Abstract

We present a multiple-input, single-output, weakly nonlinear model of spatial light modulators by use of a second-order Volterra series and describe an experimental method to measure the nonlinear transfer functions by means of sinusoidal perturbation and synchronous detection with a lock-in amplifier. We also present an application of this method to a liquid-crystal light valve.

© 1998 Optical Society of America

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  1. S. G. Batsell, T. L. Jong, J. F. Walkup, T. F. Krile, “Noise limitations in optical linear algebra processors,” Appl. Opt. 29, 2084–2090 (1990).
    [CrossRef] [PubMed]
  2. D. D. Weiner, J. F. Spina, Sinusoidal Analysis and Modeling of Weakly Nonlinear Circuits (Van Nostrand Reinhold, New York, 1980).
  3. R. J. P. De Figueriedo, “A generalized Fock space framework for nonlinear system and signal analysis,” IEEE Trans. Circuits Syst. 30, 637–647 (1983).
    [CrossRef]
  4. I. W. Sandburg, “A perspective on system theory,” IEEE Trans. Circuits Syst. 31, 88–103 (1984).
    [CrossRef]
  5. W. J. Rugh, Nonlinear System Theory: The Volterra/Wiener Approach (Johns Hopkins U. Press, Baltimore, Md., 1981).
  6. M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Wiley, New York, 1980).
  7. M. Schetzen, “Nonlinear system modeling based on the Wiener theory,” Proc. IEEE 69, 1557–1573 (1981).
    [CrossRef]
  8. E. Bedrosian, S. O. Rice, “The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs,” Proc. IEEE 59, 1688–1707 (1971).
    [CrossRef]
  9. I. W. Sandburg, “Expansion for nonlinear systems,” Bell Syst. Tech. J. 61, 159–199 (1982).
  10. A. A. M. Saleh, “Matrix analysis of mildly nonlinear, multiple-input, multiple-output systems with memory,” Bell Syst. Tech. J 61, 2221–2243 (1982).
  11. J. C. Peyton Jones, S. A. Billings, “Interpretation of non-linear frequency response functions,” Int. J. Control 52, 319–346 (1990).
    [CrossRef]
  12. J. S. Bendat, Nonlinear System Analysis and Identification from Random Data (Wiley, New York, 1990).
  13. Y. W. Lee, M. Schetzen, “Measurement of the Wiener kernels of a non-linear system by cross-correlation,” Int. J. Control 2, 237–254 (1965).
    [CrossRef]
  14. A. S. French, E. G. Butz, “Measuring the Wiener kernels of a non-linear system using the fast Fourier transform algorithm,” Int. J. Control 3, 529–539 (1973).
    [CrossRef]
  15. S. Chen, S. A. Billings, “Representations on non-linear system: the NARMAX model,” Int. J. Control 49, 1013–1032 (1989).
  16. M. J. Korenberg, “Identifying nonlinear difference equation and functional expansion representations: the fast orthogonal algorithm,” Ann. Biomed. Eng. 16, 123–142 (1988).
    [CrossRef] [PubMed]
  17. S. Boyd, Y. S. Tang, L. O. Chua, “Measuring Volterra kernels,” IEEE Trans. Circuits Syst. CAS-30, 571–577 (1983).
    [CrossRef]
  18. C. Evans, D. Rees, L. Jones, M. Weiss, “Periodic signals for measuring nonlinear Volterra kernels,” IEEE Trans. Instrum. Meas. 45, 362–371 (1996).
    [CrossRef]
  19. J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  20. W. P. Bleha, L. T. Lipton, E. Wiener-Avnear, J. Grinberg, P. G. Reif, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

1996 (1)

C. Evans, D. Rees, L. Jones, M. Weiss, “Periodic signals for measuring nonlinear Volterra kernels,” IEEE Trans. Instrum. Meas. 45, 362–371 (1996).
[CrossRef]

1990 (2)

S. G. Batsell, T. L. Jong, J. F. Walkup, T. F. Krile, “Noise limitations in optical linear algebra processors,” Appl. Opt. 29, 2084–2090 (1990).
[CrossRef] [PubMed]

J. C. Peyton Jones, S. A. Billings, “Interpretation of non-linear frequency response functions,” Int. J. Control 52, 319–346 (1990).
[CrossRef]

1989 (1)

S. Chen, S. A. Billings, “Representations on non-linear system: the NARMAX model,” Int. J. Control 49, 1013–1032 (1989).

1988 (1)

