Abstract

We investigate a full-phase-based photon-counting double-random-phase encryption (PC-DRPE) method. A PC technique is applied during the encryption process, creating sparse images. The statistical distribution of the PC decrypted data for full-phase encoding and amplitude-phase encoding are derived, and their statistical parameters are used for authentication. The performance of the full-phase PC-DRPE is compared with the amplitude-based PC-DRPE method. The PC decrypted images make it difficult to visually authenticate the input image; however, advanced correlation filters can be used to authenticate the decrypted images given the correct keys. Initial computational simulations show that the full-phase PC-DRPE has the potential to require fewer photons for authentication than the amplitude-based PC-DRPE.

© 2014 Optical Society of America

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    [CrossRef]
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  5. O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24, 762–764 (1999).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. O. Matoba and B. Javidi, “Encrypted optical storage with angular multiplexing,” Appl. Opt. 38, 7288–7293 (1999).
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  22. M. Toishi, M. Hara, K. Tanaka, T. Tanaka, and K. Watanabe, “Novel encryption method using multi reference patterns in coaxial holographic data storage,” Jpn. J. Appl. Phys. 46, 3775–3781 (2007).
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  28. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2011 (2)

2010 (3)

2009 (2)

O. Matoba, T. Nomura, E. P. Cabré, M. S. Millán, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
[CrossRef]

A. Mahalanobis and R. Muise, “Object specific image reconstruction using a compressive sensing architecture for application in surveillance systems,” IEEE Trans. Aerospace Electron. Syst. 45, 1167–1180 (2009).
[CrossRef]

2008 (1)

2007 (2)

M. Toishi, M. Hara, K. Tanaka, T. Tanaka, and K. Watanabe, “Novel encryption method using multi reference patterns in coaxial holographic data storage,” Jpn. J. Appl. Phys. 46, 3775–3781 (2007).
[CrossRef]

Y. Frauel, A. Castro, T. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15, 10253–10265 (2007).
[CrossRef]

2006 (2)

H. Suzuki, M. Yamaguchi, M. Yachida, N. Ohyama, H. Tashima, and T. Obi, “Experimental evaluation of fingerprint verification system based on double random phase encoding,” Opt. Express 14, 1755–1766 (2006).
[CrossRef]

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[CrossRef]

2005 (1)

2004 (1)

2003 (1)

2001 (1)

2000 (3)

1999 (3)

1998 (1)

1997 (1)

1996 (1)

1995 (3)

1994 (2)

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).
[CrossRef]

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[CrossRef]

1993 (1)

1991 (1)

S. Fukushima, T. Kurokawa, and Y. Sakai, “Image encipherment based on optical parallel processing using spatial light modulator,” IEEE Trans. Photon. Technol. Lett. 3, 1133–1135 (1991).

1989 (1)

1979 (1)

B. Efron, “Bootstrap methods: another look at the jackknife,” Ann. Stat. 7, 1–26 (1979).
[CrossRef]

1965 (1)

S. Shapiro and M. Wilk, “An analysis of variance test for normality (complete samples),” Biometrika 52, 591–611 (1965).

1951 (1)

F. Massey, “The Kolmogorov–Smirnov test for goodness of fit,” J. Am. Stat. Assoc. 46, 68–78 (1951).
[CrossRef]

Aloni, D.

Arcos, S.

Barrera, J. F.

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[CrossRef]

Bashaw, M.

Bolognini, N.

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[CrossRef]

Brasher, J.

Cabré, E. P.

O. Matoba, T. Nomura, E. P. Cabré, M. S. Millán, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
[CrossRef]

Carnicer, A.

Castro, A.

Chen, W.

Chen, X.

Cho, M.

Clemente, P.

Dubois, F.

Durán, V.

Efron, B.

B. Efron, “Bootstrap methods: another look at the jackknife,” Ann. Stat. 7, 1–26 (1979).
[CrossRef]

Fisher, N. I.

N. I. Fisher, Statistical Analysis of Circular Data (Cambridge University, 1996).

Frauel, Y.

Fukushima, S.

S. Fukushima, T. Kurokawa, and Y. Sakai, “Image encipherment based on optical parallel processing using spatial light modulator,” IEEE Trans. Photon. Technol. Lett. 3, 1133–1135 (1991).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2000).

Guillaume, M.

Hara, M.

