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This paper presents a comparative study of algorithms for jointly optimizing beamformers and receive filters in an interference network, where each node may have multiple antennas, each user transmits at most one data stream, and interference is treated as noise. We focus on techniques that seek good suboptimal solutions by means of iterative and distributed updates. Those include forward-backward iterative algorithms (max-signal-to-interference-plus-noise ratio (SINR) and interference leakage), weighted sum mean-squared error (MSE) algorithms, and interference pricing with incremental signal-to-noise ratio (SNR) adjustments. We compare their properties in terms of convergence and information exchange requirements, and then numerically evaluate their sum rate performance averaged over random (stationary) channel realizations. The numerical results show that the max-SINR algorithm achieves the maximum degrees of freedom (i.e., supports the maximum number of users with near-zero interference) and exhibits better convergence behavior at high SNRs than the weighted sum MSE algorithms. However, it assumes fixed power per user and achieves only a single point in the rate region whereas the weighted sum MSE criterion gives different points. In contrast, the incremental SNR algorithm adjusts the beam powers and deactivates users when interference alignment is infeasible. Furthermore, that algorithm can provide a slight increase in sum rate, relative to max-SINR, at the cost of additional iterations.

We derive the minimum mean square error (MMSE) solution to vector precoding for frequency flat multiuser scenarios with a centralised multi-antenna transmitter. The receivers employ a modulo operation, giving the transmitter the additional degree of freedom to choose a perturbation vector. Similar to existing vector precoding techniques, the optimum perturbation vector is found with a closest point search in a lattice. The proposed MMSE vector precoder does not, however, search for the perturbation vector resulting in the lowest unscaled transmit power, as proposed in all previous contributions on vector precoding, but finds an optimum compromise between noise enhancement and residual interference. We present simulation results showing that the proposed technique outperforms existing vector precoders, as well as the MMSE Tomlinson-Harashima precoder, and compare the turbo-coded performance to the capacity of the broadcast channel. Copyright © 2007 John Wiley & Sons, Ltd.

We design Tomlinson-Harashima precoding for decentralized receivers and frequency-selective channels based on the minimum mean square error criterion, where the feedforward filter is restricted to have finite length. Contrary to most other publications on Tomlinson-Harashima precoding which rely on solutions for decision feedback equalization to find the corresponding precoding filters in a heuristic manner, we deduce the optimization for Tomlinson-Harashima precoding from the optimization for the linear minimum mean square error transmit filter. Thereby, we include the precoding order explicitly in the problem formulation and thus obtain the precoding filter solutions, together with the algorithms to compute the latency time, i.e., the time difference between application of the precoder at the transmitter and detection at the receiver and the precoding order from a single optimization. Since the algorithm for THP filter computation resulting from the optimization has a high computational complexity, we present an alternative algorithm to compute the Tomlinson-Harashima precoding filters based on a Cholesky factorization with symmetric permutation, resulting in an order of complexity that is the same as for the computation of the linear transmit filters. The simulations reveal that the latency time optimization can be omitted without performance degradation for most practical channel models, i.e., the latency time can be chosen to be the order of the feedforward filter

The “max-SINR” algorithm and a number of related algorithms have been shown to be very useful for determining good strategies in MIMO interference networks and achieving spatial interference alignment at high SNR. These algorithms, however, rely on being initialized with a “proper” configuration of streams per user, as they do not have the capability of reducing a user's power or deactivating users completely. Alternatively, algorithms for MIMO interference networks can be designed to perform power control, so that interference alignment can be achieved even from an improper initialization. In this paper, we examine one particularly promising such design. We discuss in detail an update procedure based on maximizing the own rate minus a linearized cost of causing interference to other users, a technique known as "interference pricing" for simplicity, we restrict our attention to the case of one stream per user. As previously proposed, we combine the pricing updates with a gradual increase of the transmit power or SNR, which greatly improves the numerical properties. We show with numerical experiments that in an improper system our proposed pricing algorithm with incremental SNR achieves better performance than other algorithms with power control; the fixed-power algorithms, such as the max-SINR algorithm, perform poorly at high SNR.

