Abstract

The problem of recognizing targets in nonoverlapping clutter using nonlinear N-ary phase filters is addressed. Using mathematical analysis, expressions were derived for an N-ary phase filter and the intensity variance of an optical correlator output. The N-ary phase filter was shown to consist of an infinite sum of harmonic terms whose periodicity was determined by N. For the intensity variance, it was found that under certain conditions the variance was minimized due to a previously undiscovered phase quadrature effect. Comparison showed that optimal real filters produced greater signal-to-noise-ratio values than the continuous phase versions as a consequence of this effect.

© 2007 Optical Society of America

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  1. P. C. Miller, M. Royce, P. Virgo, M. Fiebig, and G. Hamlyn, "Experimental and simulated studies on the performance of an ATR system with densely cluttered imagery of natural scenes," Opt. Eng. 38, 1814-1820 (1999).
    [CrossRef]
  2. E. Watanabe and K. Kodate, "Implementation of a high-speed face recognition system that uses an optical parallel correlator," Appl. Opt. 44, 666-676 (2005).
    [CrossRef]
  3. S. Serati, X. Xia, O. Mughal, and A. Linnenberger, "High-resolution phase-only spatial light modulators with sub-millisecond response," Proc. SPIE 5106, 138-145 (2003).
    [CrossRef]
  4. T. Ewing, S. Serati, and K. Bauchert, "Optical correlator using four kilohertz analog spatial light modulators," Proc. SPIE 5437, 123-133 (2004).
    [CrossRef]
  5. D. Psaltis, E. G. Paek, and S. S. Venkatesh, "Optical image correlation with a binary spatial light modulator," Opt. Eng. 23, 698-704 (1984).
  6. M. W. Farn and J. W. Goodman, "Optimal binary phase-only matched filters," Appl. Opt. 27, 4431-4437 (1988).
  7. J. L. Horner and P. D. Gianino, "Phase-only matched Filtering," Appl. Opt. 23, 812-814 (1984).
  8. F. M. Dickey and B. D. Hansche, "Quad-phase correlation filters for pattern recognition," Appl. Opt. 28, 1611-1613 (1989).
  9. B. Javidi, "Generalization of the linear matched filter concept to nonlinear matched filters," Appl. Opt. 29, 1215-1224 (1990).
  10. A. Kozma and D. L. Kelly, "Spatial filtering for the detection of signals submerged in noise," Appl. Opt. 4, 387-392 (1965).
  11. F. M. Dickey, K. T. Stalker, and J. J. Mason, "Bandwidth considerations for binary phase-only filters," Appl. Opt. 27, 3811-3818 (1988).
  12. M. W. Farn and J. W. Goodman, "Bounds on the performance of continuous and quantised phase-only matched filters," J. Opt. Soc. Am. A 7, 66-72 (1990).
  13. R. D. Juday, B. V. K. V. Kumar, and P. K. Rajan, "Optimal real correlation filters," Appl. Opt. 30, 520-522 (1991).
  14. B. V. K. V. Kumar and R. D. Juday, "Design of phase-only, binary-phase only, and complex ternary matched filters with increased signal-to-noise ratios for colored noise," Opt. Lett. 16, 1025-1027 (1991).
  15. B. V. K. V. Kumar, R. D. Juday, and P. K. Rajan, "Saturated filters," J. Opt. Soc. Am. A 9, 405-412 (1992).
  16. R. D. Juday, "Optimal realizable filters and the minimum Euclidean distance principle," Appl. Opt. 32, 1944-1950 (1993).
  17. R. D. Juday, "Generalized Rayleigh quotient approach to filter optimisation," J. Opt. Soc. Am. A 15, 777-790 (1998).
    [CrossRef]
  18. R. D. Juday, "Generality of matched filtering and minimum Euclidean distance projection for optical pattern recognition," J. Opt. Soc. Am. A 18, 1882-1896 (2001).
    [CrossRef]
  19. B. Javidi and J. Wang, "Limitation of the classic definition of the correlation signal-to-noise ratio in optical pattern recognition with disjoint signal and scene noise," Appl. Opt. 31, 6826-6829 (1992).
  20. B. Javidi and J. Wang, "Optimum filter for detection of a target in nonoverlapping scene noise," Appl Opt. 33, 4454-4458 (1994).
  21. B. Javidi and J. Wang, "Design of filters to detect a noisy target in nonoverlapping background noise," J. Opt. Soc. Am. A 11, 2604-2612 (1994).
  22. V. Kober, M. Mozerov, and I. A. Ovseevich, "Adaptive correlation filters for pattern recognition," Pattern Recogn. Image Anal. 16, 425-431 (2006).
    [CrossRef]
  23. R. C. Young, C. R. Chatwin, and B. F. Scott, "High-speed hybrid optical/digital correlator system," Opt. Eng. 32, 2608-2615 (1993).
    [CrossRef]
  24. B. Lowans and M. F. Lewis, "Hybrid correlator employing a chirp-encoded binary phase-only filter," Opt. Lett. 25, 1195-1197 (2000).
    [CrossRef]
  25. S. Goyal, N. K. Nischal, V. K. Beri, and A. K. Gupta, "Wavelet-modified maximum average correlation height filter for rotation invariance that uses chirp encoding in a hybrid digital-optical correlator," Appl. Opt. 45, 4850-4857 (2006).
    [CrossRef]
  26. M. Savvides and B. V. K. Vijaya Kumar, "Quad-phase minimum average correlation energy filters for reduced memory illumination tolerant face authentication," in Audio- and Video-Based Biometric Person Authentication, G. Goos, J. Hartmanis, and J. van Leeuwen, eds. (SpringerLink, 2003), pp. 19-26.
  27. P. B. Chapple and D. C. Bertilone, "Stochastic simulation of infrared non-Gaussian natural terrain imagery," Opt. Commun. 150, 71-76 (1998).
    [CrossRef]
  28. R. D. Juday, R. S. Barton, and S. E. Monroe, "Experimental optical results with minimum Euclidean distance optimal filters, coupled modulators and quadratic metrics," Opt. Eng. 38, 302-312 (1999).
    [CrossRef]
  29. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1991).

