bicfit

bicfit is a lightweight Python library for fitting damped cosine and exponential functions. It combines the Generalized Pencil-of-Function method ¹ (to generate robust initial guesses) with standard optimization routines (to refine the fit).

This makes it well-suited for researchers and practitioners dealing with decaying oscillations, resonance signals, or exponential relaxations in noisy data.

Currently, bicfit supports fitting:

• Complex exponential decay $f(t) = x_0 + \sum_k A_k \exp((j\omega_k - \kappa_k) t)$
• Damped cosine $f(t) = x_0 + \sum_k A_k \exp(-\kappa_k t) \cos(\omega_k t + \phi_k)$
• Exponential decay $f(t) = x_0 + \sum_k A_k \exp(-\kappa_k t)$

Installation

You can install bicfit via pip:

pip install bicfit

Usage

Start with

import numpy as np

w = 0.2
kappa = 0.05
offset = 1 + 1j
sigma_noise = 0.03
n_points = 100

Complex exponential decay

from bicfit import fit_complex_exponential

times = np.linspace(0, 150, n_points)
noise = np.random.normal(0, sigma_noise, n_points) + 1j * np.random.normal(0, sigma_noise, n_points)
signal = offset + np.exp((1j * w - kappa) * times) + noise

result = fit_complex_exponential(times, signal)
result.plot()

# you can call `result` directly to evaluate it
fitted_signal = result(times)

result

In this example, result is an object of type ComplexResult that has the following attributes:

• offset: the offset of the data
• amplitudes: the amplitudes of the modes
• pulsations: the pulsations of the modes
• decay_rates: the decay rates of the modes
• times: the times of the data
• signal: the original signal
• frequencies: (read-only) the frequencies of the modes, computed as pulsations / (2 * np.pi)

As a convenience, the result object also has a plot() method that plots the original signal and the fitted function. It also exposes a modes property that returns a list of Mode objects, each containing the amplitude pulsation and decay rate of the mode.

The ComplexResult and ComplexMode classes (as all Result and Mode classes) are callable so you can use them to evaluate the fitted function at any time.

Damped cosine

from bicfit import fit_damped_cosine

signal = signal.real

result = fit_damped_cosine(times, signal)
result.plot()
result

Here the result is an object of type DampedCosineResult that has similar attributes to the ComplexResult class:

• offset: the offset of the data
• amplitudes: the amplitudes of the modes
• phases: the phases of the modes
• pulsations: the pulsations of the modes
• decay_rates: the decay rates of the modes
• times: the times of the data
• signal: the original signal
• frequencies: (read-only) the frequencies of the modes, computed as pulsations / (2 * np.pi)

The DampedCosineResult object also has a plot() method that plots the original signal and the fitted function. It exposes a modes property that returns a list of DampedCosineMode objects, each containing the amplitude, pulsation, decay rate, and phase of the mode.

Exponential decay

from bicfit import fit_exponential_decay

noise = np.random.normal(0, sigma_noise, n_points) + 1j * np.random.normal(0, sigma_noise, n_points)
signal = offset + np.exp(- kappa * times) + noise

result = fit_exponential_decay(times, signal, is_complex=True)
result.plot()
result

⚠️ Fitting an exponential can be tricky in general. If you do not have data long enough that the exponential plateaus, the fit will not work well. If you know where the exponential plateaus, you do not need as much data but bicfit does not exploit this knowledge yet.

Multiple modes

a1, a2 = 1.0, 0.5
w1, w2 = 0.2, 0.4
kappa1, kappa2 = 0.05, 0.01

noise = np.random.normal(0, sigma_noise, n_points) + 1j * np.random.normal(0, sigma_noise, n_points)
signal = offset + a1 * np.exp((1j * w1 - kappa1) * times) + a2 * np.exp((1j * w2 - kappa2) * times) + noise

result = fit_complex_exponential(times, signal, n_modes=2, post_fit=True)
result.plot()
result

Here the result is an object of type ExponentialDecayResultthat has similar attributes to the ComplexResult class:

• offset: the offset of the data
• amplitudes: the amplitudes of the modes
• pulsations: the decay rates of the modes
• times: the times of the data
• signal: the original signal

It also has the plot() function, modes attribute and is callable like the other result classes.

Fit with no offset

If you know that your data has no offset, you can set the post_fit parameter to bicfit.NoOffset() to force the fit to have no offset.

result = fit_complex_exponential(times, signal, post_fit=bicfit.NoOffset())

Coming features

There are a few features that are not implemented yet but could be in the future, if there is a demand for them:

• exploiting knowledge of the plateau of an exponential decay
• fitting non uniformly sampled data
• port fit functions to JAX
• [performance] amplitude fitting is probably overkill and could be done from the pencil eigenvectors

Development

You can run tests using pytest:

uv sync --dev
uv run pytest .

Feedback

We welcome any feedback, suggestions, or bug reports to help improve bicfit.

If you find a case where the fit fails, please include your data in CSV format for reproducibility.

Footnotes

1. Generalized Pencil-of-Function Method for Extracting Poles of an EM System from Its Transient Response Hua and Sarkar (IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 37, NO. 2, FEBRUARY 1989) ↩