Abstract

For multidimensional band-limited functions, the Nyquist density is defined as that density corresponding to maximally packed spectral replications. Because of the shape of the support of the spectrum, however, sampling multidimensional functions at Nyquist densities can leave gaps among these replications. In this paper we show that, when such gaps exist, the image samples can periodically be deleted or decimated without information loss. The result is an overall lower sampling density. Recovery of the decimated samples by the remaining samples is a linear interpolation process. The interpolation kernels can generally be obtained in closed form. The interpolation noise level resulting from noisy data is related to the decimation geometry. The greater the clustering of the decimated samples, the higher the interpolation noise level is.

© 1990 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. J. Marks, “Multidimensional-signal sample dependency at Nyquist densities,” J. Opt. Soc. Am. A 3, 268–273 (1986).
    [CrossRef]
  2. D. E. Dudgeon, R. M. Meresereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1984).
  3. D. P. Petersen, D. Middleton, “Sampling and reconstruction of wave-number limited functions in N-dimensional Euclidean spaces,” Inf. Control 5, 279–323 (1962).
    [CrossRef]
  4. C. E. Shannon, “Communication in the presence of noise,” IRE Proc. 37, 10–21 (1948).
    [CrossRef]
  5. E. Dubois, “The sampling and reconstruction of time-varying imagery with application in video systems,” IEEE Proc. 23, 502–522 (1985).
    [CrossRef]
  6. K. F. Cheung, “Image sampling density below that of Nyquist,” Ph.D. dissertation (Department of Electrical Engineering, University of Washington, Seattle, Wash., 1988).
  7. D. S. Chen, J. P. Allebach, “Analysis of error in reconstruction of two-dimensional signals from irregularly spaced samples,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 173–180 (1987).
    [CrossRef]
  8. J. L. Yen, “On the nonuniform sampling of bandwidth limited signals,” IRE Trans. Circuit Theory 3, 251–257 (1956).
    [CrossRef]
  9. H. K. Ching, “Truncation effects in the estimation of two-dimensional continuous bandlimited signals,” master’s thesis (Department of Electrical Engineering, University of Washington, Seattle, Wash., 1985).
  10. R. E. Crochiere, L. R. Rabiner, Multirate Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1983).
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  12. P. K. Rajan, “A study on the properties of multidimensional fourier transforms,” IEEE Trans. Circuits Syst. CAS-31, 748–750 (1984).
  13. R. N. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1965).
  14. J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978).
  15. A. V. Oppenheim, R. W. Shafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  16. A. Papoulis, “Error analysis in sampling theory,” Proc. IEEE 54, 947–955 (1966).
    [CrossRef]
  17. R. J. Marks, D. Radbel, “Error of linear estimation of lost samples in an oversampled bandlimited signal,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 648–654 (1985).
  18. D. Radbel, “Noise and truncation effects in the estimation of sampled bandlimited signals,” master’s thesis (Department of Electrical Engineering, University of Washington, Seattle, Wash., 1983).
  19. D. Radbel, R. J. Marks, “An FIR estimation filter based on the sampling theorem,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 455–460 (1985).
    [CrossRef]

1987 (1)

D. S. Chen, J. P. Allebach, “Analysis of error in reconstruction of two-dimensional signals from irregularly spaced samples,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 173–180 (1987).
[CrossRef]

1986 (1)

1985 (3)

R. J. Marks, D. Radbel, “Error of linear estimation of lost samples in an oversampled bandlimited signal,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 648–654 (1985).

D. Radbel, R. J. Marks, “An FIR estimation filter based on the sampling theorem,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 455–460 (1985).
[CrossRef]

E. Dubois, “The sampling and reconstruction of time-varying imagery with application in video systems,” IEEE Proc. 23, 502–522 (1985).
[CrossRef]

1984 (1)

P. K. Rajan, “A study on the properties of multidimensional fourier transforms,” IEEE Trans. Circuits Syst. CAS-31, 748–750 (1984).

