Abstract

Correlations of binarized data and gray-scale data in the presence of additive noise are compared using two performance criteria (mean ratio and peak-to-sidelobe ratio). General expressions are provided to evaluate these performance measures as a function of input data length, bandwidth, and signal-to-noise ratio. We find that binarized correlators outperform gray-scale correlators when the input SNRi exceeds a threshold level that depends on input bandwidth and sequence length.

© 1981 Optical Society of America

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References

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  1. J. Max, IRE Trans. Inf. Theory IT-6, 7 (1960).
    [CrossRef]
  2. P. Wintz, Proc. IEEE 60, 809 (1972).
    [CrossRef]
  3. D. Casasent, Proc. IEEE 65, 143 (1977).
    [CrossRef]
  4. B. V. K. Vijaya Kumar, D. Casasent, “Input Quantization Effects in an Image Correlator,” submitted to Appl. Opt.
  5. H. Mostafavi, F. Smith, IEEE Trans. Aerosp. Electron. Syst. AES-14, 487 (1978).
    [CrossRef]
  6. B. V. K. Vijaya Kumar, D. Casasent, J. Opt. Soc. Am. 70, 103 (1980).
    [CrossRef]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  8. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).
  9. T. W. Cole, Appl. Opt. 19, 2169 (1980).
    [CrossRef] [PubMed]
  10. D. Casasent, A. Furman, Appl. Opt. 16, 1652 (1977).
    [CrossRef] [PubMed]
  11. H. L. Van Trees, Detection, Estimation and Modulation Theory; Part 1 (Wiley, New York, 1968).
  12. F. O. Huck, N. Halyo, S. K. Park, Appl. Opt. 19, 2174 (1980).
    [CrossRef] [PubMed]

1980 (3)

1978 (1)

H. Mostafavi, F. Smith, IEEE Trans. Aerosp. Electron. Syst. AES-14, 487 (1978).
[CrossRef]

1977 (2)

1972 (1)

P. Wintz, Proc. IEEE 60, 809 (1972).
[CrossRef]

1960 (1)

J. Max, IRE Trans. Inf. Theory IT-6, 7 (1960).
[CrossRef]

Casasent, D.

B. V. K. Vijaya Kumar, D. Casasent, J. Opt. Soc. Am. 70, 103 (1980).
[CrossRef]

D. Casasent, A. Furman, Appl. Opt. 16, 1652 (1977).
[CrossRef] [PubMed]

D. Casasent, Proc. IEEE 65, 143 (1977).
[CrossRef]

B. V. K. Vijaya Kumar, D. Casasent, “Input Quantization Effects in an Image Correlator,” submitted to Appl. Opt.

Cole, T. W.

Furman, A.

Goodman, J. W.

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

Halyo, N.

Huck, F. O.

Max, J.

J. Max, IRE Trans. Inf. Theory IT-6, 7 (1960).
[CrossRef]

Mostafavi, H.

H. Mostafavi, F. Smith, IEEE Trans. Aerosp. Electron. Syst. AES-14, 487 (1978).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).

Park, S. K.

Smith, F.

H. Mostafavi, F. Smith, IEEE Trans. Aerosp. Electron. Syst. AES-14, 487 (1978).
[CrossRef]

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation and Modulation Theory; Part 1 (Wiley, New York, 1968).

Vijaya Kumar, B. V. K.

B. V. K. Vijaya Kumar, D. Casasent, J. Opt. Soc. Am. 70, 103 (1980).
[CrossRef]

B. V. K. Vijaya Kumar, D. Casasent, “Input Quantization Effects in an Image Correlator,” submitted to Appl. Opt.

Wintz, P.

P. Wintz, Proc. IEEE 60, 809 (1972).
[CrossRef]

Appl. Opt. (3)

IEEE Trans. Aerosp. Electron. Syst. (1)

H. Mostafavi, F. Smith, IEEE Trans. Aerosp. Electron. Syst. AES-14, 487 (1978).
[CrossRef]

IRE Trans. Inf. Theory (1)

J. Max, IRE Trans. Inf. Theory IT-6, 7 (1960).
[CrossRef]

J. Opt. Soc. Am. (1)

Proc. IEEE (2)

P. Wintz, Proc. IEEE 60, 809 (1972).
[CrossRef]

D. Casasent, Proc. IEEE 65, 143 (1977).
[CrossRef]

Other (4)

B. V. K. Vijaya Kumar, D. Casasent, “Input Quantization Effects in an Image Correlator,” submitted to Appl. Opt.

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

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).

H. L. Van Trees, Detection, Estimation and Modulation Theory; Part 1 (Wiley, New York, 1968).

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Figures (4)

Fig. 1
Fig. 1

Loss in binarized correlation peak I p as a function of input signal-to-noise ratio SNRi.

