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# Author: Travis Oliphant
# 1999 -- 2002
from __future__ import division, print_function, absolute_import
import warnings
from . import sigtools
from scipy.lib.six import callable
from scipy import linalg
from scipy.fftpack import fft, ifft, ifftshift, fft2, ifft2, fftn, \
ifftn, fftfreq
from numpy.fft import rfftn, irfftn
from numpy import polyadd, polymul, polydiv, polysub, roots, \
poly, polyval, polyder, cast, asarray, isscalar, atleast_1d, \
ones, real_if_close, zeros, array, arange, where, rank, \
newaxis, product, ravel, sum, r_, iscomplexobj, take, \
argsort, allclose, expand_dims, unique, prod, sort, reshape, \
transpose, dot, mean, ndarray, atleast_2d
import numpy as np
from scipy.misc import factorial
from .windows import get_window
from ._arraytools import axis_slice, axis_reverse, odd_ext, even_ext, const_ext
__all__ = ['correlate', 'fftconvolve', 'convolve', 'convolve2d', 'correlate2d',
'order_filter', 'medfilt', 'medfilt2d', 'wiener', 'lfilter',
'lfiltic', 'deconvolve', 'hilbert', 'hilbert2', 'cmplx_sort',
'unique_roots', 'invres', 'invresz', 'residue', 'residuez',
'resample', 'detrend', 'lfilter_zi', 'filtfilt', 'decimate']
_modedict = {'valid': 0, 'same': 1, 'full': 2}
_boundarydict = {'fill': 0, 'pad': 0, 'wrap': 2, 'circular': 2, 'symm': 1,
'symmetric': 1, 'reflect': 4}
def _valfrommode(mode):
try:
val = _modedict[mode]
except KeyError:
if mode not in [0, 1, 2]:
raise ValueError("Acceptable mode flags are 'valid' (0),"
" 'same' (1), or 'full' (2).")
val = mode
return val
def _bvalfromboundary(boundary):
try:
val = _boundarydict[boundary] << 2
except KeyError:
if val not in [0, 1, 2]:
raise ValueError("Acceptable boundary flags are 'fill', 'wrap'"
" (or 'circular'), \n and 'symm' (or 'symmetric').")
val = boundary << 2
return val
def _check_valid_mode_shapes(shape1, shape2):
for d1, d2 in zip(shape1, shape2):
if not d1 >= d2:
raise ValueError(
"in1 should have at least as many items as in2 in "
"every dimension for 'valid' mode.")
def correlate(in1, in2, mode='full'):
"""
Cross-correlate two N-dimensional arrays.
Cross-correlate `in1` and `in2`, with the output size determined by the
`mode` argument.
Parameters
----------
in1 : array_like
First input.
in2 : array_like
Second input. Should have the same number of dimensions as `in1`;
if sizes of `in1` and `in2` are not equal then `in1` has to be the
larger array.
mode : str {'full', 'valid', 'same'}, optional
A string indicating the size of the output:
``full``
The output is the full discrete linear cross-correlation
of the inputs. (Default)
``valid``
The output consists only of those elements that do not
rely on the zero-padding.
``same``
The output is the same size as `in1`, centered
with respect to the 'full' output.
Returns
-------
correlate : array
An N-dimensional array containing a subset of the discrete linear
cross-correlation of `in1` with `in2`.
Notes
-----
The correlation z of two arrays x and y of rank d is defined as:
z[...,k,...] = sum[..., i_l, ...]
x[..., i_l,...] * conj(y[..., i_l + k,...])
"""
in1 = asarray(in1)
in2 = asarray(in2)
val = _valfrommode(mode)
if rank(in1) == rank(in2) == 0:
return in1 * in2
elif not in1.ndim == in2.ndim:
raise ValueError("in1 and in2 should have the same rank")
if mode == 'valid':
_check_valid_mode_shapes(in1.shape, in2.shape)
ps = [i - j + 1 for i, j in zip(in1.shape, in2.shape)]
out = np.empty(ps, in1.dtype)
z = sigtools._correlateND(in1, in2, out, val)
else:
ps = [i + j - 1 for i, j in zip(in1.shape, in2.shape)]
# zero pad input
in1zpadded = np.zeros(ps, in1.dtype)
sc = [slice(0, i) for i in in1.shape]
in1zpadded[sc] = in1.copy()
if mode == 'full':
out = np.empty(ps, in1.dtype)
elif mode == 'same':
out = np.empty(in1.shape, in1.dtype)
z = sigtools._correlateND(in1zpadded, in2, out, val)
return z
def _centered(arr, newsize):
# Return the center newsize portion of the array.
newsize = asarray(newsize)
currsize = array(arr.shape)
startind = (currsize - newsize) // 2
endind = startind + newsize
myslice = [slice(startind[k], endind[k]) for k in range(len(endind))]
return arr[tuple(myslice)]
def fftconvolve(in1, in2, mode="full"):
"""Convolve two N-dimensional arrays using FFT.
Convolve `in1` and `in2` using the fast Fourier transform method, with
the output size determined by the `mode` argument.
This is generally much faster than `convolve` for large arrays (n > ~500),
but can be slower when only a few output values are needed, and can only
output float arrays (int or object array inputs will be cast to float).
Parameters
----------
in1 : array_like
First input.
in2 : array_like
Second input. Should have the same number of dimensions as `in1`;
if sizes of `in1` and `in2` are not equal then `in1` has to be the
larger array.
mode : str {'full', 'valid', 'same'}, optional
A string indicating the size of the output:
``full``
The output is the full discrete linear convolution
of the inputs. (Default)
``valid``
The output consists only of those elements that do not
rely on the zero-padding.
``same``
The output is the same size as `in1`, centered
with respect to the 'full' output.
Returns
-------
out : array
An N-dimensional array containing a subset of the discrete linear
convolution of `in1` with `in2`.
