Abstract

Signal and image and detection systems based on nonlinear operations of Fourier-transformed data often exhibit greater selectivity than standard matched-filtering techniques. One such system is the joint transform correlator. We analyze the performance of the nonlinear joint transform correlator in terms of the output signal-to-noise ratio; this signal-to-noise ratio is evaluated in terms of both output contrast (peak-to-noise floor) and output variability (peak-to-peak standard deviation). The main assumption used is that the signal energy is small relative to that of the additive noise; this assumption is defensible in practice owing to the generally small spatial extent of target images relative to scenes. With respect to the first performance measure, this study is an extension of that in an earlier paper [Appl. Opt. 34, 5218 (1995)]. The previous analysis was carried out under a restriction that the signal and noise spectra were to be similar (actually multiples of one another). In the current study there is no such constraint, and all analysis of the second measure is new. The analysis is supported by simulation. A benefit of analytical rather than simulational study is that conclusions can be drawn with greater confidence. One of the most interesting of these is that the smooth square-root Fourier plane nonlinearity, more usually known as the k-law processor with k = 0.5, offers extremely robust performance with respect to relative noise bandwidth.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. Weaver, J. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  2. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
    [CrossRef] [PubMed]
  3. B. Javidi, J. Wang, Q. Tang, “Multiple-object binary joint transform correlation using multiple-level threshold crossing,” Appl. Opt. 30, 4234–4244 (1991).
    [CrossRef] [PubMed]
  4. W. Hahn, D. Flannery, “Basic design elements of the binary joint transform correlator and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
    [CrossRef]
  5. K. Fielding, J. Horner, “1 - f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
    [CrossRef]
  6. S. Rogers, J. Kline, M. Kabrisky, J. Mills, “New binarization techniques for the joint transform correlator,” Opt. Eng. 29, 1088–1093 (1990).
    [CrossRef]
  7. P. Refregier, 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]
  8. P. Refregier, F. Goudail, “Decision theory applied to nonlinear joint transform correlation,” in Optoelectronic Information Processing, B. Javidi, P. Refregier, eds. (SPIE Press, Bellingham, Wash., 1997), pp. 137–166.
  9. P. Willett, B. Javidi, “Approximate performance of the nonlinear joint transform correlator in signal-like noise,” Appl. Opt. 34, 5218–5229 (1995).
    [CrossRef] [PubMed]
  10. S. Kassam, Signal Detection in Non-Gaussian Noise (Springer-Verlag, Berlin, 1987).
  11. H. Poor, An Introduction to Signal Detection and Estimation (Springer-Verlag, Berlin, 1987).
  12. J. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992).
    [CrossRef] [PubMed]

1995 (1)

1994 (1)

1992 (2)

W. Hahn, D. Flannery, “Basic design elements of the binary joint transform correlator and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

J. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992).
[CrossRef] [PubMed]

1991 (1)

1990 (2)

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

S. Rogers, J. Kline, M. Kabrisky, J. Mills, “New binarization techniques for the joint transform correlator,” Opt. Eng. 29, 1088–1093 (1990).
[CrossRef]

1989 (1)

1966 (1)

Fielding, K.

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

Flannery, D.

W. Hahn, D. Flannery, “Basic design elements of the binary joint transform correlator and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

Goodman, J.

Goudail, F.

P. Refregier, F. Goudail, “Decision theory applied to nonlinear joint transform correlation,” in Optoelectronic Information Processing, B. Javidi, P. Refregier, eds. (SPIE Press, Bellingham, Wash., 1997), pp. 137–166.

Hahn, W.

W. Hahn, D. Flannery, “Basic design elements of the binary joint transform correlator and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

Horner, J.

J. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992).
[CrossRef] [PubMed]

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

Javidi, B.

Kabrisky, M.

S. Rogers, J. Kline, M. Kabrisky, J. Mills, “New binarization techniques for the joint transform correlator,” Opt. Eng. 29, 1088–1093 (1990).
[CrossRef]

Kassam, S.

S. Kassam, Signal Detection in Non-Gaussian Noise (Springer-Verlag, Berlin, 1987).

Kline, J.

S. Rogers, J. Kline, M. Kabrisky, J. Mills, “New binarization techniques for the joint transform correlator,” Opt. Eng. 29, 1088–1093 (1990).
[CrossRef]

Laude, V.

Mills, J.

S. Rogers, J. Kline, M. Kabrisky, J. Mills, “New binarization techniques for the joint transform correlator,” Opt. Eng. 29, 1088–1093 (1990).
[CrossRef]

Poor, H.

