def test_odd_ext(self):
a = np.array([[1, 2, 3, 4, 5], [9, 8, 7, 6, 5]])
odd = odd_ext(a, 2, axis=1)
expected = np.array([[-1, 0, 1, 2, 3, 4, 5, 6, 7],
[11, 10, 9, 8, 7, 6, 5, 4, 3]])
assert_array_equal(odd, expected)
odd = odd_ext(a, 1, axis=0)
expected = np.array([[-7, -4, -1, 2, 5], [1, 2, 3, 4, 5],
[9, 8, 7, 6, 5], [17, 14, 11, 8, 5]])
assert_array_equal(odd, expected)
assert_raises(ValueError, odd_ext, a, 2, axis=0)
assert_raises(ValueError, odd_ext, a, 5, axis=1)
def _validate_pad(padtype, padlen, x, axis, ntaps):
"""Helper to validate padding for filtfilt"""
if padtype not in ['even', 'odd', 'constant', None]:
raise ValueError(
("Unknown value '%s' given to padtype. padtype "
"must be 'even', 'odd', 'constant', or None.") % padtype)
if padtype is None:
padlen = 0
if padlen is None:
# Original padding; preserved for backwards compatibility.
edge = ntaps * 3
else:
edge = padlen
# x's 'axis' dimension must be bigger than edge.
if x.shape[axis] <= edge:
raise ValueError("The length of the input vector x must be at least "
"padlen, which is %d." % edge)
if padtype is not None and edge > 0:
# Make an extension of length `edge` at each
# end of the input array.
if padtype == 'even':
ext = even_ext(x, edge, axis=axis)
elif padtype == 'odd':
ext = odd_ext(x, edge, axis=axis)
else:
ext = const_ext(x, edge, axis=axis)
else:
ext = x
return edge, ext
def test_odd_ext(self):
a = np.array([[1, 2, 3, 4, 5],
[9, 8, 7, 6, 5]])
odd = odd_ext(a, 2, axis=1)
expected = np.array([[-1, 0, 1, 2, 3, 4, 5, 6, 7],
[11, 10, 9, 8, 7, 6, 5, 4, 3]])
assert_array_equal(odd, expected)
odd = odd_ext(a, 1, axis=0)
expected = np.array([[-7, -4, -1, 2, 5],
[1, 2, 3, 4, 5],
[9, 8, 7, 6, 5],
[17, 14, 11, 8, 5]])
assert_array_equal(odd, expected)
assert_raises(ValueError, odd_ext, a, 2, axis=0)
assert_raises(ValueError, odd_ext, a, 5, axis=1)
def _sosfiltfilt(sos, x, axis=-1, padtype='odd', padlen=None, method='pad', irlen=None):
"""Filtfilt version using Second Order sections. Code is taken from scipy.signal.filtfilt and adapted to make it work with SOS.
Note that broadcasting does not work.
"""
from scipy.signal import sosfilt_zi
from scipy.signal._arraytools import odd_ext, axis_slice, axis_reverse
x = np.asarray(x)
if padlen is None:
edge = 0
else:
edge = padlen
# x's 'axis' dimension must be bigger than edge.
if x.shape[axis] <= edge:
raise ValueError("The length of the input vector x must be at least "
"padlen, which is %d." % edge)
if padtype is not None and edge > 0:
# Make an extension of length `edge` at each
# end of the input array.
if padtype == 'even':
ext = even_ext(x, edge, axis=axis)
elif padtype == 'odd':
ext = odd_ext(x, edge, axis=axis)
else:
ext = const_ext(x, edge, axis=axis)
else:
ext = x
# Get the steady state of the filter's step response.
zi = sosfilt_zi(sos)
# Reshape zi and create x0 so that zi*x0 broadcasts
# to the correct value for the 'zi' keyword argument
# to lfilter.
#zi_shape = [1] * x.ndim
#zi_shape[axis] = zi.size
#zi = np.reshape(zi, zi_shape)
x0 = axis_slice(ext, stop=1, axis=axis)
