Abstract

The effect of source (LED) noise on the sensitivity of fiber-optic methane sensors is discussed. Once the source is dominant, the system sensitivity cannot be improved by increasing the source power further.

© 1995 Optical Society of America

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References

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  1. Y. Shimose, T. Okamoto, A. Maruyma, M. Arizawa, H. Nagai, “Remote sensing of methane gas by differential absorption measurement using a wavelength tunable DFB LD,” IEEE Photon. Technol. Lett. 3, 86–89 (1991).
    [CrossRef]
  2. K. Uehara, H. Tai, “Remote detection of methane with a 1.66-μm diode laser,” Appl. Opt. 31, 809–813 (1992).
    [CrossRef] [PubMed]
  3. J. P. Dakin, C. A. Wade, D. Pinchbeck, J. S. Wykes, “A novel optical fiber methane sensor,” J. Opt. Sensors 2, 261–267 (1987).
  4. W. Jin, G. Stewart, B. Culshaw, S. Murray, D. Pinchbeck, “Absorption measurement of methane gas with a broadband source and interferometric signal processing,” Opt. Lett. 18, 1364–1366 (1993).
    [CrossRef] [PubMed]
  5. B. Culshaw, F. Muhammad, G. Stewart, S. Murray, D. Pinchbeck, J. Norris, S. Cassidy, M. Wilkinson, D. Williams, I. Crisp, R. V. Ewyk, A. McGhee, “Evanescent wave methane detection using optical fibre,” Electron. Lett. 28, 2232–2233 (1992).
    [CrossRef]
  6. W. Jin, G. Stewart, B. Culshaw, S. Murray, M. Wilkinson, J. Norris, A. McGhee, “A D-shaped optical fibre methane sensor,” presented at the Conference on Applied Optics and OptoElectronics, IOP, 5–8 September 1994.
  7. W. Jin, G. Stewart, B. Culshaw, S. Murray, M. Wilkinson, J. O. W. Norris, “Performance limitation of a D-fiber evanescent wave methane sensor due to interference effects,” submitted to J. Lightwave Technol.
  8. F. A. Muhammad, G. Stewart, “D-shaped optical fibre design for methane gas sensing,” Electron. Lett. 28, 1025–1026 (1993).
  9. M. Tur, E. Shafir, K. Blotekjaer, “Source-induced noise in optical systems driven by low-coherence sources,” J. Light-wave Technol. 8, 183–189 (1990).
    [CrossRef]
  10. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 3, p. 109.
  11. B. Moslehi, “Analysis of optical phase noise in fiber-optic systems employing a laser source with arbitrary coherence time,” J. Lightwave Technol. LT-4, 1334–1351 (1986).
    [CrossRef]

1993 (2)

1992 (2)

B. Culshaw, F. Muhammad, G. Stewart, S. Murray, D. Pinchbeck, J. Norris, S. Cassidy, M. Wilkinson, D. Williams, I. Crisp, R. V. Ewyk, A. McGhee, “Evanescent wave methane detection using optical fibre,” Electron. Lett. 28, 2232–2233 (1992).
[CrossRef]

K. Uehara, H. Tai, “Remote detection of methane with a 1.66-μm diode laser,” Appl. Opt. 31, 809–813 (1992).
[CrossRef] [PubMed]

1991 (1)

Y. Shimose, T. Okamoto, A. Maruyma, M. Arizawa, H. Nagai, “Remote sensing of methane gas by differential absorption measurement using a wavelength tunable DFB LD,” IEEE Photon. Technol. Lett. 3, 86–89 (1991).
[CrossRef]

1990 (1)

M. Tur, E. Shafir, K. Blotekjaer, “Source-induced noise in optical systems driven by low-coherence sources,” J. Light-wave Technol. 8, 183–189 (1990).
[CrossRef]

1987 (1)

J. P. Dakin, C. A. Wade, D. Pinchbeck, J. S. Wykes, “A novel optical fiber methane sensor,” J. Opt. Sensors 2, 261–267 (1987).

1986 (1)

B. Moslehi, “Analysis of optical phase noise in fiber-optic systems employing a laser source with arbitrary coherence time,” J. Lightwave Technol. LT-4, 1334–1351 (1986).
[CrossRef]

Arizawa, M.

