Minimum Outage Probability for the Relay Channel under Individual Power Constraints

We maximize the performance of multiple‐input multiple‐output (MIMO) multiple access channels (MAC) and broadcast channels (BC) under individual and sum power constraints. We observe that the performance metrics sum capacity and sum mean‐square error (MSE) belong to a general class of functions which are the trace of a matrix‐monotone function. The sum capacity if successive interference cancellation (SIC) is applied in the uplink or if Costa Precoding is applied in the downlink, or the normalised sum MSE if a multi‐user minimum mean‐square error (MMSE) receiver is applied at the base, belong to that class of performance functions. We use Löwner's representation of matrix monotone functions in order to characterise the optimum transmission strategies. Using the Karush‐Kuhn‐Tucker optimality conditions, we show that the mutual covariance matrix optimisation can be decomposed into power allocation and covariance matrix optimisation under individual power constraints. The covariance matrix optimisation under individual power constraints in turn can be decomposed into a kind of modified single‐user covariance matrix optimisation treating the other users as noise. The concrete structure of the single‐user program depends on the performance metric used. The proposed algorithms efficiently solve the multi‐user MIMO performance optimisation problem. These results generalise the results regarding the performance capacity optimisation recently reported for sum capacity optimisation and the results regarding the sum MSE optimisation. The optimal transmit strategy deconstructs into two independent parts: the coding and the signal processing part. The second part is adapted to the channel and performs multi‐user power control and transmit covariance matrix optimisation. Copyright © 2006 AEIT

Consider a Rayleigh flat fading relay channel under two different power constraints: sum power constraint and individual power constraints. In this paper a simple method for resource allocation based on partial channel state information at transmitter and full channel state information at receiver is introduced. It is shown that by using different time portions for the source and the relay in relaying, when the distance from relay to the source is very different from its distance to the destination, achievable rate can be enhanced and outage probability can be reduced under sum power constraint. Also in the case of individual power constraints, time allocation scheme can improve the achievable rate and outage probability.

Typical formulations of max-min weighted SIR problems involve either a total power constraint or individual power constraints. These formulations are unable to handle the complexities in multicell networks where each base station can be subject to its own sum power constraint. This paper considers the max-min weighted SIR problem subject to multiple weighted-sum power constraints, where the weights can represent relative power costs of serving different users. First, we derive the uplink-downlink duality principle by applying Lagrange duality to the single-constraint problem. Next, we apply nonlinear Perron-Frobenius theory to derive a closed-form solution for the multiple-constraint problem. Then, by exploiting the structure of the closed-form solution, we relate the multiple-constraint problem with its single-constraint subproblems and establish the dual uplink problem. Finally, we further apply nonlinear Perron-Frobenius theory to derive an algorithm which converges geometrically fast to the optimal solution.

We study the continuous frequency spectrum management problem for the two-user Gaussian interference channel. We consider a successive interference cancellation scheme between the two receivers. In this model, receiver 1 decodes user 1 and passes the decoded bits to receiver 2 which in turn cancels the interference from user 1. We analytically determine the optimal power spectral density for users 1 and 2 for an asymmetric flat fading channel. We show when frequency division multiplexing (FDM) is optimal. We also show when frequency sharing with cancellation dominates FDM. Moreover we formulate the frequency selective case as a convex optimization problem. Both the sum power constraint and the individual power constraint are analyzed. As opposed to the linear case (no cancellation), only two pure spectrum allocation schemes are optimal when we have interference cancellation: either sharing or orthogonal. No mixed scheme is optimal. The two regions are determined by the channel conditions and are independent of the power constraints imposed. We also notice that the FDM region shrinks and the spectral efficiency increases as a result of introducing cancellation.

This paper analyzes the outage performance of cognitive relay networks operating in the same frequency band of a primary transceiver. Different from previous works, opportunistic decode-and-forward (ODF) and uncoded decode-and-forward (UDF) relaying schemes are considered, in which the relay makes either a hard decision on whether or not to transmit or generates soft information of the received signal by using the maximum-likelihood (ML) detection. To facilitate the coexistence between the cognitive user (CU) and the primary user (PU), the transmit power of both the source and the relay should be optimized to minimize the outage probability of the CU, while satisfying both the outage probability constraint on the PU and their transmit power constraints. The resulting optimization problem for the ODF scheme is generally nonconvex, but the suboptimal solution can be obtained when the decoding rate threshold of the relay is sufficiently small. Whereas for the UDF scheme, the suboptimal solution is obtained by minimizing the upper bound of its outage probability. For comparison, traditional DF relaying with optimal power allocation and direct transmission are also investigated. Simulation results confirm the tightness of suboptimal solutions and the negligible performance gap for both the ODF and the UDF schemes, and demonstrate the performance improvement provided by both schemes, particularly when the relay is far from the primary receiver.

