def bandpass_timefreq(s, frequencies, sample_rate):
"""
Bandpass filter signal s at the given frequency bands, and then use the Hilber transform
to produce a complex-valued time-frequency representation of the bandpass filtered signal.
"""
freqs = sorted(frequencies)
tf_raw = np.zeros([len(frequencies), len(s)], dtype='float')
tf_freqs = list()
for k,f in enumerate(freqs):
#bandpass filter signal
if k == 0:
tf_raw[k, :] = lowpass_filter(s, sample_rate, f)
tf_freqs.append( (0.0, f) )
else:
tf_raw[k, :] = bandpass_filter(s, sample_rate, freqs[k-1], f)
tf_freqs.append( (freqs[k-1], f) )
#compute analytic signal
tf = hilbert(tf_raw, axis=1)
#print 'tf_raw.shape=',tf_raw.shape
#print 'tf.shape=',tf.shape
return np.array(tf_freqs),tf_raw,tf
def bandpass_timefreq(s, frequencies, sample_rate):
"""
Bandpass filter signal s at the given frequency bands, and then use the Hilber transform
to produce a complex-valued time-frequency representation of the bandpass filtered signal.
"""
freqs = sorted(frequencies)
tf_raw = np.zeros([len(frequencies), len(s)], dtype='float')
tf_freqs = list()
for k, f in enumerate(freqs):
#bandpass filter signal
if k == 0:
tf_raw[k, :] = lowpass_filter(s, sample_rate, f)
tf_freqs.append((0.0, f))
else:
tf_raw[k, :] = bandpass_filter(s, sample_rate, freqs[k - 1], f)
tf_freqs.append((freqs[k - 1], f))
#compute analytic signal
tf = hilbert(tf_raw, axis=1)
#print 'tf_raw.shape=',tf_raw.shape
#print 'tf.shape=',tf.shape
return np.array(tf_freqs), tf_raw, tf
def test_cross_psd(self):
np.random.seed(1234567)
sr = 1000.0
dur = 1.0
nt = int(dur * sr)
t = np.arange(nt) / sr
# create a simple signal
freqs = list()
freqs.extend(np.arange(8, 12))
freqs.extend(np.arange(60, 71))
freqs.extend(np.arange(130, 151))
s1 = np.zeros([nt])
for f in freqs:
s1 += np.sin(2 * np.pi * f * t)
s1 /= s1.max()
# create a noise corrupted, bandpassed filtered version of s1
noise = np.random.randn(nt) * 1e-1
# s2 = convolve1d(s1, filt, mode='mirror') + noise
s2 = bandpass_filter(s1, sample_rate=sr, low_freq=40., high_freq=90.)
s2 /= s2.max()
s2 += noise
# compute the signal's power spectrums
welch_freq1, welch_psd1 = welch(s1, fs=sr)
welch_freq2, welch_psd2 = welch(s2, fs=sr)
welch_psd_max = max(welch_psd1.max(), welch_psd2.max())
welch_psd1 /= welch_psd_max
welch_psd2 /= welch_psd_max
# compute the auto-correlation functions
lags = np.arange(-200, 201)
acf1 = correlation_function(s1, s1, lags, normalize=True)
acf2 = correlation_function(s2, s2, lags, normalize=True)
# compute the cross correlation functions
cf12 = correlation_function(s1, s2, lags, normalize=True)
coh12 = coherency(s1,
s2,
lags,
window_fraction=0.75,
noise_floor_db=100.)
