o Ebx7@s dZgdZddlmZmZmZmZmZmZm Z m Z ddl m Z ddl mZiZddefdd Z[iZdefd d Z[iZdefd d Z[iZefddZ[ddZiZdefddZ[iZdefddZ[iZdefddZ[iZdefddZ[iZdefddZ[dS)z1 Differential and pseudo-differential operators. ) difftilbertitilberthilbertihilbertcs_diffcc_diffsc_diffss_diffshift)piasarraysincossinhcoshtanh iscomplexobj)convolve) _datacopiedNc Cst|}|dkr |St|rt|j||dt|j||S|dur)dt|}nd}t|}||||f}|dur`t|dkrI|rI||sC||fdd}t j |||d d }|||||f<t ||} t j |||d| d S) a* 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. r ?N?cSs|r t|||SdSNr )pow)kordercr =/usr/lib/python3/dist-packages/scipy/fftpack/_pseudo_diffs.pykernelAzdiff..kernelrdZ zero_nyquistZswap_real_imag overwrite_x) r rrrealimagr lengetpopitemrinit_convolution_kernelr) xrperiod_cachetmprnomegar"r'r r r!rs0    rc Ct|}t|rt|j||dt|j||S|dur$|dt|}t|}|||f}|durUt|dkrA|rA||s;|fdd}t j ||dd}||||f<t ||}t j ||d|d S) a Return h-Tilbert transform of a periodic sequence x. If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = sqrt(-1)*coth(j*h*2*pi/period) * x_j y_0 = 0 Parameters ---------- x : array_like The input array to transform. h : float Defines the parameter of the Tilbert transform. period : float, optional The assumed period of the sequence. Default period is ``2*pi``. Returns ------- tilbert : ndarray The result of the transform. Notes ----- If ``sum(x, axis=0) == 0`` and ``n = len(x)`` is odd, then ``tilbert(itilbert(x)) == x``. If ``2 * pi * h / period`` is approximately 10 or larger, then numerically ``tilbert == hilbert`` (theoretically oo-Tilbert == Hilbert). For even ``len(x)``, the Nyquist mode of ``x`` is taken zero. rNrrcSs|r dt||SdS)Nrr rrhr r r!r"sztilbert..kernelrr%r&) r rrr(r)r r*r+r,rr-r r.r7r/r0r1r2r3r"r'r r r!rSs&$     rc Cr4) a 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`. rNrrcSs|r t|| SdSrr5r6r r r!r"r#zitilbert..kernelrr8r&) r rrr(r)r r*r+r,rr-rr9r r r!rs&      rcCst|}t|rt|jdt|jSt|}||}|dur?t|dkr/|r/||s)dd}tj ||dd}|||<t ||}tj||d|dS) a 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. rNrcSs|dkrdS|dkr dSdS)Nr rggr )rr r r!r"s zhilbert..kernelrr8r&) r rrr(r)r*r+r,rr-r)r.r0r1r2r3r"r'r r r!rs'   rcCs t| S)z 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 )r)r.r r r!rs rc Ct|}t|rt|j|||dt|j|||S|dur.|dt|}|dt|}t|}||||f}|durbt|dkrL|rL||sF||fdd}t j ||dd}|||||f<t ||} t j ||d| d S) a 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. rNrrcSs"|rt|| t||SdSr)rrrabr r r!r"@szcs_diff..kernelrr8r&) r rrr(r)r r*r+r,rr-r r.r<r=r/r0r1r2r3r"r'r r r!rs(  rc Cr:) a 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. rNrrcSs |rt||t||SdSr)rrr;r r r!r"xszsc_diff..kernelrr8r&) r rrr(r)r r*r+r,rr-rr>r r r!rPs(  rc Ct|}t|rt|j|||dt|j|||S|dur.|dt|}|dt|}t|}||||f}|dur`t|dkrL|rL||sF||fdd}t ||}|||||f<t ||} t j ||| dS)ac 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`` rNrrcSs(|rt||t||St||SN)rfloatr;r r r!r"s zss_diff..kernelr') r rr r(r)r r*r+r,rr-rr>r r r!r s(   r c Cr?)a 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`` rNrrcSst||t||Sr@)rr;r r r!r"szcc_diff..kernelrB) r rrr(r)r r*r+r,rr-rr>r r r!rs(   rc Cst|}t|rt|j||dt|j||S|dur$|dt|}t|}|||f}|durht|dkrA|rA||s;|fdd}|fdd}t j ||d d d } t j ||d d d } | | f|||f<n|\} } t ||} t j || | | d S) a Shift periodic sequence x by a: y(u) = x(u+a). If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = exp(j*a*2*pi/period*sqrt(-1)) * x_f Parameters ---------- x : array_like The array to take the pseudo-derivative from. a : float Defines the parameters of the sinh/sinh pseudo-differential period : float, optional The period of the sequences x and y. Default period is ``2*pi``. rNrrcS t||Sr@)rrr<r r r! kernel_real zshift..kernel_realcSrCr@)rrDr r r! kernel_imagrFzshift..kernel_imagr r$rrB) r rr r(r)r r*r+r,rr-rZ convolve_z) r.r<r/r0r1r2r3rErGZ omega_realZ omega_imagr'r r r!r s4        r )__doc____all__Znumpyr r rrrrrrrZscipy.fft._pocketfft.helperrr0rrrrrrrr rr r r r r!sB(  9@$ =6213/