Adaptive equalization algorithms (any two)

An adaptive equalization algorithm updates the tap weights w of the equalizer so that the error e(k) = d(k) − d̂(k) between the desired and actual output becomes small. Two common ones:

1. Least Mean Square (LMS) algorithm

• Minimises the mean square error E[e²(k)] using a stochastic gradient (steepest descent) step.
• Update equations:

 d^(k)  = wᵀ(k)·x(k)          (filter output)
 e(k)   = d(k) - d^(k)        (error)
 w(k+1) = w(k) + μ·e(k)·x(k)  (weight update)

• μ is the step size: larger μ gives faster convergence but more misadjustment; stability needs 0 < μ < 2/λ_max.
• Needs only about 2N+1 operations per iteration (N = taps), so it is simple and robust.
• Disadvantage: slow convergence when the eigenvalue spread of the input correlation matrix is large.

2. Recursive Least Squares (RLS) algorithm

• Minimises the weighted sum of past squared errors Σ λⁿ⁻ⁱ e²(i), where λ (0 < λ ≤ 1) is a forgetting factor.
• Recursively updates the inverse correlation matrix P(n) and a gain vector k(n):

 k(n)   = P(n-1)x(n) / [λ + xᵀ(n)P(n-1)x(n)]
 e(n)   = d(n) - wᵀ(n-1)x(n)
 w(n)   = w(n-1) + k(n)·e(n)
 P(n)   = [P(n-1) - k(n)xᵀ(n)P(n-1)] / λ

• Converges about an order of magnitude faster than LMS and is insensitive to eigenvalue spread.
• Needs about 2.5N² + 4.5N operations per iteration and can be numerically unstable.

(Zero forcing is a third simple algorithm that makes the combined channel–equalizer response have zero ISI.)