Abstract

We analyze the performance of the Fourier plane nonlinear filters in terms of signal-to-noise ratio (SNR). We obtain a range of nonlinearities for which SNR is robust to the variations in input-noise bandwidth. This is shown both by analytical estimates of the SNR for nonlinear filters and by experimental simulations. Specifically, we analyze the SNR when Fourier plane nonlinearity is applied to the input signal. Using the Karhunen–Loève series expansion of the noise process, we obtain precise analytic expressions of the SNR for Fourier plane nonlinear filters in the presence of various types of additive-noise processes. We find a range of nonlinearities that need to be applied that keep the output SNR of the filter stable relative to changes in the noise bandwidth.

© 2001 Optical Society of America

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References

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  1. J. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
    [CrossRef]
  2. A. Vanderlugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
  3. D. Casasent, D. Psaltis, “Position, rotation, and scale-invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
    [CrossRef] [PubMed]
  4. A. Mahalanobis, B. V. K. V. Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
    [CrossRef] [PubMed]
  5. H. J. Caufield, W. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
    [CrossRef]
  6. D. L. Flannery, J. L. Horner, “Fourier optical signal processor,” Proc. IEEE 77, 1511–1527 (1989).
    [CrossRef]
  7. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
    [CrossRef] [PubMed]
  8. B. Javidi, J. Wang, A. Fazlolahi, “Performance of the nonlinear joint transform correlator for signals with low pass characteristics,” Appl. Opt. 33, 834–848 (1994).
    [CrossRef] [PubMed]
  9. K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
    [CrossRef]
  10. Ph. Réfrégier, V. Laude, B. Javidi, “Nonlinear joint-transform correlation: an optimal solution for adaptive image discrimination and input noise robustness,” Opt. Lett. 19, 405–407 (1994).
    [PubMed]
  11. J. L. Horner, P. D. Gianino, “Phase-only matched filter,” Appl. Opt. 8, 812–816 (1984).
    [CrossRef]
  12. D. Casasent, “Unified synthetic discrimination function computational formulation,” Appl. Opt. 23, 1620–1627 (1984).
    [CrossRef]
  13. Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with Wiener approach,” Opt. Comput. Process. 1, 245–265 (1991).
  14. Ph. Réfrégier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
    [CrossRef] [PubMed]
  15. Ph. Réfrégier, “Filter design for optical pattern recognition: multicriteria approach,” Opt. Lett. 15, 854–856 (1990).
    [CrossRef]
  16. Ph. Refregier, “Optical pattern recognition: optimal trade-off circular harmonic filters,” Opt. Commun. 86, 113–118 (1991).
    [CrossRef]
  17. B. Javidi, D. Painchaud, “Distortion invariant pattern recognition using Fourier plane nonlinear filters,” Appl. Opt. 35, 318–331 (1996).
    [CrossRef] [PubMed]
  18. W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
    [CrossRef]
  19. J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1967).
  20. K. Fukunaga, Statistical Pattern Recognition, 2nd ed. (Academic, San Diego, Calif., 1990), p. 59.
  21. V. Gnedenko, The Theory of Probability (Mir, Moscow, 1976).

1996 (1)

1994 (2)

1992 (1)

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

1991 (3)

Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with Wiener approach,” Opt. Comput. Process. 1, 245–265 (1991).

Ph. Refregier, “Optical pattern recognition: optimal trade-off circular harmonic filters,” Opt. Commun. 86, 113–118 (1991).
[CrossRef]

Ph. Réfrégier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
[CrossRef] [PubMed]

1990 (2)

K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

Ph. Réfrégier, “Filter design for optical pattern recognition: multicriteria approach,” Opt. Lett. 15, 854–856 (1990).
[CrossRef]

1989 (2)

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

D. L. Flannery, J. L. Horner, “Fourier optical signal processor,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

1987 (1)

1984 (2)

D. Casasent, “Unified synthetic discrimination function computational formulation,” Appl. Opt. 23, 1620–1627 (1984).
[CrossRef]

J. L. Horner, P. D. Gianino, “Phase-only matched filter,” Appl. Opt. 8, 812–816 (1984).
[CrossRef]

1976 (1)

1969 (1)

1964 (1)

A. Vanderlugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

1960 (1)

J. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
[CrossRef]

Casasent, D.

