Abstract

Phase reconstruction is used for feedback control in adaptive optics systems. To achieve performance metrics for high actuator density or with limited processing capabilities on spacecraft, a wavelet signal processing technique is advantageous. Previous derivations of this technique have been limited to the Haar wavelet. This paper derives the relationship and algorithms to reconstruct phase with O(n) computational complexity for wavelets with the orthogonal property. This has additional benefits for performance with noise in the measurements. We also provide details on how to handle the boundary condition for telescope apertures.

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References

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  1. F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).
  2. P. Y. Bely, ed., The Design and Construction of Large Optical Telescopes (Springer, 2003).
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    [CrossRef]
  4. L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. A 19, 2100–2111 (2002).
    [CrossRef]
  5. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  6. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  7. J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  8. K. Freischlad, “Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform,” Proc. SPIE 551, 74–80 (1985).
    [CrossRef]
  9. L. Gilles, C. R. Vogel, and B. L. Ellerbroek, “Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction,” J. Opt. Soc. Am. A 19, 1817–1822 (2002).
    [CrossRef]
  10. D. G. MacMartin, “Local, hierarchic, and iterative reconstructors for adaptive optics,” J. Opt. Soc. Am. A 20, 1084–1093 (2003).
    [CrossRef]
  11. L. Gilles, “Sparse minimum-variance open-loop reconstructors for extreme adaptive optics: order N multigrid versus preordered Cholesky factorization,” Proc. SPIE 5169, 201–205 (2003).
    [CrossRef]
  12. C. R. Vogel, “Sparse matrix methods for wavefront reconstruction, revisited,” Proc. SPIE 5490, 1327–1335 (2004).
    [CrossRef]
  13. C. R. Vogel and Q. Yang, “Multigrid algorithm for least-squares wavefront reconstruction,” Appl. Opt. 45, 705–715 (2006).
    [CrossRef]
  14. C. Béchet, M. Tallon, and E. Thiébaut, “FRIM: minimum-variance reconstructor with a fractal iterative method,” Proc. SPIE 6272, 62722U (2006).
    [CrossRef]
  15. E. Thiébaut and M. Tallon, “Fast minimum variance wavefront reconstruction for extremely large telescopes,” J. Opt. Soc. Am. A 27, 1046–1059 (2010).
    [CrossRef]
  16. J. Herrmann, “Phase variance and Strehl ratio in adaptive optics,” J. Opt. Soc. Am. A 9, 2257–2258 (1992).
    [CrossRef]
  17. M. Rosensteiner, “Cumulative reconstructor: fast wavefront reconstruction algorithm for extremely large telescopes,” J. Opt. Soc. Am. A 28, 2132–2138 (2011).
    [CrossRef]
  18. C. C. de Visser and M. Verhaegen, “Wavefront reconstruction in adaptive optics systems using nonlinear multivariate splines,” J. Opt. Soc. Am. A 30, 82–95 (2013).
    [CrossRef]
  19. F. U. Dowla, “Fast Fourier and wavelet transforms for wavefront reconstruction in adaptive optics,” Proc. SPIE 4124, 118–127 (2000).
    [CrossRef]
  20. P. J. Hampton, P. Agathoklis, and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Sel. Top. Signal Process. 2, 781–792 (2008).
    [CrossRef]
  21. P. J. Hampton, “Robust order N wavelet filterbanks to perform 2-D numerical integration directly from partial difference or gradient measurements,” Ph.D. thesis (University of Victoria, 2009).
  22. P. J. Hampton, P. Agathoklis, R. Conan, and C. Bradley, “Closed-loop control of a woofer-tweeter adaptive optics system using wavelet-based phase reconstruction,” J. Opt. Soc. Am. A 27, A145–A156 (2010).
    [CrossRef]
  23. D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, 1984).
  24. W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial (SIAM, 2000).
  25. P. P. Vaidyanathan, Multirate Systems and Filter Banks(Prentice-Hall, 1993).
  26. C. Shannon, “The mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
  27. A. Haar, “Zur Theorie der orthogonalen Funktionensysteme,” Ph.D. thesis (University of Göttingen, 1909).

