Diffusion Quantum-Least Mean Square Algorithm with Steady-State Analysis

Diffusion least mean square (LMS) algorithm is a well-known algorithm for distributed estimation where estimation takes place at multiple nodes. However, it inherits slow convergence speed due to its gradient descent-based design. To deal with this challenge, we proposed a modified diffusion LMS with improved convergence performance by employing quantum-calculus-based gradient descent, and hence, we called it diffusion q-least mean square (Diff-qLMS). In the proposed design, we derive the weight update mechanism by minimizing the conventional mean square error (MSE) cost function via quantum-derivative in a distributed estimation environment. We developed two different modes of diffusion qLMS operation: combine-then-adapt (CTA) and adapt-then-combine (ATC). To improve the performance in terms of faster convergence and lower steady-state error, we also developed an efficient mechanism to obtain the optimal values of q-parameter for each tap-weight of the filter in order to achieve both faster convergence and lower steady-state error. With the aim to achieve the performance of the proposed algorithm theoretically, convergence analysis for both the transient and the steady-state scenarios is presented. Consequently, closed-form expressions governing both the transient and the steady-state behaviors in terms of mean square deviation (MSD) and excess mean square error (EMSE) for both local node and global network are derived. The theoretical claims are validated via Monte Carlo simulations. The performance of the proposed algorithm is investigated for various system noises and the results show the superiority of the proposed algorithm in terms of both the convergence speed and the steady-state error.