Caufield, H. J.

Fazlolahi, A.

Fielding, K. H.

K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

Figue, J.

Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with Wiener approach,” Opt. Comput. Process. 1, 245–265 (1991).

Flannery, D. L.

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

D. L. Flannery, J. L. Horner, “Fourier optical signal processor,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

Fukunaga, K.

K. Fukunaga, Statistical Pattern Recognition, 2nd ed. (Academic, San Diego, Calif., 1990), p. 59.

Gianino, P. D.

J. L. Horner, P. D. Gianino, “Phase-only matched filter,” Appl. Opt. 8, 812–816 (1984).
[CrossRef]

Gnedenko, V.

V. Gnedenko, The Theory of Probability (Mir, Moscow, 1976).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1967).

Hahn, W. B.

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

Horner, J. L.

K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

D. L. Flannery, J. L. Horner, “Fourier optical signal processor,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

J. L. Horner, P. D. Gianino, “Phase-only matched filter,” Appl. Opt. 8, 812–816 (1984).
[CrossRef]

Javidi, B.

Kumar, B. V. K. V.

Laude, V.

Mahalanobis, A.

Maloney, W. T.

Painchaud, D.

Psaltis, D.

Refregier, Ph.

Ph. Refregier, “Optical pattern recognition: optimal trade-off circular harmonic filters,” Opt. Commun. 86, 113–118 (1991).
[CrossRef]

Réfrégier, Ph.

Turin, J. L.

J. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
[CrossRef]

Vanderlugt, A.

A. Vanderlugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

Wang, J.

Appl. Opt. (8)

IEEE Trans. Inf. Theory (1)

A. Vanderlugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

IRE Trans. Inf. Theory (1)

J. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory IT-6, 311–329 (1960).
[CrossRef]

Opt. Commun. (1)

Ph. Refregier, “Optical pattern recognition: optimal trade-off circular harmonic filters,” Opt. Commun. 86, 113–118 (1991).
[CrossRef]

Opt. Comput. Process. (1)

Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and their comparison with Wiener approach,” Opt. Comput. Process. 1, 245–265 (1991).

Opt. Eng. (2)

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

K. H. Fielding, J. L. Horner, “1-f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
[CrossRef]

Opt. Lett. (3)

Proc. IEEE (1)

D. L. Flannery, J. L. Horner, “Fourier optical signal processor,” Proc. IEEE 77, 1511–1527 (1989).
[CrossRef]

Other (3)

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1967).

K. Fukunaga, Statistical Pattern Recognition, 2nd ed. (Academic, San Diego, Calif., 1990), p. 59.

V. Gnedenko, The Theory of Probability (Mir, Moscow, 1976).

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Figures (14)

Fig. 1
Fig. 1

Experimental (200 runs) and theoretical estimates of SNRk for white Gaussian noise, as a function of k. Noise standard deviation 0.2.

Fig. 2
Fig. 2

Theoretical estimates of SNRk for zero-mean colored Gaussian noise as a function of relative noise bandwidth and nonlinearity index k. The power spectrum is Gaussian shaped. (a) Noise standard deviation 0.2, region of stability between 0.5 and 0.6. (b) Noise standard deviation 2, region of stability k=0.75.

Fig. 3
Fig. 3

Theoretical estimation of SNRk for zero-mean colored noise as a function of relative noise bandwidth and nonlinearity index k. The power spectrum is band limited. Noise standard deviation 0.2, region of stability k=0.6.

Fig. 4
Fig. 4

Experimental result of SNRk for colored noise with a Gaussian-shaped power spectrum. Noise standard deviation 0.01. (b) is the contour plot of (a). The noise standard deviation is low in this case, and the region of stability is close to k=0.5.

