CONVEX OPTIMIZATION FOR SIGNAL PROCESSING: A PRAGMATIC FRAMEWORK FOR MODERN CHALLENGES

The field of signal processing is constantly evolving, facing new demands from data-intensive applications like 5G communication, medical imaging, and autonomous systems. Traditional signal processing techniques, often relying on simple linear models or heuristic algorithms, struggle to meet the requirements for robustness, real-time performance, and a guarantee of optimality. This paper argues that convex optimization offers a powerful and elegant framework to tackle these modern challenges. Unlike non-convex methods that may converge to suboptimal local minima, convex optimization guarantees a globally optimal solution. We explore this concept through two core applications: real-time adaptive filtering and sparse signal reconstruction in resource-constrained environments. We propose a novel formulation for an adaptive filter that minimizes a convex cost function incorporating both signal error and a regularization term for filter stability. Furthermore, we demonstrate how the principles of compressed sensing, rooted in convex optimization, can be extended to develop highly efficient reconstruction algorithms for systems with limited computational power. Our findings suggest that moving away from purely linear approaches and embracing the guarantees of convex optimization provides a path toward more reliable, efficient, and robust signal processing systems for the future.