H. Poor, An Introduction to Signal Detection and Estimation (Springer-Verlag, Berlin, 1987).

Refregier, P.

P. Refregier, 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]

P. Refregier, F. Goudail, “Decision theory applied to nonlinear joint transform correlation,” in Optoelectronic Information Processing, B. Javidi, P. Refregier, eds. (SPIE Press, Bellingham, Wash., 1997), pp. 137–166.

Rogers, S.

S. Rogers, J. Kline, M. Kabrisky, J. Mills, “New binarization techniques for the joint transform correlator,” Opt. Eng. 29, 1088–1093 (1990).
[CrossRef]

Tang, Q.

Wang, J.

Weaver, C.

Willett, P.

Appl. Opt. (1)

J. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992).
[CrossRef] [PubMed]

Appl. Opt. (4)

Opt. Eng. (1)

S. Rogers, J. Kline, M. Kabrisky, J. Mills, “New binarization techniques for the joint transform correlator,” Opt. Eng. 29, 1088–1093 (1990).
[CrossRef]

Opt. Eng. (2)

W. Hahn, D. Flannery, “Basic design elements of the binary joint transform correlator and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

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

Opt. Lett. (1)

Other (3)

P. Refregier, F. Goudail, “Decision theory applied to nonlinear joint transform correlation,” in Optoelectronic Information Processing, B. Javidi, P. Refregier, eds. (SPIE Press, Bellingham, Wash., 1997), pp. 137–166.

S. Kassam, Signal Detection in Non-Gaussian Noise (Springer-Verlag, Berlin, 1987).

H. Poor, An Introduction to Signal Detection and Estimation (Springer-Verlag, Berlin, 1987).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (16)

Fig. 1
Fig. 1

Nonlinear JTC.

Fig. 2
Fig. 2

Image and reference used for simulations, with an example of noise added.

Fig. 3
Fig. 3

Correlation peaks in the output plane, that is, h(, ŷ). This corresponds to the optimal linear architecture with θ = 0.1, and note that the large dc peak at the origin was clipped to avoid dynamic-range problems in the plot.

Fig. 4
Fig. 4

Analytical and simulated performance in terms of the SNRpnf of optimal linear (LSV) and classic (LSI) JTC’s, with a noise bandwidth parameter of W = 4, plotted against the input SNR. Solid curves are from theory; circles refer to simulation results.

Fig. 5
Fig. 5

Analytical and simulated performance in terms of the SNRpnf of optimal linear (LSV) and classic (LSI) JTC’s, with input SNR = 0.25 (θ = 0.5), plotted against noise bandwidth parameter W. Solid curves are from theory; circles refer to simulation results.

Fig. 6
Fig. 6

Analytical and simulated performance in terms of the SNRpnf of optimal linear (LSV) and optimal JTC’s, with signal-like noise (ϕ = ψ). The solid curve is from theory; circles refer to simulation results. The performances are indistinguishable.

Fig. 7
Fig. 7

Analytical and simulated performance in terms of the SNRpnf of the BSV JTC plotted against the input SNR. The noise bandwidth parameter is W = 4. The solid curve is from theory; circles refer to simulation results.

Fig. 8
Fig. 8

Analytical and simulated performance in terms of the SNRpnf of the BSV JTC plotted against the noise bandwidth parameter. The input SNR is θ2 = 0.25. The solid curve is from theory; circles refer to simulation results.

Fig. 9
Fig. 9

Analytical and simulated performance of the classic (linear) JTC, with noise bandwidth parameter W = 4, plotted against the input SNR (θ2). Solid curve, theoretical SNRpnf. Circles, SNRpnf from simulation. Dashed curve, theoretical SNRppd. Crosses (×), SNRppd from simulation.

Fig. 10
Fig. 10

Analytical and simulated performance of the classic (linear) JTC, with input SNR = 0.25 (θ = 0.5), plotted against noise bandwidth parameter W. Solid curve, theoretical SNRpnf. Circles, SNRpnf from simulation. Dashed curve, theoretical SNRppd. Crosses (×), SNRppd from simulation.

Fig. 11
Fig. 11

Analytical and simulated performance of the BSV JTC plotted against the input SNR. The noise bandwidth parameter is W = 4. Solid curve, theoretical SNRpnf. Circles, SNRpnf from simulation. Dashed curve, theoretical SNRppd. Crosses (×), SNRppd from simulation.