Fig. 5
Fig. 5

Experimental result of SNRk for colored noise with Gaussian-shaped power spectrum. Noise standard deviation 0.2. (b) is the contour plot of (a). As the noise standard deviation increases, the region of stability also increases to k=0.64.

Fig. 6
Fig. 6

Experimental result of SNRk in decibels for Gaussian colored noise. Noise standard deviation 2. (a) is the contour plot of (b). When the noise standard deviation is high, the region of stability is close to 0.75.

Fig. 7
Fig. 7

Experimental result of SNRk for colored noise with band-limited white power spectrum. Noise standard deviation 0.01, region of stability k=0.5.

Fig. 8
Fig. 8

Experimental result of SNRk for colored noise with band-limited white power spectrum. Noise standard deviation 0.2, region of stability is k=0.6.

Fig. 9
Fig. 9

Experimental result of SNRk for white noise with truncated power spectrum. Noise standard deviation 2, region of stability is k=0.75.

Fig. 10
Fig. 10

Plots of the experimental result of SNRk for colored noise that is modeled by random variable Z=0.0025Y2, where Y is the standard Gaussian random variable. Therefore Z has noise standard deviation 0.005, and region of stability is ∼0.5.

Fig. 11
Fig. 11

Plots of the experimental result of SNRk for colored noise that is modeled by random variable Z=0.75Y2, where Y the is standard Gaussian random variable. Therefore Z has noise standard deviation, 1.5, and region of stability is ∼0.75.

Fig. 12
Fig. 12

Histograms and smoothed density of the output of the nonlinear filters obtained by using 10,000 simulated runs of the filter output with colored zero-mean stationary Gaussian additive noise of standard deviation 0.2, with no target present in scene. The bandwidth of the noise in each case is 30×30 pixels.

Fig. 13
Fig. 13

Histograms and smoothed density of the output of the nonlinear filters obtained by using 10,000 simulated runs of the filter output with colored zero-mean stationary Gaussian additive noise of standard deviation 0.5, with no target present in scene. The bandwidth of the noise in each case is 30×30 pixels.

Fig. 14
Fig. 14

Histograms of the output of nonlinear filters in the presence of the target, obtained by using 10,000 simulated runs of the filter output with colored zero-mean stationary Gaussian additive noise of standard deviation 0.5. The bandwidth of the noise in each case is 30×30 pixels.

Equations (108)

