def _fft_filter_setup(
image_shape: Tuple[int, int], window: Union[np.ndarray, Window],
) -> Tuple[Tuple[int, int], np.ndarray, Tuple[int, int], Tuple[int, int]]:
window_shape = window.shape
# Optimal FFT shape
# real_fft_only = True
fft_shape = (
next_fast_len(
image_shape[0] + window_shape[0] - 1
), # , real_fft_only),
next_fast_len(
image_shape[1] + window_shape[1] - 1
), # , real_fft_only),
)
# Pad window to optimal FFT size
window_pad = _pad_window(window, fft_shape)
# Obtain the transfer function via the real valued FFT
transfer_function = rfft2(window_pad)
# Image offset before FFT and after IFFT
offset_before = _offset_before_fft(window_shape)
offset_after = _offset_after_ifft(window_shape)
return fft_shape, transfer_function, offset_before, offset_after
def shift_data_subpixel(inputs):
''' rigid shift of X by ymax and xmax '''
''' allows subpixel shifts '''
''' ** not being used ** '''
X, ymax, xmax, pad_fft = inputs
ymax = ymax.flatten()
xmax = xmax.flatten()
if X.ndim<3:
X = X[np.newaxis,:,:]
nimg, Ly0, Lx0 = X.shape
if pad_fft:
X = fft2(X.astype('float32'), (next_fast_len(Ly0), next_fast_len(Lx0)))
else:
X = fft2(X.astype('float32'))
nimg, Ly, Lx = X.shape
Ny = fft.ifftshift(np.arange(-np.fix(Ly/2), np.ceil(Ly/2)))
Nx = fft.ifftshift(np.arange(-np.fix(Lx/2), np.ceil(Lx/2)))
[Nx,Ny] = np.meshgrid(Nx,Ny)
Nx = Nx.astype('float32') / Lx
Ny = Ny.astype('float32') / Ly
dph = Nx * np.reshape(xmax, (-1,1,1)) + Ny * np.reshape(ymax, (-1,1,1))
Y = np.real(ifft2(X * np.exp((2j * np.pi) * dph)))
# crop back to original size
if Ly0<Ly or Lx0<Lx:
Lyhalf = int(np.floor(Ly/2))
Lxhalf = int(np.floor(Lx/2))
Y = Y[np.ix_(np.arange(0,nimg,1,int),
np.arange(-np.fix(Ly0/2), np.ceil(Ly0/2),1,int) + Lyhalf,
np.arange(-np.fix(Lx0/2), np.ceil(Lx0/2),1,int) + Lxhalf)]
return Y
def combine_psfs(psf_1, psf_2, are_psf=False):
"""
Function to convolve two 2D arrays.
Parameters:
psf_1 (numpy.ndarray): First 2D array
psf_2 (numpy.ndarray): Second 2D array
are_psf (bool): If true, the 2D array is a PSF and gradient of PSF otherwise.
Returns:
numpy.ndarray: Convolution of the input 2D arrays
"""
shape_1 = np.array(psf_1.shape)
shape_2 = np.array(psf_2.shape)
conv_shape = shape_1 + shape_2 - 1
fast_shape = np.array(
[next_fast_len(int(conv_shape[0])),
next_fast_len(int(conv_shape[1]))])
new_psf = np.fft.irfft2(np.fft.rfft2(psf_1, s=fast_shape) *
np.fft.rfft2(psf_2, s=fast_shape),
s=fast_shape)
new_psf = new_psf[:conv_shape[0], :conv_shape[1]]
if are_psf:
assert np.isclose(np.sum(new_psf),
1.0), "Sum of PSF is {}".format(np.sum(new_psf))
assert np.isclose(
new_psf[new_psf.shape[0] // 2, new_psf.shape[1] // 2],
np.max(new_psf)), "Difference is {}".format(
new_psf[new_psf.shape[0] // 2, new_psf.shape[1] // 2] -
np.max(new_psf))
return new_psf
def prepare_masks(refImg0, ops):
refImg = refImg0.copy()
if ops['1Preg']:
maskSlope = ops['spatial_taper'] # slope of taper mask at the edges
else:
maskSlope = 3 * ops['smooth_sigma'] # slope of taper mask at the edges
Ly, Lx = refImg.shape
maskMul = spatial_taper(maskSlope, Ly, Lx)
if ops['1Preg']:
refImg = one_photon_preprocess(refImg[np.newaxis, :, :], ops).squeeze()
maskOffset = refImg.mean() * (1. - maskMul)
# reference image in fourier domain
if ops['pad_fft']:
cfRefImg = np.conj(
fft2(refImg, (next_fast_len(ops['Ly']), next_fast_len(ops['Lx']))))
else:
cfRefImg = np.conj(fft2(refImg))
absRef = np.absolute(cfRefImg)
cfRefImg = cfRefImg / (eps0 + absRef)
# gaussian filter in space
fhg = gaussian_fft(ops['smooth_sigma'], cfRefImg.shape[0],
cfRefImg.shape[1])
cfRefImg *= fhg
maskMul = maskMul.astype('float32')
maskOffset = maskOffset.astype('float32')
cfRefImg = cfRefImg.astype('complex64')
cfRefImg = np.reshape(cfRefImg, (1, cfRefImg.shape[0], cfRefImg.shape[1]))
return maskMul, maskOffset, cfRefImg
def phasecorr_reference(refImg0: np.ndarray,
maskSlope,
smooth_sigma,
yblock,
xblock,
pad_fft: bool = False):
"""
Computes taper and fft'ed reference image for phasecorr.
Parameters
----------
refImg0: array
maskSlope
smooth_sigma
yblock
xblock
pad_fft: bool
whether to do border padding in the fft step
Returns
-------
maskMul
maskOffset
cfRefImg
"""
nb, Ly, Lx = len(
yblock), yblock[0][1] - yblock[0][0], xblock[0][1] - xblock[0][0]
dims = (nb, Ly, Lx)
cfRef_dims = (nb, next_fast_len(Ly),
next_fast_len(Lx)) if pad_fft else dims
gaussian_filter = gaussian_fft(smooth_sigma, *cfRef_dims[1:])
cfRefImg1 = np.empty(cfRef_dims, 'complex64')
maskMul = spatial_taper(maskSlope, *refImg0.shape)
maskMul1 = np.empty(dims, 'float32')
maskMul1[:] = spatial_taper(2 * smooth_sigma, Ly, Lx)
maskOffset1 = np.empty(dims, 'float32')
for yind, xind, maskMul1_n, maskOffset1_n, cfRefImg1_n in zip(
yblock, xblock, maskMul1, maskOffset1, cfRefImg1):
ix = np.ix_(
np.arange(yind[0], yind[-1]).astype('int'),
np.arange(xind[0], xind[-1]).astype('int'))
refImg = refImg0[ix]
# mask params
maskMul1_n *= maskMul[ix]
maskOffset1_n[:] = refImg.mean() * (1. - maskMul1_n)
# gaussian filter
cfRefImg1_n[:] = np.conj(fft.fft2(refImg))
cfRefImg1_n /= 1e-5 + np.absolute(cfRefImg1_n)
cfRefImg1_n[:] *= gaussian_filter
return maskMul1[:, np.
newaxis, :, :], maskOffset1[:, np.
newaxis, :, :], cfRefImg1[:,
np.
