Abstract

The basic concept of Hadamard spectroscopy is presented. General methods are given for the construction of cyclic measurement matrices. A new algorithm, which permits adaptation of the fast Hadamard transform to the calculation of spectral intensities when the measurement matrices are cyclic, is introduced. Some applications are also briefly discussed.

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  1. P. L. Richards, J. Opt. Soc. Am. 54, 1474 (1964).
  2. P. B. Fellgett, J. Phys. Radium 19, 187 (1958).
  3. N. J. A. Sloane, T. Fine, P. G. Phillips, and M. Harwit, Appl. Opt. 8, 2103 (1969).
  4. J. A. Decker, Jr. and M. Harwit, Appl. Opt. 8, 2552 (1969).
  5. J. A. Decker, Jr., Opt. Spectra 4 (4), 45 (1970).
  6. A cyclic matrix is one for which each row may be obtained from the preceding row by shifting the elements one position to the right (left); the element at the end (beginning) of a row is moved to the beginning (end) of the next row.
  7. Equation (8) gives [Equation] The inverse relationship between the {aij} and {bji} matrices, together with the Cauchy—Schwartz inequality, gives [Equation] The above yields [Equation] Finally, because |aij| ≤1, it follows that FN½.
  8. L. D. Baumert, in Digital Communications With Space Applications, edited by S. W. Golomb (Prentice-Hall, Englewood Cliffs, N. J., 1964), pp. 51–53.
  9. See Ref. 8, pp. 52, 53.
  10. See Ref. 8, pp. 166, 167.
  11. W. W. Peterson, Error-Correcting Codes (MIT Press and Wiley, New York, 1961), p. 103.
  12. See Ref. 8, pp. 53–62.
  13. See Ref. 8, p. 55.
  14. W. K. Pratt, J. Kane, and H. C. Andrews, Proc. IEEE 57, 58 (1969).
  15. E. J. Watson, Math Comp. 16, 368 (1962).

Andrews, H. C.

W. K. Pratt, J. Kane, and H. C. Andrews, Proc. IEEE 57, 58 (1969).

Baumert, L. D.

L. D. Baumert, in Digital Communications With Space Applications, edited by S. W. Golomb (Prentice-Hall, Englewood Cliffs, N. J., 1964), pp. 51–53.

Decker, Jr., J. A.

J. A. Decker, Jr. and M. Harwit, Appl. Opt. 8, 2552 (1969).

J. A. Decker, Jr., Opt. Spectra 4 (4), 45 (1970).

Fellgett, P. B.

P. B. Fellgett, J. Phys. Radium 19, 187 (1958).

Fine, T.

N. J. A. Sloane, T. Fine, P. G. Phillips, and M. Harwit, Appl. Opt. 8, 2103 (1969).

Harwit, M.

N. J. A. Sloane, T. Fine, P. G. Phillips, and M. Harwit, Appl. Opt. 8, 2103 (1969).

J. A. Decker, Jr. and M. Harwit, Appl. Opt. 8, 2552 (1969).

Kane, J.

W. K. Pratt, J. Kane, and H. C. Andrews, Proc. IEEE 57, 58 (1969).

Peterson, W. W.

W. W. Peterson, Error-Correcting Codes (MIT Press and Wiley, New York, 1961), p. 103.

Phillips, P. G.

N. J. A. Sloane, T. Fine, P. G. Phillips, and M. Harwit, Appl. Opt. 8, 2103 (1969).

Pratt, W. K.

W. K. Pratt, J. Kane, and H. C. Andrews, Proc. IEEE 57, 58 (1969).

Richards, P. L.

P. L. Richards, J. Opt. Soc. Am. 54, 1474 (1964).

Sloane, N. J. A.

N. J. A. Sloane, T. Fine, P. G. Phillips, and M. Harwit, Appl. Opt. 8, 2103 (1969).

Watson, E. J.

E. J. Watson, Math Comp. 16, 368 (1962).

Other (15)

P. L. Richards, J. Opt. Soc. Am. 54, 1474 (1964).

P. B. Fellgett, J. Phys. Radium 19, 187 (1958).

N. J. A. Sloane, T. Fine, P. G. Phillips, and M. Harwit, Appl. Opt. 8, 2103 (1969).

J. A. Decker, Jr. and M. Harwit, Appl. Opt. 8, 2552 (1969).

J. A. Decker, Jr., Opt. Spectra 4 (4), 45 (1970).

A cyclic matrix is one for which each row may be obtained from the preceding row by shifting the elements one position to the right (left); the element at the end (beginning) of a row is moved to the beginning (end) of the next row.

Equation (8) gives [Equation] The inverse relationship between the {aij} and {bji} matrices, together with the Cauchy—Schwartz inequality, gives [Equation] The above yields [Equation] Finally, because |aij| ≤1, it follows that FN½.

L. D. Baumert, in Digital Communications With Space Applications, edited by S. W. Golomb (Prentice-Hall, Englewood Cliffs, N. J., 1964), pp. 51–53.

See Ref. 8, pp. 52, 53.

See Ref. 8, pp. 166, 167.

W. W. Peterson, Error-Correcting Codes (MIT Press and Wiley, New York, 1961), p. 103.

See Ref. 8, pp. 53–62.

See Ref. 8, p. 55.

W. K. Pratt, J. Kane, and H. C. Andrews, Proc. IEEE 57, 58 (1969).

E. J. Watson, Math Comp. 16, 368 (1962).

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