Abstract

Several performance criteria are described to enable a fair comparison among the various correlation filter designs: signal-to-noise ratio, peak sharpness, peak location, light efficiency, discriminability, and distortion invariance. The trade-offs resulting between some of these criteria are illustrated with the help of a new family of filters called fractional power filters (FPFs). The classical matched filter, phase-only filter (POF), and inverse filter are special cases of FPFs. Using examples, we show that the POF appears to provide a good compromise between noise tolerance and peak sharpness.

© 1990 Optical Society of America

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References

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  1. A. VanderLugt, “Signal Detection by Complex Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
    [CrossRef]
  2. J. L. Horner, P. D. Gianino, “Phase-Only Matched Filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  3. J. L. Horner, J. R. Leger, “Pattern Recognition with Binary Phase-Only Filters,” Appl. Opt. 24, 609–611 (1985).
    [CrossRef] [PubMed]
  4. D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
    [CrossRef]
  5. F. M. Dickey, B. V. K. Vijaya Kumar, L. A. Romero, J. M. Connelly, “Complex Ternary Matched Filters Yielding High Signal-to-Noise Ratios,” Opt. Eng. 29 (Sept.1990).
  6. R. D. Juday, “Correlation with a Spatial Light Modulator Having Phase and Amplitude Cross-Coupling,” Appl. Opt. 28, 4865–4869 (1989).
    [CrossRef] [PubMed]
  7. F. M. Dickey, B. D. Hansche, “Quad-Phase Correlation Filters for Pattern Recognition,” Appl. Opt. 28, 1611–1613 (1989).
    [CrossRef] [PubMed]
  8. M. A. Kaura, W. T. Rhodes, “Optical Correlator Performance Using a Phase-with-Constrained-Magnitude Complex Spatial Filter,” Appl. Opt. 29, 2587–2593 (1990).
    [CrossRef] [PubMed]
  9. A. A. S. Awwal, M. A. Karim, S. R. Jahan, “Improved Correlation Discrimination Using an Amplitude-Modulated Phase-Only Filter,” Appl. Opt. 29, 233–236 (1990).
    [CrossRef] [PubMed]
  10. H. L. Van Trees, Detection, Estimation and Modulation Theory, Part I (Wiley, New York, 1968).
  11. D. O. North, “An Analysis of the Factors Which Determine Signal/Noise Discriminations in Pulsed-Carrier Systems,” Proc. IEEE 51, 1016–1027 (1963).
    [CrossRef]
  12. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).
  13. F. M. Dickey, K. T. Stalker, J. J. Mason, “Bandwidth Considerations for Binary Phase-Only Filters,” Appl. Opt. 27, 3811–3818 (1988).
    [CrossRef] [PubMed]
  14. D. M. Cottrell, R. A. Lilly, J. A. Davis, T. Day, “Optical Correlator Performance of Binary Phase-Only Filters Using Fourier and Hartley Transforms,” Appl. Opt. 26, 3755–3761 (1987).
    [CrossRef] [PubMed]
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  16. B. R. Brown, A. W. Lohmann, “Complex Spatial Filtering with Binary Masks,” Appl. Opt. 5, 967–970 (1966).
    [CrossRef] [PubMed]
  17. W. E. Ross, D. Psaltis, R. H. Anderson, “Two-Dimensional Magneto-Optic Spatial Light Modulator,” Opt. Eng. 22, 485–490 (1983).
    [CrossRef]
  18. D. A. Gregory, R. D. Juday, J. B. Sampsell, R. O. Gale, R. W. Cohn, S. E. Monroe, “Optical Characteristics of a Deformable Mirror Spatial Light Modulator,” Opt. Lett. 13, 10–12 (1988).
    [CrossRef] [PubMed]