2006

2005

2004

T. Ewing, S. Serati, and K. Bauchert, "Optical correlator using four kilohertz analog spatial light modulators," Proc. SPIE 5437, 123-133 (2004).
[CrossRef]

2003

S. Serati, X. Xia, O. Mughal, and A. Linnenberger, "High-resolution phase-only spatial light modulators with sub-millisecond response," Proc. SPIE 5106, 138-145 (2003).
[CrossRef]

M. Savvides and B. V. K. Vijaya Kumar, "Quad-phase minimum average correlation energy filters for reduced memory illumination tolerant face authentication," in Audio- and Video-Based Biometric Person Authentication, G. Goos, J. Hartmanis, and J. van Leeuwen, eds. (SpringerLink, 2003), pp. 19-26.

2001

2000

1999

R. D. Juday, R. S. Barton, and S. E. Monroe, "Experimental optical results with minimum Euclidean distance optimal filters, coupled modulators and quadratic metrics," Opt. Eng. 38, 302-312 (1999).
[CrossRef]

P. C. Miller, M. Royce, P. Virgo, M. Fiebig, and G. Hamlyn, "Experimental and simulated studies on the performance of an ATR system with densely cluttered imagery of natural scenes," Opt. Eng. 38, 1814-1820 (1999).
[CrossRef]

1998

P. B. Chapple and D. C. Bertilone, "Stochastic simulation of infrared non-Gaussian natural terrain imagery," Opt. Commun. 150, 71-76 (1998).
[CrossRef]

R. D. Juday, "Generalized Rayleigh quotient approach to filter optimisation," J. Opt. Soc. Am. A 15, 777-790 (1998).
[CrossRef]

1994

B. Javidi and J. Wang, "Optimum filter for detection of a target in nonoverlapping scene noise," Appl Opt. 33, 4454-4458 (1994).

B. Javidi and J. Wang, "Design of filters to detect a noisy target in nonoverlapping background noise," J. Opt. Soc. Am. A 11, 2604-2612 (1994).

