A Framework for Joint Design of Pilot Sequence and Linear Precoder

Abstract

Most performance measures of pilot-assisted multiple-input multiple-output (MIMO) systems are functions that depend on both the linear precoding filter and the pilot sequence. A framework for the optimization of these two parameters is proposed, based on a matrix-valued generalization of the concept of effective signal-to-noise ratio (SNR) introduced in a famous work by Hassibi and Hochwald. The framework applies to a wide class of utility functions of said effective SNR matrix, most notably a well-known mutual information expression for Gaussian inputs, an upper bound on the minimum mean-square error (MMSE), as well as approximations thereof. The approach consists in decomposing the joint optimization problem into three subproblems: first, we describe how to reformulate the optimization of the linear precoder subject to a fixed pilot sequence as a convex problem. Second, we do likewise for the optimization of the pilot sequence subject to a fixed precoder. Third, we describe how to generate pairs of precoders and pilot sequences that are Pareto optimal in the sense that they attain the Pareto boundary of the set of feasible effective SNR matrices. By combining these three optimization problems into an iteration, we obtain an algorithm which allows to compute jointly optimal pairs of precoders and pilot sequences with respect to some generic utility function of the effective SNR.Comment: 32 pages, 9 figures, submitted to IEEE Transactions on Information Theor

Matrix-Monotonic Optimization $-$ Part I: Single-Variable Optimization

“…According to the different levels of CSI knowledge, MIMO transceiver designs can be classified into designs relying on perfect CSI [4]- [6] and designs having partial CSI [15]- [19]. Finally, according to the system architecture, transceiver optimization can be used for point-to-point systems [10], [20], for multi-user (MU) MIMO systems [21], for distributed MIMO systems [22], [23], and for cooperative MIMO systems [24], [25].…”

Matrix-Monotonic Optimization for MIMO Systems

“…where Λ D andΛ D consist of the eigenvalues of D arranged in descending order and ascending order, while U D andŪ D contain the corresponding eigenvectors of D, respectively. Then we have the four basic matrix inequalities, ranging from (19) to (22), shown at the bottom of this page. Furthermore, in both Matrix Inequality 1 and Matrix Inequality 2, the left equality holds when U C =Ū D , and the right equality holds when U C = U D ; while in both Matrix Inequality 3 and Matrix Inequality 4, the left equality holds when U C = U D , and the right equality holds when U C =Ū D .…”

“…In wireless communication systems, the channel parameters have to be estimated. however, due to the uncertainty introduced both by noise and the time-varying nature of wireless channels, channel estimation errors inevitably exist [19], and the true channel matrix H can be expressed by the following Kronecker formula [20], [22]…”

“…where H is the estimated channel matrix and H W Ψ 1 2 is the channel estimation error, in which the elements of H W obey the independent and identical complex Gaussian distribution CN (0, 1) and the covariance matrix Ψ of the channel estimate is a function of both the training sequence and of the channel estimator [20], [22]. Based on (42), for Bayesian robust transceiver optimization, the matrix Π in the matrixmonotonic optimization can be expressed as [30]…”

Training Optimization for Hybrid MIMO Communication Systems

“…performance gains are critically dependent on the availability and accuracy or the absence of channel state information (CSI). Therefore, channel estimation becomes a critical part of various MIMO communication systems [5]- [12].…”