# Forward filter.
(y, zf) = sosfilt(sos, ext, axis=axis, zi=zi * x0)
# Backward filter.
# Create y0 so zi*y0 broadcasts appropriately.
y0 = axis_slice(y, start=-1, axis=axis)
(y, zf) = sosfilt(sos, axis_reverse(y, axis=axis), axis=axis, zi=zi * y0)
# Reverse y.
y = axis_reverse(y, axis=axis)
if edge > 0:
# Slice the actual signal from the extended signal.
y = axis_slice(y, start=edge, stop=-edge, axis=axis)
return y
def _sosfiltfilt(sos, x, axis=-1, padtype='odd', padlen=None, method='pad', irlen=None):
"""Filtfilt version using Second Order sections. Code is taken from scipy.signal.filtfilt and adapted to make it work with SOS.
Note that broadcasting does not work.
"""
from scipy.signal import sosfilt_zi
from scipy.signal._arraytools import odd_ext, axis_slice, axis_reverse
x = np.asarray(x)
if padlen is None:
edge = 0
else:
edge = padlen
# x's 'axis' dimension must be bigger than edge.
if x.shape[axis] <= edge:
raise ValueError("The length of the input vector x must be at least "
"padlen, which is %d." % edge)
if padtype is not None and edge > 0:
# Make an extension of length `edge` at each
# end of the input array.
if padtype == 'even':
ext = even_ext(x, edge, axis=axis)
elif padtype == 'odd':
ext = odd_ext(x, edge, axis=axis)
else:
ext = const_ext(x, edge, axis=axis)
else:
ext = x
# Get the steady state of the filter's step response.
zi = sosfilt_zi(sos)
# Reshape zi and create x0 so that zi*x0 broadcasts
# to the correct value for the 'zi' keyword argument
# to lfilter.
#zi_shape = [1] * x.ndim
#zi_shape[axis] = zi.size
#zi = np.reshape(zi, zi_shape)
x0 = axis_slice(ext, stop=1, axis=axis)
# Forward filter.
(y, zf) = sosfilt(sos, ext, axis=axis, zi=zi * x0)
# Backward filter.
# Create y0 so zi*y0 broadcasts appropriately.
y0 = axis_slice(y, start=-1, axis=axis)
(y, zf) = sosfilt(sos, axis_reverse(y, axis=axis), axis=axis, zi=zi * y0)
# Reverse y.
y = axis_reverse(y, axis=axis)
if edge > 0:
# Slice the actual signal from the extended signal.
y = axis_slice(y, start=edge, stop=-edge, axis=axis)
return y
def butter_bandpass_sosfiltfilt(self,
lowcut=None,
highcut=None,
accel_axis="z",
order=5,
axis=-1,
padtype="odd",
padlen=None,
method='pad',
irlen=None):
'''Filtfilt version using Second Order sections. Code is taken from scipy.signal.filtfilt and adapted to make it work with
sos. Note that broadcasting does not work'''
data = np.asarray(getattr(self.df, accel_axis[:]))
sos = self.butter_bandpass(lowcut, highcut, order)
if padlen is None:
edge = 0
else:
edge = padlen
if data.shape[axis] <= edge:
raise ValueError(
"The length of the input vector x must be at least padlen, which is %d."