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s(l)=r(l)+n(l).
P(j)=σ2.
Pg(j)=a exp(-j2/db2),
Pb(j)=(Jσ2)/b,0jb0,bjJ-1.
Yk(j)= |R(j)S(j)|kexp{i[ΦS(j)-ΦR(j)]}.
yk(0)=(1/J)j=0J-1Yk(j),
SNRk(l)=|E[yk(l)]|{Var[yk(l)]}1/2,
N(j)=[JP(j)]1/2Z(j),
E[Z(j)]=E[n(0)]δ0(j).
n(l)=1Jj=0J-1[JP(j)]Z(j)exp[(2πlj)/J].
SNRk(0)=j=0J-1|R(j)|k-1R*(j)E[|R(j)+N(j)|]k-1[R(j)+N(j)]j=0J-1|R(j)|2kE|R(j)+N(j)|2k-|R(j)|2k|E[|R(j)+N(j)|]k-1[R(j)+N(j)]|21/2.
SNRk(0)=|E[yk(0)]|12πJj=0J-1[2πE2(j)-|R(j)|2k+2c2k(j)exp-2|R(j)|2c2(j)Ak2(j)]1/2,
j=0J-1E2(j)=j=0J-1|R(j)|4kc2k(j)exp-|R(j)|2c2(j)Bk(j),
Bk(j)=l=022l2Γ2k+2l+12F[l, x(j), v(j)]Γ(l+1)c2l(j)(2l)!;
Ak(j)=l=022l+1Γk+2l+32F[l, x(j), v(j)]Γ2l+32c2l+1(j)(2l)!,
F[l, x(j), v(j)]=t=0l(2l)!(2t)!(2l-2t)! x2t(j)v(2l-2t)×(j)Γ2t+12Γ2l-2t+22,
E[yk(0)]=12πJj=0J-1|R(j)|k+1ck(j)×exp-|R(j)|2c2(j)Ak(j).
SNRk(0)=j=b+1J-1|R(j)|k+1+j=0b|R(j)|k-1R*(j)E[|R(j)+N(j)|]k-1[R(j)+N(j)]j=0b|R(j)|2kE|R(j)+N(j)|2k-|R(j)|2k|E[|R(j)+N(j)|]k-1[R(j)+N(j)]|21/2.
SNRk(0)=2πJb(k/2)|Eb[yk(0)]|Jkσk(1/J)j=0b2π|R(j)|4kexp-b|R(j)|2J2σ2Bk(j)-|R(j)|2k+2exp-2b|R(j)|2j2σ2Ak2(j)1/2,
Eb[yk(0)]=j=b+1J-1|R(j)|k+1+J1/2σk2πJb1/2j=0b|R(j)|k+1×exp-b|R(j)|2J2σ2Ak(j),
R={k : 12k34}.
r2 =j=0J-1|r(j)|21/2
N(j)=[JP(b, j)]1/2Z(j),
hk(b, j)=E(|R(j)|k-1R*(j)|R(j)+[JP(b, j)]1/2Z(j)|k-1×{R(j)+[JP(b, j)]1/2Z(j)}),
fk(b, j)=E{|R(j)|2k|R(j)+[JP(b, j)]1/2Z(j)|2k},
SNRk2(0)=j=0J-1hk(b, j)2j=0J-1fk(b, j)-j=0J-1|hk(b, j)|2.
D(k, b)=j=0J-1hk(b, j)j=0J-1fk(b, j)-j=0J-1|hk(b, j)|22;
dSNRk2(0)db=D(k, b)j=0J-1hk(b, j)j=0J-1fk(b, j)-j=0J-1hk(b, j)j=0J-1fk(b, j),
P(|Z(j)| T)1/T2.
dSNRk2(0)db-D(k, b)(2k-1)k×Jj=0J-1P(b, j)P(b, j)|R(j)|4k-2×j=0J-1|R(j)|2k+J2j=0J-1P(b, j)|R(j)|4k-2×j=0J-1P(b, j)|R(j)|2k-2,
dSNRk2(0)db
D(k, b)(4k2-7k+3)(k-1)2k2J3×-j=0J-1P(b, j)P(b, j)|R(j)|4k-4×j=0J-1P(b, j)|R(j)|2k-2+j=0J-1P2(b, j)|R(j)|4k-4j=0J-1P(b, j)|R(j)|2k-2.