  19. J. L. Horner, “Light Utilization in Optical Correlators,” Appl. Opt. 21, 4511–4514 (1982).
    [CrossRef] [PubMed]
  20. S. A. Kassam, Detection of Signals in NonGaussian Noise (Springer-Verlag, New York, 1987).
  21. B. V. K. Vijaya Kumar, Z. Bahri, “Phase-Only Filters with Improved Signal to Noise Ratio,” Appl. Opt. 28, 250–257 (1989).
    [CrossRef]
  22. C. R. Rao, Linear Statistical Inference and its Applications (Wiley, New York, 1973).
    [CrossRef]
  23. H. Mostafavi, F. Smith, “Image Correlation with Geometric Distortion, Part II: Effect of Local Accuracy,” IEEE Trans. Aerosp. Electron. Syst. 14, 496–501 (1978).
    [CrossRef]
  24. B. V. K. Vijaya Kumar, D. Casasent, “Space-Blur Bandwidth Product in Correlator Performance Evaluation,” J. Opt. Soc. Am. 70, 103–110 (1980).
    [CrossRef]
  25. D. Casasent, G. Silbershatz, B. V. K. Vijaya Kumar, “Acoustooptic Matched Filter Correlator,” Appl. Opt. 21, 2356–2364 (1982).
    [CrossRef] [PubMed]
  26. B. Javidi, “Nonlinear Joint Power Spectrum Based Optical Correlation,” Appl. Opt. 28, 2358–2367 (1989).
    [CrossRef] [PubMed]
  27. A. Mahalanobis, B. V. K. Vijaya Kumar, D. P. Casasent, “Minimum Average Correlation Energy Filters,” Appl. Opt. 26, 3633–3640 (1987).
    [CrossRef] [PubMed]
  28. F. M. Dickey, L. A. Romero, “Dual Optimality of the Phase-Only Filter,” Opt. Lett. 14, 4–5 (1989).
    [CrossRef] [PubMed]
  29. A. V. Oppenheim, A. S. Willsky, I. T. Young, Signals and Systems (Prentice-Hall, Englewood Cliffs, NJ, 1983).
  30. S. Jutamalia, G. Storti, “Architecture for a Ternary Phase Only Processor Using Amplitude Filters,” Appl. Opt. 28, 2688–2689 (1989).
    [CrossRef]
  31. R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).
  32. R. E. Blahut, Principles and Practice of Information Theory (Addison-Wesley, Reading, MA, 1987).
  33. M. Fleisher, U. Mahlab, J. Shamir, “Entropy-Optimized Filter for Pattern Recognition,” Appl. Opt. 29, 2091–2098 (1990).
    [CrossRef] [PubMed]
  34. R. R. Kallman, “Construction of Low Noise Optical Correlation Filters,” Appl. Opt. 25, 1032–1033 (1986).
    [CrossRef] [PubMed]
  35. Y.-N. Hsu, H. H. Arsenault, “Optical Pattern Recognition Using Circular Harmonic Expansion,” Appl. Opt. 21, 4016–4019 (1982).
    [CrossRef] [PubMed]
  36. C. F. Hester, D. Casasent, “Multivariant Technique for Multiclass Pattern Recognition,” Appl. Opt. 19, 1758–1761 (1980).
    [CrossRef] [PubMed]
  37. B. Javidi, C.-J. Kuo, “Joint Transform Image Correlation Using a Binary Spatial Light Modulator at the Fourier Plane,” Appl. Opt. 27, 663–665 (1988).
    [CrossRef] [PubMed]
  38. R. C. Gonzalez, P. O. Wintz, Digital Image Processing (Addison-Wesley, Reading, MA, 1987).
  39. B. V. K. Vijaya Kumar, W. Shi, C. D. Hendrix, “Phase-Only Filters with Maximally Sharp Correlation Peaks,” to appear Opt. Lett. (15July1990).
  40. B. V. K. Vijaya Kumar, Z. Bahri, “Efficient Algorithm for Designing a Ternary Valued Filter Yielding Maximum Signal to Noise Ratio,” Appl. Opt. 28, 1919–1925 (1989).
    [CrossRef]

1990

1989

1988

1987

1986

1985

1984

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

J. L. Horner, P. D. Gianino, “Phase-Only Matched Filtering,” Appl. Opt. 23, 812–816 (1984).
[CrossRef] [PubMed]

1983

W. E. Ross, D. Psaltis, R. H. Anderson, “Two-Dimensional Magneto-Optic Spatial Light Modulator,” Opt. Eng. 22, 485–490 (1983).
[CrossRef]