M. Toishi, M. Hara, K. Tanaka, T. Tanaka, and K. Watanabe, “Novel encryption method using multi reference patterns in coaxial holographic data storage,” Jpn. J. Appl. Phys. 46, 3775–3781 (2007).
[CrossRef]

Hayasaki, Y.

Heanue, J.

Henao, R.

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[CrossRef]

Hesselink, L.

Horner, J. L.

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[CrossRef]

Ide, M.

Jammalamadaka, S.

S. Jammalamadaka and A. SenGupta, Topics in Circular Statistics (World Scientific, 2001).

Javidi, B.

E. Pérez-Cabré, M. Cho, and B. Javidi, “Information authentication using photon-counting double-random-phase encrypted images,” Opt. Lett. 36, 22–24 (2011).
[CrossRef]

D. Aloni, A. Stern, and B. Javidi, “Three-dimensional photon counting integral imaging reconstruction using penalized maximum likelihood expectation maximization,” Opt. Express 19, 19681–19687 (2011).
[CrossRef]

O. Matoba, T. Nomura, E. P. Cabré, M. S. Millán, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
[CrossRef]

B. Tavakoli, B. Javidi, and E. Watson, “Three dimensional visualization by photon counting computational integral imaging,” Opt. Express 16, 4426–4436 (2008).
[CrossRef]

Y. Frauel, A. Castro, T. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15, 10253–10265 (2007).
[CrossRef]

T. Nomura, S. Mikan, Y. Morimoto, and B. Javidi, “Secure optical data storage with random phase key codes by use of a configuration of a joint transform correlator,” Appl. Opt. 42, 1508–1514 (2003).
[CrossRef]

E. Tajahuerce and B. Javidi, “Encrypting three-dimensional information with digital holography,” Appl. Opt. 39, 6595–6601 (2000).
[CrossRef]

O. Matoba and B. Javidi, “Encrypted optical storage with angular multiplexing,” Appl. Opt. 38, 7288–7293 (1999).
[CrossRef]

O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24, 762–764 (1999).
[CrossRef]

N. Towghi, B. Javidi, and Z. Luo, “Fully phase encrypted image processor,” J. Opt. Soc. Am. A 16, 1915–1927 (1999).
[CrossRef]

B. Javidi, G. Zhang, and J. Li, “Encrypted optical memory using double-random phase encoding,” Appl. Opt. 36, 1054–1058 (1997).
[CrossRef]

P. Réfrégier, V. Laude, and B. Javidi, “Basic properties of nonlinear global filtering techniques and optimal discriminant solutions,” Appl. Opt. 34, 3915–3923 (1995).
[CrossRef]

P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995).
[CrossRef]

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).
[CrossRef]

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[CrossRef]

B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
[CrossRef]

F. Sadjadi and B. Javidi, Physics of Automatic Target Recognition (Springer, 2007).

Johnson, E.

Joseph, J.

Jupp, P.

K. Mardia and P. Jupp, Directional Statistics (Academic, 1972).

Juvells, I.

Kreske, K.

Kuroda, K.

Kurokawa, T.

S. Fukushima, T. Kurokawa, and Y. Sakai, “Image encipherment based on optical parallel processing using spatial light modulator,” IEEE Trans. Photon. Technol. Lett. 3, 1133–1135 (1991).

Lancis, J.

Laude, V.

Li, J.

Li, Y.

Llebaria, A.

Luo, Z.

Mahalanobis, A.

A. Mahalanobis and R. Muise, “Object specific image reconstruction using a compressive sensing architecture for application in surveillance systems,” IEEE Trans. Aerospace Electron. Syst. 45, 1167–1180 (2009).
[CrossRef]

Mardia, K.

K. Mardia and P. Jupp, Directional Statistics (Academic, 1972).

Massey, F.

F. Massey, “The Kolmogorov–Smirnov test for goodness of fit,” J. Am. Stat. Assoc. 46, 68–78 (1951).
[CrossRef]

Matoba, O.

Matsuba, Y.

Melon, P.

Mikan, S.

Millán, M. S.

O. Matoba, T. Nomura, E. P. Cabré, M. S. Millán, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
[CrossRef]

Montes-Usategui, M.

Morimoto, Y.

Muise, R.

A. Mahalanobis and R. Muise, “Object specific image reconstruction using a compressive sensing architecture for application in surveillance systems,” IEEE Trans. Aerospace Electron. Syst. 45, 1167–1180 (2009).
[CrossRef]

Mukhopadhyay, N.

