Abstract

We present a modification of the closed-loop state space model for adaptive optics control that allows delays that are a noninteger multiple of the system frame rate. We derive the new forms of the predictive Fourier control Kalman filters for arbitrary delays and show that they are linear combinations of the whole-frame delay terms. This structure of the controller is independent of the delay. System stability margins and residual error variance both transition gracefully between integer-frame delays.

© 2008 Optical Society of America

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  1. R. N. Paschall and D. J. Anderson, “Linear quadratic Gaussian control of a deformable mirror adaptive optics system with time-delayed measurements,” Appl. Opt. 32, 6347-6358 (1993).
    [CrossRef] [PubMed]
  2. D. T. Gavel and D. Wiberg, “Towards Strehl-optimizing adaptive optics controllers,” Proc. SPIE 4839, 890-901 (2002).
    [CrossRef]
  3. B. Le Roux, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, L. M. Mugnier, and T. Fusco, “Optimal control law for classical and multiconjugate adaptive optics,” J. Opt. Soc. Am. A 21, 1261-1276 (2004).
    [CrossRef]
  4. C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. V. de Lesegno, “Optimal control, observers and integrators in adaptive optics,” Opt. Express 14, 7464-7476 (2006).
    [CrossRef] [PubMed]
  5. D. Looze, “Discrete-time model of an adaptive optics system,” J. Opt. Soc. Am. A 24, 2850-2863 (2007).
    [CrossRef]
  6. C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE 6272, 62721T (2006).
    [CrossRef]
  7. K. Hinnen, M. Verhagen, and N. Doelman, “Exploiting the spatiotemporal correlation in adaptive optics using data-driven H2-optimal control,” J. Opt. Soc. Am. A 24, 1714-1725 (2007).
    [CrossRef]
  8. L. A. Poyneer, B. A. Macintosh, and J.-P. Véran, “Fourier transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. A 24, 2645-2660 (2007).
    [CrossRef]
  9. L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wave-front control,” J. Opt. Soc. Am. A 22, 1515-1526 (2005).
    [CrossRef]
  10. P.-Y.Madec, “Control techniques,” in Adaptive Optics in Astronomy, F.Roddier, ed. (Cambridge U. Press, 1999), pp. 131-154.
    [CrossRef]
  11. B. A. Macintosh, “Gemini planet imager preliminary design document volume 2: instrument design,” Tech. Rep. UCRL-TR-230900 (Lawrence Livermore National Laboratory, 2007).
  12. M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216-231 (2005).
    [CrossRef]
  13. E. Gendron and P. Léna, “Astronomical adaptive optics I. Modal control optimization,” Astron. Astrophys. 291, 337-347 (1994).
  14. G. Clark, S. Parker, and S. Mitra, “A unified approach to time- and frequency-domain realization of FIR adaptive digital filters,” IEEE Trans. Acoust., Speech, Signal Process. 31, 1073-1083 (1983).
    [CrossRef]
  15. W. Arnold and A. Laub, “Generalized eigenproblem algorithms and software for algebraic Riccati equations,” Proc. IEEE 72, 1746-1754 (1984).
    [CrossRef]
  16. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice Hall, 1999).
  17. A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed. (Prentice Hall, 1997).
  18. C. Dessenne, P.-Y. Madec, and G. Rousset, “Optimization of a predictive controller for closed-loop adaptive optics,” Appl. Opt. 37, 4623-4633 (1998).
    [CrossRef]

2007

2006

C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE 6272, 62721T (2006).
[CrossRef]

C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. V. de Lesegno, “Optimal control, observers and integrators in adaptive optics,” Opt. Express 14, 7464-7476 (2006).
[CrossRef] [PubMed]

2005

L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wave-front control,” J. Opt. Soc. Am. A 22, 1515-1526 (2005).
[CrossRef]

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216-231 (2005).
[CrossRef]

2004

2002

D. T. Gavel and D. Wiberg, “Towards Strehl-optimizing adaptive optics controllers,” Proc. SPIE 4839, 890-901 (2002).
[CrossRef]

1999

P.-Y.Madec, “Control techniques,” in Adaptive Optics in Astronomy, F.Roddier, ed. (Cambridge U. Press, 1999), pp. 131-154.
[CrossRef]

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice Hall, 1999).

