行政院國家科學委員會專題研究計畫 期中進度報告
高速 DPSK 訊號在光纖通信系統的傳輸(1/2)
計畫類別： 個別型計畫
計畫編號： NSC92-2218-E-002-034-
執行期間： 92 年 10 月 01 日至 93 年 07 月 31 日
執行單位： 國立臺灣大學電信工程學研究所
計畫主持人： 何鏡波
報告類型： 精簡報告
處理方式： 本計畫可公開查詢
中 華 民 國 93 年 6 月 11 日
行政院國家科學委員會專題研究計畫成果報告
高速 DPSK 訊號在光纖通信系統的傳輸(1/2)
High-Speed Transmission of DPSK Signals in Lightwave Systems
計畫編號: 92-2218-E-002-034-
執行期限: 92 年 10 月 1 日 至 93 年 7 月 31 日
主持人:何鏡波 副教授 國立台灣大學電信工程學研究所
1. 中文摘要 （關鍵字：DPSK，調變，非線性相位雜訊，光纖非 線性效應，相位調變） 本計畫在研究少量光段系統下的非線性及線 性相位雜訊的統計分析。對具有非線性相位雜訊之 DPSK 訊號的確切錯誤機率進行理論推導及實驗量 測，用實驗量測結果驗證理論模型，以取得系統之最 佳化設計。並研究非線性相位雜訊的接收電場機率分 佈，以找出具有非線性相位雜訊信號的最佳檢測器， 使之具有最小的錯誤機率。此外，因非線性相位雜訊 與信號強度具有相關性，所以我們可利用接收信號補 償非線性相位雜訊，來完成一個線性補償器。 Abstract：（Keywords： DPSK, modulation, nonlinear phase noise, fiber nonlinearities, phase modulation）
In this project, tThe statistical properties of nonlinear phase noise together witbwith linear phase noise are first investigated for small number of fiber spans. The exact error probability of DPSK signals with nonlinear phase noise will beis derived and measured.
The measurement can confirm the validity of the theoretical models and its usage as a means to optimize the system design.
The joint probability density of received electric field with nonlinear phase noise, including both amplitude and phase, will be investigatedis derived the first time. .
Correlated with received intensity, nonlinear phase noise can be compensated by combining with a scale version of the received intensity. A linear compensator will be implemented in this project. Based on the probability density of the received signal, the optimal detector (or the optimal compensator) to minimize the error probability of a signal with nonlinear phase noise will also be derived analytically.
2. Introduction
Nonlinear phase noise, often called the Gordon-Mollenaer effect, or more precisely self-phase
the interaction of amplifier noise and fiber Kerr effect when optical amplifiers are used to compensate for fiber loss. Added directly to the phase, nonlinear phase noise degrades phase-modulated signal like differential phase-shift keying (DPSK) signals. Recently, DPSK signal has received renewed interests for long-haul transmission to reduce the effect of fiber nonlinearities to signal propagation.
While the effect of nonlinear phase noise is traditionally evaluated based on the variance, as non-Gaussian random variable, the probability density is necessary to better understand the statistical properties of nonlinear phase noise. Studied by us theoretically, the statistical properties of nonlinear phase noise have not been verified experimentally.
2. Result
Differential quadrature phase-shift keying (DQPSK) [1-4] modulation has renewed attention recently for spectral-efficiency transmission systems. Nonlinear phase noise, often called the Gordon-Mollenauer effect
[5], is the major degradation for phase-modulated signals [6-11]. In this paper, we derive anAn exact error probability of DQPSK signal with nonlinear phase noise
is derived the first time, taking into account the dependence between linear and nonlinear phase noise. Nonlinear phase noise is correlated with received intensity [9,11-14]. The error probability of DQPSK signal is derived with and without linearly compensated nonlinear phase noise.
