Abstract

The Bit-Error-Ratio (BER) floor caused by the laser phase noise in the optical fiber communication system with differential quadrature phase shift keying (DQPSK) and coherent detection followed by digital signal processing (DSP) is analytically evaluated. An in-phase and quadrature (I&Q) receiver with a carrier phase recovery using DSP is considered. The carrier phase recovery is based on a phase estimation of a finite sum (block) of the signal samples raised to the power of four and the phase unwrapping at transitions between blocks. It is demonstrated that errors generated at block transitions cause the dominating contribution to the system BER floor when the impact of the additive noise is negligibly small in comparison with the effect of the laser phase noise. Even the BER floor in the case when the phase unwrapping is omitted is analytically derived and applied to emphasize the crucial importance of this signal processing operation. The analytical results are verified by full Monte Carlo simulations. The BER for another type of DQPSK receiver operation, which is based on differential phase detection, is also obtained in the analytical form using the principle of conditional probability. The principle of conditional probability is justified in the case of differential phase detection due to statistical independency of the laser phase noise induced signal phase error and the additive noise contributions. Based on the achieved analytical results the laser linewidth tolerance is calculated for different system cases.

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References

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  1. L. G. Kazovsky, G. Kalogerakis, and W.-T. Shaw, “Homodyne Phase-Shift-Keying systems: Past challenges and future opportunities,” J. Lightwave Technol. (12), 4876–4884 (2006).
    [CrossRef]
  2. M. A. Grant, W. C. Michie, and M. J. Fletcher, “The performance of optical phase locked loops in the presence of nonnegligible loop propagation delay,” J. Lightwave Technol. (4), 592–597 (1987).
    [CrossRef]
  3. S. Norimatsu and K. Iwashita, “Linewidth requirements for optical synchronous detection systems with non-negligible loop delay time,” J. Lightwave Technol. (3), 341–349 (1992).
    [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(2), 674–676 (2004).
    [CrossRef]
  5. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008).
    [CrossRef] [PubMed]
  6. R. Noé, “PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I&Q baseband processing,” IEEE Photon. Technol. Lett. 17(4), 887–889 (2005).
    [CrossRef]
  7. R. Noé, “Phase noise-tolerant synchronous QPSK/BPSK baseband-type intradyne receiver concept with feedforward carrier recovery,” J. Lightwave Technol. 13(2), 802–808 (2005).
    [CrossRef]
  8. D.-S. 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(1), 12–21 (2006).
    [CrossRef]
  9. N. Kaneda, A. Leven, and Y.-K. Chen, “Block length effect on 5.0 Gbit/s real-time QPSK intradyne receivers with standard DFB laser,” Electron. Lett. 43(20), 1106–1107 (2007).
    [CrossRef]
  10. M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Lightwave Technol. 27(7), 901–914 (2009).
    [CrossRef]
  11. S. Haykin, Adaptive filter theory, Englewood Cliffs, NJ, (Prentice-Hall, Inc. 1991).
  12. S. Benedetto, E. Biglieri, and V. Castellani, Digital transmission theory, Englewood Cliffs, NJ, (Prentice-Hall, Inc. 1987), Chap. 5, Sec. 5.3.
  13. I. S. Gradshteyn, and I. M. Ryzhik, Table of Integrals, Series, and Products, Fifth Edition, Ed. A. Jeffrey (Academic Press, Inc. 1994), p. 2.
  14. C. W. Helström, Probability and Stochastic Processes for Engineers, Macmillian Publishing Company, New York (Macmillan, Inc. 1984) Ch. 1.5.
  15. I. Garrett and G. Jacobsen, “Theoretical analysis of heterodyne optical receivers for transmission systems using (semiconductor) lasers with nonnegligible linewidth,” J. Lightwave Technol. 4(3), 323–334 (1986).
    [CrossRef]
  16. S. Savory and A. Hadjifotiou, “Laser Linewidth Requirements for Optical DQPSK Systems,” IEEE Photon. Technol. Lett. 16(3), 930–932 (2004).
    [CrossRef]
  17. N. S. Avlonitis and E. M. Yeatman, “Performance Evaluation of Optical DQPSK Using Saddle Point Approximation,” J. Lightwave Technol. 24(3), 1176–1185 (2006).
    [CrossRef]
  18. G. Goldfarb and G. Li, “BER estimation of QPSK homodyne detection with carrier phase estimation using digital signal processing,” Opt. Express 14(18), 8043–8053 (2006).
    [CrossRef] [PubMed]
  19. S. Hoffmann, R. Peveling, T. Pfau, O. Adamczyk, R. Eickhoff, and R. Noé, “Multiplier-free Realtime Phase Tracking for Coherent QPSK Receivers,” IEEE Photon. Technol. Lett. 21(3), 137–139 (2009).
    [CrossRef]