1966 (1)

A. Papoulis, “Error analysis in sampling theory,” Proc. IEEE 54, 947–955 (1966).
[CrossRef]

1962 (1)

D. P. Petersen, D. Middleton, “Sampling and reconstruction of wave-number limited functions in N-dimensional Euclidean spaces,” Inf. Control 5, 279–323 (1962).
[CrossRef]

1956 (1)

J. L. Yen, “On the nonuniform sampling of bandwidth limited signals,” IRE Trans. Circuit Theory 3, 251–257 (1956).
[CrossRef]

1948 (1)

C. E. Shannon, “Communication in the presence of noise,” IRE Proc. 37, 10–21 (1948).
[CrossRef]

Allebach, J. P.

D. S. Chen, J. P. Allebach, “Analysis of error in reconstruction of two-dimensional signals from irregularly spaced samples,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 173–180 (1987).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1965).

Chen, D. S.

D. S. Chen, J. P. Allebach, “Analysis of error in reconstruction of two-dimensional signals from irregularly spaced samples,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 173–180 (1987).
[CrossRef]

Cheung, K. F.

K. F. Cheung, “Image sampling density below that of Nyquist,” Ph.D. dissertation (Department of Electrical Engineering, University of Washington, Seattle, Wash., 1988).

Ching, H. K.

H. K. Ching, “Truncation effects in the estimation of two-dimensional continuous bandlimited signals,” master’s thesis (Department of Electrical Engineering, University of Washington, Seattle, Wash., 1985).

Crochiere, R. E.

R. E. Crochiere, L. R. Rabiner, Multirate Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1983).

Dubois, E.

E. Dubois, “The sampling and reconstruction of time-varying imagery with application in video systems,” IEEE Proc. 23, 502–522 (1985).
[CrossRef]

Dudgeon, D. E.

D. E. Dudgeon, R. M. Meresereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978).

Goodman, J. W.

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

Marks, R. J.

R. J. Marks, “Multidimensional-signal sample dependency at Nyquist densities,” J. Opt. Soc. Am. A 3, 268–273 (1986).
[CrossRef]

D. Radbel, R. J. Marks, “An FIR estimation filter based on the sampling theorem,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 455–460 (1985).
[CrossRef]

R. J. Marks, D. Radbel, “Error of linear estimation of lost samples in an oversampled bandlimited signal,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 648–654 (1985).

Meresereau, R. M.

D. E. Dudgeon, R. M. Meresereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Middleton, D.

D. P. Petersen, D. Middleton, “Sampling and reconstruction of wave-number limited functions in N-dimensional Euclidean spaces,” Inf. Control 5, 279–323 (1962).
[CrossRef]

Oppenheim, A. V.

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

Papoulis, A.

A. Papoulis, “Error analysis in sampling theory,” Proc. IEEE 54, 947–955 (1966).
[CrossRef]

Petersen, D. P.

D. P. Petersen, D. Middleton, “Sampling and reconstruction of wave-number limited functions in N-dimensional Euclidean spaces,” Inf. Control 5, 279–323 (1962).
[CrossRef]

Rabiner, L. R.

R. E. Crochiere, L. R. Rabiner, Multirate Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1983).

Radbel, D.

R. J. Marks, D. Radbel, “Error of linear estimation of lost samples in an oversampled bandlimited signal,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 648–654 (1985).

D. Radbel, R. J. Marks, “An FIR estimation filter based on the sampling theorem,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 455–460 (1985).
[CrossRef]

D. Radbel, “Noise and truncation effects in the estimation of sampled bandlimited signals,” master’s thesis (Department of Electrical Engineering, University of Washington, Seattle, Wash., 1983).

Rajan, P. K.

P. K. Rajan, “A study on the properties of multidimensional fourier transforms,” IEEE Trans. Circuits Syst. CAS-31, 748–750 (1984).

Shafer, R. W.

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

Shannon, C. E.

C. E. Shannon, “Communication in the presence of noise,” IRE Proc. 37, 10–21 (1948).
[CrossRef]

Yen, J. L.

