Abstract

Detector array effects are considered for a time-integrating acoustooptic correlator used for signal detection. Effects such as detector area integration, detector element, spatial response, and the location of the correlation peak within a detector element are included. General SNR, PD, and PFA expressions are derived as a function of various system and detector parameters. Quantitative data are provided for a Gaussian-Markov signal, and initial experimental confirmation is included.

© 1985 Optical Society of America

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References

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  1. Special Issue on Acousto-Optic Signal Processing, Proc. IEEE 69, No. 1 (1981).
  2. Special Issue on Acousto-Optic Bulk Wave Devices, Proc. Soc. Photo-Opt. Instrum. Eng. 214 (1979).
  3. R. A. Sprague, C. L. Koliopoulos, “Time Integrating Acoustooptic Correlator,” Appl. Opt. 15, 89 (1976).
    [CrossRef] [PubMed]
  4. P. Kellman, “Time-Integrating Optical Signal Processors,” Ph.D. Thesis, Stanford U. (June1979).
  5. D. Psaltis, B. V. K. Vijaya Kumar, “Acoustooptic Spectral Estimation: A Statistical Analysis,” Appl. Opt. 20, 601 (1981).
    [CrossRef] [PubMed]
  6. D. Casasent, A. Goutzoulis, B. V. K. Vijaya Kumar, “Time-Integrating Acoustooptic Correlator: Error Source Modeling,” Appl. Opt. 23, 3130 (1984).
    [CrossRef] [PubMed]
  7. B. V. K. Vijaya Kumar, D. Casasent, A. Goutzoulis, “Fine Delay Estimation with Time Integrating Correlators,” Appl. Opt. 21, 3855(1982).
    [CrossRef]
  8. H. L. Van Trees, Detection, Estimation and Modulation Theory: Part 1 (Wiley, New York, 1965).
  9. H. Mostafavi, F. Smith, “Image Correlation with Geometric Distortion,” IEEE Trans. Aerosp. Electron. Syst. AES-14, 487 (1978).
    [CrossRef]
  10. B. V. K. Vijaya Kumar, D. Casasent, “Space-Blur Bandwidth Product in Correlator Performance Evaluation,” J. Opt. Soc. Am. 70, 103 (1980).
    [CrossRef]
  11. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).
  12. R. K. Hopwood, “Design Considerations for a Solid-State Image Sensing System,” Proc. Soc. Photo-Opt. Instrum. Eng. 230, 72 (1980).

1984 (1)

1982 (1)

1981 (2)

1980 (2)

B. V. K. Vijaya Kumar, D. Casasent, “Space-Blur Bandwidth Product in Correlator Performance Evaluation,” J. Opt. Soc. Am. 70, 103 (1980).
[CrossRef]

R. K. Hopwood, “Design Considerations for a Solid-State Image Sensing System,” Proc. Soc. Photo-Opt. Instrum. Eng. 230, 72 (1980).

1979 (1)

Special Issue on Acousto-Optic Bulk Wave Devices, Proc. Soc. Photo-Opt. Instrum. Eng. 214 (1979).

1978 (1)

H. Mostafavi, F. Smith, “Image Correlation with Geometric Distortion,” IEEE Trans. Aerosp. Electron. Syst. AES-14, 487 (1978).
[CrossRef]

1976 (1)

Casasent, D.

Goutzoulis, A.

Hopwood, R. K.

R. K. Hopwood, “Design Considerations for a Solid-State Image Sensing System,” Proc. Soc. Photo-Opt. Instrum. Eng. 230, 72 (1980).

Kellman, P.

P. Kellman, “Time-Integrating Optical Signal Processors,” Ph.D. Thesis, Stanford U. (June1979).

Koliopoulos, C. L.

Mostafavi, H.

H. Mostafavi, F. Smith, “Image Correlation with Geometric Distortion,” 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).

Psaltis, D.

Smith, F.

H. Mostafavi, F. Smith, “Image Correlation with Geometric Distortion,” IEEE Trans. Aerosp. Electron. Syst. AES-14, 487 (1978).
[CrossRef]

Sprague, R. A.

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation and Modulation Theory: Part 1 (Wiley, New York, 1965).

Vijaya Kumar, B. V. K.

Appl. Opt. (4)

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

H. Mostafavi, F. Smith, “Image Correlation with Geometric Distortion,” IEEE Trans. Aerosp. Electron. Syst. AES-14, 487 (1978).
[CrossRef]

J. Opt. Soc. Am. (1)

Proc. IEEE (1)

Special Issue on Acousto-Optic Signal Processing, Proc. IEEE 69, No. 1 (1981).