1998

1997

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed. (Prentice Hall, 1997).

1994

E. Gendron and P. Léna, “Astronomical adaptive optics I. Modal control optimization,” Astron. Astrophys. 291, 337-347 (1994).

1993

1984

W. Arnold and A. Laub, “Generalized eigenproblem algorithms and software for algebraic Riccati equations,” Proc. IEEE 72, 1746-1754 (1984).
[CrossRef]

1983

G. Clark, S. Parker, and S. Mitra, “A unified approach to time- and frequency-domain realization of FIR adaptive digital filters,” IEEE Trans. Acoust., Speech, Signal Process. 31, 1073-1083 (1983).
[CrossRef]

Anderson, D. J.

Arnold, W.

W. Arnold and A. Laub, “Generalized eigenproblem algorithms and software for algebraic Riccati equations,” Proc. IEEE 72, 1746-1754 (1984).
[CrossRef]

Buck, J. R.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice Hall, 1999).

Clark, G.

G. Clark, S. Parker, and S. Mitra, “A unified approach to time- and frequency-domain realization of FIR adaptive digital filters,” IEEE Trans. Acoust., Speech, Signal Process. 31, 1073-1083 (1983).
[CrossRef]

Conan, J.-M.

de Lesegno, P. V.

Dessenne, C.

Doelman, N.

Frigo, M.

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216-231 (2005).
[CrossRef]

Fusco, T.

C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE 6272, 62721T (2006).
[CrossRef]

B. Le Roux, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, L. M. Mugnier, and T. Fusco, “Optimal control law for classical and multiconjugate adaptive optics,” J. Opt. Soc. Am. A 21, 1261-1276 (2004).
[CrossRef]

Gavel, D. T.

D. T. Gavel and D. Wiberg, “Towards Strehl-optimizing adaptive optics controllers,” Proc. SPIE 4839, 890-901 (2002).
[CrossRef]

Gendron, E.

E. Gendron and P. Léna, “Astronomical adaptive optics I. Modal control optimization,” Astron. Astrophys. 291, 337-347 (1994).

Hinnen, K.

Johnson, S. G.

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216-231 (2005).
[CrossRef]

Kulcsar, C.

C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE 6272, 62721T (2006).
[CrossRef]

B. Le Roux, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, L. M. Mugnier, and T. Fusco, “Optimal control law for classical and multiconjugate adaptive optics,” J. Opt. Soc. Am. A 21, 1261-1276 (2004).
[CrossRef]

Kulcsár, C.

Laub, A.

W. Arnold and A. Laub, “Generalized eigenproblem algorithms and software for algebraic Riccati equations,” Proc. IEEE 72, 1746-1754 (1984).
[CrossRef]

Le Roux, B.

Léna, P.

E. Gendron and P. Léna, “Astronomical adaptive optics I. Modal control optimization,” Astron. Astrophys. 291, 337-347 (1994).

Looze, D.

Macintosh, B. A.

B. A. Macintosh, “Gemini planet imager preliminary design document volume 2: instrument design,” Tech. Rep. UCRL-TR-230900 (Lawrence Livermore National Laboratory, 2007).

L. A. Poyneer, B. A. Macintosh, and J.-P. Véran, “Fourier transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. A 24, 2645-2660 (2007).
[CrossRef]

Madec, P.-Y.

Mitra, S.

G. Clark, S. Parker, and S. Mitra, “A unified approach to time- and frequency-domain realization of FIR adaptive digital filters,” IEEE Trans. Acoust., Speech, Signal Process. 31, 1073-1083 (1983).
[CrossRef]

Montri, J.

C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE 6272, 62721T (2006).
[CrossRef]

Mugnier, L. M.

Nawab, S. H.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed. (Prentice Hall, 1997).

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice Hall, 1999).

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed. (Prentice Hall, 1997).

Parker, S.

G. Clark, S. Parker, and S. Mitra, “A unified approach to time- and frequency-domain realization of FIR adaptive digital filters,” IEEE Trans. Acoust., Speech, Signal Process. 31, 1073-1083 (1983).
[CrossRef]

Paschall, R. N.