"""
in1 = asarray(in1)
in2 = asarray(in2)
if rank(in1) == rank(in2) == 0: # scalar inputs
return in1 * in2
elif not in1.ndim == in2.ndim:
raise ValueError("in1 and in2 should have the same rank")
elif in1.size == 0 or in2.size == 0: # empty arrays
return array([])
s1 = array(in1.shape)
s2 = array(in2.shape)
complex_result = (np.issubdtype(in1.dtype, np.complex) or
np.issubdtype(in2.dtype, np.complex))
size = s1 + s2 - 1
if mode == "valid":
for d1, d2 in zip(s1, s2):
if not d1 >= d2:
warnings.warn("in1 should have at least as many items as in2 in "
"every dimension for 'valid' mode. In scipy "
"0.13.0 this will raise an error",
DeprecationWarning)
# Always use 2**n-sized FFT
fsize = 2 ** np.ceil(np.log2(size)).astype(int)
fslice = tuple([slice(0, int(sz)) for sz in size])
if not complex_result:
ret = irfftn(rfftn(in1, fsize) *
rfftn(in2, fsize), fsize)[fslice].copy()
ret = ret.real
else:
ret = ifftn(fftn(in1, fsize) * fftn(in2, fsize))[fslice].copy()
if mode == "full":
return ret
elif mode == "same":
return _centered(ret, s1)
elif mode == "valid":
return _centered(ret, abs(s1 - s2) + 1)
def convolve(in1, in2, mode='full'):
"""
Convolve two N-dimensional arrays.
Convolve `in1` and `in2`, with the output size determined by the
`mode` argument.
Parameters
----------
in1 : array_like
First input.
in2 : array_like
Second input. Should have the same number of dimensions as `in1`;
if sizes of `in1` and `in2` are not equal then `in1` has to be the
larger array.
mode : str {'full', 'valid', 'same'}, optional
A string indicating the size of the output:
``full``
The output is the full discrete linear convolution
of the inputs. (Default)
``valid``
The output consists only of those elements that do not
rely on the zero-padding.
``same``
The output is the same size as `in1`, centered
with respect to the 'full' output.
Returns
-------
convolve : array
An N-dimensional array containing a subset of the discrete linear
convolution of `in1` with `in2`.
"""
volume = asarray(in1)
kernel = asarray(in2)
if rank(volume) == rank(kernel) == 0:
return volume * kernel
slice_obj = [slice(None, None, -1)] * len(kernel.shape)
if np.iscomplexobj(kernel):
return correlate(volume, kernel[slice_obj].conj(), mode)
else:
return correlate(volume, kernel[slice_obj], mode)
def order_filter(a, domain, rank):
"""
Perform an order filter on an N-dimensional array.
Perform an order filter on the array in. The domain argument acts as a
mask centered over each pixel. The non-zero elements of domain are
used to select elements surrounding each input pixel which are placed
in a list. The list is sorted, and the output for that pixel is the
element corresponding to rank in the sorted list.
Parameters
----------
a : ndarray
The N-dimensional input array.
domain : array_like
A mask array with the same number of dimensions as `in`.
Each dimension should have an odd number of elements.
rank : int
A non-negative integer which selects the element from the
sorted list (0 corresponds to the smallest element, 1 is the
next smallest element, etc.).
Returns
-------
out : ndarray
The results of the order filter in an array with the same
shape as `in`.
Examples
--------
>>> from scipy import signal
>>> x = np.arange(25).reshape(5, 5)
>>> domain = np.identity(3)
>>> x
array([[ 0, 1, 2, 3, 4],
[ 5, 6, 7, 8, 9],
[10, 11, 12, 13, 14],
[15, 16, 17, 18, 19],
[20, 21, 22, 23, 24]])
>>> signal.order_filter(x, domain, 0)
array([[ 0., 0., 0., 0., 0.],
[ 0., 0., 1., 2., 0.],
[ 0., 5., 6., 7., 0.],
[ 0., 10., 11., 12., 0.],
[ 0., 0., 0., 0., 0.]])
>>> signal.order_filter(x, domain, 2)
array([[ 6., 7., 8., 9., 4.],
[ 11., 12., 13., 14., 9.],
[ 16., 17., 18., 19., 14.],
[ 21., 22., 23., 24., 19.],
[ 20., 21., 22., 23., 24.]])
"""
domain = asarray(domain)
size = domain.shape
for k in range(len(size)):
if (size[k] % 2) != 1:
raise ValueError("Each dimension of domain argument "
" should have an odd number of elements.")
return sigtools._order_filterND(a, domain, rank)
def medfilt(volume, kernel_size=None):
"""
Perform a median filter on an N-dimensional array.
Apply a median filter to the input array using a local window-size
given by `kernel_size`.
Parameters
----------
volume : array_like
An N-dimensional input array.
kernel_size : array_like, optional
A scalar or an N-length list giving the size of the median filter
window in each dimension. Elements of `kernel_size` should be odd.
If `kernel_size` is a scalar, then this scalar is used as the size in
each dimension. Default size is 3 for each dimension.
Returns
-------
out : ndarray
An array the same size as input containing the median filtered
result.
"""
volume = atleast_1d(volume)
if kernel_size is None:
kernel_size = [3] * len(volume.shape)
kernel_size = asarray(kernel_size)
if len(kernel_size.shape) == 0:
kernel_size = [kernel_size.item()] * len(volume.shape)
kernel_size = asarray(kernel_size)
for k in range(len(volume.shape)):
if (kernel_size[k] % 2) != 1:
raise ValueError("Each element of kernel_size should be odd.")
domain = ones(kernel_size)
numels = product(kernel_size, axis=0)
order = numels // 2
return sigtools._order_filterND(volume, domain, order)
def wiener(im, mysize=None, noise=None):
"""
Perform a Wiener filter on an N-dimensional array.