We consider a network of K interfering transmitter-receiver pairs, where each node has N antennas and at most one beam is transmitted per user. We investigate the asymptotic performance of different beamforming strategies, as characterized by the slope and y-axis intercept (or offset) of the high signal-to-noise-ratio (SNR) sum rate asymptote. It is known that a slope (or multiplexing gain) of 2N-1 is achievable with interference alignment. On the other hand, a strategy achieving a slope of only N might allow for a significantly higher offset. Assuming that the number of fully aligned beamformer sets that achieve a slope of 2N-1 is finite for a given channel realization, we approximate the average offset when the best out of a large number L of these sets is selected. We also derive a simple large system approximation for the sum rate of a successive beam allocation scheme when K=N. We show that both approximations accurately predict simulated results for moderate system dimensions and characterize the large-system asymptotes for different relationships between L and N.

We consider the joint optimization of beamformers and linear receivers in a MIMO interference network. Each transmitter transmits a single beam corresponding to a rank-one precoder. When the number of users K is greater than the number of antennas at each terminal N, the maximum degrees of freedom is achieved via spatial interference alignment. Interference alignment is feasible for up to K = 2N−1 users, in which case there is a finite number of solutions to the alignment conditions. This number of solutions increases rapidly with N, and the solutions depend only on the cross-channel coefficients (i. e., they are independent of the direct channels). To maximize the achievable sum rate at high SNRs we therefore wish to select an aligned solution which is best matched to the direct channels. We evaluate the performance of this scheme for large K and N, assuming that the solution is the best out of a random subset of aligned solutions. We then compare numerically this performance with the performance of previously proposed numerical (e. g., forward-backward) techniques for optimizing beams, and a new technique which tracks the local optimum as the SNR is incrementally increased, similar to a homotopy method for improving convergence properties. We observe that the incremental technique typically achieves better performance than the previously proposed methods.

We regard a MIMO network with two transmitters and two receivers, in which each transmitter sends information to both of the receivers, a scenario known as the MIMO X channel. For such a system, the recently proposed interference alignment technique is proven to achieve the maximum degrees of freedom, which cannot be reached by conventional zero-forcing. In contrast to other scenarios, for the MIMO X channel algebraic expressions to obtain interference alignment can be easily found. Additionally, the set of parameters for possible alignments is a continuous manifold rather than a discrete set, which directly raises the question of how to find the best alignment when aiming at maximizing a utility of the transmission rates. Due to the non-convexity of the problem, finding the global optimally solution is numerically exhaustive and we are willing to accept a locally optimal solution. In this work we show an efficient parametrization of the problem which allows to apply a projected gradient approach that guarantees an aligned solution. In numerical simulations we show the superiority of our method compared to existing algorithms.

We study distributed algorithms for updating transmit precoding matrices for a two-user Multi-Input/Multi-Output (MIMO) interference channel. Our objective is to maximize the sum rate with linear Minimum Mean Squared Error (MMSE) receivers, treating the interference as additive Gaussian noise. An iterative approach is considered in which given a set of precoding matrices and powers, each receiver announces an interference price (marginal decrease in rate due to an increase in interference) for each received beam, corresponding to a column of the precoding matrix. Given the interference prices from the neighboring receiver, and also knowledge of the appropriate cross-channel matrices, the transmitter can then update the beams and powers to maximize the rate minus the interference cost. Variations on this approach are presented in which beams are added sequentially (and then fixed), and in which all beams and associated powers are adjusted at each iteration. Numerical results are presented, which compare these algorithms with iterative water-filling (which requires no information exchange), and a centralized optimization algorithm, which finds locally optimal solutions. Our results show that the distributed algorithms perform close to the centralized algorithm, and by adapting the rank of the precoder matrices, achieve the optimal high-SNR slope.

To achieve the full multiplexing gain of MIMO interference networks at high SNRs, the interference from different transmitters must be aligned in lower-dimensional subspaces at the receivers. Recently a distributed “max-SINR” algorithm for precoder optimization has been proposed that achieves interference alignment for sufficiently high SNRs. We show that this algorithm can be interpreted as a variation of an algorithm that minimizes the sum Mean Squared Error (MSE). To maximize sum utility, where the utility depends on rate or SINR, a weighted sum MSE objective is used to compute the beams, where the weights are updated according to the sum utility objective. We specify a class of utility functions for which convergence of the sum utility to a local optimum is guaranteed with asynchronous updates of beams, receiver filters, and utility weights. Numerical results are presented, which show that this method achieves interference alignment at high SNRs, and can achieve different points on the boundary of the achievable rate region by adjusting the MSE weights.

