(ph@>Sr/SQrSSKrSSKJrJrJrJrJrJ r J r J r SSK J r SSKJr \R "5rSS\4SjrS\4S jrS\4S jr\4S jr\4S jrS\4S jrS\4SjrS\4SjrS\4SjrS\4Sjrg)z1 Differential and pseudo-differential operators. ) difftilbertitilberthilbertihilbertcs_diffcc_diffsc_diffss_diffshiftN)piasarraysincossinhcoshtanh iscomplexobj)convolve) _datacopiedc[U[R5(a$[US5(d0UlURn[ U5nUS:XaU$[ U5(a2[URXU5S[URXU5--$Ub S[-U- nOSn[U5nURXaU45nUcQ[U5S:a U(aUR5 U(aMX4Sjn[R"XhUSS 9nXsXaU4'[!X@5n [R"XGUS-U S 9$) 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. diff_cacher ??c.U(a[X -U5$gNr )pow)kordercs N/var/www/html/venv/lib/python3.13/site-packages/scipy/fftpack/_pseudo_diffs.pykerneldiff..kernelIs13u~%rd zero_nyquistswap_real_imag overwrite_x) isinstance threadinglocalhasattrrrrrrealimagr lengetpopitemrinit_convolution_kernelr) xr"period_cachetmpr#nomegar%r-s r$rrs8:&)//**v|,, "F "" !*C z CCHHeV4R HHeV9-6-- -  bDK  AA JJ{ #E } v;  &! 00E>?A#{c%K   Seai)4 66r'cf[U[R5(a$[US5(d0UlURn[ U5n[ U5(a2[URXU5S[URXU5--$UbUS-[-U- n[U5nURXQ45nUcO[U5S:a U(aUR5 U(aMU4Sjn[R"XWSS9nXcXQ4'[!X@5n[R"XFSUS9$) 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_cacherrrc2U(aS[X-5- $g)Nrr rr!hs r$r%tilbert..kernels49}$r'rr)r+)r.r/r0r1r?rrrr2r3r r4r5r6rr7r r8rCr9r:r;r<r=r%r-s r$rrUsH&)//**v//#%F %% !*CCsxxF3GCHHa889 9 EBJ  AA JJv E } v;  &  00a@u c%K   SaK PPr'cf[U[R5(a$[US5(d0UlURn[ U5n[ U5(a2[URXU5S[URXU5--$UbUS-[-U- n[U5nURXQ45nUcO[U5S:a U(aUR5 U(aMU4Sjn[R"XWSS9nXcXQ4'[!X@5n[R"XFSUS9$) z Return inverse h-Tilbert transform of a periodic sequence x. If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = -sqrt(-1)*tanh(j*h*2*pi/period) * x_j y_0 = 0 For more details, see `tilbert`. itilbert_cacherrrc.U(a[X-5*$grrArBs r$r%itilbert..kernelsQS z!r'rrEr+)r.r/r0r1rHrrrr2r3r r4r5r6rr7rrFs r$rrs&)//**v/00$&F !&& !*CC!V4(388Q778 8  aCF6M AA JJu E } v;  & 00A>u c%K   SaK PPr'c4[U[R5(a$[US5(d0UlURn[ U5n[ U5(a0[URU5S[URU5--$[U5nURU5nUcK[U5S:a U(aUR5 U(aMSn[R"X5SS9nXAU'[X 5n[R"X$SUS9$)a Return Hilbert transform of a periodic sequence x. If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = sqrt(-1)*sign(j) * x_j y_0 = 0 Parameters ---------- x : array_like The input array, should be periodic. _cache : dict, optional Dictionary that contains the kernel used to do a convolution with. Returns ------- y : ndarray The transformed input. See Also -------- scipy.signal.hilbert : Compute the analytic signal, using the Hilbert transform. Notes ----- If ``sum(x, axis=0) == 0`` then ``hilbert(ihilbert(x)) == x``. For even len(x), the Nyquist mode of x is taken zero. The sign of the returned transform does not have a factor -1 that is more often than not found in the definition of the Hilbert transform. Note also that `scipy.signal.hilbert` does have an extra -1 factor compared to this function. hilbert_cacherrc US:agUS:agg)Nr rgg)r!s r$r%hilbert..kernels1uQr'rrEr+)r.r/r0r1rLrrrr2r3r4r5r6rr7r)r8r:r;r<r=r%r-s r$rrsN&)//**v//#%F %% !