M. J. Korenberg, “Identifying nonlinear difference equation and functional expansion representations: the fast orthogonal algorithm,” Ann. Biomed. Eng. 16, 123–142 (1988).
[CrossRef] [PubMed]

1984 (1)

I. W. Sandburg, “A perspective on system theory,” IEEE Trans. Circuits Syst. 31, 88–103 (1984).
[CrossRef]

1983 (2)

R. J. P. De Figueriedo, “A generalized Fock space framework for nonlinear system and signal analysis,” IEEE Trans. Circuits Syst. 30, 637–647 (1983).
[CrossRef]

S. Boyd, Y. S. Tang, L. O. Chua, “Measuring Volterra kernels,” IEEE Trans. Circuits Syst. CAS-30, 571–577 (1983).
[CrossRef]

1982 (2)

I. W. Sandburg, “Expansion for nonlinear systems,” Bell Syst. Tech. J. 61, 159–199 (1982).

A. A. M. Saleh, “Matrix analysis of mildly nonlinear, multiple-input, multiple-output systems with memory,” Bell Syst. Tech. J 61, 2221–2243 (1982).

1981 (1)

M. Schetzen, “Nonlinear system modeling based on the Wiener theory,” Proc. IEEE 69, 1557–1573 (1981).
[CrossRef]

1978 (1)

W. P. Bleha, L. T. Lipton, E. Wiener-Avnear, J. Grinberg, P. G. Reif, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

1973 (1)

A. S. French, E. G. Butz, “Measuring the Wiener kernels of a non-linear system using the fast Fourier transform algorithm,” Int. J. Control 3, 529–539 (1973).
[CrossRef]

1971 (1)

E. Bedrosian, S. O. Rice, “The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs,” Proc. IEEE 59, 1688–1707 (1971).
[CrossRef]

1965 (1)

Y. W. Lee, M. Schetzen, “Measurement of the Wiener kernels of a non-linear system by cross-correlation,” Int. J. Control 2, 237–254 (1965).
[CrossRef]

Batsell, S. G.

Bedrosian, E.

E. Bedrosian, S. O. Rice, “The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs,” Proc. IEEE 59, 1688–1707 (1971).
[CrossRef]

Bendat, J. S.

J. S. Bendat, Nonlinear System Analysis and Identification from Random Data (Wiley, New York, 1990).

Billings, S. A.

J. C. Peyton Jones, S. A. Billings, “Interpretation of non-linear frequency response functions,” Int. J. Control 52, 319–346 (1990).
[CrossRef]

S. Chen, S. A. Billings, “Representations on non-linear system: the NARMAX model,” Int. J. Control 49, 1013–1032 (1989).

Bleha, W. P.

W. P. Bleha, L. T. Lipton, E. Wiener-Avnear, J. Grinberg, P. G. Reif, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

Boyd, S.

S. Boyd, Y. S. Tang, L. O. Chua, “Measuring Volterra kernels,” IEEE Trans. Circuits Syst. CAS-30, 571–577 (1983).
[CrossRef]

Brown, H. B.

W. P. Bleha, L. T. Lipton, E. Wiener-Avnear, J. Grinberg, P. G. Reif, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

Butz, E. G.

A. S. French, E. G. Butz, “Measuring the Wiener kernels of a non-linear system using the fast Fourier transform algorithm,” Int. J. Control 3, 529–539 (1973).
[CrossRef]

Casasent, D.

W. P. Bleha, L. T. Lipton, E. Wiener-Avnear, J. Grinberg, P. G. Reif, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

Chen, S.

S. Chen, S. A. Billings, “Representations on non-linear system: the NARMAX model,” Int. J. Control 49, 1013–1032 (1989).

Chua, L. O.

S. Boyd, Y. S. Tang, L. O. Chua, “Measuring Volterra kernels,” IEEE Trans. Circuits Syst. CAS-30, 571–577 (1983).
[CrossRef]

De Figueriedo, R. J. P.

R. J. P. De Figueriedo, “A generalized Fock space framework for nonlinear system and signal analysis,” IEEE Trans. Circuits Syst. 30, 637–647 (1983).
[CrossRef]

Evans, C.

C. Evans, D. Rees, L. Jones, M. Weiss, “Periodic signals for measuring nonlinear Volterra kernels,” IEEE Trans. Instrum. Meas. 45, 362–371 (1996).
[CrossRef]

French, A. S.

