An Optimal and Fair Distributed Algorithm for Power Allocation for Radios Coexisting in Unlicensed Spectra

The power allocation that maximizes the sum rate of transceivers operating in the same frequency band is a difficult non-convex problem. Lack of a convex structure excludes the direct application of Lagrangian dual techniques as the duality gap might not be zero. This paper advances current knowledge by introducing three significant steps in finding a solution. First, we show that for transceivers operating under a total power constraint, the maximum sum rate occurs at the boundary of the feasible set formed by the hyper plane representing the power constraint. This conclusion is nontrivial considering that we are dealing with an interference limited system. Second, we prove that the duality gap is zero for this problem, despite the lack of concavity of the objective. We do this by showing that the maximum sum rate is concave in the power constraint. Third, we propose an iterative algorithm that finds the optimal power allocation by solving the dual problem. Simulation results are provided to support the theorems proven in the paper as well as to demonstrate the convergence of the algorithm to the global maximum sum rate. Results of the algorithm are also compared with solutions based on Game theory.

The power allocation that maximizes the sum rate of transceivers operating in the same frequency band is a difficult non-convex problem. In our earlier work, we proved that for transceivers operating under a total power constraint, the maximum sum rate occurs at the boundary of the feasible set formed by the hyper plane representing the power constraint. This finding is nontrivial considering that we are dealing with an interference limited system. In this paper, we study the convex structure of the sum rate on the boundary of the power constraint hyper plane. For two transceivers, we prove that the sum rate is always convex on the line created by the power constraint equality. In the case of three transceivers, we identify a region in the middle of the plane created by the power constraint equality, where the sum rate is concave. This is significant because it is in the middle of the boundary plane that the power allocation can be expected to be fair to all three users. We also provide a power allocation protocol and an algorithm that distribute the power among the transceivers with fairness. Simulation results are provided to support the theorems proven in the paper as well as to demonstrate the convergence of the algorithm to the global maximum sum rate. Results of the algorithm are compared with solutions based on Game theory.

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.

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.

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

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.

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 letter, minimum outage probability for a fading relay channel is considered for long-term individual power constraints (IPC) at the source and the relay. By utilizing Lagrange multiplier theory, it is shown that solving the IPC problem is equivalent to solving a weighted sum power constrained (wSPC) problem. Opportunistic decode and forward (ODF), and simplified ODF (SODF) protocol performances are investigated and compared with a lower bound on outage probability, under both individual and sum power constraints (SPC). It is shown that, under IPC, the source or the relay does not necessarily deplete its average power constraint, but some power can remain unutilized for ODF or SODF.

We consider the problem of jointly optimizing channel pairing, channel-user assignment, and power allocation in a single-relay multiple-access system. The optimization objective is to maximize the weighted sum-rate under total and individual power constraints on the transmitters. By observing the special structure of a three-dimensional assignment problem derived from the original problem, we propose a polynomial-time algorithm based on continuity relaxation and dual minimization. The proposed method is shown to be optimal for all relaying strategies that give a concave rate function in terms of power constraints.

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.

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.

Power allocation in multi-hop OFDM transmission systems with amplify-and-forward relaying: A unified approach

A novel unified approach to deal with arbitrary transmitted power constraints for designing jointly the MMSE precoders and decoders in downlink multiuser MIMO systems is developed. This work is an extension of the iterative approach developed in (Jinfan Zhang et al., 2005) for the total power constraint. An explicit expression relating Lagrange multipliers with the per-user constraint can be obtained using the proposed approach to mitigate the time consuming root searching for the Lagrange multipliers in (Jinfan Zhang et al., 2005). Moreover, the explicit expression allows the MMSE solution subject to arbitrary transmit power constraints. In this paper, total, per-user, per-antenna, and peak power constraints are formulated in under the same format and solved using the proposed approach. It is shown in some two-user numerical examples that the raw BER performance obtained from the practical per-antenna power constraint is similar to those of the per-user and total power constraints and is 1-1.5 dB better than that of the peak power constraint.

Assessment of the Economic Value of Unlicensed Spectrum in the United States

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.