler effect in radar codes.

- Tianyu Qiu and Daniel P. Palomar, “Undersampled Sparse Phase Retrieval via Majorization-Minimization,” accepted in
*IEEE Trans. on Signal Processing*, Aug. 2017. - Zhongju Wang, Prabhu Babu, and Daniel P. Palomar, “Effective Low-Complexity Optimization Methods for Joint Phase Noise and Channel Estimation in OFDM,” accepted in
*IEEE Trans. on Signal Processing*, submitted 2016. - Licheng Zhao and Daniel P. Palomar, “Maximin Joint Optimization of Transmitting Code and Receiving Filter in Radar and Communications,”
*IEEE Trans. on Signal Processing*, vol. 65, no. 4, pp. 850-863, Feb. 2017. - Linlong Wu, Prabu Babu, and Daniel P. Palomar, “Cognitive Radar-Based Sequence Design via SINR Maximization,”
*IEEE Trans. on Signal Processing*, vol. 65, no. 3, pp. 779-793, Feb. 2017. - Licheng Zhao, Junxiao Song, Prabu Babu, and Daniel P. Palomar, “A Unified Framework for Low Autocorrelation Sequence Design via Majorization-Minimization,”
*IEEE Trans. on Signal Processing*, vol. 65, no. 2, pp. 438-453, Jan. 2017. - Zhongju Wang, Prabu Babu, and Daniel P. Palomar, “Design of PAR-Constrained Sequences for MIMO Channel Estimation via Majorization-Minimization,”
*IEEE Trans. on Signal Processing*, vol. 64, no. 23, pp. 6132-6144, Dec. 2016. - Tianyu Qiu, Prabhu Babu, and Daniel P. Palomar, “PRIME: Phase Retrieval via Majorization-Minimization,”
*IEEE Trans. on Signal Processing*, vol. 64, no. 19, pp. 5174-5186, Oct. 2016. - Junxiao Song, Prabhu Babu, and Daniel P. Palomar, “Sequence Set Design With Good Correlation Properties Via Majorization-Minimization,”
*IEEE Trans. on Signal Processing*, vol. 64, no. 11, pp. 2866-2879, June 2016. - Junxiao Song, Prabhu Babu, and Daniel P. Palomar, “Sequence Design to Minimize the Weighted Integrated and Peak Sidelobe Levels,”
*IEEE Trans. on Signal Processing*, vol. 64, no. 8, pp. 2051-2064, April 2016. - Junxiao Song, Prabhu Babu, and Daniel P. Palomar, “Optimization Methods for Designing Sequences With Low Autocorrelation Sidelobes,”
*IEEE Trans. on Signal Processing*, vol. 63, no. 15, pp. 3998-4009, Aug. 2015.

## Variational Inequality (VI) Methods for Multiuser Communication Systems and Smart Grids

The Variation Inequality (VI) problem constitutes a very general class of problems in nonlinear analysis. The VI framework embraces many different types of problems such as systems of equations, optimization problems, equilibrium programming, complementary problems, saddle-point problems, Nash equilibrium problems, and generalized Nash equilibrium problems. It specially bears strong connections with game theory. There is a well-developed theory for the analysis of solutions of VIs, as well as a wide variety of efficient algorithms with convergence properties. Therefore, it constitutes an excellent tool for analyzing the previous problems and, in particular, game problems where the classical game-theoretic tools may fall short like in advanced cognitive radio systems.