Fig. 2
Fig. 2

Binarized mean ratio MR′ as a function of the gray-scale mean ratio MR for input signal-to-noise ratios of −10, +10, and +30 dB.

Fig. 3
Fig. 3

Ratio (MR′/MR) of mean ratios as a function of input signal-to-noise ratio SNRi.

Fig. 4
Fig. 4

Minimum SNRi required for a binarized correlator to yield better SNR′ than the gray-scale correlator SNR, plotted as a function of the signal correlation coefficient ρ0 for different input data sequence lengths N = 16, 64, and 256.

Equations (27)

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y ( n ) = x ( n m 0 ) + w ( n ) ,
c ( m ) = 1 N n = 1 N x ( n ) y ( n + m ) = 1 N n = 1 N [ x ( n ) x ( n + m m 0 ) + x ( n ) w ( n + m ) ] ,
E [ C ( m ) ] = R x ( m m 0 ) ,
covar [ c ( m 1 ) , c ( m 2 ) ] = E [ c ( m 1 ) c ( m 2 ) ] E [ c ( m 1 ) ] E [ c ( m 2 ) ] = 1 N n = N N ( 1 | n | N ) [ R x ( n ) · R x ( n + m 2 m 1 ) + R x ( n + m 2 m 0 ) · R w ( n + m 1 m 0 ) + R x ( n ) · R w ( n + m 2 m 1 ) ] .
s ( n ) = { + α if s ( n ) 0 , α if s ( n ) < 0 ,
c ( m ) = 1 N n = 1 N x ( n ) y ( n + m ) .
E [ c ( m ) ] = 1 N n = 1 N E [ x ( n ) y ( n + m ) ] = α 2 N n = 1 N [ Pr · { x ( n ) y ( n + m ) 0 } Pr · { x ( n ) y ( n + m ) < 0 } ] ,
E [ x 2 ( n ) ] = R x ( 0 ) ,
E [ y 2 ( n + m ) ] = R x ( 0 ) + R w ( 0 ) ,
E [ x ( n ) y ( n + m ) ] = R x ( m m 0 ) .
E [ c ( m ) ] = 2 α 2 π sin 1 [ R x ( m m 0 ) R x 2 ( 0 ) + R x ( 0 ) · R w ( 0 ) ] .
var [ c ( m * ) ] = E { [ c ( m * ) ] 2 } = 1 N 2 n = 1 N k = 1 N E [ x ( n ) x ( k ) ] × E [ y ( k + m * ) y ( n + m * ) ] = 4 α N π 2 n = N N { ( 1 | n | N ) sin 1 [ R x ( n ) R x ( 0 ) ] × sin 1 [ R x ( n ) + R w ( n ) R x ( 0 ) + R w ( 0 ) ] } ,
I p = R x ( 0 ) ,
I p = 2 α 2 π sin 1 [ 1 1 + R w ( 0 ) R x ( 0 ) ] .
I p = ( 2 α 2 π ) sin 1 ( 1 1 + 1 SNR i ) .
MR ( m * ) = E [ c ( m 0 ) ] E [ c ( m * ) ] .
MR ( m * ) = R x ( 0 ) R x ( m * m 0 ) .
M R ( m * ) = sin 1 ( 1 / 1 + 1 SNR i ) sin 1 { 1 / [ MR ( m * ) 1 + 1 SNR i ] } .
M R ( m * ) MR ( m * ) .
M R ( m * ) MR ( m * ) [ 1 + SNR i SNR i sin 1 ( SNR i 1 + SNR i ) ]
SNR = E 2 [ c ( m 0 ) ] var [ c ( m * ) ] ,
SNR = { 1 N n = N N ( 1 | n | N ) [ R x 2 ( n ) + R x ( n ) R w ( n ) R x 2 ( 0 ) ] } 1 = { 1 N n = N N ( 1 | n | N ) R x 2 ( n ) R x 2 ( 0 ) [ 1 + R w ( n ) R x ( n ) ] } 1 = [ 1 N n = N N ( 1 | n | N ) R x 2 ( n ) R x 2 ( 0 ) ( 1 + 1 SNR i ) ] 1 .
SN R = N [ 2 π sin 1 ( SNR i 1 + SNR i ) ] 2 · { n = N N ( 1 | n | N ) × [ 2 π sin 1 R x ( n ) R x ( 0 ) ] 2 } 1 .
SN R SNR = F ( SNR i ) · G [ N , R x ( n ) ] ,
F ( x ) = [ 2 π 1 + x x sin 1 ( x 1 + x ) ] 2 ,
G [ N , R x ( n ) ] = { n = N N ( 1 | n | N ) [ R x 2 ( n ) / R x 2 ( 0 ) ] n = N N ( 1 | n | N ) [ 2 π sin 1 R x ( n ) R x ( 0 ) ] 2 } .
R x ( n ) = R x ( 0 ) exp ( a | n | ) ,

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