When the differential transmitted phase is equal to zero, the probability density function (p.d.f.) of the differential phase has a Fourier series expansion of
)
cos(
)
(
1
2
1
)
(
2 1 cm cmθ
π
π
m
m
θ
p
m∑
+∞ = Φ ∆Φ=
+
Ψ
2 + × + > Φ < Ψ = Ψ + − Φ + − Φ 2 2 2 ) ( α m, 2 1 α m, 2 1 2 1 NL 2 2 / 3 , m m,α cm λ λ ρ λ λ π α λ λ m m s m m I I m e m
m≥1 + > Φ < + > Φ < + = 1/2 2 1 NL 2 / 1 2 1 NL m α m, tan 1 s s jm jm ρ ρ α λ λ and
( )
jυ(
ρ jυ jυ)
ΨΦ
ν
=sec exp s tan + × + > Φ < Ψ = Ψ + − Φ + − Φ 2 2 2 ) ( α m, 2 1 α m, 2 1 2 1 NL 2 2 / 3 , m m,α cm λ λ ρ λ λ π λ λ α m m s m m I I m e m 1 ≥ m + > Φ < + > Φ < + = 1/2 2 1 NL 2 / 1 2 1 NL m α m, tan 1 s s jm jm ρ ρ α λ λ and
ΨΦ
( )
ν
=sec jυexp(
ρs jυtan jυ)
is the marginal characteristic function of nonlinear phasenoise that depends solely on the signal-to-noise ratio (SNR)
ρ
s [15],Φ
NL is the mean nonlinear phase shift, andα
is the linear compensation factor, andm
I (.) is the mth-order modified Bessel function of the
first kind. The usage of
α
= 0 is equivalent to no linear compensation and withλ
m=λ
m,α . From [9], the optimal compensation factor is equal to2 1 3 1 opt
2
1
+
+
=
s sρ
ρ
α
.,With Gray code, the bit-error probability is equal to
( )
( )
2 1 4 / 4 / e 4 sin 1 1 8 3 1 2 1 cm m Ψ mπ m π d p p cm Φ m∑
∫
∞ = + − ∆Φ − = − = π π θ θ .Figures 1 show the bit-error rate (BER) of a DQPSK signal with nonlinear phase noise. Figure 1a is the case without linear compensation of α= 0 in (4). Figure 1b is with linear compensation using the optimal scale factor of (6). Similar to the case of [11] for DPSK signals, the error probability with compensation is close to the performance of the error probability without compensation and half the mean nonlinear phase shift. Figures 1 also plot the error probability of DQPSK signal without nonlinear phase noise [16].
Figure 2 plot the SNR penalty as a function of nonlinear phase noise of ΦNL . The SNR penalty is calculated for a BER of 10-9. Without nonlinear phase
noise, as from Figs. 1, the required SNR for a BER of 10-9 is 18 dB. For a SNR penalty of less than 1 dB, the
14 16 18 20 22 24 10−12 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 SNR ρs (dB) Error Probability p e <Φ NL > = 0 0.5 1 1.5 2 a. 14 16 18 20 22 24 10−12 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 SNR ρ s (dB) Error Probability p e <Φ NL > = 0 0.5 1 1.5 2 b.
Fig. 1. The BER of DQPSK signal with nonlinear phase noise (a) without and (b) with linear compensation.
mean nonlinear phase shift must be less than 0.50 and 0.95 rad without and with linear compensation, respectively. The about a factor of two difference is the same as the observation of both [11, 13, 14].
The performance of a system is proportional to the SNR that is also proportional to the launched power of the signal. As shown in [14], the mean nonlinear phase shift is proportional to the launched power of the system. From Fig. 2, the SNR penalty increases with the mean nonlinear phase shift. The optimal operating point can be found by the condition that the SNR penalty is small than the SNR improvement. From Fig. 2, without compensation, the optimal operating point is for a mean nonlinear phase shift of ΦNL = 0.89 rad. With linear compensation, the optimal operating point is increased to ΦNL = 1.72 rad, slightly less than twice of that without linear compensation. The optimal operating point is about 0.1 rad less than that of 1 and 1.81 rad of DPSK signals, respectively. Because the variance of nonlinear phase noise decreases with SNR [14], requiring about 5 dB larger SNR, DQPSK signals can tolerate a mean nonlinear phase shift close to DPSK signals although the constellation is closer than DPSK signal.
Nonlinear phase noise is the major degradation for phase-modulated signals [1-8]. Non-Gaussian distributed [6, 7], the nonlinear phase noise cannot be characterized by variance or Q-factor alone [15]. The probability density function (p.d.f.) is necessary to better understand the noise properties and evaluates the system performance.