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

Special Issue on Acousto-Optic Bulk Wave Devices, Proc. Soc. Photo-Opt. Instrum. Eng. 214 (1979).

R. K. Hopwood, “Design Considerations for a Solid-State Image Sensing System,” Proc. Soc. Photo-Opt. Instrum. Eng. 230, 72 (1980).

Other (3)

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

P. Kellman, “Time-Integrating Optical Signal Processors,” Ph.D. Thesis, Stanford U. (June1979).

H. L. Van Trees, Detection, Estimation and Modulation Theory: Part 1 (Wiley, New York, 1965).

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Figures (8)

Fig. 1
Fig. 1

Schematic of a time-integrating acoustooptic correlator.

Fig. 2
Fig. 2

Effect of detector size D and integration time TI on PFA (for PD = 0.999).

Fig. 3
Fig. 3

Effects of input SNR I (amplitude) and detector size D on PFA (for PD = 0.999).

Fig. 4
Fig. 4

Effects of bandwidth BW s and detector size D on PFA (for TI = 100 μsec, SNR I = 0.1, SBR = ∞).

Fig. 5
Fig. 5

Effects of signal-to-bias ratio and detector size D on PFA for PD = 0.999.

Fig. 6
Fig. 6

Effects of the BWs D product and the location (delay) of the correlation peak within one detector (as a percent of D) on PFA (for PD = 0.999).

Fig. 7
Fig. 7

Effect of detector weighting profiles on PFA (for PD = 0.999) as a function of the detector element size D.

Fig. 8
Fig. 8

Theoretical and experimental data on the effect of the detector element size D on SNR1 and SNR2.

Equations (24)

