Abstract

A new adaptive filter utilizing acoustooptic devices in a space integrating architecture is described. Two configurations are presented; one of them, suitable for signal estimation, is shown to approximate the Wiener filter, while the other, suitable for detection, is shown to approximate the matched filter.

© 1984 Optical Society of America

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References

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  1. A. P. Appelbaum, “Adaptive Arrays,” IEEE Trans. Antennas. Propag. AP-24, 585 (1976).
    [CrossRef]
  2. B. Widrow, P. E. Mantey, T. J. Griffiths, B. B. Goode, “Adaptive Antenna Systems,” Proc. IEEE 55, 2143 (1976).
    [CrossRef]
  3. K. K. Scott, “Transversal Filter Techniques for Adaptive Array Applications,” Proc. IEE 130, Parts F+H, 29 (Feb.1983).
  4. R. Riegler, R. Compton, “An Adaptive Array for Interference Rejection,” Proc. IEEE 61, 748 (1973).
    [CrossRef]
  5. R. W. Lucky, “Adaptive Redundancy Removal in Data Transmission,” Bell Syst. Tech. J. 47, 549 (1968).
  6. M. M. Sondhi, “An Adaptive Echo Canceller,” Bell Syst. Tech. J. 46, 497 (1967).
  7. B. Widrow et al., “Adaptive Noise Cancelling: Principles and Applications,” Proc. IEEE 63, 1692 (1975).
    [CrossRef]
  8. D. R. Morgan, S. E. Craig, “Real Time Adaptive Linear Prediction Using the Least Mean Square Gradient Algorithm,” IEEE Trans. Acoust. Speech, Signal Process. ASSP-24, 494 (1976).
    [CrossRef]
  9. J. E. Bowers, G. S. Kino, D. Behar, H. Olaisen, “Adaptive Deconvolution Using SAW Storage Correlators,” IEEE Trans. Microwave Theory Tech. MTT-29, 491 (1981).
    [CrossRef]
  10. L. J. Griffiths, “Rapid Measurement of Instantaneous Frequency,” IEEE Trans. Acoust. Speech, Signal Process. ASSP-23, 209 (1975).
  11. J. Koford, G. Grover, “The Use of An Adaptive Threshold Element to Design a Linear Optimal Pattern Classifier,” IEEE Trans. Inf. Theory IT-12, 42 (1966).
    [CrossRef]
  12. A. Papoulis, Probability, Random Variables, Stochastic Processes (McGraw-Hill, New York, 1965).
  13. M. White, I. A. Mack, G. M. Borsuk, D. R. Lampe, F. J. Kub, “CCD Adaptive Discrete Analog Signal Processing,” IEEE J. Solid-State Circuits SC-14, 132 (1979).
    [CrossRef]
  14. C. F. N. Cowan, J. W. Arthur, J. Mavor, P. B. Denyer, “CCD Based Adaptive Filters: Realization and Analysis,” IEEE Trans. Acoust. Speech Signal Process ASSP-29, 220 (1981).
    [CrossRef]
  15. D. Psaltis et al., “Iterative Color-Multiplexed Electrooptical Processor,” Opt. Lett. 4, 348 (1979).
    [CrossRef] [PubMed]
  16. D. Psaltis et al., “Iterative Optical Processor for Adaptive Phase Array Radar Applications,” Proc. Soc. Photo-Opt. Instrum. Eng. 180, 114 (1979).
  17. J. F. Rhodes, “Adaptive Filter with a Time-Domain Implementation Using Correlation Cancellation Loops,” Appl. Opt. 22, 282 (1983).
    [CrossRef] [PubMed]
  18. A. VanderLugt, “Adaptive Optical Processor,” Appl. Opt. 21, 4005 (1982).
    [CrossRef]
  19. J. N. Lee, N. J. Berg, P. S. Brody, “High Speed Adaptive Filtering and Reconstruction of Broadband Signals Using Acoustooptic Techniques,” in Proceeding, Ultrasonics Symposium (1980), pp. 488–491.
  20. W. Davenport, W. Root, Introduction to Theory of Random Signals and Noise (McGraw-Hill, New York, 1967).
  21. A. Yariv, Quantum Electronics (Wiley, New York, 1967).
  22. W. T. Rhodes, “Acousto-Optic Signal Processing: Convolution and Correlation,” Proc. IEEE 69, 65 (1981).
    [CrossRef]