The joint p.d.f. of received intensity and phase
( )
θθ Φr| 0 ,
, r
pR can be modeled as a periodic function of θ with a period of 2 π and expanded as a Fourier series
( )
∑
+∞{
( )
( )}
= − Φ = + 1 | , r 0 π Re 0 1 2 ) ( , m jm m R R C r e r p r p θ θ θ π θ , r≥0, where ( ) 2 exp[ ( 2 )] 0(2 ) s s R r r r I r p = − +ρ ρis the p.d.f. of received amplitude as a Rice distribution,Cm
( )
r is the m-th Fourier coefficient as a function of the received amplitude r,the coefficients of( )
r Cm are( )
+ − Ψ = m m m m m m m m s r I s r s r r C α α 2 exp 2 2,
1
≥
m
where Im(.) is the mth-order modified Bessel function of the first kind and
+ > Φ < Ψ = Ψ Φ 2 1 s NL m m
ρ
, 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Mean Nonlinear Phase Shift <Φ
NL> (rad)
SNR Penalty (dB) for Pe = 1e−9
without compensation
with compensation
Fig. 2. The SNR penalty of DQPSK signal with nonlinear phase noise with and without linear compensation.
a.
<ΦNL >=1rad
b. <ΦNL >=2 rad
Fig. 3. The p.d.f. of received signal for a mean nonlinear phase shift of 1 and 2 rad.
−3 −2 −1 0 1 2 3 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 Received Phase Φr + <ΦNL> p.d.f. (log. unit) lin. unit <Φ NL> = 0 2 1.5 1 0.5 2 1.5 1 0.5
Fig. 4. The p.d.f of the received phase in logarithmic scale. The inset is the p.d.f. in linear scale.
4 + > Φ < = 2 / 1 2 1 sec s NL s m ρ jm ρ α , and + > Φ < + > Φ < = − 1/2 2 1 2 / 1 2 1 tan 2 1 s NL s NL m jm jm s ρ ρ ,. where
ΨΦ
( )
ν =sec( )
jν exp[
ρs jνtan jν]
is the marginal characteristic function of nonlinear phase noise [12] that depends solely on the signal-to-noise ratio (SNR) ρs, and <ΦNL> is the mean nonlinear
phase shift.
Fig. 3 shows the p.d.f. of the received signal for a mean nonlinear phase shift of 1 and 2 rad and a signal-to-noise ratio (SNR) of ρs = 18. The transmitted
phase is assumed to be θ0 = 0 with a real transmitted
electric field as ρs . With nonlinear phase shift of 1 rad, the received electric field of Fig. 1a has a mean nonlinear phase shift of 1 rad but asymmetrical with respect to the mean received electric field of ρse−j. The p.d.f. has phase spreading increasing with the mean nonlinear phase shift. However, the amplitude spreading does not change with the mean nonlinear phase shift. From the helix shape of the distribution, the received phase correlates with the received amplitude [12,13] and can be compensated using the received intensity.
Although nonlinear phase noise is uncorrelated with the phase of amplifier noise, as non-Gaussian random variable, the phase of amplifier noise and nonlinear phase noise are weakly depending on each other. As the received phase is the summation of all the linear and nonlinear phase noise, its p.d.f. is shown in Fig. 4. Shifted by the mean nonlinear phase shift, the p.d.f. is plotted in logarithmic scale to show the difference in the tail. Not shifted by the mean nonlinear phase shift, the same p.d.f. is plotted in linear scale in the inset. From Fig. 4, when the p.d.f. is broadened by the nonlinear phase noise, the broadening is not symmetrical with respect to the mean nonlinear phase shift.
4. Conclusion
We derive the exact error probability of DQPSK signals with nonlinear phase noise and with or without linear compensation. The error probability takes into
account the dependence between nonlinear phase noise and the phase of amplifier noise. Using the Fourier series expansion of the p.d.f. of the received phase, the exact error probability can be expressed as a series summation. For a SNR penalty less than 1 dB, the mean nonlinear phase shift must be less than 0.95 and 0.50 rad with and without linear compensation respectively. The optimal mean nonlinear phase shift are 1.72 and 0.89 rad with and without linear compensation, respectively. DQPSK signals can tolerate a mean nonlinear phase shift close to that of DPSK signals.
We also derive the p.d.f. of an electric signal contaminated with nonlinear phase noise. The marginal p.d.f. of the received phase is also given analytically. Those p.d.f.’s will be very helpful to find the error probability of a signal with nonlinear phase noise and derive the optimal detector to minimize the error probability.
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