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"""
Differential and pseudo-differential operators.
"""
# Created by Pearu Peterson, September 2002
__all__ = ['diff',
'tilbert','itilbert','hilbert','ihilbert',
'cs_diff','cc_diff','sc_diff','ss_diff',
'shift']
import threading
from numpy import pi, asarray, sin, cos, sinh, cosh, tanh, iscomplexobj
from . import convolve
from scipy.fft._pocketfft.helper import _datacopied
_cache = threading.local()
def diff(x,order=1,period=None, _cache=_cache):
"""
Return kth derivative (or integral) of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = pow(sqrt(-1)*j*2*pi/period, order) * x_j
y_0 = 0 if order is not 0.
Parameters
----------
x : array_like
Input array.
order : int, optional
The order of differentiation. Default order is 1. If order is
negative, then integration is carried out under the assumption
that ``x_0 == 0``.
period : float, optional
The assumed period of the sequence. Default is ``2*pi``.
Notes
-----
If ``sum(x, axis=0) = 0`` then ``diff(diff(x, k), -k) == x`` (within
numerical accuracy).
For odd order and even ``len(x)``, the Nyquist mode is taken zero.
"""
if isinstance(_cache, threading.local):
if not hasattr(_cache, 'diff_cache'):
_cache.diff_cache = {}
_cache = _cache.diff_cache
tmp = asarray(x)
if order == 0:
return tmp
if iscomplexobj(tmp):
return diff(tmp.real, order, period, _cache)+1j*diff(
tmp.imag, order, period, _cache)
if period is not None:
c = 2*pi/period
else:
c = 1.0
n = len(x)
omega = _cache.get((n,order,c))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k,order=order,c=c):
if k:
return pow(c*k,order)
return 0
omega = convolve.init_convolution_kernel(n,kernel,d=order,
zero_nyquist=1)
_cache[(n,order,c)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,omega,swap_real_imag=order % 2,
overwrite_x=overwrite_x)
def tilbert(x, h, period=None, _cache=_cache):
"""
Return h-Tilbert transform of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = sqrt(-1)*coth(j*h*2*pi/period) * x_j
y_0 = 0
Parameters
----------
x : array_like
The input array to transform.
h : float
Defines the parameter of the Tilbert transform.
period : float, optional
The assumed period of the sequence. Default period is ``2*pi``.
Returns
-------
tilbert : ndarray
The result of the transform.
Notes
-----
If ``sum(x, axis=0) == 0`` and ``n = len(x)`` is odd, then
``tilbert(itilbert(x)) == x``.
If ``2 * pi * h / period`` is approximately 10 or larger, then
numerically ``tilbert == hilbert``
(theoretically oo-Tilbert == Hilbert).
For even ``len(x)``, the Nyquist mode of ``x`` is taken zero.
"""
if isinstance(_cache, threading.local):
if not hasattr(_cache, 'tilbert_cache'):
_cache.tilbert_cache = {}
_cache = _cache.tilbert_cache
tmp = asarray(x)
if iscomplexobj(tmp):
return tilbert(tmp.real, h, period, _cache) + \
1j * tilbert(tmp.imag, h, period, _cache)
if period is not None:
h = h * 2 * pi / period
n = len(x)
omega = _cache.get((n, h))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k, h=h):
if k:
return 1.0/tanh(h*k)
return 0
omega = convolve.init_convolution_kernel(n, kernel, d=1)
_cache[(n,h)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,omega,swap_real_imag=1,overwrite_x=overwrite_x)
def itilbert(x,h,period=None, _cache=_cache):
"""
Return inverse h-Tilbert transform of a periodic sequence x.
If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = -sqrt(-1)*tanh(j*h*2*pi/period) * x_j
y_0 = 0
For more details, see `tilbert`.
"""
if isinstance(_cache, threading.local):
if not hasattr(_cache, 'itilbert_cache'):
_cache.itilbert_cache = {}
_cache = _cache.itilbert_cache
tmp = asarray(x)
if iscomplexobj(tmp):
return itilbert(tmp.real, h, period, _cache) + \
1j*itilbert(tmp.imag, h, period, _cache)
if period is not None:
h = h*2*pi/period
n = len(x)
omega = _cache.get((n,h))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k,h=h):
if k:
return -tanh(h*k)
return 0
omega = convolve.init_convolution_kernel(n,kernel,d=1)
_cache[(n,h)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,omega,swap_real_imag=1,overwrite_x=overwrite_x)
def hilbert(x, _cache=_cache):
"""
Return Hilbert transform of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = sqrt(-1)*sign(j) * x_j
y_0 = 0
Parameters
----------
x : array_like
The input array, should be periodic.
