Changxin Shi

Minimum Mean Squared Error interference alignment

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.

Distributed resource allocation schemes

In this article, we discuss distributed resource allocation schemes in which each transmitter determines its allocation autonomously, based on the exchange of interference prices. These schemes have been primarily motivated by the common model for spectrum sharing in which a user or service provider may transmit in a designated band provided that they abide by certain rules (e.g., a standard such as 802.11). An attractive property of these schemes is that they are scalable, i.e., the information exchange and overhead can be adapted according to the size of the network.

Distributed Interference Pricing for the MIMO Interference Channel

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.

Monotonic convergence of distributed interference pricing in wireless networks

We study distributed algorithms for allocating powers and/or adjusting beamforming vectors in a peer-to-peer wireless network which may have multiple-input-single-output (MISO) links. The objective is to maximize the total utility summed over all users, where each users utility is a function of the received signal-to-interference-plus-noise ratio (SINR). Each user (receiver) announces an interference price, representing the marginal cost of interference from other users. A particular user (transmitter) then updates its power and beamforming vector to maximize its utility minus the interference cost to other users, which is determined from their announced interference prices. We show that if each transmitter update is based on a current set of interference prices and the utility functions satisfy certain concavity conditions, then the total utility is non-decreasing with each update. The proof is based on the convexity of the utility functions with respect to received interference, and applies to rate utility functions, and an arbitrary number of interfering MISO links. The extension to multi-carrier links is discussed as well as algorithmic variations in which the prices are not immediately updated after power or beam updates.

Comparison of Distributed Beamforming Algorithms for MIMO Interference Networks

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.

Distributed interference pricing for OFDM wireless networks with non-separable utilities

We present a distributed algorithm for allocating power among multiple interfering transmitters in a wireless network using orthogonal frequency division multiplexing (OFDM). The algorithm attempts to maximize the sum over user utilities, where each users utility is a function of his total transmission rate. Users exchange interference prices reflecting the marginal cost of interference on each sub-channel, and then update their power allocations given the interference prices and their own channel conditions. A similar algorithm was studied earlier assuming that each users utility function is a separable function of the users rate per sub-channel. Here, we do not assume this separability. We give a different algorithm for updating each users power allocation and show that this algorithm converges monotonically. Numerical results comparing this algorithm to several others are also presented.

Distributed interference pricing with MISO channels

We study a distributed algorithm for adapting transmit beamforming vectors in a multi-antenna peer-to-peer wireless network. The algorithm attempts to maximize a sum of per-user utility functions, where each users utility is a function of his transmission rate, or equivalently the received signal-to-interference plus noise ratio (SINR). This is accomplished by exchanging interference prices, each of which represents the marginal cost of interference to a particular user. Given the interference prices, users update their beamforming vectors to maximize their utility minus the cost of interference. For a two-user system, we show that this algorithm converges for a suitable class of utility functions. Convergence of the algorithm with more than two users is illustrated numerically.

Adaptive beamforming in interference networks via bi-directional training

We study distributed algorithms for adjusting beamforming vectors and receiver filters in multiple-input multiple-output (MIMO) interference networks, with the assumption that each user uses a single beam and a linear filter at the receiver. In such a setting there have been several distributed algorithms studied for maximizing the sum-rate or sum-utility assuming perfect channel state information (CSI) at the transmitters and receivers. The focus of this paper is to study adaptive algorithms for time-varying channels, without assuming any CSI at the transmitters or receivers. Specifically, we consider an adaptive version of the recent Max-SINR algorithm for a time-division duplex system. This algorithm uses a period of bi-directional training followed by a block of data transmission. Training in the forward direction is sent using the current beam-formers and used to adapt the receive filters. Training in the reverse direction is sent using the current receive filters as beams and used to adapt the transmit beamformers. The adaptation of both receive filters and beamformers is done using a least-squares objective for the current block. In order to improve the performance when the training data is limited, we also consider using exponentially weighted data from previous blocks. Numerical results are presented that compare the performance of the algorithms in different settings.

Interference alignment in multi-carrier interference networks

We consider an interference network with multi-carrier transmission over M parallel sub-channels. There are K transmitter-receiver pairs, each transmitter transmits a single data stream with a rank-one precoding matrix, and the receivers are assumed to be linear. We show that a necessary condition for zero interference (alignment across sub-channels) is K ≤ 2M−2. In contrast, for a Multi-Input Multi-Output (MIMO) interference network with M×M spatial channels (full channel matrices) the corresponding condition is known to be K ≤ 2M − 1. We also characterize the sum rate at high Signal-to-Noise Ratios (SNR) by bounding the SNR offset (x-intercept) of the asymptote of the sum rate vs SNR curve. For a randomly chosen aligned solution as M increases, this offset shifts to the right as logM. In contrast, the SNR offset for a MIMO interference network does not increase with M. An approximation for the performance of sampling the best out of L aligned solutions is also presented. Numerical results show the analytical asymptotes accurately predict the sum rate curves at moderate to high SNRs.

Bi-Directional Training for Adaptive Beamforming and Power Control in Interference Networks

We study distributed algorithms for adapting transmit beamformers and linear receiver filters in a Time-Division Duplex Multiple-Input Multiple-Output (MIMO) interference network. Each transmitter transmits a single beam, and neither the transmitters nor receivers have a priori Channel State Information (CSI). Given a fixed set of powers, we present an adaptive version of the Max-SINR algorithm: pilot symbols are alternately transmitted in the forward direction (transmitters to receivers) and in the reverse direction (receivers to transmitters). Unlike previous channel estimation schemes, transmissions in each direction are synchronized across the source or destination nodes, and the pilots are used to update the filters/beams directly using a least squares criterion. To improve the performance with limited training, we include exponential weighting of the least squares objective across data frames. In addition, bi-directional training can be used to implement analog interference pricing for power control: training in the forward direction is used to measure received signal-to-interference plus noise ratios (SINRs) and interference prices, and those estimates combined with synchronous backward training are used to update the powers. Given sufficient training this method achieves the same performance as interference pricing updates with perfect CSI. Numerical results are presented that illustrate the performance of these methods in different settings.