1983

K. K. Scott, “Transversal Filter Techniques for Adaptive Array Applications,” Proc. IEE 130, Parts F+H, 29 (Feb.1983).

J. F. Rhodes, “Adaptive Filter with a Time-Domain Implementation Using Correlation Cancellation Loops,” Appl. Opt. 22, 282 (1983).
[CrossRef] [PubMed]

1982

1981

J. E. Bowers, G. S. Kino, D. Behar, H. Olaisen, “Adaptive Deconvolution Using SAW Storage Correlators,” IEEE Trans. Microwave Theory Tech. MTT-29, 491 (1981).
[CrossRef]

C. F. N. Cowan, J. W. Arthur, J. Mavor, P. B. Denyer, “CCD Based Adaptive Filters: Realization and Analysis,” IEEE Trans. Acoust. Speech Signal Process ASSP-29, 220 (1981).
[CrossRef]

W. T. Rhodes, “Acousto-Optic Signal Processing: Convolution and Correlation,” Proc. IEEE 69, 65 (1981).
[CrossRef]

1980

J. N. Lee, N. J. Berg, P. S. Brody, “High Speed Adaptive Filtering and Reconstruction of Broadband Signals Using Acoustooptic Techniques,” in Proceeding, Ultrasonics Symposium (1980), pp. 488–491.

1979

M. White, I. A. Mack, G. M. Borsuk, D. R. Lampe, F. J. Kub, “CCD Adaptive Discrete Analog Signal Processing,” IEEE J. Solid-State Circuits SC-14, 132 (1979).
[CrossRef]

D. Psaltis et al., “Iterative Color-Multiplexed Electrooptical Processor,” Opt. Lett. 4, 348 (1979).
[CrossRef] [PubMed]

D. Psaltis et al., “Iterative Optical Processor for Adaptive Phase Array Radar Applications,” Proc. Soc. Photo-Opt. Instrum. Eng. 180, 114 (1979).

1976

A. P. Appelbaum, “Adaptive Arrays,” IEEE Trans. Antennas. Propag. AP-24, 585 (1976).
[CrossRef]

B. Widrow, P. E. Mantey, T. J. Griffiths, B. B. Goode, “Adaptive Antenna Systems,” Proc. IEEE 55, 2143 (1976).
[CrossRef]

D. R. Morgan, S. E. Craig, “Real Time Adaptive Linear Prediction Using the Least Mean Square Gradient Algorithm,” IEEE Trans. Acoust. Speech, Signal Process. ASSP-24, 494 (1976).
[CrossRef]

1975

B. Widrow et al., “Adaptive Noise Cancelling: Principles and Applications,” Proc. IEEE 63, 1692 (1975).
[CrossRef]

L. J. Griffiths, “Rapid Measurement of Instantaneous Frequency,” IEEE Trans. Acoust. Speech, Signal Process. ASSP-23, 209 (1975).

1973

R. Riegler, R. Compton, “An Adaptive Array for Interference Rejection,” Proc. IEEE 61, 748 (1973).
[CrossRef]

1968

R. W. Lucky, “Adaptive Redundancy Removal in Data Transmission,” Bell Syst. Tech. J. 47, 549 (1968).

1967

M. M. Sondhi, “An Adaptive Echo Canceller,” Bell Syst. Tech. J. 46, 497 (1967).

1966

J. Koford, G. Grover, “The Use of An Adaptive Threshold Element to Design a Linear Optimal Pattern Classifier,” IEEE Trans. Inf. Theory IT-12, 42 (1966).
[CrossRef]

Appelbaum, A. P.

A. P. Appelbaum, “Adaptive Arrays,” IEEE Trans. Antennas. Propag. AP-24, 585 (1976).
[CrossRef]

Arthur, J. W.

C. F. N. Cowan, J. W. Arthur, J. Mavor, P. B. Denyer, “CCD Based Adaptive Filters: Realization and Analysis,” IEEE Trans. Acoust. Speech Signal Process ASSP-29, 220 (1981).
[CrossRef]

Behar, D.

J. E. Bowers, G. S. Kino, D. Behar, H. Olaisen, “Adaptive Deconvolution Using SAW Storage Correlators,” IEEE Trans. Microwave Theory Tech. MTT-29, 491 (1981).
[CrossRef]

Berg, N. J.

J. N. Lee, N. J. Berg, P. S. Brody, “High Speed Adaptive Filtering and Reconstruction of Broadband Signals Using Acoustooptic Techniques,” in Proceeding, Ultrasonics Symposium (1980), pp. 488–491.