N. Mukhopadhyay, Probability and Statistical Inference (Marcel Dekker, 2000).

Nagaoka, A.

Naughton, T.

Nishida, N.

Nomura, T.

O. Matoba, T. Nomura, E. P. Cabré, M. S. Millán, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
[CrossRef]

T. Nomura, S. Mikan, Y. Morimoto, and B. Javidi, “Secure optical data storage with random phase key codes by use of a configuration of a joint transform correlator,” Appl. Opt. 42, 1508–1514 (2003).
[CrossRef]

Obi, T.

Ohyama, N.

Okada-Shudo, Y.

Oraintara, S.

A. Vo and S. Oraintara, “On the distributions of the relative phase of complex wavelet coefficients,” in IEEE International Symposium on Circuits and Systems, J. Wang, Y. Huang, and Y. Lim, eds. (ISCAS, 2009), pp. 529–532.

Pérez-Cabré, E.

Refregier, P.

Réfrégier, P.

Rosen, J.

Sadjadi, F.

F. Sadjadi and B. Javidi, Physics of Automatic Target Recognition (Springer, 2007).

Sakai, Y.

S. Fukushima, T. Kurokawa, and Y. Sakai, “Image encipherment based on optical parallel processing using spatial light modulator,” IEEE Trans. Photon. Technol. Lett. 3, 1133–1135 (1991).

SenGupta, A.

S. Jammalamadaka and A. SenGupta, Topics in Circular Statistics (World Scientific, 2001).

Shapiro, S.

S. Shapiro and M. Wilk, “An analysis of variance test for normality (complete samples),” Biometrika 52, 591–611 (1965).

Shimura, T.

Singh, K.

Stern, A.

Suzuki, H.

Tajahuerce, E.

Takeda, M.

Tan, X.

Tanaka, K.

M. Toishi, M. Hara, K. Tanaka, T. Tanaka, and K. Watanabe, “Novel encryption method using multi reference patterns in coaxial holographic data storage,” Jpn. J. Appl. Phys. 46, 3775–3781 (2007).
[CrossRef]

Tanaka, T.

M. Toishi, M. Hara, K. Tanaka, T. Tanaka, and K. Watanabe, “Novel encryption method using multi reference patterns in coaxial holographic data storage,” Jpn. J. Appl. Phys. 46, 3775–3781 (2007).
[CrossRef]

Tashima, H.

Tavakoli, B.

Tebaldi, M.

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[CrossRef]

Toishi, M.

M. Toishi, M. Hara, K. Tanaka, T. Tanaka, and K. Watanabe, “Novel encryption method using multi reference patterns in coaxial holographic data storage,” Jpn. J. Appl. Phys. 46, 3775–3781 (2007).
[CrossRef]

Torres-Company, V.

Torroba, R.

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[CrossRef]

Towghi, N.

Unnikrishnan, G.

Vo, A.

A. Vo and S. Oraintara, “On the distributions of the relative phase of complex wavelet coefficients,” in IEEE International Symposium on Circuits and Systems, J. Wang, Y. Huang, and Y. Lim, eds. (ISCAS, 2009), pp. 529–532.

Wang, J.

Watanabe, K.

M. Toishi, M. Hara, K. Tanaka, T. Tanaka, and K. Watanabe, “Novel encryption method using multi reference patterns in coaxial holographic data storage,” Jpn. J. Appl. Phys. 46, 3775–3781 (2007).
[CrossRef]

Watson, E.

Wilk, M.

S. Shapiro and M. Wilk, “An analysis of variance test for normality (complete samples),” Biometrika 52, 591–611 (1965).

Yachida, M.

Yamaguchi, M.

Yamamoto, H.

Zhang, G.

Ann. Stat. (1)

B. Efron, “Bootstrap methods: another look at the jackknife,” Ann. Stat. 7, 1–26 (1979).
[CrossRef]

Appl. Opt. (11)

P. Réfrégier, V. Laude, and B. Javidi, “Basic properties of nonlinear global filtering techniques and optimal discriminant solutions,” Appl. Opt. 34, 3915–3923 (1995).
[CrossRef]

J. Heanue, M. Bashaw, and L. Hesselink, “Encrypted holographic data storage based on orthogonal-phase-code multiplexing,” Appl. Opt. 34, 6012–6015 (1995).
[CrossRef]