2009 (2)

S. Hoffmann, R. Peveling, T. Pfau, O. Adamczyk, R. Eickhoff, and R. Noé, “Multiplier-free Realtime Phase Tracking for Coherent QPSK Receivers,” IEEE Photon. Technol. Lett. 21(3), 137–139 (2009).
[CrossRef]

M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Lightwave Technol. 27(7), 901–914 (2009).
[CrossRef]

2008 (1)

2007 (1)

N. Kaneda, A. Leven, and Y.-K. Chen, “Block length effect on 5.0 Gbit/s real-time QPSK intradyne receivers with standard DFB laser,” Electron. Lett. 43(20), 1106–1107 (2007).
[CrossRef]

2006 (4)

2005 (2)

R. Noé, “PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I&Q baseband processing,” IEEE Photon. Technol. Lett. 17(4), 887–889 (2005).
[CrossRef]

R. Noé, “Phase noise-tolerant synchronous QPSK/BPSK baseband-type intradyne receiver concept with feedforward carrier recovery,” J. Lightwave Technol. 13(2), 802–808 (2005).
[CrossRef]

2004 (2)

M. G. Taylor, “Coherent detection method using DSP for demodulation of signal and subsequent equalization of propagation impairments,” IEEE Photon. Technol. Lett. 16(2), 674–676 (2004).
[CrossRef]

S. Savory and A. Hadjifotiou, “Laser Linewidth Requirements for Optical DQPSK Systems,” IEEE Photon. Technol. Lett. 16(3), 930–932 (2004).
[CrossRef]

1992 (1)

S. Norimatsu and K. Iwashita, “Linewidth requirements for optical synchronous detection systems with non-negligible loop delay time,” J. Lightwave Technol. (3), 341–349 (1992).
[CrossRef]

1987 (1)

M. A. Grant, W. C. Michie, and M. J. Fletcher, “The performance of optical phase locked loops in the presence of nonnegligible loop propagation delay,” J. Lightwave Technol. (4), 592–597 (1987).
[CrossRef]

1986 (1)

I. Garrett and G. Jacobsen, “Theoretical analysis of heterodyne optical receivers for transmission systems using (semiconductor) lasers with nonnegligible linewidth,” J. Lightwave Technol. 4(3), 323–334 (1986).
[CrossRef]

Adamczyk, O.

S. Hoffmann, R. Peveling, T. Pfau, O. Adamczyk, R. Eickhoff, and R. Noé, “Multiplier-free Realtime Phase Tracking for Coherent QPSK Receivers,” IEEE Photon. Technol. Lett. 21(3), 137–139 (2009).
[CrossRef]

Avlonitis, N. S.

Chen, Y.-K.

N. Kaneda, A. Leven, and Y.-K. Chen, “Block length effect on 5.0 Gbit/s real-time QPSK intradyne receivers with standard DFB laser,” Electron. Lett. 43(20), 1106–1107 (2007).
[CrossRef]

Eickhoff, R.

