^Mh@>dZgdZddlZddlmZmZmZmZmZm Z m Z m Z ddl m Z ddlmZejZddefdZdefd Zdefd Zefd Zefd Zdefd ZdefdZdefdZdefdZdefdZdS)z1 Differential and pseudo-differential operators. ) difftilbertitilberthilbertihilbertcs_diffcc_diffsc_diffss_diffshiftN)piasarraysincossinhcoshtanh iscomplexobj)convolve) _datacopiedct|tjrt|dsi|_|j}t |}|dkr|St |r2t|j|||dt|j |||zzS|dtz|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. diff_cacher ?N?c0|rt||z|SdSNr )pow)kordercs [/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/scipy/fftpack/_pseudo_diffs.pykernelzdiff..kernelIs! &1Q3u~~%1rd zero_nyquistswap_real_imag overwrite_x) isinstance threadinglocalhasattrrrrrrealimagr lengetpopitemrinit_convolution_kernelr) xr"period_cachetmpr#nomegar%r,s r$rrs:&)/**#v|,, # "F " !**C zz C-CHeVV44R HeVV9-9-6-- -  bDK  AA JJ%{ # #E } v;;   !    !!1    06E>?AAA#%{c1%%K  Seai)4 6 6 66r&crt|tjrt|dsi|_|j}t |}t |r2t|j|||dt|j |||zzS||dztz|z }t|}| ||f}|Nt|dkr|r| ||fd}tj||d}||||f<t!||}tj||d| S) a Return h-Tilbert transform of a periodic sequence x. If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = sqrt(-1)*coth(j*h*2*pi/period) * x_j y_0 = 0 Parameters ---------- x : array_like The input array to transform. h : float Defines the parameter of the Tilbert transform. period : float, optional The assumed period of the sequence. Default period is ``2*pi``. Returns ------- tilbert : ndarray The result of the transform. Notes ----- If ``sum(x, axis=0) == 0`` and ``n = len(x)`` is odd, then ``tilbert(itilbert(x)) == x``. If ``2 * pi * h / period`` is approximately 10 or larger, then numerically ``tilbert == hilbert`` (theoretically oo-Tilbert == Hilbert). For even ``len(x)``, the Nyquist mode of ``x`` is taken zero. tilbert_cacherNrrc4|rdt||zz SdS)Nrr rr!hs r$r%ztilbert..kernels# %4!99}$1r&rr(r*)r-r.r/r0r>rrrr1r2r r3r4r5rr6r r7rBr8r9r:r;r<r%r,s r$rrUseH&)/**&v// &#%F % !**CC9sxFF33GCHa8889 9 EBJ  AA JJ1v  E } v;;   !    !     0Fa@@@!u c1%%K  SaK P P PPr&crt|tjrt|dsi|_|j}t |}t |r2t|j|||dt|j |||zzS||dztz|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`. itilbert_cacherNrrc0|rt||z SdSrr@rAs r$r%zitilbert..kernels! "QqS z!1r&rrCr*)r-r.r/r0rFrrrr1r2r r3r4r5rr6rrDs r$rrsb&)/**'v/00 '$&F !& !**CC8!VV44(38Q7778 8  aCF6M AA JJ!u  E } v;;   !    !    06A>>>!u c1%%K  SaK P P PPr&c:t|tjrt|dsi|_|j}t |}t |r.t|j|dt|j |zzSt|}| |}|Jt|dkr|r| |d}tj||d}|||<t||}tj ||d|S) a Return Hilbert transform of a periodic sequence x. If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = sqrt(-1)*sign(j) * x_j y_0 = 0 Parameters ---------- x : array_like The input array, should be periodic. _cache : dict, optional Dictionary that contains the kernel used to do a convolution with. Returns ------- y : ndarray The transformed input. See Also -------- scipy.signal.hilbert : Compute the analytic signal, using the Hilbert transform. Notes ----- If ``sum(x, axis=0) == 0`` then ``hilbert(ihilbert(x)) == x``. For even len(x), the Nyquist mode of x is taken zero. The sign of the returned transform does not have a factor -1 that is more often than not found in the definition of the Hilbert transform. Note also that `scipy.signal.hilbert` does have an extra -1 factor compared to this function. hilbert_cacherNrc&|dkrdS|dkrdSdS)Nr rgg)r!s r$r%zhilbert..kernels#1uusQt3r&rrCr*)r-r.r/r0rIrrrr1r2r3r4r5rr6r)r7r9r:r;r<r%r,s r$rrs+N&)/**&v// &#%F % !