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

1982

1981

1980

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

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

1979

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

1978

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

1976

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.

IEEE Trans. Aerosp. Electron. Syst.

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

J. Opt. Soc. Am.

Proc. IEEE

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

Proc. Soc. Photo-Opt. Instrum. Eng.

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

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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