A. S. French, E. G. Butz, “Measuring the Wiener kernels of a non-linear system using the fast Fourier transform algorithm,” Int. J. Control 3, 529–539 (1973).
[CrossRef]

Goodman, J.

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Grinberg, J.

W. P. Bleha, L. T. Lipton, E. Wiener-Avnear, J. Grinberg, P. G. Reif, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

Jones, L.

C. Evans, D. Rees, L. Jones, M. Weiss, “Periodic signals for measuring nonlinear Volterra kernels,” IEEE Trans. Instrum. Meas. 45, 362–371 (1996).
[CrossRef]

Jong, T. L.

Korenberg, M. J.

M. J. Korenberg, “Identifying nonlinear difference equation and functional expansion representations: the fast orthogonal algorithm,” Ann. Biomed. Eng. 16, 123–142 (1988).
[CrossRef] [PubMed]

Krile, T. F.

Lee, Y. W.

Y. W. Lee, M. Schetzen, “Measurement of the Wiener kernels of a non-linear system by cross-correlation,” Int. J. Control 2, 237–254 (1965).
[CrossRef]

Lipton, L. T.

W. P. Bleha, L. T. Lipton, E. Wiener-Avnear, J. Grinberg, P. G. Reif, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

Markevitch, B. V.

W. P. Bleha, L. T. Lipton, E. Wiener-Avnear, J. Grinberg, P. G. Reif, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

Peyton Jones, J. C.

J. C. Peyton Jones, S. A. Billings, “Interpretation of non-linear frequency response functions,” Int. J. Control 52, 319–346 (1990).
[CrossRef]

Rees, D.

C. Evans, D. Rees, L. Jones, M. Weiss, “Periodic signals for measuring nonlinear Volterra kernels,” IEEE Trans. Instrum. Meas. 45, 362–371 (1996).
[CrossRef]

Reif, P. G.

W. P. Bleha, L. T. Lipton, E. Wiener-Avnear, J. Grinberg, P. G. Reif, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

Rice, S. O.

E. Bedrosian, S. O. Rice, “The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs,” Proc. IEEE 59, 1688–1707 (1971).
[CrossRef]

Rugh, W. J.

W. J. Rugh, Nonlinear System Theory: The Volterra/Wiener Approach (Johns Hopkins U. Press, Baltimore, Md., 1981).

Saleh, A. A. M.

A. A. M. Saleh, “Matrix analysis of mildly nonlinear, multiple-input, multiple-output systems with memory,” Bell Syst. Tech. J 61, 2221–2243 (1982).

Sandburg, I. W.

I. W. Sandburg, “A perspective on system theory,” IEEE Trans. Circuits Syst. 31, 88–103 (1984).
[CrossRef]

I. W. Sandburg, “Expansion for nonlinear systems,” Bell Syst. Tech. J. 61, 159–199 (1982).

Schetzen, M.

M. Schetzen, “Nonlinear system modeling based on the Wiener theory,” Proc. IEEE 69, 1557–1573 (1981).
[CrossRef]

Y. W. Lee, M. Schetzen, “Measurement of the Wiener kernels of a non-linear system by cross-correlation,” Int. J. Control 2, 237–254 (1965).
[CrossRef]

M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Wiley, New York, 1980).

Spina, J. F.

D. D. Weiner, J. F. Spina, Sinusoidal Analysis and Modeling of Weakly Nonlinear Circuits (Van Nostrand Reinhold, New York, 1980).

Tang, Y. S.

S. Boyd, Y. S. Tang, L. O. Chua, “Measuring Volterra kernels,” IEEE Trans. Circuits Syst. CAS-30, 571–577 (1983).
[CrossRef]

Walkup, J. F.

Weiner, D. D.

D. D. Weiner, J. F. Spina, Sinusoidal Analysis and Modeling of Weakly Nonlinear Circuits (Van Nostrand Reinhold, New York, 1980).

Weiss, M.

C. Evans, D. Rees, L. Jones, M. Weiss, “Periodic signals for measuring nonlinear Volterra kernels,” IEEE Trans. Instrum. Meas. 45, 362–371 (1996).
[CrossRef]

Wiener-Avnear, E.

W. P. Bleha, L. T. Lipton, E. Wiener-Avnear, J. Grinberg, P. G. Reif, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

Ann. Biomed. Eng. (1)

M. J. Korenberg, “Identifying nonlinear difference equation and functional expansion representations: the fast orthogonal algorithm,” Ann. Biomed. Eng. 16, 123–142 (1988).
[CrossRef] [PubMed]

Appl. Opt. (1)

Bell Syst. Tech. J (1)

A. A. M. Saleh, “Matrix analysis of mildly nonlinear, multiple-input, multiple-output systems with memory,” Bell Syst. Tech. J 61, 2221–2243 (1982).