We investigate the sum rate maximization in the MIMO broadcast channel when linear filtering is applied. Our particular interest lies in usually not considered system configurations where the base station is equipped with less antennas than the user terminals have in sum. For such scenarios, conventional block-diagonalization or zero-forcing techniques cannot be applied and even asymptotic results on the sum rate do not seem to exist. We show that the nonconvex problem at hand is closely related to a combinatorial stream selection and that choosing the number of active streams too high reduces the multiplexing gain and can even lead to a saturation of the sum rate, i.e., a multiplexing gain of zero. Besides the full featured reference algorithm probing all feasible stream allocation combinations, a reduced complexity successive version is presented as well. Both variants are based on the projected gradient algorithm.

We present a general framework for the minimization of a function which is parametrized by a set of covariance matrices over a constraint set. Since all covariance matrices have to obey the property of being positive semidefinite, this characteristic has to be reflected in the constraint set. In addition, the sum of all traces of the covariance matrices shall be upper bounded. Using a preconditioned gradient descent algorithm, we derive an orthogonal projection onto this constraint set in an easy to follow monolithic way such that it directly results from the definition of the projection. Interestingly, this projection allows for a descriptive water-spilling interpretation in the style of the well-known water-filling algorithm. Two possible applications are investigated: the sum mean-square-error minimization and the weighted sum-rate maximization for the MIMO broadcast channel. Simulations finally reveal the excellent performance of the proposed framework.

Non-linear precoding for point-to-point (P2P) multiple-input multiple-output (MIMO) systems is considered. First, the minimum mean square error (MMSE) optimal vector precoding (VP) is presented for different receiver structures, viz., weighted identity matrix, diagonal matrix, weighted unitary matrix, and matrix without particular structure. Whereas the former two structures can also be applied to the vector broadcast channel, the latter two are only realizable for cooperative receivers. Second, VP is derived that minimizes the MSE but is restricted to maximize the mutual information of the MIMO channel. Third, the corresponding Tomlinson-Harashima precoding (THP) is found by applying the nearest-plane approximation to the computation of the perturbation signal. The resulting maximum mutual information THP clearly outperforms the state-of-the-art P2P-MIMO THP based on the generalized triangular decomposition (GTD) with respect to MSE and BER.

In this paper, we examine the problem of two multiple-input-single-output (MISO) links that interfere with each other. We discuss some straightforward strategies for the users to optimize their rates and show how a certain degree of cooperation between the users can lead to an improvement of both individual links. We propose an algorithm that is based on the users exchanging 'interference prices' and is realizable in a distributed manner. The algorithm converges towards a locally sum-rate optimal solution and exhibits nearly identical performance as a non-distributed gradient ascent algorithm, outperforming the straightforward strategies over the complete range of noise powers.

In this paper, we present a novel algorithm for determining the sum-rate optimal transmit covariance matrices for the MIMO broadcast channel. Instead of optimizing the covariances directly, our algorithm operates on the preceding matrices, i.e., the square roots of the covariances. As a result, no eigenvalue decompositions are required in the iterations, and the complexity per iteration is significantly lower. A look at the convergence over the required number of computations shows a visible advantage over the state-of-the-art sum power iterative waterfilling algorithm. Also, our algorithm allows us to find the optimal sum-rate for an arbitrarily limited number of data streams per user. Finally, with a simple modification, our algorithm can also be used for sum-MSE minimization.

For the single-group multicast scenario, where K users are served with the same data by a base station equipped with N antennas, we present two beamforming algorithms which outperform state-of-the-art multicast filters and feature a drastically reduced complexity at the same time. For the power minimization problem, where QoS constraints need to be satisfied, we introduce a successive beamforming filter computation approach aiming at satisfying at least one additional SNR constraint per orthogonal filter update. As long as the number of users K is smaller than the number N of transmit antennas, this procedure delivers excellent results. Our second approach is an iterative update algorithm for the max-min problem subject to a limitation of the transmit power. Given a low-complexity initialization beamformer, we search within the local vicinity of this filter vector for a filter-update preserving the transmit power and achieving a larger minimum SNR. To this end, we improve the weakest user's SNR during each iteration and keep on applying this procedure as long as the updates increase the smallest SNR. Otherwise, we adapt the step-size and continue investigating the local vicinity. It turns out that this novel approach is superior to existing state-of-the-art multicast beamformers for an arbitrary number of users.