Apply a Wiener filter to the N-dimensional array `im`.
Parameters
----------
im : ndarray
An N-dimensional array.
mysize : int or arraylike, optional
A scalar or an N-length list giving the size of the Wiener filter
window in each dimension. Elements of mysize should be odd.
If mysize is a scalar, then this scalar is used as the size
in each dimension.
noise : float, optional
The noise-power to use. If None, then noise is estimated as the
average of the local variance of the input.
Returns
-------
out : ndarray
Wiener filtered result with the same shape as `im`.
"""
im = asarray(im)
if mysize is None:
mysize = [3] * len(im.shape)
mysize = asarray(mysize)
# Estimate the local mean
lMean = correlate(im, ones(mysize), 'same') / product(mysize, axis=0)
# Estimate the local variance
lVar = (correlate(im ** 2, ones(mysize), 'same') / product(mysize, axis=0)
- lMean ** 2)
# Estimate the noise power if needed.
if noise is None:
noise = mean(ravel(lVar), axis=0)
res = (im - lMean)
res *= (1 - noise / lVar)
res += lMean
out = where(lVar < noise, lMean, res)
return out
def convolve2d(in1, in2, mode='full', boundary='fill', fillvalue=0):
"""
Convolve two 2-dimensional arrays.
Convolve `in1` and `in2` with output size determined by `mode`, and
boundary conditions determined by `boundary` and `fillvalue`.
Parameters
----------
in1, in2 : array_like
Two-dimensional input arrays to be convolved.
mode : str {'full', 'valid', 'same'}, optional
A string indicating the size of the output:
``full``
The output is the full discrete linear convolution
of the inputs. (Default)
``valid``
The output consists only of those elements that do not
rely on the zero-padding.
``same``
The output is the same size as `in1`, centered
with respect to the 'full' output.
boundary : str {'fill', 'wrap', 'symm'}, optional
A flag indicating how to handle boundaries:
``fill``
pad input arrays with fillvalue. (default)
``wrap``
circular boundary conditions.
``symm``
symmetrical boundary conditions.
fillvalue : scalar, optional
Value to fill pad input arrays with. Default is 0.
Returns
-------
out : ndarray
A 2-dimensional array containing a subset of the discrete linear
convolution of `in1` with `in2`.
"""
in1 = asarray(in1)
in2 = asarray(in2)
if mode == 'valid':
_check_valid_mode_shapes(in1.shape, in2.shape)
val = _valfrommode(mode)
bval = _bvalfromboundary(boundary)
with warnings.catch_warnings():
warnings.simplefilter('ignore', np.ComplexWarning)
# FIXME: some cast generates a warning here
out = sigtools._convolve2d(in1, in2, 1, val, bval, fillvalue)
return out
def correlate2d(in1, in2, mode='full', boundary='fill', fillvalue=0):
"""
Cross-correlate two 2-dimensional arrays.
Cross correlate `in1` and `in2` with output size determined by `mode`, and
boundary conditions determined by `boundary` and `fillvalue`.
Parameters
----------
in1, in2 : array_like
Two-dimensional input arrays to be convolved.
mode : str {'full', 'valid', 'same'}, optional
A string indicating the size of the output:
``full``
The output is the full discrete linear cross-correlation
of the inputs. (Default)
``valid``
The output consists only of those elements that do not
rely on the zero-padding.
``same``
The output is the same size as `in1`, centered
with respect to the 'full' output.
boundary : str {'fill', 'wrap', 'symm'}, optional
A flag indicating how to handle boundaries:
``fill``
pad input arrays with fillvalue. (default)
``wrap``
circular boundary conditions.
``symm``
symmetrical boundary conditions.
fillvalue : scalar, optional
Value to fill pad input arrays with. Default is 0.
Returns
-------
correlate2d : ndarray
A 2-dimensional array containing a subset of the discrete linear
cross-correlation of `in1` with `in2`.
"""
in1 = asarray(in1)
in2 = asarray(in2)
if mode == 'valid':
_check_valid_mode_shapes(in1.shape, in2.shape)
val = _valfrommode(mode)
bval = _bvalfromboundary(boundary)
with warnings.catch_warnings():
warnings.simplefilter('ignore', np.ComplexWarning)
# FIXME: some cast generates a warning here
out = sigtools._convolve2d(in1, in2, 0, val, bval, fillvalue)
return out
def medfilt2d(input, kernel_size=3):
"""
Median filter a 2-dimensional array.
Apply a median filter to the `input` array using a local window-size
given by `kernel_size` (must be odd).
Parameters
----------
input : array_like
A 2-dimensional input array.
kernel_size : array_like, optional
A scalar or a list of length 2, giving the size of the
median filter window in each dimension. Elements of
`kernel_size` should be odd. If `kernel_size` is a scalar,
then this scalar is used as the size in each dimension.
Default is a kernel of size (3, 3).
Returns
-------
out : ndarray
An array the same size as input containing the median filtered
result.
"""
image = asarray(input)
if kernel_size is None:
kernel_size = [3] * 2
kernel_size = asarray(kernel_size)
if len(kernel_size.shape) == 0:
kernel_size = [kernel_size.item()] * 2
kernel_size = asarray(kernel_size)
for size in kernel_size:
if (size % 2) != 1:
raise ValueError("Each element of kernel_size should be odd.")
return sigtools._medfilt2d(image, kernel_size)
def lfilter(b, a, x, axis=-1, zi=None):
"""
Filter data along one-dimension with an IIR or FIR filter.
Filter a data sequence, `x`, using a digital filter. This works for many
fundamental data types (including Object type). The filter is a direct
form II transposed implementation of the standard difference equation
(see Notes).