O. Matoba and B. Javidi, “Encrypted optical storage with angular multiplexing,” Appl. Opt. 38, 7288–7293 (1999).
[CrossRef]

Y. Li, K. Kreske, and J. Rosen, “Security and encryption optical systems based on a correlator with significant output images,” Appl. Opt. 39, 5295–5301 (2000).
[CrossRef]

E. Tajahuerce and B. Javidi, “Encrypting three-dimensional information with digital holography,” Appl. Opt. 39, 6595–6601 (2000).
[CrossRef]

X. Tan, O. Matoba, Y. Okada-Shudo, M. Ide, T. Shimura, and K. Kuroda, “Secure optical memory system with polarization encryption,” Appl. Opt. 40, 2310–2315 (2001).
[CrossRef]

T. Nomura, S. Mikan, Y. Morimoto, and B. Javidi, “Secure optical data storage with random phase key codes by use of a configuration of a joint transform correlator,” Appl. Opt. 42, 1508–1514 (2003).
[CrossRef]

Y. Hayasaki, Y. Matsuba, A. Nagaoka, H. Yamamoto, and N. Nishida, “Hiding an image with a light-scattering medium and use of a contrast-discrimination method for readout,” Appl. Opt. 43, 1552–1558 (2004).
[CrossRef]

B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
[CrossRef]

F. Dubois, “Automatic spatial frequency selection algorithm for pattern recognition by correlation,” Appl. Opt. 32, 4365–4371 (1993).
[CrossRef]

B. Javidi, G. Zhang, and J. Li, “Encrypted optical memory using double-random phase encoding,” Appl. Opt. 36, 1054–1058 (1997).
[CrossRef]

Biometrika (1)

S. Shapiro and M. Wilk, “An analysis of variance test for normality (complete samples),” Biometrika 52, 591–611 (1965).

IEEE Trans. Aerospace Electron. Syst. (1)

A. Mahalanobis and R. Muise, “Object specific image reconstruction using a compressive sensing architecture for application in surveillance systems,” IEEE Trans. Aerospace Electron. Syst. 45, 1167–1180 (2009).
[CrossRef]

IEEE Trans. Photon. Technol. Lett. (1)

S. Fukushima, T. Kurokawa, and Y. Sakai, “Image encipherment based on optical parallel processing using spatial light modulator,” IEEE Trans. Photon. Technol. Lett. 3, 1133–1135 (1991).

J. Am. Stat. Assoc. (1)

F. Massey, “The Kolmogorov–Smirnov test for goodness of fit,” J. Am. Stat. Assoc. 46, 68–78 (1951).
[CrossRef]

J. Opt. Soc. Am. A (3)

Jpn. J. Appl. Phys. (1)

M. Toishi, M. Hara, K. Tanaka, T. Tanaka, and K. Watanabe, “Novel encryption method using multi reference patterns in coaxial holographic data storage,” Jpn. J. Appl. Phys. 46, 3775–3781 (2007).
[CrossRef]

Opt. Commun. (1)

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[CrossRef]

Opt. Eng. (1)

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[CrossRef]

Opt. Express (6)

Opt. Lett. (7)

Proc. IEEE (1)

O. Matoba, T. Nomura, E. P. Cabré, M. S. Millán, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
[CrossRef]

Other (7)

F. Sadjadi and B. Javidi, Physics of Automatic Target Recognition (Springer, 2007).

N. Mukhopadhyay, Probability and Statistical Inference (Marcel Dekker, 2000).

J. W. Goodman, Statistical Optics (Wiley, 2000).

N. I. Fisher, Statistical Analysis of Circular Data (Cambridge University, 1996).

A. Vo and S. Oraintara, “On the distributions of the relative phase of complex wavelet coefficients,” in IEEE International Symposium on Circuits and Systems, J. Wang, Y. Huang, and Y. Lim, eds. (ISCAS, 2009), pp. 529–532.

S. Jammalamadaka and A. SenGupta, Topics in Circular Statistics (World Scientific, 2001).

K. Mardia and P. Jupp, Directional Statistics (Academic, 1972).

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Figures (9)

Fig. 1.
Fig. 1.

(a) 256×256 pixel input image, f(x); amplitude of the encrypted image for the (b) amplitude-based DRPE, ψamp(x), and the (c) full-phase DRPE, ψfull(x); photon-limited encrypted images with 1000 photons in the scene (Np) or 0.0152photons/pixel for the (d) amplitude-based PC-DRPE, |ξamp(x)|, and the (e) full-phase PC-DRPE, |ξfull(x)|.

