Abstract

A rather large class of problems involving the determination of an object function from observation is linear-inversion problems for which unique solutions exist but that have the property that any signal-processing algorithm designed to approximate the exact solution too precisely is unstable. This is because the problems are ill posed. The precision attainable in a class of such problems is treated here abstractly in terms of a concept called a linear-precision gauge, which essentially involves an ordered family of linear estimators. Fundamental properties of linear-precision gauges are demonstrated and discussed. A major portion of the paper is given over to applying the linear-precision gauge concept and results to Fourier imaging problems that can occur, for example, in radar and tomography.

© 1984 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. L. Walker, “Range-Doppler imaging of rotating objects,” IEEE Trans. Aero. Elec. Systems. AES-16, 23–52 (1980).
    [Crossref]
  2. D. C. Munson, W. Jenkins, “A common framework for synthetic aperture radar and computer-aided tomography,” presented at the Fifteenth Asilomar Conference on Circuits, Systems and Computers, Pacific Grove, Calif., 1982.
  3. M. D. Pollak, D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty, I,” Bell Syst. Tech. J. 40, 43–64 (1961).
  4. R. W. Gerchberg, “Superresolution through error energy reduction,” Opt. Acta 21, 709–720 (1975).
    [Crossref]
  5. A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
    [Crossref]
  6. M. S. Sabri, W. Steenart, “An approach to band-limited signal extrapolation: the extrapolation matrix,” IEEE Trans. Circuits Syst. CAS-25, 74–78 (1978).
    [Crossref]
  7. J. A. Cadzow, “An extrapolation approach for band-limited signals,” IEEE Trans. Acoust. Speech Signal Proc. ASSP-27, 4–12 (1979).
    [Crossref]
  8. A. K. Jain, S. Ranganath, “Extrapolation algorithms for discrete signals with applications in spectral estimation,” IEEE Trans. Acoust. Speech Signal Proc. ASSP-21, 830–845 (1981).
    [Crossref]
  9. D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–701 (1978).
    [Crossref]
  10. L. S. Joyce, W. L. Root, “Notes on resolution enhancement, Fourier transform inversion and spectral estimation,” (1981).
  11. F. J. Beutler, W. L. Root, “The operator pseudoinverse in control and systems identification,” in Generalized Inverse and Applications, Z. Nashed, ed. (Academic, New York, 1976).
  12. E. K. Blum, Numerical Analysis and Computation: Theory and Practice (Addison-Wesley, Reading, Mass., 1972).
  13. In dynamical problems, the introduction of even a bad guess for a prior distribution, as, e.g., the initial distribution of state in a Kalman filter, often causes no trouble because its effect is swamped out by a long sequence of observed data. That is not the case here.
  14. A. Albert, Regression and the Moore–Penrose Pseunverse (Academic, New York, 1972).
  15. W. L. Root, “On the modelling and estimation of communication channels,” in Multivariate Analysis III, R. Krishnaiah, ed. (Academic, New York, 1973), pp. 61–78.
  16. A. W. Naylor, G. R. Sell, Linear Operator Theory (Holt, Rinehart and Winston, New York, 1971).
  17. N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators in Hilbert Space, translated by M. Nestell (Ungar, New York, 1961), Vol. 1.
  18. F. Riesz, B. Sz. Nagy, Functional Analysis, translated by F. Boron (Ungar, New York, 1951), Vol. 1.
  19. N. Berman, W. L. Root, “A weak stochastic integral in Banach space with application to a linear stochastic differential equation,” Appl. Math. Opt. 10, 97–125 (1983).
    [Crossref]
  20. A. Bensoussan, Filtrage Optimal des Systemes Lineaires (Dunod, Paris, 1971).
  21. W. Rudin, Real and Complex Analysis, 2nd ed. (McGraw-Hill, New York, 1974).