Y. Shimose, T. Okamoto, A. Maruyma, M. Arizawa, H. Nagai, “Remote sensing of methane gas by differential absorption measurement using a wavelength tunable DFB LD,” IEEE Photon. Technol. Lett. 3, 86–89 (1991).
[CrossRef]

Blotekjaer, K.

M. Tur, E. Shafir, K. Blotekjaer, “Source-induced noise in optical systems driven by low-coherence sources,” J. Light-wave Technol. 8, 183–189 (1990).
[CrossRef]

Cassidy, S.

B. Culshaw, F. Muhammad, G. Stewart, S. Murray, D. Pinchbeck, J. Norris, S. Cassidy, M. Wilkinson, D. Williams, I. Crisp, R. V. Ewyk, A. McGhee, “Evanescent wave methane detection using optical fibre,” Electron. Lett. 28, 2232–2233 (1992).
[CrossRef]

Crisp, I.

B. Culshaw, F. Muhammad, G. Stewart, S. Murray, D. Pinchbeck, J. Norris, S. Cassidy, M. Wilkinson, D. Williams, I. Crisp, R. V. Ewyk, A. McGhee, “Evanescent wave methane detection using optical fibre,” Electron. Lett. 28, 2232–2233 (1992).
[CrossRef]

Culshaw, B.

W. Jin, G. Stewart, B. Culshaw, S. Murray, D. Pinchbeck, “Absorption measurement of methane gas with a broadband source and interferometric signal processing,” Opt. Lett. 18, 1364–1366 (1993).
[CrossRef] [PubMed]

B. Culshaw, F. Muhammad, G. Stewart, S. Murray, D. Pinchbeck, J. Norris, S. Cassidy, M. Wilkinson, D. Williams, I. Crisp, R. V. Ewyk, A. McGhee, “Evanescent wave methane detection using optical fibre,” Electron. Lett. 28, 2232–2233 (1992).
[CrossRef]

W. Jin, G. Stewart, B. Culshaw, S. Murray, M. Wilkinson, J. Norris, A. McGhee, “A D-shaped optical fibre methane sensor,” presented at the Conference on Applied Optics and OptoElectronics, IOP, 5–8 September 1994.

W. Jin, G. Stewart, B. Culshaw, S. Murray, M. Wilkinson, J. O. W. Norris, “Performance limitation of a D-fiber evanescent wave methane sensor due to interference effects,” submitted to J. Lightwave Technol.

Dakin, J. P.

J. P. Dakin, C. A. Wade, D. Pinchbeck, J. S. Wykes, “A novel optical fiber methane sensor,” J. Opt. Sensors 2, 261–267 (1987).

Ewyk, R. V.

B. Culshaw, F. Muhammad, G. Stewart, S. Murray, D. Pinchbeck, J. Norris, S. Cassidy, M. Wilkinson, D. Williams, I. Crisp, R. V. Ewyk, A. McGhee, “Evanescent wave methane detection using optical fibre,” Electron. Lett. 28, 2232–2233 (1992).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 3, p. 109.

Jin, W.

W. Jin, G. Stewart, B. Culshaw, S. Murray, D. Pinchbeck, “Absorption measurement of methane gas with a broadband source and interferometric signal processing,” Opt. Lett. 18, 1364–1366 (1993).
[CrossRef] [PubMed]

W. Jin, G. Stewart, B. Culshaw, S. Murray, M. Wilkinson, J. Norris, A. McGhee, “A D-shaped optical fibre methane sensor,” presented at the Conference on Applied Optics and OptoElectronics, IOP, 5–8 September 1994.

W. Jin, G. Stewart, B. Culshaw, S. Murray, M. Wilkinson, J. O. W. Norris, “Performance limitation of a D-fiber evanescent wave methane sensor due to interference effects,” submitted to J. Lightwave Technol.

Maruyma, A.

Y. Shimose, T. Okamoto, A. Maruyma, M. Arizawa, H. Nagai, “Remote sensing of methane gas by differential absorption measurement using a wavelength tunable DFB LD,” IEEE Photon. Technol. Lett. 3, 86–89 (1991).
[CrossRef]

McGhee, A.

B. Culshaw, F. Muhammad, G. Stewart, S. Murray, D. Pinchbeck, J. Norris, S. Cassidy, M. Wilkinson, D. Williams, I. Crisp, R. V. Ewyk, A. McGhee, “Evanescent wave methane detection using optical fibre,” Electron. Lett. 28, 2232–2233 (1992).
[CrossRef]

W. Jin, G. Stewart, B. Culshaw, S. Murray, M. Wilkinson, J. Norris, A. McGhee, “A D-shaped optical fibre methane sensor,” presented at the Conference on Applied Optics and OptoElectronics, IOP, 5–8 September 1994.

