Abstract

The power spectral density estimate obtained by an acoustooptic (AO) system is analyzed. We find that the variance of the estimate depends on the ratio T2/T1, where T1 is the aperture of the AO cell and T2 is the integration time of the detector.

© 1981 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. M. Jenkins, D. G. Watts, Spectral Analysis and its Applications (Holden-Day, San Francisco, 1968).
  2. L. H. Koopmans, The Spectral Analysis of Time Series (Academic, New York, 1974).
  3. I. C. Chang, IEEE Trans. Sonics Ultrason. SU-23, 2 (1976).
    [CrossRef]
  4. D. L. Hecht, Opt. Eng. 16, 461 (1977).
    [CrossRef]
  5. D. B. Anderson, IEEE Spectrum 15 (12), 22 (1978).
  6. A. Papoulis, Probability, Random Variable, and Stochastic Processes (McGraw-Hill, New York, 1965).
  7. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

1978 (1)

D. B. Anderson, IEEE Spectrum 15 (12), 22 (1978).

1977 (1)

D. L. Hecht, Opt. Eng. 16, 461 (1977).
[CrossRef]

1976 (1)

I. C. Chang, IEEE Trans. Sonics Ultrason. SU-23, 2 (1976).
[CrossRef]

Anderson, D. B.

D. B. Anderson, IEEE Spectrum 15 (12), 22 (1978).

Chang, I. C.

I. C. Chang, IEEE Trans. Sonics Ultrason. SU-23, 2 (1976).
[CrossRef]

Hecht, D. L.

D. L. Hecht, Opt. Eng. 16, 461 (1977).
[CrossRef]

Jenkins, G. M.

G. M. Jenkins, D. G. Watts, Spectral Analysis and its Applications (Holden-Day, San Francisco, 1968).

Koopmans, L. H.

L. H. Koopmans, The Spectral Analysis of Time Series (Academic, New York, 1974).

Papoulis, A.

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

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Watts, D. G.

G. M. Jenkins, D. G. Watts, Spectral Analysis and its Applications (Holden-Day, San Francisco, 1968).

IEEE Spectrum (1)

D. B. Anderson, IEEE Spectrum 15 (12), 22 (1978).

IEEE Trans. Sonics Ultrason. (1)

I. C. Chang, IEEE Trans. Sonics Ultrason. SU-23, 2 (1976).
[CrossRef]

Opt. Eng. (1)

D. L. Hecht, Opt. Eng. 16, 461 (1977).
[CrossRef]

Other (4)

G. M. Jenkins, D. G. Watts, Spectral Analysis and its Applications (Holden-Day, San Francisco, 1968).

L. H. Koopmans, The Spectral Analysis of Time Series (Academic, New York, 1974).

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

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Acoustooptic spectrum analyzer (AO = acoustooptic cell, FTL = Fourier transform lens, TIDA = time integrating detector array).

Fig. 2
Fig. 2

SNR of the PSD estimate as a function of the ratio (T2/T1) and as a function of time T2 for T1 = 0.5, 5.0, and 50.0 μsec.

