Sum-Rate Maximization in Two-Way AF MIMO Relaying: Polynomial Time Solutions to a Class of DC Programming Problems

Sum-rate maximization in two-way amplify-and-forward (AF) multiple-input multiple-output (MIMO) relaying belongs to the class of difference-of-convex functions (DC) programming problems. DC programming problems occur as well in other signal processing applications and are typically solved using different modifications of the branch-and-bound method. This method, however, does not have any polynomial time complexity guarantees. In this paper, we show that a class of DC programming problems, to which the sum-rate maximization in two-way MIMO relaying belongs, can be solved very efficiently in polynomial time, and develop two algorithms. The objective function of the problem is represented as a product of quadratic ratios and parameterized so that its convex part (versus the concave part) contains only one (or two) optimization variables. One of the algorithms is called POlynomial-Time DC (POTDC) and is based on semi-definite programming (SDP) relaxation, linearization, and an iterative search over a single parameter. The other algorithm is called RAte-maximization via Generalized EigenvectorS (RAGES) and is based on the generalized eigenvectors method and an iterative search over two (or one, in its approximate version) optimization variables. We also derive an upper-bound for the optimal values of the corresponding optimization problem and show by simulations that this upper-bound can be achieved by both algorithms. The proposed methods for maximizing the sum-rate in the two-way AF MIMO relaying system are shown to be superior to other state-of-the-art algorithms.Comment: 35 pages, 10 figures, Submitted to the IEEE Trans. Signal Processing in Nov. 201

Power Allocation Based on SEP Minimization in Two-Hop Decode-and-Forward Relay Networks

The problem of optimal power allocation among the relays in a two-hop decode-and-forward cooperative relay network with independent Rayleigh fading channels is considered. It is assumed that only the relays that decode the source message correctly contribute in data transmission. Moreover, only the knowledge of statistical channel state information is available. A new simple closed-form expression for the average symbol error probability is derived. Based on this expression, a new power allocation method that minimizes the average symbol error probability and takes into account the constraints on the total average power of all the relay nodes and maximum instant power of each relay node is developed. The corresponding optimization problem is shown to be a convex problem that can be solved using interior point methods. However, an approximate closed-form solution is obtained and shown to be practically more appealing due to significant complexity reduction. The accuracy of the approximation is discussed. Moreover, the so obtained closed-form solution gives additional insights into the optimal power allocation problem. Simulation results confirm the improved performance of the proposed power allocation scheme as compared to other schemes.Comment: 27 pages, 5 figures, submitted to the IEEE Trans. Signal Processing in Feb. 201

Cooperative Precoding/Resource Allocation Games under Spectral Mask and Total Power Constraints

The use of orthogonal signaling schemes such as time-, frequency-, or code-division multiplexing (T-, F-, CDM) in multi-user systems allows for power-efficient simple receivers. It is shown in this paper that by using orthogonal signaling on frequency selective fading channels, the cooperative Nash bargaining (NB)-based precoding games for multi-user systems, which aim at maximizing the information rates of all users, are simplified to the corresponding cooperative resource allocation games. The latter provides additional practically desired simplifications to transmitter design and significantly reduces the overhead during user cooperation. The complexity of the corresponding precoding/resource allocation games, however, depends on the constraints imposed on the users. If only spectral mask constraints are present, the corresponding cooperative NB problem can be formulated as a convex optimization problem and solved efficiently in a distributed manner using dual decomposition based algorithm. However, the NB problem is non-convex if total power constraints are also imposed on the users. In this case, the complexity associate with finding the NB solution is unacceptably high. Therefore, the multi-user systems are categorized into bandwidth- and power-dominant based on a bottleneck resource, and different manners of cooperation are developed for each type of systems for the case of two-users. Such classification guarantees that the solution obtained in each case is Pareto-optimal and actually can be identical to the optimal solution, while the complexity is significantly reduced. Simulation results demonstrate the efficiency of the proposed cooperative precoding/resource allocation strategies and the reduced complexity of the proposed algorithms.Comment: 33 pages, 8 figures, Submitted to the IEEE Trans. Signal Processing in Oct. 200

On Impedance Bandwidth of Resonant Patch Antennas Implemented Using Structures with Engineered Dispersion

We consider resonant patch antennas, implemented using loaded transmission-line networks and other exotic structures having engineered dispersion. An analytical expression is derived for the ratio of radiation quality factors of such antennas and conventional patch antennas loaded with (reference) dielectrics. In the ideal case this ratio depends only on the propagation constant and wave impedance of the structure under test, and it can be conveniently used to study what kind of dispersion leads to improved impedance bandwidth. We illustrate the effect of dispersion by implementing a resonant patch antenna using a periodic network of LC elements. The analytical results predicting enhanced impedance bandwidth compared to the reference results are validated using a commercial circuit simulator. Discussion is conducted on the practical limitations for the use of the proposed expression.Comment: 4 pages, 7 figure

