| """ |
| Differential and pseudo-differential operators. |
| """ |
| |
|
|
| __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.fft._pocketfft.helper import _datacopied |
|
|
|
|
| _cache = {} |
|
|
|
|
| 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. |
| |
| """ |
| 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. |
| |
| 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. |
| |
| """ |
| 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 |
|
|