Bell Syst. Tech. J. (1)

I. W. Sandburg, “Expansion for nonlinear systems,” Bell Syst. Tech. J. 61, 159–199 (1982).

IEEE Trans. Circuits Syst. (3)

R. J. P. De Figueriedo, “A generalized Fock space framework for nonlinear system and signal analysis,” IEEE Trans. Circuits Syst. 30, 637–647 (1983).
[CrossRef]

I. W. Sandburg, “A perspective on system theory,” IEEE Trans. Circuits Syst. 31, 88–103 (1984).
[CrossRef]

S. Boyd, Y. S. Tang, L. O. Chua, “Measuring Volterra kernels,” IEEE Trans. Circuits Syst. CAS-30, 571–577 (1983).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

C. Evans, D. Rees, L. Jones, M. Weiss, “Periodic signals for measuring nonlinear Volterra kernels,” IEEE Trans. Instrum. Meas. 45, 362–371 (1996).
[CrossRef]

Int. J. Control (4)

J. C. Peyton Jones, S. A. Billings, “Interpretation of non-linear frequency response functions,” Int. J. Control 52, 319–346 (1990).
[CrossRef]

Y. W. Lee, M. Schetzen, “Measurement of the Wiener kernels of a non-linear system by cross-correlation,” Int. J. Control 2, 237–254 (1965).
[CrossRef]

A. S. French, E. G. Butz, “Measuring the Wiener kernels of a non-linear system using the fast Fourier transform algorithm,” Int. J. Control 3, 529–539 (1973).
[CrossRef]

S. Chen, S. A. Billings, “Representations on non-linear system: the NARMAX model,” Int. J. Control 49, 1013–1032 (1989).

Opt. Eng. (1)

W. P. Bleha, L. T. Lipton, E. Wiener-Avnear, J. Grinberg, P. G. Reif, D. Casasent, H. B. Brown, B. V. Markevitch, “Application of the liquid crystal light valve to real-time optical data processing,” Opt. Eng. 17, 371–384 (1978).

Proc. IEEE (2)

M. Schetzen, “Nonlinear system modeling based on the Wiener theory,” Proc. IEEE 69, 1557–1573 (1981).
[CrossRef]

E. Bedrosian, S. O. Rice, “The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs,” Proc. IEEE 59, 1688–1707 (1971).
[CrossRef]

Other (5)

D. D. Weiner, J. F. Spina, Sinusoidal Analysis and Modeling of Weakly Nonlinear Circuits (Van Nostrand Reinhold, New York, 1980).

W. J. Rugh, Nonlinear System Theory: The Volterra/Wiener Approach (Johns Hopkins U. Press, Baltimore, Md., 1981).

M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Wiley, New York, 1980).

J. S. Bendat, Nonlinear System Analysis and Identification from Random Data (Wiley, New York, 1990).

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

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Figures (18)

Fig. 1
Fig. 1

General multiple-input, multiple-output system.

Fig. 2
Fig. 2

General multiple-input, single-output system.

Fig. 3
Fig. 3

Single-input, single-output Volterra system.

Fig. 4
Fig. 4

Second-order, three-input, single-output Volterra system.

Fig. 5
Fig. 5

Synchronous measurement of a first-order Volterra transfer function.

Fig. 6
Fig. 6

Synchronous measurement of a second-order Volterra transfer function.

Fig. 7
Fig. 7

Synchronous measurement of a second-order cross-term Volterra transfer function.

Fig. 8
Fig. 8

LCLV. CdTe, cadmium telluride; CdS, cadmium sulfur; ITO, indium titanium oxide.

Fig. 9
Fig. 9

LCLV spatial light modulator with ports identified.

Fig. 10
Fig. 10

LCLV experimental setup. A.O., acousto-optic.

Fig. 11
Fig. 11

LCLV static gain.

Fig. 12
Fig. 12

LCLV ac drive first-order transfer function (H3).

Fig. 13
Fig. 13

LCLV ac drive second-order transfer function (H33).

Fig. 14
Fig. 14

LCLV read-beam first-order transfer function (H2).

Fig. 15
Fig. 15

LCLV write-beam first-order transfer function (H1).