1983 (1)

N. Berman, W. L. Root, “A weak stochastic integral in Banach space with application to a linear stochastic differential equation,” Appl. Math. Opt. 10, 97–125 (1983).
[Crossref]

1981 (1)

A. K. Jain, S. Ranganath, “Extrapolation algorithms for discrete signals with applications in spectral estimation,” IEEE Trans. Acoust. Speech Signal Proc. ASSP-21, 830–845 (1981).
[Crossref]

1980 (1)

J. L. Walker, “Range-Doppler imaging of rotating objects,” IEEE Trans. Aero. Elec. Systems. AES-16, 23–52 (1980).
[Crossref]

1979 (1)

J. A. Cadzow, “An extrapolation approach for band-limited signals,” IEEE Trans. Acoust. Speech Signal Proc. ASSP-27, 4–12 (1979).
[Crossref]

1978 (2)

M. S. Sabri, W. Steenart, “An approach to band-limited signal extrapolation: the extrapolation matrix,” IEEE Trans. Circuits Syst. CAS-25, 74–78 (1978).
[Crossref]

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–701 (1978).
[Crossref]

1975 (2)

R. W. Gerchberg, “Superresolution through error energy reduction,” Opt. Acta 21, 709–720 (1975).
[Crossref]

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[Crossref]

1961 (1)

M. D. Pollak, D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty, I,” Bell Syst. Tech. J. 40, 43–64 (1961).

Akhiezer, N. I.

N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators in Hilbert Space, translated by M. Nestell (Ungar, New York, 1961), Vol. 1.

Albert, A.

A. Albert, Regression and the Moore–Penrose Pseunverse (Academic, New York, 1972).

Bensoussan, A.

A. Bensoussan, Filtrage Optimal des Systemes Lineaires (Dunod, Paris, 1971).

Berman, N.

N. Berman, W. L. Root, “A weak stochastic integral in Banach space with application to a linear stochastic differential equation,” Appl. Math. Opt. 10, 97–125 (1983).
[Crossref]

Beutler, F. J.

F. J. Beutler, W. L. Root, “The operator pseudoinverse in control and systems identification,” in Generalized Inverse and Applications, Z. Nashed, ed. (Academic, New York, 1976).

Blum, E. K.

E. K. Blum, Numerical Analysis and Computation: Theory and Practice (Addison-Wesley, Reading, Mass., 1972).

Cadzow, J. A.

J. A. Cadzow, “An extrapolation approach for band-limited signals,” IEEE Trans. Acoust. Speech Signal Proc. ASSP-27, 4–12 (1979).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg, “Superresolution through error energy reduction,” Opt. Acta 21, 709–720 (1975).
[Crossref]

Glazman, I. M.

N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators in Hilbert Space, translated by M. Nestell (Ungar, New York, 1961), Vol. 1.

Jain, A. K.

A. K. Jain, S. Ranganath, “Extrapolation algorithms for discrete signals with applications in spectral estimation,” IEEE Trans. Acoust. Speech Signal Proc. ASSP-21, 830–845 (1981).
[Crossref]

Jenkins, W.

D. C. Munson, W. Jenkins, “A common framework for synthetic aperture radar and computer-aided tomography,” presented at the Fifteenth Asilomar Conference on Circuits, Systems and Computers, Pacific Grove, Calif., 1982.

Joyce, L. S.

L. S. Joyce, W. L. Root, “Notes on resolution enhancement, Fourier transform inversion and spectral estimation,” (1981).

Munson, D. C.

D. C. Munson, W. Jenkins, “A common framework for synthetic aperture radar and computer-aided tomography,” presented at the Fifteenth Asilomar Conference on Circuits, Systems and Computers, Pacific Grove, Calif., 1982.

Nagy, B. Sz.

F. Riesz, B. Sz. Nagy, Functional Analysis, translated by F. Boron (Ungar, New York, 1951), Vol. 1.

Naylor, A. W.

A. W. Naylor, G. R. Sell, Linear Operator Theory (Holt, Rinehart and Winston, New York, 1971).

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[Crossref]

Pollak, M. D.

M. D. Pollak, D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty, I,” Bell Syst. Tech. J. 40, 43–64 (1961).

Ranganath, S.

A. K. Jain, S. Ranganath, “Extrapolation algorithms for discrete signals with applications in spectral estimation,” IEEE Trans. Acoust. Speech Signal Proc. ASSP-21, 830–845 (1981).
[Crossref]

Riesz, F.

F. Riesz, B. Sz. Nagy, Functional Analysis, translated by F. Boron (Ungar, New York, 1951), Vol. 1.

Root, W. L.