Moslehi, B.

B. Moslehi, “Analysis of optical phase noise in fiber-optic systems employing a laser source with arbitrary coherence time,” J. Lightwave Technol. LT-4, 1334–1351 (1986).
[CrossRef]

Muhammad, F.

B. Culshaw, F. Muhammad, G. Stewart, S. Murray, D. Pinchbeck, J. Norris, S. Cassidy, M. Wilkinson, D. Williams, I. Crisp, R. V. Ewyk, A. McGhee, “Evanescent wave methane detection using optical fibre,” Electron. Lett. 28, 2232–2233 (1992).
[CrossRef]

Muhammad, F. A.

F. A. Muhammad, G. Stewart, “D-shaped optical fibre design for methane gas sensing,” Electron. Lett. 28, 1025–1026 (1993).

Murray, S.

W. Jin, G. Stewart, B. Culshaw, S. Murray, D. Pinchbeck, “Absorption measurement of methane gas with a broadband source and interferometric signal processing,” Opt. Lett. 18, 1364–1366 (1993).
[CrossRef] [PubMed]

B. Culshaw, F. Muhammad, G. Stewart, S. Murray, D. Pinchbeck, J. Norris, S. Cassidy, M. Wilkinson, D. Williams, I. Crisp, R. V. Ewyk, A. McGhee, “Evanescent wave methane detection using optical fibre,” Electron. Lett. 28, 2232–2233 (1992).
[CrossRef]

W. Jin, G. Stewart, B. Culshaw, S. Murray, M. Wilkinson, J. O. W. Norris, “Performance limitation of a D-fiber evanescent wave methane sensor due to interference effects,” submitted to J. Lightwave Technol.

W. Jin, G. Stewart, B. Culshaw, S. Murray, M. Wilkinson, J. Norris, A. McGhee, “A D-shaped optical fibre methane sensor,” presented at the Conference on Applied Optics and OptoElectronics, IOP, 5–8 September 1994.

Nagai, H.

Y. Shimose, T. Okamoto, A. Maruyma, M. Arizawa, H. Nagai, “Remote sensing of methane gas by differential absorption measurement using a wavelength tunable DFB LD,” IEEE Photon. Technol. Lett. 3, 86–89 (1991).
[CrossRef]

Norris, J.

B. Culshaw, F. Muhammad, G. Stewart, S. Murray, D. Pinchbeck, J. Norris, S. Cassidy, M. Wilkinson, D. Williams, I. Crisp, R. V. Ewyk, A. McGhee, “Evanescent wave methane detection using optical fibre,” Electron. Lett. 28, 2232–2233 (1992).
[CrossRef]

W. Jin, G. Stewart, B. Culshaw, S. Murray, M. Wilkinson, J. Norris, A. McGhee, “A D-shaped optical fibre methane sensor,” presented at the Conference on Applied Optics and OptoElectronics, IOP, 5–8 September 1994.

Norris, J. O. W.

W. Jin, G. Stewart, B. Culshaw, S. Murray, M. Wilkinson, J. O. W. Norris, “Performance limitation of a D-fiber evanescent wave methane sensor due to interference effects,” submitted to J. Lightwave Technol.

Okamoto, T.

Y. Shimose, T. Okamoto, A. Maruyma, M. Arizawa, H. Nagai, “Remote sensing of methane gas by differential absorption measurement using a wavelength tunable DFB LD,” IEEE Photon. Technol. Lett. 3, 86–89 (1991).
[CrossRef]

Pinchbeck, D.

W. Jin, G. Stewart, B. Culshaw, S. Murray, D. Pinchbeck, “Absorption measurement of methane gas with a broadband source and interferometric signal processing,” Opt. Lett. 18, 1364–1366 (1993).
[CrossRef] [PubMed]

B. Culshaw, F. Muhammad, G. Stewart, S. Murray, D. Pinchbeck, J. Norris, S. Cassidy, M. Wilkinson, D. Williams, I. Crisp, R. V. Ewyk, A. McGhee, “Evanescent wave methane detection using optical fibre,” Electron. Lett. 28, 2232–2233 (1992).
[CrossRef]

J. P. Dakin, C. A. Wade, D. Pinchbeck, J. S. Wykes, “A novel optical fiber methane sensor,” J. Opt. Sensors 2, 261–267 (1987).