Parameters
----------
b : array_like
The numerator coefficient vector in a 1-D sequence.
a : array_like
The denominator coefficient vector in a 1-D sequence. If ``a[0]``
is not 1, then both `a` and `b` are normalized by ``a[0]``.
x : array_like
An N-dimensional input array.
axis : int
The axis of the input data array along which to apply the
linear filter. The filter is applied to each subarray along
this axis. Default is -1.
zi : array_like, optional
Initial conditions for the filter delays. It is a vector
(or array of vectors for an N-dimensional input) of length
``max(len(a),len(b))-1``. If `zi` is None or is not given then
initial rest is assumed. See `lfiltic` for more information.
Returns
-------
y : array
The output of the digital filter.
zf : array, optional
If `zi` is None, this is not returned, otherwise, `zf` holds the
final filter delay values.
Notes
-----
The filter function is implemented as a direct II transposed structure.
This means that the filter implements::
a[0]*y[n] = b[0]*x[n] + b[1]*x[n-1] + ... + b[nb]*x[n-nb]
- a[1]*y[n-1] - ... - a[na]*y[n-na]
using the following difference equations::
y[m] = b[0]*x[m] + z[0,m-1]
z[0,m] = b[1]*x[m] + z[1,m-1] - a[1]*y[m]
...
z[n-3,m] = b[n-2]*x[m] + z[n-2,m-1] - a[n-2]*y[m]
z[n-2,m] = b[n-1]*x[m] - a[n-1]*y[m]
where m is the output sample number and n=max(len(a),len(b)) is the
model order.
The rational transfer function describing this filter in the
z-transform domain is::
-1 -nb
b[0] + b[1]z + ... + b[nb] z
Y(z) = ---------------------------------- X(z)
-1 -na
a[0] + a[1]z + ... + a[na] z
"""
if isscalar(a):
a = [a]
if zi is None:
return sigtools._linear_filter(b, a, x, axis)
else:
return sigtools._linear_filter(b, a, x, axis, zi)
def lfiltic(b, a, y, x=None):
"""
Construct initial conditions for lfilter.
Given a linear filter (b, a) and initial conditions on the output `y`
and the input `x`, return the inital conditions on the state vector zi
which is used by `lfilter` to generate the output given the input.
Parameters
----------
b : array_like
Linear filter term.
a : array_like
Linear filter term.
y : array_like
Initial conditions.
If ``N=len(a) - 1``, then ``y = {y[-1], y[-2], ..., y[-N]}``.
If `y` is too short, it is padded with zeros.
x : array_like, optional
Initial conditions.
If ``M=len(b) - 1``, then ``x = {x[-1], x[-2], ..., x[-M]}``.
If `x` is not given, its initial conditions are assumed zero.
If `x` is too short, it is padded with zeros.
Returns
-------
zi : ndarray
The state vector ``zi``.
``zi = {z_0[-1], z_1[-1], ..., z_K-1[-1]}``, where ``K = max(M,N)``.
See Also
--------
lfilter
"""
N = np.size(a) - 1
M = np.size(b) - 1
K = max(M, N)
y = asarray(y)
zi = zeros(K, y.dtype.char)
if x is None:
x = zeros(M, y.dtype.char)
else:
x = asarray(x)
L = np.size(x)
if L < M:
x = r_[x, zeros(M - L)]
L = np.size(y)
if L < N:
y = r_[y, zeros(N - L)]
for m in range(M):
zi[m] = sum(b[m + 1:] * x[:M - m], axis=0)
for m in range(N):
zi[m] -= sum(a[m + 1:] * y[:N - m], axis=0)
return zi
def deconvolve(signal, divisor):
"""Deconvolves `divisor` out of `signal`.
"""
num = atleast_1d(signal)
den = atleast_1d(divisor)
N = len(num)
D = len(den)
if D > N:
quot = []
rem = num
else:
input = ones(N - D + 1, float)
input[1:] = 0
quot = lfilter(num, den, input)
rem = num - convolve(den, quot, mode='full')
return quot, rem
def hilbert(x, N=None, axis=-1):
"""
Compute the analytic signal.
The transformation is done along the last axis by default.
Parameters
----------
x : array_like
Signal data
N : int, optional
Number of Fourier components. Default: ``x.shape[axis]``
axis : int, optional
Axis along which to do the transformation. Default: -1.
Returns
-------
xa : ndarray
Analytic signal of `x`, of each 1-D array along `axis`
Notes
-----
The analytic signal `x_a(t)` of `x(t)` is::
x_a = F^{-1}(F(x) 2U) = x + i y
where ``F`` is the Fourier transform, ``U`` the unit step function,
and ``y`` the Hilbert transform of ``x``. [1]_
`axis` argument is new in scipy 0.8.0.
References
----------
.. [1] Wikipedia, "Analytic signal".
http://en.wikipedia.org/wiki/Analytic_signal
"""
x = asarray(x)
if iscomplexobj(x):
raise ValueError("x must be real.")
if N is None:
N = x.shape[axis]
if N <= 0:
raise ValueError("N must be positive.")
Xf = fft(x, N, axis=axis)
h = zeros(N)
if N % 2 == 0:
h[0] = h[N // 2] = 1
h[1:N // 2] = 2
else:
h[0] = 1
h[1:(N + 1) // 2] = 2
if len(x.shape) > 1:
ind = [newaxis] * x.ndim
ind[axis] = slice(None)
h = h[ind]
x = ifft(Xf * h, axis=axis)
return x
def hilbert2(x, N=None):
"""
Compute the '2-D' analytic signal of `x`
Parameters
----------
x : array_like
2-D signal data.
N : int or tuple of two ints, optional
Number of Fourier components. Default is ``x.shape``
Returns
-------
xa : ndarray
Analytic signal of `x` taken along axes (0,1).