Fig. 16
Fig. 16

LCLV write-beam second-order transfer function (H11).

Fig. 17
Fig. 17

Estimated LCLV complete write-beam second-order transfer function (H11).

Fig. 18
Fig. 18

Estimated LCLV write-beam–read-beam second-order transfer function (H12).

Equations (43)

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x t = x 1 t x 2 t x N t ,     y t = y 1 t y 2 t y M t ,     y t = T x t .
y x 1 ,   x 2 , ,   x N = m 1 = 0 m 2 = 0     m N = 0 m 1 + m 2 + + m N f x 10 ,   x 20 , ,   x N 0 x 1 m 1 x 2 m 2 x N m N × x 1 - x 10 m 1 x 2 - x 20 m 2 x N - x N 0 m N m 1 ! m 2 ! m N ! ,
y x 1 ,   x 2 , ,   x N = m 1 = 0 m 2 = 0     m N = 0   a m 1 , m 2 , ,   m N × x 1 - x 10 m 1 x 2 - x 20 m 2     x N - x N 0 m N ,
a m 1 , m 2 , , m N = m 1 + m 2 + + m N f x 10 ,   x 20 , ,   x N 0 x 1 m 1 x 2 m 2 x N m N m 1 ! m 2 ! m N ! .
y t = n = 1 N   y n t ,
y 1 t = -   h 1 τ 1 x t - τ 1 d τ 1 ,
y 2 t = -   h 2 τ 1 ,   τ 2 x t - τ 1 × x t - τ 2 d τ 1 d τ 2
y n t = -       h n τ 1 ,   τ 2 , ,   τ n × x t - τ 1 x t - τ 2     x t - τ n × d τ 1 d τ 2     d τ n .
h 2 τ 1 ,   τ 2 = h 2 τ 2 ,   τ 1 .
h n τ 1 ,   τ 2 , ,   τ n = 0 ,     for   any   τ i < 0 ,   i = 1 , ,   n .
-       | h n τ 1 ,   τ 2 , ,   τ n | d τ 1 d τ 2 , ,   d τ n < .
y 12 t = -   h 12 τ 1 ,   τ 2 x 1 t - τ 1 × x 2 t - τ 2 d τ 1 d τ 2 .
H 1 f 1 = -   h 1 τ 1 exp - j 2 π f 1 τ 1 d τ 1 ,
H 2 f 1 ,   f 2 = -   h 2 τ 1 ,   τ 2 × exp - j 2 π f 1 τ 1 + f 2 τ 2 d τ 1 d τ 2 ,
H n f 1 ,   f 2 , ,   f n = -       h n τ 1 ,   τ 2 , ,   τ n × exp - j 2 π f 1 τ 1 + f 2 τ 2 + + f n τ n d τ 1 d τ 2     d τ n .
H 2 f 1 ,   f 2 = H 2 f 2 ,   f 1 .
H 2 * f 1 ,   f 2 = H 2 - f 1 ,   - f 2 .
H 2 * f ,   - f = H 2 - f ,   f = H 2 f ,   - f .
h n τ 1 ,   τ 2 , ,   τ n = -     H n f 1 ,   f 2 , ,   f n × exp j 2 π f 1 τ 1 + f 2 τ 2 + + f n τ n d τ 1 d τ 2     d τ n .
H 12 f 1 ,   f 2 = -   h 12 τ 1 ,   τ 2 × exp - j 2 π f 1 τ 1 + f 2 τ 2 d τ 1 d τ 2 ,
y n t = -     H n f 1 ,   f 2 , ,   f n X f 1 × exp j 2 π f 1 t X f 2 exp j 2 π f 2 t × X f n exp j 2 π f n t d f 1 d f 2     d f n ,
x t = A   cos 2 π f 0 t ,
X f = A 2 δ f - f 0 + δ f + f 0 ,
y 1 t = - H 1 f 1 A 2 δ f 1 - f 0 + δ f 1 + f 0 × exp j 2 π f 1 t d f 1 .
y 1 t = A 2 H 1 f 0 exp j 2 π f 0 t + A 2 H 1 - f 0 × exp - j 2 π f 0 t .
y 1 t = A 2 H 1 f 0 exp j 2 π f 0 t + A 2 H 1 * f 0 × exp - j 2 π f 0 t .
H 1 f 0 = | H 1 f 0 | exp j ϕ 1 ,