1993

R. C. Young, C. R. Chatwin, and B. F. Scott, "High-speed hybrid optical/digital correlator system," Opt. Eng. 32, 2608-2615 (1993).
[CrossRef]

R. D. Juday, "Optimal realizable filters and the minimum Euclidean distance principle," Appl. Opt. 32, 1944-1950 (1993).

1992

1991

1990

1989

1988

1984

J. L. Horner and P. D. Gianino, "Phase-only matched Filtering," Appl. Opt. 23, 812-814 (1984).

D. Psaltis, E. G. Paek, and S. S. Venkatesh, "Optical image correlation with a binary spatial light modulator," Opt. Eng. 23, 698-704 (1984).

1965

Barton, R. S.

R. D. Juday, R. S. Barton, and S. E. Monroe, "Experimental optical results with minimum Euclidean distance optimal filters, coupled modulators and quadratic metrics," Opt. Eng. 38, 302-312 (1999).
[CrossRef]

Bauchert, K.

T. Ewing, S. Serati, and K. Bauchert, "Optical correlator using four kilohertz analog spatial light modulators," Proc. SPIE 5437, 123-133 (2004).
[CrossRef]

Beri, V. K.

Bertilone, D. C.

P. B. Chapple and D. C. Bertilone, "Stochastic simulation of infrared non-Gaussian natural terrain imagery," Opt. Commun. 150, 71-76 (1998).
[CrossRef]

Chapple, P. B.

P. B. Chapple and D. C. Bertilone, "Stochastic simulation of infrared non-Gaussian natural terrain imagery," Opt. Commun. 150, 71-76 (1998).
[CrossRef]

Chatwin, C. R.

R. C. Young, C. R. Chatwin, and B. F. Scott, "High-speed hybrid optical/digital correlator system," Opt. Eng. 32, 2608-2615 (1993).
[CrossRef]

Dickey, F. M.

Ewing, T.

T. Ewing, S. Serati, and K. Bauchert, "Optical correlator using four kilohertz analog spatial light modulators," Proc. SPIE 5437, 123-133 (2004).
[CrossRef]

Farn, M. W.

Fiebig, M.

P. C. Miller, M. Royce, P. Virgo, M. Fiebig, and G. Hamlyn, "Experimental and simulated studies on the performance of an ATR system with densely cluttered imagery of natural scenes," Opt. Eng. 38, 1814-1820 (1999).
[CrossRef]

Gianino, P. D.

Goodman, J. W.

Goyal, S.

Gupta, A. K.

Hamlyn, G.

P. C. Miller, M. Royce, P. Virgo, M. Fiebig, and G. Hamlyn, "Experimental and simulated studies on the performance of an ATR system with densely cluttered imagery of natural scenes," Opt. Eng. 38, 1814-1820 (1999).
[CrossRef]

Hansche, B. D.

Horner, J. L.

Javidi, B.

Juday, R. D.

Kelly, D. L.

Kober, V.

V. Kober, M. Mozerov, and I. A. Ovseevich, "Adaptive correlation filters for pattern recognition," Pattern Recogn. Image Anal. 16, 425-431 (2006).
[CrossRef]

Kodate, K.

Kozma, A.

Kumar, B. V. K. V.

Lewis, M. F.

Linnenberger, A.

S. Serati, X. Xia, O. Mughal, and A. Linnenberger, "High-resolution phase-only spatial light modulators with sub-millisecond response," Proc. SPIE 5106, 138-145 (2003).
[CrossRef]

Lowans, B.

Mason, J. J.

Miller, P. C.

P. C. Miller, M. Royce, P. Virgo, M. Fiebig, and G. Hamlyn, "Experimental and simulated studies on the performance of an ATR system with densely cluttered imagery of natural scenes," Opt. Eng. 38, 1814-1820 (1999).
[CrossRef]

Monroe, S. E.

R. D. Juday, R. S. Barton, and S. E. Monroe, "Experimental optical results with minimum Euclidean distance optimal filters, coupled modulators and quadratic metrics," Opt. Eng. 38, 302-312 (1999).
[CrossRef]

Mozerov, M.

V. Kober, M. Mozerov, and I. A. Ovseevich, "Adaptive correlation filters for pattern recognition," Pattern Recogn. Image Anal. 16, 425-431 (2006).
[CrossRef]

Mughal, O.

