Abstract

Shannon’s theory of information for communication channels is used to assess the performance of line-scan and sensor-array imaging systems and to optimize the design trade-offs involving sensitivity, spatial response, and sampling intervals. Formulations and computational evaluations account for spatial responses typical of line-scan and sensor-array mechanisms, lens diffraction and transmittance shading, defocus blur, and square and hexagonal sampling lattices.

© 1984 Optical Society of America

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  1. P. B. Fellgett, E. H. Linfoot, Philos. Trans. R. Soc. London 247, 269 (1955).
    [CrossRef]
  2. E. H. Linfoot, J. Opt. Soc. Am. 45, 808 (1955).
    [CrossRef]
  3. C. Shannon, Bell Syst. Tech. J. 27, 379 (1978); or C. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, 1964).
  4. F. O. Huck, S. K. Park, Appl. Opt. 14, 2508 (1975).
    [CrossRef] [PubMed]
  5. F. O. Huck, N. Halyo, S. K. Park, Appl. Opt. 20, 1990 (1981).
    [CrossRef] [PubMed]
  6. F. O. Huck et al., Opt. Laser Technol. 15, 21 (1982).
    [CrossRef]
  7. O. H. Schade, J. Soc. Motion Pict. Telev. Eng.56, 131 (1951); J. Soc. Motion Pict. Telev. Eng. 58, 181 (1952); J. Soc. Motion Pict. Telev. Eng. 61, 97 (1953); J. Soc. Motion Pict. Telev. Eng. 64, 593 (1955); J. Soc. Motion Pict. Telev. Eng. 73, 81 (1964).
  8. L. M. Biberman, Ed., Perception of Displayed Information (Plenum, New York, 1973).
    [CrossRef]
  9. D. P. Peterson, D. Middleton, Inf. Control 5, 279 (1962).
    [CrossRef]
  10. R. M. Mersereau, Proc. IEEE 67, 930 (1979).
    [CrossRef]
  11. F. O. Huck, N. Halyo, S. K. Park, Appl. Opt. 19, 2174 (1980).
    [CrossRef] [PubMed]
  12. S. K. Park, R. A. Schowengerdt, Appl. Opt. 21, 3142 (1982).
    [CrossRef] [PubMed]
  13. Y. Itakura et al., Infrared Phys. 14, 17 (1974).
    [CrossRef]
  14. H. H. Hopkins, Proc. R. Soc. London 231, 91 (1955).
    [CrossRef]
  15. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).
  16. M. Mino, Y. Okano, Appl. Opt. 10, 2219 (1971). Expressions for p2 and q2 in Eq. (9) contain a typographical error.
    [CrossRef] [PubMed]
  17. Y. L. Lee, Statistical Theory of Communications (John Wiley and Sons, New York, 1964).
  18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  19. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

1982

1981

1980

1979

R. M. Mersereau, Proc. IEEE 67, 930 (1979).
[CrossRef]

1978

C. Shannon, Bell Syst. Tech. J. 27, 379 (1978); or C. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, 1964).

1975

1974

Y. Itakura et al., Infrared Phys. 14, 17 (1974).
[CrossRef]

1971

1962

D. P. Peterson, D. Middleton, Inf. Control 5, 279 (1962).
[CrossRef]

1955

P. B. Fellgett, E. H. Linfoot, Philos. Trans. R. Soc. London 247, 269 (1955).
[CrossRef]

E. H. Linfoot, J. Opt. Soc. Am. 45, 808 (1955).
[CrossRef]

H. H. Hopkins, Proc. R. Soc. London 231, 91 (1955).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

Fellgett, P. B.

P. B. Fellgett, E. H. Linfoot, Philos. Trans. R. Soc. London 247, 269 (1955).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Halyo, N.

Hopkins, H. H.

H. H. Hopkins, Proc. R. Soc. London 231, 91 (1955).
[CrossRef]

Huck, F. O.

Itakura, Y.

Y. Itakura et al., Infrared Phys. 14, 17 (1974).
[CrossRef]

Lee, Y. L.

Y. L. Lee, Statistical Theory of Communications (John Wiley and Sons, New York, 1964).

Linfoot, E. H.

