Vectfit: Python Vector Fitting for Rational Function Approximation

Vectfit is a Python implementation of the Fast Relaxed Vector Fitting algorithm originally developed by Gustavsen & Semlyen, commonly used in electrical engineering and physics to approximate frequency-domain responses with rational functions. This repository, maintained by the Caltech Experimental Gravity group, provides a robust and accessible implementation suitable for graduate-level research and practical applications.

What is Vector Fitting?

Vector Fitting (Vectfit) approximates a complex-valued frequency-domain function ( H(s) ) by a rational function:

[ H(s) \approx \sum_{m=1}^{N} \frac{r_m}{s - p_m} + d + s e ]

where:

• ( p_m ): Poles (complex frequencies).
• ( r_m ): Residues corresponding to each pole.
• ( d, e ): Constant and linear terms for proper/non-proper rational approximations.

Vectfit is efficient for modeling systems characterized by measured frequency response data, including electronic circuits, mechanical resonances, optical cavities, and gravitational-wave detectors.

Features of this Implementation

• Pure-Python implementation (no compiled dependencies).
• Handles proper and non-proper rational approximations.
• Automatic initial pole placement, pole relocation, and iterative fitting.
• Supports weighted fitting to prioritize frequency ranges.
• Well-documented code and easy-to-follow examples.

Repository Structure

.
├── vectfit.py                # Main algorithm implementation
├── test_vectfit_auto.py      # Comprehensive example script
├── test_vectfit_NPRO.py      # Example for non-proper rational fits
├── test_vectfit_w_weight.py  # Weighted fitting example
├── data/                     # Example data files
├── Figures/                  # Output figures from example scripts
├── BodePlot.mplstyle         # Matplotlib style for clear Bode plots
├── LICENSE                   # GPL-2.0 License
└── README.md                 # This file

Getting Started

Requirements

• Python 3.x
• NumPy, SciPy, Matplotlib

Install with pip:

pip install numpy scipy matplotlib

Installation

Clone this repository to your local machine:

git clone https://github.com/CaltechExperimentalGravity/Vectfit.git
cd Vectfit

Quick Example: Fitting Data with Vectfit

Suppose you have measured frequency-domain data (freq, H_data) and want to fit it:

import numpy as np
import vectfit
import matplotlib.pyplot as plt

# Example frequency data (rad/s)
freq = np.linspace(1e2, 1e5, 500)

# Example measured response data
H_data = np.exp(-1j * freq * 1e-4) / (1 + 1j * freq * 1e-3)

# Initial pole guess (log-spaced complex poles)
initial_poles = vectfit.generate_initial_poles(freq, n_poles=10)

# Perform vector fitting
poles, residues, d, fit_result, rms_error = vectfit.vectfit(H_data, freq, initial_poles)

# Plot original vs fitted data
plt.figure()
plt.semilogx(freq, 20 * np.log10(np.abs(H_data)), label='Original Data')
plt.semilogx(freq, 20 * np.log10(np.abs(fit_result)), '--', label='Vectfit Approximation')
plt.xlabel('Frequency (rad/s)')
plt.ylabel('Magnitude (dB)')
plt.legend()
plt.title('Vector Fitting Example')
plt.grid(True, which='both')
plt.show()

Choosing Initial Poles

Selecting initial poles significantly affects convergence:

• Use evenly spaced complex poles spanning your data range.
• Utilize vectfit.generate_initial_poles(freq, n_poles) for convenience.

Weighted Fitting

Weights allow emphasis or suppression of specific frequency ranges:

weights = 1 / np.abs(H_data)  # Example weight emphasizing weaker signals
poles, residues, d, fit_result, rms_error = vectfit.vectfit(H_data, freq, initial_poles, weights=weights)

Non-Proper Rational Fitting

Non-proper rational fitting adds linear frequency dependence:

poles, residues, d, e, fit_result, rms_error = vectfit.vectfit(H_data, freq, initial_poles, n_polynomial=1)

n_polynomial=1 adds a linear frequency term (e) to the rational approximation.

Testing and Validation

Use included scripts to test implementation and learn usage patterns:

• Basic test: test_vectfit_auto.py
• Weighted fitting example: test_vectfit_w_weight.py
• Non-proper rational example: test_vectfit_NPRO.py

Run any of these scripts directly from your terminal:

python test_vectfit_auto.py

References

• Primary Papers:
  • B. Gustavsen and A. Semlyen, "Rational approximation of frequency domain responses by Vector Fitting," IEEE Trans. Power Delivery, vol. 14, no. 3, pp. 1052-1061, 1999.
  • B. Gustavsen, "Improving the pole relocating properties of vector fitting," IEEE Trans. Power Delivery, vol. 21, no. 3, pp. 1587-1592, 2006.

License

Distributed under the GPL-2.0 License.