Mathematics Research Projects

Explosion of millimeter-wave (mm-wave) bandwidth opens up applications in 5G wireless systems spanning communications, localization, imaging, and radar. This project addresses challenges in mathematics, engineering, and science in developing efficient wideband beamformers based on sparse factorizations of the matrix called-delay Vandermonde matrices (DVM). The proposed highly integrated approach is attractive for mobile applications including 5G smart devices, the internet of things, mobile robotics, unmanned aerial vehicles, and other emerging applications focused on mm-waves.

A multi-beam array receiver is deeply difficult to realize in integrated circuit (IC) form due to the underlying complexity of its signal flow graph. Through the proposed work, mathematical methods based on the theories of i) sparse factorization and complexity of the structured complex DVM with the introduction of a super class for the discrete Fourier transform(which is DVM), and ii) approximation transforms are proved to solve this problem.

The resulting matrices are realized with multi-GHz bandwidths using analog ICs. The novel DVM algorithm solves the longstanding "beam squint" problem, i.e., the fact that the beam direction changes with input frequency, making true wideband operation impossible. Moreover, the proposed multi-beamforming networks in analog IC form will be realized efficiently while addressing precision circuit design, digital calibration, built-in self-test, etc. Besides scientific merits, both minority students and female students will be mentored to pursue careers in the STEM disciplines through the proposed project.

This project was funded by the National Science Foundation (the division of Electrical, Communications, and Cyber Systems) with award numbers 1711625 and 1711395.