Fig. 12
Fig. 12

Analytical and simulated performance of the BSV JTC plotted against the noise bandwidth parameter. The input SNR is θ2 = 0.25. Solid curve, theoretical SNRpnf. Circles, SNRpnf from simulation. Dashed curve, theoretical SNRppd. Crosses (×), SNRppd from simulation.

Fig. 13
Fig. 13

Examples of k-law nonlinearities used. The threshold [see Eq. (59)] is in this case a = 1.5.

Fig. 14
Fig. 14

Analytical and simulated performance of the k-law JTC plotted against k. The input SNR is θ2 = 0.25, and the relative noise bandwidth is W = 4. Solid curve, theoretical SNRpnf. Circles, SNRpnf from simulation. Dashed curve, theoretical SNRppd. Crosses (×), SNRppd from simulation.

Fig. 15
Fig. 15

Analytical performance surface of the k-law nonlinear JTC in terms of the SNRpnf plotted as a function of k (the power law) and W (the relative noise bandwidth). A large W means that the noise is more white than the signal.

Fig. 16
Fig. 16

Analytical performance surface of the k-law nonlinear JTC in terms of the SNRppd plotted as a function of k (the power law) and W (the relative noise bandwidth). A large W means that the noise is more white than the signal.

Equations (60)

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

a ( x ,   y ) = s ( x ,   y ) + r ( x - x 0 ,   y ) ,
s ( x ,   y ) = r ( x + x 0 ,   y ) + ν ( x ,   y ) .
A ( α ,   β ) = x = 0 N - 1 y = 0 N - 1   a ( x ,   y ) exp [ - j 2 π ( α x + β y ) / N ] .
h x ˆ ,   y ˆ = α = 0 N - 1 β = 0 N - 1   g α , β B α ,   β × exp j 2 π x ˆ α + y ˆ β / N 2 ,
SNR pnf = { h ( 2 x 0 ,   0 ) } { h ( x ˆ ,   y ˆ ) } | ( x ˆ , y ˆ ) { ( - 2 x 0 ,   0 ) ,   ( 0 ,   0 ) ,   ( 2 x 0 ,   0 ) } ,
SNR ppd = { h ( 2 x 0 ,   0 ) } ( Var { h ( 2 x 0 ,   0 ) } ) 1 / 2 ,
a x ,   y = ν x ,   y + r x + x 0 ,   y + r x - x 0 ,   y ,
A α ,   β = N α ,   β + R α ,   β exp j 2 π α x 0 / N + R α ,   β exp - j 2 π α x 0 / N ,
R α ,   β = θ ϕ α ,   β 1 / 2 exp j Φ r α ,   β .
N α ,   β = z α ,   β ψ α ,   β 1 / 2 exp j Φ n α ,   β ,
a θ x ,   y = ν x ,   y + r x + x 0 ,   y + r x - x 0 ,   y .
A θ α ,   β = N α ,   β + R α ,   β exp 2 π j α x 0 / N + R α ,   β exp - j 2 π α x 0 / N ,
B θ = z 2 ψ + 2 θ 2 ϕ 1 + cos 4 π α x 0 / N + 4 θ z ϕ ψ cos Φ r - Φ n cos 2 π α x 0 / N ,