S. Serati, X. Xia, O. Mughal, and A. Linnenberger, "High-resolution phase-only spatial light modulators with sub-millisecond response," Proc. SPIE 5106, 138-145 (2003).
[CrossRef]

Nischal, N. K.

Ovseevich, I. A.

V. Kober, M. Mozerov, and I. A. Ovseevich, "Adaptive correlation filters for pattern recognition," Pattern Recogn. Image Anal. 16, 425-431 (2006).
[CrossRef]

Paek, E. G.

D. Psaltis, E. G. Paek, and S. S. Venkatesh, "Optical image correlation with a binary spatial light modulator," Opt. Eng. 23, 698-704 (1984).

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1991).

Psaltis, D.

D. Psaltis, E. G. Paek, and S. S. Venkatesh, "Optical image correlation with a binary spatial light modulator," Opt. Eng. 23, 698-704 (1984).

Rajan, P. K.

Royce, M.

P. C. Miller, M. Royce, P. Virgo, M. Fiebig, and G. Hamlyn, "Experimental and simulated studies on the performance of an ATR system with densely cluttered imagery of natural scenes," Opt. Eng. 38, 1814-1820 (1999).
[CrossRef]

Savvides, M.

M. Savvides and B. V. K. Vijaya Kumar, "Quad-phase minimum average correlation energy filters for reduced memory illumination tolerant face authentication," in Audio- and Video-Based Biometric Person Authentication, G. Goos, J. Hartmanis, and J. van Leeuwen, eds. (SpringerLink, 2003), pp. 19-26.

Scott, B. F.

R. C. Young, C. R. Chatwin, and B. F. Scott, "High-speed hybrid optical/digital correlator system," Opt. Eng. 32, 2608-2615 (1993).
[CrossRef]

Serati, S.

T. Ewing, S. Serati, and K. Bauchert, "Optical correlator using four kilohertz analog spatial light modulators," Proc. SPIE 5437, 123-133 (2004).
[CrossRef]

S. Serati, X. Xia, O. Mughal, and A. Linnenberger, "High-resolution phase-only spatial light modulators with sub-millisecond response," Proc. SPIE 5106, 138-145 (2003).
[CrossRef]

Stalker, K. T.

Venkatesh, S. S.

D. Psaltis, E. G. Paek, and S. S. Venkatesh, "Optical image correlation with a binary spatial light modulator," Opt. Eng. 23, 698-704 (1984).

Vijaya Kumar, B. V. K.

M. Savvides and B. V. K. Vijaya Kumar, "Quad-phase minimum average correlation energy filters for reduced memory illumination tolerant face authentication," in Audio- and Video-Based Biometric Person Authentication, G. Goos, J. Hartmanis, and J. van Leeuwen, eds. (SpringerLink, 2003), pp. 19-26.

Virgo, P.

P. C. Miller, M. Royce, P. Virgo, M. Fiebig, and G. Hamlyn, "Experimental and simulated studies on the performance of an ATR system with densely cluttered imagery of natural scenes," Opt. Eng. 38, 1814-1820 (1999).
[CrossRef]

Wang, J.

Watanabe, E.

Xia, X.

S. Serati, X. Xia, O. Mughal, and A. Linnenberger, "High-resolution phase-only spatial light modulators with sub-millisecond response," Proc. SPIE 5106, 138-145 (2003).
[CrossRef]

Young, R. C.

R. C. Young, C. R. Chatwin, and B. F. Scott, "High-speed hybrid optical/digital correlator system," Opt. Eng. 32, 2608-2615 (1993).
[CrossRef]

Appl Opt.

B. Javidi and J. Wang, "Optimum filter for detection of a target in nonoverlapping scene noise," Appl Opt. 33, 4454-4458 (1994).

Appl. Opt.

B. Javidi and J. Wang, "Limitation of the classic definition of the correlation signal-to-noise ratio in optical pattern recognition with disjoint signal and scene noise," Appl. Opt. 31, 6826-6829 (1992).