2013 (1)

2011 (1)

2010 (2)

2008 (1)

P. J. Hampton, P. Agathoklis, and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Sel. Top. Signal Process. 2, 781–792 (2008).
[CrossRef]

2006 (2)

C. Béchet, M. Tallon, and E. Thiébaut, “FRIM: minimum-variance reconstructor with a fractal iterative method,” Proc. SPIE 6272, 62722U (2006).
[CrossRef]

C. R. Vogel and Q. Yang, “Multigrid algorithm for least-squares wavefront reconstruction,” Appl. Opt. 45, 705–715 (2006).
[CrossRef]

2004 (1)

C. R. Vogel, “Sparse matrix methods for wavefront reconstruction, revisited,” Proc. SPIE 5490, 1327–1335 (2004).
[CrossRef]

2003 (2)

D. G. MacMartin, “Local, hierarchic, and iterative reconstructors for adaptive optics,” J. Opt. Soc. Am. A 20, 1084–1093 (2003).
[CrossRef]

L. Gilles, “Sparse minimum-variance open-loop reconstructors for extreme adaptive optics: order N multigrid versus preordered Cholesky factorization,” Proc. SPIE 5169, 201–205 (2003).
[CrossRef]

2002 (2)

2000 (1)

F. U. Dowla, “Fast Fourier and wavelet transforms for wavefront reconstruction in adaptive optics,” Proc. SPIE 4124, 118–127 (2000).
[CrossRef]

1992 (1)

1985 (1)

K. Freischlad, “Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform,” Proc. SPIE 551, 74–80 (1985).
[CrossRef]

1980 (2)

1977 (2)

1948 (1)

C. Shannon, “The mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).

Agathoklis, P.

P. J. Hampton, P. Agathoklis, R. Conan, and C. Bradley, “Closed-loop control of a woofer-tweeter adaptive optics system using wavelet-based phase reconstruction,” J. Opt. Soc. Am. A 27, A145–A156 (2010).
[CrossRef]

P. J. Hampton, P. Agathoklis, and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Sel. Top. Signal Process. 2, 781–792 (2008).
[CrossRef]

Béchet, C.

C. Béchet, M. Tallon, and E. Thiébaut, “FRIM: minimum-variance reconstructor with a fractal iterative method,” Proc. SPIE 6272, 62722U (2006).
[CrossRef]

Bradley, C.

P. J. Hampton, P. Agathoklis, R. Conan, and C. Bradley, “Closed-loop control of a woofer-tweeter adaptive optics system using wavelet-based phase reconstruction,” J. Opt. Soc. Am. A 27, A145–A156 (2010).
[CrossRef]

P. J. Hampton, P. Agathoklis, and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Sel. Top. Signal Process. 2, 781–792 (2008).
[CrossRef]

Brase, J. M.

Briggs, W. L.

W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial (SIAM, 2000).

Conan, R.

de Visser, C. C.

Dowla, F. U.

F. U. Dowla, “Fast Fourier and wavelet transforms for wavefront reconstruction in adaptive optics,” Proc. SPIE 4124, 118–127 (2000).
[CrossRef]

Dudgeon, D. E.

D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, 1984).

Ellerbroek, B. L.

Freischlad, K.

K. Freischlad, “Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform,” Proc. SPIE 551, 74–80 (1985).
[CrossRef]

Fried, D. L.

Gavel, D. T.

Gilles, L.

L. Gilles, “Sparse minimum-variance open-loop reconstructors for extreme adaptive optics: order N multigrid versus preordered Cholesky factorization,” Proc. SPIE 5169, 201–205 (2003).
[CrossRef]

L. Gilles, C. R. Vogel, and B. L. Ellerbroek, “Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction,” J. Opt. Soc. Am. A 19, 1817–1822 (2002).
[CrossRef]

Haar, A.

A. Haar, “Zur Theorie der orthogonalen Funktionensysteme,” Ph.D. thesis (University of Göttingen, 1909).

Hampton, P. J.

