def check_IFT_continuous(a, t0, f0, method, f):
H = sinegauss_FT(f, t0, f0, a)
t, h = IFT_continuous(f, H, method=method)
assert_allclose(h, sinegauss(t, t0, f0, a), atol=1E-12)
def check_wavelets(t0, f0, Q, t):
h = sinegauss(t, t0, f0, Q)
f, H = FT_continuous(t, h)
H2 = sinegauss_FT(f, t0, f0, Q)
assert_allclose(H, H2, atol=1E-8)
def check_PSD_continuous(a, t0, f0, method, t):
h = sinegauss(t, t0, f0, a)
f, P = PSD_continuous(t, h, method=method)
assert_allclose(P, sinegauss_PSD(f, t0, f0, a), atol=1E-12)
# "Statistics, Data Mining, and Machine Learning in Astronomy" (2013)
# For more information, see http://astroML.github.com
import numpy as np
from matplotlib import pyplot as plt
from astroML.fourier import FT_continuous, IFT_continuous, sinegauss
#------------------------------------------------------------
# Set up the wavelets
t0 = 0
t = np.linspace(-0.4, 0.4, 10000)
f0 = np.array([5, 5, 10, 10])
Q = np.array([1, 0.5, 1, 0.5])
# compute wavelets all at once
W = sinegauss(t, t0, f0[:, None], Q[:, None])
#------------------------------------------------------------
# Plot the wavelets
fig = plt.figure()
fig.subplots_adjust(hspace=0.05, wspace=0.05)
# in each panel, plot and label a different wavelet
for i in range(4):
ax = fig.add_subplot(221 + i)
ax.plot(t, W[i].real, '-k')
ax.plot(t, W[i].imag, '--k')
ax.text(0.02, 0.98, "$f_0 = %i$\n$Q = %.1f$" % (f0[i], Q[i]),
ha='left', va='top', transform=ax.transAxes, size=14)
def check_FT_continuous(a, t0, f0, method, t):
h = sinegauss(t, t0, f0, a)
f, H = FT_continuous(t, h, method=method)
assert_allclose(H, sinegauss_FT(f, t0, f0, a), atol=1E-12)
# you can set usetex to False. from astroML.plotting import setup_text_plots setup_text_plots(fontsize=8, usetex=True) #------------------------------------------------------------ # Choose parameters for the wavelet N = 10000 t0 = 5 f0 = 2 Q = 2 #------------------------------------------------------------ # Compute the wavelet on a grid of times Dt = 0.01 t = t0 + Dt * (np.arange(N) - N / 2) h = sinegauss(t, t0, f0, Q) #------------------------------------------------------------ # Approximate the continuous Fourier Transform f, H = FT_continuous(t, h) rms_err = np.sqrt(np.mean(abs(H - sinegauss_FT(f, t0, f0, Q)) ** 2)) #------------------------------------------------------------ # Plot the results fig = plt.figure(figsize=(5, 3.75)) fig.subplots_adjust(hspace=0.25) # plot the wavelet ax = fig.add_subplot(211) ax.plot(t, h.real, '-', c='black', label='$Re[h]$', lw=1)
from scipy import fftpack from astroML.fourier import FT_continuous, sinegauss, sinegauss_FT #------------------------------------------------------------ # Choose parameters for the wavelet N = 10000 t0 = 5 f0 = 2 Q = 2 #------------------------------------------------------------ # Compute the wavelet on a grid of times Dt = 0.01 t = t0 + Dt * (np.arange(N) - N / 2) h = sinegauss(t, t0, f0, Q) #------------------------------------------------------------ # Approximate the continuous Fourier Transform f, H = FT_continuous(t, h) rms_err = np.sqrt(np.mean(abs(H - sinegauss_FT(f, t0, f0, Q))**2)) #------------------------------------------------------------ # Plot the results fig = plt.figure() fig.subplots_adjust(hspace=0.25) # plot the wavelet ax = fig.add_subplot(211) ax.plot(t, h.real, '-', c='black', label='$Re[h]$', lw=1)