Fig. 2.
Fig. 2.

Decrypted images for the (a) amplitude-based and the (b) full-phase PC-DRPE at Np=1000 or 0.0152photons/pixel.

Fig. 3.
Fig. 3.

Histogram of (a) R{fphamp(x)}, which is N(0,0.0159), and (b) I{fphamp(x)}, which is N(0,0.0146), for image shown in Fig. 2(a).

Fig. 4.
Fig. 4.

Histogram of the decrypted amplitude-based PC-DRPE image [Fig. 2(a)], which follows a sum of Gamma distributions, i=12Γi(1/2,2σi,amp2), with σ^1,amp2=0.0159 and σ^2,amp2=0.0146 at Np=1000.

Fig. 5.
Fig. 5.

Histogram of (a) Acos[πfphfull(x)], which is N(0,0.0075), and (b) Asin[πfphfull(x)], which is N(0,0.0076), of the full phase PC-DRPE image shown in Fig. 2(b).

Fig. 6.
Fig. 6.

Unwrapped circular histogram of the decrypted image from the full-phase PC-DRPE using the image shown in Fig. 1(a) as the input at (a) Np=10 or 1.52e4photons/pixel, which follows a uniform distribution U(0,1) and (b) Np=1000 [Fig. 2(b)], which follows the absolute value of the wrapped Cauchy distribution |WC(0,0.1030)|/π.

Fig. 7.
Fig. 7.

(a) 128×128 pixel binary input image; photon-limited decrypted images with 1000 photons in the scene (Np) or 0.061photon/pixel for the (b) amplitude-based DRPE, fphamp(x) and the (c) full-phase PC-DRPE, fphfull(x); output of optimum filter for the (d) amplitude-based PC-DPRE and the (e) full-phase PC-DRPE.

Fig. 8.
Fig. 8.

Log of the POE versus the number of photons in the scene (Np) for the amplitude-based and full-phase PC-DRPE for a binary image using the optimum filter [Eq. (48)].

Fig. 9.
Fig. 9.

(a) 128×128 pixel false class binary image, g(x); (b) optimum filter output for the full-phase PC-DRPE with g(x) at Np=1000 or 0.061photon/pixel, which has a maximum correlation peak value of 0.360.

Equations (77)