- Gesualdo Scutari, Francisco Facchinei, Jong-Shi Pang, and Daniel P. Palomar, “Real and Comple Monotone Communication Games,”
*IEEE Trans. on Information Theory*, vol. 60, no. 7, pp. 4197-4231, July 2014. - Italo Atzeni, Luis G. Ordóñez, Gesualdo Scutari, Daniel P. Palomar, and Javier R. Fonollosa, “Noncooperative Day-Ahead Bidding Strategies for Demand-Side Expected Cost Minimization with Real-Time Adjustments: A GNEP Approach,”
*IEEE Trans. on Signal Processing*, vol. 62, no. 9, pp. 2397-2412, May 2014. - Italo Atzeni, Luis G. Ordóñez, Gesualdo Scutari, Daniel P. Palomar, and Javier R. Fonollosa, “Demand-Side Management via Distributed Energy Generation and Storage Optimization,”
*IEEE Trans. on Smart Grids*, vol. 4, no. 2, pp. 866-876, June 2013. - Italo Atzeni, Luis G. Ordóñez, Gesualdo Scutari, Daniel P. Palomar, and Javier R. Fonollosa, “Noncooperative and Cooperative Optimization of Distributed Energy Generation and Storage in the Demand-Side of the Smart Grid,”
*IEEE Trans. on Signal Processing*, vol. 61, no. 10, pp. 2454-2472, May 2013. - Gesualdo Scutari, Daniel P. Palomar, Francisco Facchinei, and Jong-Shi Pang, “Monotone Games for Cognitive Radio Systems,” in
*Distributed Decision-Making and Control*, Ch. 4, Eds. Anders Rantzer and Rolf Johansson, Lecture Notes in Control and Information Sciences Series, Springer Verlag, 2011. [book] - Jong-Shi Pang, Gesualdo Scutari, Daniel P. Palomar, and Francisco Facchinei, “Design of Cognitive Radio Systems Under Temperature-Interference Constraints: A Variational Inequality Approach,”
*IEEE Trans. on Signal Processing*, vol. 58, no. 6, pp. 3251-3271, June 2010. - Gesualdo Scutari, Daniel P. Palomar, Francisco Facchinei, and Jong-Shi Pang, “Convex Optimization, Game Theory, and Variational Inequality Theory,”
*IEEE Signal Processing Magazine*, vol. 27, no. 3, pp. 35-49, May 2010. - Gesualdo Scutari, Daniel P. Palomar, Jong-Shi Pang, and Francisco Facchinei, “Flexible Design for Cognitive Wireless Systems: From Game Theory to Variational Inequality Theory,”
*IEEE Signal Processing Magazine*, vol. 26, no. 5, pp. 107-123, Sept. 2009.

## Rank-Constrained Semidefinite Programming for Beamforming and MIMO Radar

Semidefinite programming (SDP) is a class of convex optimization problems with a rich theory that can be efficiently solved in polynomial time. Many problems in wireless communications and radar systems can be formulated as SDPs with additional rank constraints. Unfortunately, rank-constrained SDPs are nonconvex and are hard to solve in general (some of them are in fact NP-hard, but not all of them). One important example is beamforming design in the downlink of a cellular network with multi-antenna base stations transmitting to single-antenna users. Such a problem can be formulated as a rank-constrained SDP. We have developed a framework to quantify exactly what low-rank solutions can be achieved, as well as algorithms to obtain such solutions. Another example is multiple-input multiple-output (MIMO) radar, which resembles the problem of MIMO communication systems. We have addressed the problem of design and optimization of the transmitted waveform for optimal performance.

- Maria Gregori, Miquel Payaró, and Daniel P. Palomar, “Sum-Rate Maximization for Energy Harvesting Nodes With a Generalized Power Consumption Model,”
*IEEE Trans. on Wireless Comm.*, vol. 15, no. 8, pp. 5341-5354, Aug. 2016. - Ying Sun, Arnaud Breloy, Prabhu Babu, Daniel P. Palomar, Frédéric Pascal, and Guillaume Ginolhac, “Low-Complexity Algorithms for Low Rank Clutter Parameter Estimation in Radar Systems,”
*IEEE Trans. on Signal Processing*, vol. 64, no. 8, pp. 1986-1998, April 2016. - Yongwei Huang and Daniel P. Palomar, “Randomized Algorithms for Optimal Solutions of Double-Sided QCQP with Applications in Signal Processing,”
*IEEE Trans. on Signal Processing*, vol. 62, no. 5, pp. 1093-1108, March 2014. - Benjamín Béjar, Santiago Zazo, and Daniel P. Palomar, “Energy Efficient Collaborative Beamforming in Wireless Sensor Networks,”
*IEEE Trans. on Signal Processing*, vol. 62, no. 2, pp. 496-510, Jan. 2014. - Yongwei Huang and Daniel P. Palomar, “A Dual Perspective on Separable Semideﬁnite Programming with Applications to Optimal Downlink Beamforming,”
*IEEE Trans. on Signal Processing*, vol. 58, no. 8, pp. 4254-4271, Aug. 2010. - Yongwei Huang and Daniel P. Palomar, “Rank-Constrained Separable Semidefinite Programming With Applications to Optimal Beamforming,”
*IEEE Trans. on Signal Processing*, vol. 58, no. 2, pp. 664-678, Feb. 2010. - Antonio De Maio, Yongwei Huang, Daniel P. Palomar, Shuzhong Zhang, and Alfonso Farina, “Fractional QCQP with Applications in ML Steering Direction Estimation for Radar Detection,”
*IEEE Trans. on Signal Processing*, vol. 59, no. 1, pp. 172-185, Jan. 2011. - Antonio De Maio, Silvio De Nicola, Yongwei Huang, Daniel P. Palomar, Shuzhong Zhang, and Alfonso Farina, “Code Design for Radar STAP via Optimization Theory,”
*IEEE Trans. on Signal Processing*, vol. 58, no. 2, pp. 679-694, Feb. 2010.

