a BCCfx7@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|r.kernelrdZ zero_nyquistZswap_real_imag overwrite_x) r rrrealimagr lengetpopitemrinit_convolution_kernelr) xrperiod_cachetmprnomegar"r&r r r!rs.     rc Cst|}t|r0t|j||dt|j||S|durH|dt|}t|}|||f}|durt|dkr|r|qr|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|rdt||SdS)Nrr rrhr r r!r"sztilbert..kernelrr$r%) r rrr'r(r r)r*r+rr,r r-r5r.r/r0r1r2r"r&r r r!rSs$$      rc Cst|}t|r0t|j||dt|j||S|durH|dt|}t|}|||f}|durt|dkr|r|qr|fdd}t j ||dd}||||f<t ||}t j ||d|d S) 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|rt|| SdSrr3r4r r r!r"szitilbert..kernelrr6r%) r rrr'r(r r)r*r+rr,rr7r r r!rs$       rcCst|}t|r(t|jdt|jSt|}||}|dur|t|dkr\|r\|qNdd}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|dkr dS|dkrdSdS)Nr rggr )rr r r!r"s zhilbert..kernelrr6r%) r rrr'r(r)r*r+rr,r)r-r/r0r1r2r"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 Cst|}t|r4t|j|||dt|j|||S|dur\|dt|}|dt|}t|}||||f}|durt|dkr|r|q||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..kernelrr6r%) r rrr'r(r r)r*r+rr,r r-r9r:r.r/r0r1r2r"r&r r r!rs&   rc Cst|}t|r4t|j|||dt|j|||S|dur\|dt|}|dt|}t|}||||f}|durt|dkr|r|q||fdd}t j ||dd}|||||f<t ||} t j ||d| d S) 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)rrr8r r r!r"xszsc_diff..kernelrr6r%) r rrr'r(r r)r*r+rr,rr;r r r!rPs&   rc Cst|}t|r4t|j|||dt|j|||S|dur\|dt|}|dt|}t|}||||f}|durt|dkr|r|q||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)rfloatr8r r r!r"szss_diff..kernelr&) r rr r'r(r r)r*r+rr,rr;r r r!r s&    r c Cst|}t|r4t|j|||dt|j|||S|dur\|dt|}|dt|}t|}||||f}|durt|dkr|r|q||fdd}t ||}|||||f<t ||} t j ||| dS)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<)rr8r r r!r"szcc_diff..kernelr>) r rrr'r(r r)r*r+rr,rr;r r r!rs&    rc Cst|}t|r0t|j||dt|j||S|durH|dt|}t|}|||f}|durt|dkr|r|qr|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``. rNrrcSs t||Sr<)rrr9r r r! kernel_realszshift..kernel_realcSs t||Sr<)rr?r r r! kernel_imagszshift..kernel_imagr r#rr>) r rr r'r(r r)r*r+rr,rZ convolve_z) r-r9r.r/r0r1r2r@rAZ omega_realZ omega_imagr&r r r!r s2         r )__doc____all__numpyr r rrrrrrrZscipy.fft._pocketfft.helperrr/rrrrrrrr rr r r r r!s@(  9@$ =6213/