数值拉普拉斯变换python

Question

I have a time series of experimental data x = x(t) in two numpy arrays, x for the observable and t for the time values of the observations. Is there a numpy function or a way that can evaluate the laplace transform of the timeseries? Thank you in advance.

Answer 1

I think you should have to consider the Laplace Transform of f(x) as the Fourier Transform of Gamma(x)f(x)e^(bx), in which Gamma is a step function that delete the negative part of the integral and e^(bx) constitute the real part of the complex exponential. There is a well known algorithm for Fourier Transform known as "Fast Fourier Transform" (FFT), for which you can find a lot of tutorials on both Python and Matlab websites.

Here I give you a short code that calculate the Fourier transform of a step function such as y = 0 for |x| > 1 y = 1 for |x| < 1

for which the Fourier Transform can be analitically calculated as sin(pi x) / (pi x).

import matplotlib.pyplot as plt
import scipy
from scipy.fftpack import fftshift
import numpy as np

x = np.arange(-3, 3, 0.01)
y = np.zeros(len(x))
y[150:450] = 1
plt.plot(x, y) # plot of the step function

yShift = fftshift(y) # shift of the step function
Fourier = scipy.fft(yShift) # Fourier transform of y implementing the FFT
Fourier = fftshift(Fourier) # inverse shift of the Fourier Transform
plt.plot(Fourier) # plot of the Fourier transform

Note that before and after applying the Fast Fourier Transform you have to use the fftshift command that provide a shift of the left side of the plot to the right side and viceversa. This is not the complete answer to your question, but I believe that is a good start.

Answer 2

You may use the Trapezoidal rule to calculate numerically the integral for the Laplace transform. One paper which describes this method is Edward H. Hellen: Padé –Laplace analysis of signal averaged voltage decays obtained from a simple circuit (Equation 2)

Notes:

1) "The summation approximates the integral in Eq. 1 [in the article] as long as the data are close to zero by the last data point."

2) "A good choice for p0 is the inverse of the time it takes for the data to decay to 1/2 its initial value."

3) Since your data is not spaced equally, I would use interpolation (from scipy import interpolate) for the data first. The Trepozoidal rule needs equally spaced data.