N. Berman, W. L. Root, “A weak stochastic integral in Banach space with application to a linear stochastic differential equation,” Appl. Math. Opt. 10, 97–125 (1983).
[Crossref]

W. L. Root, “On the modelling and estimation of communication channels,” in Multivariate Analysis III, R. Krishnaiah, ed. (Academic, New York, 1973), pp. 61–78.

L. S. Joyce, W. L. Root, “Notes on resolution enhancement, Fourier transform inversion and spectral estimation,” (1981).

F. J. Beutler, W. L. Root, “The operator pseudoinverse in control and systems identification,” in Generalized Inverse and Applications, Z. Nashed, ed. (Academic, New York, 1976).

Rudin, W.

W. Rudin, Real and Complex Analysis, 2nd ed. (McGraw-Hill, New York, 1974).

Sabri, M. S.

M. S. Sabri, W. Steenart, “An approach to band-limited signal extrapolation: the extrapolation matrix,” IEEE Trans. Circuits Syst. CAS-25, 74–78 (1978).
[Crossref]

Sell, G. R.

A. W. Naylor, G. R. Sell, Linear Operator Theory (Holt, Rinehart and Winston, New York, 1971).

Slepian, D.

M. D. Pollak, D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty, I,” Bell Syst. Tech. J. 40, 43–64 (1961).

Steenart, W.

M. S. Sabri, W. Steenart, “An approach to band-limited signal extrapolation: the extrapolation matrix,” IEEE Trans. Circuits Syst. CAS-25, 74–78 (1978).
[Crossref]

Walker, J. L.

J. L. Walker, “Range-Doppler imaging of rotating objects,” IEEE Trans. Aero. Elec. Systems. AES-16, 23–52 (1980).
[Crossref]

Youla, D. C.

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–701 (1978).
[Crossref]

Appl. Math. Opt. (1)

N. Berman, W. L. Root, “A weak stochastic integral in Banach space with application to a linear stochastic differential equation,” Appl. Math. Opt. 10, 97–125 (1983).
[Crossref]

Bell Syst. Tech. J. (1)

M. D. Pollak, D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty, I,” Bell Syst. Tech. J. 40, 43–64 (1961).

IEEE Trans. Acoust. Speech Signal Proc. (2)

J. A. Cadzow, “An extrapolation approach for band-limited signals,” IEEE Trans. Acoust. Speech Signal Proc. ASSP-27, 4–12 (1979).
[Crossref]

A. K. Jain, S. Ranganath, “Extrapolation algorithms for discrete signals with applications in spectral estimation,” IEEE Trans. Acoust. Speech Signal Proc. ASSP-21, 830–845 (1981).
[Crossref]

IEEE Trans. Aero. Elec. Systems. (1)

J. L. Walker, “Range-Doppler imaging of rotating objects,” IEEE Trans. Aero. Elec. Systems. AES-16, 23–52 (1980).
[Crossref]

IEEE Trans. Circuits Syst. (3)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[Crossref]

M. S. Sabri, W. Steenart, “An approach to band-limited signal extrapolation: the extrapolation matrix,” IEEE Trans. Circuits Syst. CAS-25, 74–78 (1978).
[Crossref]

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–701 (1978).
[Crossref]

Opt. Acta (1)

R. W. Gerchberg, “Superresolution through error energy reduction,” Opt. Acta 21, 709–720 (1975).
[Crossref]

Other (12)

D. C. Munson, W. Jenkins, “A common framework for synthetic aperture radar and computer-aided tomography,” presented at the Fifteenth Asilomar Conference on Circuits, Systems and Computers, Pacific Grove, Calif., 1982.

A. Bensoussan, Filtrage Optimal des Systemes Lineaires (Dunod, Paris, 1971).

W. Rudin, Real and Complex Analysis, 2nd ed. (McGraw-Hill, New York, 1974).

L. S. Joyce, W. L. Root, “Notes on resolution enhancement, Fourier transform inversion and spectral estimation,” (1981).

F. J. Beutler, W. L. Root, “The operator pseudoinverse in control and systems identification,” in Generalized Inverse and Applications, Z. Nashed, ed. (Academic, New York, 1976).