P. B. Fellgett, E. H. Linfoot, Philos. Trans. R. Soc. London 247, 269 (1955).
[CrossRef]

E. H. Linfoot, J. Opt. Soc. Am. 45, 808 (1955).
[CrossRef]

Mersereau, R. M.

R. M. Mersereau, Proc. IEEE 67, 930 (1979).
[CrossRef]

Middleton, D.

D. P. Peterson, D. Middleton, Inf. Control 5, 279 (1962).
[CrossRef]

Mino, M.

Okano, Y.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Park, S. K.

Peterson, D. P.

D. P. Peterson, D. Middleton, Inf. Control 5, 279 (1962).
[CrossRef]

Schade, O. H.

O. H. Schade, J. Soc. Motion Pict. Telev. Eng.56, 131 (1951); J. Soc. Motion Pict. Telev. Eng. 58, 181 (1952); J. Soc. Motion Pict. Telev. Eng. 61, 97 (1953); J. Soc. Motion Pict. Telev. Eng. 64, 593 (1955); J. Soc. Motion Pict. Telev. Eng. 73, 81 (1964).

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Schowengerdt, R. A.

Shannon, C.

C. Shannon, Bell Syst. Tech. J. 27, 379 (1978); or C. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, 1964).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

Appl. Opt.

Bell Syst. Tech. J.

C. Shannon, Bell Syst. Tech. J. 27, 379 (1978); or C. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, 1964).

Inf. Control

D. P. Peterson, D. Middleton, Inf. Control 5, 279 (1962).
[CrossRef]

Infrared Phys.

Y. Itakura et al., Infrared Phys. 14, 17 (1974).
[CrossRef]

J. Opt. Soc. Am.

Opt. Laser Technol.

F. O. Huck et al., Opt. Laser Technol. 15, 21 (1982).
[CrossRef]

Philos. Trans. R. Soc. London

P. B. Fellgett, E. H. Linfoot, Philos. Trans. R. Soc. London 247, 269 (1955).
[CrossRef]

Proc. IEEE

R. M. Mersereau, Proc. IEEE 67, 930 (1979).
[CrossRef]

Proc. R. Soc. London

H. H. Hopkins, Proc. R. Soc. London 231, 91 (1955).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

Y. L. Lee, Statistical Theory of Communications (John Wiley and Sons, New York, 1964).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

O. H. Schade, J. Soc. Motion Pict. Telev. Eng.56, 131 (1951); J. Soc. Motion Pict. Telev. Eng. 58, 181 (1952); J. Soc. Motion Pict. Telev. Eng. 61, 97 (1953); J. Soc. Motion Pict. Telev. Eng. 64, 593 (1955); J. Soc. Motion Pict. Telev. Eng. 73, 81 (1964).

L. M. Biberman, Ed., Perception of Displayed Information (Plenum, New York, 1973).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Model of imaging process.

Fig. 2
Fig. 2

Sampling lattices and passbands. Equal sampling density occurs (i.e., B ^ s = B ^ r) when the sampling intervals Xs of the square lattice and the sampling intervals Xr of the regular hexagonal lattice are such that X s = 3 / 2 X r = 0.93 X r.

Fig. 3
Fig. 3

Wiener spectrum of radiance field.

Fig. 4
Fig. 4

Optical transfer function of diffraction-limited lens with clear and shaded apertures for a coherent cutoff frequency 1/2λF = 1 and several values of defocus u. The effective lens transmittance is k = 0.33 for α = 2, and k = 0.17 for α = 1.

Fig. 5
Fig. 5

Sensor-array patterns. Sensor apertures are of equal area when γ s = 3 / 2 γ r = 0.93 γ r.

Fig. 6
Fig. 6

Spatial response of photosensor apertures and spot intensity profile.

Fig. 7
Fig. 7

Information density hi vs SNR L/σN for sensor-array imaging systems. Results are given for an infinite lens cutoff frequency (i.e., 1/2λF = ∞), contiguous photosensor apertures (i.e., γs = 0.93 γr = 1), and several radiance fields with different mean spatial detail μr.