Borsuk, G. M.

M. White, I. A. Mack, G. M. Borsuk, D. R. Lampe, F. J. Kub, “CCD Adaptive Discrete Analog Signal Processing,” IEEE J. Solid-State Circuits SC-14, 132 (1979).
[CrossRef]

Bowers, J. E.

J. E. Bowers, G. S. Kino, D. Behar, H. Olaisen, “Adaptive Deconvolution Using SAW Storage Correlators,” IEEE Trans. Microwave Theory Tech. MTT-29, 491 (1981).
[CrossRef]

Brody, P. S.

J. N. Lee, N. J. Berg, P. S. Brody, “High Speed Adaptive Filtering and Reconstruction of Broadband Signals Using Acoustooptic Techniques,” in Proceeding, Ultrasonics Symposium (1980), pp. 488–491.

Compton, R.

R. Riegler, R. Compton, “An Adaptive Array for Interference Rejection,” Proc. IEEE 61, 748 (1973).
[CrossRef]

Cowan, C. F. N.

C. F. N. Cowan, J. W. Arthur, J. Mavor, P. B. Denyer, “CCD Based Adaptive Filters: Realization and Analysis,” IEEE Trans. Acoust. Speech Signal Process ASSP-29, 220 (1981).
[CrossRef]

Craig, S. E.

D. R. Morgan, S. E. Craig, “Real Time Adaptive Linear Prediction Using the Least Mean Square Gradient Algorithm,” IEEE Trans. Acoust. Speech, Signal Process. ASSP-24, 494 (1976).
[CrossRef]

Davenport, W.

W. Davenport, W. Root, Introduction to Theory of Random Signals and Noise (McGraw-Hill, New York, 1967).

Denyer, P. B.

C. F. N. Cowan, J. W. Arthur, J. Mavor, P. B. Denyer, “CCD Based Adaptive Filters: Realization and Analysis,” IEEE Trans. Acoust. Speech Signal Process ASSP-29, 220 (1981).
[CrossRef]

Goode, B. B.

B. Widrow, P. E. Mantey, T. J. Griffiths, B. B. Goode, “Adaptive Antenna Systems,” Proc. IEEE 55, 2143 (1976).
[CrossRef]

Griffiths, L. J.

L. J. Griffiths, “Rapid Measurement of Instantaneous Frequency,” IEEE Trans. Acoust. Speech, Signal Process. ASSP-23, 209 (1975).

Griffiths, T. J.

B. Widrow, P. E. Mantey, T. J. Griffiths, B. B. Goode, “Adaptive Antenna Systems,” Proc. IEEE 55, 2143 (1976).
[CrossRef]

Grover, G.

J. Koford, G. Grover, “The Use of An Adaptive Threshold Element to Design a Linear Optimal Pattern Classifier,” IEEE Trans. Inf. Theory IT-12, 42 (1966).
[CrossRef]

Kino, G. S.

J. E. Bowers, G. S. Kino, D. Behar, H. Olaisen, “Adaptive Deconvolution Using SAW Storage Correlators,” IEEE Trans. Microwave Theory Tech. MTT-29, 491 (1981).
[CrossRef]

Koford, J.

J. Koford, G. Grover, “The Use of An Adaptive Threshold Element to Design a Linear Optimal Pattern Classifier,” IEEE Trans. Inf. Theory IT-12, 42 (1966).
[CrossRef]

Kub, F. J.

M. White, I. A. Mack, G. M. Borsuk, D. R. Lampe, F. J. Kub, “CCD Adaptive Discrete Analog Signal Processing,” IEEE J. Solid-State Circuits SC-14, 132 (1979).
[CrossRef]

Lampe, D. R.

M. White, I. A. Mack, G. M. Borsuk, D. R. Lampe, F. J. Kub, “CCD Adaptive Discrete Analog Signal Processing,” IEEE J. Solid-State Circuits SC-14, 132 (1979).
[CrossRef]

Lee, J. N.

J. N. Lee, N. J. Berg, P. S. Brody, “High Speed Adaptive Filtering and Reconstruction of Broadband Signals Using Acoustooptic Techniques,” in Proceeding, Ultrasonics Symposium (1980), pp. 488–491.

Lucky, R. W.

R. W. Lucky, “Adaptive Redundancy Removal in Data Transmission,” Bell Syst. Tech. J. 47, 549 (1968).

Mack, I. A.