**CCJsx((2&0I0I+III AA JJqMME } v;;   !    !    06A>>>q c1%%K  SaK P P PPr&ct|tjrt|dsi|_|j}t || S)z Return inverse Hilbert transform of a periodic sequence x. If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = -sqrt(-1)*sign(j) * x_j y_0 = 0 ihilbert_cache)r-r.r/r0rMr)r7r9s r$rrsN&)/**'v/00 '$&F !& Av   r&c t|tjrt|dsi|_|j}t |}t |r4t|j||||dt|j ||||zzS| |dztz|z }|dztz|z }t|}| |||f}|Pt|dkr|r| |||fd}tj||d}|||||f<t!||} tj||d| S) a Return (a,b)-cosh/sinh pseudo-derivative of a periodic sequence. If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = -sqrt(-1)*cosh(j*a*2*pi/period)/sinh(j*b*2*pi/period) * x_j y_0 = 0 Parameters ---------- x : array_like The array to take the pseudo-derivative from. a, b : float Defines the parameters of the cosh/sinh pseudo-differential operator. period : float, optional The period of the sequence. Default period is ``2*pi``. Returns ------- cs_diff : ndarray Pseudo-derivative of periodic sequence `x`. Notes ----- For even len(`x`), the Nyquist mode of `x` is taken as zero. cs_diff_cacherNrrcV|r&t||z t||zz SdSr)rrr!abs r$r%zcs_diff..kernelHs0 ,QqS z$qs))++1r&rrCr*)r-r.r/r0rOrrrr1r2r r3r4r5rr6r r7rRrSr8r9r:r;r<r%r,s r$rrs|<&)/**&v// &#%F % !**CC:sxAvv66'#(Aq&&999: :  aCF6M aCF6M AA JJ!Aw  E } v;;   !    !1    06A>>>!Awc1%%K  SaK P P PPr&c t|tjrt|dsi|_|j}t |}t |r4t|j||||dt|j ||||zzS| |dztz|z }|dztz|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. sc_diff_cacherNrrcT|r%t||zt||zz SdSr)rrrQs r$r%zsc_diff..kernels. +AaCyyac**1r&rrCr*)r-r.r/r0rVrrr r1r2r r3r4r5rr6rrTs r$r r Rs|4&)/**&v// &#%F % !**CC<sxAvv66GCHaFF;;;< <  aCF6M aCF6M AA JJ!Aw  E } v;;   !    !1    06A>>>!Awc1%%K  SaK P P PPr&c t|tjrt|dsi|_|j}t |}t |r4t|j||||dt|j ||||zzS| |dztz|z }|dztz|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`` ss_diff_cacherNrrct|r%t||zt||zz St||z SN)rfloatrQs r$r%zss_diff..kernels9 +AaCyyac**88A: r&r,)r-r.r/r0rYrrr r1r2r r3r4r5rr6rrTs r$r r su2&)/**&v// &#%F % !**CC:sxAvv66'#(Aq&&999: :  aCF6M aCF6M AA JJ!Aw  E } v;;   !    !1    06::!Awc1%%K  S; ? ? ??r&c t|tjrt|dsi|_|j}t |}t |r4t|j||||dt|j ||||zzS| |dztz|z }|dztz|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`` cc_diff_cacherNrrcLt||zt||zz Sr[)rrQs r$r%zcc_diff..kernels!!99T!A#YY& &r&r])r-r.r/r0r_rrrr1r2r r3r4r5rr6rrTs r$rrss:&)/**&v// &#%F % !**CC<sxAvv66GCHaFF;;;< <  aCF6M aCF6M AA JJ!Aw  E } v;;   !    !1 ' ' ' '06::!Awc1%%K  S; ? ? ??r&ct|tjrt|dsi|_|j}t |}t |r2t|j|||dt|j |||zzS||dztz|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``. shift_cacherNrrc&t||zSr[)rr!rRs r$ kernel_realzshift..kernel_realqs88Or&c&t||zSr[)rrds r$ kernel_imagzshift..kernel_imagrfr&r r'rr])r-r.r/r0rbrrr r1r2r r3r4r5rr6r convolve_z) r7rRr8r9r:r;r<rerh omega_real omega_imagr,s r$r r s$&)/**$v}-- $!#F # !**CC)SXq&&11B Ha:):)5)) )  aCF6M AA JJ!u  E } v;;   !    !        5a aCDFFF 5a aCDFFF ":-!u % :c1%%K  s:j+6 8 8 88r&)__doc____all__r.numpyr rrrrrrrrscipy.fft._pocketfft.helperrr/r9rrrrrrr r rr rKr&r$rqs     GGGGGGGGGGGGGGGGGGGG333333   $v<6<6<6<6~fBQBQBQBQJV&Q&Q&Q&QR?Q?Q?Q?QD$!8Q8Q8Q8Qv!4Q4Q4Q4Qn!3@3@3@3@l!5@5@5@5@pF282828282828r&