% edge)
if padtype is not None and edge > 0:
if padtype == "even":
ext = even_ext(data, edge, axis=axis)
elif padtype == "odd":
ext = odd_ext(data, edge, axis=axis)
else:
ext = const_ext(data, edge, axis=axis)
else:
ext = data
# Get the steady state of the filter's first step resopnse
zi = sosfilt_zi(sos)
# Reshape zi and create x0 so that zi*x0 broadcasts to the correct value for the zi keyword argument to lfilter
x0 = axis_slice(ext, stop=1, axis=axis)
# Forward filter
(y, zf) = sosfilt(sos, ext, axis=axis, zi=zi * x0)
y0 = axis_slice(y, start=-1, axis=axis)
# Backward filter
(y, zf) = sosfilt(sos,
axis_reverse(y, axis=axis),
axis=axis,
zi=zi * y0)
y = axis_reverse(y, axis=axis)
if edge > 0:
y = axis_slice(y, start=edge, stop=-edge, axis=axis)
return y
def sosfiltfilt(sos, x, axis=-1, padtype="odd", padlen=0):
x = asarray(x)
# `method` is "pad"
if padtype not in ["even", "odd", "constant", None]:
raise ValueError(
("Unknown value '%s' given to padtype. padtype " "must be 'even', 'odd', 'constant', or None.") % padtype
)
if padtype is None:
padlen = 0
if padlen is None:
edge = sos.shape[0] * 6
else:
edge = padlen
# x's 'axis' dimension must be bigger than edge.
if x.shape[axis] <= edge:
raise ValueError("The length of the input vector x must be at least " "padlen, which is %d." % edge)
if padtype is not None and edge > 0:
# Make an extension of length `edge` at each
# end of the input array.
if padtype == "even":
ext = even_ext(x, edge, axis=axis)
elif padtype == "odd":
ext = odd_ext(x, edge, axis=axis)
else:
ext = const_ext(x, edge, axis=axis)
else:
ext = x
# Get the steady state of the filter's step response.
zi = sosfilt_zi(sos)
# Reshape zi and create x0 so that zi*x0 broadcasts
# to the correct value for the 'zi' keyword argument
# to lfilter.
zi_shape = [1] * x.ndim
zi_shape[axis] = zi.size
zi = reshape(zi, zi_shape)
x0 = axis_slice(ext, stop=1, axis=axis)
zix0 = reshape(zi * x0, (sos.shape[0], 2))
# Forward filter
(y, zf) = sosfilt(sos, ext, axis=axis, zi=zix0)
# Backward filter
# Create y0 so zi*y0 broadcasts appropriately.
y0 = axis_slice(y, start=-1, axis=axis)
ziy0 = reshape(zi * y0, (sos.shape[0], 2))
(y, zf) = sosfilt(sos, axis_reverse(y, axis=axis), axis=axis, zi=ziy0)
# Reverse y
y = axis_reverse(y, axis=axis)
if edge > 0:
# Slice the actual signal from the extended signal.
y = axis_slice(y, start=edge, stop=-edge, axis=axis)
return y
def cogve(COP, freq, mass, height, show=False, ax=None):
"""COGv estimation using COP data based on the inverted pendulum model.
This function estimates the center of gravity vertical projection (COGv)
displacement from the center of pressure (COP) displacement at the
anterior-posterior direction during quiet upright standing. COP and COGv
displacements are measurements useful to quantify the postural sway of a
person while standing.
The COGv displacement is estimated by low-pass filtering the COP
displacement in the frequency domain according to the person's moment
of rotational inertia as a single inverted pendulum [1]_.