J. L. Yen, “On the nonuniform sampling of bandwidth limited signals,” IRE Trans. Circuit Theory 3, 251–257 (1956).
[CrossRef]

IEEE Proc. (1)

E. Dubois, “The sampling and reconstruction of time-varying imagery with application in video systems,” IEEE Proc. 23, 502–522 (1985).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (3)

R. J. Marks, D. Radbel, “Error of linear estimation of lost samples in an oversampled bandlimited signal,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 648–654 (1985).

D. S. Chen, J. P. Allebach, “Analysis of error in reconstruction of two-dimensional signals from irregularly spaced samples,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 173–180 (1987).
[CrossRef]

D. Radbel, R. J. Marks, “An FIR estimation filter based on the sampling theorem,” IEEE Trans. Acoust. Speech Signal Process. ASSP-33, 455–460 (1985).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

P. K. Rajan, “A study on the properties of multidimensional fourier transforms,” IEEE Trans. Circuits Syst. CAS-31, 748–750 (1984).

Inf. Control (1)

D. P. Petersen, D. Middleton, “Sampling and reconstruction of wave-number limited functions in N-dimensional Euclidean spaces,” Inf. Control 5, 279–323 (1962).
[CrossRef]

IRE Proc. (1)

C. E. Shannon, “Communication in the presence of noise,” IRE Proc. 37, 10–21 (1948).
[CrossRef]

IRE Trans. Circuit Theory (1)

J. L. Yen, “On the nonuniform sampling of bandwidth limited signals,” IRE Trans. Circuit Theory 3, 251–257 (1956).
[CrossRef]

J. Opt. Soc. Am. A (1)

Proc. IEEE (1)

A. Papoulis, “Error analysis in sampling theory,” Proc. IEEE 54, 947–955 (1966).
[CrossRef]

Other (9)

D. E. Dudgeon, R. M. Meresereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1984).

H. K. Ching, “Truncation effects in the estimation of two-dimensional continuous bandlimited signals,” master’s thesis (Department of Electrical Engineering, University of Washington, Seattle, Wash., 1985).

R. E. Crochiere, L. R. Rabiner, Multirate Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1983).

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

R. N. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1965).

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978).

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

D. Radbel, “Noise and truncation effects in the estimation of sampled bandlimited signals,” master’s thesis (Department of Electrical Engineering, University of Washington, Seattle, Wash., 1983).

K. F. Cheung, “Image sampling density below that of Nyquist,” Ph.D. dissertation (Department of Electrical Engineering, University of Washington, Seattle, Wash., 1988).

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 (19)

Fig. 1
Fig. 1

(a) Hexagonal sampling lattice corresponding to the Ny-quist-density sampling of a circularly band-limited function, (b) resulting spectral replications.

Fig. 2
Fig. 2

(a) Example of a sampling lattice and (b) its corresponding spectral replications. Each dashed-line parallelogram is a cell C.

Fig. 3
Fig. 3

Two different cells corresponding to the same periodicity matrix. In this example, the puzzle pieces A, B, C, and D are simply rearranged.

Fig. 4
Fig. 4

Region of integration, , of the interpolation kernel in Eq. (8).

Fig. 5
Fig. 5

Example of a spectral replication caused by the oversampling of a function.

Fig. 6
Fig. 6

(a) Example of a decimation geometry imposed upon the samples in Fig. 2, (b) corresponding subcell pattern (dashed-line squares).

Fig. 7
Fig. 7

Spectral replications corresponding to the rectangular sampling at the minimum density of a circularly band-limited function.

Fig. 8
Fig. 8

Decimation geometry for example 1. A decimation period is outlined by a dashed-line square.

Fig. 9
Fig. 9

Example 1: a subcell is totally embedded within a gap region.

Fig. 10
Fig. 10

Hexagonal support of the Fourier transform of a hinc function.

Fig. 11
Fig. 11

Decimation geometry for example 2. A decimation period is outlined by a dashed-line rectangle.

Fig. 12
Fig. 12

Example 2: a subcell is totally embedded within a gap region.

Fig. 13
Fig. 13

Three-layer window corresponding to example 1.

Fig. 14
Fig. 14

Plot of the estimate x ( 0 ) versus the number of layers for example 1.

