Research Projects

The use of multiple-input multiple-output (MIMO) techniques has become the norm in modern wireless communication systems, since they provide increased data rates and enhanced reliability. One essential ingredient in realizing these benefits is by the use of matrix precoders that are typically unitary matrices that can be used to beamform the transmissions for optimal reception. With the advent of 5G and beyond systems, the very wide band nature of the communication channel necessitates that the precoder be fed back for a wide range of frequencies from the receiver to the transmitter, thereby increasing the feedback overhead in such systems, despite their being more significant line of sight components that result in more sparse time-domain representations. To address this issue, we propose to use time-domain matrix precoders, since they capture the unitary structure of the precoding matrices while providing realizable time domain filters. Recent work has shown that precoders can be viewed as matrix all-pass filters, with the coefficients being easier to feedback owing to the sparse time-domain channel realization. In this project we aim to construct such time-domain all-pass filters as matrix precoders from a limited data set given by the desired filter's matrix-valued phases at a finite set of frequencies. Our approach towards solving this is through interpolation of the data by a matrix of rational function entries. This is a well-known open problem from complex analysis: the boundary case of subspace Nevanlinna-Pick interpolation problem (B-SNIP). In our recent work we have shown how the open problem of B-SNIP can be solved by compensating the data-set with derivative-like additional data. However, this solution raises a number of crucial questions that needs answering for a comprehensive solution of the matrix precoder design via B-SNIP. The proposed project is aimed at resolving these issues. We briefly state these issues here: 1. B-SNIP admits non-unique solutions. Our existing method chooses a solution in an ad hoc manner. A full characterization of all solutions of B-SNIP is needed to choose the most desirable precoder that is also stable in the time domain. 2. We have already shown that a key parameter in the solution is the Schwarz-Pick matrix, which in turn is parametrized by the above-mentioned derivative-like parameters that are additionally supplied as input. These additional parameters allow us to vary the solution, and thereby obtain a precoder that outperforms others. This optimization problem needs to fully understood and formalized. 3. Since the precoder design is based on the frequency domain data, it is of utmost importance that the method be immune to unwanted perturbation in the data. The extent of robustness of our solution to B-SNIP needs to be investigated and quantified; necessary mitigation technique to this effect needs to be devised. 4. The devised techniques' performance needs to be verified experimentally.