# do an FFT shift to the lags and the window, otherwise the FFT of the ACFs is not equal to the power
# spectrum for some numerical reason
shift_lags = fftshift(lags)
if len(lags) % 2 == 1:
# shift zero from end of shift_lags to beginning
shift_lags = np.roll(shift_lags, 1)
acf1_shift = correlation_function(s1, s1, shift_lags)
acf2_shift = correlation_function(s2, s2, shift_lags)
# compute the power spectra from the auto-spectra
ps1 = fft(acf1_shift)
ps1_freq = fftfreq(len(acf1), d=1.0 / sr)
fi = ps1_freq > 0
ps1 = ps1[fi]
assert np.sum(
np.abs(ps1.imag) > 1e-8
) == 0, "Nonzero imaginary part for fft(acf1) (%d)" % np.sum(
np.abs(ps1.imag) > 1e-8)
ps1_auto = np.abs(ps1.real)
ps1_auto_freq = ps1_freq[fi]
ps2 = fft(acf2_shift)
ps2_freq = fftfreq(len(acf2), d=1.0 / sr)
fi = ps2_freq > 0
ps2 = ps2[fi]
assert np.sum(np.abs(ps2.imag) > 1e-8
) == 0, "Nonzero imaginary part for fft(acf2)"
ps2_auto = np.abs(ps2.real)
ps2_auto_freq = ps2_freq[fi]
assert np.sum(ps1_auto < 0) == 0, "negatives in ps1_auto"
assert np.sum(ps2_auto < 0) == 0, "negatives in ps2_auto"
# compute the cross spectral density from the correlation function
cf12_shift = correlation_function(s1, s2, shift_lags, normalize=True)
psd12 = fft(cf12_shift)
psd12_freq = fftfreq(len(cf12_shift), d=1.0 / sr)
fi = psd12_freq > 0
psd12 = np.abs(psd12[fi])
psd12_freq = psd12_freq[fi]
# compute the cross spectral density from the power spectra
psd12_welch = welch_psd1 * welch_psd2
psd12_welch /= psd12_welch.max()
# compute the coherence from the cross spectral density
cfreq,coherence,coherence_var,phase_coherence,phase_coherence_var,coh12_freqspace,coh12_freqspace_t = \
coherence_jn(s1, s2, sample_rate=sr, window_length=0.100, increment=0.050, return_coherency=True)
coh12_freqspace /= np.abs(coh12_freqspace).max()
# weight the coherence by one minus the normalized standard deviation
coherence_std = np.sqrt(coherence_var)
# cweight = coherence_std / coherence_std.sum()
# coherence_weighted = (1.0 - cweight)*coherence
coherence_weighted = coherence - coherence_std
coherence_weighted[coherence_weighted < 0] = 0
# compute the coherence from the fft of the coherency
coherence2 = fft(fftshift(coh12))
coherence2_freq = fftfreq(len(coherence2), d=1.0 / sr)
fi = coherence2_freq > 0
coherence2 = np.abs(coherence2[fi])
coherence2_freq = coherence2_freq[fi]
"""
plt.figure()
ax = plt.subplot(2, 1, 1)
plt.plot(ps1_auto_freq, ps1_auto*ps2_auto, 'c-', linewidth=2.0, alpha=0.75)
plt.plot(psd12_freq, psd12, 'g-', linewidth=2.0, alpha=0.9)
plt.plot(ps1_auto_freq, ps1_auto, 'k-', linewidth=2.0, alpha=0.75)
plt.plot(ps2_auto_freq, ps2_auto, 'r-', linewidth=2.0, alpha=0.75)
plt.axis('tight')
plt.legend(['denom', '12', '1', '2'])
ax = plt.subplot(2, 1, 2)
plt.plot(psd12_freq, coherence, 'b-')
plt.axis('tight')
plt.show()
"""
# normalize the cross-spectral density and power spectra
psd12 /= psd12.max()
ps_auto_max = max(ps1_auto.max(), ps2_auto.max())
ps1_auto /= ps_auto_max
ps2_auto /= ps_auto_max
# make some plots
plt.figure()
nrows = 2
ncols = 2
# plot the signals
ax = plt.subplot(nrows, ncols, 1)
plt.plot(t, s1, 'k-', linewidth=2.0)
plt.plot(t, s2, 'r-', alpha=0.75, linewidth=2.0)