References
----------
.. [1] Wikipedia, "Analytic signal",
http://en.wikipedia.org/wiki/Analytic_signal
"""
x = atleast_2d(x)
if len(x.shape) > 2:
raise ValueError("x must be rank 2.")
if iscomplexobj(x):
raise ValueError("x must be real.")
if N is None:
N = x.shape
elif isinstance(N, int):
if N <= 0:
raise ValueError("N must be positive.")
N = (N, N)
elif len(N) != 2 or np.any(np.asarray(N) <= 0):
raise ValueError("When given as a tuple, N must hold exactly "
"two positive integers")
Xf = fft2(x, N, axes=(0, 1))
h1 = zeros(N[0], 'd')
h2 = zeros(N[1], 'd')
for p in range(2):
h = eval("h%d" % (p + 1))
N1 = N[p]
if N1 % 2 == 0:
h[0] = h[N1 // 2] = 1
h[1:N1 // 2] = 2
else:
h[0] = 1
h[1:(N1 + 1) // 2] = 2
exec("h%d = h" % (p + 1), globals(), locals())
h = h1[:, newaxis] * h2[newaxis, :]
k = len(x.shape)
while k > 2:
h = h[:, newaxis]
k -= 1
x = ifft2(Xf * h, axes=(0, 1))
return x
def cmplx_sort(p):
"sort roots based on magnitude."
p = asarray(p)
if iscomplexobj(p):
indx = argsort(abs(p))
else:
indx = argsort(p)
return take(p, indx, 0), indx
def unique_roots(p, tol=1e-3, rtype='min'):
"""
Determine unique roots and their multiplicities from a list of roots.
Parameters
----------
p : array_like
The list of roots.
tol : float, optional
The tolerance for two roots to be considered equal. Default is 1e-3.
rtype : {'max', 'min, 'avg'}, optional
How to determine the returned root if multiple roots are within
`tol` of each other.
- 'max': pick the maximum of those roots.
- 'min': pick the minimum of those roots.
- 'avg': take the average of those roots.
Returns
-------
pout : ndarray
The list of unique roots, sorted from low to high.
mult : ndarray
The multiplicity of each root.
Notes
-----
This utility function is not specific to roots but can be used for any
sequence of values for which uniqueness and multiplicity has to be
determined. For a more general routine, see `numpy.unique`.
Examples
--------
>>> from scipy import signal
>>> vals = [0, 1.3, 1.31, 2.8, 1.25, 2.2, 10.3]
>>> uniq, mult = signal.unique_roots(vals, tol=2e-2, rtype='avg')
Check which roots have multiplicity larger than 1:
>>> uniq[mult > 1]
array([ 1.305])
"""
if rtype in ['max', 'maximum']:
comproot = np.maximum
elif rtype in ['min', 'minimum']:
comproot = np.minimum
elif rtype in ['avg', 'mean']:
comproot = np.mean
p = asarray(p) * 1.0
tol = abs(tol)
p, indx = cmplx_sort(p)
pout = []
mult = []
indx = -1
curp = p[0] + 5 * tol
sameroots = []
for k in range(len(p)):
tr = p[k]
if abs(tr - curp) < tol:
sameroots.append(tr)
curp = comproot(sameroots)
pout[indx] = curp
mult[indx] += 1
else:
pout.append(tr)
curp = tr
sameroots = [tr]
indx += 1
mult.append(1)
return array(pout), array(mult)
def invres(r, p, k, tol=1e-3, rtype='avg'):
"""
Compute b(s) and a(s) from partial fraction expansion: r,p,k
If ``M = len(b)`` and ``N = len(a)``::
b(s) b[0] x**(M-1) + b[1] x**(M-2) + ... + b[M-1]
H(s) = ------ = ----------------------------------------------
a(s) a[0] x**(N-1) + a[1] x**(N-2) + ... + a[N-1]
r[0] r[1] r[-1]
= -------- + -------- + ... + --------- + k(s)
(s-p[0]) (s-p[1]) (s-p[-1])
If there are any repeated roots (closer than tol), then the partial
fraction expansion has terms like::
r[i] r[i+1] r[i+n-1]
-------- + ----------- + ... + -----------
(s-p[i]) (s-p[i])**2 (s-p[i])**n
Parameters
----------
r : ndarray
Residues.
p : ndarray
Poles.
k : ndarray
Coefficients of the direct polynomial term.
tol : float, optional
The tolerance for two roots to be considered equal. Default is 1e-3.
rtype : {'max', 'min, 'avg'}, optional
How to determine the returned root if multiple roots are within
`tol` of each other.
'max': pick the maximum of those roots.
'min': pick the minimum of those roots.
'avg': take the average of those roots.
See Also
--------
residue, unique_roots
"""
extra = k
p, indx = cmplx_sort(p)
r = take(r, indx, 0)
pout, mult = unique_roots(p, tol=tol, rtype=rtype)
p = []
for k in range(len(pout)):
p.extend([pout[k]] * mult[k])
a = atleast_1d(poly(p))
if len(extra) > 0:
b = polymul(extra, a)
else:
b = [0]
indx = 0
for k in range(len(pout)):
temp = []
for l in range(len(pout)):
if l != k:
temp.extend([pout[l]] * mult[l])
for m in range(mult[k]):
t2 = temp[:]
t2.extend([pout[k]] * (mult[k] - m - 1))
b = polyadd(b, r[indx] * poly(t2))
indx += 1
b = real_if_close(b)
while allclose(b[0], 0, rtol=1e-14) and (b.shape[-1] > 1):
b = b[1:]
return b, a
def residue(b, a, tol=1e-3, rtype='avg'):
"""
Compute partial-fraction expansion of b(s) / a(s).