Petit, C.

C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE 6272, 62721T (2006).
[CrossRef]

C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. V. de Lesegno, “Optimal control, observers and integrators in adaptive optics,” Opt. Express 14, 7464-7476 (2006).
[CrossRef] [PubMed]

Poyneer, L. A.

Rabaud, D.

C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE 6272, 62721T (2006).
[CrossRef]

Raynaud, H.-F.

Rousset, G.

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice Hall, 1999).

Véran, J.-P.

Verhagen, M.

Wiberg, D.

D. T. Gavel and D. Wiberg, “Towards Strehl-optimizing adaptive optics controllers,” Proc. SPIE 4839, 890-901 (2002).
[CrossRef]

Willsky, A. S.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed. (Prentice Hall, 1997).

Appl. Opt.

Astron. Astrophys.

E. Gendron and P. Léna, “Astronomical adaptive optics I. Modal control optimization,” Astron. Astrophys. 291, 337-347 (1994).

IEEE Trans. Acoust., Speech, Signal Process.

G. Clark, S. Parker, and S. Mitra, “A unified approach to time- and frequency-domain realization of FIR adaptive digital filters,” IEEE Trans. Acoust., Speech, Signal Process. 31, 1073-1083 (1983).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Proc. IEEE

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216-231 (2005).
[CrossRef]

W. Arnold and A. Laub, “Generalized eigenproblem algorithms and software for algebraic Riccati equations,” Proc. IEEE 72, 1746-1754 (1984).
[CrossRef]

Proc. SPIE

D. T. Gavel and D. Wiberg, “Towards Strehl-optimizing adaptive optics controllers,” Proc. SPIE 4839, 890-901 (2002).
[CrossRef]

C. Petit, J.-M. Conan, C. Kulcsar, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE 6272, 62721T (2006).
[CrossRef]

Other

P.-Y.Madec, “Control techniques,” in Adaptive Optics in Astronomy, F.Roddier, ed. (Cambridge U. Press, 1999), pp. 131-154.
[CrossRef]

B. A. Macintosh, “Gemini planet imager preliminary design document volume 2: instrument design,” Tech. Rep. UCRL-TR-230900 (Lawrence Livermore National Laboratory, 2007).

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice Hall, 1999).

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed. (Prentice Hall, 1997).

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Figures (8)

Fig. 1
Fig. 1

Block diagram of hybrid continuous-time/discrete-time AO control loop for a single Fourier mode. The phase aberration ϕ ( t ) is corrected in closed loop in the presence of measurement noise v [ t ] . The WFS dynamics are represented by W ( s ) , the DM dynamics by D ( s ) , and the controller delay by exp ( s τ ) . The discrete-time control law is C ( z ) . A/D and D/A conversion surround the control computer.

Fig. 2
Fig. 2

Comparison of the discrete-time state space model for delay τ = T (row 1) to the new model, which assumes the signals are constant over an interval of the sampling period (row 2). When τ T , the DM signal shifts to the left and becomes asynchronous (row 3). When T τ , the DM signal shifts to the right (row 4).

Fig. 3
Fig. 3

Flow diagram illustrating the implementation of the predictive filter for 0 τ 2 T . The coefficients used in the filter depend on Δ and are taken from Eqs. (39, 40, 41). Note that when τ T , D 2 = 0 .

Fig. 4
Fig. 4

Simulink model of the hybrid continuous-time/discrete-time AO control loop for a complex-valued Fourier modal coefficient. The WFS module implements the standard transfer function W ( s ) , while the DM module is a Simulink zero-order hold block. The integral controller and Kalman filters are implemented with discrete-time blocks of gains and delays. (The Kalman block is implemented exactly as in Fig. 3.) A switch allows use of either controller.

Fig. 5
Fig. 5

Error transfer function for a predictive controller with τ = 1.5 T , based on the Laplace transform model or as determined by running white noise through the Simulink model. There are some small discrepancies due to the modeling of the DM.

Fig. 6
Fig. 6

Phase margins for a high-SNR case for the optimized-gain integrator and the predictor. For most delays τ, the optimal gain is limited to be the maximum gain set by stability. The predictor can be more aggressive and has larger margins for all τ. Note how the margin gracefully transitions between integer time step delays.

