Abstract

Matched filtering is described as a spatial filtering operation. A technique for producing a matched filter, wherein the filter transfer function is modulated onto a spatial carrier and the resulting function is hard-clipped allowing a filter construction of completely opaque and transparent lines, is given. The effect of this nonlinearity on the S/N is shown to be small. The effects of extraneous frequencies in the filter is shown to be negligible if the spatial carrier is sufficiently high. Experimental results are presented showing the detectability of the signal in the presence of various levels of additive noise.

© 1965 Optical Society of America

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References

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  1. P. M. Duffieux, l’Integrale de Fourier et ses Applications à l’Optique, chez l’Auteur (Université de Besançon, Besançon, France, 1946).
  2. T. P. Cheatham, A. Kohlenberg, Inst. Radio Engrs. Intern. Conv. Record, Part 4, 6 (1954).
  3. E. O’Neill, Inst. Radio Engrs. Trans. Inform. Theory IT-2, 55 (1956).
  4. A. Maréchal, in Optical Processing of Information, D. K. Pollack, C. J. Koester, J. T. Tippett, eds. (Spartan, Baltimore, 1963).
  5. L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, Inst. Radio Engrs. Trans. Inform. Theory IT-6, 391 (1960).
  6. W. M. Brown, Analysis of Linear Time-Invariant Systems (McGraw-Hill, New York, 1963), Sec. 6-2, p. 171.
  7. G. L. Turin, Inst. Radio Engrs. Trans. Inform. Theory IT-6, 386 (1960).
  8. E. Leith, J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962).
    [CrossRef]
  9. A. Vander Lugt, Trans. IEEE IT-10, 139 (1964).
  10. W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), p. 282. [The details of the expansion Eq. (2) are worked out in this reference.]

1964 (1)

A. Vander Lugt, Trans. IEEE IT-10, 139 (1964).

1962 (1)

1960 (2)

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, Inst. Radio Engrs. Trans. Inform. Theory IT-6, 391 (1960).

G. L. Turin, Inst. Radio Engrs. Trans. Inform. Theory IT-6, 386 (1960).

1956 (1)

E. O’Neill, Inst. Radio Engrs. Trans. Inform. Theory IT-2, 55 (1956).

1954 (1)

T. P. Cheatham, A. Kohlenberg, Inst. Radio Engrs. Intern. Conv. Record, Part 4, 6 (1954).

Brown, W. M.

W. M. Brown, Analysis of Linear Time-Invariant Systems (McGraw-Hill, New York, 1963), Sec. 6-2, p. 171.

Cheatham, T. P.

T. P. Cheatham, A. Kohlenberg, Inst. Radio Engrs. Intern. Conv. Record, Part 4, 6 (1954).

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, Inst. Radio Engrs. Trans. Inform. Theory IT-6, 391 (1960).

Davenport, W. B.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), p. 282. [The details of the expansion Eq. (2) are worked out in this reference.]

Duffieux, P. M.

P. M. Duffieux, l’Integrale de Fourier et ses Applications à l’Optique, chez l’Auteur (Université de Besançon, Besançon, France, 1946).

Kohlenberg, A.

T. P. Cheatham, A. Kohlenberg, Inst. Radio Engrs. Intern. Conv. Record, Part 4, 6 (1954).

Leith, E.

Leith, E. N.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, Inst. Radio Engrs. Trans. Inform. Theory IT-6, 391 (1960).

Maréchal, A.

A. Maréchal, in Optical Processing of Information, D. K. Pollack, C. J. Koester, J. T. Tippett, eds. (Spartan, Baltimore, 1963).

O’Neill, E.

E. O’Neill, Inst. Radio Engrs. Trans. Inform. Theory IT-2, 55 (1956).

Palermo, C. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, Inst. Radio Engrs. Trans. Inform. Theory IT-6, 391 (1960).

Porcello, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, Inst. Radio Engrs. Trans. Inform. Theory IT-6, 391 (1960).

Root, W. L.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), p. 282. [The details of the expansion Eq. (2) are worked out in this reference.]

Turin, G. L.

G. L. Turin, Inst. Radio Engrs. Trans. Inform. Theory IT-6, 386 (1960).

Upatnieks, J.

Vander Lugt, A.