Fig. 8
Fig. 8

Aliasing σa, blurring σb, and information density hi vs lens coherent cutoff frequency 1/2λF for sensor-array imaging systems. Results are given for a clear lens, contiguous photosensor apertures (i.e:, γs = 0.93 γr = 1), a SNR L/σN = 128, and several radiance fields with different mean spatial details μr.

Fig. 9
Fig. 9

Aliasing σa, blurring σb, and information density hi vs defocus u for sensor-array imaging systems. Results are given for a clear lens with coherent cutoff frequency 1/2λF = 1, contiguous photosensor apertures (i.e., γs = 0.93 γr = 1), a SNR L/σN = 128, and several radiance fields with different mean spatial details μr.

Fig. 10
Fig. 10

Aliasing σa, blurring σb, and information density hi vs sampling intervals Xs = Ys for line-scan imaging systems. Results are given for an infinite lens cutoff frequency (i.e., 1/2λF = ∞), a SNR L/σN = 128, and several radiance fields with different mean spatial details μr.

Fig. 11
Fig. 11

Information density hi vs lens coherent cutoff frequency 1/2λF for sensor-array imaging systems, and relationship between OTF and sampling passband which maximizes information density.

Fig. 12
Fig. 12

Information density hi vs defocus u for sensor-array imaging systems with a clear lens and square array, and relationship between OTF and sampling passband which maximizes information density.

Fig. 13
Fig. 13

Same as Fig. 12 except for a shaded instead of a clear lens.

Fig. 14
Fig. 14

Information density hi and efficiency hi| B ^ |−1 vs sampling interval Xs for line-scan imaging systems, and relationship between OTF and range of sampling passbands which provide a favorable compromise between high information density and efficiency. The lens coherent cutoff frequency is 1/2λF = 1, and the SNR is L/σN = 128. The reconstruction passband r ^ s is equal to the sampling passband B ^ s. (i) for all sampling intervals, and (ii) for all sampling intervals larger than 0.47. The sampling passband exceeds the OTF zero crossing when Xs < 0.47.

Equations (103)

