The Linear Model under Mixed Gaussian Inputs: Designing the Transfer Matrix

Suppose a linear model y = Hx + n, where inputs x, n are independent Gaussian mixtures. The problem is to design the transfer matrix H so as to minimize the mean square error (MSE) when estimating x from y. This problem has important applications, but faces at least three hurdles. Firstly, even for a fixed H, the minimum MSE (MMSE) has no analytical form. Secondly, the MMSE is generally not convex in H. Thirdly, derivatives of the MMSE w.r.t. H are hard to obtain. This paper casts the problem as a stochastic program and invokes gradient methods. The study is motivated by two applications in signal processing. One concerns the choice of error-reducing precoders; the other deals with selection of pilot matrices for channel estimation. In either setting, our numerical results indicate improved estimation accuracy - markedly better than those obtained by optimal design based on standard linear estimators. Some implications of the non-convexities of the MMSE are noteworthy, yet, to our knowledge, not well known. For example, there are cases in which more pilot power is detrimental for channel estimation. This paper explains why

Matrix-Monotonic Optimization for MIMO Systems

For MIMO systems, due to the deployment of multiple antennas at both the transmitter and the receiver, the design variables e.g., precoders, equalizers, training sequences, etc. are usually matrices. It is well known that matrix operations are usually more complicated compared to their vector counterparts. In order to overcome the high complexity resulting from matrix variables, in this paper we investigate a class of elegant multi-objective optimization problems, namely matrix-monotonic optimization problems (MMOPs). In our work, various representative MIMO optimization problems are unified into a framework of matrix-monotonic optimization, which includes linear transceiver design, nonlinear transceiver design, training sequence design, radar waveform optimization, the corresponding robust design and so on as its special cases. Then exploiting the framework of matrix-monotonic optimization the optimal structures of the considered matrix variables can be derived first. Based on the optimal structure, the matrix-variate optimization problems can be greatly simplified into the ones with only vector variables. In particular, the dimension of the new vector variable is equal to the minimum number of columns and rows of the original matrix variable. Finally, we also extend our work to some more general cases with multiple matrix variables.Comment: 37 Pages, 5 figures, IEEE Transactions on Signal Processing, Final Versio

On training optimization for estimation of correlated MIMO channels in the presence of multiuser interference

In this paper, the problem of estimating multiple-input multiple-output (MIMO) channels in a realistic environment involving correlated channel fading and multiuser interference is considered. Four estimation schemes are studied, including the linear minimum mean squared error (LMMSE), least squares (LS), and Gauss-Markov (GM) estimators, as well as a novel scheme which is derived here as an alternative to LMMSE estimation. The MSE-optimal training sequences for each of them are provided and their requirements for side information feedback are assessed. The new scheme is shown to exhibit a performance comparable to or even better than LMMSE, at a significantly lower feedback and computational cost. The analytical comparison of the estimation schemes is supported by numerous simulation results that cover a wide range of antenna configurations, relative interference power, and channel correlation strengths. The results of this paper provide a complete picture for a palette of estimation schemes, with their relative performance and costs of training. © 2008 IEEE