If ``M = len(b)`` and ``N = len(a)``, then the partial-fraction
expansion H(s) is defined as::
b(s) b[0] s**(M-1) + b[1] s**(M-2) + ... + b[M-1]
H(s) = ------ = ----------------------------------------------
a(s) a[0] s**(N-1) + a[1] s**(N-2) + ... + a[N-1]
r[0] r[1] r[-1]
= -------- + -------- + ... + --------- + k(s)
(s-p[0]) (s-p[1]) (s-p[-1])
If there are any repeated roots (closer together than `tol`), then H(s)
has terms like::
r[i] r[i+1] r[i+n-1]
-------- + ----------- + ... + -----------
(s-p[i]) (s-p[i])**2 (s-p[i])**n
Returns
-------
r : ndarray
Residues.
p : ndarray
Poles.
k : ndarray
Coefficients of the direct polynomial term.
See Also
--------
invres, numpy.poly, unique_roots
"""
b, a = map(asarray, (b, a))
rscale = a[0]
k, b = polydiv(b, a)
p = roots(a)
r = p * 0.0
pout, mult = unique_roots(p, tol=tol, rtype=rtype)
p = []
for n in range(len(pout)):
p.extend([pout[n]] * mult[n])
p = asarray(p)
# Compute the residue from the general formula
indx = 0
for n in range(len(pout)):
bn = b.copy()
pn = []
for l in range(len(pout)):
if l != n:
pn.extend([pout[l]] * mult[l])
an = atleast_1d(poly(pn))
# bn(s) / an(s) is (s-po[n])**Nn * b(s) / a(s) where Nn is
# multiplicity of pole at po[n]
sig = mult[n]
for m in range(sig, 0, -1):
if sig > m:
# compute next derivative of bn(s) / an(s)
term1 = polymul(polyder(bn, 1), an)
term2 = polymul(bn, polyder(an, 1))
bn = polysub(term1, term2)
an = polymul(an, an)
r[indx + m - 1] = polyval(bn, pout[n]) / polyval(an, pout[n]) \
/ factorial(sig - m)
indx += sig
return r / rscale, p, k
def residuez(b, a, tol=1e-3, rtype='avg'):
"""
Compute partial-fraction expansion of b(z) / a(z).
If ``M = len(b)`` and ``N = len(a)``::
b(z) b[0] + b[1] z**(-1) + ... + b[M-1] z**(-M+1)
H(z) = ------ = ----------------------------------------------
a(z) a[0] + a[1] z**(-1) + ... + a[N-1] z**(-N+1)
r[0] r[-1]
= --------------- + ... + ---------------- + k[0] + k[1]z**(-1) ...
(1-p[0]z**(-1)) (1-p[-1]z**(-1))
If there are any repeated roots (closer than tol), then the partial
fraction expansion has terms like::
r[i] r[i+1] r[i+n-1]
-------------- + ------------------ + ... + ------------------
(1-p[i]z**(-1)) (1-p[i]z**(-1))**2 (1-p[i]z**(-1))**n
See also
--------
invresz, unique_roots
"""
b, a = map(asarray, (b, a))
gain = a[0]
brev, arev = b[::-1], a[::-1]
krev, brev = polydiv(brev, arev)
if krev == []:
k = []
else:
k = krev[::-1]
b = brev[::-1]
p = roots(a)
r = p * 0.0
pout, mult = unique_roots(p, tol=tol, rtype=rtype)
p = []
for n in range(len(pout)):
p.extend([pout[n]] * mult[n])
p = asarray(p)
# Compute the residue from the general formula (for discrete-time)
# the polynomial is in z**(-1) and the multiplication is by terms
# like this (1-p[i] z**(-1))**mult[i]. After differentiation,
# we must divide by (-p[i])**(m-k) as well as (m-k)!
indx = 0
for n in range(len(pout)):
bn = brev.copy()
pn = []
for l in range(len(pout)):
if l != n:
pn.extend([pout[l]] * mult[l])
an = atleast_1d(poly(pn))[::-1]
# bn(z) / an(z) is (1-po[n] z**(-1))**Nn * b(z) / a(z) where Nn is
# multiplicity of pole at po[n] and b(z) and a(z) are polynomials.
sig = mult[n]
for m in range(sig, 0, -1):
if sig > m:
# compute next derivative of bn(s) / an(s)
term1 = polymul(polyder(bn, 1), an)
term2 = polymul(bn, polyder(an, 1))
bn = polysub(term1, term2)
an = polymul(an, an)
r[indx + m - 1] = (polyval(bn, 1.0 / pout[n]) /
polyval(an, 1.0 / pout[n]) /
factorial(sig - m) / (-pout[n]) ** (sig - m))
indx += sig
return r / gain, p, k
def invresz(r, p, k, tol=1e-3, rtype='avg'):
"""
Compute b(z) and a(z) from partial fraction expansion: r,p,k
If ``M = len(b)`` and ``N = len(a)``::
b(z) b[0] + b[1] z**(-1) + ... + b[M-1] z**(-M+1)
H(z) = ------ = ----------------------------------------------
a(z) a[0] + a[1] z**(-1) + ... + a[N-1] z**(-N+1)
r[0] r[-1]
= --------------- + ... + ---------------- + k[0] + k[1]z**(-1) ...
