RLS Algorithm

The Recursive Least Squares (RLS) algorithm is a sophisticated adaptive filter technique that excels in environments where rapid convergence and tracking are essential. Unlike its simpler cousin, the LMS algorithm, RLS offers faster convergence at the cost of increased computational complexity.

Core Principles

The algorithm operates on the following key principles:

1. Recursive Operation: Updates filter coefficients with each new sample
2. Exponential Weighting: Applies more weight to recent samples through a forgetting factor
3. Matrix Operations: Utilizes the matrix inversion lemma for efficient computation

Mathematical Foundation

The RLS algorithm minimizes the cost function:

J(n) = Σ λ^(n-i) |e(i)|^2

Where:

• λ is the forgetting factor (typically 0.98 to 0.9999)
• e(i) represents the error at time i
• n is the current time index

Implementation Steps

1. Initialization:

   • Set initial filter coefficients
   • Initialize the correlation matrix
   • Choose forgetting factor λ

2. For each new sample:

   • Compute filter output
   • Calculate error
   • Update Kalman gain vector
   • Update filter coefficients
   • Update inverse correlation matrix

Applications

RLS finds widespread use in:

• Channel equalization
• Echo cancellation
• System identification
• Adaptive beamforming
• Noise cancellation

Advantages and Limitations

Advantages

• Fast convergence rate
• Excellent tracking capability
• Superior performance in non-stationary signals
• Low steady-state error

Limitations

• Higher computational complexity (O(N²))
• More sensitive to numerical precision
• Requires more memory than simpler algorithms
• Potential numerical instability issues

Variants

Several modifications exist to address specific challenges:

• QR-RLS: Improved numerical stability
• Fast RLS: Reduced computational complexity
• Square-root RLS: Better numerical properties
• Lattice RLS: Enhanced stability

Comparison with Other Methods

The RLS algorithm sits within a broader family of adaptive algorithms, each with distinct characteristics:

• More complex than LMS algorithm
• Faster convergence than NLMS algorithm
• More general than Kalman filter

Implementation Considerations

When implementing RLS, practitioners should consider:

1. Choice of forgetting factor
2. Initialization of correlation matrix
3. Numerical precision requirements
4. Available computational resources
5. Real-time processing constraints