M. White, I. A. Mack, G. M. Borsuk, D. R. Lampe, F. J. Kub, “CCD Adaptive Discrete Analog Signal Processing,” IEEE J. Solid-State Circuits SC-14, 132 (1979).
[CrossRef]

Mantey, P. E.

B. Widrow, P. E. Mantey, T. J. Griffiths, B. B. Goode, “Adaptive Antenna Systems,” Proc. IEEE 55, 2143 (1976).
[CrossRef]

Mavor, J.

C. F. N. Cowan, J. W. Arthur, J. Mavor, P. B. Denyer, “CCD Based Adaptive Filters: Realization and Analysis,” IEEE Trans. Acoust. Speech Signal Process ASSP-29, 220 (1981).
[CrossRef]

Morgan, D. R.

D. R. Morgan, S. E. Craig, “Real Time Adaptive Linear Prediction Using the Least Mean Square Gradient Algorithm,” IEEE Trans. Acoust. Speech, Signal Process. ASSP-24, 494 (1976).
[CrossRef]

Olaisen, H.

J. E. Bowers, G. S. Kino, D. Behar, H. Olaisen, “Adaptive Deconvolution Using SAW Storage Correlators,” IEEE Trans. Microwave Theory Tech. MTT-29, 491 (1981).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, Stochastic Processes (McGraw-Hill, New York, 1965).

Psaltis, D.

D. Psaltis et al., “Iterative Color-Multiplexed Electrooptical Processor,” Opt. Lett. 4, 348 (1979).
[CrossRef] [PubMed]

D. Psaltis et al., “Iterative Optical Processor for Adaptive Phase Array Radar Applications,” Proc. Soc. Photo-Opt. Instrum. Eng. 180, 114 (1979).

Rhodes, J. F.

Rhodes, W. T.

W. T. Rhodes, “Acousto-Optic Signal Processing: Convolution and Correlation,” Proc. IEEE 69, 65 (1981).
[CrossRef]

Riegler, R.

R. Riegler, R. Compton, “An Adaptive Array for Interference Rejection,” Proc. IEEE 61, 748 (1973).
[CrossRef]

Root, W.

W. Davenport, W. Root, Introduction to Theory of Random Signals and Noise (McGraw-Hill, New York, 1967).

Scott, K. K.

K. K. Scott, “Transversal Filter Techniques for Adaptive Array Applications,” Proc. IEE 130, Parts F+H, 29 (Feb.1983).

Sondhi, M. M.

M. M. Sondhi, “An Adaptive Echo Canceller,” Bell Syst. Tech. J. 46, 497 (1967).

VanderLugt, A.

White, M.

M. White, I. A. Mack, G. M. Borsuk, D. R. Lampe, F. J. Kub, “CCD Adaptive Discrete Analog Signal Processing,” IEEE J. Solid-State Circuits SC-14, 132 (1979).
[CrossRef]

Widrow, B.

B. Widrow, P. E. Mantey, T. J. Griffiths, B. B. Goode, “Adaptive Antenna Systems,” Proc. IEEE 55, 2143 (1976).
[CrossRef]

B. Widrow et al., “Adaptive Noise Cancelling: Principles and Applications,” Proc. IEEE 63, 1692 (1975).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1967).

Appl. Opt.

Bell Syst. Tech. J.

R. W. Lucky, “Adaptive Redundancy Removal in Data Transmission,” Bell Syst. Tech. J. 47, 549 (1968).

M. M. Sondhi, “An Adaptive Echo Canceller,” Bell Syst. Tech. J. 46, 497 (1967).

IEEE J. Solid-State Circuits

M. White, I. A. Mack, G. M. Borsuk, D. R. Lampe, F. J. Kub, “CCD Adaptive Discrete Analog Signal Processing,” IEEE J. Solid-State Circuits SC-14, 132 (1979).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process

C. F. N. Cowan, J. W. Arthur, J. Mavor, P. B. Denyer, “CCD Based Adaptive Filters: Realization and Analysis,” IEEE Trans. Acoust. Speech Signal Process ASSP-29, 220 (1981).
[CrossRef]

IEEE Trans. Acoust. Speech, Signal Process.

L. J. Griffiths, “Rapid Measurement of Instantaneous Frequency,” IEEE Trans. Acoust. Speech, Signal Process. ASSP-23, 209 (1975).