Shafir, E.

M. Tur, E. Shafir, K. Blotekjaer, “Source-induced noise in optical systems driven by low-coherence sources,” J. Light-wave Technol. 8, 183–189 (1990).
[CrossRef]

Shimose, Y.

Y. Shimose, T. Okamoto, A. Maruyma, M. Arizawa, H. Nagai, “Remote sensing of methane gas by differential absorption measurement using a wavelength tunable DFB LD,” IEEE Photon. Technol. Lett. 3, 86–89 (1991).
[CrossRef]

Stewart, G.

W. Jin, G. Stewart, B. Culshaw, S. Murray, D. Pinchbeck, “Absorption measurement of methane gas with a broadband source and interferometric signal processing,” Opt. Lett. 18, 1364–1366 (1993).
[CrossRef] [PubMed]

F. A. Muhammad, G. Stewart, “D-shaped optical fibre design for methane gas sensing,” Electron. Lett. 28, 1025–1026 (1993).

B. Culshaw, F. Muhammad, G. Stewart, S. Murray, D. Pinchbeck, J. Norris, S. Cassidy, M. Wilkinson, D. Williams, I. Crisp, R. V. Ewyk, A. McGhee, “Evanescent wave methane detection using optical fibre,” Electron. Lett. 28, 2232–2233 (1992).
[CrossRef]

W. Jin, G. Stewart, B. Culshaw, S. Murray, M. Wilkinson, J. Norris, A. McGhee, “A D-shaped optical fibre methane sensor,” presented at the Conference on Applied Optics and OptoElectronics, IOP, 5–8 September 1994.

W. Jin, G. Stewart, B. Culshaw, S. Murray, M. Wilkinson, J. O. W. Norris, “Performance limitation of a D-fiber evanescent wave methane sensor due to interference effects,” submitted to J. Lightwave Technol.

Tai, H.

Tur, M.

M. Tur, E. Shafir, K. Blotekjaer, “Source-induced noise in optical systems driven by low-coherence sources,” J. Light-wave Technol. 8, 183–189 (1990).
[CrossRef]

Uehara, K.

Wade, C. A.

J. P. Dakin, C. A. Wade, D. Pinchbeck, J. S. Wykes, “A novel optical fiber methane sensor,” J. Opt. Sensors 2, 261–267 (1987).

Wilkinson, M.

B. Culshaw, F. Muhammad, G. Stewart, S. Murray, D. Pinchbeck, J. Norris, S. Cassidy, M. Wilkinson, D. Williams, I. Crisp, R. V. Ewyk, A. McGhee, “Evanescent wave methane detection using optical fibre,” Electron. Lett. 28, 2232–2233 (1992).
[CrossRef]

W. Jin, G. Stewart, B. Culshaw, S. Murray, M. Wilkinson, J. O. W. Norris, “Performance limitation of a D-fiber evanescent wave methane sensor due to interference effects,” submitted to J. Lightwave Technol.

W. Jin, G. Stewart, B. Culshaw, S. Murray, M. Wilkinson, J. Norris, A. McGhee, “A D-shaped optical fibre methane sensor,” presented at the Conference on Applied Optics and OptoElectronics, IOP, 5–8 September 1994.

Williams, D.

B. Culshaw, F. Muhammad, G. Stewart, S. Murray, D. Pinchbeck, J. Norris, S. Cassidy, M. Wilkinson, D. Williams, I. Crisp, R. V. Ewyk, A. McGhee, “Evanescent wave methane detection using optical fibre,” Electron. Lett. 28, 2232–2233 (1992).
[CrossRef]

Wykes, J. S.

J. P. Dakin, C. A. Wade, D. Pinchbeck, J. S. Wykes, “A novel optical fiber methane sensor,” J. Opt. Sensors 2, 261–267 (1987).

Appl. Opt. (1)

Electron. Lett. (2)

B. Culshaw, F. Muhammad, G. Stewart, S. Murray, D. Pinchbeck, J. Norris, S. Cassidy, M. Wilkinson, D. Williams, I. Crisp, R. V. Ewyk, A. McGhee, “Evanescent wave methane detection using optical fibre,” Electron. Lett. 28, 2232–2233 (1992).
[CrossRef]

F. A. Muhammad, G. Stewart, “D-shaped optical fibre design for methane gas sensing,” Electron. Lett. 28, 1025–1026 (1993).