1982

1980

1978

H. Mostafavi, F. Smith, “Image Correlation with Geometric Distortion, Part II: Effect of Local Accuracy,” IEEE Trans. Aerosp. Electron. Syst. 14, 496–501 (1978).
[CrossRef]

1966

1964

A. VanderLugt, “Signal Detection by Complex Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

1963

D. O. North, “An Analysis of the Factors Which Determine Signal/Noise Discriminations in Pulsed-Carrier Systems,” Proc. IEEE 51, 1016–1027 (1963).
[CrossRef]

Anderson, R. H.

W. E. Ross, D. Psaltis, R. H. Anderson, “Two-Dimensional Magneto-Optic Spatial Light Modulator,” Opt. Eng. 22, 485–490 (1983).
[CrossRef]

Arsenault, H. H.

Awwal, A. A. S.

Bahri, Z.

Blahut, R. E.

R. E. Blahut, Principles and Practice of Information Theory (Addison-Wesley, Reading, MA, 1987).

Brown, B. R.

Casasent, D.

Casasent, D. P.

Cohn, R. W.

Connelly, J. M.

F. M. Dickey, B. V. K. Vijaya Kumar, L. A. Romero, J. M. Connelly, “Complex Ternary Matched Filters Yielding High Signal-to-Noise Ratios,” Opt. Eng. 29 (Sept.1990).

Cottrell, D. M.

Davis, J. A.

Day, T.

Dickey, F. M.

Duda, R. O.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Fleisher, M.

Gale, R. O.

Gianino, P. D.

Gonzalez, R. C.

R. C. Gonzalez, P. O. Wintz, Digital Image Processing (Addison-Wesley, Reading, MA, 1987).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gregory, D. A.

Hansche, B. D.

Hart, P. E.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Hendrix, C. D.

B. V. K. Vijaya Kumar, W. Shi, C. D. Hendrix, “Phase-Only Filters with Maximally Sharp Correlation Peaks,” to appear Opt. Lett. (15July1990).

Hester, C. F.

Horner, J. L.

Hsu, Y.-N.

Jahan, S. R.

Javidi, B.

Juday, R. D.

Jutamalia, S.

Kallman, R. R.

Karim, M. A.

Kassam, S. A.

S. A. Kassam, Detection of Signals in NonGaussian Noise (Springer-Verlag, New York, 1987).

Kaura, M. A.

Kuo, C.-J.

Leger, J. R.

Lilly, R. A.

Lohmann, A. W.

Mahalanobis, A.

Mahlab, U.

Mason, J. J.

Monroe, S. E.

Mostafavi, H.

H. Mostafavi, F. Smith, “Image Correlation with Geometric Distortion, Part II: Effect of Local Accuracy,” IEEE Trans. Aerosp. Electron. Syst. 14, 496–501 (1978).
[CrossRef]

North, D. O.

D. O. North, “An Analysis of the Factors Which Determine Signal/Noise Discriminations in Pulsed-Carrier Systems,” Proc. IEEE 51, 1016–1027 (1963).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, A. S. Willsky, I. T. Young, Signals and Systems (Prentice-Hall, Englewood Cliffs, NJ, 1983).

Paek, E. G.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).

Psaltis, D.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

W. E. Ross, D. Psaltis, R. H. Anderson, “Two-Dimensional Magneto-Optic Spatial Light Modulator,” Opt. Eng. 22, 485–490 (1983).
[CrossRef]

Rao, C. R.

C. R. Rao, Linear Statistical Inference and its Applications (Wiley, New York, 1973).
[CrossRef]

Rhodes, W. T.

Romero, L. A.

F. M. Dickey, B. V. K. Vijaya Kumar, L. A. Romero, J. M. Connelly, “Complex Ternary Matched Filters Yielding High Signal-to-Noise Ratios,” Opt. Eng. 29 (Sept.1990).

F. M. Dickey, L. A. Romero, “Dual Optimality of the Phase-Only Filter,” Opt. Lett. 14, 4–5 (1989).
[CrossRef] [PubMed]

Ross, W. E.