P. J. Hampton, P. Agathoklis, R. Conan, and C. Bradley, “Closed-loop control of a woofer-tweeter adaptive optics system using wavelet-based phase reconstruction,” J. Opt. Soc. Am. A 27, A145–A156 (2010).
[CrossRef]

P. J. Hampton, P. Agathoklis, and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Sel. Top. Signal Process. 2, 781–792 (2008).
[CrossRef]

P. J. Hampton, “Robust order N wavelet filterbanks to perform 2-D numerical integration directly from partial difference or gradient measurements,” Ph.D. thesis (University of Victoria, 2009).

Henson, V. E.

W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial (SIAM, 2000).

Herrmann, J.

Hudgin, R. H.

MacMartin, D. G.

McCormick, S. F.

W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial (SIAM, 2000).

Mersereau, R. M.

D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, 1984).

Poyneer, L. A.

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).

Rosensteiner, M.

Shannon, C.

C. Shannon, “The mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).

Southwell, W. H.

Tallon, M.

E. Thiébaut and M. Tallon, “Fast minimum variance wavefront reconstruction for extremely large telescopes,” J. Opt. Soc. Am. A 27, 1046–1059 (2010).
[CrossRef]

C. Béchet, M. Tallon, and E. Thiébaut, “FRIM: minimum-variance reconstructor with a fractal iterative method,” Proc. SPIE 6272, 62722U (2006).
[CrossRef]

Thiébaut, E.

E. Thiébaut and M. Tallon, “Fast minimum variance wavefront reconstruction for extremely large telescopes,” J. Opt. Soc. Am. A 27, 1046–1059 (2010).
[CrossRef]

C. Béchet, M. Tallon, and E. Thiébaut, “FRIM: minimum-variance reconstructor with a fractal iterative method,” Proc. SPIE 6272, 62722U (2006).
[CrossRef]

Vaidyanathan, P. P.

P. P. Vaidyanathan, Multirate Systems and Filter Banks(Prentice-Hall, 1993).

Verhaegen, M.

Vogel, C. R.

Yang, Q.

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

C. Shannon, “The mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).

IEEE J. Sel. Top. Signal Process. (1)

P. J. Hampton, P. Agathoklis, and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Sel. Top. Signal Process. 2, 781–792 (2008).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (8)

Proc. SPIE (5)

K. Freischlad, “Wavefront reconstruction from noisy slope or difference data using the discrete Fourier transform,” Proc. SPIE 551, 74–80 (1985).
[CrossRef]

L. Gilles, “Sparse minimum-variance open-loop reconstructors for extreme adaptive optics: order N multigrid versus preordered Cholesky factorization,” Proc. SPIE 5169, 201–205 (2003).
[CrossRef]

C. R. Vogel, “Sparse matrix methods for wavefront reconstruction, revisited,” Proc. SPIE 5490, 1327–1335 (2004).
[CrossRef]

C. Béchet, M. Tallon, and E. Thiébaut, “FRIM: minimum-variance reconstructor with a fractal iterative method,” Proc. SPIE 6272, 62722U (2006).
[CrossRef]

F. U. Dowla, “Fast Fourier and wavelet transforms for wavefront reconstruction in adaptive optics,” Proc. SPIE 4124, 118–127 (2000).
[CrossRef]

Other (7)

P. J. Hampton, “Robust order N wavelet filterbanks to perform 2-D numerical integration directly from partial difference or gradient measurements,” Ph.D. thesis (University of Victoria, 2009).

D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, 1984).

W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial (SIAM, 2000).

P. P. Vaidyanathan, Multirate Systems and Filter Banks(Prentice-Hall, 1993).

A. Haar, “Zur Theorie der orthogonalen Funktionensysteme,” Ph.D. thesis (University of Göttingen, 1909).

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).

P. Y. Bely, ed., The Design and Construction of Large Optical Telescopes (Springer, 2003).

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Figures (16)

Fig. 1.
Fig. 1.

Noble identities in a block diagram showing the equivalency of Eq. (5) for both dimensions. The serial combination of two operations that both occur in the same dimension cannot change order without changing the filtering operation.

Fig. 2.
Fig. 2.

Tree structure of a QMF for a single dimensional signal x[n]. Tree structures with the perfect reconstruction property result in y[n] being equivalent to a shifted x[n]. The channel with G˜(z) is a low-pass filter, and the channel with H˜(z) is a high-pass filter.

