import numpy as np
import matplotlib.pyplot as plt

# on fabrique un signal sinusoïdal exponentiellement amorti
tau = 10.
f0 = 1.
omega0 = 2*np.pi*f0
t = np.arange(0.,50.,0.01)
x = np.sin(omega0*t)*np.exp(-t/tau)

# on l'affiche
plt.figure()
plt.plot(t,x)
plt.xlabel("t")
plt.ylabel("x")
plt.show()

# on calcule la transformée de Fourier, puis le spectre
tf = np.fft.fft(x)
f = np.fft.fftfreq(n=t.shape[-1], d=np.mean(np.diff(t)))
psd = np.square(np.abs(tf))
psdn = psd/psd.max()

# on affiche le spectre
plt.figure()
plt.plot(f, psdn)
plt.xlabel("f")
plt.ylabel("TF[x]")
plt.xlim(0.,2.)
plt.show()