Comparative Simulation of Learning Curves for Basic LMS Algorithm and Normalized LMS Algorithm

In this paper, we explore the comparative simulation of learning curves for both the basic LMS algorithm and the normalized LMS algorithm. First, we introduce the background and principles of LMS and normalized LMS algorithms to help readers better understand their theoretical foundations. The implementation typically involves MATLAB code with key functions like LMS filtering using weight update equations w(n+1) = w(n) + μ*e(n)*x(n), where μ represents the step size parameter. For normalized LMS, the code incorporates normalization by the input power ||x(n)||² to improve stability.

Next, we discuss the implementation approaches for both algorithms, including their advantages, limitations, and application scenarios. The basic LMS algorithm implementation requires careful selection of step size parameters to balance convergence speed and stability, while normalized LMS automatically adjusts the step size based on input signal power. Code implementation typically involves iterative weight updates and error calculation loops with performance monitoring.

After implementing the algorithms, we conduct comparative simulations of their learning curves, analyzing convergence behavior, steady-state error, and computational complexity. The simulation results demonstrate that normalized LMS generally provides faster convergence and better stability in scenarios with varying input signal statistics. We analyze these results using performance metrics like Mean Square Error (MSE) plots and convergence rate comparisons.

Finally, we summarize the research findings and propose future research directions, including potential extensions to variable step-size implementations and applications in real-time adaptive filtering systems. The code structure allows for easy modification to test different signal conditions and algorithm variations.