Parameters
----------
COP : 1D array_like
center of pressure data [cm]
freq : float
sampling frequency of the COP data
mass : float
body mass of the subject [kg]
height : float
height of the subject [cm]
show : bool, optional (default = False)
True (1) plots data and results in a matplotlib figure
False (0) to not plot
ax : matplotlib.axes.Axes instance, optional (default = None)
Returns
-------
COGv : 1D array
center of gravity vertical projection data [cm]
References
----------
.. [1] http://nbviewer.ipython.org/github/demotu/BMC/blob/master/notebooks/IP_Model.ipynb
Examples
--------
>>> from cogve import cogve
>>> y = np.cumsum(np.random.randn(3000))/50
>>> cogv = cogve(y, freq=100, mass=70, height=170, show=True)
"""
from scipy.signal._arraytools import odd_ext
import scipy.fftpack
COP = np.asarray(COP)
height = height / 100 # cm to m
g = 9.8 # gravity acceleration in m/s2
# height of the COG w.r.t. ankle (McGinnis, 2005; Winter, 2005)
hcog = 0.56 * height - 0.039 * height
# body moment of inertia around the ankle
# (Breniere, 1996), (0.0572 for the ml direction)
I = mass * 0.0533 * height**2 + mass * hcog**2
# Newton-Euler equation of motion for the inverted pendulum
# COGv'' = w02*(COGv - COP)
# where w02 is the squared pendulum natural frequency
w02 = mass * g * hcog / I
# add (pad) data and remove mean to avoid problems at the extremities
COP = odd_ext(COP, n=freq)
COPm = np.mean(COP)
COP = COP - COPm
# COGv is estimated by filtering the COP data in the frequency domain
# using the transfer function for the inverted pendulum equation of motion
N = COP.size
COPfft = scipy.fftpack.fft(COP, n=N) / N # COP fft
w = 2 * np.pi * scipy.fftpack.fftfreq(n=N, d=1 / freq) # angular frequency
# transfer function
TF = w02 / (w02 + w**2)
COGv = np.real(scipy.fftpack.ifft(TF * COPfft) * N)
COGv = COGv[0:N]
# get back the mean and pad off data
COP, COGv = COP + COPm, COGv + COPm
COP, COGv = COP[freq:-freq], COGv[freq:-freq]
if show:
_plot(COP, COGv, freq, ax)
return COGv
def sosfiltfilt(sos, x, axis=-1, padtype='odd', padlen=0):
x = asarray(x)
# `method` is "pad"
if padtype not in ['even', 'odd', 'constant', None]:
raise ValueError(("Unknown value '%s' given to padtype. padtype "
"must be 'even', 'odd', 'constant', or None.") %
padtype)
if padtype is None:
padlen = 0
if padlen is None:
edge = sos.shape[0] * 6
else:
edge = padlen
# x's 'axis' dimension must be bigger than edge.
if x.shape[axis] <= edge:
raise ValueError("The length of the input vector x must be at least "
"padlen, which is %d." % edge)
if padtype is not None and edge > 0:
# Make an extension of length `edge` at each
# end of the input array.
if padtype == 'even':
ext = even_ext(x, edge, axis=axis)
elif padtype == 'odd':
ext = odd_ext(x, edge, axis=axis)
else:
ext = const_ext(x, edge, axis=axis)
else:
ext = x
# Get the steady state of the filter's step response.
zi = sosfilt_zi(sos)
# Reshape zi and create x0 so that zi*x0 broadcasts
# to the correct value for the 'zi' keyword argument
# to lfilter.
zi_shape = [1] * x.ndim
zi_shape[axis] = zi.size
zi = reshape(zi, zi_shape)
x0 = axis_slice(ext, stop=1, axis=axis)
zix0 = reshape(zi * x0, (sos.shape[0], 2))
# Forward filter
(y, zf) = sosfilt(sos, ext, axis=axis, zi=zix0)
# Backward filter
# Create y0 so zi*y0 broadcasts appropriately.
y0 = axis_slice(y, start=-1, axis=axis)
ziy0 = reshape(zi * y0, (sos.shape[0], 2))
(y, zf) = sosfilt(sos, axis_reverse(y, axis=axis), axis=axis, zi=ziy0)
# Reverse y
y = axis_reverse(y, axis=axis)
if edge > 0:
# Slice the actual signal from the extended signal.
y = axis_slice(y, start=edge, stop=-edge, axis=axis)
return y
def filtfilt(b, a, x, axis=-1, padtype='odd', padlen=None):
"""
A forward-backward filter.
This function applies a linear filter twice, once forward
and once backwards. The combined filter has linear phase.
Before applying the filter, the function can pad the data along the
given axis in one of three ways: odd, even or constant. The odd
and even extensions have the corresponding symmetry about the end point
of the data. The constant extension extends the data with the values
at end points. On both the forward and backwards passes, the
initial condition of the filter is found by using `lfilter_zi` and
scaling it by the end point of the extended data.