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s 2 ( t ) = B 2 + s ( t τ 0 ) + n ( t ) .
I ( n ) = 1 T 1 T I / 2 T I / 2 ( n 1 / 2 ) D ( n + 1 / 2 ) D w n ( τ ) [ B 1 + s ( t τ ) ] × [ B 2 + s ( t τ 0 ) + n ( t ) ] d τ d t ,
P D = 1 2 π E 2 [ C ( 0 ) ] / SNR 1 θ exp ( SNR 1 { x E [ C ( 0 ) ] } 2 2 E 2 [ C ( 0 ) ] ) d x ,
P F A = 1 2 π E 2 [ C ( 0 ) ] / SNR 2 θ exp ( SNR 2 { x E [ C ( τ ) ] } 2 2 E 2 [ C ( 0 ) ] ) d x ,
I ( n ) = B T I T I / 2 T I / 2 ( n 1 / 2 ) D ( n + 1 / 2 ) D s ( t τ ) d τ d t + B T I T I / 2 T I / 2 ( n 1 / 2 ) D ( n + 1 / 2 ) D s ( t τ 0 ) d τ d t + B T I T I / 2 T I / 2 ( n 1 / 2 ) D ( n + 1 / 2 ) D n ( t ) d τ d t + 1 T I T I / 2 T I / 2 ( n 1 / 2 ) D ( n + 1 / 2 ) D s ( t τ ) n ( t ) d τ d t + 1 T I T I / 2 T I / 2 ( n 1 / 2 ) D ( n + 1 / 2 ) D s ( t τ ) s ( t τ 0 ) d τ d t .
E 2 [ I ( n ) ] = [ ( n 1 / 2 ) D ( n + 1 / 2 ) D R s ( τ τ 0 ) d τ ] 2 ,
var [ I ( n ) ] = E [ I ( n ) ] 2 E 2 [ I ( n ) ] = B 2 T I 2 T I / 2 T I / 2 ( n 1 / 2 ) D ( n + 1 / 2 ) D R s ( t u τ + τ ) d τ d τ dtdu + B 2 D 2 T I 2 T I / 2 T I / 2 R s ( t u ) dtdu + B 2 D 2 T I 2 T I / 2 T I / 2 R n ( t u ) dtdu + 1 T I 2 T I / 2 T I / 2 ( n 1 / 2 ) D ( n + 1 / 2 ) D R s ( t u τ + τ ) R n ( t u ) d τ d τ dtdu + 1 T I 2 T I / 2 T I / 2 ( n 1 / 2 ) D ( n + 1 / 2 ) D R s ( t u ) R s ( t u τ + τ ) d τ d τ dtdu + 1 T I 2 T I / 2 T I / 2 ( n 1 / 2 ) D ( n + 1 / 2 ) D R s ( t u τ + τ 0 ) R s ( t u τ + τ τ 0 ) d τ d τ dtdu + 2 B 2 D T I 2 T I / 2 T I / 2 ( n 1 / 2 ) D ( n + 1 / 2 ) D R s ( t u τ + τ 0 ) d τ dtdu ,
var [ I ( n ) ] = B 2 T I 2 T I T I ( n 1 / 2 ) D ( n + 1 / 2 ) D ( T I | z | ) R s ( z τ + τ ) d τ d τ d z + B 2 D 2 T I 2 ( 1 + 1 SNR I ) T I T I ( T I | z | ) R s ( z ) d z + 1 T I 2 ( 1 + 1 SNR I ) T I T I ( n 1 / 2 ) D ( n + 1 / 2 ) D ( T I | z | ) R s ( z ) R s ( z τ + τ ) d τ d τ d z + 1 T I 2 T I T I ( n 1 / 2 ) D ( n + 1 / 2 ) D ( T I | z | ) R s ( z τ + τ 0 ) R s ( z + τ τ 0 ) d τ d τ d z + 2 B 2 D T I 2 T I T I ( n 1 / 2 ) D ( n + 1 / 2 ) D ( T I | z | ) R s ( z τ + τ 0 ) d τ d z ,
var [ I ( n ) ] = B 2 T I T I T I D D ( D | q | ) R s ( z + q ) dqdz + B 2 D 2 T I ( 1 + 1 SNR I ) T I T I R s ( z ) d z + 1 T I ( 1 + 1 SNR I ) T I T I D D ( D | q | ) R s ( z ) × R s ( z + q ) dqdz + 1 T I T I T I ( n 1 / 2 ) D ( n + 1 / 2 ) D R s ( z τ + τ 0 ) × R s ( z + τ τ 0 ) d τ d τ d z + 2 B 2 D 2 T I T I T I ( n 1 / 2 ) D ( n + 1 / 2 ) D R s ( z τ + τ 0 ) d τ d z .
E 2 [ I ( 0 ) ] = [ D / 2 D / 2 R s ( τ τ 0 ) d τ ] 2 ,
R s ( z ) = R 0 exp ( β | z | ) ,
E 2 { [ I ( 0 ) ] = R 0 2 2 [ 2 exp [ β ( D / 2 + τ 0 ) ] exp [ β ( D / 2 + τ 0 ) ] } 2 .
var [ I ( 0 ) ] = R 0 B 2 D 2 T I β ( 8 + 2 SNR I ) + 4 R 0 2 D T I β 2 ( 2 + 1 SNR I ) + R 0 2 T I β 2 A 1 + R 0 2 T I β 2 ( 1 + 1 SNR I ) A 2
var [ I ( n ) ] = R 0 B 2 D 2 T I β ( 8 + 2 SNR I ) + R 0 2 T I β 2 ( 2 + 1 SNR I ) A 3 ,
A 1 = 8 τ 0 6 β exp ( 2 β τ 0 ) 4 τ 0 exp ( 2 β τ 0 ) + exp ( β D ) [ exp ( 2 β τ 0 ) + exp ( 2 β τ 0 ) ] ( D + 3 β ) exp ( β D ) [ exp ( 2 β τ 0 ) exp ( 2 β τ 0 ) ] 2 τ 0 ,
A 2 = 6 β + 6 β exp ( β D ) + 2 D exp ( β D ) ,
A 3 = 4 D 6 β + 6 β exp ( β D ) + 2 D exp ( β D ) .
SNR 1 = { 2 exp [ β ( D / 2 + τ 0 ) ] exp [ β ( D / 2 + τ 0 ) ] } 2 D 2 β T I ( SBR ) 2 ( 8 + 2 SNR I ) + 4 D T I ( 2 + 1 SNR I ) + A 1 T I + A 2 T I ( 1 + 1 SNR I ) ,
SNR 2 = { 2 exp [ β ( D / 2 + τ 0 ) ] exp [ β ( D / 2 + τ 0 ) ] } 2 D 2 β T I ( SBR ) 2 ( 8 + 2 SNR I ) + A 3 T I ( 1 + 1 SNR I ) ,
SNR 1 = T I β ( 2 + 1 SNR I ) + ( 8 + 2 SNR I ) 1 ( SBR ) 2 ,
SNR 2 = T I β ( 1 + 1 SNR I ) + ( 8 + 2 SNR I ) 1 ( SBR ) 2 .
P FAT = 1 ( 1 P FA ) N + 1
w n ( τ ) = { 2 τ D d 2 ( n 1 / 2 ) D D d n D D / 2 < τ < n D d / 2 , 1 n D d / 2 < τ < n D + d / 2 , 2 τ D d + 2 ( n + 1 / 2 ) D D d n D + d / 2 < τ < n D + D / 2 ,
E 2 ( I ( 0 ) ] = [ D / 2 D / 2 w 0 ( τ ) R s ( τ τ 0 ) d τ ] 2 .

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