IEEE Photon. Technol. Lett. (1)

Y. Shimose, T. Okamoto, A. Maruyma, M. Arizawa, H. Nagai, “Remote sensing of methane gas by differential absorption measurement using a wavelength tunable DFB LD,” IEEE Photon. Technol. Lett. 3, 86–89 (1991).
[CrossRef]

J. Light-wave Technol. (1)

M. Tur, E. Shafir, K. Blotekjaer, “Source-induced noise in optical systems driven by low-coherence sources,” J. Light-wave Technol. 8, 183–189 (1990).
[CrossRef]

J. Lightwave Technol. (1)

B. Moslehi, “Analysis of optical phase noise in fiber-optic systems employing a laser source with arbitrary coherence time,” J. Lightwave Technol. LT-4, 1334–1351 (1986).
[CrossRef]

J. Opt. Sensors (1)

J. P. Dakin, C. A. Wade, D. Pinchbeck, J. S. Wykes, “A novel optical fiber methane sensor,” J. Opt. Sensors 2, 261–267 (1987).

Opt. Lett. (1)

Other (3)

W. Jin, G. Stewart, B. Culshaw, S. Murray, M. Wilkinson, J. Norris, A. McGhee, “A D-shaped optical fibre methane sensor,” presented at the Conference on Applied Optics and OptoElectronics, IOP, 5–8 September 1994.

W. Jin, G. Stewart, B. Culshaw, S. Murray, M. Wilkinson, J. O. W. Norris, “Performance limitation of a D-fiber evanescent wave methane sensor due to interference effects,” submitted to J. Lightwave Technol.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 3, p. 109.

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Figures (2)

Fig. 1
Fig. 1

Experimental methane detection system.

Fig. 2
Fig. 2

Noise levels in terms of minimum detectable methane concentration versus power level I T ; ●, experimental result.

Equations (28)