plt.xlabel('Time (s)')
plt.ylabel('Signal')
plt.axis('tight')
# plot the spectra
ax = plt.subplot(nrows, ncols, 2)
plt.plot(welch_freq1, welch_psd1, 'k-', linewidth=2.0, alpha=0.85)
plt.plot(ps1_auto_freq, ps1_auto, 'k--', linewidth=2.0, alpha=0.85)
plt.plot(welch_freq2, welch_psd2, 'r-', alpha=0.75, linewidth=2.0)
plt.plot(ps2_auto_freq, ps2_auto, 'r--', linewidth=2.0, alpha=0.75)
plt.axis('tight')
plt.xlabel('Frequency (Hz)')
plt.ylabel('Power')
# plot the correlation functions
ax = plt.subplot(nrows, ncols, 3)
plt.axhline(0, c='k')
plt.plot(lags, acf1, 'k-', linewidth=2.0)
plt.plot(lags, acf2, 'r-', alpha=0.75, linewidth=2.0)
plt.plot(lags, cf12, 'g-', alpha=0.75, linewidth=2.0)
plt.plot(lags, coh12, 'b-', linewidth=2.0, alpha=0.75)
plt.plot(coh12_freqspace_t * 1e3,
coh12_freqspace,
'm-',
linewidth=1.0,
alpha=0.95)
plt.xlabel('Lag (ms)')
plt.ylabel('Correlation Function')
plt.axis('tight')
plt.ylim(-0.5, 1.0)
handles = custom_legend(['k', 'r', 'g', 'b', 'c'],
['acf1', 'acf2', 'cf12', 'coh12', 'coh12_f'])
plt.legend(handles=handles)
# plot the cross spectral density
ax = plt.subplot(nrows, ncols, 4)
handles = custom_legend(['g', 'k', 'b'],
['CSD', 'Coherence', 'Weighted'])
plt.axhline(0, c='k')
plt.axhline(1, c='k')
plt.plot(psd12_freq, psd12, 'g-', linewidth=3.0)
plt.errorbar(cfreq,
coherence,
yerr=np.sqrt(coherence_var),
fmt='k-',
ecolor='r',
linewidth=3.0,
elinewidth=5.0,
alpha=0.8)
plt.plot(cfreq, coherence_weighted, 'b-', linewidth=3.0, alpha=0.75)
plt.xlabel('Frequency (Hz)')
plt.ylabel('Cross-spectral Density/Coherence')
plt.legend(handles=handles)
"""
plt.figure()
plt.axhline(0, c='k')
plt.plot(lags, cf12, 'k-', alpha=1, linewidth=2.0)
plt.plot(lags, coh12, 'b-', linewidth=3.0, alpha=0.75)
plt.plot(coh12_freqspace_t*1e3, coh12_freqspace, 'r-', linewidth=2.0, alpha=0.95)
plt.xlabel('Lag (ms)')
plt.ylabel('Correlation Function')
plt.axis('tight')
plt.ylim(-0.5, 1.0)
handles = custom_legend(['k', 'b', 'r'], ['cf12', 'coh12', 'coh12_f'])
plt.legend(handles=handles)
"""
plt.show()
def test_cross_psd(self):
np.random.seed(1234567)
sr = 1000.0
dur = 1.0
nt = int(dur*sr)
t = np.arange(nt) / sr
# create a simple signal
freqs = list()
freqs.extend(np.arange(8, 12))
freqs.extend(np.arange(60, 71))
freqs.extend(np.arange(130, 151))
s1 = np.zeros([nt])
for f in freqs:
s1 += np.sin(2*np.pi*f*t)
s1 /= s1.max()
# create a noise corrupted, bandpassed filtered version of s1
noise = np.random.randn(nt)*1e-1
# s2 = convolve1d(s1, filt, mode='mirror') + noise
s2 = bandpass_filter(s1, sample_rate=sr, low_freq=40., high_freq=90.)
s2 /= s2.max()
s2 += noise
# compute the signal's power spectrums
welch_freq1,welch_psd1 = welch(s1, fs=sr)
welch_freq2,welch_psd2 = welch(s2, fs=sr)
welch_psd_max = max(welch_psd1.max(), welch_psd2.max())
welch_psd1 /= welch_psd_max
welch_psd2 /= welch_psd_max
# compute the auto-correlation functions
lags = np.arange(-200, 201)
acf1 = correlation_function(s1, s1, lags, normalize=True)
acf2 = correlation_function(s2, s2, lags, normalize=True)
# compute the cross correlation functions
cf12 = correlation_function(s1, s2, lags, normalize=True)
coh12 = coherency(s1, s2, lags, window_fraction=0.75, noise_floor_db=100.)