## Cognitive Radio Systems via Game Theory

Radio regulatory bodies are recently recognizing that rigid spectrum assignment granting exclusive use to licensed services is highly inefficient. A more efficient way to utilize the scarce spectrum resources is with a dynamic spectrum access, depending on the real spectrum usage and traffic demands. The concept of cognitive radio has recently received great attention from the research community as a promising paradigm to achieve efficient use of the frequency resource by allowing the coexistence of licensed (primary) and unlicensed (secondary) users in the same bandwidth.

We consider underlay/interweave multi-antenna networks, where primary users establish proper null and/or soft shaping constraints on the transmit covariance matrix of secondary users, so that the interference generated by secondary users is conﬁned within the interference-temperature limits. The secondary users compete then for the resource allocation, which can be formally modeled with game theory to obtain a completely decentralized approach.

We consider underlay/interweave multi-antenna networks, where primary users establish proper null and/or soft shaping constraints on the transmit covariance matrix of secondary users, so that the interference generated by secondary users is conﬁned within the interference-temperature limits. The secondary users compete then for the resource allocation, which can be formally modeled with game theory to obtain a completely decentralized approach.

- Yang Yang, Gesualdo Scutari, Peiran Song, and Daniel P. Palomar, “Robust MIMO Cognitive Radio Systems Under Interference Temperature Constraints,”
*IEEE Journal on Selected Areas in Communications*, vol. 31, no. 11, pp. 2465-2482, Nov. 2013. - Jiaheng Wang, Gesualdo Scutari, and Daniel P. Palomar, “Robust MIMO Cognitive Radio via Game Theory,”
*IEEE Trans. on Signal Processing*, vol. 59, no. 3, pp. 1183-1201, March 2011. - Gesualdo Scutari and Daniel P. Palomar, “MIMO Cognitive Radio: A Game Theoretical Approach,”
*IEEE Trans. on Signal Processing*, vol. 58, no. 2, pp. 761-780, Feb. 2010. - Gesualdo Scutari, Daniel P. Palomar, and Sergio Barbarossa, “Competitive Optimization of Cognitive Radio MIMO Systems via Game Theory,” in
*Convex Optimization in Signal Processing and Communications*, Cambridge Univ. Press, 2009. [book] - Gesualdo Scutari, Daniel P. Palomar, and Sergio Barbarossa, “Cognitive MIMO Radio: Competitive Optimality Design Based on Subspace Projections,”
*IEEE Signal Processing Magazine*, vol. 25, no. 6, pp. 46-59, Nov. 2008.

## Robust Designs

The design of communication systems depends strongly on the degree of knowledge of the channel state information (CSI). The best spectral efﬁciency and/or performance is obviously achieved when perfect CSI is available at both sides of the link. However, in practical communication systems, imperfect CSI arises from a variety of sources such as channel estimation errors, quantization of the channel estimate in the feedback channel, and outdated channel estimates with respect to the current channel (for time-varying channels). When the CSI is imperfect, it is necessary to model such imperfections or uncertainties and develop robust designs that take them into account.

There are two main philosophies for the design of systems robust to uncertainties: the worst-case approach, which considers that the uncertainty is within a given set around the nominal estimated value, and the Bayesian approach, which models the uncertainty statistically. The worst-case design guarantees a certain system performance for any possible channel sufﬁciently close to the estimated one, whereas the Bayesian design guarantees a certain system performance averaged over the channel realizations. We consider both perspectives in the design of robust MIMO communication systems.

There are two main philosophies for the design of systems robust to uncertainties: the worst-case approach, which considers that the uncertainty is within a given set around the nominal estimated value, and the Bayesian approach, which models the uncertainty statistically. The worst-case design guarantees a certain system performance for any possible channel sufﬁciently close to the estimated one, whereas the Bayesian design guarantees a certain system performance averaged over the channel realizations. We consider both perspectives in the design of robust MIMO communication systems.

- Yang Yang, Gesualdo Scutari, Peiran Song, and Daniel P. Palomar, “Robust MIMO Cognitive Radio Systems Under Interference Temperature Constraints,”
*IEEE Journal on Selected Areas in Communications*, vol. 31, no. 11, pp. 2465-2482, Nov. 2013.

