Abstract

A number of binary cyclic coding schemes for multiplex spectrometry are discussed and evaluated in terms of a linear, least mean square, unbiased estimate. The optical realization of such codes in dispersion instruments is briefly discussed. We show that there are many advantages both in the construction of the instrument and in its operation which accrue from cyclic codes.

© 1969 Optical Society of America

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References

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  1. R. N. Ibbett, D. Aspinall, J. F. Grainger, Appl. Opt. 7, 1089 (1968).
    [CrossRef] [PubMed]
  2. J. A. Decker, M. O. Harwit, Appl. Opt. 7, 2205 (1968).
    [CrossRef] [PubMed]
  3. F. Gottlieb, IEEE Trans. Inform. Theory IT-14, 428 (1968).
    [CrossRef]
  4. H. Cramér, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1946).
  5. R. Deutsch, Estimation Theory (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1965).
  6. Marshall Hall, Combinatorial Theory (Blaisdell Publishing Co., Waltham, Massachusetts, 1967), p. 204.
  7. S. W. Golomb, Ed., Digital Communications with Space Applications (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).
  8. R. Thoene, S. W. Golomb, California Institute of Technology, Jet Propulsion Laboratory, Space Programs Summary No. 37-40, IV, 207 (1966).
  9. L. D. Baumert, California Institute of Technology, Jet Propulsion Laboratory, Space Programs Summary No. 37-43, IV, 311 (1967).

1968

Aspinall, D.

Baumert, L. D.

L. D. Baumert, California Institute of Technology, Jet Propulsion Laboratory, Space Programs Summary No. 37-43, IV, 311 (1967).

Cramér, H.

H. Cramér, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1946).

Decker, J. A.

Deutsch, R.

R. Deutsch, Estimation Theory (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1965).

Golomb, S. W.

R. Thoene, S. W. Golomb, California Institute of Technology, Jet Propulsion Laboratory, Space Programs Summary No. 37-40, IV, 207 (1966).

Gottlieb, F.

F. Gottlieb, IEEE Trans. Inform. Theory IT-14, 428 (1968).
[CrossRef]

Grainger, J. F.

Hall, Marshall

Marshall Hall, Combinatorial Theory (Blaisdell Publishing Co., Waltham, Massachusetts, 1967), p. 204.

Harwit, M. O.

Ibbett, R. N.

Thoene, R.

R. Thoene, S. W. Golomb, California Institute of Technology, Jet Propulsion Laboratory, Space Programs Summary No. 37-40, IV, 207 (1966).

Appl. Opt.

IEEE Trans. Inform. Theory

F. Gottlieb, IEEE Trans. Inform. Theory IT-14, 428 (1968).
[CrossRef]

Other

H. Cramér, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1946).

R. Deutsch, Estimation Theory (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1965).

Marshall Hall, Combinatorial Theory (Blaisdell Publishing Co., Waltham, Massachusetts, 1967), p. 204.

S. W. Golomb, Ed., Digital Communications with Space Applications (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).

R. Thoene, S. W. Golomb, California Institute of Technology, Jet Propulsion Laboratory, Space Programs Summary No. 37-40, IV, 207 (1966).

L. D. Baumert, California Institute of Technology, Jet Propulsion Laboratory, Space Programs Summary No. 37-43, IV, 311 (1967).

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Figures (2)

Fig. 1
Fig. 1

Mask with 2N − 1 (N = 7) slots for encoding the spectral distribution. Shaded areas are opaqe and represent 0’s. The clear areas represent 1’s.

Fig. 2
Fig. 2

Mask configuration for the reflecting mode in conjunction with the HT and GT matrices.

Tables (1)

Tables Icon

Table I First Row of the G Matrix for Low Values of N − 1

Equations (19)

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W = 1 0 0 1 0 1 1 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 1 1 1 0 0 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1
x i = n i + j = 1 N ω i j E j , i = 1 , 2 , . . . , M .
x = E W + n .
E ̂ = x A ,
E ̂ = x A = x A .
x = E W + n = E W .
A = W T ( W W T ) 1 .
H H T = N I .
H = [ 1 1 1 1 G 1 ] .
row i has + 1 . . . + 1 . . . 1 . . . 1 and row j has + 1 in N / 4 places . . . 1 . . . . . . + 1 . . . . . . 1 . . .
G = + + + + + + + + + + + + + + + + + + + + + ,
In H : row i row j = { 0 i j N i = j In G : row i row j = { 1 i j N 1 i = j
G J = J G T = [ ( N 2 1 ) 1 + N 2 ( 1 ) ] J = J
T r [ ( G 1 ) T G 1 ] = T r 1 N 2 [ ( G J ) ( G T J ) ] = T r 1 N ( I + J ) = 2 2 N .
In S : row i row j = { N / 4 i j N / 2 i = j .
S S T = ( N / 4 ) ( I + J ) , S J = J S T = ( N / 2 ) J , S 1 = ( 2 / N ) ( 2 S T J ) . T r [ ( S 1 ) T S 1 ] = 4 ( 8 / N ) + ( 4 / N 2 ) = [ 2 ( 2 / N ) ] 2 .
2 2 N
4 8 N + 4 N 2
E ̂ = x ( S T ) 1 .

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