E. K. Blum, Numerical Analysis and Computation: Theory and Practice (Addison-Wesley, Reading, Mass., 1972).

In dynamical problems, the introduction of even a bad guess for a prior distribution, as, e.g., the initial distribution of state in a Kalman filter, often causes no trouble because its effect is swamped out by a long sequence of observed data. That is not the case here.

A. Albert, Regression and the Moore–Penrose Pseunverse (Academic, New York, 1972).

W. L. Root, “On the modelling and estimation of communication channels,” in Multivariate Analysis III, R. Krishnaiah, ed. (Academic, New York, 1973), pp. 61–78.

A. W. Naylor, G. R. Sell, Linear Operator Theory (Holt, Rinehart and Winston, New York, 1971).

N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators in Hilbert Space, translated by M. Nestell (Ungar, New York, 1961), Vol. 1.

F. Riesz, B. Sz. Nagy, Functional Analysis, translated by F. Boron (Ungar, New York, 1951), Vol. 1.

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 (2)

Fig. 1
Fig. 1

Relation between the dFt and the Nyquist-rate-sampled, continuous transform.

Fig. 2
Fig. 2

Relation between the truncated dFt and the oversampled, continuous transform.

Tables (1)

Tables Icon

Table 1 Numerical Behavior of the Lower Bound of the Error Numbers eνν [Eq. (4.25)]

Equations (157)