Fig.15
Fig.15

Plot of the estimate x ( 0 ) versus the number of layers for example 2.

Fig. 16
Fig. 16

Decimation geometry specified in example 3.

Fig. 17
Fig. 17

Example 3: a subcell is totally embedded in a gap region.

Fig. 18
Fig. 18

Example 4: two subcells are totally embedded in two gap regions.

Fig. 19
Fig. 19

Example 4: an illustration of the partitioning of each of the two subcells into four partitions.

Tables (1)

Tables Icon

Table 1 Weights and Displacements Corresponding to the Four Partitions of the Subcell in Example 4

Equations (123)

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

X ( ν ) = t x ( t ) exp ( j 2 π ν T t ) d t ,
ν = ( ν 1 , ν 2 , , ν N ) T
t d t = t 1 d t 1 t 2 d t 2 t N d t N .
x ( t ) = ν A X ( ν ) exp ( j 2 π ν T t ) d ν .
x ̂ ( t ) = n x ( t + V k s ) δ ( t Vn ) = n x [ V ( n + k s ) ] δ ( t Vn ) ,
V = [ v 1 | v 2 | | v N ]
n = n 1 n 2 n N .
X ̂ ( ν ) = D n X ( ν Un ) ,
U = [ u 1 | u 2 | | u N ] ,
D = | U | = 1 | V | ( samples per area ) ,
U T = V 1 .
x ( t ) = n x ( Vn ) f ( t Vn ) ,
f ( t ) = 1 D exp ( j 2 π ν T t ) d ν
X ̂ ( ν ) = 2 S n = X ( ν 2 n S ) .
X ̂ ( ν ) 0 , ν D .
H ( ν ) X ̂ ( ν ) 0 ,
n = x ( n 2 S ) h ( t n 2 S ) 0 .
m x ( Vm ) h ( t Vm ) = n x ( Vn ) h ( t Vn ) .
m x ( Vm ) h [ V ( I m ) ] = n x ( Vn ) h [ V ( I m ) ] , I .
H x = H R x R ,
x = H 1 H R x R .
V d = VM = [ v d 1 | v d 2 | | v d N ] ,
D d = 1 | V d | = D | M | ,
r d = 1 | M |
M = [ 2 0 1 2 ]
k d = ( 1 , 1 ) T .
L = | M | .
L = j = 1 N L j .
x i ( V d n ) = x ( V d n + V k i ) , i = 0 to L 1 ,
{ x ( Vn ) } = i = 0 L 1 { x i ( V d n V k i ) } .
X ̂ ( ν ) = i = 0 L 1 X ̂ i ( ν ) exp ( j 2 π ν T V k i ) .
U d T = V d 1 .
x α ( V d n ) = x ( V d n + V k α ) .
= { m | m = Mn + k α } ,
X ̂ ( ν ) = i α X ̂ i ( ν ) exp ( j 2 π ν T V k i ) + X ̂ α ( ν ) exp ( j 2 π ν T V k α ) .
X ̂ ( ν ) 0 , ν C ̂ d .
X ̂ α ( ν ) = i α X ̂ i ( ν ) exp [ j 2 π ν T V ( k i k α ) ] , ν C ̂ d .
x α ( V d m ) = 1 D d C ̂ d X ̂ α ( ν ) exp ( j 2 π ν T V d m ) d ν .
x α ( V d m ) = i α n x i ( V d n ) f d [ V d ( m n ) V ( k i k α ) ] ,
f d ( t ) = 1 D d C ̂ d exp ( j 2 π ν T t ) d ν
V = [ 1 0 0 1 ] .
