o `^h@>@sdZgdZddlZddlmZmZmZmZmZm Z m Z m Z ddl m Z ddlmZeZddefdd Zdefd d Zdefd d ZefddZefddZdefddZdefddZdefddZdefddZdefddZdS)z1 Differential and pseudo-differential operators. ) difftilbertitilberthilbertihilbertcs_diffcc_diffsc_diffss_diffshiftN)piasarraysincossinhcoshtanh iscomplexobj)convolve) _datacopiedc Cst|tjrt|dsi|_|j}t|}|dkr|St|r1t|j|||dt|j |||S|dur.kernelrd zero_nyquistswap_real_imag overwrite_x) isinstance threadinglocalhasattrrr rrrealimagr lengetpopitemrinit_convolution_kernelr) xrperiod_cachetmpr nomegar#r*r!r!r"rs<      rc Ct|tjrt|dsi|_|j}t|}t|r+t|j|||dt|j |||S|dur7|dt |}t |}| ||f}|durht |dkrT|rT| |sN|fdd}tj||dd }||||f<t||}tj||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. tilbert_cacherNrrcSs|r dt||SdS)Nrr rrhr!r!r"r#sztilbert..kernelrr&r()r+r,r-r.r<r rrr/r0r r1r2r3rr4r r5r?r6r7r8r9r:r#r*r!r!r"rUs. $     rc Cr;) 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`. itilbert_cacherNrrcSs|r t|| SdSrr=r>r!r!r"r#r$zitilbert..kernelrr@r()r+r,r-r.rBr rrr/r0r r1r2r3rr4rrAr!r!r"rs.     rcCst|tjrt|dsi|_|j}t|}t|r't|j|dt|j |St |}| |}|durRt |dkrB|rB| |s.kernelrr@r()r+r,r-r.rCr rrr/r0r1r2r3rr4r)r5r7r8r9r:r#r*r!r!r"rs& '    rcCs.t|tjrt|dsi|_|j}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 ihilbert_cache)r+r,r-r.rDr)r5r7r!r!r"rs  rc Ct|tjrt|dsi|_|j}t|}t|r-t|j||||dt|j ||||S|durA|dt |}|dt |}t |}| |||f}|durut |dkr_|r_| |sY||fdd}tj||dd }|||||f<t||} tj||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. cs_diff_cacherNrrcSs"|rt|| t||SdSr)rrrabr!r!r"r#Hszcs_diff..kernelrr@r()r+r,r-r.rFr rrr/r0r r1r2r3rr4r r5rHrIr6r7r8r9r:r#r*r!r!r"rs0    rc CrE) 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. sc_diff_cacherNrrcSs |rt||t||SdSr)rrrGr!r!r"r#szsc_diff..kernelrr@r()r+r,r-r.rKr rrr/r0r r1r2r3rr4rrJr!r!r"rRs0    rc Ct|tjrt|dsi|_|j}t|}t|r-t|j||||dt|j ||||S|durA|dt |}|dt |}t |}| |||f}|durst |dkr_|r_| |sY||fdd}t||}|||||f<t||} tj||| 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`` ss_diff_cacherNrrcSs(|rt||t||St||SN)rfloatrGr!r!r"r#s zss_diff..kernelr*)r+r,r-r.rMr rr r/r0r r1r2r3rr4rrJr!r!r"r s0     r c CrL) 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`` cc_diff_cacherNrrcSst||t||SrN)rrGr!r!r"r#szcc_diff..kernelrP)r+r,r-r.rQr rrr/r0r r1r2r3rr4rrJr!r!r"rs0     rc Cst|tjrt|dsi|_|j}t|}t|r+t|j|||dt|j |||S|dur7|dt |}t |}| ||f}|dur{t |dkrT|rT| |sN|fdd}|fdd }tj||d d d } tj||d d d } | | f|||f<n|\} } t||} tj|| | | 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``. shift_cacherNrrcS t||SrN)rrrHr!r!r" kernel_real zshift..kernel_realcSrSrN)rrTr!r!r" kernel_imagrVzshift..kernel_imagr r%rrP)r+r,r-r.rRr rr r/r0r r1r2r3rr4r convolve_z) r5rHr6r7r8r9r:rUrW omega_real omega_imagr*r!r!r"r s@          r )__doc____all__r,numpyr r rrrrrrrscipy.fft._pocketfft.helperrr-r7rrrrrrrr rr r!r!r!r"s"(  ?E ) B;768