# 1D Fourier transformΒΆ

[ ]:

import numpy as np
import matplotlib.pyplot as plt

from deconvoluted import fourier_transform



Suppose we want to compute the 1D fourier transform $$F(\nu)$$ of a function $$f(t)$$. Let us generate a signal which is a superposition of a signal with $$\nu_1 = 1$$ Hz and $$\nu_2 = 3$$ Hz:

[9]:

t = np.linspace(0, 20, 201)  # 20 seconds
nu_1 = 1
nu_2 = 3
f_t = np.sin(nu_1 * 2 * np.pi * t) + np.sin(nu_2 * 2 * np.pi * t)

plt.plot(t, f_t)
plt.xlabel(r'$t$ / s')
plt.ylabel(r'$f(t)$')
plt.show()



Taking the transform is now simply a matter of calling fourier_transform:

[10]:

F_nu, nu = fourier_transform(f_t, t)

[11]:

plt.plot(nu, F_nu)
plt.xlabel(r'$\nu$ / $s^{-1}$')
plt.ylabel(r'$F(\nu)$')
plt.show()


As expected, we find resonances at $$\nu_1 = 1$$ Hz and $$\nu_2 = 3$$ Hz.

We could also perform the transform using angular frequency instead:

[12]:

F_omega, omega = fourier_transform(f_t, t, convention=(1, -1))

[13]:

plt.plot(omega, F_omega)
plt.xlabel(r'$\omega$ / rad $s^{-1}$')
plt.ylabel(r'$F(\omega)$')
plt.show()


Now our resonances are at $$\omega_1 = 2 \pi$$ and $$\omega_2 = 6 \pi$$ instead.