tKgx7dZgdZddlmZmZmZmZmZmZm Z m Z ddl m Z ddl mZiZddefdZ[iZdefd Z[iZdefd Z[iZefd Z[d ZiZdefd Z[iZdefdZ[iZdefdZ[iZdefdZ[iZdefdZ[y)z1 Differential and pseudo-differential operators. ) difftilbertitilberthilbertihilbertcs_diffcc_diffsc_diffss_diffshift)piasarraysincossinhcoshtanh iscomplexobj)convolve) _datacopiedNct|}|dk(r|St|r2t|j||dt|j||zzS| dt z|z }nd}t |}|j|||f}|Jt |dkDr|r|j|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 ??c&|rt||z|SyNr )pow)kordercs _/home/alanp/www/video.onchill/myenv/lib/python3.12/site-packages/scipy/fftpack/_pseudo_diffs.pykernelzdiff..kernelAs1Q3u~%rd zero_nyquistswap_real_imag overwrite_x) rrrrealimagr lengetpopitemrinit_convolution_kernelr) xr!period_cachetmpr"nomegar$r+s r#rrs: !*C z CCHHU6*2d388E&.I+III  bDK  AA JJ%{ #E } v;  !1 006E>?A#%{c1%K   Seai)4 66r%ct|}t|r2t|j||dt|j||zzS||dzt z|z }t |}|j||f}|Gt |dkDr|r|j|r|fd}tj||d}||||f<t||}tj||d|S)a Return h-Tilbert transform of a periodic sequence x. If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = sqrt(-1)*coth(j*h*2*pi/period) * x_j y_0 = 0 Parameters ---------- x : array_like The input array to transform. h : float Defines the parameter of the Tilbert transform. period : float, optional The assumed period of the sequence. Default period is ``2*pi``. Returns ------- tilbert : ndarray The result of the transform. Notes ----- If ``sum(x, axis=0) == 0`` and ``n = len(x)`` is odd, then ``tilbert(itilbert(x)) == x``. If ``2 * pi * h / period`` is approximately 10 or larger, then numerically ``tilbert == hilbert`` (theoretically oo-Tilbert == Hilbert). For even ``len(x)``, the Nyquist mode of ``x`` is taken zero. rrrc*|rdt||zz Sy)Nrr rr hs r#r$ztilbert..kernels4!9}$r%rr'r)) rrrr,r-r r.r/r0rr1r r2r<r3r4r5r6r7r$r+s r#rrSsH !*CCsxxF+GCHHa001 1 EBJ  AA JJ1v E } v;    00Fa@!u c1%K   SaK PPr%ct|}t|r2t|j||dt|j||zzS||dzt z|z }t |}|j||f}|Gt |dkDr|r|j|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`. rrrc&|rt||z Syrr:r;s r#r$zitilbert..kernelsQqS z!r%rr=r)) rrrr,r-r r.r/r0rr1rr>s r#rrs !*CC6*(388Af--. .  aCF6M AA JJ!u E } v;   006A>!u c1%K   SaK PPr%ct|}t|r.t|jdt|jzzSt |}|j |}|Ct |dkDr|r|j|rd}tj||d}|||<t||}tj||d|S)a Return Hilbert transform of a periodic sequence x. If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = sqrt(-1)*sign(j) * x_j y_0 = 0 Parameters ---------- x : array_like The input array, should be periodic. _cache : dict, optional Dictionary that contains the kernel used to do a convolution with. Returns ------- y : ndarray The transformed input. See Also -------- scipy.signal.hilbert : Compute the analytic signal, using the Hilbert transform. Notes ----- If ``sum(x, axis=0) == 0`` then ``hilbert(ihilbert(x)) == x``. For even len(x), the Nyquist mode of x is taken zero. The sign of the returned transform does not have a factor -1 that is more often than not found in the definition of the Hilbert transform. Note also that `scipy.signal.hilbert` does have an extra -1 factor compared to this function. rrc|dkDry|dkryy)Nr rgg)r s r#r$zhilbert..kernels1uQr%rr=r)) rrrr,r-r.r/r0rr1r)r2r4r5r6r7r$r+s r#rrsN !