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

W ( f ) = - R ( τ ) exp ( - j 2 π f τ ) d τ ,
S ( f ) = 1 T 1 | - T 1 / 2 T 1 / 2 s ( x ) exp ( - j 2 π f x ) d x | 2 ,
F ( f , t ) = T 1 s ( t + x ) exp ( - j 2 π f x ) d x ,
S ( f , t ) = 1 T 1 F ( f , t ) 2 = 1 T 1 | - T 1 / 2 T 1 / 2 s ( t + x ) exp ( - j 2 π f x ) d x | 2 ,
P ( f ) = 1 T 2 - T 2 / 2 T 2 / 2 S ( f , t ) d t .
SNR = E 2 P ( f ) Var [ P ( f ) ] .
E { P ( f ) } = 1 T 2 - T 2 / 2 T 2 / 2 E { S ( f , t ) } d t ,
E { S ( f , t ) } = 1 T 1 - T 1 / 2 T 1 / 2 E { s ( x + t ) s * ( y + t ) } exp [ - j 2 π f ( x - y ) ] d x d y = 1 T 1 - T 1 / 2 T 1 / 2 R s ( y - x ) exp [ j 2 π f ( y - x ) ] d x d y = - T 1 T 1 ( 1 - τ T 1 ) R s ( τ ) exp ( j 2 π f τ ) d τ = W ( f ) * { T 1 sinc 2 ( f T 1 ) } .
E { P ( f ) } = E { S ( f , t ) } = W ( f ) * { T 1 sinc 2 ( f T 1 ) } ,
Var { P ( f ) } = E { P ( f ) P * ( f ) } - E { P ( f ) } 2 = E { 1 T 2 2 - T 2 / 2 + T 2 / 2 S ( f , t ) S * ( f , u ) d t d u } - E { S ( f ) } 2 = 1 T 2 2 - T 2 / 2 + T 2 / 2 [ E { S ( f , t ) S * ( f , u ) } - E { S ( f ) } 2 ] d t d u .
E { S ( f , t ) S * ( f , u ) } - E { S ( f ) } 2 = 1 T 1 2 - T 1 / 2 + T 1 / 2 E { s ( x + t ) s ( y + t ) s ( p + u ) s ( q + u ) } · exp [ - j 2 π f ( x - y + p - q ) ] d x d y d p d q - E { S ( f ) } 2 .
Var { P ( f ) } = 1 T 2 2 - T 2 / 2 + T 2 / 2 d t d u [ 1 / T 1 2 - T 1 / 2 + T 1 / 2 × exp [ - j 2 π f ( x - y + p - q ) ] · { R s ( p - x + u - t ) R s ( q - y + u - t ) + R s ( q - x + u - t ) R s ( p - y + u - t ) } d x d y d p d q ] .
E { P ( f ) } = E { S ( f ) } = R 0 .
Var { S ( f ) = R 0 2 { 1 + sinc 2 ( 2 f T 1 ) } .
SNR { S ( f ) = E { S ( f ) } 2 Var { S ( f ) } = R 0 2 R 0 2 { 1 + sinc 2 ( 2 f T 1 ) } = [ 1 + sinc 2 ( 2 f T 1 ) ] - 1 .
Var { P ( f ) } { R 0 2 [ 1 - 2 / 3 ( T 2 / T 1 ) + 1 / 6 ( T 2 / T 1 ) 2 ] for T 1 > T 2 R 0 2 [ 2 / 3 ( T 1 / T 2 ) - 1 / 6 ( T 1 / T 2 ) 2 ] for T 2 > T 1 .
SNR { P ( f ) } { [ 1 - 2 / 3 ( T 2 / T 1 ) + 1 / 6 ( T 2 / T 1 ) 2 ] - 1 for T 1 > T 2 , [ 2 / 3 ( T 1 / T 2 ) - 1 / 6 ( T 1 / T 2 ) 2 ] - 1 for T 2 > T 1 .
SNR { P ( f ) } 3 / 2 ( T 2 / T 1 ) ,
| Δ W ( f ) Δ f | < W max Δ τ = > Δ W ( f ) < W max Δ τ T 1 ,
Var { P ( f ) } = 1 T 2 - T 2 T 2 ( 1 - γ T 2 ) | - W ( f - f ) exp [ j 2 π γ ( f - f ) ] × T 1 sinc 2 ( f T 1 ) d f | 2 d γ + 1 T 2 - T 2 T 2 ( 1 - γ T 2 ) | - T 1 T 1 × R s ( α + γ ) sinc [ 2 f ( T 1 - α ) ] ( 1 - α T 1 ) d α | 2 d γ .
