Performance of DPSK signals with nonlinear phase noise for systems with small number of fiber spans

Performance of DPSK Signals with Nonlinear Phase Noise

for Systems with Small Number of Fiber Spans

Keang-Po Ho

Institute of Communication Engineering and Department of Electrical Engineering

National Taiwan University, Taipei 106, Taiwan

E-mail: [email protected]

Abstract

When the dependence between linear and nonlinear phase noise is taken into account, the exact error prob-ability of DPSK signals with nonlinear phase noise is derived analytically for a fiber system with finite number of fiber spans. For the same mean nonlinear phase shift, the SNR penalty is reduced with the number of fiber spans. The discrepancy between the exact error probability and independence approximation increases with the number of fiber spans.

I. Introduction

Nonlinear phase noise, often called Gordon-Mollenauer effect [1], or more precisely, self-phase modulation induced nonlinear phase noise, adds directly to the phase of a signal and degrades differential phase-shift keying (DPSK) signals [1]–[6] that has received renewed attention for either long-haul [7]–[10] or spectrally efficiency [11]–[13] transmission.

Nonlinear phase noise is found to be non-Gaussian distributed both experimentally [5] and theoretically [14], [15]. While uncorrelated to the linear phase noise, as non-Gaussian random variable, nonlinear phase noise is weakly depending on the linear phase noise. For systems with more than 32 fiber spans, the dependence between linear and nonlinear phase noise increases the error probability [4], [16]. Recently, DPSK signals have been used in systems with small number of fiber spans [10], [17], [18]. When DPSK signal is used in typical terrestrial systems with small (< 32) number of fiber spans, other than the approximation that linear and non-linear phase noise is independent [6], an accurate model of the nonlinear phase noise must take into account the dependence between linear and nonlinear phase noise.

II. Joint Statistics of Linear and Nonlinear

Phase Noise

For an N-span systems, for simplicity and without loss of generality, the overall quadratic nonlinear phase noise is [1], [15]

ΦNL = |
E0+
n1|2+ |
E0+
n1+
n2|2

+ · · · + |
E0+
n1+ · · · +
nN|2, (1)

where
E₀ = (A, 0) is a two-dimensional vector repre-senting the transmitted electric field,
n_k, k = 1, . . . , N, are independent identically distributed (i.i.d.) zero-mean circular Gaussian random complex number as the optical amplifier noise introduced into the system at the kth fiber span. The noise variance is E{|
nk|2} = 2σ02, k = 1, . . . , N, where σ2

0 is the noise variance per span per dimension. Without affected the SNR, both signal and noise in (1) can be scaled by the same ratio for different mean nonlinear phase shift of <Φ_NL>=

NA2_{+ N(N + 1)σ}2 0.

In the linear regime, the signal received afterN spans is

EN =
E0+
n1+
n2+ · · · +
nN (2) with an instantaneous power of P_N = |
E_N|2 and SNR of ρs= A2/(2Nσ20).

The joint characteristic function of the nonlinear phase noise and electric field is

Ψ_Φ,
E(ν,
ω) = E
exp(jνΦNL+ j
ω ·
EN



, (3)

where
ω = (ω₁, ω₂). Without going into detail, after some algebra, we obtain

Ψ_Φ,
E(ν,
ω) = ΨNL(ν) exp  jω1mN(ν) − σ2N(ν)|
ω| 2 2  , (4) where ΨΦNL(ν) = N  k=1 expjνA2(
vTkw)
2/λk 1−2jνσ2 0λk  1 − 2jνσ2 0λk , (5) mN(ν) = A N  k=1 (
vT k
w)(
vkT
wI)/λk 1 − 2jνσ2 0λk , (6) σ2 N(ν) = σ02 N  k=1 (
vT k
wI)2 1 − 2jνσ2 0λk, (7) where
w = (N, N − 1, . . . , 2, 1)T,
w_I = (1, 1, . . . , 1)T, and λ_k,
v_k, k = 1, 2, . . . , N are the eigenvalues and eigenvectors of the covariance matrix C, respectively. The covariance matrix isC = MTM with

ThC4

2:30 pm – 2:45 pm

10 11 12 13 14 15 16 17 10−12 10−11 10−10 10−9 10−8 10−7 10−6 10−5 SNRρ_s (dB) Error Probability N = 1 2 8 32 ∞ <Φ NL > = 0 Exact Approx.