_cache : dict, optional
Dictionary that contains the kernel used to do a convolution with.
Returns
-------
y : ndarray
The transformed input.
See Also
--------
scipy.signal.hilbert : Compute the analytic signal, using the Hilbert
transform.
Notes
-----
If ``sum(x, axis=0) == 0`` then ``hilbert(ihilbert(x)) == x``.
For even len(x), the Nyquist mode of x is taken zero.
The sign of the returned transform does not have a factor -1 that is more
often than not found in the definition of the Hilbert transform. Note also
that `scipy.signal.hilbert` does have an extra -1 factor compared to this
function.
"""
if isinstance(_cache, threading.local):
if not hasattr(_cache, 'hilbert_cache'):
_cache.hilbert_cache = {}
_cache = _cache.hilbert_cache
tmp = asarray(x)
if iscomplexobj(tmp):
return hilbert(tmp.real, _cache) + 1j * hilbert(tmp.imag, _cache)
n = len(x)
omega = _cache.get(n)
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k):
if k > 0:
return 1.0
elif k < 0:
return -1.0
return 0.0
omega = convolve.init_convolution_kernel(n,kernel,d=1)
_cache[n] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,omega,swap_real_imag=1,overwrite_x=overwrite_x)
def ihilbert(x, _cache=_cache):
"""
Return inverse Hilbert transform of a periodic sequence x.
If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = -sqrt(-1)*sign(j) * x_j
y_0 = 0
"""
if isinstance(_cache, threading.local):
if not hasattr(_cache, 'ihilbert_cache'):
_cache.ihilbert_cache = {}
_cache = _cache.ihilbert_cache
return -hilbert(x, _cache)
def cs_diff(x, a, b, period=None, _cache=_cache):
"""
Return (a,b)-cosh/sinh pseudo-derivative of a periodic sequence.
If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = -sqrt(-1)*cosh(j*a*2*pi/period)/sinh(j*b*2*pi/period) * x_j
y_0 = 0
Parameters
----------
x : array_like
The array to take the pseudo-derivative from.
a, b : float
Defines the parameters of the cosh/sinh pseudo-differential
operator.
period : float, optional
The period of the sequence. Default period is ``2*pi``.
Returns
-------
cs_diff : ndarray
Pseudo-derivative of periodic sequence `x`.
Notes
-----
For even len(`x`), the Nyquist mode of `x` is taken as zero.
"""
if isinstance(_cache, threading.local):
if not hasattr(_cache, 'cs_diff_cache'):
_cache.cs_diff_cache = {}
_cache = _cache.cs_diff_cache
tmp = asarray(x)
if iscomplexobj(tmp):
return cs_diff(tmp.real, a, b, period, _cache) + \
1j*cs_diff(tmp.imag, a, b, period, _cache)
if period is not None:
a = a*2*pi/period
b = b*2*pi/period
n = len(x)
omega = _cache.get((n,a,b))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k,a=a,b=b):
if k:
return -cosh(a*k)/sinh(b*k)
return 0
omega = convolve.init_convolution_kernel(n,kernel,d=1)
_cache[(n,a,b)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,omega,swap_real_imag=1,overwrite_x=overwrite_x)
def sc_diff(x, a, b, period=None, _cache=_cache):
"""
Return (a,b)-sinh/cosh pseudo-derivative of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = sqrt(-1)*sinh(j*a*2*pi/period)/cosh(j*b*2*pi/period) * x_j
y_0 = 0
Parameters
----------
x : array_like
Input array.
a,b : float
Defines the parameters of the sinh/cosh pseudo-differential
operator.
period : float, optional
The period of the sequence x. Default is 2*pi.
Notes
-----
``sc_diff(cs_diff(x,a,b),b,a) == x``
For even ``len(x)``, the Nyquist mode of x is taken as zero.