(1-p[0]z**(-1)) (1-p[-1]z**(-1))
If there are any repeated roots (closer than tol), then the partial
fraction expansion has terms like::
r[i] r[i+1] r[i+n-1]
-------------- + ------------------ + ... + ------------------
(1-p[i]z**(-1)) (1-p[i]z**(-1))**2 (1-p[i]z**(-1))**n
See Also
--------
residuez, unique_roots
"""
extra = asarray(k)
p, indx = cmplx_sort(p)
r = take(r, indx, 0)
pout, mult = unique_roots(p, tol=tol, rtype=rtype)
p = []
for k in range(len(pout)):
p.extend([pout[k]] * mult[k])
a = atleast_1d(poly(p))
if len(extra) > 0:
b = polymul(extra, a)
else:
b = [0]
indx = 0
brev = asarray(b)[::-1]
for k in range(len(pout)):
temp = []
# Construct polynomial which does not include any of this root
for l in range(len(pout)):
if l != k:
temp.extend([pout[l]] * mult[l])
for m in range(mult[k]):
t2 = temp[:]
t2.extend([pout[k]] * (mult[k] - m - 1))
brev = polyadd(brev, (r[indx] * poly(t2))[::-1])
indx += 1
b = real_if_close(brev[::-1])
return b, a
def resample(x, num, t=None, axis=0, window=None):
"""
Resample `x` to `num` samples using Fourier method along the given axis.
The resampled signal starts at the same value as `x` but is sampled
with a spacing of ``len(x) / num * (spacing of x)``. Because a
Fourier method is used, the signal is assumed to be periodic.
Parameters
----------
x : array_like
The data to be resampled.
num : int
The number of samples in the resampled signal.
t : array_like, optional
If `t` is given, it is assumed to be the sample positions
associated with the signal data in `x`.
axis : int, optional
The axis of `x` that is resampled. Default is 0.
window : array_like, callable, string, float, or tuple, optional
Specifies the window applied to the signal in the Fourier
domain. See below for details.
Returns
-------
resampled_x or (resampled_x, resampled_t)
Either the resampled array, or, if `t` was given, a tuple
containing the resampled array and the corresponding resampled
positions.
Notes
-----
The argument `window` controls a Fourier-domain window that tapers
the Fourier spectrum before zero-padding to alleviate ringing in
the resampled values for sampled signals you didn't intend to be
interpreted as band-limited.
If `window` is a function, then it is called with a vector of inputs
indicating the frequency bins (i.e. fftfreq(x.shape[axis]) ).
If `window` is an array of the same length as `x.shape[axis]` it is
assumed to be the window to be applied directly in the Fourier
domain (with dc and low-frequency first).
For any other type of `window`, the function `scipy.signal.get_window`
is called to generate the window.
The first sample of the returned vector is the same as the first
sample of the input vector. The spacing between samples is changed
from dx to:
dx * len(x) / num
If `t` is not None, then it represents the old sample positions,
and the new sample positions will be returned as well as the new
samples.
"""
x = asarray(x)
X = fft(x, axis=axis)
Nx = x.shape[axis]
if window is not None:
if callable(window):
W = window(fftfreq(Nx))
elif isinstance(window, ndarray) and window.shape == (Nx,):
W = window
else:
W = ifftshift(get_window(window, Nx))
newshape = ones(len(x.shape))
newshape[axis] = len(W)
W.shape = newshape
X = X * W
sl = [slice(None)] * len(x.shape)
newshape = list(x.shape)
newshape[axis] = num
N = int(np.minimum(num, Nx))
Y = zeros(newshape, 'D')
sl[axis] = slice(0, (N + 1) // 2)
Y[sl] = X[sl]
sl[axis] = slice(-(N - 1) // 2, None)
Y[sl] = X[sl]
y = ifft(Y, axis=axis) * (float(num) / float(Nx))
if x.dtype.char not in ['F', 'D']:
y = y.real
if t is None:
return y
else:
new_t = arange(0, num) * (t[1] - t[0]) * Nx / float(num) + t[0]
return y, new_t
def detrend(data, axis=-1, type='linear', bp=0):
"""
Remove linear trend along axis from data.
Parameters
----------
data : array_like
The input data.
axis : int, optional
The axis along which to detrend the data. By default this is the
last axis (-1).
type : {'linear', 'constant'}, optional
The type of detrending. If ``type == 'linear'`` (default),
the result of a linear least-squares fit to `data` is subtracted
from `data`.
If ``type == 'constant'``, only the mean of `data` is subtracted.
bp : array_like of ints, optional
A sequence of break points. If given, an individual linear fit is
performed for each part of `data` between two break points.
Break points are specified as indices into `data`.
Returns
-------
ret : ndarray
The detrended input data.
Examples
--------
>>> from scipy import signal
>>> randgen = np.random.RandomState(9)
>>> npoints = 1e3
>>> noise = randgen.randn(npoints)
>>> x = 3 + 2*np.linspace(0, 1, npoints) + noise
>>> (signal.detrend(x) - noise).max() < 0.01
True
"""
if type not in ['linear', 'l', 'constant', 'c']:
raise ValueError("Trend type must be 'linear' or 'constant'.")
data = asarray(data)
dtype = data.dtype.char
if dtype not in 'dfDF':
dtype = 'd'
if type in ['constant', 'c']:
ret = data - expand_dims(mean(data, axis), axis)
return ret
else:
dshape = data.shape
N = dshape[axis]
bp = sort(unique(r_[0, bp, N]))
if np.any(bp > N):
raise ValueError("Breakpoints must be less than length "
"of data along given axis.")
Nreg = len(bp) - 1
# Restructure data so that axis is along first dimension and
# all other dimensions are collapsed into second dimension
rnk = len(dshape)
if axis < 0:
axis = axis + rnk
newdims = r_[axis, 0:axis, axis + 1:rnk]
newdata = reshape(transpose(data, tuple(newdims)),
(N, prod(dshape, axis=0) // N))
newdata = newdata.copy() # make sure we have a copy
if newdata.dtype.char not in 'dfDF':
newdata = newdata.astype(dtype)
# Find leastsq fit and remove it for each piece
for m in range(Nreg):
Npts = bp[m + 1] - bp[m]
A = ones((Npts, 2), dtype)
A[:, 0] = cast[dtype](arange(1, Npts + 1) * 1.0 / Npts)
sl = slice(bp[m], bp[m + 1])
coef, resids, rank, s = linalg.lstsq(A, newdata[sl])
newdata[sl] = newdata[sl] - dot(A, coef)
# Put data back in original shape.
tdshape = take(dshape, newdims, 0)
ret = reshape(newdata, tuple(tdshape))
vals = list(range(1, rnk))
olddims = vals[:axis] + [0] + vals[axis:]
ret = transpose(ret, tuple(olddims))
return ret
def lfilter_zi(b, a):
"""
Compute an initial state `zi` for the lfilter function that corresponds
to the steady state of the step response.
