RLS Algorithm and Lattice Structures

RLS Algorithm (Recursive Least Squares)

Minimizes weighted sum of squared errors: \(J[n] = \sum_{k=0}^{n} \lambda^{n-k} e^2[k]\)

$\lambda$ = forgetting factor ($0 < \lambda \le 1$, typically $0.98-0.999$)

Update Equations:

Variable	Update
Gain vector	$\mathbf{k}[n] = \frac{\lambda^{-1} \mathbf{P}[n-1] \mathbf{x}[n]}{1 + \lambda^{-1} \mathbf{x}^T[n] \mathbf{P}[n-1] \mathbf{x}[n]}$
Error	$e[n] = d[n] - \mathbf{w}^T[n-1] \mathbf{x}[n]$
Weights	$\mathbf{w}[n] = \mathbf{w}[n-1] + \mathbf{k}[n] e[n]$
Inverse correlation	$\mathbf{P}[n] = \lambda^{-1} \mathbf{P}[n-1] - \lambda^{-1} \mathbf{k}[n] \mathbf{x}^T[n] \mathbf{P}[n-1]$

Initialization: $\mathbf{P}[0] = \delta^{-1} \mathbf{I}$ ($\delta \approx 0.01$)

RLS vs LMS Comparison

Aspect	LMS	RLS
Complexity	$O(M)$	$O(M^2)$
Convergence	Slow (~$200M$ samples)	Fast (~$2M$ samples)
Steady-state MSE	Higher (misadjustment)	Optimal (matches Wiener)
Tracking	Moderate	Excellent
Numerical issues	Robust	Needs stabilization
Memory	$O(M)$	$O(M^2)$

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Lattice-Ladder RLS (Proakis 13.4)

Numerically stable RLS using lattice structure:

• Orthogonalizes input → better conditioning
• Modular structure
• Lower complexity than direct-form RLS for large $M$

Key Advantages:

• No matrix inversion
• Natural parameter constraints
• Good for fixed-point implementation

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