newaxis, :, :]
def phasecorr_reference(refImg0, ops):
""" computes masks and fft'ed reference image for phasecorr
Parameters
----------
refImg0 : int16
reference image
ops : dictionary
requires 'smooth_sigma'
(if ```ops['1Preg']```, need 'spatial_taper', 'spatial_hp', 'pre_smooth')
Returns
-------
maskMul : float32
mask that is multiplied to spatially taper frames
maskOffset : float32
shifts in x from cfRefImg to data for each frame
cfRefImg : complex64
reference image fft'ed and complex conjugate and multiplied by gaussian
filter in the fft domain with standard deviation 'smooth_sigma'
"""
refImg = refImg0.copy()
if '1Preg' in ops and ops['1Preg']:
maskSlope = ops['spatial_taper'] # slope of taper mask at the edges
else:
maskSlope = 3 * ops['smooth_sigma'] # slope of taper mask at the edges
Ly, Lx = refImg.shape
maskMul = utils.spatial_taper(maskSlope, Ly, Lx)
if ops['1Preg']:
refImg = utils.one_photon_preprocess(refImg[np.newaxis, :, :],
ops).squeeze()
maskOffset = refImg.mean() * (1. - maskMul)
# reference image in fourier domain
if 'pad_fft' in ops and ops['pad_fft']:
cfRefImg = np.conj(fft2(refImg,
(next_fast_len(Ly), next_fast_len(Lx))))
else:
cfRefImg = np.conj(fft2(refImg))
absRef = np.absolute(cfRefImg)
cfRefImg = cfRefImg / (1e-5 + absRef)
# gaussian filter in space
fhg = utils.gaussian_fft(ops['smooth_sigma'], cfRefImg.shape[0],
cfRefImg.shape[1])
cfRefImg *= fhg
maskMul = maskMul.astype('float32')
maskOffset = maskOffset.astype('float32')
cfRefImg = cfRefImg.astype('complex64')
cfRefImg = np.reshape(cfRefImg, (1, cfRefImg.shape[0], cfRefImg.shape[1]))
return maskMul, maskOffset, cfRefImg
def phasecorr_reference(refImg1, ops):
""" create blocked reference image and take its fft and multiply by gaussian smoothing mask """
if 'yblock' not in ops:
ops = utils.make_blocks(ops)
refImg0=refImg1.copy()
if ops['1Preg']:
maskSlope = ops['spatial_taper']
else:
maskSlope = 3 * ops['smooth_sigma'] # slope of taper mask at the edges
Ly,Lx = refImg0.shape
maskMul = utils.spatial_taper(maskSlope, Ly, Lx)
if ops['1Preg']:
refImg0 = utils.one_photon_preprocess(refImg0[np.newaxis,:,:], ops).squeeze()
# split refImg0 into multiple parts
cfRefImg1 = []
maskMul1 = []
maskOffset1 = []
nb = len(ops['yblock'])
#patch taper
Ly = ops['yblock'][0][1] - ops['yblock'][0][0]
Lx = ops['xblock'][0][1] - ops['xblock'][0][0]
if ops['pad_fft']:
cfRefImg1 = np.zeros((nb,1,next_fast_len(Ly), next_fast_len(Lx)),'complex64')
else:
cfRefImg1 = np.zeros((nb,1,Ly,Lx),'complex64')
maskMul1 = np.zeros((nb,1,Ly,Lx),'float32')
maskOffset1 = np.zeros((nb,1,Ly,Lx),'float32')
for n in range(nb):
yind = ops['yblock'][n]
yind = np.arange(yind[0],yind[-1]).astype('int')
xind = ops['xblock'][n]
xind = np.arange(xind[0],xind[-1]).astype('int')
refImg = refImg0[np.ix_(yind,xind)]
maskMul2 = utils.spatial_taper(2 * ops['smooth_sigma'], Ly, Lx)
maskMul1[n,0,:,:] = maskMul[np.ix_(yind,xind)].astype('float32')
maskMul1[n,0,:,:] *= maskMul2.astype('float32')
maskOffset1[n,0,:,:] = (refImg.mean() * (1. - maskMul1[n,0,:,:])).astype(np.float32)
cfRefImg = np.conj(fft.fft2(refImg))
absRef = np.absolute(cfRefImg)
cfRefImg = cfRefImg / (1e-5 + absRef)
# gaussian filter
fhg = utils.gaussian_fft(ops['smooth_sigma'], cfRefImg.shape[0], cfRefImg.shape[1])
cfRefImg *= fhg
cfRefImg1[n,0,:,:] = (cfRefImg.astype('complex64'))
return maskMul1, maskOffset1, cfRefImg1
def standard_mask(mask, extend_ratio=EXTEND_RATIO, smooth=True, taper=True):
'''``mask`` modified in-place
'''
# _smooth_window might change mask.shape if mode == 'full'
n_x = next_fast_len(mask.shape[0] * EXTEND_RATIO)
# tapering need to be done before smoothing, because the smoothing will change the boundary
if taper:
mask *= _get_taper(mask > 0., WIDTH)
if smooth:
if taper:
# if taperring, then smoothing beyond the boundary doesn't make sense
mask_boolean = mask > 0.
temp = np.zeros_like(mask)
temp[mask_boolean] = _smooth_window(mask,
mode='same')[mask_boolean]
mask = temp
del mask_boolean, temp
else:
mask = _smooth_window(mask, mode='full')
# this normalization doesn't matter
# mask is used in 2 places, mode-coupling, and pseudo-spectra
# in mode-coupling, norm_fft is used
# in pseudo-spectra as a weight, the absolute scale doesn't matter
# normalize_max(mask)
return zero_padding(mask, (n_x, n_x))
def get_ns(wf1,source_conf,insta):
# Nr of time steps in traces
if insta:
# get path to instaseis db
#ToDo: ugly.
dbpath = json.load(open(os.path.join(source_conf['project_path'],
'config.json')))['wavefield_path']
# open
db = instaseis.open_db(dbpath)
# get a test seismogram to determine...
stest = db.get_seismograms(source=instaseis.ForceSource(latitude=0.0,
longitude=0.0),receiver=instaseis.Receiver(latitude=10.,
longitude=0.0),dt=1./source_conf['sampling_rate'])[0]
nt = stest.stats.npts
Fs = stest.stats.sampling_rate
else:
with WaveField(wf1) as wf1:
nt = int(wf1.stats['nt'])
Fs = round(wf1.stats['Fs'],8)
# Necessary length of zero padding for carrying out frequency domain correlations/convolutions
n = next_fast_len(2*nt-1)
# Number of time steps for synthetic correlation
n_lag = int(source_conf['max_lag'] * Fs)
if nt - 2*n_lag <= 0:
click.secho('Resetting maximum lag to %g seconds: Synthetics are too\
short for a maximum lag of %g seconds.' %(nt//2/Fs,n_lag/Fs))
n_lag = nt // 2
n_corr = 2*n_lag + 1
return nt,n,n_corr,Fs
def crop_tails(self):
"""Crops out tails after every impulse response has decayed to noise floor."""
if self.fs != self.estimator.fs:
raise ValueError(
'Refusing to crop tails because HRIR\'s sampling rate doesn\'t match impulse response '
'estimator\'s sampling rate.')
# Find indices after which there is only noise in each track
tail_indices = []
lengths = []
for speaker, pair in self.irs.items():
for side, ir in pair.items():
_, tail_ind, _, _ = ir.decay_params()
tail_indices.append(tail_ind)
lengths.append(len(ir))
# Crop all tracks by last tail index
seconds_per_octave = len(
self.estimator) / self.estimator.fs / self.estimator.n_octaves
fade_out = 2 * int(self.fs * seconds_per_octave *
(1 / 24)) # Duration of 1/24 octave in the sweep
window = signal.hanning(fade_out)[fade_out // 2:]
fft_len = fftpack.next_fast_len(max(tail_indices))
tail_ind = min(np.min(lengths), fft_len)
for speaker, pair in self.irs.items():
for ir in pair.values():
ir.data = ir.data[:tail_ind]
ir.data *= np.concatenate(
[np.ones(len(ir.data) - len(window)), window])
def real_stream_cc_output_dict(real_templates, real_multichannel_stream):
""" return a dict of outputs from all stream_xcorr functions """
out = {}
fft_len = next_fast_len(real_templates[0][0].stats.npts +
real_multichannel_stream[0].stats.npts + 1)
short_fft_len = 2**8
for name, func in stream_funcs.items():
for cores in [1, cpu_count()]:
print("Running {0} with {1} cores".format(name, cores))
cc_out = time_func(func,
name,
real_templates,
real_multichannel_stream,
cores=cores,
fft_len=short_fft_len)
out["{0}.{1}".format(name, cores)] = cc_out
if "fftw" in name:
print("Running fixed fft-len: {0}".format(fft_len))
# Make sure that running with a pre-defined fft-len works
cc_out = time_func(func,
name,
real_templates,
real_multichannel_stream,
cores=cores,
fft_len=fft_len)
out["{0}.{1}_fixed_fft".format(name, cores)] = cc_out
return out
def spectral_whitening(data, sr=None, smooth=None, filter=None,
waterlevel=1e-8):
"""
Apply spectral whitening to data
Data is divided by its smoothed (Default: None) amplitude spectrum.