W. E. Ross, D. Psaltis, R. H. Anderson, “Two-Dimensional Magneto-Optic Spatial Light Modulator,” Opt. Eng. 22, 485–490 (1983).
[CrossRef]

Sampsell, J. B.

Shamir, J.

Shi, W.

B. V. K. Vijaya Kumar, W. Shi, C. D. Hendrix, “Phase-Only Filters with Maximally Sharp Correlation Peaks,” to appear Opt. Lett. (15July1990).

Silbershatz, G.

Smith, F.

H. Mostafavi, F. Smith, “Image Correlation with Geometric Distortion, Part II: Effect of Local Accuracy,” IEEE Trans. Aerosp. Electron. Syst. 14, 496–501 (1978).
[CrossRef]

Stalker, K. T.

Storti, G.

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation and Modulation Theory, Part I (Wiley, New York, 1968).

VanderLugt, A.

A. VanderLugt, “Signal Detection by Complex Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Venkatesh, S. S.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

Vijaya Kumar, B. V. K.

Willsky, A. S.

A. V. Oppenheim, A. S. Willsky, I. T. Young, Signals and Systems (Prentice-Hall, Englewood Cliffs, NJ, 1983).

Wintz, P. O.

R. C. Gonzalez, P. O. Wintz, Digital Image Processing (Addison-Wesley, Reading, MA, 1987).

Young, I. T.

A. V. Oppenheim, A. S. Willsky, I. T. Young, Signals and Systems (Prentice-Hall, Englewood Cliffs, NJ, 1983).

Appl. Opt.

J. L. Horner, P. D. Gianino, “Phase-Only Matched Filtering,” Appl. Opt. 23, 812–816 (1984).
[CrossRef] [PubMed]

J. L. Horner, J. R. Leger, “Pattern Recognition with Binary Phase-Only Filters,” Appl. Opt. 24, 609–611 (1985).
[CrossRef] [PubMed]

R. D. Juday, “Correlation with a Spatial Light Modulator Having Phase and Amplitude Cross-Coupling,” Appl. Opt. 28, 4865–4869 (1989).
[CrossRef] [PubMed]

F. M. Dickey, B. D. Hansche, “Quad-Phase Correlation Filters for Pattern Recognition,” Appl. Opt. 28, 1611–1613 (1989).
[CrossRef] [PubMed]

M. A. Kaura, W. T. Rhodes, “Optical Correlator Performance Using a Phase-with-Constrained-Magnitude Complex Spatial Filter,” Appl. Opt. 29, 2587–2593 (1990).
[CrossRef] [PubMed]

A. A. S. Awwal, M. A. Karim, S. R. Jahan, “Improved Correlation Discrimination Using an Amplitude-Modulated Phase-Only Filter,” Appl. Opt. 29, 233–236 (1990).
[CrossRef] [PubMed]

F. M. Dickey, K. T. Stalker, J. J. Mason, “Bandwidth Considerations for Binary Phase-Only Filters,” Appl. Opt. 27, 3811–3818 (1988).
[CrossRef] [PubMed]

D. M. Cottrell, R. A. Lilly, J. A. Davis, T. Day, “Optical Correlator Performance of Binary Phase-Only Filters Using Fourier and Hartley Transforms,” Appl. Opt. 26, 3755–3761 (1987).
[CrossRef] [PubMed]

B. R. Brown, A. W. Lohmann, “Complex Spatial Filtering with Binary Masks,” Appl. Opt. 5, 967–970 (1966).
[CrossRef] [PubMed]

J. L. Horner, “Light Utilization in Optical Correlators,” Appl. Opt. 21, 4511–4514 (1982).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, Z. Bahri, “Phase-Only Filters with Improved Signal to Noise Ratio,” Appl. Opt. 28, 250–257 (1989).
[CrossRef]

D. Casasent, G. Silbershatz, B. V. K. Vijaya Kumar, “Acoustooptic Matched Filter Correlator,” Appl. Opt. 21, 2356–2364 (1982).
[CrossRef] [PubMed]