M = [ 1 3 3 0 ] ,
U d = [ 0 1 / 3 1 / 3 1 / 9 ] .
f d ( t 1 , t 2 ) = 1 2 3 hinc ( t 1 3 3 + t 2 18 3 + t 2 6 ) exp [ j 2 π ( t 1 + t 2 ) ] ,
hinc ( t 1 , t 2 ) = 4 3 { sinc 2 3 t 1 sinc 2 t 2 + 1 2 π t 2 × [ sinc 1 3 ( t 1 3 t 2 ) sin π ( 3 t 1 + t 2 ) sinc 1 3 t 1 sin π 3 t 1 ] } ,
sinc ( x ) = sin π x π x .
M = [ 2 2 2 2 ] .
U d = [ 1 / 8 1 / 16 0 3 / 16 ] ,
f d ( t 1 , t 2 ) = sinc ( t 1 + t 2 4 ) sinc ( t 1 + t 2 4 ) exp [ j 2 π ( t 1 + t 2 ) ]
jinc ( t 1 , t 2 ) = 2 J 1 [ 2 π ( t 1 2 + t 2 2 ) 1 / 2 ] π ( t 1 2 + t 2 2 ) ,
x d ( V d m ) = i d n x i ( V d n ) f d [ V d ( m n ) V ( k i k d ) ] ,
n = n 1 = M M n 2 = M M n N = M M
V = [ 1 0 1 / 3 2 / 3 ] .
U d = 1 8 U = [ 1 / 8 1 / 16 0 3 / 16 ] .
f d ( t 1 , t 2 ) = sinc t 1 8 sinc ( t 1 + 3 t 2 16 ) × exp [ j π ( 3 + 16 3 16 t 1 + 3 16 t 2 ) ]
i = 0 L 1 X ̂ i ( ν ) exp ( 2 π V k i ) = 0 , ν C ̂ d , 1 , i = 0 L 1 X ̂ i ( ν ) exp ( 2 π V k i ) = 0 , ν C ̂ d , 2 .
= 1 2 ,
1 = { m | m = Mn + k 1 } , 2 = { m | m = Mn + k 2 } .
i = 1 , 2 exp ( 2 π ν T V k i ) X ̂ i ( ν ) = G ( ν ) , ν C ̂ d , 1 , i = 1 , 2 exp [ j 2 π ( ν + d ) T V k i ] X ̂ i ( ν + d ) = G ( ν + d ) , ν C ̂ d , 1 ,
G ( ν ) = j 1 , 2 X ̂ j ( ν ) exp ( j 2 π ν T V k j ) .
H d X d = G , ν C ̂ d , 1
H d = [ exp ( j 2 π ν T V k 1 ) exp ( j 2 π ν T V k 2 ) exp [ j 2 π ν T ( V + d ) k 1 ] exp [ j 2 π ν T ( V + d ) k 2 ] ] , X d = [ X ̂ k 1 ( ν ) X ̂ k 2 ( ν ) ] , G = [ G ( ν ) G ( ν + d ) ] ,
G ( ν + d ) = j 1 , 2 X ̂ j ( ν ) exp [ j 2 π ( ν + d ) T V k j ] .
G = H r X r ,
X d = H d 1 H r X r
X ̂ j ( ν ) = j 1 , 2 H i , j ( ν ) X ̂ j ( ν ) , i = 1 , 2 ,
H i , j ( ν ) = C i , j ( d ) exp [ j 2 π ν T V ( k j k i ) ] ,
C 1 , j ( d ) = sin [ π d T V ( k j k 2 ) ] sin [ π d T V ( k 1 k 2 ) ] exp [ j π d T V ( k j k 1 ) ] ,
H 2 , j = C 2 , j ( d ) exp [ j 2 π ν T V ( k j k 2 ) ] ,
C 2 , j ( d ) = sin [ π d T V ( k j k 1 ) ] sin [ π d T V ( k 1 k 2 ) ] exp [ j π d T V ( k j k 2 ) ] .
x i ( V d m ) = j 1 , 2 n x j ( V d n ) h i , j [ V d ( m n ) V ( k j k i ) ] , i = 1 , 2 ,
h i , j ( t ) = 1 D d C i , j ( d ) C ̂ d , 1 exp ( j 2 π ν T t ) d ν
d = ( 7 16 , 16 7 3 16 ) T 2.38 u d 1 + 2.24 u d 2 .