In this paper, we study the channel estimation and the optimal training design for relay networks that operate under the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">decode-and-forward</i> (DF) strategy with the knowledge of the interference covariance. In addition to the total power constraint on all the relays, we introduce individual power constraint for each relay, which reflects the practical scenario where all relays are separated from one another. Considering the individual power constraint for the relay networks is the major difference from that in the traditional point-to-point communication systems where only a total power constraint exists for all colocated antennas. Two types of channel estimation are involved: maximum likelihood (ML) and minimum mean square error (MMSE). For ML channel estimation, the channels are assumed as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">deterministic</i> and the optimal training results from an efficient multilevel waterfilling type solution that is derived from the majorization theory. For MMSE channel estimation, however, the second-order statistics of the channels are assumed known and the general optimization problem turns out to be nonconvex. We instead consider three special yet reasonable scenarios. The problem in the first scenario is convex and could be efficiently solved by state-of-the-art optimization tools. Closed-form waterfilling type solutions are found in the remaining two scenarios, of which the first one has an interesting physical interpretation as pouring water into caves.

In this paper, we investigate distributed beamforming design for non-regenerative MIMO relay networks, in which all nodes are equipped with multiple antennas. We first give an unified formulation for SNR maximizing beamforming algorithm, which is suitable for both MIMO relay and SISO relay networks, and for both instantaneous channel state information and stochastic channel state information. Based on the general formulation, we design two distributed SNR maximizing beamforming algorithms. First, we design a beamforming algorithm for MIMO relay based on instantaneous channel state information with individual power constraints, using positive semidefinite programming (SDP). In order to reduce network cost for channel estimation, second, we also deign a beamforming algorithm for MIMO relay networks based on stochastic channel state information. We prove that in a symmetric network for the beamforming algorithm based on stochastic channel state information with individual power constraints, there exists a closed-form solution that has almost the same performance as that of the numerical methods. This result can be easily extended to a general beamforming algorithm based on stochastic channel state information, but with total power constraint.

We study the channel quantization problem for amplify-and-forward (AF) relay networks and our target is to design a quantizer to minimize the outage probability. It is priorly known that any fixed-length quantizer with a finite-cardinality codebook cannot attain the same minimum outage probability as the case where all nodes in the AF relay networks have access to perfect channel state information (CSI). We propose variable-length quantizers with random infinite-cardinality codebooks for the sum and individual power constraints. We provide theoretical proofs and numerical simulations to validate that the proposed quantizers can achieve the full-CSI outage probabilities with finite average feedback rates.

Bit and power loading (BPL) techniques have been intensively investigated for the single-link communications. In this paper, we propose a margin-adaptive BPL approach for orthogonal frequency-division multiplexing (OFDM) systems assisted by a single cooperative relay. This orthogonal half-duplex relay operates either in the selection detection-and-forward (SDF) mode or in the amplify-and-forward (AF) mode. Maximum-ratio combining is employed at the destination to attain the achievable distributed spatial diversity-gain. Assuming perfect channel knowledge is available at all nodes, the proposed approach is to minimize the transmit-power consumption at the target throughput (average number of bits/symbol) and the target link performance. With respect to various power-constraint conditions, we investigate two distributed resource-allocation strategies, namely flexible power ratio (FLPR) and fixed power ratio (FIPR). The FLPR strategy is proposed for scenarios without individual local power constraint. The source power and relay power have a flexible ratio for each subcarrier. The FIPR strategy is proposed for scenarios with individual local power constraint. The source power and relay power have a fixed ratio for each subcarrier. Computer simulations are carried out to evaluate the proposed approach with respect to the relay location. Significant performance improvement is observed in terms of both the symbol-error-rate and the transmit-power efficiency.

We consider distributed estimation in wireless sensor networks (WSNs). Using linear minimum mean squared error (LMMSE) estimator, we derive mean squared error (MSE) to measure the quality of estimation. We introduce a global signal to noise ratio (SNR), where we can derive capacity of data collection in terms of mutual information as reciprocity of MSE. Based on the global SNR, we derive equal power allocation and optimal power allocation in orthogonal multiple access channel (MAC) models. We also derive asymptotic behavior of the global SNR when the power and the number of sensors become unlimited. We minimize MSE as well as maximize mutual information by considering total and individual power constraints. We show that MSE and mutual information of equal power allocation outperforms optimal power allocation. Moreover, system with individual power constraint is worse than the system without that because the suggested power to be allocated is constrained by maximum transmit power of the sensors.