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

s ( η ) = E exp [ - j 2 π ( x , η ) ] σ ( x ) d x ,
( x , η ) = i = 1 n x i η i .
s F ( η ) = { s ( η ) for η F 0 otherwise
E = { - a 1 x 1 a 1 , - a 2 x 2 a 2 } F = { - b 1 η b 1 , - b 2 η 2 b 2 } .
s F ( η 1 , η 2 ) = - a 1 a 1 - a 2 a 2 exp [ - j 2 π ( x 1 η 1 + x 2 η 2 ) ] × σ ( x 1 , x 2 ) d x d x 2 ,
z ( η 1 , η 2 ) = s F ( η 1 , η 2 ) + w ( η 1 , η 2 ) .
q m 1 m 2 s F ( m 1 2 a 1 , m 2 2 a 2 ) ,
μ 1 4 a 1 b 1 , μ 2 4 a 2 b 2 .
r k 1 k 2 = 1 4 a 1 a 2 m 1 = - K 1 K 1 - 1 m 2 = - K 2 K 2 - 1 q m 1 m 2 exp [ j 2 π ( k 1 m 1 μ 1 + k 2 m 2 μ 2 ) ] ,
q m 1 m 2 = 1 4 b 1 b 2 k 1 = - K 1 K 1 - 1 k 2 = - K 2 K 2 - 1 r k 1 k 2 exp [ - j 2 π ( k 1 m 1 μ 2 + k 2 m 2 μ 2 ) ] .
r = D - 1 q ,
q = D r ,
r ˜ = D ˜ - 1 q ˜ ,
q ˜ = D ˜ r ˜
( P v ˜ ) k 1 k 2 = v ˜ k 1 k 2 for - K i k i K i - 1 , i = 1 , 2. = 0 otherwise
q m 1 m 2 s F ( m 1 2 a ˜ 1 , m 2 2 a ˜ 2 ) for m i = - K i , - K i + 1 , , K i - 1 , i = 1 , 2 , 0 for m i = - K ˜ i , , K ˜ i - 1
q = P q ˜ ,
P q ˜ = P D ˜ P r ˜
q = ( P D ˜ P ) r ,
P q ˜ = P D ˜ r ˜
P D ˜ [ I - P ] r ˜ = 0.
q = ( P D ˜ P ) r .
q ˜ = D ˜ r ˜ .
r ˜ = D ˜ - 1 q ˜ .
q = P q ˜ = P D ˜ r ˜ P D ˜ P r ˜ = P D ˜ P r ˜ .
s = B σ ,
z = s + w = B σ + w ,
z ( η ) = E b ( η , x ) σ ( x ) d x + w ( η ) ,             η F ,
F E b ( η , x ) 2 d x d η < ,
z = B σ + w ,
E σ = σ = true value .
E σ ^ - σ 2 E C z - σ 2
E C w 2 E C w 2 .
σ ^ = ( B * Γ - 1 B ) - 1 B * Γ - 1 z .
σ ^ = ( B * B ) - 1 B * z .
E σ - σ ^ 2 = T r [ ( B * Γ B ) - 1 ] ,
E σ - σ ^ 2 = c T r [ ( B * B ) - 1 ] ,
E σ - σ ^ 2 = c k = 1 N μ k - 1 .
E σ - σ ^ 2 = c k = 1 μ k - 1 = + .
P k z = P k B σ + P k w ,
z k P k z = P k B ( Q k σ + Q k σ ) + w k = B σ k + P k B Q k σ + w k ,
z k = B σ k + w k = B k σ k + w k ,
σ ^ k = B k - 1 z k .
e k 1 c d k var σ ^ k = 1 d k T r [ ( B k * B k ) - 1 ] ,
B * B β n = λ n β n , B B * ϕ n = λ n ϕ n , B β n = λ n 1 / 2 ϕ n , B * ϕ n = λ n 1 / 2 β n ,
e k 0 = 1 k n = 1 k 1 λ n
B k - 1 P k B Q k σ B k - 1 Q k σ B k - 1 σ .
z 2 = G z 1 = G B σ + G w .
s ( η ) = E b ( η , x ) σ ( x ) d x ,             η F ,
s ( η ) = I [ - b , b ] ( η ) - a a exp ( - j 2 π η x ) σ ( x ) d x ,
I [ - b , b ] ( η ) = 1 for - b η b = 0 otherwise
σ k ( 1 ) 2 b k / 2 b ( k + 1 ) / 2 b σ ( x ) d x = average of σ over k th cell ,
ρ k ( 1 ) ( x ) σ k ( 1 ) for k / 2 b < x ( k + 1 ) / 2 b 0 otherwise
σ ( 1 ) ( x ) k = - K 1 K 1 - 1 ρ k ( 1 ) ( x )