Fig. 1. The error probability of DPSK signal as a function of SNR forN =1, 2, 4, 8, 32, and infinite number of fiber spans and mean nonlinear phase shift of<Φ_NL>= 0.5 rad.

M =      1 0 0 · · · 0 1 1 0 · · · 0 .. . ... ... . .. ... 1 1 1 · · · 1     . (8)

III. Exact Error Probability

Similar to the approaches of [3], [4], [16], the exact error probability is pe=1₂ −1₂ ∞  k=0 (−1)kλ_ke−λk 2k + 1 |ΨΦNL(2k + 1)|2 ×Ik  λk 2  + Ik+1  λk 2  2(9) where Ik(·) is the kth-order modified Bessel function of the first kind and the “angular frequency” depending SNR is λk= m 2 N(2k + 1) 2σ2 N(2k + 1). (10) The error probability of (9) is the same as that in [4], [16] but with different parameter of λ_k from (10) with (6) and (7).

Figure 1 shows the exact error probability as a func-tion of SNR for<Φ_NL>= 0.5 rad. Figure 2 shows the SNR penalty for an error probability of10−9 as a func-tion of mean nonlinear phase shift<ΦNL>. Both Figs. 1 and 2 are calculated using (9) and the independence approximation of [6]. The independence approximation of [6] underestimates and the SNR penalty of a DPSK signal with quadratic phase noise of (1). The exact and approximated error probability for N = ∞ are the distributed model from [16] and [19], respectively. From Figs. 1 and 2, for the same mean nonlinear phase shift of

0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Mean Nonlinear Phase Shift (rad)

SNR Penalty (dB)

N = 1 2 8 32 ∞

Exact Approx.

Fig. 2. The SNR penalty vs. mean nonlinear phase shift<Φ_NL>.

<ΦNL>, the SNR penalty is larger for smaller number of fiber spans. The independence approximation of [15] is closer to the exact error probability for small number of fiber spans. In all cases, the independence assump-tion of [6], [19] underestimates the error probability of the system, contradicting to the conservative principle of system design. The dependence between linear and nonlinear phase noise increases the SNR penalty up to 0.23 dB. The distributed model of [16], [19] can be used when the number of fiber spans is larger than 32.

IV. Conclusions

For a system with small number of fiber spans, the exact error probability of a DPSK signal with nonlinear phase noise is derived analytically the first time when the dependence between linear and nonlinear phase noise is taking into account. For the mean nonlinear phase shift, the error probability increases for small number of fiber spans.

References

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[4] K.-P. Ho, “Statistical properties of nonlinear phase noise,” in

Advances in Optics and Laser Research 3 (W. T. Arkin, ed.),

2003.

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[6] K.-P. Ho, IEEE Photon. Technol. Lett. 15, 1216 (2003). [7] A. H. Gnauck et al. OFC ’02, paper FC2.

[8] C. Rasmussen et al. OFC ’03, paper PD18. [9] J.-X. Cai et al. OFC ’04, paper PDP34. [10] A. H. Gnauck et al. OFC ’04, paper PDP35.

[11] P. S. Cho et al. IEEE Photon. Technol. Lett. 15, 473 (2003). [12] C. Wree et al. IEEE Photon. Technol. Lett., 15, 1303 (2003). [13] N. Yoshikane and I. Morita, OFC ’04, paper PDP38. [14] K.-P. Ho, Opt. Lett. 28, 1350–1352 (2003). [15] K.-P. Ho, J. Opt. Soc. Am. B 20, 1875 (2003).

[16] K.-P. Ho, “Exact error probability of phase-modulated signals with nonlinear phase noise,” submitted to J. Lightwave Technol., 2003.

[17] Y. Miyamoto et al. Electron. Lett. 38, 1569 (2002). [18] H. Bissessur et al. Electron. Lett. 39, 192 (2003). [19] K.-P. Ho, IEEE Photon. Technol. Lett. 15, 1213 (2003).