Abstract

An approximate analytical expression for the bit error rate of a QPSK homodyne receiver employing digital signal processing for carrier recovery is derived. BER estimated using the analytical expression is in excellent agreement with Monte-Carlo simulations. The analytical approximation leads to an intuitive understanding of the trade off in such systems and allows optimization of system parameters without resorting to Monte-Carlo simulations.

© 2006 Optical Society of America

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References

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  1. L. Kazovsky, S. Benedetto, and A. Willner, Optical Fiber Communication Systems (Artech House Inc., 1996).
  2. R. Noe, "PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I & Q baseband processing," IEEE Photon. Technol. Lett. 17, 887-889 (2005).
    [CrossRef]
  3. D. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, "Coherent detection of Optical Quadrature Phase-shift keying signals with carrier phase estimation," J. Lightwave Technol. 24, 12-21 (2006).
    [CrossRef]
  4. M. G. Taylor, "Coherent detection method using DSP for demodulation of signal and subsequent equalization of propagation impairments," IEEE Photon. Technol. Lett. 16, 674-676 (2004).
    [CrossRef]
  5. W. C. Lindsey, and M. K. Simon, Telecommunication Systems Engineering (Prentice-Hall Inc., 1973).
  6. M. K. Simon, "On the Bit-Error probability of differentially encoded QPSK and offset QPSK in the presence of carrier synchronization," IEEE Transactions on Comm. 54, 806-812 (2006).
    [CrossRef]
  7. J. Salz, "Modulation and detection for Coherent Lightwave Communications," IEEE Comm. Mag. 24, 38-49 (1986).
    [CrossRef]
  8. G. Casella, and R. L. Berger, Statistical Inference, 2nd Ed. (Pacific Grove, CA: Thomson Learning, 2002).

2006

D. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, "Coherent detection of Optical Quadrature Phase-shift keying signals with carrier phase estimation," J. Lightwave Technol. 24, 12-21 (2006).
[CrossRef]

M. K. Simon, "On the Bit-Error probability of differentially encoded QPSK and offset QPSK in the presence of carrier synchronization," IEEE Transactions on Comm. 54, 806-812 (2006).
[CrossRef]

2005

R. Noe, "PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I & Q baseband processing," IEEE Photon. Technol. Lett. 17, 887-889 (2005).
[CrossRef]

2004

M. G. Taylor, "Coherent detection method using DSP for demodulation of signal and subsequent equalization of propagation impairments," IEEE Photon. Technol. Lett. 16, 674-676 (2004).
[CrossRef]

1986

J. Salz, "Modulation and detection for Coherent Lightwave Communications," IEEE Comm. Mag. 24, 38-49 (1986).
[CrossRef]

Katoh, K.

Kikuchi, K.

Ly-Gagnon, D.

Noe, R.

R. Noe, "PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I & Q baseband processing," IEEE Photon. Technol. Lett. 17, 887-889 (2005).
[CrossRef]

Salz, J.

J. Salz, "Modulation and detection for Coherent Lightwave Communications," IEEE Comm. Mag. 24, 38-49 (1986).
[CrossRef]

Simon, M. K.

M. K. Simon, "On the Bit-Error probability of differentially encoded QPSK and offset QPSK in the presence of carrier synchronization," IEEE Transactions on Comm. 54, 806-812 (2006).
[CrossRef]

Taylor, M. G.

M. G. Taylor, "Coherent detection method using DSP for demodulation of signal and subsequent equalization of propagation impairments," IEEE Photon. Technol. Lett. 16, 674-676 (2004).
[CrossRef]

Tsukamoto, S.

IEEE Comm. Mag.

J. Salz, "Modulation and detection for Coherent Lightwave Communications," IEEE Comm. Mag. 24, 38-49 (1986).
[CrossRef]

IEEE Photon. Technol. Lett.

R. Noe, "PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I & Q baseband processing," IEEE Photon. Technol. Lett. 17, 887-889 (2005).
[CrossRef]

M. G. Taylor, "Coherent detection method using DSP for demodulation of signal and subsequent equalization of propagation impairments," IEEE Photon. Technol. Lett. 16, 674-676 (2004).
[CrossRef]

IEEE Transactions on Comm.