σ ( 2 ) ( x ) k = - K 2 K 2 - 1 ρ k ( 2 ) ( x )
s ( η ) = I [ - b , b ] ( η ) - a a exp ( - j 2 π x η ) σ ( 1 ) ( x ) d x = 1 2 b exp ( - j π η / 2 b ) sinc ( η / 2 b ) × k = - K 1 K 1 - 1 σ k ( 1 ) exp ( - j 2 π k η / 2 b ) ,             - b η b ,
s ( η ) = 1 2 b n exp ( - j π η / 2 b n ) sinc ( η / 2 b n ) × k = - K n K n - 1 σ k ( n ) exp ( - j 2 π k η / 2 b n ) ,             - b η b ,
s ( m / 2 a ) = 1 2 b exp ( - j π m / μ ) sinc ( m / μ ) × k = - K 1 K 1 - 1 σ k ( 1 ) exp ( - j 2 π k m μ ) .
σ k ( 1 ) = 1 2 a m = - K 1 K 1 - 1 θ ( π m / μ ) s ( m / 2 a ) exp ( + j 2 π k m / μ ) .
σ ( 1 ) 2 = - a a σ ( 1 ) ( x ) 2 d x = 1 2 b k = - K 1 K 1 - 1 σ k ( 1 ) 2 .
s 1 2 = 1 2 a m = - K 1 K 1 - 1 | s ( m 2 a ) | 2 ,
2 / π = sinc ( 1 / 2 ) θ ( π m / μ ) - 1 1             for all m .
z ( m 2 a ) = s ( m 2 a ) + w m ,             - K 1 m K 1 - 1 ,
e 1 ( 1 ) = 1 d ( N 1 ) Tr { [ ( G 1 B 1 ) * ( G 1 B 1 ) ] - 1 } = 1 d 1 Tr { [ ( A 1 D 1 ) * ( A 1 D 1 ) ] - 1 } = 1 d 1 Tr { [ D 1 * A 1 * A 1 D 1 ] - 1 } = 1 d 1 Tr { D 1 * [ A 1 * A 1 ] - 1 D 1 }
I ( A 1 * A 1 ) - 1 ( π 2 ) 2 I ,
d 1 = Tr [ D 1 * D ] Tr [ D 1 * ( A 1 * A 1 ) - 1 D 1 ] < π 2 4 Tr [ D 1 * D 1 ] ,
1 e 1 ( 1 ) π 2 / 4.
z ( η 0 ) 1 Δ η 0 - Δ / 2 η 0 + Δ / 2 s ( η ) d η + ( noise random variable )
( noise random variable ) = w ( η 0 ) = 1 Δ η 0 - Δ / 2 η 0 + Δ / 2 w ( η ) d η ,
E [ w ( η 0 ) ] 2 = N 0 / Δ ,
q m ( ν ) θ ( π m / μ ν ) s ( m / 2 a ν ) = 1 2 b ν k = - K ν K ν - 1 σ k ( ν ) × exp ( - j 2 π k m / μ ν ) ,             m = - K ν , , K ν - 1.
Δ ν = P ˜ ν D ν P ˜ ν ,
σ ( ν ) 2 = - a a σ ( ν ) ( x ) 2 d x = 1 2 b ν k = - K ν K ν - 1 σ k ( ν ) 2 .
s ν 2 = 1 2 a m = - K ν K ν - 1 | s ( m 2 a ν ) | 2 .
e ν ( ν ) = 1 d ( N ν ) T r { [ G ν B ν ) * ( G ν B ν ) ] - 1 } = 1 d ν T r { [ ( A ν Δ ν ) * ( A ν Δ ν ) ] - 1 } = 1 2 ν - 1 μ T r ( [ Δ ν * A ν * A ν Δ ν ] - 1 ) = 1 2 ν - 1 μ T r ( [ D ν * P ˜ ν A ν * A ν P ˜ ν D ν ] r - 1 ) ,
γ ν 2 I < A ν * A ν I ,
γ ν = sinc ( K ν / μ ν ) > sinc | 1 2 ( 2 ν - 1 ) | .
γ ν 2 D ν * P ˜ ν D ν < D ν * P ˜ ν A ν * A ν P ˜ ν D ν D ν * P ˜ ν D ν ,
1 2 ν - 1 μ Tr ( [ D ν * P ˜ r D r ] - 1 ) e ν ( ν ) < 1 2 ν - 1 μ γ ν 2 Tr ( [ D ν * P ˜ ν D ν ] - 1 ) .
1 2 ν - 1 μ Tr ( [ D ν * P ˜ ν D ν ] r - 1 ) = 1 μ ν Tr ( [ D ν - 1 P ˜ ν D ν ] r - 1 ) .
ρ ( n - m ) = 1 μ ν exp [ j π ( n - m ) / μ ν ] sin [ π ( n - m ) d ν / μ ν ] sin [ π ( n - m ) / μ ν ] ,
