Implementation combining THRML discrete sampling with gradient refinement for adaptive chirplet decomposition.

Note: This implementation has only been tested on Mac M series laptops. Compatibility with other platforms is not guaranteed. By default, the code runs on CPU for maximum compatibility.

# Create the virtual environment
uv venv .venv

# Sync dependencies
uv sync

# Activate the virtual environment
source .venv/bin/activate

# Run the visualization
python run_visualization.py

# Run the test suite
python act-challenging-test.py
python act-profile-test.py
python act-synth-test.py

from act import decompose_signal

components, parameters, residual = decompose_signal(
    signal,
    sampling_rate=256.0,
    n_components=10,
    residual_threshold=0.01,
    use_thrml=True
)

from act import THRMLChirpletOptimizer

optimizer = THRMLChirpletOptimizer(signal, sampling_rate=256.0)
results = optimizer.optimize(
    n_components=5,
    use_coarse_sampling=True,
    n_refinement_candidates=10
)

• n_levels: Discretization resolution (default: 32)
• beta: Sampling temperature (default: 0.5-1.0)
• n_samples: Number of MCMC samples (default: 5000)
• n_warmup: Burn-in iterations (default: 1000)
• n_refinement_candidates: Number of candidates for BFGS refinement (default: 10)

Decompose a signal into chirplet components.

Parameters

• signal: Input signal array
• sampling_rate: Sampling frequency (Hz)
• n_components: Maximum number of components
• residual_threshold: Stop when residual < threshold * signal energy
• use_thrml: Whether to use THRML sampling (vs pure BFGS)

Returns: (components, parameters, residual)

Dataclass containing chirplet parameters

• tc: Time center (samples)
• fc: Frequency center (Hz)
• logDt: Log duration scale
• c: Chirp rate (Hz/s)

The Adaptive Chirplet Transform uses four continuous parameters to define Gaussian chirplet atoms.

$$g_{t_c,f_c,\log\Delta t,c}(t) = \frac{1}{\sqrt{2\pi}\Delta t} \exp\left(-\frac{1}{2}\left(\frac{t-t_c}{\Delta t}\right)^2\right) \exp\left(j2\pi\left[c(t-t_c)^2 + f_c(t-t_c)\right]\right)$$

Note: Some references use $\exp(-\frac{1}{2}(t/\Delta t)^2)$ for the Gaussian envelope, but this would center the atom at $t=0$ regardless of $t_c$.

Components

• Gaussian envelope: $\frac{1}{\sqrt{2\pi}\Delta t} \exp\left(-\frac{1}{2}\left(\frac{t-t_c}{\Delta t}\right)^2\right)$ provides time localization centered at $t_c$
• Chirp modulation: $\exp\left(j2\pi\left[c(t-t_c)^2 + f_c(t-t_c)\right]\right)$ encodes frequency sweep
  • Linear term $f_c(t-t_c)$: base frequency
  • Quadratic term $c(t-t_c)^2$: frequency chirp rate

The implementation uses a hybrid coarse-to-fine optimization strategy.

1. THRML Coarse Sampling: Explores discrete parameter space (32 levels per parameter) using probabilistic graphical model with 20 binary spin nodes. Samples 5000 configurations to find promising regions.
2. BFGS Refinement: Gradient-based optimization from top THRML candidates (typically 10 candidates refined with L-BFGS).
3. Matching Pursuit: Extracts components iteratively from residual signal until convergence threshold is met.

THRML provides maximum benefit when

• Highly multimodal signals (overlapping chirps with similar characteristics)
• Noisy measurements (SNR < 10 dB) where gradient estimates are unreliable

Skip THRML when,

• Real-time processing (< 10ms latency) - use pure BFGS
• High-SNR signals (> 30 dB) with well-separated components

• Discretization: 32 levels (5 bits) per parameter provides decent balance between resolution and tractability
• Sampling: Requires 500-1000 warmup iterations for convergence
• Memory: On-the-fly chirplet generation recommended for large signals to avoid 4GB+ GPU memory usage

The reference C++ implementation uses exhaustive dictionary search

// GEMV: scores = A^T * signal  (600k atoms × signal length)
act::blas::gemv_colmajor_trans(m, n, alpha, A, x, ...);
int best_idx = cblas_isamax(n, scores, 1);  // Find max correlation

This implementation uses probabilistic sampling via THRML

# Sample parameter space probabilistically
sampled_params = sample_chirplet_parameters(...)  # 3000 samples via MCMC
correlations = [compute_correlation(residual, generate_chirplet(p)) 
                for p in sampled_params]
top_candidates = top_k(correlations)  # Select best 10-12

However, the search strategy is fundamentally different

• C++ ACT: Exhaustive dictionary search (optimized via BLAS/GPU)
• Python THRML: Probabilistic sampling (optimized via THRML+BFGS hybrid)

The Python implementation prioritizes sample efficiency (fewer evaluations) over raw throughput (GPU-accelerated exhaustive search), making it suitable for exploration and research rather than real-time processing.

• THRML: https://github.com/extropic-ai/thrml
• ACT Implementation: https://byron-the-bulb.github.io/act/