def _fit_edge(x, window_start, window_stop, interp_start, interp_stop,
axis, polyorder, deriv, delta, y):
"""
Given an n-d array `x` and the specification of a slice of `x` from
`window_start` to `window_stop` along `axis`, create an interpolating
polynomial of each 1-d slice, and evaluate that polynomial in the slice
from `interp_start` to `interp_stop`. Put the result into the
corresponding slice of `y`.
"""
# Get the edge into a (window_length, -1) array.
x_edge = axis_slice(x, start=window_start, stop=window_stop, axis=axis)
if axis == 0 or axis == -x.ndim:
xx_edge = x_edge
swapped = False
else:
xx_edge = x_edge.swapaxes(axis, 0)
swapped = True
xx_edge = xx_edge.reshape(xx_edge.shape[0], -1)
# Fit the edges. poly_coeffs has shape (polyorder + 1, -1),
# where '-1' is the same as in xx_edge.
poly_coeffs = np.polyfit(np.arange(0, window_stop - window_start),
xx_edge, polyorder)
if deriv > 0:
poly_coeffs = _polyder(poly_coeffs, deriv)
# Compute the interpolated values for the edge.
i = np.arange(interp_start - window_start, interp_stop - window_start)
values = np.polyval(poly_coeffs, i.reshape(-1, 1)) / (delta ** deriv)
# Now put the values into the appropriate slice of y.
# First reshape values to match y.
shp = list(y.shape)
shp[0], shp[axis] = shp[axis], shp[0]
values = values.reshape(interp_stop - interp_start, *shp[1:])
if swapped:
values = values.swapaxes(0, axis)
# Get a view of the data to be replaced by values.
y_edge = axis_slice(y, start=interp_start, stop=interp_stop, axis=axis)
y_edge[...] = values
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 : 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.
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.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)
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 : 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.
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.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)
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, 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), 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