y 1 t = A | H 1 f 0 | cos 2 π f 0 t + ϕ 1 .
y 2 t = - H 2 f 1 ,   f 2 A 2 δ f 1 - f 0 + δ f 1 + f 0 A 2 δ f 2 - f 0 + δ f 2 + f 0 × exp j 2 π f 1 t exp j 2 π f 2 t d f 1 d f 2 ,
y 2 t = A 2 4 H 2 f 0 ,   f 0 exp j 2 π f 0 t exp j 2 π f 0 t + A 2 4 H 2 - f 0 ,   - f 0 exp - j 2 π f 0 t exp - j 2 π f 0 t + A 2 4 H 2 - f 0 ,   f 0 exp - j 2 π f 0 t exp j 2 π f 0 t + A 2 4 H 2 f 0 ,   - f 0 exp j 2 π f 0 t exp - j 2 π f 0 t .
y 2 t = A 2 4   | H 2 f 0 ,   f 0 | cos 2 π 2 f 0 t + ϕ 2 + A 2 2   | H 2 f 0 ,   - f 0 | .
y t = | H 1 f 0 | A   cos 2 π f 0 t + ϕ 1 + A 2 2   | H 2 f 0 ,   f 0 | cos 2 π 2 f 0 t + ϕ 2 + A 2 2   | H 2 f 0 ,   - f 0 | .
x t = A   cos 2 π f 10 t + B   cos 2 π f 20 t .
X f = A 2 δ f - f 10 + δ f + f 10 + B 2 δ f - f 20 + δ f + f 20 .
y 12 t = - H 11 f 1 ,   f 2 A 2 δ f - f 10 + δ f + f 10 + B 2 δ f - f 20 + δ f + f 20 × A 2 δ f - f 10 + δ f + f 10 + B 2 δ f - f 20 + δ f + f 20 × exp j 2 π f 1 t exp j 2 π f 2 t d f 1 d f 2 .
y 11 t = A 2 2   | H 11 f 10 ,   f 10 | cos 2 π 2 f 10 t + ϕ 10 + A 2 4   | H 11 f 10 ,   f 10 | + AB | H 11 f 10 ,   f 20 | cos 2 π f 10 + f 20 t + ϕ 12 + AB | H 2 f 10 ,   - f 20 | cos 2 π f 10 - f 20 t + ϕ 12 + B 2 2   | H 11 f 20 ,   f 20 | cos 2 π 2 f 20 t + ϕ 20 + B 2 4   | H 11 f 20 ,   f 20 | .
y t = A | H 1 f 10 | cos 2 π f 10 t + ϕ 1 + B | H 1 f 20 | cos 2 π f 20 t + ϕ 2 + A 2 2   | H 11 f 10 ,   f 10 | cos 2 π 2 f 10 t + ϕ 10 + A 2 4   | H 11 f 10 ,   f 10 | + AB | H 11 f 10 ,   f 20 | cos 2 π f 10 + f 20 t + ϕ 12 + AB | H 2 f 10 ,   - f 20 | cos 2 π f 10 - f 20 t + ϕ 12 + B 2 2   | H 11 f 20 ,   f 20 | cos 2 π 2 f 20 t + ϕ 20 + B 2 4   | H 11 f 20 ,   f 20 | .
x 1 t = A   cos 2 π f 10 t   and   x 2 t = B   cos 2 π f 20 t .
y 12 t = - H 12 f 1 ,   f 2 A 2 δ f 1 - f 10 + δ f 1 + f 10 B 2 δ f 2 - f 20 + δ f 2 + f 20 × exp j 2 π f 1 t exp j 2 π f 2 t d f 1 d f 2 .
y 12 t = AB 4 H 12 f 10 ,   f 20 exp j 2 π f 10 t exp j 2 π f 20 t + AB 4 H 12 - f 10 ,   - f 10 exp - j 2 π f 10 t × exp - j 2 π f 20 t + AB 4 H 12 - f 10 ,   f 20 exp - j 2 π f 10 t × exp j 2 π f 20 t + AB 4 H 12 f 10 ,   - f 10 exp j 2 π f 10 t × exp - j 2 π f 20 t .
y 12 t = AB 2   | H 12 f 10 ,   f 20 | cos 2 π f 10 + f 20 t + ϕ 12 + AB 2   | H 12 f 10 ,   f 20 | cos 2 π f 10 - f 20 t + ϕ 12 .
H 11 f 1 ,   f 2 = H f 1 + f 2 ,
H 12 f 1 ,   f 2 = H 1 f 1 H 2 f 2 .

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