Parameters
----------
b : (N,) array_like
The numerator coefficient vector of the filter.
a : (N,) array_like
The denominator coefficient vector of the filter. If a[0]
is not 1, then both a and b are normalized by a[0].
x : array_like
The array of data to be filtered.
axis : int, optional
The axis of `x` to which the filter is applied.
Default is -1.
padtype : str or None, optional
Must be 'odd', 'even', 'constant', or None. This determines the
type of extension to use for the padded signal to which the filter
is applied. If `padtype` is None, no padding is used. The default
is 'odd'.
padlen : int or None, optional
The number of elements by which to extend `x` at both ends of
`axis` before applying the filter. This value must be less than
`x.shape[axis]-1`. `padlen=0` implies no padding.
The default value is 3*max(len(a),len(b)).
Returns
-------
y : ndarray
The filtered output, an array of type numpy.float64 with the same
shape as `x`.
See Also
--------
lfilter_zi, lfilter
Examples
--------
First we create a one second signal that is the sum of two pure sine
waves, with frequencies 5 Hz and 250 Hz, sampled at 2000 Hz.
>>> t = np.linspace(0, 1.0, 2001)
>>> xlow = np.sin(2 * np.pi * 5 * t)
>>> xhigh = np.sin(2 * np.pi * 250 * t)
>>> x = xlow + xhigh
Now create a lowpass Butterworth filter with a cutoff of 0.125 times
the Nyquist rate, or 125 Hz, and apply it to x with filtfilt. The
result should be approximately xlow, with no phase shift.
>>> from scipy import signal
>>> b, a = signal.butter(8, 0.125)
>>> y = filtfilt(b, a, x, padlen=150)
>>> print('%.5g' % np.abs(y - xlow).max())
9.1086e-06
We get a fairly clean result for this artificial example because
the odd extension is exact, and with the moderately long padding,
the filter's transients have dissipated by the time the actual data
is reached. In general, transient effects at the edges are
unavoidable.
"""
if padtype not in ['even', 'odd', 'constant', None]:
raise ValueError(("Unknown value '%s' given to padtype. padtype must "
"be 'even', 'odd', 'constant', or None.") %
padtype)
b = np.asarray(b)
a = np.asarray(a)
x = np.asarray(x)
ntaps = max(len(a), len(b))
if padtype is None:
padlen = 0
if padlen is None:
# Original padding; preserved for backwards compatibility.
edge = ntaps * 3
else:
edge = padlen
# x's 'axis' dimension must be bigger than edge.
if x.shape[axis] <= edge:
raise ValueError("The length of the input vector x must be at least "
"padlen, which is %d." % edge)
if padtype is not None and edge > 0:
# Make an extension of length `edge` at each
# end of the input array.
if padtype == 'even':
ext = even_ext(x, edge, axis=axis)
elif padtype == 'odd':
ext = odd_ext(x, edge, axis=axis)
else:
ext = const_ext(x, edge, axis=axis)
else:
ext = x
# Get the steady state of the filter's step response.
zi = lfilter_zi(b, a)
# Reshape zi and create x0 so that zi*x0 broadcasts
# to the correct value for the 'zi' keyword argument
# to lfilter.
zi_shape = [1] * x.ndim
zi_shape[axis] = zi.size
zi = np.reshape(zi, zi_shape)
x0 = axis_slice(ext, stop=1, axis=axis)
# Forward filter.
(y, zf) = lfilter(b, a, ext, axis=axis, zi=zi * x0)
# Backward filter.
# Create y0 so zi*y0 broadcasts appropriately.
y0 = axis_slice(y, start=-1, axis=axis)
(y, zf) = lfilter(b, a, axis_reverse(y, axis=axis), axis=axis, zi=zi * y0)
# Reverse y.
y = axis_reverse(y, axis=axis)
if edge > 0:
# Slice the actual signal from the extended signal.
y = axis_slice(y, start=edge, stop=-edge, axis=axis)
return y
def filtfilt(b, a, x, axis=-1, padtype='odd', padlen=None):
"""
A forward-backward filter.