# do an FFT shift to the lags and the window, otherwise the FFT of the ACFs is not equal to the power
# spectrum for some numerical reason
shift_lags = fftshift(lags)
if len(lags) % 2 == 1:
# shift zero from end of shift_lags to beginning
shift_lags = np.roll(shift_lags, 1)
acf1_shift = correlation_function(s1, s1, shift_lags)
acf2_shift = correlation_function(s2, s2, shift_lags)
# compute the power spectra from the auto-spectra
ps1 = fft(acf1_shift)
ps1_freq = fftfreq(len(acf1), d=1.0/sr)
fi = ps1_freq > 0
ps1 = ps1[fi]
assert np.sum(np.abs(ps1.imag) > 1e-8) == 0, "Nonzero imaginary part for fft(acf1) (%d)" % np.sum(np.abs(ps1.imag) > 1e-8)
ps1_auto = np.abs(ps1.real)
ps1_auto_freq = ps1_freq[fi]
ps2 = fft(acf2_shift)
ps2_freq = fftfreq(len(acf2), d=1.0/sr)
fi = ps2_freq > 0
ps2 = ps2[fi]
assert np.sum(np.abs(ps2.imag) > 1e-8) == 0, "Nonzero imaginary part for fft(acf2)"
ps2_auto = np.abs(ps2.real)
ps2_auto_freq = ps2_freq[fi]
assert np.sum(ps1_auto < 0) == 0, "negatives in ps1_auto"
assert np.sum(ps2_auto < 0) == 0, "negatives in ps2_auto"
# compute the cross spectral density from the correlation function
cf12_shift = correlation_function(s1, s2, shift_lags, normalize=True)
psd12 = fft(cf12_shift)
psd12_freq = fftfreq(len(cf12_shift), d=1.0/sr)
fi = psd12_freq > 0
psd12 = np.abs(psd12[fi])
psd12_freq = psd12_freq[fi]
# compute the cross spectral density from the power spectra
psd12_welch = welch_psd1*welch_psd2
psd12_welch /= psd12_welch.max()
# compute the coherence from the cross spectral density
cfreq,coherence,coherence_var,phase_coherence,phase_coherence_var,coh12_freqspace,coh12_freqspace_t = \
coherence_jn(s1, s2, sample_rate=sr, window_length=0.100, increment=0.050, return_coherency=True)
coh12_freqspace /= np.abs(coh12_freqspace).max()
# weight the coherence by one minus the normalized standard deviation
coherence_std = np.sqrt(coherence_var)
# cweight = coherence_std / coherence_std.sum()
# coherence_weighted = (1.0 - cweight)*coherence
coherence_weighted = coherence - coherence_std
coherence_weighted[coherence_weighted < 0] = 0
# compute the coherence from the fft of the coherency
coherence2 = fft(fftshift(coh12))
coherence2_freq = fftfreq(len(coherence2), d=1.0/sr)
fi = coherence2_freq > 0
coherence2 = np.abs(coherence2[fi])
coherence2_freq = coherence2_freq[fi]
"""
plt.figure()
ax = plt.subplot(2, 1, 1)
plt.plot(ps1_auto_freq, ps1_auto*ps2_auto, 'c-', linewidth=2.0, alpha=0.75)
plt.plot(psd12_freq, psd12, 'g-', linewidth=2.0, alpha=0.9)
plt.plot(ps1_auto_freq, ps1_auto, 'k-', linewidth=2.0, alpha=0.75)