Fig. 3.
Fig. 3.

Two-dimensional QMF for the analysis section.

Fig. 4.
Fig. 4.

Fried geometry relationship between phase points and their measured slope lattice for a single lenslet. A Shack–Hartmann sensor will have an array of these lenslets.

Fig. 5.
Fig. 5.

Block diagram relationship between the phase points and the Hudgin and Fried geometries.

Fig. 6.
Fig. 6.

2D QMF diagram of the channels at the second iteration. The upper left and lower right are each divided into four channels.

Fig. 7.
Fig. 7.

2D QMF performed iteratively on the upper left and lower right until scalar values remain.

Fig. 8.
Fig. 8.

Synthesis section of the 2D QMF.

Fig. 9.
Fig. 9.

Eigenvalues of the 64×64 circular aperture in monotonic order.

Fig. 10.
Fig. 10.

Coefficients of the G˜0(z) factored polynomials for the Daubechies family.

Fig. 11.
Fig. 11.

Coefficients of the H˜0(z) factored polynomials for the Daubechies family.

Fig. 12.
Fig. 12.

(a) Digital frequency response of G˜(z) for the Daubechies family. The frequency response for H˜(z) would be mirrored at π/2. (b) The digital frequency response of G˜0(z) for the Daubechies family. The frequency response for H˜0(z) would be mirrored at π/2. The filtering improvement can be readily seen.

Fig. 13.
Fig. 13.

(a) Original 128×128 wavefront; the remaining images are all reconstructed with (b) the Haar wavelet (the result is the same as it would be for [20]), (c) the 10 dB SNR Haar wavelet, (d) the 3 dB SNR Haar wavelet, (e) the 10 dB SNR with the Daubechies 3 wavelet, (f) the 10 dB SNR with the Daubechies 3 wavelet, (g) the 10 dB SNR with the Daubechies 9 wavelet, and (h) the 3 dB SNR with the Daubechies 9 wavelet.

Fig. 14.
Fig. 14.

(a) Original 256×256 wavefront with a telescope mask applied and (b) the reconstructed wavefront using the Daubechies 3 wavelet.

Fig. 15.
Fig. 15.

Rows (a) 220, (b) 177, and (c) 90 from Fig. 14 are shown. In each plot, the dashed curve shows the original wavefront, compared against the reconstructed wavefront shown by the solid curve for Daubechies 3 and by the dotted curve for Daubechies 9.

Fig. 16.
Fig. 16.

(a)–(c) Three rows from Fig. 14 are shown after applying the boundary correction. In each plot, the dashed curve shows the original wavefront, compared against the reconstructed wavefront shown by the dotted curve for Daubechies 3 and by the solid curve for Daubechies 3 that has the boundary correction applied. The comparison of these results with Fig. 15 shows improvement in the result inside the aperture boundary.

Equations (49)