This function applies a linear filter twice, once forward
and once backwards. The combined filter has linear phase.
Before applying the filter, the function can pad the data along the
given axis in one of three ways: odd, even or constant. The odd
and even extensions have the corresponding symmetry about the end point
of the data. The constant extension extends the data with the values
at end points. On both the forward and backwards passes, the
initial condition of the filter is found by using `lfilter_zi` and
scaling it by the end point of the extended data.
Parameters
----------
b : (N,) array_like
The numerator coefficient vector of the filter.
a : (N,) array_like
The denominator coefficient vector of the filter. If a[0]
is not 1, then both a and b are normalized by a[0].
x : array_like
The array of data to be filtered.
axis : int, optional
The axis of `x` to which the filter is applied.
Default is -1.
padtype : str or None, optional
Must be 'odd', 'even', 'constant', or None. This determines the
type of extension to use for the padded signal to which the filter
is applied. If `padtype` is None, no padding is used. The default
is 'odd'.
padlen : int or None, optional
The number of elements by which to extend `x` at both ends of
`axis` before applying the filter. This value must be less than
`x.shape[axis]-1`. `padlen=0` implies no padding.
The default value is 3*max(len(a),len(b)).
Returns
-------
y : ndarray
The filtered output, an array of type numpy.float64 with the same
shape as `x`.
See Also
--------
lfilter_zi, lfilter
Examples
--------
First we create a one second signal that is the sum of two pure sine
waves, with frequencies 5 Hz and 250 Hz, sampled at 2000 Hz.
>>> t = np.linspace(0, 1.0, 2001)
>>> xlow = np.sin(2 * np.pi * 5 * t)
>>> xhigh = np.sin(2 * np.pi * 250 * t)
>>> x = xlow + xhigh
Now create a lowpass Butterworth filter with a cutoff of 0.125 times
the Nyquist rate, or 125 Hz, and apply it to x with filtfilt. The
result should be approximately xlow, with no phase shift.
>>> from scipy import signal
>>> b, a = signal.butter(8, 0.125)
>>> y = filtfilt(b, a, x, padlen=150)
>>> print('%.5g' % np.abs(y - xlow).max())
9.1086e-06
We get a fairly clean result for this artificial example because
the odd extension is exact, and with the moderately long padding,
the filter's transients have dissipated by the time the actual data
is reached. In general, transient effects at the edges are
unavoidable.
"""
if padtype not in ['even', 'odd', 'constant', None]:
raise ValueError(
("Unknown value '%s' given to padtype. padtype must "
"be 'even', 'odd', 'constant', or None.") % padtype)
b = np.asarray(b)
a = np.asarray(a)
x = np.asarray(x)
ntaps = max(len(a), len(b))
if padtype is None:
padlen = 0
if padlen is None:
# Original padding; preserved for backwards compatibility.
edge = ntaps * 3
else:
edge = padlen
# x's 'axis' dimension must be bigger than edge.
if x.shape[axis] <= edge:
raise ValueError(
"The length of the input vector x must be at least "
"padlen, which is %d." % edge)
if padtype is not None and edge > 0:
# Make an extension of length `edge` at each
# end of the input array.
if padtype == 'even':
ext = even_ext(x, edge, axis=axis)
elif padtype == 'odd':
ext = odd_ext(x, edge, axis=axis)
else:
ext = const_ext(x, edge, axis=axis)
else:
ext = x
# Get the steady state of the filter's step response.
zi = lfilter_zi(b, a)
# Reshape zi and create x0 so that zi*x0 broadcasts
# to the correct value for the 'zi' keyword argument
# to lfilter.
zi_shape = [1] * x.ndim
zi_shape[axis] = zi.size
zi = np.reshape(zi, zi_shape)
x0 = axis_slice(ext, stop=1, axis=axis)