S. Hoffmann, R. Peveling, T. Pfau, O. Adamczyk, R. Eickhoff, and R. Noé, “Multiplier-free Realtime Phase Tracking for Coherent QPSK Receivers,” IEEE Photon. Technol. Lett. 21(3), 137–139 (2009).
[CrossRef]

Fletcher, M. J.

M. A. Grant, W. C. Michie, and M. J. Fletcher, “The performance of optical phase locked loops in the presence of nonnegligible loop propagation delay,” J. Lightwave Technol. (4), 592–597 (1987).
[CrossRef]

Garrett, I.

I. Garrett and G. Jacobsen, “Theoretical analysis of heterodyne optical receivers for transmission systems using (semiconductor) lasers with nonnegligible linewidth,” J. Lightwave Technol. 4(3), 323–334 (1986).
[CrossRef]

Goldfarb, G.

Grant, M. A.

M. A. Grant, W. C. Michie, and M. J. Fletcher, “The performance of optical phase locked loops in the presence of nonnegligible loop propagation delay,” J. Lightwave Technol. (4), 592–597 (1987).
[CrossRef]

Hadjifotiou, A.

S. Savory and A. Hadjifotiou, “Laser Linewidth Requirements for Optical DQPSK Systems,” IEEE Photon. Technol. Lett. 16(3), 930–932 (2004).
[CrossRef]

Hoffmann, S.

S. Hoffmann, R. Peveling, T. Pfau, O. Adamczyk, R. Eickhoff, and R. Noé, “Multiplier-free Realtime Phase Tracking for Coherent QPSK Receivers,” IEEE Photon. Technol. Lett. 21(3), 137–139 (2009).
[CrossRef]

Iwashita, K.

S. Norimatsu and K. Iwashita, “Linewidth requirements for optical synchronous detection systems with non-negligible loop delay time,” J. Lightwave Technol. (3), 341–349 (1992).
[CrossRef]

Jacobsen, G.

I. Garrett and G. Jacobsen, “Theoretical analysis of heterodyne optical receivers for transmission systems using (semiconductor) lasers with nonnegligible linewidth,” J. Lightwave Technol. 4(3), 323–334 (1986).
[CrossRef]

Kalogerakis, G.

L. G. Kazovsky, G. Kalogerakis, and W.-T. Shaw, “Homodyne Phase-Shift-Keying systems: Past challenges and future opportunities,” J. Lightwave Technol. (12), 4876–4884 (2006).
[CrossRef]

Kaneda, N.

N. Kaneda, A. Leven, and Y.-K. Chen, “Block length effect on 5.0 Gbit/s real-time QPSK intradyne receivers with standard DFB laser,” Electron. Lett. 43(20), 1106–1107 (2007).
[CrossRef]

Katoh, K.

Kazovsky, L. G.

L. G. Kazovsky, G. Kalogerakis, and W.-T. Shaw, “Homodyne Phase-Shift-Keying systems: Past challenges and future opportunities,” J. Lightwave Technol. (12), 4876–4884 (2006).
[CrossRef]

Kikuchi, K.

Leven, A.

N. Kaneda, A. Leven, and Y.-K. Chen, “Block length effect on 5.0 Gbit/s real-time QPSK intradyne receivers with standard DFB laser,” Electron. Lett. 43(20), 1106–1107 (2007).
[CrossRef]

Li, G.

Ly-Gagnon, D.-S.

Michie, W. C.

M. A. Grant, W. C. Michie, and M. J. Fletcher, “The performance of optical phase locked loops in the presence of nonnegligible loop propagation delay,” J. Lightwave Technol. (4), 592–597 (1987).
[CrossRef]

Noé, R.