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ψamp(x)={f(x)×exp[j2πn(x)]}*h(x),
ψfull(x)={exp[jπf(x)]×exp[j2πn(x)]}*h(x),
P(li;λi)=[λi]lieλili!,forλi>0,li{0,1,2,},
|fphfull(x)|=|Arg{Aexp[jπfphfull(x)]}/π|,
fphamp(x)=R{fphamp(x)}+jI{fphamp(x)},
Ho:data comes fromN(μ,σ2)versusHa:data does not come fromN(μ,σ2),
x1x2xn.
W=[i=1kai(xni+1xi)]2/i=1n(xix¯)2,fori=1,2,,k,
k={(n+1)/2,ifnis oddn/2,ifnis even.
σ^2=1ni=1nxi2,
R{fphamp(x)}N(0,σ^1,amp2),
{fphamp(x)}N(0,σ^2,amp2).
|fphamp(x)|2=[R{fphamp(x)}]2+[{fphamp(x)}]2.
[R{fphamp(x)}]2Γ(12,2σ^1,amp2),
[{fphamp(x)]2Γ(12,2σ^2,amp2).
|fphamp(x)|2Γ(12,2σ^1,amp2)+Γ(12,2σ^2,amp2),
i=12Γi(12,2σ^i,amp2).
F(x)=1ni=1nI[Xix],
Ho:Famp(x)=F1,t(x)versusHa:Famp(x)F1,t(x),
maxx|Famp(x)F1,t(x)|>Kα,
E[i=12Γi(12,2σi,amp2)]=σ1,amp2+σ2,amp2,
Var[i=12Γi(12,2σi,amp2)]=2σ1,amp4+2σ2,amp4.
Aexp[jπfphfull(x)]=Acos[πfphfull(x)]+jAsin[πfphfull(x)],
Acos[πfphfull(x)]N(0,σ^1,full2),
Asin[πfphfull(x)]N(0,σ^2,full2).
|Arg{Aexp[jπfphfull(x)]}/π|=|Arg{Acos(πfphfull)+jAsin(πfphfull)}/π|.
sin(πfphfull)cos(πfphfull)N(0,σ^2,full2)N(0,σ^1,full2)=Cauchy(0,σ^2,fullσ^1,full),
Z12π1ρ21+ρ22ρcos(zγ),π<γπ,0<ρ<1.
γ=Arg{i=1ncos(zi)+ji=1nsin(zi)},
ρ=[(i=1ncos(zi)/n)2+(i=1nsin(zi)/n)2]1/2
γ1=2ρcosγ1+ρ2,γ2=2ρsinγ1+ρ2.
Z12π1c(1γ1coszγ2sinz),
c=c(γ1,γ2)=11γ12γ22.
η1=cγ1andη2=cγ2.
c=1+η12+η22.
Z12π1(1+η12+η22η1coszη2sinz).
1ci=1nwi[cosziγ1]=0,
1ci=1nwi[sinziγ2]=0,
γ1={i=1nwicoszi}/i=1nwi,
γ2={i=1nwisinzi}/i=1nwi.
γ^=tan1[γ^2γ^1],
ρ^=11γ^1γ^2γ^12+γ^22,
|Arg{Aexp[jπfphfull(x)]}/π|U(0,1)asρ0.
h(y)1π1ρ21+ρ22ρcos(y),0yπ,0<ρ<1.
1πE[y]=1π0πy1π1ρ21+ρ22ρcos(y)dy,
E[y2]=0πy21π1ρ21+ρ22ρcos(y)dy.
1π2Var[y]=1π2{E[y2]E2[y]}.
Ho:Ffull(x)=F2,t(x)versusHa:Ffull(x)F2,t(x),
Hopt*(υ)=R(υ)+mbW1(υ)+mrWr(υ)|R(υ)+mbW1(υ)+mrWr(υ)|2+12πW2(υ)*Nb0(υ)+12π|Wr(υ)|2*Nr0(υ)+mb2[W2(υ)|W1(υ)|2],
W1(υ)=|W0(υ)|2/dWr(υ),
W2(υ)=|W0(υ)|2+|Wr(υ)|22|W0(υ)|2real[Wr(υ)]/d,
POE=|E[a(0,0)]|2(E{1LHx=0L1ς=0H1|a(x,ς)|2})1,
|fphamp(x)|2=[R{fphamp(x)}]2+[I{fphamp(x)}]2,
|fphamp(x)|2Γ(12,2σ^1,amp2)+Γ(12,2σ^2,amp2),
i=12Γi(12,2σ^i,amp2),
[N(0,σ2)]2=[σN(0,1)]2=σ2[N(0,1)]2.
χ2(1)=Γ(12,2),
σ2χ2(1)=Γ(12,2σ2).
|fphamp(x)|2Γ(12,2σ^1,amp2)+Γ(12,2σ^2,amp2),
i=12Γi(12,2σ^i,amp2).
Z=Arg{Aexp[jπfphfull(x)]}12π1c(1γ1coszγ2sinz),
γ1=2ρcosγ1+ρ2,γ2=2ρsinγ1+ρ2,
c=c(γ1,γ2)=11γ12γ22,
Z12π1γ12γ22(1γ1coszγ2sinz).
Z12π1(2ρcosγ1+ρ2)2(2ρsinγ1+ρ2)2(12ρcosγcosz1+ρ22ρsinγsinz1+ρ2).
Z12π1(2ρ1+ρ2)2[(cosγ)2+(sinγ)2](1+ρ22ρcosγcosz2ρsinγsinz1+ρ2).
Z12π1(2ρ1+ρ2)2(1+ρ22ρcos(zγ)1+ρ2).
Z12πρ4+12ρ2(1+ρ2)2(1+ρ22ρcos(zγ)1+ρ2).
Z12π(ρ21)2(1+ρ2)2(1+ρ22ρcos(zγ)1+ρ2).
Z12π1ρ(1+ρ2ρcos(zγ)).
c=c(γ1,γ2)=11γ12γ22,
η1=cγ1,andη2=cγ2,
c=11η12c2η22c2.
1η12c2η22c2=1c.
c=1+η12+η22.
h(y)12π1ρ21+ρ22ρcos(y)|1|+12π1ρ21+ρ22ρcos(y)|1|,=1π1ρ21+ρ22ρcos(y),0yπ,0<ρ<1,
0π1π1ρ21+ρ22ρcos(y)dy=1.

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