def test_FT_continuous(a, t0, f0, method):
t = np.linspace(-9, 10, 10000)
h = sinegauss(t, t0, f0, a)
f, H = FT_continuous(t, h, method=method)
assert_allclose(H, sinegauss_FT(f, t0, f0, a), atol=1E-12)
def test_PSD_continuous(a, t0, f0, method):
t = np.linspace(-9, 10, 10000)
h = sinegauss(t, t0, f0, a)
f, P = PSD_continuous(t, h, method=method)
assert_allclose(P, sinegauss_PSD(f, t0, f0, a), atol=1E-12)
def check_IFT_continuous(a, t0, f0, method, f):
H = sinegauss_FT(f, t0, f0, a)
t, h = IFT_continuous(f, H, method=method)
assert_allclose(h, sinegauss(t, t0, f0, a), atol=1E-12)
def test_wavelets(t0, f0, Q):
t = np.linspace(-10, 10, 10000)
h = sinegauss(t, t0, f0, Q)
f, H = FT_continuous(t, h)
H2 = sinegauss_FT(f, t0, f0, Q)
assert_allclose(H, H2, atol=1E-8)
def check_wavelets(t0, f0, Q, t):
h = sinegauss(t, t0, f0, Q)
f, H = FT_continuous(t, h)
H2 = sinegauss_FT(f, t0, f0, Q)
assert_allclose(H, H2, atol=1E-8)
def check_PSD_continuous(a, t0, f0, method, t):
h = sinegauss(t, t0, f0, a)
f, P = PSD_continuous(t, h, method=method)
assert_allclose(P, sinegauss_PSD(f, t0, f0, a), atol=1E-12)
def check_FT_continuous(a, t0, f0, method, t):
h = sinegauss(t, t0, f0, a)
f, H = FT_continuous(t, h, method=method)
assert_allclose(H, sinegauss_FT(f, t0, f0, a), atol=1E-12)
setup_text_plots(fontsize=8, usetex=True) #------------------------------------------------------------ # Sample the function: localized noise np.random.seed(0) N = 1024 t = np.linspace(-5, 5, N) x = np.ones(len(t)) h = np.random.normal(0, 1, len(t)) h *= np.exp(-0.5 * (t / 0.5)**2) #------------------------------------------------------------ # Compute an example wavelet W = sinegauss(t, 0, 1.5, Q=1.0) #------------------------------------------------------------ # Compute the wavelet PSD f0 = np.linspace(0.5, 7.5, 100) wPSD = wavelet_PSD(t, h, f0, Q=1.0) #------------------------------------------------------------ # Plot the results fig = plt.figure(figsize=(5, 5)) fig.subplots_adjust(hspace=0.05, left=0.12, right=0.95, bottom=0.08, top=0.95) # First panel: the signal ax = fig.add_subplot(311) ax.plot(t, h, '-k', lw=1) ax.text(0.02,
def check_IFT_continuous(a, t0, f0, method):
f = np.linspace(-9, 10, 10000)
H = sinegauss_FT(f, t0, f0, a)
t, h = IFT_continuous(f, H, method=method)
assert_allclose(h, sinegauss(t, t0, f0, a), atol=1E-12)
FT_continuous, IFT_continuous, sinegauss, sinegauss_FT, wavelet_PSD
#------------------------------------------------------------
# Sample the function: localized noise
np.random.seed(0)
N = 1024
t = np.linspace(-5, 5, N)
x = np.ones(len(t))
h = np.random.normal(0, 1, len(t))
h *= np.exp(-0.5 * (t / 0.5) ** 2)
#------------------------------------------------------------
# Compute an example wavelet
W = sinegauss(t, 0, 1.5, Q=1.0)
#------------------------------------------------------------
# Compute the wavelet PSD
f0 = np.linspace(0.5, 7.5, 100)
wPSD = wavelet_PSD(t, h, f0, Q=1.0)
#------------------------------------------------------------
# Plot the results
fig = plt.figure(figsize=(8, 8))
fig.subplots_adjust(hspace=0.05, left=0.12, right=0.95, bottom=0.08, top=0.95)
# First panel: the signal
ax = fig.add_subplot(311)
ax.plot(t, h, '-k', lw=1)
ax.text(0.02, 0.95, ("Input Signal:\n"
def compute_Gaussian():