"""
if isinstance(_cache, threading.local):
if not hasattr(_cache, 'sc_diff_cache'):
_cache.sc_diff_cache = {}
_cache = _cache.sc_diff_cache
tmp = asarray(x)
if iscomplexobj(tmp):
return sc_diff(tmp.real, a, b, period, _cache) + \
1j * sc_diff(tmp.imag, a, b, period, _cache)
if period is not None:
a = a*2*pi/period
b = b*2*pi/period
n = len(x)
omega = _cache.get((n,a,b))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k,a=a,b=b):
if k:
return sinh(a*k)/cosh(b*k)
return 0
omega = convolve.init_convolution_kernel(n,kernel,d=1)
_cache[(n,a,b)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,omega,swap_real_imag=1,overwrite_x=overwrite_x)
def ss_diff(x, a, b, period=None, _cache=_cache):
"""
Return (a,b)-sinh/sinh pseudo-derivative of a periodic sequence x.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = sinh(j*a*2*pi/period)/sinh(j*b*2*pi/period) * x_j
y_0 = a/b * x_0
Parameters
----------
x : array_like
The array to take the pseudo-derivative from.
a,b
Defines the parameters of the sinh/sinh pseudo-differential
operator.
period : float, optional
The period of the sequence x. Default is ``2*pi``.
Notes
-----
``ss_diff(ss_diff(x,a,b),b,a) == x``
"""
if isinstance(_cache, threading.local):
if not hasattr(_cache, 'ss_diff_cache'):
_cache.ss_diff_cache = {}
_cache = _cache.ss_diff_cache
tmp = asarray(x)
if iscomplexobj(tmp):
return ss_diff(tmp.real, a, b, period, _cache) + \
1j*ss_diff(tmp.imag, a, b, period, _cache)
if period is not None:
a = a*2*pi/period
b = b*2*pi/period
n = len(x)
omega = _cache.get((n,a,b))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k,a=a,b=b):
if k:
return sinh(a*k)/sinh(b*k)
return float(a)/b
omega = convolve.init_convolution_kernel(n,kernel)
_cache[(n,a,b)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,omega,overwrite_x=overwrite_x)
def cc_diff(x, a, b, period=None, _cache=_cache):
"""
Return (a,b)-cosh/cosh pseudo-derivative of a periodic sequence.
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = cosh(j*a*2*pi/period)/cosh(j*b*2*pi/period) * x_j
Parameters
----------
x : array_like
The array to take the pseudo-derivative from.
a,b : float
Defines the parameters of the sinh/sinh pseudo-differential
operator.
period : float, optional
The period of the sequence x. Default is ``2*pi``.
Returns
-------
cc_diff : ndarray
Pseudo-derivative of periodic sequence `x`.
Notes
-----
``cc_diff(cc_diff(x,a,b),b,a) == x``
"""
if isinstance(_cache, threading.local):
if not hasattr(_cache, 'cc_diff_cache'):
_cache.cc_diff_cache = {}
_cache = _cache.cc_diff_cache
tmp = asarray(x)
if iscomplexobj(tmp):
return cc_diff(tmp.real, a, b, period, _cache) + \
1j * cc_diff(tmp.imag, a, b, period, _cache)
if period is not None:
a = a*2*pi/period
b = b*2*pi/period
n = len(x)
omega = _cache.get((n,a,b))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel(k,a=a,b=b):
return cosh(a*k)/cosh(b*k)
omega = convolve.init_convolution_kernel(n,kernel)
_cache[(n,a,b)] = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve(tmp,omega,overwrite_x=overwrite_x)
def shift(x, a, period=None, _cache=_cache):
"""
Shift periodic sequence x by a: y(u) = x(u+a).
If x_j and y_j are Fourier coefficients of periodic functions x
and y, respectively, then::
y_j = exp(j*a*2*pi/period*sqrt(-1)) * x_f
Parameters
----------
x : array_like
The array to take the pseudo-derivative from.
a : float
Defines the parameters of the sinh/sinh pseudo-differential
period : float, optional
The period of the sequences x and y. Default period is ``2*pi``.
"""
if isinstance(_cache, threading.local):
if not hasattr(_cache, 'shift_cache'):
_cache.shift_cache = {}
_cache = _cache.shift_cache
tmp = asarray(x)
if iscomplexobj(tmp):
return shift(tmp.real, a, period, _cache) + 1j * shift(
tmp.imag, a, period, _cache)
if period is not None:
a = a*2*pi/period
n = len(x)
omega = _cache.get((n,a))
if omega is None:
if len(_cache) > 20:
while _cache:
_cache.popitem()
def kernel_real(k,a=a):
return cos(a*k)
def kernel_imag(k,a=a):
return sin(a*k)
omega_real = convolve.init_convolution_kernel(n,kernel_real,d=0,
zero_nyquist=0)
omega_imag = convolve.init_convolution_kernel(n,kernel_imag,d=1,
zero_nyquist=0)
_cache[(n,a)] = omega_real,omega_imag
else:
omega_real,omega_imag = omega
overwrite_x = _datacopied(tmp, x)
return convolve.convolve_z(tmp,omega_real,omega_imag,
overwrite_x=overwrite_x)
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