:param data: numpy array with data to manipulate
:param sr: sampling rate (only needed for smoothing)
:param smooth: length of smoothing window in Hz
(default None -> no smoothing)
:param filter: filter spectrum with bandpass after whitening
(tuple with min and max frequency)
:param waterlevel: waterlevel relative to mean of spectrum
:return: whitened data
"""
data = _fill_array(data, fill_value=0.)
mask = np.ma.getmask(data)
nfft = next_fast_len(len(data))
spec = fft(data, nfft)
spec_ampl = np.abs(spec)
spec_ampl /= np.max(spec_ampl)
if smooth:
smooth = int(smooth * nfft / sr)
spec_ampl = ifftshift(smooth_func(fftshift(spec_ampl), smooth))
# save guard against division by 0
spec_ampl[spec_ampl < waterlevel] = waterlevel
spec /= spec_ampl
if filter is not None:
spec *= _filter_resp(*filter, sr=sr, N=len(spec), whole=True)[1]
ret = np.real(ifft(spec, nfft)[:len(data)])
return _fill_array(ret, mask=mask, fill_value=0.)
def FFT(data: list, samplingFrequency: int,
zeroPadding: bool) -> (List[float], List[float], List[float]):
signalLength = len(data)
if zeroPadding:
fftLength = fftpack.next_fast_len(signalLength)
else:
fftLength = signalLength
amplitudePhase = fftpack.fft(data, fftLength) / signalLength
if fftLength % 2 == 0:
binCount = int(fftLength / 2 + 1)
amplitude = abs(amplitudePhase[0:binCount])
amplitude[1:-1] = 2 * amplitude[1:-1]
else:
binCount = int((fftLength + 1) / 2)
amplitude = abs(amplitudePhase[0:binCount])
amplitude[1:] = 2 * amplitude[1:]
phase = numpy.angle(amplitudePhase[0:binCount])
frequency = samplingFrequency / 2 * numpy.linspace(0, 1, binCount)
return (frequency, amplitude, phase)
def _make_trace_df(self, phase_df):
"""
Make the data arrays.
"""
# get time rep. in dataframe
sampling_rate = phase_df['sampling_rate'].unique()[0]
num_samples = (phase_df['tw_end'] - phase_df['tw_start']) * sampling_rate
array_len = next_fast_len(int(num_samples.max() + 1))
time = np.arange(0, array_len) * (1. / sampling_rate)
# apply preprocessing to each trace
traces = phase_df.apply(self.process_trace, axis=1)
# create numpy array, fill with data
values = np.zeros((len(phase_df.index), len(time)))
for i, trace in enumerate(traces.values):
values[i, 0:len(trace.data)] = trace.data
# init df from filled values
df = pd.DataFrame(values, index=phase_df.index, columns=time)
# set name of columns
df.columns.name = 'time'
# add original lengths to the channel_info
lens = traces.apply(lambda x: len(x.data))
self.channel_info.data['sample_count'] = lens
# apply time-domain pre-processing
df = self.process_trace_dataframe_hook(df)
return df
def stack(self, data, stack_type='Linear Stack', order=2):
"""
Stack data by first axis.
:type stack_type: str or tuple
:param stack_type: Type of stack, one of the following:
``'linear'``: average stack (default),
``('pw', order)``: phase weighted stack of given order
(see [Schimmel1997]_, order 0 corresponds to linear stack),
``('root', order)``: root stack of given order
(order 1 corresponds to linear stack).
"""
if stack_type == 'Linear Stack':
stack = np.mean(data, axis=0)
elif stack_type == 'Phase Weigth Stack':
npts = np.shape(data)[1]
nfft = next_fast_len(npts)
anal_sig = hilbert(data, N=nfft)[:, :npts]
phase_stack = np.abs(np.mean(anal_sig, axis=0))**order
stack = np.mean(data, axis=0) * phase_stack
elif stack_type == 'root':
r = np.mean(np.sign(data) * np.abs(data)**(1 / order), axis=0)
stack = np.sign(r) * np.abs(r)**order
else:
raise ValueError('stack type is not valid.')
return stack
def correlate_phase(data1, data2, shift, demean=True, normalize=True):
from scipy.signal import hilbert
from scipy.fftpack import next_fast_len
assert len(data1) == len(data2)
nfft = next_fast_len(len(data1))
if demean:
data1 = data1 - np.mean(data1)
data2 = data2 - np.mean(data2)
sig1 = hilbert(data1, N=nfft)[:len(data1)]
sig2 = hilbert(data2, N=nfft)[:len(data2)]
phi1 = np.angle(sig1)
phi2 = np.angle(sig2)
def phase_stack(phi1, phi2, shift):
s1 = max(0, shift)
s2 = min(0, shift)
N = len(phi1)
assert len(phi2) == N
return np.sum(np.abs(np.cos((phi1[s1:N+s2] - phi2[-s2:N-s1]) / 2)) -
np.abs(np.sin((phi1[s1:N+s2] - phi2[-s2:N-s1]) / 2)))
cc = [phase_stack(phi1, phi2, s) for s in range(-shift, shift + 1)]
cc = np.array(cc)
if normalize:
cc = cc / len(data1)
return cc
def array_ccs_low_amp(array_template, array_stream, pads):
""" Use each function stored in the normxcorr cache to correlate the
templates and arrays, return a dict with keys as func names and values
as the cc calculated by said function.
This specifically tests low amplitude streams as raised in issue #181."""
out = {}
arr_stream = array_stream * 10e-8
for name in list(corr.XCORR_FUNCS_ORIGINAL.keys()):
func = corr.get_array_xcorr(name)
print("Running {0} with low-variance".format(name))
_log_handler.reset()
cc, _ = time_func(func, name, array_template, arr_stream, pads)
out[name] = (cc, copy.deepcopy(log_messages['warning']))
if "fftw" in name:
print("Running fixed len fft")
_log_handler.reset()
fft_len = next_fast_len(
max(len(array_stream) // 4, len(array_template)))
cc, _ = time_func(func,
name,
array_template,
array_stream,
pads,
fft_len=fft_len)
out[name + "_fixed_len"] = (cc,
copy.deepcopy(log_messages['warning']))
return out
def autocov(samples, axis=-1):
"""Compute autocovariance estimates for every lag for the input array.
