def test_periodogram_spectral_normalization():
"""
Check that the spectral estimators are normalized in the
correct Watts/Hz fashion
"""
x = np.random.randn(1024)
f1, Xp1 = tsa.periodogram(x)
f2, Xp2 = tsa.periodogram(x, Fs=100)
f3, Xp3 = tsa.periodogram(x, N=2**12)
p1 = np.sum(Xp1) * 2 * np.pi / 2**10
p2 = np.sum(Xp2) * 100 / 2**10
p3 = np.sum(Xp3) * 2 * np.pi / 2**12
npt.assert_( np.abs(p1 - p2) < 1e-14,
'Inconsistent frequency normalization in periodogram (1)' )
npt.assert_( np.abs(p3 - p2) < 1e-8,
'Inconsistent frequency normalization in periodogram (2)' )
td_var = np.var(x)
# assure that the estimators are at least in the same
# order of magnitude as the time-domain variance
npt.assert_( np.abs(np.log10(p1/td_var)) < 1,
'Incorrect frequency normalization in periodogram' )
# check the freq vector while we're here
npt.assert_( f2.max() == 50, 'Periodogram returns wrong frequency bins' )
def test_periodogram_spectral_normalization():
"""
Check that the spectral estimators are normalized in the
correct Watts/Hz fashion
"""
x = np.random.randn(1024)
f1, Xp1 = tsa.periodogram(x)
f2, Xp2 = tsa.periodogram(x, Fs=100)
f3, Xp3 = tsa.periodogram(x, N=2**12)
p1 = np.sum(Xp1) * 2 * np.pi / 2**10
p2 = np.sum(Xp2) * 100 / 2**10
p3 = np.sum(Xp3) * 2 * np.pi / 2**12
nt.assert_true(
np.abs(p1 - p2) < 1e-14,
'Inconsistent frequency normalization in periodogram (1)')
nt.assert_true(
np.abs(p3 - p2) < 1e-8,
'Inconsistent frequency normalization in periodogram (2)')
td_var = np.var(x)
# assure that the estimators are at least in the same
# order of magnitude as the time-domain variance
nt.assert_true(
np.abs(np.log10(p1 / td_var)) < 1,
'Incorrect frequency normalization in periodogram')
# check the freq vector while we're here
nt.assert_true(f2.max() == 50, 'Periodogram returns wrong frequency bins')
def test_periodogram():
arsig, _, _ = ut.ar_generator(N=512)
avg_pwr = (arsig * arsig.conjugate()).mean()
f, psd = tsa.periodogram(arsig, N=2048)
df = 2. * np.pi / 2048
avg_pwr_est = np.trapz(psd, dx=df)
npt.assert_almost_equal(avg_pwr, avg_pwr_est, decimal=1)
def test_periodogram():
arsig, _, _ = ut.ar_generator(N=512)
avg_pwr = (arsig * arsig.conjugate()).mean()
f, psd = tsa.periodogram(arsig, N=2048)
df = 2. * np.pi / 2048
avg_pwr_est = np.trapz(psd, dx=df)
npt.assert_almost_equal(avg_pwr, avg_pwr_est, decimal=1)
def test_periodogram():
arsig,_,_ = ut.ar_generator(N=512)
avg_pwr = (arsig*arsig.conjugate()).mean()
f, psd = tsa.periodogram(arsig, N=2048)
# for efficiency, let's leave out the 2PI in the numerator and denominator
# for the following integral
dw = 1./2048
avg_pwr_est = np.trapz(psd, dx=dw)
npt.assert_almost_equal(avg_pwr, avg_pwr_est, decimal=1)
def test_periodogram():
arsig, _, _ = ut.ar_generator(N=512)
avg_pwr = (arsig * arsig.conjugate()).mean()
f, psd = tsa.periodogram(arsig, N=2048)
# for efficiency, let's leave out the 2PI in the numerator and denominator
# for the following integral
dw = 1. / 2048
avg_pwr_est = np.trapz(psd, dx=dw)
npt.assert_almost_equal(avg_pwr, avg_pwr_est, decimal=1)
def test_periodogram():
"""Test some of the inputs to periodogram """
arsig, _, _ = utils.ar_generator(N=1024)
Sk = fftpack.fft(arsig)
f1, c1 = tsa.periodogram(arsig)
f2, c2 = tsa.periodogram(arsig, Sk=Sk)
npt.assert_equal(c1, c2)
# Check that providing a complex signal does the right thing
# (i.e. two-sided spectrum):
N = 1024
r, _, _ = utils.ar_generator(N=N)
c, _, _ = utils.ar_generator(N=N)
arsig = r + c * scipy.sqrt(-1)
f, c = tsa.periodogram(arsig)
npt.assert_equal(f.shape[0], N) # Should be N, not the one-sided N/2 + 1
def test_periodogram():
"""Test some of the inputs to periodogram """
arsig, _, _ = utils.ar_generator(N=1024)
Sk = np.fft.fft(arsig)
f1, c1 = tsa.periodogram(arsig)
f2, c2 = tsa.periodogram(arsig, Sk=Sk)
npt.assert_equal(c1, c2)
# Check that providing a complex signal does the right thing
# (i.e. two-sided spectrum):
N = 1024
r, _, _ = utils.ar_generator(N=N)
c, _, _ = utils.ar_generator(N=N)
arsig = r + c * scipy.sqrt(-1)
f, c = tsa.periodogram(arsig)
npt.assert_equal(f.shape[0], N) # Should be N, not the one-sided N/2 + 1
def periodogram(self):
"""
This is the spectrum estimated as the FFT of the time-series
Returns
-------
(f,spectrum): f is an array with the frequencies and spectrum is the
complex-valued FFT.