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K = k A l Ω p 0 L ( λ ) τ ( λ ) d λ ,
B ^ s = 1 / X s 2 ,
B ^ r = 2 / 3 X r 2 .
r ^ ( υ , ω ) = s ^ ( υ , ω ) Π ^ ( υ , ω ) ,
Π ^ s ( υ , ω ) = { 1 , υ < 1 2 X s , ω < 1 2 X s , 0 , elsewhere ,
Π ^ r ( υ , ω ) = { 1 , υ , υ 2 + 3 ω 2 < 1 3 X r , 0 , elsewhere .
σ 2 = σ b 2 + ( υ , ω ) B ^ Φ ^ L ( υ , ω ) d υ d ω + σ a 2 + σ N 2 ,
σ b 2 = B ^ Φ ^ L ( υ , ω ) 1 - τ ^ ( υ , ω ) 2 d υ d ω .
σ L 2 = - Φ ^ L ( υ , ω ) d υ d ω .
h i B ^ - 1 = X s 2 h i ,
h i B ^ - 1 = 3 X r 2 h i / 2
Φ L ( x , y ) = σ L 2 exp ( - r / μ r ) ,
Φ ^ L ( υ , ω ) = 2 π μ r 2 σ L 2 [ 1 + ( 2 π μ r ρ ) 2 ] 3 / 2 ,
τ ^ l ( υ , ω ) = - P l ( υ + ρ ¯ / 2 , w ) P l * ( υ - ρ ¯ / 2 , w ) exp ( i u ρ ¯ υ ) d υ d w - P l ( υ , w ) P l * ( υ , w ) d υ d w ,
u π 2 λ Δ l ( D l i ) 2 π Δ l 2 λ F 2 ,
T ( v , w ) = 1 - ( v 2 + w 2 ) α / 2
k = 2 0 1 z T ( z ) 2 d z ,
τ s ( x , y ) = { 1 γ s 2 , x < γ s 2 , y < γ s 2 , 0 ,             elsewhere ,
τ ^ s ( υ , ω ) = sinc γ s υ sinc γ s ω .
τ r ( x , y ) = { 2 3 γ r 2 , y < γ 2 , 3 2 x + 1 2 y < γ 2 , 0 ,             elsewhere ,
τ ^ r ( υ , ω ) = 1 3 { sinc ( γ r υ 3 ) [ cos π 2 γ r ( υ 3 - ω ) sinc 1 2 γ r ( υ 3 + ω ) + cos π 2 γ r ( υ 3 + ω ) sinc 1 2 γ r ( υ 3 - ω ) ] + cos ( π γ r υ 3 ) sinc 1 2 γ r ( υ 3 - ω ) sinc 1 2 γ r ( υ 3 + ω ) } .
τ c ( x , y ) = { 2 3 γ r 2 , r < c γ r 2 , 0 ,             elsewhere ,
τ ^ c ( υ , ω ) = 2 J 1 ( π c γ r ρ ) π c γ r ρ .
τ g ( x , y ) = 1 γ s 2 exp ( - π r 2 / γ s 2 ) ,
τ ^ g ( υ , ω ) = exp ( - π γ s 2 ρ 2 ) ,
Ω P A p l i 2 = ( γ s l i ) 2 = π 4 ( c γ r l i ) 2 .
λ = λ [ m ] = λ γ , υ = υ [ m - 1 ] = γ υ .
1 2 υ τ ^ = 0.05 X s 1 2 υ τ ^ = 0.4 ,
P ( x , y ) = λ 2 - L ( x , y ) h l ( x - x , y - y ) 2 d x d y ,
h l ( x , y ) = - P l ( λ l i υ , λ l i ω ) exp [ i W ρ 2 + i 2 π ( υ x + ω y ) ] d υ d ω .
P l ( υ , w ) P l ( λ l i ρ c υ , λ l i ρ c w ) P i ( υ D 2 , w D 2 ) .
h ^ l ( υ , ω ) = P l ( λ l i υ , λ l i ω ) exp ( i W ρ 2 ) .
h ^ l ( υ , ω ) = h ^ l ( ρ ¯ ) = { exp ( + i u ρ ¯ 2 / 2 ) , ρ ¯ 1 , 0 , ρ ¯ > 1 ,
u = 2 W ρ c 2 π 2 λ Δ l ( D l i ) 2 π Δ l 2 λ F 2 ,
ρ c = D / 2 λ l i 1 / 2 λ F .
P ( x , y ) = λ 2 L ( x , y ) * h l ( x , y ) 2 .
τ l ( x , y ) = h l ( x , y ) 2 - h l ( x , y ) 2 d x d y .
- h l ( x , y ) 2 d x d y = - h ^ l ( υ , ω ) 2 d υ ω ,
τ ^ l ( υ , ω ) = h ^ l ( υ , ω ) * h ^ l * ( - υ , - ω ) - h ^ l ( υ , ω ) 2 d υ d ω .
- h ^ l ( υ , ω ) 2 d υ d ω = - P l ( λ l i υ , λ l i ω ) 2 d υ d ω = k A l λ 2 l i 2 ,