B. Javidi, “Nonlinear Joint Power Spectrum Based Optical Correlation,” Appl. Opt. 28, 2358–2367 (1989).
[CrossRef] [PubMed]

A. Mahalanobis, B. V. K. Vijaya Kumar, D. P. Casasent, “Minimum Average Correlation Energy Filters,” Appl. Opt. 26, 3633–3640 (1987).
[CrossRef] [PubMed]

M. Fleisher, U. Mahlab, J. Shamir, “Entropy-Optimized Filter for Pattern Recognition,” Appl. Opt. 29, 2091–2098 (1990).
[CrossRef] [PubMed]

R. R. Kallman, “Construction of Low Noise Optical Correlation Filters,” Appl. Opt. 25, 1032–1033 (1986).
[CrossRef] [PubMed]

Y.-N. Hsu, H. H. Arsenault, “Optical Pattern Recognition Using Circular Harmonic Expansion,” Appl. Opt. 21, 4016–4019 (1982).
[CrossRef] [PubMed]

C. F. Hester, D. Casasent, “Multivariant Technique for Multiclass Pattern Recognition,” Appl. Opt. 19, 1758–1761 (1980).
[CrossRef] [PubMed]

B. Javidi, C.-J. Kuo, “Joint Transform Image Correlation Using a Binary Spatial Light Modulator at the Fourier Plane,” Appl. Opt. 27, 663–665 (1988).
[CrossRef] [PubMed]

S. Jutamalia, G. Storti, “Architecture for a Ternary Phase Only Processor Using Amplitude Filters,” Appl. Opt. 28, 2688–2689 (1989).
[CrossRef]

B. V. K. Vijaya Kumar, Z. Bahri, “Efficient Algorithm for Designing a Ternary Valued Filter Yielding Maximum Signal to Noise Ratio,” Appl. Opt. 28, 1919–1925 (1989).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst.

H. Mostafavi, F. Smith, “Image Correlation with Geometric Distortion, Part II: Effect of Local Accuracy,” IEEE Trans. Aerosp. Electron. Syst. 14, 496–501 (1978).
[CrossRef]

IEEE Trans. Inf. Theory

A. VanderLugt, “Signal Detection by Complex Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

J. Opt. Soc. Am.

Opt. Eng.

W. E. Ross, D. Psaltis, R. H. Anderson, “Two-Dimensional Magneto-Optic Spatial Light Modulator,” Opt. Eng. 22, 485–490 (1983).
[CrossRef]

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698–704 (1984).
[CrossRef]

F. M. Dickey, B. V. K. Vijaya Kumar, L. A. Romero, J. M. Connelly, “Complex Ternary Matched Filters Yielding High Signal-to-Noise Ratios,” Opt. Eng. 29 (Sept.1990).

Opt. Lett.

Proc. IEEE

D. O. North, “An Analysis of the Factors Which Determine Signal/Noise Discriminations in Pulsed-Carrier Systems,” Proc. IEEE 51, 1016–1027 (1963).
[CrossRef]

Other

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

S. A. Kassam, Detection of Signals in NonGaussian Noise (Springer-Verlag, New York, 1987).

H. L. Van Trees, Detection, Estimation and Modulation Theory, Part I (Wiley, New York, 1968).

A. V. Oppenheim, A. S. Willsky, I. T. Young, Signals and Systems (Prentice-Hall, Englewood Cliffs, NJ, 1983).

C. R. Rao, Linear Statistical Inference and its Applications (Wiley, New York, 1973).
[CrossRef]

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

R. E. Blahut, Principles and Practice of Information Theory (Addison-Wesley, Reading, MA, 1987).

R. C. Gonzalez, P. O. Wintz, Digital Image Processing (Addison-Wesley, Reading, MA, 1987).

B. V. K. Vijaya Kumar, W. Shi, C. D. Hendrix, “Phase-Only Filters with Maximally Sharp Correlation Peaks,” to appear Opt. Lett. (15July1990).

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Figures (7)

Fig. 1
Fig. 1

Simplified block diagram of the simple binary detection problem.

Fig. 2
Fig. 2

Aircraft (MIG) image used in numerical experiments.