ν 2 = ν 1 + d k , k = 1 , 2 , 3 , 4 .
d k = U d p k , k = 1 , 2 , 3 , 4 ,
p 1 = ( 3 , 3 ) T , p 2 = ( 2 , 3 ) T , p 3 ( 3 , 2 ) T , p 4 = ( 2 , 2 ) T .
H d k X d k = G k , v C ̂ d , 1 , k , k = 1 , 2 , 3 , 4 .
X d = k = 1 4 X d k ,
X d = k = 1 4 H d k 1 G k .
h i , j ( t ) = 1 D d k = 1 4 C i , j ( d k ) h k ( t ) , i = 1 , 2 ,
h k ( t ) = C ̂ d , 1 , k exp ( j 2 π ν T t ) d ν .
k ( t 1 , t 2 ) = 3 128 sinc ( t 1 8 ) sinc ( t 1 16 + 3 t 2 16 ) × exp [ j π ( 3 t 1 16 + 3 t 2 16 ) ] .
S k = [ s k 1 0 0 s k 2 ] , k = 1 , 2 , 3 , 4 ,
C ̂ d , 1 , k ( ν ) = C d 0 ( S k ν + r k ) , k = 1 , 2 , 3 , 4 ,
h k ( t ) = s k 1 s k 2 k ( S k T t ) exp ( j 2 π r k T t ) ,
h i , j ( t ) = 1 D d k = 1 4 C i , j s k 1 s k 2 k ( S k T t ) exp ( j 2 π r k T t ) .
f i , j ( t ) = k = 1 4 a i , j , k f k ( t ) .
a i , j , k = C i , j ( d k )
f k ( t ) = 1 D d s k 1 s k 2 ( S k T t ) exp ( j 2 π r k T t ) .
C ̂ d , j = { ν | ν { C ̂ d , 1 d j } } , j = 2 , , q ,
C ̂ d , j , k = { ν | ν { C ̂ d , 1 , k d j , k } } , j = 2 , , q , k = 1 , , q N
d j , k = U d p j , k ,
H d , k X d = G , ν C ̂ d , 1 , k ,
k ( t ) = C ̂ d , 1 exp ( j 2 π ν T t ) d ν .
f i , j ( t ) = k = 1 q N a i , j , k f k ( t ) , i ̂ , j ̂ ,
f k ( t ) = 1 D d ( Π n = 1 N S k , n ) k ( S k T t ) exp ( j 2 π r k T t ) .
x i ( V d m ) = j ̂ n x j ( V d n ) h i , j [ V d ( m n ) V ( k j k i ) ] , i ̂ .
E { ξ ( Vn ) ξ ( Vm ) } = ξ 2 ¯ δ [ n m ] ,
η ( V d n ) = i d m ξ ( V d m + V k i ) f d [ V d ( n m ) ( k i k α ) ] , n .
η ( 0 ) = i 0 m ξ ( V d m + V k i ) f d ( V d m V k i )
E { ξ ( V d m + V k r ) ξ * ( V d n + Vs ) } = ξ 2 ¯ δ [ n [ n + M 1 ( k r k s ) ] ] ,
η 2 ¯ = ξ 2 ¯ r 0 s 0 m n δ [ n [ m + M 1 ( k r k s ) ] ] × f d ( V d m V k r ) f d * ( V d V k s ) = ξ 2 ¯ r 0 [ m | f d ( V d m V k r ) | 2 ] .
m | f d ( V d m V k r ) | 2 = D d C ̂ d | F d ( ν ) | 2 d ν ,
| F d ( ν ) | = { 1 / D d ν C ̂ d 0 otherwise .
m | f d ( V d m V k r ) | 2 = 1 ,
η 2 ¯ = ( L 1 ) ξ 2 ¯ .
η 2 ¯ ξ 2 ¯ = j k = 1 q N | a i , j , k | 2 ( n = 1 N s k n ) , i ̂ .
10 3 16 2 3
16 9 3 3
7 16
7 3 8 10
8 4 3 3
16 9 3 3
17 3 16 16 3
7 3 8 10
5 3 8 3
10 3 16 3
8 3 8 3
4 3 3 8 3
8 4 3 3
10 3 16 3
1 2
4 3 3 8 3

Metrics