M. K. Simon, "On the Bit-Error probability of differentially encoded QPSK and offset QPSK in the presence of carrier synchronization," IEEE Transactions on Comm. 54, 806-812 (2006).
[CrossRef]

J. Lightwave Technol.

Other

W. C. Lindsey, and M. K. Simon, Telecommunication Systems Engineering (Prentice-Hall Inc., 1973).

L. Kazovsky, S. Benedetto, and A. Willner, Optical Fiber Communication Systems (Artech House Inc., 1996).

G. Casella, and R. L. Berger, Statistical Inference, 2nd Ed. (Pacific Grove, CA: Thomson Learning, 2002).

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Figures (6)

Fig. 1.
Fig. 1.

Schematic of feedforward carrier recovery using DSP

Fig. 2.
Fig. 2.

Processing unit block diagram

Fig. 3.
Fig. 3.

Top: Generated and approximated PDFs of Δφ, Bottom: Accumulation of BER integral for the two PDFs

Fig. 4.
Fig. 4.

log(stdφ)) vs. SNR and block size with optimal Nb superimposed. Left: Beat LW of 600kHz, Right: 2MHz.

Fig. 5.
Fig. 5.

Comparison of var(Δφ) from MC simulation and analytical expression

Fig. 6.
Fig. 6.

MC simulation and approximated analytical BER. Left: Beat LW of 600kHz, Right: 2MHz.

Equations (20)

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φ est PU 1 = 1 4 arg { w = 1 Nb Z w 4 }
P e = 1 2 π 4 π 4 erfc ( 2 γ b · cos ( π 4 + ε 1 ) ) · P Δφ ( ε 1 ) · d ε 1 .
· π 4 π 4 erfc ( 2 γ b · sin ( π 4 + ε 2 ) ) · P Δφ ( ε 2 ) · d ε 2
π 4 π 4 erfc ( 2 γ b ( cos ( ε ) sin ( ε ) ) ) · P Δφ ( ε ) · d ε
Δ φ k = φ k 1 4 arg { w = 1 Nb Z w 4 }
Z w 4 = ( e j ( φ d w + φ w ) + n w ) 4 = e j 4 φ w + 4 · ρ w + o ( n w 3 )
Where ρ w = n w · e j 3 ( φ d w + φ w ) ( 1 + 1 . 5 · e j ( φ d w + φ w ) n w ) .
Δ φ k φ k 1 4 arg { w = 1 Nb [ e j 4 φ w 4 · ρ w ] } .
Δ φ k φ 4 1 4 arg { w = 1 Nb e j 4 φ w } = φ k φ 1 1 4 arg { 1 + e j 4 ( φ 2 φ 1 ) + + e j 4 ( φ Nb φ 1 ) } .
φ m φ m n = q = 1 m δ q q = 1 m n δ q = q = m n + 1 m δ q
Δ φ k = φ k φ 1 1 4 arg { B ( Nb ) }
B ( Nb ) = 1 + w = 2 Nb exp { j · 4 p = 2 w δ p } .
B ( Nb ) Nb· { j · 4 · θ ( Nb ) }
where θ ( Nb ) = 1 Nb p = 0 Nb 2 ( p + 1 ) δ Nb p ,
[ Δ φ 1 Δ φ Nb ] 1 Nb [ 1 Nb 2 Nb 2 1 1 2 Nb 2 2 Nb 2 1 1 2 Nb 2 Nb 1 ] [ δ 2 δ Nb ] . M
Δφ ~ N ( 0 , σ δ 2 Nb · p = 1 Nb q = 1 Nb 1 M p . q 2 = Nb 2 1 6 · Nb · σ δ 2 ) .
Δ φ k = φ k φ 1 1 4 arg { B ( Nb ) + 4 · e j 4 φ 1 · w = 1 Nb ρ w } q = 2 Nb M k , q · δ q 1 4 arg { 1 + 4 Nb w = 1 Nb ρ w } .
Δ φ k q = 2 Nb M k , q · δ q 1 Nb w = 1 Nb Im { ρ w }
Δφ ~ N ( 0 , Nb 2 1 6 · Nb · σ δ 2 + σ n 2 ( 1 + 4.5 · σ n 2 ) 2 · Nb ) .
Nb opt = round ( 3 σ n 2 ( 1 + 4.5 · σ n 2 ) σ δ 2 1 )

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