D. R. Morgan, S. E. Craig, “Real Time Adaptive Linear Prediction Using the Least Mean Square Gradient Algorithm,” IEEE Trans. Acoust. Speech, Signal Process. ASSP-24, 494 (1976).
[CrossRef]

IEEE Trans. Antennas. Propag.

A. P. Appelbaum, “Adaptive Arrays,” IEEE Trans. Antennas. Propag. AP-24, 585 (1976).
[CrossRef]

IEEE Trans. Inf. Theory

J. Koford, G. Grover, “The Use of An Adaptive Threshold Element to Design a Linear Optimal Pattern Classifier,” IEEE Trans. Inf. Theory IT-12, 42 (1966).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

J. E. Bowers, G. S. Kino, D. Behar, H. Olaisen, “Adaptive Deconvolution Using SAW Storage Correlators,” IEEE Trans. Microwave Theory Tech. MTT-29, 491 (1981).
[CrossRef]

Opt. Lett.

Proc. IEE

K. K. Scott, “Transversal Filter Techniques for Adaptive Array Applications,” Proc. IEE 130, Parts F+H, 29 (Feb.1983).

Proc. IEEE

R. Riegler, R. Compton, “An Adaptive Array for Interference Rejection,” Proc. IEEE 61, 748 (1973).
[CrossRef]

B. Widrow, P. E. Mantey, T. J. Griffiths, B. B. Goode, “Adaptive Antenna Systems,” Proc. IEEE 55, 2143 (1976).
[CrossRef]

B. Widrow et al., “Adaptive Noise Cancelling: Principles and Applications,” Proc. IEEE 63, 1692 (1975).
[CrossRef]

W. T. Rhodes, “Acousto-Optic Signal Processing: Convolution and Correlation,” Proc. IEEE 69, 65 (1981).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

D. Psaltis et al., “Iterative Optical Processor for Adaptive Phase Array Radar Applications,” Proc. Soc. Photo-Opt. Instrum. Eng. 180, 114 (1979).

Proceeding, Ultrasonics Symposium

J. N. Lee, N. J. Berg, P. S. Brody, “High Speed Adaptive Filtering and Reconstruction of Broadband Signals Using Acoustooptic Techniques,” in Proceeding, Ultrasonics Symposium (1980), pp. 488–491.

Other

W. Davenport, W. Root, Introduction to Theory of Random Signals and Noise (McGraw-Hill, New York, 1967).

A. Yariv, Quantum Electronics (Wiley, New York, 1967).

A. Papoulis, Probability, Random Variables, Stochastic Processes (McGraw-Hill, New York, 1965).

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Figures (8)

Fig. 1
Fig. 1

Passive processor.

Fig. 2
Fig. 2

Active processor.

Fig. 3
Fig. 3

Space integrating optical convolver.

Fig. 4
Fig. 4

Space integrating optical correlator.

Fig. 5
Fig. 5

Adaptive optical processor (passive).

Fig. 6
Fig. 6

Passive processor response to two jammers: f1 = 9.8 MHz, f2 = 10.2 MHz, T = 10 μsec.

Fig. 7
Fig. 7

Passive processor response to two jammers: f1 = 9.8 MHz, f2 = 10.2 MHz, T = 50 μsec.

Fig. 8
Fig. 8

Adaptive optical processor (active).