B 0 = z 2 ψ ,   B ˙ 0 = 4 z ϕ ψ cos Φ r - Φ n cos 2 π α x 0 / N ,   B ¨ 0 = 4 ϕ 1 + cos 4 π α x 0 / N .
h x ˆ ,   y ˆ = | IDFT { g α , β ( B θ ) } | 2 ,
{ h ( x ˆ ,   y ˆ ) } = α = 0 N - 1 β = 0 N - 1 [ { g α , β 2 ( B θ ) } - ( { g α , β ( B θ ) } ) 2 ] + α = 0 N - 1 β = 0 N - 1   { g α , β ( B θ ) } exp [ j 2 π ( α x ˆ + β y ˆ ) / N ] 2 .
g α , β ( B θ ) g α , β ( B 0 ) + θ B ˙ 0 g ˙ α , β ( B 0 ) + ( 1 / 2 ) θ 2 B ¨ 0 g ˙ α , β ( B 0 ) + ( 1 / 2 ) θ 2 ( B ˙ 0 ) 2 g ¨ α , β ( B 0 )
{ g α , β [ B θ ( α ,   β ) ] } { g } + 2 θ 2 ϕ [ 1 + cos ( 2 α x 0 ) ] × ( { g ˙ } + ψ { z 2 g ¨ } ) ,
{ g α , β 2 ( B θ ) } { g 2 } + 4 θ 2 ϕ [ 1 + cos ( 2 α x 0 ) ] × ( { g g ˙ } + ψ { z 2 g g ¨ } + ψ { ( z g ˙ ) 2 } ) .
{ h n x ˆ ,   y ˆ } = α = 0 N - 1 β = 0 N - 1 [ { g α , β 2 ( B θ ) } - ( { g α , β ( B θ ) } ) 2 ] ,
h n x ˆ ,   y ˆ α = 0 N - 1 β = 0 N - 1 [ { g 2 } + 4 θ 2 ϕ ( g g ˙ + ψ { z 2 g g ¨ } + ψ { z 2 g ˙ 2 } ) - ( { g ˙ } ) 2 - 4 θ 2 ϕ g × ( g ˙ + ψ { z 2 g ¨ } ) ] ,
h s x ˆ ,   y ˆ = α = 0 N - 1 β = 0 N - 1   { g α , β ( B θ ) } exp j 2 π α x ˆ + β y ˆ / N 2 .
h s 2 x 0 ,   0 θ 4 α = 0 N - 1 β = 0 N - 1   ϕ ( { g ˙ } + ψ { z 2 g ¨ } ) 2 .
ψ g ˙ = - g 0 f μ 0 - 0   g μ ψ f ˙ μ ( μ ) d μ , ψ 2 { z 2 g ¨ } = g 0 f μ 0 + 0   g μ ψ [ 2 f ˙ μ ( μ ) + μ f ¨ μ μ ] d μ , ψ g g ˙ = - 1 2   g 2 0 f μ 0 - 1 2 0 × g 2 μ ψ f ˙ μ ( μ ) d μ , ψ 2 { z 2 g g ¨ + z 2 g ˙ 2 } = 1 2   g 2 ( 0 ) f μ ( 0 ) + 1 2 0 g 2 ( μ ψ ) [ 2 f ˙ μ ( μ ) + μ f ¨ μ ( μ ) ] d μ .
SNR pnf = θ 4 α = 0 N - 1 β = 0 N - 1 ϕ α ,   β ψ α ,   β   I 1 α ,   β 2 α = 0 N - 1 β = 0 N - 1 I 2 α ,   β - I 3 α ,   β 2 + 4 θ 2 ϕ α ,   β ψ α ,   β 1 2   I 4 α ,   β - I 3 α ,   β I 1 α ,   β ,
I 1 ( α ,   β ) = 0   g [ μ ψ ( α ,   β ) ] [ f ˙ μ ( μ ) + μ f ¨ μ ( μ ) ] d μ , I 2 ( α ,   β ) = 0   g 2 [ μ ψ ( α ,   β ) ] f μ ( μ ) d μ , I 3 ( α ,   β ) = 0   g [ μ ψ ( α ,   β ) ] f μ ( μ ) d μ , I 4 ( α ,   β ) = 0   g 2 [ μ ψ ( α ,   β ) ] [ f ˙ μ ( μ ) + μ f ¨ μ ( μ ) ] d μ .