Parameters
----------
samples : `numpy.ndarray(n_chains, n_iters)`
An array containing samples
Returns
-------
acov: `numpy.ndarray`
Autocovariance of samples that has same size as the input array
"""
axis = axis if axis > 0 else len(samples.shape) + axis
n = samples.shape[axis]
m = next_fast_len(2 * n)
samples = samples - samples.mean(axis, keepdims=True)
# added to silence tuple warning for a submodule
with warnings.catch_warnings():
warnings.simplefilter("ignore")
ifft_samp = np.fft.rfft(samples, n=m, axis=axis)
ifft_samp *= np.conjugate(ifft_samp)
shape = tuple(
slice(None) if dim_len != axis else slice(0, n) for dim_len, _ in enumerate(samples.shape)
)
cov = np.fft.irfft(ifft_samp, n=m, axis=axis)[shape]
cov /= n
return cov
def xc_trace(x_trace, x_design, pix_per_mm):
"""
Use a cross-correlation to find the offset
"""
# _x_design = -pix_per_mm*x_design[::-1]
_x_design = pix_per_mm * x_design
size = slit_function_length(_x_design, oversample=10)[1]
fftsize = fftpack.next_fast_len(size)
design_offset, design_slits_x, design_slits_y \
= build_slit_function(_x_design, oversample=10, size=fftsize)
trace_offset, trace_slits_x, trace_slits_y \
= build_slit_function(x_trace, oversample=10, size=fftsize)
# pyplot.plot(trace_slits_x, trace_slits_y)
pyplot.plot(design_slits_x, design_slits_y)
pyplot.scatter(_x_design,
numpy.ones(_x_design.size),
color='C3',
marker='.',
s=50,
lw=0)
pyplot.show()
xc = signal.correlate(numpy.roll(design_slits_y, (fftsize - size) // 2),
numpy.roll(trace_slits_y, (fftsize - size) // 2),
mode='full',
method='fft')
pyplot.plot(numpy.arange(xc.size), xc)
max_xc = numpy.argmax(xc)
pyplot.scatter(max_xc, xc[max_xc], marker='x', color='C3')
pyplot.show()
import pdb
pdb.set_trace()
def calcFFT(self):
#Länge des zu transformierenden Vectors bestimmen
N = len(self.volt)
#Optimale FFT-Länge bestimmen und ggf Zeropadding
N_fast = next_fast_len(N)
#Kaiser-Bessel-Fenster bestimmen
w = win.kaiser(N,12)
w = w / w.sum() * N #Normierung damit Amplituden wieder stimmen
#Transformieren
yf = fft(self.volt*w, N_fast)
yf = fftshift(yf)
#Abschneiden
yf = yf[N_fast//2:]
#Skalieren
yf = 2/N_fast*abs(yf)
#Frequenzachse erzeugen
xf = fftfreq(N_fast)
xf = fftshift(xf)
#Abschneiden
xf = xf[N_fast//2:]
#Skalieren
xf = xf*self.samplingrate
#Veröffentlichen der Werte
self.spectrum = yf
self.freq = xf
return '-DONE-'
def _make_trace_df(self, st_array: np.ndarray) -> pd.DataFrame:
"""
Make the dataframe containing time series data.
"""
sampling_rate = self.sampling_rate
assert sampling_rate is not None
# figure out with streams are fractured and drop them
good_st, new_ind = self._filter_stream_array(st_array)
# get lens and create array empty array with next fast fft length
lens = [len(x[0]) for x in good_st]
max_fast = next_fast_len(
max(lens)) # + 1) # What is the point of the +1?
values = np.empty((len(new_ind), max_fast)) * np.NaN
# iterate each stream and fill array
for i, stream in enumerate(good_st):
values[i, 0:len(stream[0].data)] = stream[0].data
# init df from filled values
time = np.arange(0,
float(max_fast) / sampling_rate, 1.0 / sampling_rate)
df = pd.DataFrame(values, index=new_ind, columns=time)
# set name of columns
df.columns.name = "time"
# add original lengths to the channel_info
self.stats.loc[new_ind, "npts"] = lens
return df
def gaussian(logLam_temp, line_wave, FWHM_gal, pixel=True):
"""
Instrumental Gaussian line spread function (LSF), optionally integrated
within the pixels. The function is normalized in such a way that
line.sum() = 1
When the LSF is not severey undersampled, and when pixel=False, the output
of this function is nearly indistinguishable from a normalized Gaussian:
x = (logLam_temp[:, None] - np.log(line_wave))/dx
gauss = np.exp(-0.5*(x/xsig)**2)
gauss /= np.sqrt(2*np.pi)*xsig
However, to deal rigorously with the possibility of severe undersampling,
this Gaussian is defined analytically in frequency domain and transformed
numerically to time domain. This makes the convolution within pPXF exact
to machine precision regardless of sigma (including sigma=0).
:param logLam_temp: np.log(wavelength) in Angstrom
:param line_wave: Vector of lines wavelength in Angstrom
:param FWHM_gal: FWHM in Angstrom. This can be a scalar or the name of
a function wich returns the instrumental FWHM for given wavelength.
In this case the sigma returned by pPXF will be the intrinsic one,
namely the one corrected for instrumental dispersion, in the same
way as the stellar kinematics is returned.
- To measure the *observed* dispersion, ignoring the instrumental
dispersison, one can set FWHM_gal=0. In this case the Gaussian
line templates reduce to Dirac delta functions. The sigma returned
by pPXF will be the same one would measure by fitting a Gaussian
to the observed spectrum (exept for the fact that this function
accurately deals with pixel integration).
:param pixel: set to True to perform integration over the pixels.
:return: LSF computed for every logLam_temp
"""
line_wave = np.asarray(line_wave)
if callable(FWHM_gal):
FWHM_gal = FWHM_gal(line_wave)
n = logLam_temp.size
npad = fftpack.next_fast_len(n)
nl = npad // 2 + 1 # Expected length of rfft
dx = (logLam_temp[-1] - logLam_temp[0]) / (n - 1)
x0 = (np.log(line_wave) - logLam_temp[0]) / dx
xsig = FWHM_gal / 2.355 / line_wave / dx # sigma in pixels units
w = np.linspace(0, np.pi, nl)[:, None]
# Gaussian with sigma=xsig and center=x0,
# optionally convolved with an unitary pixel UnitBox[]
# analytically defined in frequency domain
# and numerically transformed to time domain
rfft = np.exp(-0.5 * (w * xsig)**2 - 1j * w * x0)
if pixel:
rfft *= np.sinc(w / (2 * np.pi))
line = np.fft.irfft(rfft, n=npad, axis=0)
return line[:n, :]
def hires_power_spectrum(signal, oversampling=1):
"""
Return a high resolution power spectrum (compare :func:`power_spectrum`)
Resolution is enhanced by feeding the FFT a n times larger, zero-padded signal,
which will yield frequency values of higher precision.
:param signal: input signal
:type signal: numpy.array
:param oversampling: oversampling factor
:type oversampling: int
:return: frequencies and fourier transformed values
:rtype: tuple(numpy.array, numpy.array)
"""
arr_len = len(signal)
fast_size = next_fast_len(oversampling * arr_len)
tmp_data = np.zeros(fast_size)
tmp_data[:arr_len] = signal
frequencies, fourier_values = power_spectrum(tmp_data)
fourier_values[0] = 0
fourier_values = fourier_values[frequencies < arr_len]
frequencies = frequencies[frequencies < arr_len]
return frequencies, fourier_values
def autocov(ary, axis=-1):
"""Compute autocovariance estimates for every lag for the input array.
Parameters
----------
ary : Numpy array
An array containing MCMC samples
Returns
-------
acov: Numpy array same size as the input array
"""
axis = axis if axis > 0 else len(ary.shape) + axis
n = ary.shape[axis]
m = next_fast_len(2 * n)
ary = ary - ary.mean(axis, keepdims=True)
# added to silence tuple warning for a submodule
with warnings.catch_warnings():
warnings.simplefilter("ignore")
ifft_ary = np.fft.rfft(ary, n=m, axis=axis)
ifft_ary *= np.conjugate(ifft_ary)
shape = tuple(
slice(None) if dim_len != axis else slice(0, n)
for dim_len, _ in enumerate(ary.shape))
cov = np.fft.irfft(ifft_ary, n=m, axis=axis)[shape]
cov /= n
return cov
def get_envelope(data, axis=0):
'''Extracts the instantaneous amplitude (envelope) of an analytic
signal using the Hilbert transform'''
n_samples = data.shape[axis]
instantaneous_amplitude = np.abs(
hilbert(data, N=next_fast_len(n_samples), axis=axis))
return np.take(instantaneous_amplitude, np.arange(n_samples), axis=axis)
def hilbert(darray, N=None, dim=None):
"""
Compute the analytic signal, using the Hilbert transform.
The transformation is done along the selected dimension.
Parameters
----------
darray : xarray
Signal data. Must be real.
N : int, optional
Number of Fourier components. Defaults to size along dim.
dim : string, optional
Axis along which to do the transformation.
Uses the only dimension of darray is 1D.
Returns
-------
darray : xarray
Analytic signal of the Hilbert transform of 'darray' along selected axis.