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

u ( t ) = - i 1 ( t - τ ) i 2 ( τ ) d τ ,
w ( t ) = - i 1 ( t + τ ) i 2 * ( τ ) d τ ,
z ( t ) = x ( t ) - G - z ( τ ) x * ( α - σ ) x ( t + α - τ - σ ) d α d τ = x ( t ) - G - z ( τ ) [ s * ( α - σ ) + n * ( α - σ ) ] × [ s ( t + α - τ - σ ) + n ( t + α - τ - σ ) ] d α d τ ,
z ( t ) x ( t ) - G - z ( τ ) n * ( α - σ ) n ( t + α - τ - σ ) d α d τ .
Z ( ω ) X ( ω ) - G Z ( ω ) N ( ω ) 2 ,
Z ( ω ) X ( ω ) 1 + G N ( ω ) 2 = 1 / G 1 / G + N ( ω ) 2 X ( ω ) .
e ( t ) = - x ( t - τ ) s * ( - τ ) d τ - G - x ( t - τ ) x * ( α ) e ( τ + α ) d α d τ ,
e ( t ) - x ( t - τ ) s * ( - τ ) d τ - G - n ( t - τ ) n * ( α ) e ( τ + α ) d α d τ .
E ( ω ) X ( ω ) S * ( ω ) 1 + G N ( ω ) 2 X ( ω ) S * ( ω ) G N ( ω ) 2 ,
sin θ B = λ ω 0 / 4 π v ,
A ( x , t ) { exp ( - j 2 π sin θ B x / λ ) + j ( m / 2 ) a 1 * ( t - x / v ) exp [ - j ω 0 ( t - x / v ) ] × exp ( - j 2 π sin θ B x / λ ) } rect ( x / W ) [ exp ( - j 2 π sin θ B x / λ ) + j ( m / 2 ) a 1 * ( t - x / v ) exp ( - j ω 0 t ) × exp ( j 2 π sin θ B x / λ ) ] rect ( x / W ) ,
A 2 ( x , t ) [ a 2 ( t - x / v ) exp ( - j 2 π sin θ B x / λ ) exp ( j ω 0 t ) + a 1 * ( t + x / v ) exp ( - j 2 π sin θ B x / λ ) × exp ( - j ω 0 t ) ] rect ( x / W ) .
I 1 ( t ) - T / 2 T / 2 a 1 ( t + τ ) 2 d τ + - T / 2 T / 2 a 2 ( t - τ ) 2 d τ + 2 Re { exp ( j 2 ω 0 t ) - T 2 T / 2 a 1 ( t + τ ) a 2 ( t - τ ) d τ } ,
I 2 ( t ) - | - T / 4 T / 4 [ a 1 ( t + 2 τ ) exp ( j ω 0 t ) + a 2 ( t + τ ) exp ( j 2 ω 0 t ) ] exp ( j k τ ) d τ | 2 d k - T / 4 T / 4 a 1 ( t + 2 τ ) 2 d τ + - T / 4 T / 4 a 2 ( t + τ ) 2 d τ + 2 Re { exp ( j ω 0 t ) - T / 4 T / 4 a 1 * ( t + 2 τ ) a 2 ( t + τ ) d τ } ,
= 2 Re [ exp ( j ω 0 t ) t - T / 2 t + T / 2 a 1 * ( α ) a 3 ( t + α ) d α ] ,
u ( t ) = ( 1 / T ) - T / 4 T / 4 i 1 ( t + τ ) i 2 ( t - τ ) d τ ,
w ( t ) = ( 1 / T ) - T / 4 T / 4 i 1 * ( t + 2 τ ) i 2 ( t + τ ) d τ .
z ( t ) = x ( t ) - ( G / T 2 ) - T / 4 T / 4 x * ( t + 2 v - σ ) × x ( t + τ + v - σ ) z ( t - τ + v ) d τ d v ,
z ( t ) = x ( t ) - ( G / T 2 ) - T / 4 T / 4 n * ( t + 2 v - σ ) × n ( t + τ + v - σ ) z ( t - τ + v ) d τ d v .
Z ( ω ) X ( ω ) - ( G / 4 π 2 ) - N * ( α ) × N ( β ) exp [ - j ( β - α ) σ ] Z ( ω + α - β ) × sinc [ ( ω - α ) T / 4 π ] sinc [ ( ω + α - 2 β ) T / 4 π ] d α d β .
Z ( ω ) X ( ω ) - ( G / 4 π 2 ) Z ( ω ) - N ( α ) 2 × sinc 2 [ ( ω - α ) T / 4 π ] d α .
Z ( ω ) X ( ω ) 1 + ( G / 4 π 2 ) - N ( α ) 2 sinc 2 [ ( ω - α ) T / 4 π ] d α .
e ( t ) = ( 1 / T ) - T / 4 T / 4 x ( t + τ ) s * ( τ - t ) d τ - ( G / T 2 ) - T / 4 T / 4 x ( t + τ ) x * ( t - τ + 2 v ) e ( t - τ + v ) d v d τ ,
e ( t ) ( 1 / T ) - T / 4 T / 4 x ( t + τ ) s * ( τ - t ) d τ - ( G / T 2 ) - T / 4 T / 4 n ( t + τ ) n * ( t - τ + 2 v ) e ( t - τ + v ) d v d τ ,
E ( ω ) ( 1 / 2 π ) - X ( α ) S * ( ω - α ) sinc [ ( ω - 2 α ) T / 4 π ] a 1 + ( G / 4 π 2 ) - N ( α ) 2 sinc 2 [ ( ω - 2 α ) T / 4 π ] d α .

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