Twenty-Five Years of Advances in Beamforming: From convex and nonconvex optimization to learning techniques

Beamforming is a signal processing technique to steer, shape, and focus an electromagnetic (EM) wave using an array of sensors toward a desired direction. It has been used in many engineering applications, such as radar, sonar, acoustics, astronomy, seismology, medical imaging, and communications. With the advent of multiantenna technologies in, say, radar and communication, there has been a great interest in designing beamformers by exploiting convex or nonconvex optimization methods. Recently, machine learning (ML) is also leveraged for obtaining attractive solutions to more complex beamforming scenarios. This article captures the evolution of beamforming in the last 25 years from convex to nonconvex optimization and optimization to learning approaches. It provides a glimpse into these important signal processing algorithms for a variety of transmit-receive architectures, propagation zones, propagation paths, and multidisciplinary applications.