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X(z1,z2)=n1=n2=x[n1,n2]z1n1z2n2.
y=H(z1,z2)x[n1,n2]=H1(z1)H2(z2)x
y1=D1xy1[n1,n2]=x[2n1,n2],y2=D2xy2[n1,n2]=x[n1,2n2],y3=D1D2x=D2D1xy3[n1,n2]=x[2n1,2n2].
y1=U1x{y1[2n1,n2]=x[n1,n2]y1[2n1+1,n2]=0,y2=U2x{y2[n1,2n2]=x[n1,n2]y2[n1,2n2+1]=0
y1=D1H(z12)x=H(z1)D1x,y2=D2H(z22)x=H(z2)D2x.
y=(G(z)UDG˜(z)+H(z)UDH˜(z))x.
G(z)UDG˜(z)+H(z)UDH˜(z)=zL,
g(z)12(1+z1),g(z)12(1z1).
g(zN)=(=0N1z)g(z)N1=(=0N21z2)2g(z)g(z)ifNis even,
G(z)=g(z)G0(z),H(z)=g(z)H0(z),G˜(z)=g(z)G˜0(z),H˜(z)=g(z)H˜0(z).
XF[m,n]=12(Φ[m,n1]Φ[m1,n1]+Φ[m,n]Φ[m1,n]),YF[m,n]=12(Φ[m1,n]Φ[m1,n1]+Φ[m,n]Φ[m,n1]).
XF=g(z2)g(z1)Φ,YF=g(z1)g(z2)Φ.
ϕLL1=D2D1G˜(z2)G˜(z1)Φ,ϕLH1=D2D1H˜(z2)G˜(z1)Φ,ϕHL1=D2D1G˜(z2)H˜(z1)Φ,ϕHH1=D2D1H˜(z2)H˜(z1)Φ.
ϕLH1=D2D1H˜0(z2)G˜0(z1)(g(z2)g(z1)Φ)=D2D1H˜0(z2)G˜0(z1)XF,ϕHL1=D2D1G˜0(z2)H˜0(z1)(g(z2)g(z1)Φ)=D2D1G˜0(z2)H˜0(z1)YF.
ϕLL/L2=D2D1G˜(z2)G˜(z1)ϕLL1=D2D1G˜(z2)G˜(z1)D2D1G˜(z2)G˜(z1)Φ,ϕLH/L2=D2D1H˜(z2)G˜(z1)ϕLL1=D2D1H˜(z2)G˜(z1)D2D1G˜(z2)G˜(z1)Φ,ϕHL/L2=D2D1G˜(z2)H˜(z1)ϕLL1=D2D1G˜(z2)H˜(z1)D2D1G˜(z2)G˜(z1)Φ,ϕHH/L2=D2D1H˜(z2)H˜(z1)ϕLL1=D2D1H˜(z2)H˜(z1)D2D1G˜(z2)G˜(z1)Φ.
ϕLH/L2=D2D1H˜0(z2)g(z2)G˜(z1)D2D1G˜(z2)G˜0(z1)g(z1)Φ=D2D1H˜0(z2)G˜(z1)D2D1G˜(z2)G˜0(z1)(g(z22)g(z1)Φ)=D2D1H˜0(z2)G˜(z1)D2D1G˜(z2)G˜0(z1)g(z2)(2XF),
ϕHL/L2=D2D1G˜(z2)H˜0(z1)g(z1)D2D1G˜0(z2)g(z2)G˜(z1)Φ=D2D1G˜(z2)H˜0(z1)D2D1G˜0(z2)G˜(z1)(g(z12)g(z2)Φ)=D2D1G˜(z2)H˜0(z1)D2D1G˜0(z2)G˜(z1)g(z1)(2YF).
ϕHH/L2=D2D1H˜0(z2)g(z2)H˜(z1)D2D1G˜(z2)G˜0(z1)g(z1)Φ=D2D1H˜0(z2)H˜(z1)D2D1G˜(z2)G˜0(z1)(g(z22)g(z1)Φ)
=D2D1H˜0(z1)H˜(z2)D2D1G˜(z2)G˜0(z1)(g(z12)g(z2)Φ).
ϕHH/L2=12D2D1H˜0(z2)H˜(z1)D2D1G˜(z2)G˜0(z1)g(z2)(2XF)+12D2D1H˜(z2)H˜0(z1)D2D1G˜0(z2)G˜(z1)g(z1)(2YF).
ϕLH/H2=D2D1H˜0(z2)G˜(z1)D2D1H˜(z2)H˜0(z1)g(z2)(2YF),ϕHL/H2=D2D1G˜(z2)H˜0(z1)D2D1H˜(z1)H˜0(z2)g(z1)(2XF),ϕHH/H2=12D2D1H˜0(z2)H˜(z1)D2D1H˜(z2)H˜0(z1)g(z2)(2YF)+12D2D1H˜(z2)H˜0(z1)D2D1H˜0(z2)H˜(z1)g(z1)(2XF).