Var { P ( f ) } 1 T 2 - T 2 T 2 ( 1 - γ T 2 ) | - W ( f - f ) exp [ j 2 π γ ( f - f ) × T 1 sinc 2 ( f T 1 ) d f | 2 d γ 1 T 2 - T 2 T 2 ( 1 - γ T 2 ) | W ( f ) exp ( j 2 π γ f ) × - T 1 sinc 2 ( f T 1 ) exp ( - j 2 π γ f ) d f | 2 d γ .
Var { P ( f ) } { W ( f ) 2 [ 1 - ( 2 / 3 ) ( T 2 / T 1 ) + ( 1 / 6 ) ( T 2 / T 1 ) 2 ] for T 2 < T 1 , W ( f ) 2 [ ( 2 / 3 ) ( T 1 / T 2 ) - ( 1 / 6 ) ( T 1 / T 2 ) 2 ] for T 1 < T 2 .
Var { P ( f ) } = T 1 2 T 2 T 2 T 2 ( 1 - γ T 2 ) | f 0 - Δ f 2 f 0 + Δ f 2 sinc 2 [ T 1 ( f - f ) ] × exp ( j 2 π γ f ) d f | 2 d γ .
Var { P } = 1 + sinc 4 ( Δ f T 1 2 ) + 2 sinc 2 ( Δ f T 1 2 ) × { 2 [ 1 - cos ( π T 2 Δ f / 2 ) ] ( π T 2 Δ f / 2 ) 2 } .
lim T 2 [ Var { P } ] = 1 + sinc 4 ( Δ f T 1 2 ) .
SNR = E { P ( f ) } 2 Var { P ( f ) } + Var { n } .
SNR = E { P 1 ( f ) } 2 Var { P 1 ( f ) } + Var { P 2 ( f ) } { [ W 1 ( f ) 2 W 1 ( f ) 2 + W 2 ( f ) 2 ] [ 1 - 2 / 3 ( T 2 / T 1 + 1 / 6 ( T 2 / T 1 ) 2 ] - 1 for T 2 < T 1 [ W 1 ( f ) 2 W 1 ( f ) 2 + W 2 ( f ) 2 ] [ 2 / 3 ( T 1 / T 2 ) - 1 / 6 ( T 1 / T 2 ) 2 ] - 1 for T 1 < T 2 .
Var { P ( f ) } = R 0 2 T 1 2 T 2 2 - T 2 / 2 + T 2 / 2 × d t d u [ - T 1 / 2 T 1 / 2 exp [ - j 2 π f ( x - y + p - q ) ] · { δ ( p - x + u - t ) δ ( q - y + u - t ) + δ ( q - x + u - t ) δ ( p - y + u - t ) } d x d y d p d q ] .
Var { P ( f ) } = R 0 2 T 1 2 T 2 - T 2 T 2 ( 1 - τ T 2 ) × | T 1 - T 1 T 1 ( 1 - α T 1 ) δ ( α + τ ) exp ( - j 2 π f α ) d α | 2 d τ + R 0 2 T 1 2 T 2 - T 2 T 2 ( 1 - τ T 2 ) × | - T 1 / 2 T 1 / 2 δ ( p - x + τ ) × exp [ - j 2 π f ( p + x ) ] d p d x | 2 d τ ,
Var { P ( f ) } = 1 T 2 - T 2 T 2 ( 1 - τ T 2 ) × | - T 1 T 1 ( 1 - α T 1 ) δ ( α + τ ) exp ( - j 2 π f α ) d α | 2 d τ + 1 4 T 1 2 T 2 - T 2 T 2 ( 1 - τ T 2 ) × | - T 1 T 1 sin 2 π f ( T 1 - u ) π f δ ( u + τ ) d u | 2 d τ .
Var { P ( f ) } = 1 T 2 - T 2 T 2 ( 1 - τ T 2 ) ( 1 - τ T 1 ) 2 d τ + 1 T 1 2 T 2 - T 2 T 2 ( 1 - τ T 2 ) [ sin 2 π f ( T 1 - τ ) 2 π f ] 2 d τ = [ 1 - ( 2 / 3 ) ( T 2 T 1 ) + ( 1 / 6 ) ( T 2 T 1 ) 2 ] + 1 4 π 2 f 2 T 1 2 T 2 2 × { T 2 2 2 + T 2 4 π f sin 4 π f T 1 - T 2 4 π f sin 4 π ( T 1 - T 2 ) + cos 4 π f ( T 1 - T 2 ) ( 4 π f ) 2 - cos 4 π f T 1 ( 4 π f ) 2 } .
Var { P ( f ) } 1 - ( 2 / 3 ) ( T 2 T 1 ) + ( 1 / 6 ) ( T 2 T 1 ) 2             for             T 2 < T 1 .
Var { P ( f ) } ( 2 / 3 ) ( T 1 T 2 ) - ( 1 / 6 ) ( T 1 T 2 ) 2             for             T 1 < T 2 ,

Metrics