R. D. Juday, B. V. K. V. Kumar, and P. K. Rajan, "Optimal real correlation filters," Appl. Opt. 30, 520-522 (1991).

R. D. Juday, "Optimal realizable filters and the minimum Euclidean distance principle," Appl. Opt. 32, 1944-1950 (1993).

E. Watanabe and K. Kodate, "Implementation of a high-speed face recognition system that uses an optical parallel correlator," Appl. Opt. 44, 666-676 (2005).
[CrossRef]

M. W. Farn and J. W. Goodman, "Optimal binary phase-only matched filters," Appl. Opt. 27, 4431-4437 (1988).

J. L. Horner and P. D. Gianino, "Phase-only matched Filtering," Appl. Opt. 23, 812-814 (1984).

F. M. Dickey and B. D. Hansche, "Quad-phase correlation filters for pattern recognition," Appl. Opt. 28, 1611-1613 (1989).

B. Javidi, "Generalization of the linear matched filter concept to nonlinear matched filters," Appl. Opt. 29, 1215-1224 (1990).

A. Kozma and D. L. Kelly, "Spatial filtering for the detection of signals submerged in noise," Appl. Opt. 4, 387-392 (1965).

F. M. Dickey, K. T. Stalker, and J. J. Mason, "Bandwidth considerations for binary phase-only filters," Appl. Opt. 27, 3811-3818 (1988).

S. Goyal, N. K. Nischal, V. K. Beri, and A. K. Gupta, "Wavelet-modified maximum average correlation height filter for rotation invariance that uses chirp encoding in a hybrid digital-optical correlator," Appl. Opt. 45, 4850-4857 (2006).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

P. B. Chapple and D. C. Bertilone, "Stochastic simulation of infrared non-Gaussian natural terrain imagery," Opt. Commun. 150, 71-76 (1998).
[CrossRef]

Opt. Eng.

R. D. Juday, R. S. Barton, and S. E. Monroe, "Experimental optical results with minimum Euclidean distance optimal filters, coupled modulators and quadratic metrics," Opt. Eng. 38, 302-312 (1999).
[CrossRef]

R. C. Young, C. R. Chatwin, and B. F. Scott, "High-speed hybrid optical/digital correlator system," Opt. Eng. 32, 2608-2615 (1993).
[CrossRef]

P. C. Miller, M. Royce, P. Virgo, M. Fiebig, and G. Hamlyn, "Experimental and simulated studies on the performance of an ATR system with densely cluttered imagery of natural scenes," Opt. Eng. 38, 1814-1820 (1999).
[CrossRef]

D. Psaltis, E. G. Paek, and S. S. Venkatesh, "Optical image correlation with a binary spatial light modulator," Opt. Eng. 23, 698-704 (1984).

Opt. Lett.

Pattern Recogn. Image Anal.

V. Kober, M. Mozerov, and I. A. Ovseevich, "Adaptive correlation filters for pattern recognition," Pattern Recogn. Image Anal. 16, 425-431 (2006).
[CrossRef]

Proc. SPIE

S. Serati, X. Xia, O. Mughal, and A. Linnenberger, "High-resolution phase-only spatial light modulators with sub-millisecond response," Proc. SPIE 5106, 138-145 (2003).
[CrossRef]

T. Ewing, S. Serati, and K. Bauchert, "Optical correlator using four kilohertz analog spatial light modulators," Proc. SPIE 5437, 123-133 (2004).
[CrossRef]

Other

M. Savvides and B. V. K. Vijaya Kumar, "Quad-phase minimum average correlation energy filters for reduced memory illumination tolerant face authentication," in Audio- and Video-Based Biometric Person Authentication, G. Goos, J. Hartmanis, and J. van Leeuwen, eds. (SpringerLink, 2003), pp. 19-26.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 1991).

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Figures (10)

Fig. 1
Fig. 1

One-dimensional amplitude slices of correlation planes obtained with an off-centered impulse function. The top and middle slices are obtained with BPOFs with β = 0 and π / 2 , respectively. The bottom is obtained by combining the top two to generate the correlation obtained with a QPOF.

Fig. 2
Fig. 2

Variation in exponential sum term with beta for N = 2 , 4, and 8.

Fig. 3
Fig. 3

(a) Reference F16 model aircraft signature embedded in uniform background. (b) Input containing F16 reference embedded in nonoverlapping clutter.

Fig. 4
Fig. 4

(a) Variation in correlation peak intensity with beta for different NPOFs and (b) for NBPFs with different values of k calculated using explicit phase encoding (solid curve) and by analysis (dotted curve).

