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