Fast directional spatially localized spherical harmonic transform

We propose a transform for signals defined on the sphere that reveals their localized directional content in the spatio-spectral domain when used in conjunction with an asymmetric window function. We call this transform the directional spatially localized spherical harmonic transform (directional SLSHT) which extends the SLSHT from the literature whose usefulness is limited to symmetric windows. We present an inversion relation to synthesize the original signal from its directional-SLSHT distribution for an arbitrary window function. As an example of an asymmetric window, the most concentrated band-limited eigenfunction in an elliptical region on the sphere is proposed for directional spatio-spectral analysis and its effectiveness is illustrated on the synthetic and Mars topographic data-sets. Finally, since such typical data-sets on the sphere are of considerable size and the directional SLSHT is intrinsically computationally demanding depending on the band-limits of the signal and window, a fast algorithm for the efficient computation of the transform is developed. The floating point precision numerical accuracy of the fast algorithm is demonstrated and a full numerical complexity analysis is presented.Comment: 12 pages, 5 figure

Cooperative Jamming for Secure Communications in MIMO Relay Networks

Secure communications can be impeded by eavesdroppers in conventional relay systems. This paper proposes cooperative jamming strategies for two-hop relay networks where the eavesdropper can wiretap the relay channels in both hops. In these approaches, the normally inactive nodes in the relay network can be used as cooperative jamming sources to confuse the eavesdropper. Linear precoding schemes are investigated for two scenarios where single or multiple data streams are transmitted via a decode-and-forward (DF) relay, under the assumption that global channel state information (CSI) is available. For the case of single data stream transmission, we derive closed-form jamming beamformers and the corresponding optimal power allocation. Generalized singular value decomposition (GSVD)-based secure relaying schemes are proposed for the transmission of multiple data streams. The optimal power allocation is found for the GSVD relaying scheme via geometric programming. Based on this result, a GSVD-based cooperative jamming scheme is proposed that shows significant improvement in terms of secrecy rate compared to the approach without jamming. Furthermore, the case involving an eavesdropper with unknown CSI is also investigated in this paper. Simulation results show that the secrecy rate is dramatically increased when inactive nodes in the relay network participate in cooperative jamming.Comment: 30 pages, 7 figures, to appear in IEEE Transactions on Signal Processin

Robust Beamforming for Security in MIMO Wiretap Channels with Imperfect CSI

In this paper, we investigate methods for reducing the likelihood that a message transmitted between two multiantenna nodes is intercepted by an undetected eavesdropper. In particular, we focus on the judicious transmission of artificial interference to mask the desired signal at the time it is broadcast. Unlike previous work that assumes some prior knowledge of the eavesdropper's channel and focuses on maximizing secrecy capacity, we consider the case where no information regarding the eavesdropper is available, and we use signal-to-interference-plus-noise-ratio (SINR) as our performance metric. Specifically, we focus on the problem of maximizing the amount of power available to broadcast a jamming signal intended to hide the desired signal from a potential eavesdropper, while maintaining a prespecified SINR at the desired receiver. The jamming signal is designed to be orthogonal to the information signal when it reaches the desired receiver, assuming both the receiver and the eavesdropper employ optimal beamformers and possess exact channel state information (CSI). In practice, the assumption of perfect CSI at the transmitter is often difficult to justify. Therefore, we also study the resulting performance degradation due to the presence of imperfect CSI, and we present robust beamforming schemes that recover a large fraction of the performance in the perfect CSI case. Numerical simulations verify our analytical performance predictions, and illustrate the benefit of the robust beamforming schemes.Comment: 10 pages, 5 figures; to appear, IEEE Transactions on Signal Processing, 201

Separation of Reliability and Secrecy in Rate-Limited Secret-Key Generation

For a discrete or a continuous source model, we study the problem of secret-key generation with one round of rate-limited public communication between two legitimate users. Although we do not provide new bounds on the wiretap secret-key (WSK) capacity for the discrete source model, we use an alternative achievability scheme that may be useful for practical applications. As a side result, we conveniently extend known bounds to the case of a continuous source model. Specifically, we consider a sequential key-generation strategy, that implements a rate-limited reconciliation step to handle reliability, followed by a privacy amplification step performed with extractors to handle secrecy. We prove that such a sequential strategy achieves the best known bounds for the rate-limited WSK capacity (under the assumption of degraded sources in the case of two-way communication). However, we show that, unlike the case of rate-unlimited public communication, achieving the reconciliation capacity in a sequential strategy does not necessarily lead to achieving the best known bounds for the WSK capacity. Consequently, reliability and secrecy can be treated successively but not independently, thereby exhibiting a limitation of sequential strategies for rate-limited public communication. Nevertheless, we provide scenarios for which reliability and secrecy can be treated successively and independently, such as the two-way rate-limited SK capacity, the one-way rate-limited WSK capacity for degraded binary symmetric sources, and the one-way rate-limited WSK capacity for Gaussian degraded sources.Comment: 18 pages, two-column, 9 figures, accepted to IEEE Transactions on Information Theory; corrected typos; updated references; minor change in titl