Fig. 5
Fig. 5

(a) Aerial infrared image of natural vegetative clutter, (b) 3D intensity plot of the original clutter, and (c) the filtered clutter.

Fig. 6
Fig. 6

Variation in the clutter intensity variance with beta calculated by analytic expression, (solid curve), and from the clutter data, (dashed curve).

Fig. 7
Fig. 7

Variation in intensity variance ratio with threshold line angle for different even∕odd ratios with (a) N = 2 , (b) 4, and (c) 8. The dashed lines are the analysis bounds. (The arrow indicates the direction of increasing oddness in the target for each curve.)

Fig. 8
Fig. 8

Variation in intensity variance ratio with threshold line angle for (a) k = 0.5 and (b) 1.0. The dashed lines are the analysis bounds. (The arrow indicates the direction of increasing oddness in the target for each curve.)

Fig. 9
Fig. 9

Variation in optimal SNR with sensor noise for different reference even∕odd ratios and NPOFs with (a) N = 2 , (b) 4, and (c) 8. (The result obtained for the continuous POF is very similar to that obtained with N = 8 .)

Fig. 10
Fig. 10

Variation in optimal SNR with sensor noise for different reference even–odd ratios and NNPFs with (a) N = 2 and (b) continuous phase.

Equations (63)

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

y ( x ) = 1 2 π S ( ω ) H ( ω ) exp ( i ω x ) d ω ,
s ( x ) = r ( x ) + n o ( x ) ,
SNR = | E { y ( 0 ) } | 2 var { y ( x ) } ,
var { y ( x ) } = 1 2 π P o ( ω ) | H ( ω ) | 2 d ω ,
H m f ( ω ) = R * ( ω ) / 1 2 π P o ( ω ) ,
H 2 ( ω ) = { 1 , i f Re { R ( ω ) exp ( i β ) 0 } 1 , otherwise ,
H 2 ( ω ) = 2 π p = ( p   odd ) ( 1 ) ( | p | 1 ) / 2 | p | exp { i p [ ϕ R ( ω ) ] } ,
[ sin ( π / N ) π / N ] 2 SNR NPOF SNR POF 1.
n ˜ ( x ) = n ( x ) [ 1 w r ( x ) ] ,
SNR = | E { y ( 0 ) } | 2 var { y ( x ) } ¯ ,
SNR = L 2 π | E [ S ( ω ) ] H ( ω ) d ω | 2 var [ S ( ω ) ] | H ( ω ) | 2 d ω ,
H g m f * ( ω ) = R ( ω ) + m W ( ω ) 1 2 π P ˜ o ( ω ) ,
P ˜ o ( ω ) = | W ( ω ) | 2 P o ( ω ) ,
SNR = | E { y ( 0 ) } | 2 var { y ( x ) } + σ d 2 .
H mf ( ω ) = [ G exp ( i β ) ] | R ( ω ) | 1 2 π N o ( ω ) exp [ i ϕ R ( ω ) ] .
H ( ω ) = closest_to { H mf ( ω ) } .
J = ( I Λ I Ψ ) 2 σ Λ T 2 + σ Ψ T 2 ,
J = ( I Λ ) 2 σ n 2 + σ d 2 .