We address the problem of jointly optimizing the pre coder and the receivers in a multi-user broadcast system for a linear and nonlinear transmitter under sum mean square error (MSE) minimization. By means of alternating optimization (AO), we find an iterative algorithm which always converges to the global optimum. In addition to an elegant initialization of the receivers' weights, we come up with expressions for the speed of convergence. Our algorithm is applicable to both single-antenna and multi-antenna receivers and can be combined with nonlinear Tomlinson-Harashima precoding (THP). For THP, the precoding order can easily be included in the AO. Moreover, we show that for sum MSE minimization the user-wise channels are not necessarily diagonalized and give some ideas how this fact can be exploited

We investigate the problem of finding the linear precoder that maximizes the sum-rate for the vector broadcast channel. In contrast to previous work, we examine the dual uplink problem, where the optimal power allocation must be found. This point of view leads to the observation that considering only uniform power allocations on a user subset yields nearly optimal performance, simplifying the problem to a combinatorial problem. We furthermore present two algorithms with complexity growing only quadratically with the number of users in the system, but with performance comparable to more complex existing schemes.

A 2-year-old male Labrador Retriever was presented to the University of Missouri Veterinary Teaching Hospital with the primary complaints of polydipsia, polyuria, and joint or muscle pain. Low blood urea nitrogen concentration, hyperchloremia, and marked proteinuria were the only abnormalities in a serum biochemical profile and urinalysis. Decreased creatinine clearance and increased renal fractional excretion of sodium, potassium, calcium, and phosphorus were detected by renal clearance studies. Increased excretion of most amino acids was found by amino acid analysis of urine, but not all amino acids were lost with equal magnitude. Amino acids with secondary amino groups or basic side chains were lost at increased rates, whereas those with acidic side chains were not. These differences could be related to defects in specific renal amino acid transport mechanisms. Identification of these transport mechanisms may allow for pharmacologic intervention at the point of renal loss to alleviate clinical signs of disease.

Canine granulocytic ehrlichial organisms were transmitted from an infected dog from Missouri to two male, 10-month-old dogs by an intravenous injection of whole blood. Physical or behavioral abnormalities were not detected during the 98 days of evaluation other than a mild pyrexia from Day 18 to 20. Ehrlichial morulae were found in blood granulocytes of Dog 1 from Day 13 to 44 and of Dog 2 from Day 14 to 34 with the peak rickettsemia occurring on Day 16 for both dogs. By Day 21 after inoculaiuon, both dogs had positive titers to Ehrlichia canis. The highest titers for both dogs were found 63 days after inoculation, after which the titers decreased. Most of the hematologic abnormalities (i.e., neutropenia, lymphocytosis, thrombocytopenia) and fever occurred between 18 and 24 days after inoculation. The pathologic bases of these abnormalities were not investigated but their concurrent presence suggested an association with the dogs' immunologic responses to the granulocytic ehrlichial agent. Results from the study indicated that the canine granulocytic ehrlichial agent of Missouri may produce subclinical infections and suggested that dogs may be able to clear the organism without antimicrobial therapy.

Nineteen dogs were identified that had mastocytemia (mast cells in venous blood samples) not associated with mast cell neoplasia. The first 10 dogs were identified by examination of blood films from dogs with suspected parvovirus enteritis (8), fibrinous pericarditis and pleuritis secondary to thoracic lacerations (1), and renal insufficiency of unknown cause (1). Because of the apparent association with acute enteritis, blood films from 52 suspected canine parvovirus cases were examined retrospectively and 8 mastocytemic dogs were found. An additional 52 canine blood films were randomly selected from the same retrospective time period and 1 mastocytemic dog was found that had pneumothorax, pelvic fractures, and hemorrhagic septic abdominal effusion secondary to renal hemorrhage and traumatized intestines. All mastocytemic dogs had acute inflammatory leukograms the day that mast cells were first detected: neutropenia with toxic neutrophils (4), neutropenia with a left shift (8), total neutrophil count within reference interval but with a left shift (5), or neutrophilia with a left shift (2). All dogs except the renal insufficiency case had circulating toxic neutrophils. Five dogs were mastocytemic on more than 1 day. The pathogenesis of the mastocytemia associated with the acute inflammatory disease was not determined.