"""
dim = get_maybe_only_dim(darray, dim)
n_orig = darray.shape[axis]
N_unspecified = N is None
if N_unspecified:
N = next_fast_len(n_orig)
out = xarray.apply_ufunc(_hilbert_wraper,
darray,
input_core_dims=[[dim]],
output_core_dims=[[dim]],
kwargs=dict(N=N,
n_orig=n_orig,
N_unspecified=N_unspecified))
return out
def __init__(self, file, sourcegrid=None, w='r'):
self.w = w
try:
self.file = h5py.File(file, self.w)
except IOError:
msg = 'Unable to open input file ' + file
raise IOError(msg)
self.stats = dict(self.file['stats'].attrs)
self.fdomain = self.stats['fdomain']
self.sourcegrid = self.file['sourcegrid']
try:
self.data = self.file['data']
except KeyError:
self.data = self.file['data_z']
if self.fdomain:
self.freq = np.fft.rfftfreq(self.stats['npad'],
d=1. / self.stats['Fs'])
if 'npad' not in self.stats:
if self.fdomain:
self.stats['npad'] = 2 * self.stats['nt'] - 2
else:
self.stats['npad'] = next_fast_len(2 * self.stats['nt'] - 1)
def _regen_kernel(self, XYZ):
"""
Compute kernel.
Parameters
----------
XYZ : :py:class:`~numpy.ndarray`
(N_antenna, 3) Cartesian instrument geometry.
`XYZ` must be given in BFSF.
"""
N_samples = fftpack.next_fast_len(self._NFS)
lon_smpl = fourier.ffs_sample(self._T, self._NFS, self._Tc, N_samples)
px_x, px_y, px_z = sph.pol2cart(1, self._grid_colat,
lon_smpl.reshape(1, -1))
pix_smpl = np.stack([px_x, px_y, px_z], axis=0)
N_antenna = len(XYZ)
N_height = len(self._grid_colat)
# `self._NFS` assumes imaging is performed with `XYZ` centered at the origin.
XYZ_c = XYZ - XYZ.mean(axis=0)
window = func.Tukey(self._T, self._Tc, self._alpha_window)
k_smpl = np.zeros((N_antenna, N_height, N_samples), dtype=self._cp)
ne.evaluate('exp(A * B) * C',
dict(A=1j * 2 * np.pi / self._wl,
B=np.tensordot(XYZ_c, pix_smpl, axes=1),
C=window(lon_smpl)),
out=k_smpl,
casting='same_kind') # Due to limitations of NumExpr2
self._FSk = fourier.ffs(k_smpl, self._T, self._Tc, self._NFS, axis=2)
self._XYZk = XYZ
def stack(data, stack_type='linear'):
"""
Stack data by first axis.
:type stack_type: str or tuple
:param stack_type: Type of stack, one of the following:
``'linear'``: average stack (default),
``('pw', order)``: phase weighted stack of given order
(see [Schimmel1997]_, order 0 corresponds to linear stack),
``('root', order)``: root stack of given order
(order 1 corresponds to linear stack).
"""
if stack_type == 'linear':
stack = np.mean(data, axis=0)
elif stack_type[0] == 'pw':
from scipy.signal import hilbert
try:
from scipy.fftpack import next_fast_len
except ImportError: # scipy < 0.18
next_fast_len = next_pow_2
npts = np.shape(data)[1]
nfft = next_fast_len(npts)
anal_sig = hilbert(data, N=nfft)[:, :npts]
norm_anal_sig = anal_sig / np.abs(anal_sig)
phase_stack = np.abs(np.mean(norm_anal_sig, axis=0)) ** stack_type[1]
stack = np.mean(data, axis=0) * phase_stack
elif stack_type[0] == 'root':
r = np.mean(np.sign(data) * np.abs(data)
** (1 / stack_type[1]), axis=0)
stack = np.sign(r) * np.abs(r) ** stack_type[1]
else:
raise ValueError('stack type is not valid.')
return stack
def colgin2009(self):
"""colgin
Returns:
[type]: [description]
References
------------
1) Colgin, L. L., Denninger, T., Fyhn, M., Hafting, T., Bonnevie, T., Jensen, O., ... & Moser, E. I. (2009). Frequency of gamma oscillations routes flow of information in the hippocampus. Nature, 462(7271), 353-357.
2) Tallon-Baudry, C., Bertrand, O., Delpuech, C., & Pernier, J. (1997). Oscillatory γ-band (30–70 Hz) activity induced by a visual search task in humans. Journal of Neuroscience, 17(2), 722-734.
"""
t_wavelet = np.arange(-4, 4, 1 / self.sampfreq)
freqs = self.freqs
n = len(self.lfp)
fastn = next_fast_len(n)
signal = np.pad(self.lfp, (0, fastn - n), "constant", constant_values=0)
# signal = np.tile(np.expand_dims(signal, axis=0), (len(freqs), 1))
# wavelet_at_freqs = np.zeros((len(freqs), len(t_wavelet)), dtype=complex)
conv_val = np.zeros((len(freqs), n), dtype=complex)
for i, freq in enumerate(freqs):
sigma = 7 / (2 * np.pi * freq)
A = (sigma * np.sqrt(np.pi)) ** -0.5
wavelet_at_freq = (
A
* np.exp(-(t_wavelet ** 2) / (2 * sigma ** 2))
* np.exp(2j * np.pi * freq * t_wavelet)
)
conv_val[i, :] = sg.fftconvolve(
signal, wavelet_at_freq, mode="same", axes=-1
)[:n]
return np.abs(conv_val) ** 2
def fft(x,
axes=None,
center=True,
normalize=False,
backend=None,
inverse=False,
fft_backend=None,
pad=False,
y=None,
allow_c2r=False):
"""Perform the FFT of an input."""
# Get backend
if backend is None:
backend = getBackend(x)
# Determine optimal size to pad, if desired
original_size = shape(x)
if pad:
padded_size = list(original_size)
for ind, d in enumerate(original_size):
if next_fast_len(d) != d:
padded_size[ind] = next_fast_len(d)
else:
padded_size = original_size
# Get FFT functions
fft_fun, ifft_fun = fftfuncs(padded_size,
axes,
center,
normalize,
getDatatype(x),
backend,
fft_backend,
allow_c2r=allow_c2r)
# Select FFT inverse
FFT = fft_fun if not inverse else ifft_fun
# Set correct byte order
x[:] = setByteOrder(x, 'f')
if padded_size is not original_size:
return crop(FFT(pad(x, padded_size, center=True), y=y),
original_size,
center=True)
else:
return FFT(x, y=y)
def THDN(signal, fs, weight=None):
"""Measure the THD+N for a signal and print the results
Prints the estimated fundamental frequency and the measured THD+N. This is
calculated from the ratio of the entire signal before and after
notch-filtering.
This notch-filters by nulling out the frequency coefficients ±10% of the
fundamental
TODO: Make R vs F reference a parameter (currently is R)
TODO: Or report all of the above in a dictionary?
"""
# Get rid of DC and window the signal
signal = np.asarray(signal) + 0.0 # Float-like array
# TODO: Do this in the frequency domain, and take any skirts with it?
signal -= mean(signal)
window = general_cosine(len(signal), flattops['HFT248D'])
windowed = signal * window
del signal
# Zero pad to nearest power of two
new_len = next_fast_len(len(windowed))
windowed = concatenate((windowed, zeros(new_len - len(windowed))))
# Measure the total signal before filtering but after windowing
total_rms = rms_flat(windowed)
# Find the peak of the frequency spectrum (fundamental frequency)
f = rfft(windowed)
i = argmax(abs(f))
true_i = parabolic(log(abs(f)), i)[0]
frequency = fs * (true_i / len(windowed))
# Filter out fundamental by throwing away values ±10%
lowermin = int(true_i * 0.9)
uppermin = int(true_i * 1.1)
f[lowermin: uppermin] = 0
# TODO: Zeroing FFT bins is bad
# Transform noise back into the time domain and measure it
noise = irfft(f)
# TODO: RMS and A-weighting in frequency domain? Parseval?
if weight is None:
pass
elif weight == 'A':
# Apply A-weighting to residual noise (Not normally used for
# distortion, but used to measure dynamic range with -60 dBFS signal,
# for instance)
noise = A_weight(noise, fs)
# TODO: filtfilt? tail end of filter?
else:
raise ValueError('Weighting not understood')