# Forward filter.
(y, zf) = lfilter(b, a, ext, axis=axis, zi=zi * x0)
# Backward filter.
# Create y0 so zi*y0 broadcasts appropriately.
y0 = axis_slice(y, start=-1, axis=axis)
(y, zf) = lfilter(b,
a,
axis_reverse(y, axis=axis),
axis=axis,
zi=zi * y0)
# Reverse y.
y = axis_reverse(y, axis=axis)
if edge > 0:
# Slice the actual signal from the extended signal.
y = axis_slice(y, start=edge, stop=-edge, axis=axis)
return y
def filt_FFTWEAVE(b, x, padtype='odd', padlen=None):
"""A forward-backward filter.
This function applies a linear filter twice, once forward
and once backwards. The combined filter has linear phase.
Before applying the filter, the function can pad the data along the
given axis in one of three ways: odd, even or constant. The odd
and even extensions have the corresponding symmetry about the end point
of the data. The constant extension extends the data with the values
at end points. On both the forward and backwards passes, the
initial condition of the filter is found by using lfilter_zi and
scaling it by the end point of the extended data.
Parameters
----------
b : array_like, 1-D
The numerator coefficient vector of the filter.
a : array_like, 1-D
The denominator coefficient vector of the filter. If a[0]
is not 1, then both a and b are normalized by a[0].
x : array_like
The array of data to be filtered.
padtype : str or None, optional
Must be 'odd', 'even', 'constant', or None. This determines the
type of extension to use for the padded signal to which the filter
is applied. If `padtype` is None, no padding is used. The default
is 'odd'.
padlen : int or None, optional
The number of elements by which to extend `x` at both ends of
`axis` before applying the filter. This value must be less than
`x.shape[axis]-1`. `padlen=0` implies no padding.
The default value is 3*max(len(a),len(b)).
Returns
-------
y : ndarray
The filtered output, an array of type numpy.float64 with the same
shape as `x`.
See Also
--------
lfilter_zi
lfilter
Examples
--------
First we create a one second signal that is the sum of two pure sine
waves, with frequencies 5 Hz and 250 Hz, sampled at 2000 Hz.
>>> t = np.linspace(0, 1.0, 2001)
>>> xlow = np.sin(2 * np.pi * 5 * t)
>>> xhigh = np.sin(2 * np.pi * 250 * t)
>>> x = xlow + xhigh
Now create a lowpass Butterworth filter with a cutoff of 0.125 times
the Nyquist rate, or 125 Hz, and apply it to x with filtfilt. The
result should be approximately xlow, with no phase shift.
>>> from scipy.signal import butter
>>> b, a = butter(8, 0.125)
>>> y = filtfilt(b, a, x, padlen=150)
>>> np.abs(y - xlow).max()
9.1086182074789912e-06
We get a fairly clean result for this artificial example because
the odd extension is exact, and with the moderately long padding,
the filter's transients have dissipated by the time the actual data
is reached. In general, transient effects at the edges are
unavoidable.
"""
if padtype not in ['even', 'odd', 'constant', None]:
raise ValueError(("Unknown value '%s' given to padtype. padtype must "
"be 'even', 'odd', 'constant', or None.") % padtype)
b = np.asarray(b)
x = np.asarray(x)
ntaps = len(b)
if padtype is None:
padlen = 0
if padlen is None:
# Original padding; preserved for backwards compatibility.
edge = ntaps * 3
else:
edge = padlen
# x's 'axis' dimension must be bigger than edge.
#if x.shape[axis] <= edge:
if len(x) <= edge:
raise ValueError(
"The length of the input vector x must be larger than "
"padlen, which is %d." % edge)
if padtype is not None and edge > 0:
# Make an extension of length `edge` at each
# end of the input array.
if padtype == 'even':
ext = even_ext(x, edge) #, axis=axis)
elif padtype == 'odd':
ext = odd_ext(x, edge) #, axis=axis)
else:
ext = const_ext(x, edge) #, axis=axis)
else:
ext = x
# Get the steady state of the filter's step response.
#zi = lfilter_zi(b, a)
# Reshape zi and create x0 so that zi*x0 broadcasts
# to the correct value for the 'zi' keyword argument
# to lfilter.
#zi_shape = [1] * x.ndim
#zi_shape[axis] = zi.size
#zi = np.reshape(zi, zi_shape)
#x0 = axis_slice(ext, stop=1, axis=axis)
# Forward filter.
ext = convolve_cython(ext, b)
"""
if filtConf.useWeave:
try :
ext = convolve_weave(ext, b)
except (CompileError, WindowsError) as e:
ext = convolve_noWeave(ext, b)
warnings.warn("Weave failed at runtime. Setting useWeave to False. This will results in slower filtering.", RuntimeWarning)
filtConf.useWeave = False
else:
ext = convolve_noWeave(ext, b)
"""
if edge > 0:
# Slice the actual signal from the extended signal. Reverse and return y.
return ext[edge:-edge]
else:
# Reverse and return y.
return ext
def cogve(COP, freq, mass, height, show=False, ax=None):
"""COGv estimation using COP data based on the inverted pendulum model.
This function estimates the center of gravity vertical projection (COGv)
displacement from the center of pressure (COP) displacement at the
anterior-posterior direction during quiet upright standing. COP and COGv
displacements are measurements useful to quantify the postural sway of a
person while standing.
The COGv displacement is estimated by low-pass filtering the COP
displacement in the frequency domain according to the person's moment
of rotational inertia as a single inverted pendulum [1]_.
Parameters
----------
COP : 1D array_like
center of pressure data [cm]
freq : float
sampling frequency of the COP data
mass : float
body mass of the subject [kg]
height : float
height of the subject [cm]
show : bool, optional (default = False)
True (1) plots data and results in a matplotlib figure
False (0) to not plot
ax : matplotlib.axes.Axes instance, optional (default = None)
Returns
-------
COGv : 1D array
center of gravity vertical projection data [cm]
References
----------
.. [1] http://nbviewer.ipython.org/github/demotu/BMC/blob/master/notebooks/IP_Model.ipynb
Examples
--------
>>> from cogve import cogve
>>> y = np.cumsum(np.random.randn(3000))/50
>>> cogv = cogve(y, freq=100, mass=70, height=170, show=True)
"""
from scipy.signal._arraytools import odd_ext
import scipy.fftpack
COP = np.asarray(COP)
height = height / 100 # cm to m
g = 9.8 # gravity acceleration in m/s2
# height of the COG w.r.t. ankle (McGinnis, 2005; Winter, 2005)
hcog = 0.56 * height - 0.039 * height
# body moment of inertia around the ankle
# (Breniere, 1996), (0.0572 for the ml direction)
I = mass * 0.0533 * height ** 2 + mass * hcog ** 2
# Newton-Euler equation of motion for the inverted pendulum
# COGv'' = w02*(COGv - COP)
# where w02 is the squared pendulum natural frequency
w02 = mass * g * hcog / I
# add (pad) data and remove mean to avoid problems at the extremities
COP = odd_ext(COP, n=freq)
COPm = np.mean(COP)
COP = COP - COPm
# COGv is estimated by filtering the COP data in the frequency domain
# using the transfer function for the inverted pendulum equation of motion
N = COP.size
COPfft = scipy.fftpack.fft(COP, n=N) / N # COP fft
w = 2 * np.pi * scipy.fftpack.fftfreq(n=N, d=1 / freq) # angular frequency
# transfer function
TF = w02 / (w02 + w ** 2)
COGv = np.real(scipy.fftpack.ifft(TF * COPfft) * N)
COGv = COGv[0: N]
# get back the mean and pad off data
COP, COGv = COP + COPm, COGv + COPm
COP, COGv = COP[freq: -freq], COGv[freq: -freq]
if show:
_plot(COP, COGv, freq, ax)
return COGv
