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# scipy/scipy

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 """ Differential and pseudo-differential operators. """ # Created by Pearu Peterson, September 2002 from __future__ import division, print_function, absolute_import __all__ = ['diff', 'tilbert','itilbert','hilbert','ihilbert', 'cs_diff','cc_diff','sc_diff','ss_diff', 'shift'] from numpy import pi, asarray, sin, cos, sinh, cosh, tanh, iscomplexobj from . import convolve from scipy.fftpack.basic import _datacopied import atexit atexit.register(convolve.destroy_convolve_cache) del atexit _cache = {} def diff(x,order=1,period=None, _cache=_cache): """ Return k-th 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. """ tmp = asarray(x) if order == 0: return tmp if iscomplexobj(tmp): return diff(tmp.real,order,period)+1j*diff(tmp.imag,order,period) 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) del _cache _cache = {} 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. """ tmp = asarray(x) if iscomplexobj(tmp): return tilbert(tmp.real, h, period) + \ 1j * tilbert(tmp.imag, h, period) 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) del _cache _cache = {} 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`. """ tmp = asarray(x) if iscomplexobj(tmp): return itilbert(tmp.real,h,period) + \ 1j*itilbert(tmp.imag,h,period) 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) del _cache _cache = {} 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. 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. """ tmp = asarray(x) if iscomplexobj(tmp): return hilbert(tmp.real)+1j*hilbert(tmp.imag) 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) del _cache def ihilbert(x): """ 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 """ 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. """ tmp = asarray(x) if iscomplexobj(tmp): return cs_diff(tmp.real,a,b,period) + \ 1j*cs_diff(tmp.imag,a,b,period) 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) del _cache _cache = {} 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. """ tmp = asarray(x) if iscomplexobj(tmp): return sc_diff(tmp.real,a,b,period) + \ 1j*sc_diff(tmp.imag,a,b,period) 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) del _cache _cache = {} 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`` """ tmp = asarray(x) if iscomplexobj(tmp): return ss_diff(tmp.real,a,b,period) + \ 1j*ss_diff(tmp.imag,a,b,period) 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) del _cache _cache = {} 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`` """ tmp = asarray(x) if iscomplexobj(tmp): return cc_diff(tmp.real,a,b,period) + \ 1j*cc_diff(tmp.imag,a,b,period) 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) del _cache _cache = {} 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``. """ tmp = asarray(x) if iscomplexobj(tmp): return shift(tmp.real,a,period)+1j*shift(tmp.imag,a,period) 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) del _cache
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