ρ ( n - m ) = 1 μ ν sin [ π ( n - m ) / 2 ν - 1 ] sin [ π ( n - m ) / μ ν ] ,
[ z 0 z 1 ] = [ A ν ( 0 ) Δ n 0 0 A ν ( 1 ) Δ n ] [ σ ( n ) U 1 σ ( n ) ] + [ w 0 w 1 ] = [ A ν ( 0 ) Δ n A ν ( 1 ) Δ n U 1 ] [ σ ( n ) ] + [ w 0 w 1 ] .
z = [ z 0 z 1 ] = [ B 0 B 1 ] [ σ ( n ) ] + [ w 1 w 2 ] = B ˜ σ ( n ) + w ,
σ ^ ( n ) = [ B ˜ * B ˜ ] - 1 B ˜ * z = [ B 0 * B 0 + B 1 * B 1 ] - 1 [ B 0 * B 1 * ] z .
E σ ( n ) - σ ^ ( n ) 2 = c T r ( [ B 0 * B 0 + B 1 * B 1 ] - 1 ) .
σ ^ 0 = σ ( n ) + B 0 - 1 w 0 , σ ^ 1 = σ ( n ) + B 1 - 1 w 1 .
σ ^ ( n ) = 1 2 ( B 0 - 1 z 0 + B 1 - 1 z 1 ) = σ ( n ) + 1 2 ( B 0 - 1 w 0 + B 1 - 1 w 1 ) .
v = 1 2 ( B 0 - 1 w 0 + U 1 * B 0 - 1 w 1 )
E ( vv * ) = c 4 [ ( B 0 * B 0 ) - 1 + U 1 * ( B 0 * B 0 ) - 1 U 1 ] .
E σ ( n ) - σ ^ ( n ) 2 = Tr [ E ( vv * ) ] = 1 2 Tr [ ( B 0 * B 0 ) - 1 ] .
e n ( n + 1 ) 1 2 e n ( n ) ,             ν = n + 1.
e n ( ν ) 1 2 ν - n e n ( n ) .
B 0 * B 0 + B 1 * B 1 = D 1 * A ν * ( 0 ) A ν ( 0 ) D 1 + U * D 1 * A ν * ( 1 ) A ν ( 1 ) D 1 U 1 .
( 2 / π ) 2 I A ν * ( i ) A ν ( i ) I ,             i = 0 , 1 , ( 2 π ) 2 [ D 1 * D 1 + U 1 * D 1 * D 1 U 1 ] B 0 * B 0 + B 1 * B 1 D 1 * D 1 + U 1 * D 1 * D 1 U 1 ,
( 2 π ) 2 2 I B 0 * B 0 + B 1 * B 1 2 I
d n 2 Tr ( [ B 0 * B 0 + B 1 * B 1 ] - 1 ) ( π 2 ) 2 d n 2 ,
1 2 e 1 ( 2 ) 1 2 ( π 2 ) 2 .
1 2 ν - 1 e 1 ( ν ) 1 2 ν - 1 ( π 2 ) 2 ,
s ( η ) = R n exp [ - j 2 π ( x , η ) ] σ ( x ) d x ,
σ ( x ) = R n exp [ j 2 π ( x , η ) ] s ( η ) d η ,
( x , η ) = i = 1 n x i η i .
R n s ( η ) 2 d η = R n σ ( x ) 2 d x .
r k = 1 c m = - K μ - 1 - K q m exp ( j 2 π m k / μ ) ,
q m = c μ k = - K μ - 1 - K r k exp ( - j 2 π m k / μ ) .
m = - K μ - 1 - K q m 2 = c 2 μ k = - K μ - 1 - K r k 2 .
r k 1 k 2 = 1 c 1 c 2 m 1 = - K 1 K 1 - 1 m 2 = - K 2 K 2 - 1 q m 1 m 2 × exp [ j 2 π ( m 1 k 1 + m 2 k 2 ) / μ 1 μ 2 ] , q m 1 m 2 = c 1 c 2 μ 1 μ 2 k 1 = - K 1 K 1 - 1 k 2 = - K 2 K 2 - 1 r k 1 k 2 × exp [ - j 2 π ( m 1 k 1 + m 2 k 2 ) / μ 1 μ 2 ] .
m 1 , m 2 q m 1 m 2 2 = c 1 2 c 2 2 μ 1 μ 2 k 1 , k 2 r k 1 k 2 2 .
s F ( η ) = s ( η ) - b η b = 0 otherwise .
r k = 1 2 a m = - K K - 1 q m exp ( j 2 π m k / μ ) ,
q m = 1 2 b k = - K K - 1 r k exp ( - j 2 π m k / μ ) ,
1 2 a m = - K K - 1 q m 2 = 1 2 b k = - K K - 1 r k 2 .
σ ( x ) = 1 2 a m = - a m exp ( j 2 π m x / 2 a ) ,
a m = - a a σ ( x ) exp ( - j 2 π m x / 2 a ) d x .
σ ( k / 2 b ) = 1 2 a m = - q m exp ( j 2 π m k / μ ) ,
f = n = 1 f , ϕ n ϕ n
lim N f - n = 1 N f , ϕ n ϕ n = 0.
f - k = 1 n ( ) f , ϕ k ϕ k