# TODO: Return a dict or list of frequency, THD+N?
return rms_flat(noise) / total_rms
def LSDecompFW(wav, width= 16384, max_nnz_rate=8000.0/262144.0, sparsify = 0.01, taps = 10,
level = 3, wl_weight = 1, verbose = False,fc=120):
MaxiterA = 60
length = len(wav)
n = sft.next_fast_len(length)
signal = np.zeros((n))
signal[0:length] = wav[0:length]
h0,h1 = daubcqf(taps,'min')
L = level
#print(n)
original_signal = lambda s: sft.idct(s[0:n]) + (1.0)*(wl_weight)*idwt(s[n+1:],h0,h1,L)[0]
LSseparate = lambda x: np.concatenate([sft.dct(x),(1.0)*(wl_weight)*dwt(x,h0,h1,L)[0]],axis=0)
#measurment
y = signal
#FISTA
###############################
cnnz = float("Inf")
c = signal
temp = LSseparate(y)
temp2 = original_signal(temp)
print('aaa'+str(temp2.shape))
maxabsThetaTy = max(abs(temp))
while cnnz > max_nnz_rate * n:
#FISTA
tau = sparsify * maxabsThetaTy
tolA = 1.0e-7
fh = (original_signal,LSseparate)
c = relax.fista(A=fh, b=y,x=LSseparate(c),tol=tolA,l=tau,maxiter=MaxiterA )[0]
cnnz = np.size(np.nonzero(original_signal(c)))
print('nnz = '+ str(cnnz)+ ' / ' + str(n) +' at tau = '+str(tau))
sparsify = sparsify * 2
if sparsify == 0.166:
sparsify = 0.1
signal_dct = sft.idct(c[0:n])
signal_dct = signal_dct[0:length]
signal_wl = (1.0) * float(wl_weight) * idwt(c[n+1:],h0,h1,level)[0]
signal_wl = signal_wl[0:length]
return signal_dct,signal_wl
import numpy as np
import os
import matplotlib.pyplot as plt
import h5py
from noisi.my_classes.basisfunction import BasisFunction
from noisi.util.source_masks import get_source_mask
from noisi.util.plot import plot_grid
try:
from scipy.fftpack import next_fast_len
except ImportError:
from noisi.borrowed_functions.scipy_next_fast_len import next_fast_len
from obspy.signal.invsim import cosine_taper
import json
n = next_fast_len(2*n_samples-1)
freq = np.fft.rfftfreq(n,d=1./sampling_rate)
print(freq.shape)
taper = cosine_taper(len(freq),0.005)
grd = np.load(os.path.join(projectpath,'sourcegrid.npy'))
source_config = json.load(open(os.path.join(sourcepath,'source_config.json')))
bfunc_type = source_config['spectra_decomposition']
bfunc_K = source_config['spectra_nr_parameters']
b = BasisFunction(bfunc_type,bfunc_K,N=len(freq))
print('Found wavefield.')
with WaveField(wfs[0]) as wf:
df = wf.stats['Fs']
nt = wf.stats['nt']
else:
df = float(input('Sampling rate of synthetic Greens functions in Hz?\n'))
nt = int(input('Nr of time steps in synthetic Greens functions?\n'))
#s for the fft is larger due to zeropadding --> apparent higher frequency sampling\n",
# n = next_fast_len(2*nt-1)
n = next_fast_len(2*nt-1)
freq = np.fft.rfftfreq(n,d=1./df)
taper = cosine_taper(len(freq),0.01)
print('Determined frequency axis.')
def get_distance(grid,location):
def f(lat,lon,location):
return abs(gps2dist_azimuth(lat,lon,location[0],location[1])[0])
dist = np.array([f(lat,lon,location) for lat,lon in zip(grid[1],grid[0])])
return dist
# Use Basemap to figure out where ocean is
def get_ocean_mask():
print('Getting ocean mask...')
from mpl_toolkits.basemap import Basemap
latmin = grd[1].min()
def cross_correlate_masked(arr1, arr2, m1, m2, mode='full', axes=(-2, -1),
overlap_ratio=3 / 10):
"""
Masked normalized cross-correlation between arrays.
Parameters
----------
arr1 : ndarray
First array.
arr2 : ndarray
Seconds array. The dimensions of `arr2` along axes that are not
transformed should be equal to that of `arr1`.
m1 : ndarray
Mask of `arr1`. The mask should evaluate to `True`
(or 1) on valid pixels. `m1` should have the same shape as `arr1`.
m2 : ndarray
Mask of `arr2`. The mask should evaluate to `True`
(or 1) on valid pixels. `m2` should have the same shape as `arr2`.
mode : {'full', 'same'}, optional
'full':
This returns the convolution at each point of overlap. At
the end-points of the convolution, the signals do not overlap
completely, and boundary effects may be seen.
'same':
The output is the same size as `arr1`, centered with respect
to the `‘full’` output. Boundary effects are less prominent.
axes : tuple of ints, optional
Axes along which to compute the cross-correlation.
overlap_ratio : float, optional
Minimum allowed overlap ratio between images. The correlation for
translations corresponding with an overlap ratio lower than this
threshold will be ignored. A lower `overlap_ratio` leads to smaller
maximum translation, while a higher `overlap_ratio` leads to greater
robustness against spurious matches due to small overlap between
masked images.
Returns
-------
out : ndarray
Masked normalized cross-correlation.
Raises
------
ValueError : if correlation `mode` is not valid, or array dimensions along
non-transformation axes are not equal.
References
----------
.. [1] Dirk Padfield. Masked Object Registration in the Fourier Domain.
IEEE Transactions on Image Processing, vol. 21(5),
pp. 2706-2718 (2012). :DOI:`10.1109/TIP.2011.2181402`
.. [2] D. Padfield. "Masked FFT registration". In Proc. Computer Vision and
Pattern Recognition, pp. 2918-2925 (2010).
:DOI:`10.1109/CVPR.2010.5540032`
"""
if mode not in {'full', 'same'}:
raise ValueError("Correlation mode {} is not valid.".format(mode))
fixed_image = np.array(arr1, dtype=np.float)
fixed_mask = np.array(m1, dtype=np.bool)
moving_image = np.array(arr2, dtype=np.float)
moving_mask = np.array(m2, dtype=np.bool)
eps = np.finfo(np.float).eps
# Array dimensions along non-transformation axes should be equal.
all_axes = set(range(fixed_image.ndim))
for axis in (all_axes - set(axes)):
if fixed_image.shape[axis] != moving_image.shape[axis]:
raise ValueError(
"Array shapes along non-transformation axes should be "
"equal, but dimensions along axis {a} not".format(a=axis))
# Determine final size along transformation axes
# Note that it might be faster to compute Fourier transform in a slightly
# larger shape (`fast_shape`). Then, after all fourier transforms are done,
# we slice back to`final_shape` using `final_slice`.
final_shape = list(arr1.shape)
for axis in axes:
final_shape[axis] = fixed_image.shape[axis] + \
moving_image.shape[axis] - 1
final_shape = tuple(final_shape)
final_slice = tuple([slice(0, int(sz)) for sz in final_shape])
# Extent transform axes to the next fast length (i.e. multiple of 3, 5, or
# 7)
fast_shape = tuple([next_fast_len(final_shape[ax]) for ax in axes])
# We use numpy's fft because it allows to leave transform axes unchanged,
# which is not possible with SciPy's fftn/ifftn
# E.g. arr shape (2,3,7), transform along axes (0, 1) with shape (4,4)
# results in arr_fft shape (4,4, 7)
fft = partial(np.fft.fftn, s=fast_shape, axes=axes)
ifft = partial(np.fft.ifftn, s=fast_shape, axes=axes)
fixed_image[np.logical_not(fixed_mask)] = 0.0
moving_image[np.logical_not(moving_mask)] = 0.0
# N-dimensional analog to rotation by 180deg is flip over all relevant axes.
# See [1] for discussion.
rotated_moving_image = _flip(moving_image, axes=axes)
rotated_moving_mask = _flip(moving_mask, axes=axes)
fixed_fft = fft(fixed_image)
rotated_moving_fft = fft(rotated_moving_image)
fixed_mask_fft = fft(fixed_mask)
rotated_moving_mask_fft = fft(rotated_moving_mask)