- Jiaheng Wang, Mats Bengtsson, Björn Ottersten, and Daniel P. Palomar, “Robust MIMO Precoding for Several Classes of Channel Uncertainty,”
*IEEE Trans. on Signal Processing*, vol. 61, no. 12, pp. 3056-3070, June 2013. - Yongwei Huang, Daniel P. Palomar, and Shuzhong Zhang, “Lorentz-Positive Maps and Quadratic Matrix Inequalities with Applications to Robust MISO Transmit Beamforming,”
*IEEE Trans. on Signal Processing*, vol. 61, no. 5, pp. 1121-1130, March 2013. - Jiaheng Wang, Gesualdo Scutari, and Daniel P. Palomar, “Robust MIMO Cognitive Radio via Game Theory,”
*IEEE Trans. on Signal Processing*, vol. 59, no. 3, pp. 1183-1201, March 2011. - Jiaheng Wang and Daniel P. Palomar, “Robust MMSE Precoding in MIMO Channels with Pre-Fixed Receivers,”
*IEEE Trans. on Signal Processing*, vol. 58, no. 11, pp. 5802-5818, Nov. 2010.

- Jiaheng Wang and Daniel P. Palomar, “Worst-Case Robust MIMO Transmission with Imperfect Channel Knowledge,”
*IEEE Trans. on Signal Processing*, vol. 57, no. 8, pp. 3086-3100, Aug. 2009. - Xi Zhang, Daniel P. Palomar, and Björn Ottersten, “Statistically Robust Design of Linear MIMO Transceivers,”
*IEEE Trans. on Signal Processing*, vol. 56, no. 8, pp. 3678-3689, Aug. 2008. - A. Pascual-Iserte, Daniel P. Palomar, Ana I. Pérez-Neira, and Miguel A. Lagunas, “A Robust Maximin Approach for MIMO Communications with Partial Channel State Information Based on Convex Optimization,”
*IEEE Trans. on Signal Processing*, vol. 54, no. 1, pp. 346-360, Jan. 2006. - Daniel P. Palomar, John M. Cioffi, and Miguel Angel Lagunas, “Uniform Power Allocation in MIMO Channels: A Game-Theoretic Approach,”
*IEEE Trans. on Information Theory*, vol. 49, no. 7, pp. 1707-1727, July 2003.

## Game Theory for CompetitiveAd-Hoc Communications

Many communication systems of interest contain multiple uncoordinated users that share a common medium (e.g., wireless ad-hoc networks). These systems can be mathematically modeled as the so-called interference channel, for which the capacity region is still unknown. In these scenarios, centralized solutions are to be avoided and distributed algorithms play a central role. In fact, in many cases, the right model is to consider selfish users that compete with each other for the resources. Game theory is the right framework to study such competitive networks of users, leading to the concept of Nash equilibrium (NE) as solution of the game. This results in fully distributed algorithms with no signaling required among the users. Within the context of game theory, there are three main aspects to be studied: i) existence of NE, ii) uniqueness of NE, and iii) development of practical distributed algorithms with provable convergence to the NE.

This problem has been studied since 2001 by different researchers in the frequency-selective case. Among other things, we provide the state-of-the-art conditions for convergence of a class of iterative algorithms that contain as special cases the popular sequential iterative waterfilling algorithms as well as the novel asynchronous iterative waterfilling algorithm (more realistic in practice). We then extend the analysis to the MIMO case which provides a unified view of the problem. As a side result, we provide a novel interpretation of the (MIMO) waterfilling operator as a projection onto a proper convex set (this result is in fact fundamental to prove contraction mapping and convergence of algorithms).

This problem has been studied since 2001 by different researchers in the frequency-selective case. Among other things, we provide the state-of-the-art conditions for convergence of a class of iterative algorithms that contain as special cases the popular sequential iterative waterfilling algorithms as well as the novel asynchronous iterative waterfilling algorithm (more realistic in practice). We then extend the analysis to the MIMO case which provides a unified view of the problem. As a side result, we provide a novel interpretation of the (MIMO) waterfilling operator as a projection onto a proper convex set (this result is in fact fundamental to prove contraction mapping and convergence of algorithms).

- Gesualdo Scutari, Daniel P. Palomar, and Sergio Barbarossa, “The MIMO Iterative Waterfilling Algorithm,”
*IEEE Trans. on Signal Processing*, vol. 57, no. 5, pp. 1917-1935, May 2009. - Gesualdo Scutari, Daniel P. Palomar, and Sergio Barbarossa, “Competitive Design of Multiuser MIMO Systems based on Game Theory: A Unified View,”
*IEEE Journal on Selected Areas in Communications: Special Issue on Game Theory*, vol. 25, no. 7, pp. 1089-1103, Sept. 2008. - Gesualdo Scutari, Daniel P. Palomar, and Sergio Barbarossa, “Asynchronous Iterative Water-Filling for Gaussian Frequency-Selective Interference Channels,”
*IEEE Trans. on Information Theory*, vol. 54, no. 7, pp. 2868-2878, July 2008. - Gesualdo Scutari, Daniel P. Palomar, and Sergio Barbarossa, “Optimal Linear Precoding Strategies for Wideband Noncooperative Systems Based on Game Theory – Part I: Nash Equilibria,”
*IEEE Trans. on Signal Processing*, vol. 56, no. 3, pp. 1230-1249, March 2008. - Gesualdo Scutari, Daniel P. Palomar, and Sergio Barbarossa, “Optimal Linear Precoding Strategies for Wideband Noncooperative Systems Based on Game Theory – Part II: Algorithms,”
*IEEE Trans. on Signal Processing*, vol. 56, no. 3, pp. 1250-1267, March 2008.