from astroML.fourier import\
FT_continuous, IFT_continuous, sinegauss, sinegauss_FT, wavelet_PSD
#----------------------------------------------------------------------
# This function adjusts matplotlib settings for a uniform feel in the textbook.
# Note that with usetex=True, fonts are rendered with LaTeX. This may
# result in an error if LaTeX is not installed on your system. In that case,
# you can set usetex to False.
from astroML.plotting import setup_text_plots
setup_text_plots(fontsize=8, usetex=False)
#------------------------------------------------------------
# Sample the function: localized noise
np.random.seed(0)
N = 1024
t = np.linspace(-5, 5, N)
x = np.ones(len(t))
h = np.random.normal(0, 1, len(t))
h *= np.exp(-0.5 * (t / 0.5) ** 2)
#------------------------------------------------------------
# Compute an example wavelet
W = sinegauss(t, 0, 1.5, Q=1.0)
#------------------------------------------------------------
# Compute the wavelet PSD
f0 = np.linspace(0.5, 7.5, 100)
wPSD = wavelet_PSD(t, h, f0, Q=1.0)
#------------------------------------------------------------
# Plot the results
fig = plt.figure(figsize=(5, 5))
fig.subplots_adjust(hspace=0.05, left=0.12, right=0.95, bottom=0.08, top=0.95)
# First panel: the signal
ax = fig.add_subplot(311)
ax.plot(t, h, '-k', lw=1)
ax.text(0.02, 0.95, ("Input Signal:\n"
"Localized Gaussian noise"),
ha='left', va='top', transform=ax.transAxes)
ax.set_xlim(-4, 4)
ax.set_ylim(-2.9, 2.9)
ax.xaxis.set_major_formatter(plt.NullFormatter())
ax.set_ylabel('$h(t)$')
# Second panel: an example wavelet
ax = fig.add_subplot(312)
ax.plot(t, W.real, '-k', label='real part', lw=1)
ax.plot(t, W.imag, '--k', label='imag part', lw=1)
ax.text(0.02, 0.95, ("Example Wavelet\n"
"$t_0 = 0$, $f_0=1.5$, $Q=1.0$"),
ha='left', va='top', transform=ax.transAxes)
ax.text(0.98, 0.05,
(r"$w(t; t_0, f_0, Q) = e^{-[f_0 (t - t_0) / Q]^2}"
"e^{2 \pi i f_0 (t - t_0)}$"),
ha='right', va='bottom', transform=ax.transAxes)
ax.legend(loc=1)
ax.set_xlim(-4, 4)
ax.set_ylim(-1.4, 1.4)
ax.set_ylabel('$w(t; t_0, f_0, Q)$')
ax.xaxis.set_major_formatter(plt.NullFormatter())
# Third panel: the spectrogram
ax = plt.subplot(313)
ax.imshow(wPSD, origin='lower', aspect='auto', cmap=plt.cm.jet,
extent=[t[0], t[-1], f0[0], f0[-1]])
ax.text(0.02, 0.95, ("Wavelet PSD"), color='w',
ha='left', va='top', transform=ax.transAxes)
ax.set_xlim(-4, 4)
ax.set_ylim(0.5, 7.5)
ax.set_xlabel('$t$')
ax.set_ylabel('$f_0$')
plt.show()
