Abstract

Synchronous detection of randomly phase-modulated interferograms is examined for the cases of low and high SNR. Long-average phase estimates exhibit (time × bandwidth)−1 dependence and give useful results with low SNR and σϕ < π. Optimum averaging times are determined for phase tracking in the case of relatively high SNR.

© 1981 Optical Society of America

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References

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  1. M. Vinokur, Ann. Astrophys. 28, 412 (1965).
  2. H. R. Raemer, Statistical Communications Theory and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1969), Chap. 8.
  3. M. A. Johnson, A. L. Betz, C. H. Townes, Phys. Rev. Lett. 33, 1617 (1974).
    [CrossRef]
  4. P. Assus et al., J. Optics (Paris) 10, 345 (1979).
    [CrossRef]
  5. D. W. McCarthy, F. J. Low, R. Howell, Opt. Eng. 16, 569 (1977).
    [CrossRef]
  6. N. Abramson, IEEE Trans. Commun. C-11, 407 (1963).
    [CrossRef]
  7. D. J. Sakrison, Communications Theory: Transmission of Waveforms and Digital Information (Wiley, New York, 1968), Chap. 7.

1979 (1)

P. Assus et al., J. Optics (Paris) 10, 345 (1979).
[CrossRef]

1977 (1)

D. W. McCarthy, F. J. Low, R. Howell, Opt. Eng. 16, 569 (1977).
[CrossRef]

1974 (1)

M. A. Johnson, A. L. Betz, C. H. Townes, Phys. Rev. Lett. 33, 1617 (1974).
[CrossRef]

1965 (1)

M. Vinokur, Ann. Astrophys. 28, 412 (1965).

1963 (1)

N. Abramson, IEEE Trans. Commun. C-11, 407 (1963).
[CrossRef]

Abramson, N.

N. Abramson, IEEE Trans. Commun. C-11, 407 (1963).
[CrossRef]

Assus, P.

P. Assus et al., J. Optics (Paris) 10, 345 (1979).
[CrossRef]

Betz, A. L.

M. A. Johnson, A. L. Betz, C. H. Townes, Phys. Rev. Lett. 33, 1617 (1974).
[CrossRef]

Howell, R.

D. W. McCarthy, F. J. Low, R. Howell, Opt. Eng. 16, 569 (1977).
[CrossRef]

Johnson, M. A.

M. A. Johnson, A. L. Betz, C. H. Townes, Phys. Rev. Lett. 33, 1617 (1974).
[CrossRef]

Low, F. J.

D. W. McCarthy, F. J. Low, R. Howell, Opt. Eng. 16, 569 (1977).
[CrossRef]

McCarthy, D. W.

D. W. McCarthy, F. J. Low, R. Howell, Opt. Eng. 16, 569 (1977).
[CrossRef]

Raemer, H. R.

H. R. Raemer, Statistical Communications Theory and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1969), Chap. 8.

Sakrison, D. J.

D. J. Sakrison, Communications Theory: Transmission of Waveforms and Digital Information (Wiley, New York, 1968), Chap. 7.

Townes, C. H.

M. A. Johnson, A. L. Betz, C. H. Townes, Phys. Rev. Lett. 33, 1617 (1974).
[CrossRef]

Vinokur, M.

M. Vinokur, Ann. Astrophys. 28, 412 (1965).

Ann. Astrophys. (1)

M. Vinokur, Ann. Astrophys. 28, 412 (1965).

IEEE Trans. Commun. (1)

N. Abramson, IEEE Trans. Commun. C-11, 407 (1963).
[CrossRef]

J. Optics (Paris) (1)

P. Assus et al., J. Optics (Paris) 10, 345 (1979).
[CrossRef]

Opt. Eng. (1)

D. W. McCarthy, F. J. Low, R. Howell, Opt. Eng. 16, 569 (1977).
[CrossRef]

Phys. Rev. Lett. (1)

M. A. Johnson, A. L. Betz, C. H. Townes, Phys. Rev. Lett. 33, 1617 (1974).
[CrossRef]

Other (2)

D. J. Sakrison, Communications Theory: Transmission of Waveforms and Digital Information (Wiley, New York, 1968), Chap. 7.

H. R. Raemer, Statistical Communications Theory and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1969), Chap. 8.

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Figures (3)

Fig. 1
Fig. 1

Output phase-fluctuation noise power as a function of BϕT for the values of σϕ noted beside the curve. The inner, left-hand scale gives β2/2 for the values of σϕ indicated.

Fig. 2
Fig. 2

Attenuation factor as a function of T/Tc.

Fig. 3
Fig. 3

Normalized MSE as a function of T/Tc. The dashed lines give the MSE when additive noise is zero.