## Quaternions

The use of complex numbers allows for a compact notation in many areas such as in baseband representation of communication systems. Quaternions constitute a further step: they are four-dimensional hypercomplex numbers. Quaternions have already found applications in image processing, wind modeling, processing of polarized waves, and design of space-time block codes.

- Javier Vía, Daniel P. Palomar, Luis Vielva, and Ignacio Santamaría, “Quaternion ICA from Second-Order Statistics,”
*IEEE Trans. on Signal Processing*, vol. 59, no. 4, pp. 1586-1600, April 2011. - Javier Vía, Daniel P. Palomar, and Luis Vielva, “Generalized Likelihood Ratios for Testing the Properness of Quaternion Gaussian Vectors,”
*IEEE Trans. on Signal Processing*, vol. 59, no. 4, pp. 1356-1370, April 2011.

## Information Theory and Estimation Theory

Information theory and estimation theory have generally been regarded as two separate theories with little overlap. Recently, however, it has been recognized that the relations between the two theories are fundamental (e.g., relating the mutual information with the minimum mean-square error) and can indeed be very useful to transfer results from one area to the other. In addition to the intrinsic theoretical interest of such relations, they have already found several applications such as the mercury/waterﬁlling optimal power allocation over a set of parallel Gaussian channels, a simple proof for the entropy power inequality, a simple proof of the monotonicity of the non-Gaussianness of independent random variables, and the study of extrinsic information of good codes.

We have further explored these connections in the vector Gaussian and arbitrary (non-Gaussian) settings. One interesting application of such a characterization is the efficient computation of the mutual information achieved by a given code over a channel via the symbolwise a posteriori probabilities (which previously could not be computed). We have also considered an alternative definition of information termed lautum information (different from mutual information).

We have further explored these connections in the vector Gaussian and arbitrary (non-Gaussian) settings. One interesting application of such a characterization is the efficient computation of the mutual information achieved by a given code over a channel via the symbolwise a posteriori probabilities (which previously could not be computed). We have also considered an alternative definition of information termed lautum information (different from mutual information).

- Ronit Bustin, Miquel Payaró, Daniel P. Palomar, and Shlomo Shamai, “On MMSE Crossing Properties and Implications in Parallel Vector Gaussian Channels,”
*IEEE Trans. on Information Theory*, vol. 59, no. 2, pp. 818-844, Feb. 2013. - Eduard Calvo, Daniel P. Palomar, Javier R. Fonollosa, and Josep Vidal, “On the Computation of the Capacity Region of the Discrete MAC,”
*IEEE Trans. on Communications*, vol. 58, no. 12, pp. 3512-3525, Dec. 2010. - Miquel Payaró and Daniel P. Palomar, “Hessian and Concavity of Mutual Information, Differential Entropy, and Entropy Power in Linear Vector Gaussian Channels,”
*IEEE Trans. on Information Theory*, vol. 55, no. 8, pp. 3613-3628, Aug. 2009. - Daniel P. Palomar and Sergio Verdú, “Lautum Information,”
*IEEE Trans. on Information Theory*, vol. 54, no. 3, pp. 964-975, March 2008. - Daniel P. Palomar and Sergio Verdú, “Representation of Mutual Information via Input Estimates,”
*IEEE Trans. on Information Theory*, vol. 53, no. 2, pp. 453-470, Feb. 2007. - Daniel P. Palomar and Sergio Verdú, “Gradient of Mutual Information in Linear Vector Gaussian Channels,”
*IEEE Trans. on Information Theory*, vol. 52, no. 1, pp. 141-154, Jan. 2006.

## Random Matrix Theory for Communication Systems

The performance of multiple-input multiple-output (MIMO) communication systems is related to the eigenstructure of the channel matrix H (channel eigenmodes) or, more exactly, to the non-zero eigenvalues of HH†. Therefore, the probabilistic characterization of these eigenvalues is necessary in order to derive analytical expressions for the average and outage performance measures of the system. In MIMO wireless communications, the channel matrix H is commonly modeled with Gaussian distributed entries. This results in HH† being a Wishart random matrix. The Wishart distribution and some closely related distributions have been widely studied during the sixties and seventies in the mathematical literature, due to its importance in various areas of research such as the analysis of time series or nuclear physics. More recently, the statistical properties of the eigenvalues of Wishart matrices have been investigated and effectively applied to analyze the information theoretical limits of MIMO channels as well as the performance of practical MIMO systems.

