Ugx7dZgdZddlmZmZmZmZmZmZm Z m Z ddl m Z ddl mZiZddefdZ[iZdefd Z[iZdefd Z[iZefd Z[d ZiZdefd Z[iZdefdZ[iZdefdZ[iZdefdZ[iZdefdZ[dS)z1 Differential and pseudo-differential operators. ) difftilbertitilberthilbertihilbertcs_diffcc_diffsc_diffss_diffshift)piasarraysincossinhcoshtanh iscomplexobj)convolve) _datacopiedNct|}|dkr|St|r0t|j||dt|j||zzS|dt z|z }nd}t |}||||f}|Qt |dkr|r||||fd}tj |||d }|||||f<t||} tj |||dz| 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?c0|rt||z|SdSNr )pow)kordercs Z/var/www/surfInsights/venv3-11/lib/python3.11/site-packages/scipy/fftpack/_pseudo_diffs.pykernelzdiff..kernelAs! &1Q3u~~%1rd zero_nyquistswap_real_imag overwrite_x) rrrrealimagr lengetpopitemrinit_convolution_kernelr) xr!period_cachetmpr"nomegar$r+s r#rrsT: !**C zz CJCHU6**2d38E&.I.I+III  bDK  AA JJ%{ # #E } v;;   !    !!1    06E>?AAA#%{c1%%K  Seai)4 6 6 66r%ct|}t|r0t|j||dt|j||zzS||dzt z|z }t |}|||f}|Nt |dkr|r|||fd}tj ||d}||||f<t||}tj ||d|S) a Return h-Tilbert transform of a periodic sequence x. If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = sqrt(-1)*coth(j*h*2*pi/period) * x_j y_0 = 0 Parameters ---------- x : array_like The input array to transform. h : float Defines the parameter of the Tilbert transform. period : float, optional The assumed period of the sequence. Default period is ``2*pi``. Returns ------- tilbert : ndarray The result of the transform. Notes ----- If ``sum(x, axis=0) == 0`` and ``n = len(x)`` is odd, then ``tilbert(itilbert(x)) == x``. If ``2 * pi * h / period`` is approximately 10 or larger, then numerically ``tilbert == hilbert`` (theoretically oo-Tilbert == Hilbert). For even ``len(x)``, the Nyquist mode of ``x`` is taken zero. rNrrc4|rdt||zz SdS)Nrr rr hs r#r$ztilbert..kernels# %4!99}$1r%rr'r)) rrrr,r-r r.r/r0rr1r r2r<r3r4r5r6r7r$r+s r#rrSs+H !**CC1sxF++GCHa0001 1 EBJ  AA JJ1v  E } v;;   !    !     0Fa@@@!u c1%%K  SaK P P PPr%ct|}t|r0t|j||dt|j||zzS||dzt z|z }t |}|||f}|Nt |dkr|r|||fd}tj ||d}||||f<t||}tj ||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`. rNrrc0|rt||z SdSrr:r;s r#r$zitilbert..kernels! "QqS z!1r%rr=r)) rrrr,r-r r.r/r0rr1rr>s r#rrs' !**CC.6**(38Af---. .  aCF6M AA JJ!u  E } v;;   !    !    06A>>>!u c1%%K  SaK P P PPr%ct|}t|r,t|jdt|jzzSt |}||}|Jt |dkr|r||d}tj ||d}|||<t||}tj||d|S)a Return Hilbert transform of a periodic sequence x. If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = sqrt(-1)*sign(j) * x_j y_0 = 0 Parameters ---------- x : array_like The input array, should be periodic. _cache : dict, optional Dictionary that contains the kernel used to do a convolution with. Returns ------- y : ndarray The transformed input. See Also -------- scipy.signal.hilbert : Compute the analytic signal, using the Hilbert transform. Notes ----- If ``sum(x, axis=0) == 0`` then ``hilbert(ihilbert(x)) == x``. For even len(x), the Nyquist mode of x is taken zero. The sign of the returned transform does not have a factor -1 that is more often than not found in the definition of the Hilbert transform. Note also that `scipy.signal.hilbert` does have an extra -1 factor compared to this function. rNrc&|dkrdS|dkrdSdS)Nr rgg)r s r#r$zhilbert..kernels#1uusQt3r%rr=r)) rrrr,r-r.r/r0rr1r)r2r4r5r6r7r$r+s r#rrsN !**CC6sx  GCH$5$5!555 AA JJqMME } v;;   !    !    