Abstract

Integrated-optical All-Pass Filters are of interest for their potential compactness and economy of production. For broadband applications, the number of APFs involved can be as large as 50. To find optima for all the large number of parameters involved, we need a fast and efficient algorithm based on recursive equations. APF design algorithms based on complex cepstrum are proposed in digital signal processing. In this paper, we enhance these algorithms to efficiently fit the differential phase profile required for in-line broadband Polarization Mode Dispersion and Polarization Dependent Loss compensation.

© 2004 Optical Society of America

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References

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    [CrossRef] [PubMed]

Electron. Lett.

G.J. Dolecek, J.D. Carmona, �??Digital All-Pass filter design method based on Complex Cepstrums,�?? Electron. Lett. 39, 695-697 (2003).
[CrossRef]

IEEE Int. Conf. Acoustics, Speech 1990

G.R. Reddy, M.N.S. Swamy, �??Digital All-Pass Filter Design Through Discrete Hilbert Transform,�?? Proc. IEEE Int. Conf. Acoustics, Speech Signal Processing, 646-649 (1990)

IEEE Photon. Technol. Lett.

C.K. Madsen, G. Lenz, A.J. Bruce, M.A. Capuzzo, L.T. Gomez, and R.E. Scotti, �??Integrated All-Pass Filters for Tunable Dispersion and Dispersion Slope Compensation,�?? IEEE Photon. Technol. Lett. 11, 1623-1625 (1999).
[CrossRef]

IEEE Signal Processing Lett.

K. Rajamani, Y.S. Lai, �??A Novel Method for Designing Allpass Digital Filters,�?? IEEE Signal Processing Lett. 6, 207-209 (1999)
[CrossRef]

J. Lightwave Technol.

P.B. Phua and E. P. Ippen, �??A Deterministic Broadband Polarization-Dependent-Loss Compensator,�?? To appear in J. Lightwave Technol..

J.of Lightwave Technol.

P.B. Phua, H. A. Haus and E. P. Ippen, �??All-Frequency PMD Compensator In Feed-Forward Scheme,�?? J.of Lightwave Technol. 22, 1280-1289 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. Symposium Time Series Analysis

B.P. Bogert, M.J.R. Healy and J.W. Tukey, �??The Quefrency Analysis of Time Series for Echoes: Cepstrum, Pseudoautocovariance, Cross-Cepstrum, and Saphe Cracking," Proc. Symposium Time Series Analysis, M. Rosenblatt, Ed., John Wiley and Sons, New York, 209-243 (1963).

Other

C. Madsen and J. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach. New York, Wiley (1999).

A.V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, Prentice Hall (1989)

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Figures (3)

Fig. 1.
Fig. 1.

A common building block in the broadband PMD compensator

Fig. 2.
Fig. 2.

A random sample of the rotation angle profile θ(ω) required for one of the frequency-dependent polarization rotators used in the broadband PMD compensator. The desired rotation angle profile of θ(ω) is shown by the solid curve while the profile approximated by the APFs is shown by the dashed-curve. The number of APFs used for each polarization arms is 10 for (a), 20 for (b) and 30 for (c).

Fig. 3.
Fig. 3.

The cumulative probability distribution of the Bit Error Rate (BER) curve with and without PMD compensation.

Equations (23)

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H ( z ) = z N N ( z ) D ( z ) = z N n = 0 N a n * z n n = 0 N a n z n
H ( e j ω ) = e j ω N n = 0 N a n * e j ω n n = 0 N a n e j ω n
ϕ H ( ω ) = N ω + ϕ N ( ω ) ϕ D ( ω ) = N ω 2 ϕ D ( ω )
τ H ( ω ) = d ϕ H ( ω ) d ω = N + 2 ϕ D ( ω )
τ D ( ω ) = τ H ( ω ) 2 + N 2
Φ Vert ( ω ) Φ Hor ( ω ) = θ ( ω )
τ Vert ( ω ) τ Hor ( ω ) = τ DGD ( ω )
τ Vert ( ω ) + τ Hor ( ω ) = N Vert + N Hor
τ Vert ( ω ) = N Vert + N Hor 2 + τ DGD ( ω ) 2
τ Hor ( ω ) = N Vert + N Hor 2 τ DGD ( ω ) 2
τ DV ( ω ) = τ Vert ( ω ) 2 + N Vert 2
τ DV even ( ω ) = τ DV ( ω ) + τ DV ( ω ) 2
τ DV odd ( ω ) = τ DV ( ω ) τ DV ( ω ) 2
D ̂ ( ω ) = ln D ( ω ) = c ( 0 ) + k = 1 c ( k ) e j k ω
ϕ ( ω ) + 2 υ π = k = 1 Re [ c ( k ) ] sin k ω + k = 1 Im [ c ( k ) ] cos k ω
τ ( ω ) = k = 1 k Re [ c ( k ) ] cos k ω + k = 1 k Im [ c ( k ) ] sin k ω
ln D ( ω ) = c ( 0 ) + k = 1 Re [ c ( k ) ] cos k ω + k = 1 Im [ c ( k ) ] sin k ω
τ DV even ( ω ) = k = 1 k Re [ c ( k ) ] cos k ω
τ DV odd ( ω ) = k = 1 k Im [ c ( k ) ] sin k ω
a n = k = 0 n ( k n ) c ( k ) a n k n > 0
d D ̂ ( z ) d z = 1 D ( z ) d D ( z ) d z
n x [ n ] z Transform z d X ( z ) d z
x 1 [ n ] * x 2 [ n ] z Transform X 1 ( Z ) . X 2 ( Z )

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