We consider a class of Hermitian random matrices that contains as particular cases the classical Wishart, the correlated central Wishart, the correlated central Pseudo-Wishart, and the noncentral Wishart. We first obtain expressions for the distribution of the ordered eigenvalues; in particular, for i) the joint cdf, ii) the marginal cdf’s, and iii) the marginal pdf’s. Then, for simpler tractability, we develop first-order Taylor expansions that translate into convenient SNR gain and diversity gain characterizations of the performance in communication systems.

We consider a class of Hermitian random matrices that contains as particular cases the classical Wishart, the correlated central Wishart, the correlated central Pseudo-Wishart, and the noncentral Wishart. We first obtain expressions for the distribution of the ordered eigenvalues; in particular, for i) the joint cdf, ii) the marginal cdf’s, and iii) the marginal pdf’s. Then, for simpler tractability, we develop first-order Taylor expansions that translate into convenient SNR gain and diversity gain characterizations of the performance in communication systems.

- Luis G. Ordóñez, Daniel P. Palomar, and Javier R. Fonollosa, “Array Gain in the DMT Framework for MIMO Channels,”
*IEEE Trans. on Information Theory*, vol. 58, no. 7, pp. 4577-4593, July 2012. - Luis G. Ordóñez, Daniel P. Palomar, Alba Pagès-Zamora, and Javier R. Fonollosa, “Minimum BER Linear MIMO Transceivers With Adaptive Number of Substreams,”
*IEEE Trans. on Signal Processing*, vol. 57, no. 6, pp. 2336-2353, June 2009. - Luis G. Ordóñez, Daniel P. Palomar, and Javier R. Fonollosa, “Ordered Eigenvalues of a General Class of Hermitian Random Matrices With Application to the Performance Analysis of MIMO Systems,”
*IEEE Trans. on Signal Processing*, vol. 57, no. 2, pp. 672-689, Feb. 2009. - Luis García-Ordoñez, Daniel P. Palomar, Alba Pagès-Zamora, and Javier R. Fonollosa, “High-SNR Analytical Performance of Spatial Multiplexing MIMO Systems with CSI,”
*IEEE Trans. on Signal Processing*, vol. 55, no. 11, pp. 5447-5463, Nov. 2007.

## Cross-Layer Network Optimization

During the last decade, it has been widely recognized that an independent optimization of the different OSI layers in a communication system is a limiting design factor. Instead, a cross-layer design is necessary. In this sense, network utility maximization (NUM) problem formulations provide an important approach to conduct network resource allocation and to view layering as optimization decomposition.

In the previous existing literature, distributed implementations were typically achieved by means of the so-called dual decomposition technique. However, the span of decomposition possibilities includes many other elements that had not been fully exploited, such as the use of the primal decomposition technique, the versatile introduction of auxiliary variables, and the potential of multilevel decompositions. We have developed a systematic framework to exploit alternative decomposition structures as a way to obtain different distributed algorithms, each with a different tradeoff among convergence speed, message passing amount and asymmetry, and distributed computation architecture. Some illustrative applications include resource-constrained and direct-control rate allocation, and rate allocation among QoS classes with multipath routing.

In the previous existing literature, distributed implementations were typically achieved by means of the so-called dual decomposition technique. However, the span of decomposition possibilities includes many other elements that had not been fully exploited, such as the use of the primal decomposition technique, the versatile introduction of auxiliary variables, and the potential of multilevel decompositions. We have developed a systematic framework to exploit alternative decomposition structures as a way to obtain different distributed algorithms, each with a different tradeoff among convergence speed, message passing amount and asymmetry, and distributed computation architecture. Some illustrative applications include resource-constrained and direct-control rate allocation, and rate allocation among QoS classes with multipath routing.

- Chee Wei Tan, Daniel P. Palomar, and Mung Chiang, “Energy-Robustness Tradeoff in Cellular Network Power Control,”
*IEEE/ACM Trans. on Networking*, vol. 17, no. 3, pp. 912-925, June 2009. - Daniel P. Palomar and Mung Chiang, “Alternative Distributed Algorithms for Network Utility Maximization: Framework and Applications,”
*IEEE Trans. on Automatic Control*, vol. 52, no. 12, pp. 2254-2269, Dec. 2007. - Mung Chiang, Chee Wei Tan, Daniel P. Palomar, Daniel O’Neill, and David Julian, “Power Control by Geometric Programming,”
*IEEE Trans. on Wireless Communications*, vol. 6, no. 7, pp. 2640-2651, July 2007.

