Adaptive Filter Lab

A compact, reproducible implementation of adaptive filtering algorithms for finite impulse response system identification.

Algorithms

The package implements:

• Least mean squares (LMS)
• Normalized least mean squares (NLMS)
• Affine projection algorithm (APA)
• Steepest descent on diagonalized quadratic cost surfaces

For an input vector $\mathbf{x}(n)$, desired response $d(n)$, coefficient vector $\mathbf{w}(n)$, and error $e(n)=d(n)-\mathbf{w}^{T}(n)\mathbf{x}(n)$, the LMS update is

$$ \mathbf{w}(n+1)=\mathbf{w}(n)+\mu e(n)\mathbf{x}(n). $$

NLMS normalizes the update by the instantaneous input power:

$$ \mathbf{w}(n+1)=\mathbf{w}(n)+ \frac{\mu e(n)\mathbf{x}(n)} {\delta+\mathbf{x}^{T}(n)\mathbf{x}(n)}. $$

APA uses the most recent $P$ regressors in $\mathbf{U}(n)$:

$$ \mathbf{w}(n+1)=\mathbf{w}(n)+ \mu\mathbf{U}^{T}(n) \left(\mathbf{U}(n)\mathbf{U}^{T}(n)+\delta\mathbf{I}\right)^{-1} \mathbf{e}(n). $$

Engineering Highlights

• A shared, validated API for all adaptive filters
• Deterministic experiments using fixed random seeds
• Full coefficient histories for convergence analysis
• Numerical regularization for NLMS and APA
• Explicit stability validation for quadratic steepest descent
• Automated tests for convergence, invalid inputs, determinism, and zero-energy signals
• No plotting, notebook, or proprietary MATLAB dependency in the core package

Repository Layout

.
├── examples/
│   └── steepest_descent.py
├── src/adaptive_filters/
│   ├── benchmark.py
│   ├── filters.py
│   └── steepest_descent.py
├── tests/
│   ├── test_filters.py
│   └── test_steepest_descent.py
└── pyproject.toml

Quick Start

python3 -m venv .venv
source .venv/bin/activate
python3 -m pip install -e .
python3 -m unittest discover -s tests -v
adaptive-filter-benchmark

The benchmark identifies the unknown system $\mathbf{h}=[0.8,-0.45,0.3,0.1]^{T}$ from colored input data with additive Gaussian noise.

Reproducible Benchmark

The default benchmark uses $4000$ samples, noise standard deviation $\sigma=0.03$, and random seed $29$.

Algorithm	Coefficient error $\lVert\hat{\mathbf{h}}-\mathbf{h}\rVert_2$	Final-window MSE
LMS	$0.00298$	$0.000910$
NLMS	$0.01528$	$0.001368$
APA	$0.01804$	$0.001487$

Run the benchmark with another seed:

adaptive-filter-benchmark --seed 42 --output results/seed-42.json

Validation Scope

The automated validation assumes a real-valued, time-invariant unknown FIR system, finite one-dimensional inputs, and additive zero-mean noise. The benchmark evaluates coefficient recovery and steady-state mean-square error; it is not a universal ranking of the algorithms.