|R(j)|[JP(b, j)]1/2|R(j)|Jσ.
yk(0)=1Jj=0J-1{|R(j)[R(j)+N(j)]|}k-1×[R(j)+N(j)]R*(j).
E[yk(0)]=1Jj=0J-1|R(j)|k-1R*(j)E×{|R(j)[R(j)+N(j)]|k-1×[R(j)+N(j)]R*(j)}.
E{|R(j)+c(j)Z(j)|k-1[R(j)+c(j)Z(j)]}.
E{|R(j)+c(j)Z(j)|(k-1)/2[R(j)+c(j)Z(j)]}
=EI(j)+iEII(j),
EI(j)=EI=E{[(x+cu)2+(v+cw)2](k-1)/2(x+cu)}
EII(j)=EII=E{[(x+cu)2+(v+cw)2](k-1)/2(v+cw)}.
EI=--[(x+cu)2+(v+cw)2](k-1)/2×(x+cu)f(u, w)dudw=1c2--(u2+w2)(k-1)/2ufu-xc, w-vcdudw=1c2002πρk+1cos(θ)fρ cos(θ)-xc, ρ sin(θ)-vc×dθdρ.
EI(j)=exp(-|R(j)|2/c2)π002πρk+1exp(-ρ2/c2)×cos(θ)exp2ρ[x cos(θ)+v sin(θ)]c2dθdρ.
G(ρ, x, v)=02πcos(θ)exp2ρ[x cos(θ)+v sin(θ)]c2dθ.
02πcos(θ)[x cos(θ)+v sin(θ)]ldθ,
0π/2cosm θ sinn θdθ=Γm+12Γn+122Γm+n+22,
12Γ[(l+2)/2]t=0ll!t!(l-t)! xty(l-t)[1+(-1)t+1
+(-1)l-t+(-1)l+1]Γt+22Γl-t+12.
G(ρ, x, v)=2l=0(2ρ)2l+1F(l, x, v)(c2)2l+1(2l)!Γ[(2l+3)/2],
F(l, x, v)=t=0l(2l)!(2t)!(2l-2t)! x2tv2l-2tΓ×2t+12Γ2l-2t+22.
Lk(l)=1πc20ρk+1+(2l+1)exp(-ρ2/c2)dρ.
Lk(l)=ck+2l+12π Γk+2l+32.
EI(j)=xckexp-|R(j)|2c22π×l=022l+1Γk+2l+32F(l, x, v)c2l+1(2l)!.
EII(j)=vckexp-|R(j)|2c22π×l=022l+1Γk+2l+32F(l, x, v)c2l+1(2l)!.
E[yk(0)]=12πJj=0J-1|R(j)|k+1ck(j)exp-|R(j)|2c2(j)Ak(j),
Ak(j)=l=022l+1Γk+2l+32F[l,x(j),v(j)]Γ2l+32c2l+1(j)(2l)!.
j=0J-1E[|R(j)|2kE|R(j)+c(j)Z(j)|2k],
E|R(j)+c(j)Z(j)|2k=12π c2k(j)exp-|R(j)|2c(j)2Bk(j),
Bk(j)=l=022l2Γ2k+2l+12F[l, x(j), v(j)]Γ(l+1)c2l(j)(2l)!.
SNRk(0)=|E[yk(0)]|12πj=0J-12πE2(j)-|R(j)|2k+2c2k(j)exp-2|R(j)|2c2(j)Ak2(j)1/2 ,
j=0J-1E2(j)=j=0J-1|R(j)|4kc2k(j)exp-|R(j)|2c2(j)Bk(j).
SNRk2(0)=j=0J-1hk(b, j)2j=0J-1fk(b, j)-j=0J-1|hk(b, j)|2,
hk(b, j)=E(|R(j)|k-1R*(j)|R(j)+[JP(b, j)]1/2Z(j)|k-1×{R(j)+[JP(b, j)]1/2Z(j)}),
fk(b, j)=E{|R(j)|2k|R(j)+[JP(b, j)]1/2Z(j)|2k}.
D(k, b)=2j=0J-1hk(b, j)j=0J-1fk(b, j)|-j=0J-1|hk(b, j)|22.
dSNRk2(0)db=D(k, b)j=0J-1fk(b, j)j=0J-1dhk(b, j)db-j=0J-1hk(b, j)j=0J-1dfk(b, j)db.
dSNRk2(0)db
-D(k, b)(4k2-7k+3)(k-1)kJ2