k = 1 A l - P ( x , y ) 2 d x d y .
P ( x , y ) = k A l l i 2 L ( x , y ) * τ l ( x , y ) .
P ( x , y ; λ ) = k A l l i 2 L ( x , y ; λ ) * τ l ( x , y ; λ ) ,
L ( x , y ; λ ) = L ( x , y ) L ( λ ) ,
P ( x , y ; λ ) h c / λ Δ λ .
Δ J ( x , y ; λ ) = e ( λ h c ) η ( λ ) P ( x , y ; λ ) Δ λ ,
J ( x , y ) = 0 e ( λ h c ) η ( λ ) P ( x , y ; λ ) d λ .
s = - P p ( x , y ) J ( x , y ) d x d y .
τ p ( x , y ) = P p ( x , y ) - P p ( x , y ) d x d y = P p ( x , y ) A p ,
τ ^ p ( υ , ω ) = - τ p ( x , y ) exp [ - i 2 π ( υ x + ω y ) ] d x d y , τ ^ p ( 0 , 0 ) = - τ p ( x , y ) d x d y = 1.
τ l ( x , y ) 0 e ( λ h c ) η ( λ ) L ( λ ) τ l ( x , y ; λ ) d λ 0 e ( λ h c ) η ( λ ) L ( λ ) d λ .
s = K - τ p ( x , y ) [ L ( x , y ) * τ l ( x , y ) ] d x d y ,
K k A l A p l i 2 0 L ( λ ) τ ( λ ) d λ ,
τ ( λ ) e ( λ h c ) η ( λ ) .
K = k A l Ω p 0 L ( λ ) τ ( λ ) d λ .
s ( x , y ) = K L ( x , y ) * τ l ( x , y ) * τ p ( x , y )
= K L ( x , y ) * τ ( x , y ) .
s ( m X , n Y ; t ) = - [ L ( x , y ) * τ l ( x , y ) ] τ p , m n ( x , y ) d x d y + N e ( m X , n Y ; t ) * τ f ( t ) ,
s ( m X , n Y ; t ) = L ( m X , n Y ) * τ ( m X , n Y ) + N e ( m X , n Y ; t ) * τ f ( t ) ,
s ( m X , n Y ) = L ( m X , n Y ) * τ ( m X , n Y ) + N e ( m X , n Y )
s ( m , n ) = L s ( m , n ) + N e ( m , n ) .
R N e ( m , n ) = E { N e ( m + m , n + n ) N e ( m , n ) } = δ ( m , n ) σ N e 2 ,
δ ( m , n ) = 1 , m , n = 0 = 0 , m , n 0 ,
σ N e 2 E { N e 2 ( m , n ) } .
X ^ ( υ , ω ) = m , n X ( m , n ) exp [ - i 2 π ( m υ X + n ω Y ) ] ,
X ( m , n ) = B ^ - 1 - 1 / 2 X 1 / 2 X - 1 / 2 Y 1 / 2 Y X ^ ( υ , ω ) × exp [ i 2 π ( υ m X + ω n Y ] d υ d ω ,
J ^ ( υ , ω ) = m , n R ( m , n ) exp [ - i 2 π ( υ m X + ω n Y ] ,
R ( m , n ) = X Y - 1 / 2 X 1 / 2 X - 1 / 2 Y 1 / 2 Y J ^ ( υ , ω ) × exp [ i 2 π ( υ m X + ω n Y ) ] d υ d ω .
E { X 2 ( m , n ) } = R ( 0 , 0 ) = - 1 / 2 X 1 / 2 X - 1 / 2 Y 1 / 2 Y [ X Y J ^ ( υ , ω ) ] d υ d ω ,
Φ ^ ( υ , ω ) X Y J ^ ( υ , ω ) .
Φ ^ N e ( υ , ω ) = B ^ - 1 σ N e 2 = X Y σ N e 2 .
R ( m , n ) = R a ( m X , n Y ) ,
Φ ^ ( υ , ω ) = m , n Φ ^ a ( υ - m X , ω - n Y ) ,
Φ ^ a ( υ , ω ) = - R a ( x , y ) exp [ - i 2 π ( υ x + ω y ) ] d x d y .
m ( N x - 1 2 ) , n ( N y - 1 2 )
X ( m , n ) = 1 N x N y k = - ( N x - 1 ) / 2 ( N x - 1 ) / 2 l = - ( N y - 1 ) / 2 ( N y - 1 ) / 2 X ^ ( υ k , ω l ) × exp ( i 2 π υ k m X + i 2 π ω l n Y ) ,
X ^ ( υ k . ω l ) = m = - ( N x - 1 ) / 2 ( N x - 1 ) / 2 n = - ( N y - 1 ) / 2 ( N y - 1 ) / 2 X ( m , n ) × exp ( - i 2 π υ k m X - i 2 π ω l n Y ) ,