{ h x ˆ ,   y ˆ 2 } = α 1 , β 1 α 2 , β 2 α 3 , β 3 α 4 , β 4   g B θ α 1 ,   β 1 g B θ α 2 ,   β 2 × g B θ α 3 ,   β 3 g B θ α 4 ,   β 4 × exp j 2 π x ˆ α 1 - α 2 + α 3 - α 4 / N + j 2 π y ˆ β 1 - β 2 + β 3 - β 4 / N = α 1 , β 1 α 2 , β 2 α 3 , β 3 α 4 , β 4   g B θ α 1 ,   β 1 g B θ α 2 ,   β 2 × g B θ α 3 ,   β 3 g B θ α 4 ,   β 4 × exp j 2 π x ˆ α 1 - α 2 + α 3 - α 4 / N + j 2 π y ˆ β 1 - β 2 + β 3 - β 4 / N + 4   α 1 , β 1 α 2 , β 2 α 3 , β 3 ( { g 2 β θ α 1 ,   β 1 } - g B θ α 1 ,   β 1 2 ) × g B θ α 2 ,   β 2 g B θ α 3 ,   β 3 × exp j 2 π x ˆ α 2 - α 3 / N + j 2 π y ˆ β 2 - β 3 / N + 2 α 1 , β 1 α 2 , β 2 α 3 , β 3 ( { g 2 β θ α 1 ,   β 1 } - g B θ α 1 ,   β 1 2 ) × g B θ α 2 ,   β 2 g B θ α 3 ,   β 3 × exp j 2 π x ˆ 2 α 1 - α 2 - α 3 / N + j 2 π y ˆ × 2 β 1 - β 2 - β 3 / N + 4 α 1 , β 1 α 2 , β 2 ( { g 3 B θ α 1 ,   β 1 } - g B θ α 1 ,   β 1 3 ) × g B θ α 2 ,   β 2 × exp j 2 π x ˆ α 1 - α 2 / N + j 2 π y ˆ β 1 - β 2 / N + α 1 , β 1 α 2 , β 2 ( { g 2 B θ α 1 ,   β 1 } { g 2 B θ α 2 ,   β 2 } - g B θ α 1 ,   β 1 2 g B θ α 2 ,   β 2 2 ) × exp j 4 π x ˆ α 1 - α 2 / N + j 4 π y ˆ β 1 - β 2 / N + 2   α 1 , β 1 α 2 , β 2 ( { g 2 B θ α 1 ,   β 1 } { g 2 B θ α 2 ,   β 2 } - g B θ α 1 ,   β 1 2 g B θ α 2 ,   β 2 2 ) + α 1 , β 1 ( { g 4 B θ α 1 ,   β 1 } - g B θ α 1 ,   β 1 4 ) ,
{ g α , β B θ α ,   β } g + 2 θ 2 ϕ 1 + cos 4 π α x 0 / N × ( g ˙ + ψ { z 2 g ¨ } ) ,
{ g α , β 2 B θ α ,   β } { g 2 } + 4 θ 2 ϕ 1 + cos 4 π α x 0 / N × ( { g g ˙ } + ψ { z 2 g g ¨ } + ψ { z g ˙ 2 } ) ,
{ g α , β 3 B θ α ,   β } { g 3 } + 6 θ 2 ϕ 1 + cos 4 π α x 0 / N × ( { g 2 g ˙ } + ψ { z 2 g 2 g ¨ } + ψ { z 2 g g ˙ 2 } ) ,
Var h 2 x 0 ,   0 = { h 2 2 x 0 ,   0 } - h 2 x 0 ,   0 2 = 4   α 1 , β 1 α 2 , β 2 α 3 , β 3 ( { g 2 B θ α 1 ,   β 1 } - g B θ α 1 ,   β 1 2 ) × g B θ α 2 ,   β 2 g B θ α 3 ,   β 3 × exp j 4 π x 0 α 2 - α 3 / N + 4 α 1 , β 1 α 2 , β 2 ( { g 3 B θ α 1 ,   β 1 } - g B θ α 1 ,   β 1 3 ) × g B θ α 2 ,   β 2 exp j 4 π x 0 α 1 - α 2 / N ,
κ = α 1 , β 1 ( g 3 B θ α 1 ,   β 1 - g B θ α 1 ,   β 1 3 ) cos 4 π x 0 α 1 / N .
κ = θ 2 α = 0 N - 1 β = 0 N - 1 ϕ α ,   β ψ α ,   β I 5 α ,   β - I 3 α ,   β 2 I 1 α ,   β ,
h s 2 x 0 ,   0 = θ 4 α = 0 N - 1 β = 0 N - 1 ϕ α ,   β ψ α ,   β   I 1 α ,   β 2 ,
h n x ˆ ,   y ˆ = α = 0 N - 1 β = 0 N - 1 I 2 α ,   β - I 3 α ,   β 2 , + 4 θ 2 ϕ α ,   β ψ α ,   β × 1 2   I 4 α ,   β - I 3 α ,   β I 1 α ,   β ,
I 5 α ,   β = 0   g 3 μ ψ α ,   β [ f ˙ μ ( μ ) + μ f ¨ μ ( μ ) ] d μ
