Abstract

Quadratically nonlinear systems may be analyzed and synthesized by linear methods by exchanging an <i>N</i>-dimensional nonlinear problem for a 2<i>N</i>-dimensional linear formulation. This paper describes the basis for such linearization of a quadratic functional, and applies the method to partially coherent transilluminated optical systems.

© 1969 Optical Society of America

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  1. Quadratic-filter theory is appropriate whenever a system to be studied includes an energy-flux measurement, or has available as the input signal only a second moment of the observed process, a class of problems in which optical systems include perhaps the most important examples.
  2. J. L. Dobb, Stochastic Processes (John Wiley & Sons, Inc., New York, 1953).
  3. W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw—Hill Book Co., New York, 1958).
  4. R. J. Becherer and G. B. Parrent, Jr., J. Opt. Soc. Am. 57, 1479 (1967).
  5. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).
  6. E. L. O'Neill, IRE Trans. IT-2, 56 (1956).
  7. L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, IRE Trans. IT-6, 386 (1960).
  8. C. W. Helstrom, J. Opt. Soc. Am. 57, 297 (1967).
  9. B. R. Frieden, J. Opt. Soc. Am. 58, 1272 (1968).
  10. C. W. Helstrom, J. Opt. Soc. Am. 59, 164 (1969).
  11. R. K. Raney, Ph.D. thesis, Report No. 69–12, 218, University Microfilms, Inc., Ann Arbor, Mich.
  12. A more precise development of Eq. (16) is available in Ref. 11.
  13. G is not bilinear in (u,v). The terminology "dilinear" is introduced to represent the specific two-variable extension of a single variable function that allows the quadratically nonlinear filter of this discussion to be analyzed and synthesized by linear-transfer-function techniques.
  14. The linearity of Eq. (20) is not dependent on the infinite-limit assumption.
  15. M. Beran and G. B. Parrent, Jr., Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964).
  16. H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).

Becherer, R. J.

R. J. Becherer and G. B. Parrent, Jr., J. Opt. Soc. Am. 57, 1479 (1967).

Beran, M.

M. Beran and G. B. Parrent, Jr., Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, IRE Trans. IT-6, 386 (1960).

Davenport, W. B.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw—Hill Book Co., New York, 1958).

Dobb, J. L.

J. L. Dobb, Stochastic Processes (John Wiley & Sons, Inc., New York, 1953).

Frieden, B. R.

B. R. Frieden, J. Opt. Soc. Am. 58, 1272 (1968).

Helstrom, C. W.

C. W. Helstrom, J. Opt. Soc. Am. 59, 164 (1969).

C. W. Helstrom, J. Opt. Soc. Am. 57, 297 (1967).

Hopkins, H. H.

H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).

Leith, E. N.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, IRE Trans. IT-6, 386 (1960).

O’Neill, E. L.

E. L. O'Neill, IRE Trans. IT-2, 56 (1956).

Palermo, C. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, IRE Trans. IT-6, 386 (1960).

Parrent, Jr., G. B.

M. Beran and G. B. Parrent, Jr., Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964).

R. J. Becherer and G. B. Parrent, Jr., J. Opt. Soc. Am. 57, 1479 (1967).

Porcello, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, IRE Trans. IT-6, 386 (1960).

Raney, R. K.

R. K. Raney, Ph.D. thesis, Report No. 69–12, 218, University Microfilms, Inc., Ann Arbor, Mich.

Root, W. L.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw—Hill Book Co., New York, 1958).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).

Other (16)

Quadratic-filter theory is appropriate whenever a system to be studied includes an energy-flux measurement, or has available as the input signal only a second moment of the observed process, a class of problems in which optical systems include perhaps the most important examples.

J. L. Dobb, Stochastic Processes (John Wiley & Sons, Inc., New York, 1953).

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw—Hill Book Co., New York, 1958).

R. J. Becherer and G. B. Parrent, Jr., J. Opt. Soc. Am. 57, 1479 (1967).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1959).

E. L. O'Neill, IRE Trans. IT-2, 56 (1956).

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, IRE Trans. IT-6, 386 (1960).

C. W. Helstrom, J. Opt. Soc. Am. 57, 297 (1967).

B. R. Frieden, J. Opt. Soc. Am. 58, 1272 (1968).

C. W. Helstrom, J. Opt. Soc. Am. 59, 164 (1969).

R. K. Raney, Ph.D. thesis, Report No. 69–12, 218, University Microfilms, Inc., Ann Arbor, Mich.

A more precise development of Eq. (16) is available in Ref. 11.

G is not bilinear in (u,v). The terminology "dilinear" is introduced to represent the specific two-variable extension of a single variable function that allows the quadratically nonlinear filter of this discussion to be analyzed and synthesized by linear-transfer-function techniques.

The linearity of Eq. (20) is not dependent on the infinite-limit assumption.

M. Beran and G. B. Parrent, Jr., Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964).

H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).

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