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Signal

Textbook guide to signal helpers shared by transport and spectroscopy: unbiased FFT autocorrelations, windowing, and frequency grids.

These are array primitives — not frame analyses. Spectra and Green–Kubo routes compose them with Fits (CumulativeTrapezoid, PowerSpectrum, …).

1. Autocorrelation via FFT (Wiener–Khinchin)

For a real series \(x_t\),

\[ C(\tau) = \frac{1}{N-\tau}\sum_{t=0}^{N-\tau-1} x_t\, x_{t+\tau} \]

(unbiased origin average). signal.acf_fft evaluates this in \(\mathcal{O}(N\log N)\). For multi-entity vector series \((T,N,3)\) prefer Acf().compute(...) which contracts components and averages entities consistently.

2. Windowing and spectral resolution

Truncating \(C(\tau)\) before it decays produces sinc ringing in frequency space. Apply Hann/Blackman (signal.apply_window) before the FFT. Resolution \(\Delta\tilde\nu \sim 1/(c T_\mathrm{ACF})\); Nyquist \(\tilde\nu_\max \approx 16678/(\Delta t/\mathrm{fs})\) cm⁻¹.

3. Usage

import numpy as np
from molpy.compute import signal, Acf, PowerSpectrum

rng = np.random.default_rng(0)
x = np.ascontiguousarray(rng.standard_normal(256))
C = signal.acf_fft(x, max_lag=64)
C_w = signal.apply_window(C, "hann")
vdos = PowerSpectrum()(C_w, dt_fs=1.0)

vel = np.ascontiguousarray(rng.standard_normal((128, 20, 3)))
vacf = Acf().compute(vel, max_lag=32)

4. Pitfalls

  1. Biased (\(1/N\)) vs unbiased (\(1/(N-\tau)\)) normalization mismatch.
  2. No window on a truncated ACF → spectral ringing.
  3. Averaging atoms before correlating when you wanted per-particle memory.

See also