# Calculate overlap of masks at every point in the convolution.
# Locations with high overlap should not be taken into account.
number_overlap_masked_px = np.real(
ifft(rotated_moving_mask_fft * fixed_mask_fft))
number_overlap_masked_px[:] = np.round(number_overlap_masked_px)
number_overlap_masked_px[:] = np.fmax(number_overlap_masked_px, eps)
masked_correlated_fixed_fft = ifft(rotated_moving_mask_fft * fixed_fft)
masked_correlated_rotated_moving_fft = ifft(
fixed_mask_fft * rotated_moving_fft)
numerator = ifft(rotated_moving_fft * fixed_fft)
numerator -= masked_correlated_fixed_fft * \
masked_correlated_rotated_moving_fft / number_overlap_masked_px
fixed_squared_fft = fft(np.square(fixed_image))
fixed_denom = ifft(rotated_moving_mask_fft * fixed_squared_fft)
fixed_denom -= np.square(masked_correlated_fixed_fft) / \
number_overlap_masked_px
fixed_denom[:] = np.fmax(fixed_denom, 0.0)
rotated_moving_squared_fft = fft(np.square(rotated_moving_image))
moving_denom = ifft(fixed_mask_fft * rotated_moving_squared_fft)
moving_denom -= np.square(masked_correlated_rotated_moving_fft) / \
number_overlap_masked_px
moving_denom[:] = np.fmax(moving_denom, 0.0)
denom = np.sqrt(fixed_denom * moving_denom)
# Slice back to expected convolution shape.
numerator = numerator[final_slice]
denom = denom[final_slice]
number_overlap_masked_px = number_overlap_masked_px[final_slice]
if mode == 'same':
_centering = partial(_centered,
newshape=fixed_image.shape, axes=axes)
denom = _centering(denom)
numerator = _centering(numerator)
number_overlap_masked_px = _centering(number_overlap_masked_px)
# Pixels where `denom` is very small will introduce large
# numbers after division. To get around this problem,
# we zero-out problematic pixels.
tol = 1e3 * eps * np.max(np.abs(denom), axis=axes, keepdims=True)
nonzero_indices = denom > tol
out = np.zeros_like(denom)
out[nonzero_indices] = numerator[nonzero_indices] / denom[nonzero_indices]
np.clip(out, a_min=-1, a_max=1, out=out)
# Apply overlap ratio threshold
number_px_threshold = overlap_ratio * np.max(number_overlap_masked_px,
axis=axes, keepdims=True)
out[number_overlap_masked_px < number_px_threshold] = 0.0
return out
def circular_resample(x, Fs, tau, pad_to_next_fast_len=True):
'''Circularly resample (i.e. shift) signal `x` by delay `tau`.
The resampling is "circular" in that it uses the FFT, which maps
a finite `N`-length sequence onto the unit circle with the implicit
assumption of periodic boundary conditions. Resampling `x` with a
time shift `tau` then corresponds to a `Fs * tau`-rotation of the
sampling points about the unit circle. Because of the periodicity
assumption, this rotation can lead to "wrap-around" artifacts.
Accounting for trigger offsets, for example, typically requires
small rotations such that the wrap-around effects are negligible.
Regardless, wrap-around effects can be minimized by appropriately
zero-padding `x` prior to calling this function.
Input parameters:
-----------------
x - array_like, `(N,)`
The uniformly sampled, real sequence to be circularly resampled.
If `x` is not real, a ValueError is raised. (Note that this
algorithm could readily be extended to work with complex signals,
but the `random_data` package is optimized for real-valued
signals in the time domain).
[x] = arbitrary units
Fs - float
Sample rate of sequence `x`.
[Fs] = arbitrary units
tau - float
If the original sequence `x` has corresponding sample times `t`,
then the circularly resampled sequence will correspond to sample
times `t + tau`. Thus, positive `tau` corresponds to signal delay,
and negative `tau` corresponds to signal advancement. Note that
`tau` need *not* be an integer multiple of the sample spacing.
[tau] = 1 / [Fs]
pad_to_next_fast_len - bool
If True, prior to computing the FFT, pad `x` with zeros up to the
next-largest "5-smooth" number, for which Numpy's FFTPACK routines
can efficiently compute the FFT.
Returns:
--------
xr - array_like, `(N,)`
The circularly resampled, real sequence.
[xr] = [x]
'''
if np.iscomplexobj(x):
raise ValueError('`x` must be a real-valued array')
N = len(x)
if pad_to_next_fast_len:
N = next_fast_len(N)
xhat = np.fft.rfft(x, n=N)
f = np.fft.rfftfreq(N, d=(1. / Fs))
# Apply linear phase rotation corresponding to delay `tau`
xhat *= np.exp(1j * 2 * np.pi * f * tau)
# Per the documentation, `n` must be specified if an odd number
# of output points is desired (i.e. if the `N` used in the forward
# FFT computation above is odd)
xr = np.fft.irfft(xhat, n=N)
# Finally, if `x` was zero padded prior to computing the FFT,
# the above computed `xr` does *not* have the same length as `x`.
# Return only first `len(x)` values of circularly resampled sequence.
return xr[:len(x)]
def test_circular_resample():
# Complex input should raise ValueError:
# ======================================
x = np.zeros(10, dtype='complex')
tools.assert_raises(
ValueError,
rd.signals.sampling.circular_resample,
*[x, 1., 0])
# Null (i.e. no shift):
# =====================
# N even, 5-smooth number:
# ------------------------
N = 10
x = np.arange(N)
Fs = 1.
tau = 0
xr = rd.signals.sampling.circular_resample(
x, Fs, tau, pad_to_next_fast_len=True)
np.testing.assert_almost_equal(xr, x)
# N even, *not* a 5-smooth number:
# --------------------------------
N = 14
x = np.arange(N)
Fs = 1.
tau = 0
xr = rd.signals.sampling.circular_resample(
x, Fs, tau, pad_to_next_fast_len=True)
np.testing.assert_almost_equal(xr, x)
# N odd, 5-smooth number:
# -----------------------
N = 9
x = np.arange(N)
Fs = 1.
tau = 0
xr = rd.signals.sampling.circular_resample(
x, Fs, tau, pad_to_next_fast_len=True)
np.testing.assert_almost_equal(xr, x)
# N odd, *not* a 5-smooth number:
# -------------------------------
N = 11
x = np.arange(N)
Fs = 1.
tau = 0
xr = rd.signals.sampling.circular_resample(
x, Fs, tau, pad_to_next_fast_len=True)
np.testing.assert_almost_equal(xr, x)
# Positive, integer shift:
# ========================
# N even, 5-smooth number:
# ------------------------
N = 10
x = np.arange(N)
Fs = 1.
tau = 1
# No zero padding of `x`, so simply moves `x[0]` to end of array
xr = rd.signals.sampling.circular_resample(
x, Fs, tau, pad_to_next_fast_len=True)
np.testing.assert_almost_equal(
xr,
np.concatenate((x[1:], [x[0]])))
# N even, *not* a 5-smooth number:
# --------------------------------
N = 14
x = np.arange(N)
Fs = 1.
tau = 1
# Zero padding of `x` means that `x[0]` is cycled out of the
# viewing window and a `0` is cycled into the end of `x`
xr = rd.signals.sampling.circular_resample(
x, Fs, tau, pad_to_next_fast_len=True)
np.testing.assert_almost_equal(
xr,
np.concatenate((x[1:], [0])))
# N odd, 5-smooth number:
# -----------------------
N = 9
x = np.arange(N)
Fs = 1.
tau = 1
# No zero padding of `x`, so simply moves `x[0]` to end of array
xr = rd.signals.sampling.circular_resample(
x, Fs, tau, pad_to_next_fast_len=True)
np.testing.assert_almost_equal(
xr,
np.concatenate((x[1:], [x[0]])))
# N odd, *not* a 5-smooth number:
# -------------------------------
N = 11
x = np.arange(N)
Fs = 1.
tau = 1
# Zero padding of `x` means that `x[0]` is cycled out of the
# viewing window and a `0` is cycled into the end of `x`
xr = rd.signals.sampling.circular_resample(
x, Fs, tau, pad_to_next_fast_len=True)
np.testing.assert_almost_equal(
xr,
np.concatenate((x[1:], [0])))
# Negative, integer shift:
# ========================
# N even, 5-smooth number:
# ------------------------
N = 10
x = np.arange(N)
Fs = 1.