S. Hoffmann, R. Peveling, T. Pfau, O. Adamczyk, R. Eickhoff, and R. Noé, “Multiplier-free Realtime Phase Tracking for Coherent QPSK Receivers,” IEEE Photon. Technol. Lett. 21(3), 137–139 (2009).
[CrossRef]

R. Noé, “PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I&Q baseband processing,” IEEE Photon. Technol. Lett. 17(4), 887–889 (2005).
[CrossRef]

R. Noé, “Phase noise-tolerant synchronous QPSK/BPSK baseband-type intradyne receiver concept with feedforward carrier recovery,” J. Lightwave Technol. 13(2), 802–808 (2005).
[CrossRef]

Norimatsu, S.

S. Norimatsu and K. Iwashita, “Linewidth requirements for optical synchronous detection systems with non-negligible loop delay time,” J. Lightwave Technol. (3), 341–349 (1992).
[CrossRef]

Peveling, R.

S. Hoffmann, R. Peveling, T. Pfau, O. Adamczyk, R. Eickhoff, and R. Noé, “Multiplier-free Realtime Phase Tracking for Coherent QPSK Receivers,” IEEE Photon. Technol. Lett. 21(3), 137–139 (2009).
[CrossRef]

Pfau, T.

S. Hoffmann, R. Peveling, T. Pfau, O. Adamczyk, R. Eickhoff, and R. Noé, “Multiplier-free Realtime Phase Tracking for Coherent QPSK Receivers,” IEEE Photon. Technol. Lett. 21(3), 137–139 (2009).
[CrossRef]

Savory, S.

S. Savory and A. Hadjifotiou, “Laser Linewidth Requirements for Optical DQPSK Systems,” IEEE Photon. Technol. Lett. 16(3), 930–932 (2004).
[CrossRef]

Savory, S. J.

Shaw, W.-T.

L. G. Kazovsky, G. Kalogerakis, and W.-T. Shaw, “Homodyne Phase-Shift-Keying systems: Past challenges and future opportunities,” J. Lightwave Technol. (12), 4876–4884 (2006).
[CrossRef]

Taylor, M. G.

M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Lightwave Technol. 27(7), 901–914 (2009).
[CrossRef]

M. G. Taylor, “Coherent detection method using DSP for demodulation of signal and subsequent equalization of propagation impairments,” IEEE Photon. Technol. Lett. 16(2), 674–676 (2004).
[CrossRef]

Tsukamoto, S.

Yeatman, E. M.

Electron. Lett. (1)

N. Kaneda, A. Leven, and Y.-K. Chen, “Block length effect on 5.0 Gbit/s real-time QPSK intradyne receivers with standard DFB laser,” Electron. Lett. 43(20), 1106–1107 (2007).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

M. G. Taylor, “Coherent detection method using DSP for demodulation of signal and subsequent equalization of propagation impairments,” IEEE Photon. Technol. Lett. 16(2), 674–676 (2004).
[CrossRef]

R. Noé, “PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I&Q baseband processing,” IEEE Photon. Technol. Lett. 17(4), 887–889 (2005).
[CrossRef]

S. Savory and A. Hadjifotiou, “Laser Linewidth Requirements for Optical DQPSK Systems,” IEEE Photon. Technol. Lett. 16(3), 930–932 (2004).
[CrossRef]

S. Hoffmann, R. Peveling, T. Pfau, O. Adamczyk, R. Eickhoff, and R. Noé, “Multiplier-free Realtime Phase Tracking for Coherent QPSK Receivers,” IEEE Photon. Technol. Lett. 21(3), 137–139 (2009).
[CrossRef]

J. Lightwave Technol. (8)

D.-S. 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(1), 12–21 (2006).
[CrossRef]

N. S. Avlonitis and E. M. Yeatman, “Performance Evaluation of Optical DQPSK Using Saddle Point Approximation,” J. Lightwave Technol. 24(3), 1176–1185 (2006).
[CrossRef]

I. Garrett and G. Jacobsen, “Theoretical analysis of heterodyne optical receivers for transmission systems using (semiconductor) lasers with nonnegligible linewidth,” J. Lightwave Technol. 4(3), 323–334 (1986).
[CrossRef]