"""
return tsa.periodogram(self.input.data, Fs=self.input.sampling_rate)
def periodogram(self):
"""
This is the spectrum estimated as the FFT of the time-series
Returns
-------
(f,spectrum): f is an array with the frequencies and spectrum is the
complex-valued FFT.
"""
return tsa.periodogram(self.input.data, Fs=self.input.sampling_rate)
def test_hermitian_periodogram_csd():
"""
Make sure CSD matrices returned by various methods have
Hermitian symmetry.
"""
sig = np.random.randn(4, 256)
_, csd1 = tsa.periodogram_csd(sig)
for i in range(4):
for j in range(i + 1):
xc1 = csd1[i, j]
xc2 = csd1[j, i]
npt.assert_equal(xc1,
xc2.conj(),
err_msg='Periodogram CSD not Hermitian')
_, psd = tsa.periodogram(sig)
for i in range(4):
npt.assert_almost_equal(
psd[i],
csd1[i, i].real,
err_msg='Periodgram CSD diagonal inconsistent with real PSD')
def test_hermitian_periodogram_csd():
"""
Make sure CSD matrices returned by various methods have
Hermitian symmetry.
"""
sig = np.random.randn(4,256)
_, csd1 = tsa.periodogram_csd(sig)
for i in range(4):
for j in range(i+1):
xc1 = csd1[i,j]
xc2 = csd1[j,i]
npt.assert_equal(
xc1, xc2.conj(), err_msg='Periodogram CSD not Hermitian'
)
_, psd = tsa.periodogram(sig)
for i in range(4):
npt.assert_almost_equal(
psd[i], csd1[i,i].real,
err_msg='Periodgram CSD diagonal inconsistent with real PSD'
)
def get_psd_v2(data, fs):
freq, psd = tsa.periodogram(data,fs,normalize=True)
return freq, psd
"""
psd *= 2
dB(psd, psd)
"""
We begin by using the naive periodogram function (:func:`tsa.periodogram` in
order to calculate the PSD and compare that to the true PSD calculated above.
"""
freqs, d_psd = tsa.periodogram(ar_seq)
dB(d_psd, d_psd)
fig03 = plot_spectral_estimate(freqs, psd, (d_psd,), elabels=("Periodogram",))
"""
.. image:: fig/multi_taper_spectral_estimation_03.png
Next, we use Welch's periodogram, by applying :func:`tsa.get_spectra`. Note
that we explicitely provide the function with a 'method' dict, which specifies
the method used in order to calculate the PSD, but the default method is 'welch'.
"""
This is a one-sided spectrum, so we double the power:
"""
psd *= 2
dB(psd, psd)
"""
We begin by using the naive periodogram function (:func:`tsa.periodogram` in
order to calculate the PSD and compare that to the true PSD calculated above.
"""
freqs, d_psd = tsa.periodogram(ar_seq)
dB(d_psd, d_psd)
fig03 = plot_spectral_estimate(freqs,
psd, (d_psd, ),
elabels=("Periodogram", ))
"""
.. image:: fig/multi_taper_spectral_estimation_03.png
Next, we use Welch's periodogram, by applying :func:`tsa.get_spectra`. Note
that we explicitly provide the function with a 'method' dict, which specifies
the method used in order to calculate the PSD, but the default method is 'welch'.
"""
ar_seq = foo["arr_0"]
nz = foo["arr_1"]
alpha = foo["arr_2"]
else:
ar_seq, nz, alpha = utils.ar_generator(N=N, drop_transients=10)
ar_seq -= ar_seq.mean()
np.savez("example_arrs", ar_seq, nz, alpha)
# --- True SDF
fgrid, hz = alg.my_freqz(1.0, a=np.r_[1, -alpha], Nfreqs=N)
sdf = (hz * hz.conj()).real
# onesided spectrum, so double the power
sdf[1:-1] *= 2
dB(sdf, sdf)
# --- Direct Spectral Estimator
freqs, d_sdf = alg.periodogram(ar_seq)
dB(d_sdf, d_sdf)
# --- Welch's Overlapping Periodogram Method via mlab
mlab_sdf, mlab_freqs = pp.mlab.psd(ar_seq, NFFT=N)
mlab_freqs *= np.pi / mlab_freqs.max()
mlab_sdf = mlab_sdf.squeeze()
dB(mlab_sdf, mlab_sdf)
### Taper Bandwidth Adjustments
NW = 4
# --- Regular Multitaper Estimate
f, sdf_mt, nu = alg.multi_taper_psd(ar_seq, width=NW, adaptive=False, jackknife=False)
dB(sdf_mt, sdf_mt)