Quantum Calculus-Based Two-Dimensional Least Mean Square Algorithm

Recent work of the one-dimensional q-LMS algorithm give insight that the Quantum Calculus based gradient called the q-gradient is capable of improving the convergence behavior in contrast to the standard LMS algorithm. The reason for this is the fact that the q-derivative takes larger steps in minimizing a cost function as it evaluates the secant in contrast to the tangent of the cost function. Motivated by this, we propose a two-dimensional version of the q-LMS algorithm. More precisely, we employ Quantum Calculus based q-gradient in steepest descent optimization of mean-square-error (MSE) cost function in 2D adaptive filtering scenario to obtain two-dimensional q-LMS (2D q-LMS) algorithm. A thorough analytical investigation of the proposed 2D q-LMS algorithm in both mean and mean-square sense is provided. Both the transient and steady-state convergence behaviours are examined. Consequently, excess MSE (EMSE) learning curve and its steady-state expression are evaluated in closed form. Simulation results are presented for the application of noise cancelation in images to show the superiority of the proposed 2D q-LMS algorithm.