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I 2kHz = - 4 C 1 - R 1 + R ρ n e L m - 1 ( - 1 ) m R m J 2 ( 2 m X ) ( i I i ) ,
E T ( t ) = ( 1 - R ) i = 0 R i E ( t - i τ ) ,
I T = E T ( t ) E T * ( t ) = ( 1 - R ) 2 i = 0 j = 0 R i + j E ( t - i τ ) E * ( t - j τ ) ,
I T = ( 1 - R ) 2 I 0 i = 0 R 2 i = 1 - R 1 + R I 0 .
C I ( t 1 , t 2 ) = E T ( t 1 ) E T * ( t 1 ) E T ( t 2 ) E T * ( t 2 ) - I T 2 .
C I ( t 1 , t 2 ) = ( 1 - R ) 4 i = 0 j = 0 m = 0 n = 0 R i + j + m + n × E ( t 1 - i τ ) E * ( t 1 - j τ ) E ( t 2 - m τ ) × E * ( t 2 - n τ ) - I T 2 .
C I ( t 1 , t 2 ) = ( 1 - R ) 4 i = 0 j = 0 m = 0 n = 0 R i + j + m + n × { Γ [ ( j - i ) τ ] Γ ( n - m ) τ ] + Γ [ t 1 - t 2 - ( i - n ) τ ] × Γ [ t 2 - t 1 - ( m - j ) τ ] } - I T 2 ,
C I ( t 1 , t 2 ) = ( 1 - R ) 4 i = 0 j = 0 m = 0 n = 0 R i + j + m + n × { Γ ( 0 ) Γ ( 0 ) δ ( i - j ) δ ( m - n ) + Γ ( T ) 2 δ [ ( i - n ) - ( j - m ) ] } - I T 2 ,
Γ [ ( j - i ) τ ] Γ [ ( n - m ) τ ] = Γ 2 ( 0 ) for j - i = n - m = 0 , 0 for others ,
Γ ( t 1 - t 2 - ( i - n ) τ ] Γ [ t 2 - t 1 - ( m - j ) τ ] = Γ ( T ) 2 for i - n = j - m , 0             for others .
C I ( t 1 , t 2 ) = ( 1 - R ) 4 i = 0 j = 0 m = 0 n = 0 R i + j + m + n × Γ ( T ) 2 δ [ i - n - ( j - m ) ] .
S ( f ) = I T 2 [ 1 - R 4 1 + R 4 - 2 R 2 cos ( 2 π f τ ) ] S i ( f ) I 0 2 ,
S ( f ) I T 2 ( 1 + R 2 1 - R 2 ) S i ( f ) I 0 2 ,
S i ( f ) S i ( 0 ) .
S i ( 0 ) I 0 2 τ c ,
S ( f ) S ( 0 ) = I T 2 ( 1 + R 2 1 - R 2 ) τ c ,
S T ( f ) S ( 0 ) + shot + thermal = I T 2 ( 1 + R 2 1 - R 2 τ c + 2 e R D I T + 1 R D 2 I T 2 4 k T R L ) ,
C min = 1 4 L ρ / n e I 0 i I i 1 m = 1 ( - 1 ) m R m J 2 ( 2 m X ) × [ ( 1 + R 2 1 - R 2 τ c + 2 e R D I T + 1 R D 2 I T 2 4 k T R L ) B ] 1 / 2 .
S ( f ) = F [ C I ( t 1 - t 2 ) ] = - + C I ( t 1 - t 2 ) exp [ - j 2 π f ( t 1 - t 2 ) ] d ( t 1 - t 2 ) .
S ( f ) = ( 1 - R ) 4 i = 0 j = 0 m = 0 n = 0 R i + j + m + n F [ Γ ( T ) 2 ] × δ [ i - n - ( j - m ) ] ,
F [ Γ ( t - ξ ) ] = F [ Γ ( t ) ] exp ( 2 π f ξ ) ,
S ( f ) = ( 1 - R ) 4 S i ( f ) j = 0 m = 0 i = 0 n = 0 R i + j + m + n × exp [ j 2 π f ( i - n ) τ ] δ [ i - n - ( j - m ) ] ,
F [ Γ ( t ) 2 ] = S i ( f ) ,
i = 0 n = 0 R i + m + n exp [ j 2 π f ( i - n ) τ ] δ [ i - n - ( - m ) ] = i = m R 2 i exp ( j 2 π f m τ ) = ( 1 1 - R 2 ) × R 2 m exp ( - j 2 π m f τ ) .
m = 0 i = 0 n = 0 R i + m + n exp [ j 2 π f ( i - n ) τ ] δ [ i - n + m ) ] = m = 0 1 1 - R 2 R 2 m exp ( - j 2 π f m τ ) = 1 1 - R 2 1 1 - R 2 exp ( - j 2 π f τ ) .
m = 0 i = 0 n = 0 R i + 1 + m + n exp [ j 2 π f ( i - n ) τ ] δ [ i - n - ( 1 - m ) ] = R 2 1 - R 2 [ 1 1 - R 2 exp ( - j 2 π f τ ) + exp j 2 π f τ ] .
m = 0 i = 0 n = 0 R i + p + m + n exp [ j 2 π f ( i - n ) τ ] δ [ i - n - ( p - m ) ] = R 2 p 1 - R 2 [ 1 1 - R 2 exp ( - j 2 π f τ ) + exp ( j 2 π f τ ) + exp ( j 2 × 2 π f τ ) + + exp ( j p × 2 π f τ ) ]
j = 0 m = 0 i = 0 n = 0 R i + j + m + n exp [ j 2 π f ( i - n ) τ ] × δ [ i - n - ( j - m ) ] = 1 ( 1 - R 2 ) [ 1 - R 2 exp ( - j 2 π f τ ) ] ( 1 + R 2 + R 4 + ) + exp ( j 2 π f τ ) 1 - R 2 ( R 2 + R 4 + R 6 + ) + + exp ( j p 2 π f τ ) 1 - R 2 [ R 2 p + R 2 ( p + 1 ) + ] + = 1 ( 1 - R 2 ) 2 [ 1 - R 2 exp ( - j 2 π f τ ) ] + 1 ( 1 - R 2 ) 2 [ R 2 exp ( j 2 π f τ ) + R 4 exp ( j 2 × 2 π f τ ) + ] + 1 ( 1 - R 2 ) 2 [ 1 1 - R 2 exp ( - j 2 π f τ ) + R 2 exp ( j 2 π f τ ) 1 - R 2 exp ( j 2 π f τ ) ] = 1 ( 1 - R 2 ) 2 [ 1 - R 4 1 + R 4 - 2 R 4 cos ( 2 π f τ ) ] .

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