Fig. 3
Fig. 3

SNR, PCE, and light efficiency of fractional power filters as a function of the fractional power p when the aircraft image is used.

Fig. 4
Fig. 4

Character used numerical experiments.

Fig. 5
Fig. 5

SNR, PCE, and light efficiency of fractional power filters as a function of the fractional power p when the character is used.

Fig. 6
Fig. 6

Truck image used in numerical experiments.

Fig. 7
Fig. 7

SNR, PCE, and light efficiency of fractional power filters as a function of the fractional power p when the truck image is used.

Equations (31)

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

r ( x ) = s ( x ) + n ( x ) .
H ( ω ) = α S * ( ω ) P n ( ω ) ,
h ( x ) = α s ( - x ) .
y ( x ) = r ( x ) * s ( - x ) = r ( x )     s ( x ) = - r ( u ) s ( u + x ) d u ,
Y ( ω ) = R ( ω ) S * ( ω ) ,
SNR = E { y ( 0 ) } 2 var { y ( 0 ) } .
SNR = 1 2 π | - H ( ω ) S ( ω ) d ω | 2 - P n ( ω ) H ( ω ) 2 d ω .
PSR = E { y ( 0 ) } 2 var { y ( τ ) } ,
PRMSR = y ( 0 ) 2 y rms 2 ,
y rms = [ 1 N Ω i Ω y ( i ) 2 ] 1 / 2 .
PCE = y ( 0 ) 2 E y ,
E y = - y ( x ) 2 d x .
E y = 1 2 π - S ( ω ) H ( ω ) 2 d ω .
PCE = β | - S ( ω ) H ( ω ) d ω | 2 - S ( ω ) 2 H ( ω ) 2 d ω ,
η = total light intensity in output total light intensity in input = - y ( x ) 2 d x - s ( x ) 2 d x .
η = - S ( ω ) 2 H ( ω ) 2 d ω - S ( ω ) 2 d ω .
H ( ω ) = S 1 * ( ω ) - S 2 * ( ω ) .
FR = E { y 1 ( 0 ) } - E { y 2 ( 0 ) } 2 [ var { y 1 ( 0 ) } + var { y 2 ( 0 ) } ] / 2 ,
DR = min i Ω 1 y i ( 0 ) max i Ω 2 y i ( 0 ) ,
S ( ω ) = S ( ω ) exp [ j θ ( ω ) ] ,
H CMF ( ω ) = S ( ω ) exp [ - j θ ( ω ) ] ,
H POF ( ω ) = exp [ - j θ ( ω ) ] .
H IF ( ω ) = S ( ω ) - 1 exp [ - j θ ( ω ) ] .
H FPF p ( ω ) = { S ( ω ) p exp [ - j θ ( ω ) ]     if S ( ω ) 0 , 0     if S ( ω ) = 0 ,
y ( x ) = IFT { S ( ω ) H FPF p ( ω ) } = 1 2 π - S ( ω ) ( p + 1 ) exp ( j ω x ) d ω ,
y ( x ) = | 1 2 π - S ( ω ) ( p + 1 ) exp ( j ω x ) d ω | 1 2 π - S ( ω ) ( p + 1 ) d ω = y ( 0 ) .
SNR = 1 2 π [ - S ( ω ) ( p + 1 ) d ω ] 2 N 0 - S ( ω ) 2 p d ω ,
- S ( ω ) 2 p d ω - S ( ω ) ( p + 1 ) d ω = - S ( ω ) 2 p ln [ S ( ω ) ] d ω - S ( ω ) ( p + 1 ) ln [ S ( ω ) ] d ω ,
PCE = [ - S ( ω ) ( p + 1 ) d ω ] 2 - S ( ω ) 2 ( p + 1 ) d ω .
- S ( ω ) 2 ( 1 + p ) d ω - S ( ω ) ( p + 1 ) d ω = - S ( ω ) 2 ( 1 + p ) ln [ S ( ω ) ] d ω - S ( ω ) ( p + 1 ) ln [ S ( ω ) ] d ω .
η = δ - S ( ω ) 2 ( 1 + p ) d ω - S ( ω ) 2 d ω ,

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