- Daniel P. Palomar and Mung Chiang, “A Tutorial on Decomposition Methods for Network Utility Maximization,”
*IEEE Journal on Selected Areas in Communications: Special Issue on Nonlinear Optimization of Communication Systems*, vol. 24, no. 8, pp. 1439-1451, Aug. 2006.

- Mung Chiang, Chee Wei Tan, Daniel P. Palomar, Daniel O’Neill, and David Julian, “Power Control by Geometric Programming,” in
*Resource Allocation in Next Generation Wireless Networks*, vol. 5, Chapter 13, pp. 289-313, W. Li, Y. Pan, Editors, Nova Sciences Publishers, ISBN 1-59554-583-9, 2005. [book]

## MIMO Communication Systems via Convex Optimisation and Majorization Theory

Multiple-input multiple-output (MIMO) channels provide an abstract and uniﬁed representation of different physical communication systems, ranging from multi-antenna wireless channels to wireless digital subscriber line systems. They have the key property that several data streams can be simultaneously established. In general, the design of communication systems for MIMO channels is quite involved. The ﬁrst diffculty lies on how to measure the global performance of such systems given the tradeoﬀ on the performance among the diﬀerent data streams. Once the problem formulation is deﬁned, the resulting mathematical problem is typically too complicated to be optimally solved as it is a matrix-valued nonconvex optimization problem.

This design problem has been studied for the past three decades (the ﬁrst papers dating back to the 1970s) motivated initially by cable systems and more recently by wireless multi-antenna systems. The approach was to choose a speciﬁc global measure of performance and then to design the system accordingly, either optimally or suboptimally, depending on the difficulty of the problem. We develop a novel uniﬁed mathematical framework for the design of point-to-point MIMO transceivers with channel state information at both sides of the link according to an arbitrary cost function as a measure of the system performance. Majorization theory is the underlying mathematical theory on which the framework hinges. It allows the transformation of the originally complicated matrix-valued nonconvex problem into a simple scalar problem. In particular, the additive majorization relation plays a key role in the design of linear MIMO transceivers (i.e., a linear precoder at the transmitter and a linear equalizer at the receiver), whereas the multiplicative majorization relation is the basis for nonlinear decision- feedback MIMO transceivers (i.e., a linear precoder at the transmitter and a decision-feedback equalizer at the receiver).

This design problem has been studied for the past three decades (the ﬁrst papers dating back to the 1970s) motivated initially by cable systems and more recently by wireless multi-antenna systems. The approach was to choose a speciﬁc global measure of performance and then to design the system accordingly, either optimally or suboptimally, depending on the difficulty of the problem. We develop a novel uniﬁed mathematical framework for the design of point-to-point MIMO transceivers with channel state information at both sides of the link according to an arbitrary cost function as a measure of the system performance. Majorization theory is the underlying mathematical theory on which the framework hinges. It allows the transformation of the originally complicated matrix-valued nonconvex problem into a simple scalar problem. In particular, the additive majorization relation plays a key role in the design of linear MIMO transceivers (i.e., a linear precoder at the transmitter and a linear equalizer at the receiver), whereas the multiplicative majorization relation is the basis for nonlinear decision- feedback MIMO transceivers (i.e., a linear precoder at the transmitter and a decision-feedback equalizer at the receiver).

- Javier Rubio, Antonio Pascual-Iserte, Daniel P. Palomar, and Andrea Goldsmith, “Joint Optimization of Power and Data Transfer in Multiuser MIMO Systems,”
*IEEE Trans. on Signal Processing*, vol. 65, no. 1, pp. 212-227, Jan. 2017. - Daniel P. Palomar and Yi Jiang, MIMO Transceiver Design via Majorization Theory, Foundations and Trends® in Communications and Information Theory, Now Publishers, vol. 3, no. 4-5, 2007. [pdf] [typos]
- Jiaheng Wang and Daniel P. Palomar, “Majorization Theory with Applications in Signal Processing and Communication Systems,” in
*Mathematical Foundations for Signal Processing, Communications and Networking*, Ch. 16, Eds. Thomas Chen, Dinesh Rajan, and Erchin Serpedin, CRC Press, 2011. [book] - Antonio A. D’Amico, Luca Sanguinetti, and Daniel P. Palomar, “Convex Separable Problems with Linear Constraints in Signal Processing and Communications,”
*IEEE Trans. on Signal Processing*, vol. 62, no. 22, pp. 6045-6058, Nov. 2014. - Svante Bergman, Daniel P. Palomar, and Björn Ottersten, “Joint Bit Allocation and Precoding for MIMO Systems with Decision Feedback Detection,”
*IEEE Trans. on Signal Processing*, vol. 57, no. 11, pp. 4509-4521, Nov. 2009. - Daniel P. Palomar, A. Pascual-Iserte, John M. Cioffi, and Miguel A. Lagunas, “Convex Optimization Theory Applied to Joint Transmitter-Receiver Design in MIMO Channels,” in
*Space-Time Processing for MIMO Communications*, Chapter 8, pp. 269-318, A. B. Gershman and N. Sidiropoulos, Editors, John Wiley & Sons, ISBN 0-470-01002-9, April 2005. [book] - Daniel P. Palomar, “Unified Design of Linear Transceivers for MIMO Channels,” in
*Smart Antennas – State-of-the-Art*, vol. 3, Chapter 18, EURASIP Hindawi Book Series on SP&C, T. Kaiser, A. Bourdoux, H. Boche, J. R. Fonollosa, J. B. Andersen, and W. Utschick, Editors, ISBN 977-5945-09-7, 2005. [book]