06A>>>q c1%%K  SaK P P PPr%c"t| S)z Return inverse Hilbert transform of a periodic sequence x. If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = -sqrt(-1)*sign(j) * x_j y_0 = 0 )r)r2s r#rrs AJJ;r%c(t|}t|r2t|j|||dt|j|||zzS| |dzt z|z }|dzt z|z }t |}||||f}|Pt |dkr|r||||fd}tj ||d}|||||f<t||} tj ||d| S) a Return (a,b)-cosh/sinh pseudo-derivative of a periodic sequence. If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = -sqrt(-1)*cosh(j*a*2*pi/period)/sinh(j*b*2*pi/period) * x_j y_0 = 0 Parameters ---------- x : array_like The array to take the pseudo-derivative from. a, b : float Defines the parameters of the cosh/sinh pseudo-differential operator. period : float, optional The period of the sequence. Default period is ``2*pi``. Returns ------- cs_diff : ndarray Pseudo-derivative of periodic sequence `x`. Notes ----- For even len(`x`), the Nyquist mode of `x` is taken as zero. rNrrcV|r&t||z t||zz SdSr)rrr abs r#r$zcs_diff..kernel@s0 ,QqS z$qs))++1r%rr=r)) rrrr,r-r r.r/r0rr1r r2rHrIr3r4r5r6r7r$r+s r#rrsB< !**CC/sx!F++'#(1Qv.../ /  aCF6M aCF6M AA JJ!Aw  E } v;;   !    !1    06A>>>!Awc1%%K  SaK P P PPr%c(t|}t|r2t|j|||dt|j|||zzS| |dzt z|z }|dzt z|z }t |}||||f}|Pt |dkr|r||||fd}tj ||d}|||||f<t||} tj ||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. rNrrcT|r%t||zt||zz SdSr)rrrGs r#r$zsc_diff..kernelxs. +AaCyyac**1r%rr=r)) rrr r,r-r r.r/r0rr1rrJs r#r r PsB4 !**CC/sx!F++'#(1Qv.../ /  aCF6M aCF6M AA JJ!Aw  E } v;;   !    !1    06A>>>!Awc1%%K  SaK P P PPr%c"t|}t|r2t|j|||dt|j|||zzS| |dzt z|z }|dzt z|z }t |}||||f}|Nt |dkr|r||||fd}tj ||}|||||f<t||} tj ||| S)ac Return (a,b)-sinh/sinh pseudo-derivative of a periodic sequence x. If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = sinh(j*a*2*pi/period)/sinh(j*b*2*pi/period) * x_j y_0 = a/b * x_0 Parameters ---------- x : array_like The array to take the pseudo-derivative from. a,b Defines the parameters of the sinh/sinh pseudo-differential operator. period : float, optional The period of the sequence x. Default is ``2*pi``. Notes ----- ``ss_diff(ss_diff(x,a,b),b,a) == x`` rNrrct|r%t||zt||zz St||z SN)rfloatrGs r#r$zss_diff..kernels9 +AaCyyac**88A: r%r+) rrr r,r-r r.r/r0rr1rrJs r#r r s;2 !**CC/sx!F++'#(1Qv.../ /  aCF6M aCF6M AA JJ!Aw  E } v;;   !    !1    06::!Awc1%%K  S; ? ? ??r%c"t|}t|r2t|j|||dt|j|||zzS| |dzt z|z }|dzt z|z }t |}||||f}|Nt |dkr|r||||fd}tj ||}|||||f<t||} tj ||| S)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`` rNrrcLt||zt||zz SrO)rrGs r#r$zcc_diff..kernels!!99T!A#YY& &r%rQ) rrrr,r-r r.r/r0rr1rrJs r#rrs9: !**CC/sx!F++'#(1Qv.../ /  aCF6M aCF6M AA JJ!Aw  E } v;;   !    !1 ' ' ' '06::!Awc1%%K  S; ? ? ??r%cJt|}t|r0t|j||dt|j||zzS||dzt z|z }t |}|||f}|ot |dkr|r|||fd}|fd}tj ||dd} tj ||d d} | | f|||f<n|\} } t||} tj || | | S) a Shift periodic sequence x by a: y(u) = x(u+a). If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = exp(j*a*2*pi/period*sqrt(-1)) * x_f Parameters ---------- x : array_like The array to take the pseudo-derivative from. a : float Defines the parameters of the sinh/sinh pseudo-differential period : float, optional The period of the sequences x and y. Default period is ``2*pi``. rNrrc&t||zSrO)rr rHs r# kernel_realzshift..kernel_realqs88Or%c&t||zSrO)rrVs r# kernel_imagzshift..kernel_imagrXr%r r&rrQ) rrr r,r-r r.r/r0rr1r convolve_z) r2rHr3r4r5r6r7rWrZ omega_real omega_imagr+s r#r r s$ !**CCDSXa''5!F+C+C(CCC  aCF6M AA JJ!u  E } v;;   !    !        5a aCDFFF 5a aCDFFF ":-!u % :c1%%K  s:j+6 8 8 88r%)__doc____all__numpyr rrrrrrrrscipy.fft._pocketfft.helperrr4rrrrrrr r rr rCr%r#rcs     HGGGGGGGGGGGGGGGGGGG333333 $v66666666r  f=Q=Q=Q=Q@  V!Q!Q!Q!QH  :Q:Q:Q:Qz     !3Q3Q3Q3Ql  !/Q/Q/Q/Qd  !.@.@.@.@b  !0@0@0@0@f  F,8,8,8,8^ FFr%