υ k = k / N x X k / l x , ω l = l / N y Y l / l y , X ^ ( υ k , ω l ) = 0 for k > N x - 1 2 , l > N y - 1 2 .
k N x - 1 2 , l 0 ,
y ^ ( υ k , ω l ) = X Y N x N y X ^ ( υ k , ω l ) ,
1 2 k = - ( N x - 1 ) / 2 ( N x - 1 ) / 2 l = - ( N y - 1 ) / 2 ( N y - 1 ) / 2 log 2 Φ ^ ( υ k , ω l ) ,
A 2 - 1 / 2 X 1 / 2 X - 1 / 2 Y 1 / 2 Y log 2 Φ ^ ( υ , ω ) d υ d ω .
N ( x , y ) = m , n N e ( m , n ) sinc ( x X - m ) sinc ( y Y - n ) ,
Φ ^ N ( υ , ω ) = { Φ ^ N e ( υ , ω ) , υ 1 2 X , ω 1 2 Y , 0 , otherwise .
R N ( x , y ) = - Φ ^ N ( υ , ω ) exp [ i 2 π ( υ x + ω y ) ] d υ d ω = σ N e 2 sinc ( x X ) sinc ( y Y ) .
s [ x ( t ) , y ( t ) ; t ] = L [ x ( t ) , y ( t ) ] * τ [ x ( t ) , y ( t ) ] + N e ( t ) ,
s ( t ) s ( v t ) = - { L s [ x ( t ) , y ( t ) ] + N e ( v t ) } × τ f ( v t - v t ) d ( v t ) ,
q = n + m M y , n = 0 , ± 1 , , ± ( M y - 1 2 ) , m = 0 , ± 1 , , ± ( M x - 1 2 ) ,
s ( m X , n Y ) = m = - ( M x - 1 ) / 2 ( M x - 1 ) / 2 - l y / 2 l y / 2 [ L s ( m X , y ) + N e ( y + m l y ) ] τ f [ n Y - y + ( m - m ) l y ] d y .
s ( m X , n Y ) = L s ( m X , n Y ) + N ( m , n ) ,
L s ( x , y ) L ( x , y ) * τ ( x , y ) , τ ( x , y ) τ ( x , y ) * τ f ( y ) = τ l ( x , y ) * τ p ( x , y ) * τ f ( y ) , N ( m , n ) [ N e ( m X , y ) * τ f ( y ) ] y = n Y , N e ( m X , y ) N e ( y + m l y ) .
R ¯ ( m , n ) = 1 M x M y m , n E { N ( m + m , n + n ) N ( m , n ) }
R ¯ ( m , n ) = { ( M x - m ) ( M y - n ) M x M y - l x / 2 l x / 2 - l y / 2 l y / 2 R N e ( y - y + n Y + m M y Y ) τ f ( y ) τ f ( y ) d y d y ,             m < M x , n < N y 0 ,             m M x , n M y .
R ( m , n ) = δ ( m ) - R N e ( y - y + n Y ) τ f ( y ) τ f ( y ) d y d y .
R N e ( z ) = - Φ ^ N e ( ω ) exp ( - i 2 π ω z ) d ω ,
Φ ^ N e ( υ , ω ) = X Y m , n R ( m , n ) exp [ - i 2 π ( m υ X + n ω Y ) ]
= X n Φ ^ N e ( ω - n Y ) | τ ^ f ( ω - n Y ) | 2 .
Φ ^ N ( υ , ω ) = { X n Φ ^ N e ( ω - n Y ) | τ ^ f ( ω - n Y ) | 2 ,             υ 1 2 X , ω 1 2 Y , 0 ,             otherwise .
σ N e 2 = - 1 / 2 Y 1 / 2 Y Φ ^ N e ( ω ) d ω = Φ ^ N e Y , Φ ^ N ( υ , ω ) = { X Y σ N e 2 , υ 1 2 X , ω 1 2 Y , 0 , otherwise .
s q ( m , n ) = s ( m , n ) + N q ( m , n ) ,
R N q ( m , n ) = E { N q ( m + m , n + n ) N q ( m , n ) } = δ ( m , n ) σ N q 2 ,
Φ ^ N q ( υ , ω ) = B ^ - 1 σ N q 2 = X Y σ N q 2 ,
Φ ^ q ( υ , ω ) = { X Y σ N q 2 ,             υ 1 2 X ,             ω 1 2 Y , 0 ,             otherwise .

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