Fig. 7
Fig. 7

Residual error performance of the integral controller and the predictive filter for a variety of gains. The optimized gain (as determined by Laplace analysis) results in the best performance in Simulink. The predictor was manipulated by feeding an incorrect WFS noise level into the ARE solve, producing a range of predictors. Again, the predictor produced by the model results in the best performance in Simulink.

Fig. 8
Fig. 8

Residual error for the predictor and the optimized-gain integrator. Input atmospheric power of 1; WFS noise set by SNRs of 1 and 10. Simulink results are the mean (data points) and standard deviation (error bars) of 32 different 8 s Simulink runs. Where error bars are not visible, the standard deviation was smaller than the data point marker size. Theoretical results from the Laplace model are given as dashed curves. The predictor performs gracefully for arbitrary delays, providing more of an advantage over the integral controller for longer delays.

Equations (47)

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y [ t ] = ϕ [ t 1 ] d [ t 1 ] .
d [ t + 1 ] = ϕ ̂ [ t + 1 t ] .
y [ t ] = ϕ [ t 1 ] { Δ d [ t ] + ( 1 + Δ ) d [ t 1 ] } .
d [ t + 1 ] = Δ ϕ ̂ [ t t ] + ( 1 + Δ ) ϕ ̂ [ t + 1 t ] .
y [ t ] = ϕ [ t 1 ] { ( 1 Δ ) d [ t 1 ] + Δ d [ t 2 ] } ,
d [ t + 1 ] = ( 1 Δ ) ϕ ̂ [ t + 1 t ] + Δ ϕ ̂ [ t + 2 t ] .
a [ t ] = α a [ t 1 ] + w [ t ] .
a = ( a 0 , a 1 , , a L ) ,
R = Diag ( α 0 , α 1 , , α L ) .
P w = Diag ( σ a 0 2 , σ a 1 2 , , σ a L 2 ) .
x [ t ] = ( a [ t ] , ϕ [ t 1 ] ) T .
A = ( R 0 1 × ( L + 1 ) 1 ( L + 1 ) × 1 0 ) .
B = ( I ( L + 1 ) × ( L + 1 ) 0 ( L + 1 ) × 1 ) .
C = ( 0 1 × ( L + 1 ) , 1 ) ,
D = { ( 1 ) ( Δ , 1 Δ ) ( 1 ) ( Δ 1 , Δ ) ( 1 ) ,
u [ t ] = { ( d [ t ] ) if Δ = 1 , ( d [ t ] , d [ t 1 ] ) T if 1 Δ 0 , ( d [ t 1 ] ) if Δ = 0 , ( d [ t 1 ] , d [ t 2 ] ) T if 0 Δ 1 , ( d [ t 2 ] ) if Δ = 1 .
x ̂ [ t t ] = ( I K s C ) A x ̂ [ t 1 t 1 ] + K s ( y [ t ] Du [ t ] ) .
K s = P s C H ( CP s C H + P v ) 1 .
P s = AP s A H + BP w B H AP s C H ( CP s C H + P v ) 1 CP s A H .
C ( z ) = z D ( z ) Y ( z ) .
A k ( z ) = α k z 1 A k ( z ) + Q 1 p L + 1 , k [ Y ( z ) + D ( z ) z 1 z 1 l = 0 L A l ( z ) ] ,
ϕ ̂ [ t + 1 t ] = k = 0 L α k a ̂ k [ t t ] ,
z D ( z ) = k = 0 L α k A k ( z ) .