M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Lightwave Technol. 27(7), 901–914 (2009).
[CrossRef]

R. Noé, “Phase noise-tolerant synchronous QPSK/BPSK baseband-type intradyne receiver concept with feedforward carrier recovery,” J. Lightwave Technol. 13(2), 802–808 (2005).
[CrossRef]

L. G. Kazovsky, G. Kalogerakis, and W.-T. Shaw, “Homodyne Phase-Shift-Keying systems: Past challenges and future opportunities,” J. Lightwave Technol. (12), 4876–4884 (2006).
[CrossRef]

M. A. Grant, W. C. Michie, and M. J. Fletcher, “The performance of optical phase locked loops in the presence of nonnegligible loop propagation delay,” J. Lightwave Technol. (4), 592–597 (1987).
[CrossRef]

S. Norimatsu and K. Iwashita, “Linewidth requirements for optical synchronous detection systems with non-negligible loop delay time,” J. Lightwave Technol. (3), 341–349 (1992).
[CrossRef]

Opt. Express (2)

Other (4)

S. Haykin, Adaptive filter theory, Englewood Cliffs, NJ, (Prentice-Hall, Inc. 1991).

S. Benedetto, E. Biglieri, and V. Castellani, Digital transmission theory, Englewood Cliffs, NJ, (Prentice-Hall, Inc. 1987), Chap. 5, Sec. 5.3.

I. S. Gradshteyn, and I. M. Ryzhik, Table of Integrals, Series, and Products, Fifth Edition, Ed. A. Jeffrey (Academic Press, Inc. 1994), p. 2.

C. W. Helström, Probability and Stochastic Processes for Engineers, Macmillian Publishing Company, New York (Macmillan, Inc. 1984) Ch. 1.5.

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Figures (7)

Fig. 1
Fig. 1

Variance of the phase error versus symbol position within the block. The block size: 3 (crosses), 5 (dots), 10 (triangles) and 15 (squares), respectively. Dashed lines are plotted in order to guide the eyes.

Fig. 2
Fig. 2

Maximum (squares) and minimum (crosses) value of the phase error variance within the block of QPSK symbols and variance of the estimated phase difference (dots) at block transitions, respectively, versus the block size parameter.

Fig. 3
Fig. 3

The DQPSK system BER floor (I&Q receiver with carrier phase recovery) versus phase noise variance (or total laser linewidth) for the block size parameter equal to 1, 2, 3, 5, 10, and 15, respectively. Symbols represent the results of Monte Carlo simulations. Solid lines are plotted using the analytical expression given by Eq. (13).

Fig. 4
Fig. 4

Maximum tolerable laser linewidth versus the block size parameter for the DQPSK system BER floor (I&Q receiver with carrier phase recovery, the symbol rate of 10Gsymbol/s) at 10−4, 10−6, and 10−9, respectively.

Fig. 5
Fig. 5

The DQPSK system BER when estimated phase slips are not corrected (I&Q receiver with carrier phase recovery) versus phase noise variance (or total laser linewidth) for the block size parameter equal to 1, 2, 3, 5, 10, and 15, respectively. Symbols represent the results of Monte Carlo simulations. Solid lines are plotted using Eq. (14).

Fig. 6
Fig. 6

The DQPSK system BER versus SNR for the differential phase receiver (for the total laser linewidth of 4 MHz, 10 MHz, 20 MHz and 40 MHz and in the absence of the laser phase noise, respectively) and for the ideal I&Q receiver.

Fig. A1
Fig. A1

The estimated phase PDF when the phase slips are not corrected (top) and corrected (bottom). Shaded areas indicate the intervals contributing to symbol errors.