H 2 , k ( ω , β ) = | R ( ω ) | k 2 π p = ( p   odd ) ( 1 ) ( | p | 1 ) / 2 | p | × exp { i p [ ϕ R ( ω ) β ] } ,
H 4 , k ( ω , β ) = | R ( ω ) | k | 1 + exp ( i π 2 ) | [ H 2 , 0 ( ω , β ) + H 2 , 0 ( ω , β + π / 2 ) exp ( i π 2 ) ] ,
H N , k ( ω , β ) = | R ( ω ) | k | 1 + exp ( i π 2 m 1 ) | [ H 2 m 1 , 0 ( ω , β ) + H 2 m 1 , 0 ( ω , β + π / 2 ) exp ( i π 2 m 1 ) ] ,
H N , k ( ω , β ) = | R ( ω ) | k q ( N ) p = ± l N + 1 ( 1 ) ( | p | 1 ) / 2 | p | × exp { i p [ ϕ R ( ω ) β ] } ,
q ( N ) = { N π for   N = 2 N π m = 2 log 2 N > 1 | exp [ i ( π / 2 π / 2 m 1 ) ] + exp ( i π / 2 ) | otherwise ,
E { s ( x ) } = r ( x ) + m [ 1 w R ( x ) ] .
E { I N , k , β ( 0 ) } = q 2 ( N ) | p = ± l N + 1 ( 1 ) ( | p | 1 ) / 2 | p | × exp ( i p β ) E { y k , p ( 0 ) } | 2 ,
E { y k , p ( 0 ) } = 1 2 π | R ( ω ) + m W ( ω ) | k + 1 × exp [ i ( p 1 ) ϕ R ( ω ) ] d ω ,
E { I N , k , max β ( 0 ) } = q 2 ( N ) | E { y k ,1 ( 0 ) } | 2 .
q 2 ( N ) | p = ± l N + 1 ( 1 ) ( | p | 1 ) / 2 | p | exp ( i p β ) E { y k , p ( 0 ) } | 2
= | E { y k , 1 ( 0 ) } | 2 ,
| p = ± l N + 1 ( 1 ) ( | p | 1 ) / 2 | p | exp ( i p β ) | = 1 q ( N ) .
| p = ± l N + 1 ( 1 ) ( | p | 1 ) / 2 | p | exp ( i p β ) | = | 1 2 l = 1 1 ( l N ) 2 1 | .
q ( N ) 2 E { I N , k , max β ( 0 ) } E { I , k ( 0 ) } 1.
n ( x ) = m + n o ( x ) ,
H N ( ω ) = { 1 , if  ω = 0 exp ( i β ) , otherwise ,
y ( x ) = m + n o ( x ) exp ( i β ) .
var { I ( x ) } = 4 m 2 cos 2 ( β ) σ 2 + 2 σ 4 .
var { I ( x ) } = 2 m 2 σ 2 + σ 4 .
var { I ( x ) } ¯ = 4 y r R ( x ) 2 ¯ E { y n ˜ o R ( x ) 2 } ¯ + 2 E { y n ˜ o R ( x ) 2 } 2 ¯ × 4 y r I ( x ) 2 ¯ E { y n ˜ o I ( x ) 2 } ¯ + 2 E { y n ˜ o I ( x ) 2 } 2 ¯
E { y n ˜ o R ( x ) 2 } 2 ¯ = 1 4 π 2 L { P o ( ω ) [ H e R ( ω , β ) 2 + H o I ( ω , β ) 2 ] } 2 | W ( ω ) | 2 d ω ,
y r R ( x ) 2 ¯ E { y n ˜ o R ( x ) 2 } ¯ = 1 4 π 2 L | R ( ω ) + m W ( ω ) | 2 × [ H e R ( ω , β ) 2 + H o I ( ω , β ) 2 ] d ω × 1 4 π 2 L [ P o ( ω ) | W ( ω ) | 2 ] × [ H e R ( ω , β ) 2 + H o I ( ω , β ) 2 ] d ω ,
H N , k ( ω , β ) = H e R ( ω , β ) + H o R ( ω , β ) + i [ H e I ( ω , β ) + H e I ( ω , β ) ] ,
H e ( ω ) = | R ( ω ) | k q ( N ) p = ± l N + 1 ( 1 ) ( | p | 1 ) / 2 ( 1 ) l | p | × cos [ p ϕ R ( ω ) ] ,
H o ( ω ) = | R ( ω ) | k q ( N ) p = ± l N + 1 ( 1 ) ( | p | 1 ) / 2 ( 1 ) l | p | × sin [ p ϕ R ( ω ) ] .