C ( z ) = ( Q 1 k = 0 L p L + 1 , k α k 1 α k z 1 ) × ( 1 + z 1 Q 1 k = 0 L p L + 1 , k ) 1 .
A k ( z ) = α k z 1 A k ( z ) + Q 1 p L + 1 , k ( Y ( z ) + D ( z ) z 1 l = 0 L A l ( z ) ) .
z D ( z ) = k = 0 L A k ( z ) .
C ( z ) = Q 1 k = 0 L p L + 1 , k α k 1 α k z 1 .
A k ( z ) = α k z 1 A k ( z ) + Q 1 p L + 1 , k ( Y ( z ) + D ( z ) z 2 z 1 l = 0 L A l ( z ) ) .
z D ( z ) = k = 0 L α k 2 A k ( z ) .
C ( z ) = ( Q 1 k = 0 L p L + 1 , k α k 1 α k z 1 ) × ( 1 + z 1 Q 1 k = 0 L p L + 1 , k + z 2 Q 1 k = 0 L p L + 1 , k α k ) 1 .
A k ( z ) = α k z 1 A k ( z ) + Q 1 p L + 1 , k ( Y ( z ) + D ( z ) [ Δ + ( 1 + Δ ) z 1 ] z 1 l = 0 L A l ( z ) ) .
z D ( z ) = k = 0 L ( Δ + [ 1 + Δ ] α k ) A k ( z ) .
C ( z ) = ( Q 1 k = 0 L p L + 1 , k 1 α k z 1 [ Δ + ( 1 + Δ ) α k ] ) ( 1 + z 1 Q 1 k = 0 L p L + 1 , k [ 1 ( Δ + [ 1 + Δ ] α k ) ( Δ + [ 1 + Δ ] z 1 ) 1 α k z 1 ] ) 1 .
( 1 + Δ ) [ ( 1 Δ ) + Δ α k ] + ( 1 + Δ ) Δ z 1 ( 1 α k ) 2 1 α k z 1 .
C ( z ) = ( Q 1 k = 0 L p L + 1 , k 1 α k z 1 [ Δ + ( 1 + Δ ) α k ] ) ( 1 + z 1 Q 1 ( 1 + Δ ) k = 0 L p L + 1 , k [ ( 1 Δ ) + Δ α k ] ) 1 .
A k ( z ) = α k z 1 A k ( z ) + Q 1 p L + 1 , k ( Y ( z ) + D ( z ) z 1 [ ( 1 Δ ) + Δ z 1 ] z 1 l = 0 L A l ( z ) ) .
z D ( z ) = k = 0 L ( [ 1 Δ ] + Δ α k ) α k A k ( z ) .
C ( z ) = ( Q 1 k = 0 L p L + 1 , k α k 1 α k z 1 [ ( 1 Δ ) + Δ α k ] ) ( 1 + z 1 Q 1 k = 0 L p L + 1 , k [ 1 α k z 1 ( 1 Δ + Δ z 1 ) ( 1 Δ + Δ α k ) 1 α k z 1 ] ) 1 .
C ( z ) = ( Q 1 k = 0 L p L + 1 , k α k 1 α k z 1 [ ( 1 Δ ) + Δ α k ] ) ( 1 + z 1 Q 1 k = 0 L p L + 1 , k + z 2 Q 1 Δ k = 0 L p L + 1 , k α k ( 2 Δ + [ Δ 1 ] α k ) ) 1 .
K k = { Q 1 p L + 1 , k [ Δ + ( 1 + Δ ) α k ] if 1 Δ 0 , Q 1 p L + 1 , k α k [ ( 1 Δ ) + Δ α k ] if 0 Δ 1 .
D 1 = { Q 1 ( 1 + Δ ) k = 0 L p L + 1 , k [ ( 1 Δ ) + Δ α k ] if 1 Δ 0 , Q 1 k = 0 L p L + 1 , k if 0 Δ 1 ,
D 2 = { 0 if 1 Δ 0 , Q 1 Δ k = 0 L p L + 1 , k α k [ 2 Δ + ( Δ 1 ) α k ] if 0 Δ 1 .
g = p 1 , 1 p 1 , 1 + σ v 2 = Q 1 p 1 , 1 .
K = { g α if Δ = 1 , g α ( Δ + [ 1 + Δ ] α ) if 1 Δ 0 , g α 2 if Δ = 0 , g α 2 ( [ 1 Δ ] + Δ α ) if 0 Δ 1 , g α 3 if Δ = 1 .
H e ( f ) = 1 1 + W ( s ) C ( z ) exp ( s τ ) D ( s ) .
H n ( f ) = C ( z ) exp ( s τ ) D ( s ) 1 + W ( s ) C ( z ) exp ( s τ ) D ( s ) .
H ̂ e ( f ) 2 = P ̂ ε ( f ) P ̂ ϕ ( f ) ,

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