tau = -1
# No zero padding of `x`, so simply moves `x[-1]` to start of array
xr = rd.signals.sampling.circular_resample(
x, Fs, tau, pad_to_next_fast_len=True)
np.testing.assert_almost_equal(
xr,
np.concatenate(([x[-1]], x[:-1])))
# N even, *not* a 5-smooth number:
# --------------------------------
N = 14
x = np.arange(N)
Fs = 1.
tau = -1
# Zero padding of `x` means that `x[-1]` is cycled out of the
# viewing window and a `0` is cycled into the start of `x`
xr = rd.signals.sampling.circular_resample(
x, Fs, tau, pad_to_next_fast_len=True)
np.testing.assert_almost_equal(
xr,
np.concatenate(([0], x[:-1])))
# N odd, 5-smooth number:
# -----------------------
N = 9
x = np.arange(N)
Fs = 1.
tau = -1
# No zero padding of `x`, so simply moves `x[-1]` to start of array
xr = rd.signals.sampling.circular_resample(
x, Fs, tau, pad_to_next_fast_len=True)
np.testing.assert_almost_equal(
xr,
np.concatenate(([x[-1]], x[:-1])))
# N odd, *not* a 5-smooth number:
# -------------------------------
N = 11
x = np.arange(N)
Fs = 1.
tau = -1
# Zero padding of `x` means that `x[-1]` is cycled out of the
# viewing window and a `0` is cycled into the start of `x`
xr = rd.signals.sampling.circular_resample(
x, Fs, tau, pad_to_next_fast_len=True)
np.testing.assert_almost_equal(
xr,
np.concatenate(([0], x[:-1])))
# Positive, non-integer shift:
# ============================
#
# Note that `circular_resample` uses the FFT, which maps a finite
# `N`-length sequence onto the unit circle with the implicit assumption
# of periodic boundary conditions. The resulting periodicity allows
# representation of the signal with an `N`-term Fourier series.
#
# Now, after being mapped onto the unit circle, the ramping sequence
# used in the above computations (`x = np.arange(N)`) corresponds to a
# periodic sawtooth signal. It is well-known that the Fourier series
# does a poor job of representing the underlying signal at sharp edges
# (i.e. the Gibbs phenomenon), such as the vertices of the sawteeth.
# The Gibbs phenomenon becomes readily apparent when shifting the
# sawtooth signal by a non-integer number of samples.
#
# Because of this, it makes more sense to use a simpler signal, such
# as a pure sinusoid with period equal to the sequence length, to check
# that `circular_resample` is performing as expected for non-integer
# shifts. Note that this test only works well if we do *not* zero pad
# the input signal to a 5-smooth number (i.e. `len(x)` needs to already
# be a 5-smooth number, or the `pad_to_next_fast_len` keyword argument
# must be `False`).
def unity_frequency_sine(t):
'Get unity frequency sine wave.'
f0 = 1
return np.sin(2 * np.pi * f0 * t)
def period_N_sine_wrapper(N):
'Get `N`-length sine wave containing exactly *one* period.'
t = np.arange(0, N, dtype='float') / N
Fs = 1. / (t[1] - t[0])
x = unity_frequency_sine(t)
return t, Fs, x
# N even, 5-smooth number:
# ------------------------
N = 10
t, Fs, x = period_N_sine_wrapper(N)
tau = 0.5 / Fs
xr = rd.signals.sampling.circular_resample(
x, Fs, tau, pad_to_next_fast_len=True)
np.testing.assert_almost_equal(
xr,
unity_frequency_sine(t + tau))
# N odd, 5-smooth number:
# -----------------------
N = 9
t, Fs, x = period_N_sine_wrapper(N)
tau = 0.5 / Fs
xr = rd.signals.sampling.circular_resample(
x, Fs, tau, pad_to_next_fast_len=True)
np.testing.assert_almost_equal(
xr,
unity_frequency_sine(t + tau))
# Negative, non-integer shift:
# ============================
# N even, 5-smooth number:
# ------------------------
N = 10
t, Fs, x = period_N_sine_wrapper(N)
tau = -0.5 / Fs
xr = rd.signals.sampling.circular_resample(
x, Fs, tau, pad_to_next_fast_len=True)
np.testing.assert_almost_equal(
xr,
unity_frequency_sine(t + tau))
# N odd, 5-smooth number:
# -----------------------
N = 9
t, Fs, x = period_N_sine_wrapper(N)
tau = -0.5 / Fs
xr = rd.signals.sampling.circular_resample(
x, Fs, tau, pad_to_next_fast_len=True)
np.testing.assert_almost_equal(
xr,
unity_frequency_sine(t + tau))
# Spectrum for non-integer shift, N *not* a 5-smooth number:
# ==========================================================
# Construct signal:
# -----------------
# Desired number of points in signal; *not* a 5-smooth number
N = 63331
tools.assert_not_equal(
N,
next_fast_len(N),
msg='`N` *is* a 5-smooth number; testing requires non-5-smooth number')
# Translate `N` into values usable by `random_data` routines
N_power_of_2 = np.int(2 ** np.ceil(np.log2(N)))
Fs = 1.
t0 = 0
T = N_power_of_2 / Fs
# Parameters of broadband spectrum
f0_broad = 0.
tau_broad = 2.
G0 = 1.
noise_floor = 1e-2
seed = None
# Broadband signal, where `len(sig.x)` is a power of 2
sig = rd.signals.RandomSignal(
Fs=Fs, t0=t0, T=T,
f0=f0_broad, tau=tau_broad, G0=G0,
noise_floor=noise_floor, seed=seed)
# Only take the first `N` points of `sig.x`, producing
# a signal with a length that is *not* a 5-smooth number
x = sig.x[:N]
t = sig.t()[:N]
tools.assert_not_equal(
len(x),
next_fast_len(len(x)),
msg='`len(x)` *is* a 5-smooth number; non-5-smooth number required')
# Circularly resample signal:
# ---------------------------
# Integer shift
tau_1 = 1. / Fs
x_1 = rd.signals.sampling.circular_resample(
x, Fs, tau_1, pad_to_next_fast_len=True)
# Non-integer shift
tau_05 = 0.5 / Fs
x_05 = rd.signals.sampling.circular_resample(
x, Fs, tau_05, pad_to_next_fast_len=True)
# Compare the spectra of the original and time-shifted signals:
# -------------------------------------------------------------
# Spectral-estimation parameters
Nreal_per_ens = 1000
Tens = t[-1] - t[0] # Define ensemble to be full length of `x`
# Estimate spectra
asd = rd.spectra.AutoSpectralDensity(
x, Fs=Fs, t0=t0,
Tens=Tens, Nreal_per_ens=Nreal_per_ens)
asd_1 = rd.spectra.AutoSpectralDensity(
x_1, Fs=Fs, t0=t0,
Tens=Tens, Nreal_per_ens=Nreal_per_ens)
asd_05 = rd.spectra.AutoSpectralDensity(
x_05, Fs=Fs, t0=t0,
Tens=Tens, Nreal_per_ens=Nreal_per_ens)
# Relative error of time-shifted spectra relative to original
relerr_1 = np.squeeze((asd_1.Gxx - asd.Gxx) / asd.Gxx)
relerr_05 = np.squeeze((asd_05.Gxx - asd.Gxx) / asd.Gxx)
# Relative random error of an autospectral density estimate
# is 1 / sqrt(Nreal_per_ens). Check that the relative errors
# of the time-shifted spectra are "not too much larger" than
# the random error expected for the unshifted spectral estimate.
# The "not too much larger" is quantified by `fudge_factor`.
# Note that a fudge factor of 2 is larger than typically needed,
# but it ensures robust testing in the presence of rare
# statistical events that more severely affect the shift
# calculation.
fudge_factor = 2.
max_relerr = fudge_factor / np.sqrt(Nreal_per_ens)
tools.assert_less_equal(
np.max(np.abs(relerr_1)),
max_relerr)
tools.assert_less_equal(
np.max(np.abs(relerr_05)),
max_relerr)
return
