How does the LMS adaptive filter algorithm work?

LMS (Least Mean Squares) adapts filter coefficients to minimize mean squared error between output and desired signal. Algorithm: y[n] = w^T * x[n] (filter output), e[n] = d[n] - y[n] (error), w[n+1] = w[n] + mu * e[n] * x[n] (update). Parameter mu (step size) controls convergence speed and stability (mu < 2/(lambda_max) for convergence). Trade-offs: larger mu = faster adaptation but more noise, smaller mu = slower but steadier. Applications: Echo cancellation, noise cancellation, channel equalization, and system identification. Variants: NLMS (normalized), RLS (faster but more complex).