Equations (22)

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Z k = exp { i ( Q k + 1 2 ) π 2 + i ϕ k } + n k ,
B E R ideal = erfc ( S N R ) ( 1 1 2 erfc ( S N R ) ) .
Φ m = 1 4 arg ( k = 1 + ( m 1 ) N b m N b Z k 4 ) .
Δ ϕ k = ϕ k 1 4 arg ( q = 1 N b e 4 i ϕ q ) = ϕ k ϕ 1 1 4 arg ( 1 + q = 2 N b e 4 i ( ϕ q ϕ 1 ) ) .
1 + q = 2 N b e 4 i ( ϕ q ϕ 1 ) 1 + q = 2 N b ( 1 + 4 i ( ϕ q ϕ 1 ) ) N b exp ( 4 i N b q = 2 N b ( ϕ q ϕ 1 ) ) ,
Δ ϕ k ϕ k ϕ 1 1 N b q = 2 N b ( ϕ q ϕ 1 ) .
ϕ k ϕ 1 = p = 2 k δ p .
q = 2 N b p = 2 q δ p = ( N b + 1 ) q = 2 N b δ p q = 2 N b p δ p
Δ ϕ k 1 N b ( p = 2 k ( p 1 ) δ p + p = k + 1 N b ( p N b 1 ) δ p ) .
σ k 2 = 1 N b 2 ( P = 2 k ( p 1 ) 2 + P = k + 1 N b ( p N b 1 ) 2 ) σ ϕ 2 .
σ k 2 = σ ϕ 2 6 N b 2 ( 2 ( k 1 ) 3 + 3 ( k 1 ) 2 + 2 ( N b k ) 3 + 3 ( N b k ) 2 + N b 1 ) , k = 1 , ... , N b .
σ Δ Φ 2 = 2 σ max 2 + σ ϕ 2 = σ ϕ 2 2 N b 2 + 1 3 N b .
B E R I & Q , f l o o r 1 2 N b erfc ( π 4 2 σ Δ Φ ) , when σ Δ Φ < < π 4 2 , i .e . σ ϕ 2 < < N b 2 N b 2 + 1 .
B E R I & Q , f l o o r u n c o r r 2 σ Δ Φ π N b 0 π 4 2 σ Δ Φ erfc ( ξ ) d ξ .
B E R I & Q , f l o o r u n c o r r 2 σ ϕ π 3 π N b = 2 π 2 ( Δ f s i g n a l + Δ f L O ) T s y m b o l 3 N b .
Ψ = exp { i ( Q k Q k + 1 + 1 2 ) π 2 + i ( ϕ k ϕ k + 1 ) } ( 1 + n ˜ k ) ( 1 + n ˜ k + 1 * ) .
B E R diff 1 2 2 π σ ϕ + exp { ( ε ) 2 / ( 2 σ ϕ 2 ) } erfc ( η ( ε ) S N R ) d ε ,
Δ Φ c o r r = { π 2 , 0 , π 2 , Φ m + 1 > Φ m + π 4 | Φ m + 1 Φ m | < π 4 Φ m + 1 < Φ m π 4
S E R I & Q , f l o o r u n c o r r 2 π π 4 π 4     | Φ m + 1 Φ m | > π 4 f Φ m ( Φ m + 1 ) d Φ m + 1 d Φ m
f Φ m ( Φ m + 1 ) 1 2 π σ Δ Φ 2 n = n = + exp { ( Φ m + 1 Φ m + π n / 2 ) 2 2 σ Δ Φ 2 } .
S E R I & Q , f l o o r u n c o r r 1 π π 4 π 4 erf ( ( π / 4 + | Φ m | ) / ( 2 σ Δ Φ ) ) erf ( ( π / 4 + | Φ m | ) / ( 2 σ Δ Φ ) ) d Φ m .
S E R I & Q , f l o o r 2 π π 4 π 4 | Φ ˜ m + 1 Φ m | > π 4 f ˜ Φ m ( Φ ˜ m + 1 ) d Φ ˜ m + 1 d Φ m .

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