A typical use of this function is to set the initial state so that the
output of the filter starts at the same value as the first element of
the signal to be filtered.
Parameters
----------
b, a : array_like (1-D)
The IIR filter coefficients. See `lfilter` for more
information.
Returns
-------
zi : 1-D ndarray
The initial state for the filter.
Notes
-----
A linear filter with order m has a state space representation (A, B, C, D),
for which the output y of the filter can be expressed as::
z(n+1) = A*z(n) + B*x(n)
y(n) = C*z(n) + D*x(n)
where z(n) is a vector of length m, A has shape (m, m), B has shape
(m, 1), C has shape (1, m) and D has shape (1, 1) (assuming x(n) is
a scalar). lfilter_zi solves::
zi = A*zi + B
In other words, it finds the initial condition for which the response
to an input of all ones is a constant.
Given the filter coefficients `a` and `b`, the state space matrices
for the transposed direct form II implementation of the linear filter,
which is the implementation used by scipy.signal.lfilter, are::
A = scipy.linalg.companion(a).T
B = b[1:] - a[1:]*b[0]
assuming `a[0]` is 1.0; if `a[0]` is not 1, `a` and `b` are first
divided by a[0].
Examples
--------
The following code creates a lowpass Butterworth filter. Then it
applies that filter to an array whose values are all 1.0; the
output is also all 1.0, as expected for a lowpass filter. If the
`zi` argument of `lfilter` had not been given, the output would have
shown the transient signal.
>>> from numpy import array, ones
>>> from scipy.signal import lfilter, lfilter_zi, butter
>>> b, a = butter(5, 0.25)
>>> zi = lfilter_zi(b, a)
>>> y, zo = lfilter(b, a, ones(10), zi=zi)
>>> y
array([1., 1., 1., 1., 1., 1., 1., 1., 1., 1.])
Another example:
>>> x = array([0.5, 0.5, 0.5, 0.0, 0.0, 0.0, 0.0])
>>> y, zf = lfilter(b, a, x, zi=zi*x[0])
>>> y
array([ 0.5 , 0.5 , 0.5 , 0.49836039, 0.48610528,
0.44399389, 0.35505241])
Note that the `zi` argument to `lfilter` was computed using
`lfilter_zi` and scaled by `x[0]`. Then the output `y` has no
transient until the input drops from 0.5 to 0.0.
"""
# FIXME: Can this function be replaced with an appropriate
# use of lfiltic? For example, when b,a = butter(N,Wn),
# lfiltic(b, a, y=numpy.ones_like(a), x=numpy.ones_like(b)).
#
# We could use scipy.signal.normalize, but it uses warnings in
# cases where a ValueError is more appropriate, and it allows
# b to be 2D.
b = np.atleast_1d(b)
if b.ndim != 1:
raise ValueError("Numerator b must be rank 1.")
a = np.atleast_1d(a)
if a.ndim != 1:
raise ValueError("Denominator a must be rank 1.")
while len(a) > 1 and a[0] == 0.0:
a = a[1:]
if a.size < 1:
raise ValueError("There must be at least one nonzero `a` coefficient.")
if a[0] != 1.0:
# Normalize the coefficients so a[0] == 1.
a = a / a[0]
b = b / a[0]
n = max(len(a), len(b))
# Pad a or b with zeros so they are the same length.
if len(a) < n:
a = np.r_[a, np.zeros(n - len(a))]
elif len(b) < n:
b = np.r_[b, np.zeros(n - len(b))]
IminusA = np.eye(n - 1) - linalg.companion(a).T
B = b[1:] - a[1:] * b[0]
# Solve zi = A*zi + B
zi = np.linalg.solve(IminusA, B)
# For future reference: we could also use the following
# explicit formulas to solve the linear system:
#
# zi = np.zeros(n - 1)
# zi[0] = B.sum() / IminusA[:,0].sum()
# asum = 1.0
# csum = 0.0
# for k in range(1,n-1):
# asum += a[k]
# csum += b[k] - a[k]*b[0]
# zi[k] = asum*zi[0] - csum
return zi
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 = signal.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
from scipy.signal.filter_design import cheby1
from scipy.signal.fir_filter_design import firwin
def decimate(x, q, n=None, ftype='iir', axis=-1):
"""
Downsample the signal by using a filter.
By default, an order 8 Chebyshev type I filter is used. A 30 point FIR
filter with hamming window is used if `ftype` is 'fir'.
Parameters
----------
x : ndarray
The signal to be downsampled, as an N-dimensional array.
q : int
The downsampling factor.
n : int, optional
The order of the filter (1 less than the length for 'fir').
ftype : str {'iir', 'fir'}, optional
The type of the lowpass filter.
axis : int, optional
The axis along which to decimate.
Returns
-------
y : ndarray
The down-sampled signal.
See also
--------
resample
"""
if not isinstance(q, int):
raise TypeError("q must be an integer")
if n is None:
if ftype == 'fir':
n = 30
else:
n = 8
if ftype == 'fir':
b = firwin(n + 1, 1. / q, window='hamming')
a = 1.
else:
b, a = cheby1(n, 0.05, 0.8 / q)
y = lfilter(b, a, x, axis=axis)
sl = [slice(None)] * y.ndim
sl[axis] = slice(None, None, q)
return y[sl]
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