SNR ppd = h s 2 x 0 ,   0 Var h 2 x 0 ,   0 1 / 2 = 1 2   θ 2 α = 0 N - 1 β = 0 N - 1 ϕ ψ   I 1 α = 0 N - 1 β = 0 N - 1 I 2 - I 3 2 + 4 θ 2 ϕ ψ 1 2   I 4 - I 3 I 1 + α = 0 N - 1 β = 0 N - 1 ϕ ψ I 5 - I 3 2 I 1 α = 0 N - 1 β = 0 N - 1 ϕ ψ   I 1 1 / 2 ,
f μ ( μ ) = exp ( - μ ) u ( μ ) ,
ψ α ,   β = κ 1 + α 2 + β 2 WW 0 2 ,
α = 0 N - 1 β = 0 N - 1 α 2 + β 2 ϕ α ,   β α = 0 N - 1 β = 0 N - 1   ϕ α ,   β = α = 0 N - 1 β = 0 N - 1 α 2 + β 2 ψ α ,   β α = 0 N - 1 β = 0 N - 1   ψ α ,   β ,
g μ ψ = μ γ
SNR pnf θ 4 α = 0 N - 1 β = 0 N - 1 ϕ ψ 2 1 + 4 θ 2 ϕ ψ ,
γ = κ   ϕ ψ 1 + 4 θ 2 ϕ ψ
g α , β x = x ϕ α ,   β ψ α ,   β ψ α ,   β + 4 θ 2 ϕ α ,   β .
SNR pnf = θ 4 α = 0 N - 1 β = 0 N - 1   ϕ α ,   β 2 α = 0 N - 1 β = 0 N - 1 ψ 2 α ,   β + 4 θ 2 ϕ α ,   β ψ α ,   β ,
SNR pnf α = 0 N - 1 β = 0 N - 1 ϕ α ,   β 2 ψ α ,   β 2 + 4 θ 2 ϕ α ,   β ψ α ,   β ,
SNR pnf α = 0 N - 1 β = 0 N - 1 0 ϕ ψ μ - 1 2 1 + 2 θ 2 ϕ ψ μ - 1 d μ ,
g α ,   β x = ϕ α ,   β ψ α ,   β x ψ α ,   β - 1 1 + 2 θ 2 ϕ α ,   β ψ α ,   β x ψ α ,   β - 1 .
g α , β ( x ) = u x - τ γ α ,   β ,
I 1 α ,   β = 0   u μ ψ - τ γ μ - 1 exp ( - μ ) d μ = τ γ ψ exp ( - τ γ / ψ ) ,
SNR pnf = θ 4 α = 0 N - 1 β = 0 N - 1 ϕ ψ τ γ ψ exp - τ γ / ψ 2 α = 0 N - 1 β = 0 N - 1 exp - τ γ / ψ - exp - 2 τ γ / ψ + 2 θ 2 ϕ ψ τ ϕ ψ exp - τ γ / ψ - 2   exp - τ γ / ψ .
g μ ψ = μ γ
γ     ϕ ψ 1 + 4 θ 2 ϕ ψ ,
g α , β x = x   ϕ α ,   β ψ α ,   β ψ α ,   β + 4 θ 2 ϕ α ,   β .
SNR ppd = 1 2   θ 2 α = 0 N - 1 β = 0 N - 1 ϕ γ ψ α = 0 N - 1 β = 0 N - 1 γ 2 + 4 θ 2 ϕ γ 2 ψ + α = 0 N - 1 β = 0 N - 1 ϕ γ 3 ψ α = 0 N - 1 β = 0 N - 1 ϕ γ ψ 1 / 2 ,
SNR ppd = 1 2   θ 2 α = 0 N - 1 β = 0 N - 1   ϕ α = 0 N - 1 β = 0 N - 1 ψ 2 + 4 θ 2 ϕ ψ + α = 0 N - 1 β = 0 N - 1   ϕ ψ 2 α = 0 N - 1 β = 0 N - 1   ϕ 1 / 2 .
g α , β x = u x - τ γ α ,   β ,
I 1 α ,   β = 0   u μ ψ - τ γ μ - 1 exp ( - μ ) d μ = τ γ ψ exp ( - τ γ / ψ ) ,
SNR ppd = 1 2   θ 2 α = 0 N - 1 β = 0 N - 1 τ ϕ γ ψ 2 exp   - τ γ ψ × α = 0 N - 1 β = 0 N - 1 exp - τ γ ψ - exp   - 2 τ γ ψ + 4 θ 2 τ ϕ γ ψ 2 1 2 exp - τ γ ψ - exp - 2 τ γ ψ + α = 0 N - 1 β = 0 N - 1 τ γ ψ exp - τ γ ψ - exp - 2 τ γ ψ α = 0 N - 1 β = 0 N - 1 τ ϕ γ ψ 2 exp - τ γ ψ - 1 / 2 .
g α , β x = c 1 sgn x - a α ,   β | x - a α ,   β | k + c 2 ,

Metrics