B B * ϕ n = λ n ϕ n .
( B * B B * ϕ n ) = λ n ( B * ϕ n ) .
f n , f m = B * ϕ n , B * ϕ m = B B * ϕ n , ϕ m = λ n ϕ n , ϕ m = λ n δ n m .
B * B β n = λ n β n , B B * ϕ n = λ n ϕ n ,
B β n = λ n 1 / 2 ϕ n , B * ϕ n = λ n 1 / 2 β n .
R ( B * B ) R ( B * ) R ( B * B ) ¯
R ( B B * ) R ( B ) B B * ) ¯ ,
σ 1 , σ 2 E σ 1 ( x ) σ 2 ( x ) ¯ d x , σ 2 σ , σ = E σ ( x ) 2 d x .
s ( η ) = E b ( η , x ) σ ( x ) d x
F E b ( η , x ) 2 d x d η < .
b ( η , x ) = b ˜ ( η - x ) ,
Z = { u : u = η - x , η F , x E } ,
Z b ˜ ( u ) 2 d u < .
s ( η ) - s ( η + h ) 2 E b ( η , x ) - b ( η + h , x ) 2 d x E σ ( x ) 2 d x = σ 2 E b ( η , x ) - b ( η + h , x ) 2 d x .
E b ( η , x ) - b ( η + h , x ) 2 d x = E b ˜ ( η - x ) - b ˜ ( η + h - x ) 2 d x Z b ˜ ( u ) - b ˜ ( u + h ) 2 d u ,
F E b ˜ ( η - x ) 2 d x d η F Z b ˜ ( u ) 2 d u d η = F d η Z b ˜ ( u ) 2 d u < ,
B * B β n = λ n β n , B B * ϕ n = λ n ϕ n , B β n = λ n 1 / 2 ϕ n , B * ϕ n = λ n 1 / 2 β n ,
B n * B n η k = μ k η k , B n B n * θ k = μ k θ k , B n η k = μ k 1 / 2 θ k , B n * θ k = μ k 1 / 2 η k ,
f = k = 1 d n f , η k η k .
min f = 1 B n f = B n η d n = μ d n 1 / 2 .
η 1 = a 11 β 1 + a 12 β 2 + + a 1 d n β d n + a 1 ( d n + 1 ) β d n + 1 + , η 2 = a 21 β 1 + a 22 β 2 + , η d n = a d n 1 β 1 + a d n 2 β 2 +
min f = 1 B n f = μ d n 1 / 2 ,
y = k = d n c k β k , k = d n c k 2 = 1 ,
B n y = B y = k = d n c k B β k = k = d n c k λ k 1 / 2 ϕ k
B n y 2 = k = d n c k λ k 1 / 2 2 λ d n .
e n = 1 d n Tr [ ( B n * B n ) - 1 ] = 1 d n k = 1 d n 1 μ k
Q n y , z = B * P n x , Q n z = x , P n B Q n z = 0.
u = u 1 + u 2 ,
0 = u 2 , B * P n x = P n B u 2 , x .
B n * B n f = Q n B * P n P n B Q n f = B * B f ,
s ( η ) = I F ( η ) E exp ( - j 2 π x η ) σ ( x ) d x ,
σ ( x ) = I E ( x ) F exp ( j 2 π x η ) s ( η ) d η .
s ( η ) = a 1 a 2 exp ( - j 2 π x η ) σ ( x ) d x ,
s ( ρ , η ) = 0 a 0 a exp [ - j 2 π ( x ρ + y η ) ] σ ( x , y ) d x d y .
g ( ρ , y ) 0 a exp ( - j 2 π x ρ ) σ ( x , y ) d x
s ( ρ , η ) = 0 a g ( ρ , y ) exp ( - j 2 π y η ) d y .
h ( x ) 0 a σ ( x , y ) 2 d y < for a . e . x
0 a h ( x ) d x < .
0 a g ( ρ , y ) 2 d y = 0 a 0 a 0 a σ ( x , y ) σ ( x y ) × exp [ j 2 π ρ ( x - x ) ] d x d x d y 0 a 0 a 0 a σ ( x , y ) σ ( x , y ) d y d x d x 0 a 0 a ( 0 a σ ( x , y ) 2 d y ) 1 / 2 × ( 0 a σ ( x , y ) 2 d y ) 1 / 2 d x d x = 0 a 0 a h 1 / 2 ( x ) h 1 / 2 ( x ) d x d x = ( 0 a h 1 / 2 ( x ) d x ) 2 a 0 a h ( x ) d x < .

Metrics