plt.plot(ps2_auto_freq, ps2_auto, 'r-', linewidth=2.0, alpha=0.75)
plt.axis('tight')
plt.legend(['denom', '12', '1', '2'])
ax = plt.subplot(2, 1, 2)
plt.plot(psd12_freq, coherence, 'b-')
plt.axis('tight')
plt.show()
"""
# normalize the cross-spectral density and power spectra
psd12 /= psd12.max()
ps_auto_max = max(ps1_auto.max(), ps2_auto.max())
ps1_auto /= ps_auto_max
ps2_auto /= ps_auto_max
# make some plots
plt.figure()
nrows = 2
ncols = 2
# plot the signals
ax = plt.subplot(nrows, ncols, 1)
plt.plot(t, s1, 'k-', linewidth=2.0)
plt.plot(t, s2, 'r-', alpha=0.75, linewidth=2.0)
plt.xlabel('Time (s)')
plt.ylabel('Signal')
plt.axis('tight')
# plot the spectra
ax = plt.subplot(nrows, ncols, 2)
plt.plot(welch_freq1, welch_psd1, 'k-', linewidth=2.0, alpha=0.85)
plt.plot(ps1_auto_freq, ps1_auto, 'k--', linewidth=2.0, alpha=0.85)
plt.plot(welch_freq2, welch_psd2, 'r-', alpha=0.75, linewidth=2.0)
plt.plot(ps2_auto_freq, ps2_auto, 'r--', linewidth=2.0, alpha=0.75)
plt.axis('tight')
plt.xlabel('Frequency (Hz)')
plt.ylabel('Power')
# plot the correlation functions
ax = plt.subplot(nrows, ncols, 3)
plt.axhline(0, c='k')
plt.plot(lags, acf1, 'k-', linewidth=2.0)
plt.plot(lags, acf2, 'r-', alpha=0.75, linewidth=2.0)
plt.plot(lags, cf12, 'g-', alpha=0.75, linewidth=2.0)
plt.plot(lags, coh12, 'b-', linewidth=2.0, alpha=0.75)
plt.plot(coh12_freqspace_t*1e3, coh12_freqspace, 'm-', linewidth=1.0, alpha=0.95)
plt.xlabel('Lag (ms)')
plt.ylabel('Correlation Function')
plt.axis('tight')
plt.ylim(-0.5, 1.0)
handles = custom_legend(['k', 'r', 'g', 'b', 'c'], ['acf1', 'acf2', 'cf12', 'coh12', 'coh12_f'])
plt.legend(handles=handles)
# plot the cross spectral density
ax = plt.subplot(nrows, ncols, 4)
handles = custom_legend(['g', 'k', 'b'], ['CSD', 'Coherence', 'Weighted'])
plt.axhline(0, c='k')
plt.axhline(1, c='k')
plt.plot(psd12_freq, psd12, 'g-', linewidth=3.0)
plt.errorbar(cfreq, coherence, yerr=np.sqrt(coherence_var), fmt='k-', ecolor='r', linewidth=3.0, elinewidth=5.0, alpha=0.8)
plt.plot(cfreq, coherence_weighted, 'b-', linewidth=3.0, alpha=0.75)
plt.xlabel('Frequency (Hz)')
plt.ylabel('Cross-spectral Density/Coherence')
plt.legend(handles=handles)
"""
plt.figure()
plt.axhline(0, c='k')
plt.plot(lags, cf12, 'k-', alpha=1, linewidth=2.0)
plt.plot(lags, coh12, 'b-', linewidth=3.0, alpha=0.75)
plt.plot(coh12_freqspace_t*1e3, coh12_freqspace, 'r-', linewidth=2.0, alpha=0.95)
plt.xlabel('Lag (ms)')
plt.ylabel('Correlation Function')
plt.axis('tight')
plt.ylim(-0.5, 1.0)
handles = custom_legend(['k', 'b', 'r'], ['cf12', 'coh12', 'coh12_f'])
plt.legend(handles=handles)
"""
plt.show()