*CCsxx(2&0I+III AA JJqME } v;  &  00A>q c%K   SaK PPr'c[U[R5(a$[US5(d0UlURn[ X5*$)z Return inverse Hilbert transform of a periodic sequence x. If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = -sqrt(-1)*sign(j) * x_j y_0 = 0 ihilbert_cache)r.r/r0r1rQr)r8r:s r$rrsD&)//**v/00$&F !&& A  r'c [U[R5(a$[US5(d0UlURn[ U5n[ U5(a2[URXX45S[URXX45--$UbUS-[-U- nUS-[-U- n[U5nURXaU45nUcP[U5S:a U(aUR5 U(aMX4Sjn[R"XhSS9nXtXaU4'[!XP5n [R"XWSU S9$) 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_cacherrrcJU(a[X-5*[X -5- $gr)rrr!abs r$r%cs_diff..kernelHs!QS z$qs)++r'rrEr+)r.r/r0r1rSrrrr2r3r r4r5r6rr7r r8rVrWr9r:r;r<r=r%r-s r$rrs(<&)//**v//#%F %% !*CCsxxv6'#((A&99: :  aCF6M aCF6M AA JJAw E } v;  & 00A>Awc%K   SaK PPr'c [U[R5(a$[US5(d0UlURn[ U5n[ U5(a2[URXX45S[URXX45--$UbUS-[-U- nUS-[-U- n[U5nURXaU45nUcP[U5S:a U(aUR5 U(aMX4Sjn[R"XhSS9nXtXaU4'[!XP5n [R"XWSU S9$) 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_cacherrrcHU(a[X-5[X -5- $gr)rrrUs r$r%sc_diff..kernelsACyac**r'rrEr+)r.r/r0r1r[rrr r2r3r r4r5r6rr7rrYs r$r r Rs(4&)//**v//#%F %% !*CCsxxv6GCHHaF;;< <  aCF6M aCF6M AA JJAw E } v;  & 00A>Awc%K   SaK PPr'c [U[R5(a$[US5(d0UlURn[ U5n[ U5(a2[URXX45S[URXX45--$UbUS-[-U- nUS-[-U- n[U5nURXaU45nUcQ[U5S:a U(aUR5 U(aMX4Sjn[R"Xh5nXtXaU4'[!XP5n [R"XWU S9$)a 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_cacherrrcbU(a[X-5[X -5- $[U5U- $N)rfloatrUs r$r%ss_diff..kernels*ACyac**8A: r'r-)r.r/r0r1r_rrr r2r3r r4r5r6rr7rrYs r$r r s$2&)//**v//#%F %% !*CCsxxv6'#((A&99: :  aCF6M aCF6M AA JJAw E } v;  & 00:Awc%K   S; ??r'c [U[R5(a$[US5(d0UlURn[ U5n[ U5(a2[URXX45S[URXX45--$UbUS-[-U- nUS-[-U- n[U5nURXaU45nUcQ[U5S:a U(aUR5 U(aMX4Sjn[R"Xh5nXtXaU4'[!XP5n [R"XWU S9$)ab 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_cacherrrc8[X-5[X -5- $ra)rrUs r$r%cc_diff..kernels9T!#Y& &r'rd)r.r/r0r1rfrrrr2r3r r4r5r6rr7rrYs r$rrs":&)//**v//#%F %% !*CCsxxv6GCHHaF;;< <  aCF6M aCF6M AA JJAw E } v;  & '00:Awc%K   S; ??r'c[U[R5(a$[US5(d0UlURn[ U5n[ U5(a2[URXU5S[URXU5--$UbUS-[-U- n[U5nURXQ45nUco[U5S:a U(aUR5 U(aMU4SjnU4Sjn[R"XWSSS9n [R"XXS SS9n X4X5U4'OUup[!X@5n [R""XIU U S 9$) 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_cacherrrc[X-5$ra)rr!rVs r$ kernel_realshift..kernel_real qs8Or'c[X-5$ra)rrls r$ kernel_imagshift..kernel_imagror'r r(rrd)r.r/r0r1rjrrr r2r3r r4r5r6rr7r convolve_z) r8rVr9r:r;r<r=rmrq omega_real omega_imagr-s r$r r sO$&)//**v}--!#F ## !*CCSXXq&1B HHa:)5)) )  aCF6M AA JJu E } v;  &  55aaCDF 55aaCDF "-!u % c%K   sj+6 88r')__doc____all__r/numpyr rrrrrrrrscipy.fft._pocketfft.helperrr0r:rrrrrrr r rr rNr'r$r{s   GGG3  $v<6~fBQJV&QR?QD$!8Qv!4Qn!3@l!5@pF28r'