×j=0J-1P(b, j)P(b, j)|R(j)|4k-4
×j=0J-1|R(j)|2k-D(k, b)(4k2-7k+3)(k-1)2k2J3
×j=0J-1P(b, j)P(b, j)|R(j)|4k-4
×j=0J-1P(b, j)|R(j)|2k-2-D(k, b)(2k-1)kJ
×j=0J-1P(b, j)P(b, j)×|R(j)|4k-2
×j=0J-1|R(j)|2k-D(k, b)(2k-1)kJ2
×j=0J-1P(b, j)|R(j)|4k-2j=0J-1P(b, j)|R(j)|2k-2
+D(k, b)(4k2-7k+3)(k-1)2k2J3
×j=0J-1P2(b, j)|R(j)|4k-4j=0J-1P(b, j)|R(j)|2k-2
+D(k, b)(2k-1)k2(k-1)J2j=0J-1P(b, j)
×|R(j)|4k-2j=0J-1P(b, j)|R(j)|2k-2+D(k, b)(k-1)kJ
×j=0J-1|R(j)|4k×j=0J-1P(b, j)|R(j)|2k-2.
σ2=j=0J-1P(b, j)J
Jασr21,
dSNRk2(0)db-D(k, b)(4k2-7k+3)(k-1)2k2J3×j=0J-1P(b, j)P(b, j)|R(j)|4k-4×j=0J-1P(b, j)|R(j)|2k-2+D(k, b)(4k2-7k+3)(k-1)2k2J3×j=0J-1P2(b, j)|R(j)|4k-4×j=0J-1P(b, j)|R(j)|2k-2.
E(|R(j)+[JP(b, j)]1/2Z(j)|k-1
×{R(j)+[JP(b, j)]1/2Z(j)})
IT(w)=1if|Z(j, w)| T,
IT(w)=0if|Z(j, w)| <T;
limT E(IT|R(j)+[JP(b, j)]1/2Z(j)|k-1
×{R(j)+[JP(b, j)]1/2Z(j)})=0.
E[IT|R(j)+[JP(b, j)]1/2Z(j)|k-1{R(j)
+[JP(b, j)]1/2Z(j)}]{E[IT]2E|R(j)
+[JP(b, j)]1/2Z(j)|2k}1/2.
{E|R(j)+[JP(b, j)]1/2Z(j)|2k}1/2
{E|R(j)+P(b, j)1/2Z(j)|2}k/2.
{E|R(j)+[JP(b, j)]1/2Z(j)|2}k/2
[|R(j)|2+JP(b, j)]k/2.
(E[IT]2)1/2=[P(|Z(j)| T)]1/2.
[P(|Z(j)| T)]1/21/T.
H1(J)=1Jj=0J-1|R(j)|k-1R*(j)×|R(j)+[JP(b, j)]1/2Z(j)|k-1×{R(j)+[JP(b, j)]1/2Z(j)},
H0(J)=1Jj=0J-1|R(j)|k-1R*(j)×|[JP(b, j)]1/2×Z(j)|k-1{[JP(b, j)]1/3Z(j)}.
limJ ProbHi(J)-Mi(J)Di(J)>x=12π-xexp -(x2/2)dz,
Mi(J)=E[Hi(J)],Di2(J)=E|Hi(J)|2,
limJ 1Di2(J)j=0J{|x-αj(i)|>τDi(J)}
×(x-E[aj(i)])dF(j, i)(x)=0,
αj(0)=|R(j)|k-1R*(j)|[JP(b, j)]1/2Z(j)|k-1[JP(b, j)]1/2Z(j)J,
αj(1)=|R(j)|k-1R*(j)|R(j)+[JP(b, j)]1/2Z(j)|k-1{R(j)+[JP(b, j)]1/2Z(j)}J,
Pd=1-Φτ-M1(J)D1(J),
|y(l0)|2=1Jj=0J-1|R(j)+N(j)|k|R(j)|k×exp{ΦR(j)-Φ[R(j)+N(j)]}exp[(2πijl0)J]2,
E|y(l0)|2=1J2j=0J-1|R(j)|2{E|R(j)+N(j)|2k-[E|R(j)+N(j)|kexp(ΦR(j)+N(j))]2}+1Jj=0J-1E(|R(j)+N(j)|k|R(j)|k×exp{ΦR(j)-Φ[R(j)+N(j)]})×exp[(2πijl0)/J2]=I+II(l0).

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