*CCsxx GCHH$5!555 AA JJqME } v;    006A>q c1%K   SaK PPr%ct| S)z Return inverse Hilbert transform of a periodic sequence x. If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = -sqrt(-1)*sign(j) * x_j y_0 = 0 )r)r2s r#rrs AJ;r%ct|}t|r4t|j|||dt|j|||zzS||dzt z|z }|dzt z|z }t |}|j|||f}|It |dkDr|r|j|r||fd}tj||d}|||||f<t||} tj||d| S)a Return (a,b)-cosh/sinh pseudo-derivative of a periodic sequence. If ``x_j`` and ``y_j`` are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = -sqrt(-1)*cosh(j*a*2*pi/period)/sinh(j*b*2*pi/period) * x_j y_0 = 0 Parameters ---------- x : array_like The array to take the pseudo-derivative from. a, b : float Defines the parameters of the cosh/sinh pseudo-differential operator. period : float, optional The period of the sequence. Default period is ``2*pi``. Returns ------- cs_diff : ndarray Pseudo-derivative of periodic sequence `x`. Notes ----- For even len(`x`), the Nyquist mode of `x` is taken as zero. rrrcD|rt||z t||zz Syr)rrr abs r#r$zcs_diff..kernel@s%QqS z$qs)++r%rr=r)) rrrr,r-r r.r/r0rr1r r2rHrIr3r4r5r6r7r$r+s r#rrs< !*CCsxx!F+'#((1Qv../ /  aCF6M aCF6M AA JJ!Aw E } v;  1 006A>!Awc1%K   SaK PPr%ct|}t|r4t|j|||dt|j|||zzS||dzt z|z }|dzt z|z }t |}|j|||f}|It |dkDr|r|j|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. rrrcB|rt||zt||zz Syr)rrrGs r#r$zsc_diff..kernelxs#AaCyac**r%rr=r)) rrr r,r-r r.r/r0rr1rrJs r#r r Ps4 !*CCsxx!F+'#((1Qv../ /  aCF6M aCF6M AA JJ!Aw E } v;  1 006A>!Awc1%K   SaK PPr%ct|}t|r4t|j|||dt|j|||zzS||dzt z|z }|dzt z|z }t |}|j|||f}|Gt |dkDr|r|j|r||fd}tj||}|||||f<t||} tj||| S)ac Return (a,b)-sinh/sinh pseudo-derivative of a periodic sequence x. If x_j and y_j are Fourier coefficients of periodic functions x and y, respectively, then:: y_j = sinh(j*a*2*pi/period)/sinh(j*b*2*pi/period) * x_j y_0 = a/b * x_0 Parameters ---------- x : array_like The array to take the pseudo-derivative from. a,b Defines the parameters of the sinh/sinh pseudo-differential operator. period : float, optional The period of the sequence x. Default is ``2*pi``. Notes ----- ``ss_diff(ss_diff(x,a,b),b,a) == x`` rrrc\|rt||zt||zz St||z SN)rfloatrGs r#r$zss_diff..kernels.AaCyac**8A: r%r+) rrr r,r-r r.r/r0rr1rrJs r#r r s2 !*CCsxx!F+'#((1Qv../ /  aCF6M aCF6M AA JJ!Aw E } v;  1 006:!Awc1%K   S; ??r%ct|}t|r4t|j|||dt|j|||zzS||dzt z|z }|dzt z|z }t |}|j|||f}|Gt |dkDr|r|j|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`` rrrc<t||zt||zz SrO)rrGs r#r$zcc_diff..kernels!9T!A#Y& &r%rQ) rrrr,r-r r.r/r0rr1rrJs r#rrs: !*CCsxx!F+'#((1Qv../ /  aCF6M aCF6M AA JJ!Aw E } v;  1 '006:!Awc1%K   S; ??r%ct|}t|r2t|j||dt|j||zzS||dzt z|z }t |}|j||f}|it |dkDr|r|j|r|fd}|fd}tj||dd} tj||dd} | | 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``. rrrct||zSrO)rr rHs r# kernel_realzshift..kernel_realqs8Or%ct||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 $ !*CCSXXa'5!F+C(CCC  aCF6M AA JJ!u E } v;    55a aCDF 55a aCDF ":-!u % :c1%K   s:j+6 88r%)__doc____all__numpyr rrrrrrrrscipy.fft._pocketfft.helperrr4rrrrrrr r rr rCr%r#rcs   HGG3 $v66r  f=Q@  V!QH  :Qz   !3Ql  !/Qd  !.@b  !0@f  F,8^ r%