A. Vander Lugt, Trans. IEEE IT-10, 139 (1964).

Inst. Radio Engrs. Intern. Conv. Record (1)

T. P. Cheatham, A. Kohlenberg, Inst. Radio Engrs. Intern. Conv. Record, Part 4, 6 (1954).

Inst. Radio Engrs. Trans. Inform. Theory (3)

E. O’Neill, Inst. Radio Engrs. Trans. Inform. Theory IT-2, 55 (1956).

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, Inst. Radio Engrs. Trans. Inform. Theory IT-6, 391 (1960).

G. L. Turin, Inst. Radio Engrs. Trans. Inform. Theory IT-6, 386 (1960).

J. Opt. Soc. Am. (1)

Trans. IEEE (1)

A. Vander Lugt, Trans. IEEE IT-10, 139 (1964).

Other (4)

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), p. 282. [The details of the expansion Eq. (2) are worked out in this reference.]

P. M. Duffieux, l’Integrale de Fourier et ses Applications à l’Optique, chez l’Auteur (Université de Besançon, Besançon, France, 1946).

W. M. Brown, Analysis of Linear Time-Invariant Systems (McGraw-Hill, New York, 1963), Sec. 6-2, p. 171.

A. Maréchal, in Optical Processing of Information, D. K. Pollack, C. J. Koester, J. T. Tippett, eds. (Spartan, Baltimore, 1963).

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Figures (12)

Fig. 1
Fig. 1

A coherent optical system used for spatial filtering.

Fig. 2
Fig. 2

Matched filtering with a coherent optical system.

Fig. 3
Fig. 3

A spatial carrier modulated filter used for matched filtering.

Fig. 4
Fig. 4

A nonlinear system producing a hard-clipped transfer function.

Fig. 5
Fig. 5

Signal derived from a pseudo-random binary code.

Fig. 6
Fig. 6

Hard-clipped matched filter.

Fig. 7
Fig. 7

Result of filtering signal.

Fig. 8
Fig. 8

Noise used to obscure signal.

Fig. 9
Fig. 9

(a) Signal plus noise: S/N = −1.5 dB. (b) Result of filtering: S/N = −1.5 dB.

Fig. 10
Fig. 10

(a) Signal plus noise: S/N = −4.5 dB. (b) Result of filtering: S/N = −4.5 dB.

Fig. 11
Fig. 11

(a) Signal plus noise: S/N = −7.5 dB. (b) Result of filtering: S/N = −7.5 dB.

Fig. 12
Fig. 12

(a) Signal plus noise: S/N = −10.5 dB. (b) Result of filtering: S/N = −10.5 dB.

Equations (10)

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T = 1 2 + 1 2 A ( ω ) cos [ α ω + ϕ ( ω ) ] A ( ω ) cos [ α ω + ϕ ( ω ) ] ,
T = m = 0 C ( 0 , m ) cos [ m α ω + m ϕ ( ω ) ] ,
C ( 0 , m ) = m 2 Γ ( 1 - m / 2 ) Σ ( 1 + m / 2 ) ;
m = { 1 , m = 0 2 , m = 1 , 2 ,
T = 1 2 + m = 1 , 3 , 5 , 2 ( - 1 ) m - 1 2 m π cos [ m α ω + m ϕ ( ω ) ] .
T = m = 0 , ± 1 , ± 3 , C ( 0 , m ) m e i m [ α ω + ϕ ( ω ) ] .
F ( ω ) = { k 1 sin a ω ω e i ϕ ( ω ) , for - π / a ω π / a 0             otherwise .
T = { S b + k 2 ( sin a ω a ω ) cos [ α ω + ϕ ( ω ) ] , for - π / a ω π / a 0             otherwise ,
ρ m = | 1 2 π - π / a π / a k 1 k 2 sin a ω ω sin a ω a ω d ω | 2 N 0 2 π - π / a π / a k 2 2 ( sin a ω a ω ) 2 d ω , ρ m = 0.452 k 1 2 a N 0 .
ρ = | 1 2 π - π / a π / a k 1 π sin a ω ω d ω | 2 N 0 2 π - π / a π / a ( 1 π ) 2 d ω , ρ = 0.348 k 1 2 a N 0 .

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