- Daniel P. Palomar, “Convex Primal Decomposition for Multicarrier Linear MIMO Transceivers,”
*IEEE Trans. on Signal Processing*, vol. 53, no. 12, pp. 4661-4674, Dec. 2005. - Daniel P. Palomar and Sergio Barbarossa, “Designing MIMO Communication Systems: Constellation Choice and Linear Transceiver Design,”
*IEEE Trans. on Signal Processing*, vol. 53, no. 10, pp. 3804-3818, Oct. 2005. - Daniel P. Palomar, Mats Bengtsson, and Björn Ottersten, “Minimum BER Linear Transceivers for MIMO Channels via Primal Decomposition,”
*IEEE Trans. on Signal Processing*, vol. 53, no. 8, pp. 2866-2882, Aug. 2005. - Daniel P. Palomar and Javier Rodriguez Fonollosa, “Practical Algorithms for a Family of Waterfilling Solutions,”
*IEEE Trans. on Signal Processing*, vol. 53, no. 2, pp. 686-695, Feb. 2005. - Daniel P. Palomar, “Unified Framework for Linear MIMO Transceivers with Shaping Constraints,”
*IEEE Communications Letters*, vol. 8, no. 12, pp. 697-699, Dec. 2004. - Daniel P. Palomar, Miguel Angel Lagunas, and John M. Cioffi, “Optimum Linear Joint Transmit-Receive Processing for MIMO Channels with QoS Constraints,”
*IEEE Trans. on Signal Processing*, vol. 52, no. 5, pp. 1179-1197, May 2004. - Daniel P. Palomar, John M. Cioffi, and Miguel Angel Lagunas, “Joint Tx-Rx Beamforming Design for Multicarrier MIMO Channels: A Unified Framework for Convex Optimization,”
*IEEE Trans. on Signal Processing*, vol. 51, no. 9, pp. 2381-2401, Sept. 2003.

- Daniel P. Palomar and Miguel Angel Lagunas, “Joint Transmit-Receive Space-Time Equalization in Spatially Correlated MIMO channels: A Beamforming Approach,”
*IEEE Journal on Selected Areas in Communications: Special Issue on MIMO Systems and Applications*, vol. 21, no. 5, pp. 730-743, June 2003.

## Blind Beamforming in CDMA Systems

Beamforming in wireless communication systems with multiple receive antennas has been studied for the last three decades. Traditionally, the design of the beamformer is based on either the knowledge of the spatial signature of the signal of interest or the availability of a training sequence. It is, however, possible to use blind techniques to design the beamformer without any spatial reference or training sequence. This requires knowledge of some structural property of the desired signal (e.g., the constant modulus blind techniques).

We consider Spread Spectrum signals, which contain a rich time/frequency structure, for blind beamforming. One example is CDMA systems, where the temporal structure is given by the codes. It is possible to have coexisting users using different codes and still implement blind beamforming for each of the users.

We consider Spread Spectrum signals, which contain a rich time/frequency structure, for blind beamforming. One example is CDMA systems, where the temporal structure is given by the codes. It is possible to have coexisting users using different codes and still implement blind beamforming for each of the users.

- Daniel P. Palomar and Miguel Angel Lagunas, “Temporal diversity on DS-CDMA communication systems for blind array signal processing,”
*EURASIP Signal Processing*, vol. 81, no. 8, pp. 1625-1640, Aug. 2001.

- Daniel P. Palomar, Montse Nájar, and Miguel Angel Lagunas, “Self-reference Spatial Diversity Processing for Spread Spectrum Communications,”
*AEÜ International Journal of Electronics and Communications*, vol. 54, no. 5, pp. 267-276, Nov. 2000.