var { I ( x ) } ¯ = 4 y r R ( x ) 2 ¯ σ ˜ 2 + 2 σ ˜ 4 ,
σ ˜ 2 = 1 4 π 2 L [ P o ( ω ) | W ( ω ) | 2 ] [ H e ( ω ) 2 + H o ( ω ) 2 ] d ω .
var { I ( x ) } ¯ = 4 cos 2 ( π N ) y r R ( x ) 2 ¯ σ ˜ 2 + 4 sin 2 ( π N ) y r I ( x ) 2 ¯ σ ˜ 2 + 2 cos 4 ( π N ) σ ˜ 4 + 2 sin 4 ( π N ) σ ˜ 4 .
cos 2 ( π N ) var { I ( x ) } N ¯ var { I ( x ) } ¯ 1 ,
R ( 0 ) 2 + 4 k P ˜ o ( 0 ) π 2 + 4 y r I ( x ) 2 ¯ σ ˜ 2 + 2 σ ˜ 4 4 y r ( x ) 2 ¯ σ ˜ 2 + 2 σ ˜ 4 var { I ( x ) } 2 ¯ var { I ( x ) } ¯ 1.
2 σ ˜ 2 4 y r ( x ) 2 ¯ + 2 σ ˜ 2 var { I ( x ) } 2 ¯ var { I ( x ) } ¯ 1.
var { I ( x ) } ¯ = E { I ( x ) 2 } ¯ E { I ( x ) } 2 ¯ .
I ( x ) = y R ( x ) 2 + y I ( x ) 2 ,
var { I ( x ) } ¯ E { y R ( x ) 4 } ¯ + E { y I ( x ) 4 } ¯ E { y R ( x ) 2 } 2 ¯ E { y I ( x ) 2 } 2 ¯ .
y R ( x ) = y r R ( x ) + y n ˜ o R ( x ) ,
y I ( x ) = y r I ( x ) + y n ˜ o I ( x )
H N , k ( ω , β ) = | R ( ω ) | k q ( N ) p = ± l N + 1 ( 1 ) ( | p | 1 ) / 2 | p | × { cos [ p ϕ R ( ω ) ] cos ( p β ) sin [ p ϕ R ( ω ) ] × sin ( p β ) i cos [ p ϕ R ( ω ) ] sin ( p β ) i sin [ p ϕ R ( ω ) ] cos ( p π N ) } ,
H N , k ( ω , β ) = H e R ( ω , β ) + H o R ( ω , β ) + i [ H e I ( ω , β ) + H e I ( ω , β ) ] ,
y r R ( x ) = 1 2 π [ R ( ω ) + m W ( ω ) ] [ H e R ( ω , β ) + i H o I ( ω , β ) ] exp ( i ω x ) d ω ,
y n ˜ o R ( x ) = 1 2 π N ˜ o ( ω ) [ H e R ( ω , β ) + i H o I ( ω , β ) ] exp ( i ω x ) d ω ,
E { [ y r R ( x ) + y n ˜ o R ( x ) ] 4 } ¯ E { y n ˜ o R ( x ) 4 } ¯ + 4 y r R ( x ) 2 ¯ E { y n ˜ o R ( x ) 2 } ¯ + y r R ( x ) 4 ¯ + 2 y r R ( x ) 2 ¯ E { y n ˜ o R ( x ) 2 } ¯ ,
E { [ y r R ( x ) + y n ˜ o R ( x ) ] 2 } 2 ¯ = y n ˜ o R ( x ) 4 ¯ + 2 E { y n ˜ o R ( x ) 2 } y r R ( x ) 2 ¯ + E { y n ˜ o R ( x ) 2 } 2 ¯ .
var { I ( x ) } ¯ = 4 y r R ( x ) 2 ¯ E { y n ˜ o R ( x ) 2 } ¯ + 2 E { y n ˜ o R ( x ) 2 } 2 ¯ 4 y r I ( x ) 2 ¯ E { y n ˜ o I ( x ) 2 } ¯ + 2 E { y n ˜ o I ( x ) 2 } 2 ¯ .
E { y n ˜ o R ( x ) 2 } 2 ¯ = 1 4 π 2 L { P o ( ω ) [ H e R ( ω , β ) 2 + H o I ( ω , β ) 2 ] } 2 | W ( ω ) | 2 d ω ,
y r R ( x ) 2 ¯ E { y n ˜ o R ( x ) 2 } ¯ = 1 4 π 2 L | R ( ω ) + m W ( ω ) | 2 × [ H e R ( ω , β ) 2 + H o I ( ω , β ) 2 ] d ω × 1 4 π 2 L [ P o ( ω ) | W ( ω ) | 2 ] × [ H e R ( ω , β ) 2 + H o I ( ω , β ) 2 ] d ω ,

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