This paper provides a novel low complexity and provably convergent algorithm to find the sum capacity for vector Gaussian broadcast channels. We formulate the problem in the context of a MIMO-OFDM system and discuss the simplifications provided by the block diagonal channel structure. The algorithm also provides the optimum set of covariance matrices for the dual multiple access channel, from which the transmit covariance matrices for the broadcast channel are easy to compute. The key idea of the algorithm is to iteratively maximize the sum capacity for randomly selected pairs of users whilst considering the other users' signals as noise. The proposed algorithm can be considered dual to the iterative water filling (IWF) algorithm used to maximize the sum capacity for the multiple access channel with individual power constraints, but unlike the recently proposed sum power constraint IWF, our proposed algorithm has lower complexity and does not require additional precautions to ensure the convergence. We have proved analytically that the proposed algorithm is convergent in probability and the computer simulations clearly show that the proposed algorithm converges faster than other known algorithms. Index Terms—Broadcast channels, MIMO systems, orthogonal frequency division multiplexing, iterative water-filling.

In this article, we adopt outage probability (λ-capacity) in fast-fading channels as a pay-off function in a zero-sum game between a legitimate transceiver pair and an uncorrelated Gaussian jammer. The transmitter aims at minimising the outage probability, while the jammer attempts to maximise the outage probability. We consider both peak (over each codeword) and average (over all codewords) power constraints. For peak power constraints, a transmission rate is either supported by the system, or if too large, causes the whole transmission to fail. By imposing average power constraints, large rates can be supported at the cost of positive probability of codeword error. Maxmin (where the jammer is assumed to play first) and minimax (where the transmitter is assumed to play first) power control strategies are developed under each power constraint, which show that no Nash equilibrium of pure strategies exists under the average power constraint when the follower has perfect knowledge about the randomisation outcome of the first player.

A two-way relay channel (TWRC) in which two terminals <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> exchange orthogonal frequency division multiplexing (OFDM) signals with the help of an amplify-and-forward (AF) relay <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> is considered here, and an efficient technique for allocating powers to <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> parallel tones of OFDM is developed. A sum rate maximization problem is formulated by replacing the individual power constraints of the conventional sum rate maximization problem, which limit the power of each terminal, with the total power constraint limiting the sum of powers of all terminals. The maximization problem with the total power constraint yields a more efficient power allocation policy than the conventional problem with individual power constraints. It is shown that the closed-form solution of the maximization problem under the total power constraint can be obtained for a single-tone system ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> =1). Based on this result, a two-step suboptimal approach is proposed in which the power is optimally assigned to each tone first, and then at each tone the assigned power is distributed to the three terminals. The proposed method is shown to assign 50% of the total power to relay <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> irrespective of the channels. It is demonstrated that the proposed method is considerably simpler to implement than the conventional dual-decomposition method (DDM), yet the performance of the former is almost identical to that of the latter.

In this paper, we study the joint allocation of three types of resources, namely, power, subcarriers and relay nodes, in multi-relay assisted dual-hop cooperative OFDM systems. All the relays adopt the amplify-and-forward protocol and assist the transmission from the source to destination simultaneously but on orthogonal subcarriers. The objective is to maximize the system transmission rate subject to individual power constraints on each node or a total network power constraint. We formulate such a problem as a subcarrier-pair based resource allocation that seeks the joint optimization of subcarrier pairing, subcarrier-pair-to-relay assignment, and power allocation. Using a dual approach, we solve this problem efficiently in an asymptotically optimal manner. Specifically, for the optimization problem with individual power constraints, the computational complexity is polynomial in the number of subcarriers and relay nodes, whereas the complexity of the problem with a total power constraint is polynomial in the number of subcarriers.We further propose two suboptimal algorithms for the former to trade off performance for complexity. Simulation studies are conducted to evaluate the average transmission rate and outage probability of the proposed algorithms. The impact of relay location is also discussed.

In this paper, we study the outage behavior of a decode-and-forward multi-hop free-space optical (FSO) system over a Poisson channel degraded by atmospheric turbulence. We assume that perfect channel side information (CSI) is available at the receiver side and consider both cases of perfect CSI and no CSI at the transmitter side. We solve the outage probability minimization problem subject to a peak power constraint as well as a short- or long-term average sum power constraint. As a result, optimal power control strategies are presented for different scenarios under consideration. A sub-optimal yet low-complexity solution is further proposed under the short-term power constraint. Our results demonstrate that multi-hop relaying yields significant performance improvements which are particularly important for long-range FSO links.