w[n̲]={0outside aperture1inside aperture,
F(z̲)=[g(z1)g(z2)g(z1)g(z2)],
B={n̲|Fw[n̲]0},W={n̲|Fw[n̲]=0andw[n̲]=1}.
Φ[n̲]=H(FΦ[n̲])
F(w[n̲]Φ[n̲])=w[n̲]FΦ[n̲]+[F(w[n̲]Φ[n̲])w[n̲]FΦ[n̲]]=w[n̲][XF[n̲]YF[n̲]]+E[n̲],
E[n̲]=0n̲B,
E[n̲]=̲B[X˜̲Y˜̲]δ[n̲̲],
w[n̲]Φ[n̲]=H(w[n̲][XF[n̲]YF[n̲]])+̲BX˜̲H([δ[n̲̲]0])+̲BY˜̲H([0δ[n̲̲]])n̲.
[XF[n̲]YF[n̲]]=FH(w[n̲][XF[n̲]YF[n̲]])+̲BΓXX˜̲+̲BΓYY˜̲.
ΓX[n̲,̲]=FH([δ[n̲̲]0])R2×1,ΓY[n̲,̲]=FH([0δ[n̲̲]])R2×1
[XF[n̲]YF[n̲]]FH(w[n̲][XF[n̲]YF[n̲]])=̲BΓX[n̲,̲]X˜̲+̲BΓY[n̲,̲]Y˜̲
z̲W=Γz˜̲B,
ΓTΓ=UΛUT,
ΓTΓ=U¯Λ¯U¯T,
α̲=Λ¯1U¯TΓ¯Tz̲W,z˜̲B=U¯α̲,
X[n̲]=XF[n̲]+̲BX˜̲δ[n̲̲],Y[n̲]=YF[n̲]+̲BY˜̲δ[n̲̲].
G˜(z)=zg(z),H˜(z)=zg(z),G(z)=g(z),H(z)=g(z),
a[m]=12(x[2m+1]+x[2m]),d[m]=12(x[2m+1]x[2m]),
y[2m]=12(a[m]d[m])=x[2m],y[2m+1]=12(a[m]+d[m])=x[2m+1].
g(zN)1zN2
=0N1z1zN1z1.
g(zN)=1zN2=(=0N1z)(1z1)2=(=0N1z)g(z).
(=0N1z)g(z)=(=0N21z2+z21)g(z)=((1+z1)=0N21z2)g(z)=(=0N21z2)2g(z)g(z).
ϕLH/Lk=(D2D1H˜(z2)G˜(z1))(D2D1G˜(z2)G˜(z1))k2(D2D1G˜(z2)G˜(z1))Φ,ϕHL/Lk=(D2D1G˜(z2)H˜(z1))(D2D1G˜(z2)G˜(z1))k2(D2D1G˜(z2)G˜(z1))Φ,ϕHH/Lk=(D2D1H˜(z2)H˜(z1))(D2D1G˜(z2)G˜(z1))k2(D2D1G˜(z2)G˜(z1))Φ.
ϕLH/Lk=(D2D1H˜0(z2)G˜(z1))(D2D1G˜(z2)G˜(z1))k2(D2D1G˜(z2)G˜0(z1))((=02k11z2)XF),ϕHL/Lk=(D2D1G˜(z2)H˜0(z1))(D2D1G˜(z2)G˜(z1))k2(D2D1G˜0(z2)G˜(z1))((=02k11z1)YF),
ϕHH/Lk=12(D2D1H˜0(z2)H˜(z1))(D2D1G˜(z2)G˜(z1))k2(D2D1G˜(z2)G˜0(z1))((=02k11z2)XF)+12(D2D1H˜(z2)H˜0(z1))(D2D1G˜(z2)G˜(z1))k2(D2D1G˜0(z2)G˜(z1))((=02k11z1)YF).
ϕLH/Hk=(D2D1H˜0(z2)G˜(z1))(D2D1G˜(z2)G˜(z1))k2(D2D1H˜(z2)H˜0(z1))((=02k21z22)2g(z2)YF),ϕHL/Hk=(D2D1G˜(z2)H˜0(z1))(D2D1G˜(z2)G˜(z1))k2(D2D1H˜0(z2)H˜(z1))((=02k21z12)2g(z1)XF),
ϕHH/Hk=12(D2D1H˜0(z2)H˜(z1))(D2D1G˜(z2)G˜(z1))k2(D2D1H˜(z2)H˜0(z1))((=02k21z22)2g(z2)YF)+12(D2D1H˜(z2)H˜0(z1))(D2D1G˜(z2)G˜(z1))k2(D2D1H˜0(z2)H˜(z1))((=02k21z12)2g(z1)XF).

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