Equations (30)

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v ( t ) = A cos [ 2 π f 0 t + α + ϕ ( t ) ] + n ( t ) ,
ϕ ( t ) = 0 , ϕ 2 ( t ) = σ ϕ 2 , ϕ ( t ) ϕ ( t + τ ) = R ϕ ( τ ) = σ ϕ 2 exp ( τ / T c ) ,
n ( t ) = 0 , n 2 ( t ) = σ n 2 , n ( t ) n ( t + τ ) = R n ( τ ) = ( σ n 2 / 2 B n ) δ ( τ ) ,
Φ ( f ) = ( σ ϕ 2 / π B ) [ 1 + ( f / B ϕ ) 2 ] - 1 ,
C ^ = 1 T - T / 2 T / 2 { A cos [ α + ϕ ( t ) ] + n c ( t ) } d t ,
S ^ = 1 T - T / 2 T / 2 { A sin [ α + ϕ ( t ) ] + n s ( t ) } d t ,
A ^ = ( C ^ 2 + S ^ 2 ) 1 / 2             and             α ^ = tan - 1 ( - S ^ / C ^ ) .
C = C ^ = A β cos α             and             S = S ^ = A β sin α .
σ c 2 = C ^ 2 - C ^ 2             and             σ s 2 = S ^ 2 - S ^ 2 ,
C ^ 2 = ( 1 T - T / 2 T / 2 { A cos [ α + ϕ ( t ) ] + n c ( t ) } d t ) 2 = 1 T 2 - T T ( T - τ ) [ R c ( τ ) + R n c ( τ ) ] d τ ,
S ^ 2 = 1 T 2 - T T ( T - τ ) [ R s ( τ ) + R n s ( τ ) ] d τ .
σ c n 2 = σ s n 2 = σ n 2 / B n T .
R c ( τ ) = A 2 2 ( cos θ 1 + cos θ 2 cos 2 α )
R c ( τ ) = A 2 2 ( exp { - [ σ ϕ 2 - R ϕ ( τ ) ] } + exp { - [ σ ϕ 2 + R ϕ ( τ ) ] } cos 2 α ) ,
R s ( τ ) = A 2 2 ( exp { - [ σ ϕ 2 - R ϕ ( τ ) ] } - exp { - [ σ ϕ 2 + R ϕ ] } cos 2 α )
σ c ϕ 2 = 1 T 2 - T T ( T - τ ) R c ( τ ) d τ - A 2 β 2 cos 2 α ,
σ s ϕ 2 = 1 T 2 - T T ( T - τ ) R s ( τ ) d τ - A 2 β 2 sin 2 α .
σ 0 2 = ( K A 2 / B ϕ T ) + ( σ n 2 / B n T ) .
σ ϕ < 0.5 rad , K σ ϕ 2 / 2 π σ ϕ 1 rad , K ( 2 π σ ϕ 2 ) - 1 .
σ α 2 = [ ( K A 2 / B ϕ ) + ( σ n 2 / B n ) ] ( A 2 β 2 T ) - 1 .
C ^ ( t 0 ) = 1 T t 0 - T / 2 t 0 + T / 2 { A cos [ θ ( t 0 ) + ϕ ( t ) - ϕ ( t 0 ) ] + n c ( t ) } d t ,
C ( t 0 ) = A t 0 - T / 2 t 0 + T / 2 [ cos ( α + ϕ 0 ) cos ( ϕ - ϕ 0 ) - sin ( α + ϕ 0 ) sin ( ϕ - ϕ 0 ) ] d t ,
p ( ϕ / τ , ϕ 0 ) = 1 { 2 π [ 1 - ρ 2 ( τ ) ] } 1 / 2 σ ϕ · exp { [ ϕ - ρ ( τ ) ϕ 0 ] 2 2 σ ϕ 2 [ 1 - ρ 2 ( τ ) ] } ,
cos ( ϕ - ϕ 0 ) = cos { ϕ 0 [ 1 - ρ ( τ ) ] } exp { - σ ϕ 2 [ 1 - ρ 2 ( τ ) ] / 2 } , sin ( ϕ - ϕ 0 ) = sin { ϕ 0 [ ρ ( τ ) - 1 ] } exp { - σ ϕ 2 [ 1 - ρ 2 ( τ ) ] / 2 } .
A ^ = A - T / 2 T / 2 exp { - σ ϕ 2 [ 1 - ρ 2 ( τ ) ] / 2 } d τ = A D ( σ ϕ , T ) ,
2 = - [ 1 - H a ( f ) 2 G ( f ) + N ( f ) H a ( f ) 2 ] d f ,
c , s 2 = - [ R A R c , s ( τ ) + R n ( τ ) R a ( τ ) ] d τ ,
h a ( t ) = 1 T t T / 2 = 0 t > T / 2 , R a ( τ ) = T - τ T 2 τ T = 0 τ > T .
n 2 = σ n 2 / B n T .
ϕ 2 = A 2 { 0.5 - 2 γ - 1 [ 1 - exp ( - γ / 2 ) ] + γ - 1 - γ - 2 [ 1 - exp ( - γ ) ] } .

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