# Power Penalty Due to First-order PMD in Optical OFDM/QAM and FBMC/OQAM
Transmission System

Jianping Wang, Ke Zhang, Xianyu Du, He Zhen, Jing Yan  Department of
communication Engineering,30 Xueyuan Road, Haidian District, Beijing 100083 P.
R.China

###### Abstract

Polarization mode dispersion (PMD) is a challenge for high-data-rate optical-
communication systems. More researches are desirable for impairments that is
induced by PMD in high-speed optical orthogonal frequency division
multiplexing (OFDM) transmission system. In this paper, an approximately
analytical method for evaluating the power penalty due to first-order PMD in
optical OFDM with quadrature amplitude modulation (OFDM/QAM) and filter bank
based multi-carrier with offset quadrature amplitude modulation (FBMC/OQAM)
transmission system is presented. The simulation results show that, compared
with the single carrier with quadrature phase shift keying(SC-QPSK), both the
OFDM/QAM and the FBMC/OQAM can decrease the power penalty caused by PMD by
half. Furthermore, the FBMC/OQAM shows better power penalty immunity than the
OFDM/QAM under the influence of first order PMD.

###### keywords:

polarization mode dispersion , OFDM/QAM , FBMC/OQAM , power penalty

##  1 Introduction

Optical fiber system has become a hot spot because of its ultra high speed,
huge capacity and long haul transmission ability. Nowadays optical signal
amplification and fiber dispersion compensation techniques are increasingly
developed, and PMD has become the key limitation of transmission speed and
distance[1, 2, 3, 4]. PMD is a physical phenomenon caused by the birefringence
of the optical fiber. When transmitting optical signal, PMD will cause
different delays for different polarizations and the group delay difference
between the slow and the fast modes is called differential group delay(DGD).
When DGD is getting larger and can not be neglected compared with the signal
bit duration, it will cause the pulse broadening and Inter-symbol
Interference(ISI)[1], then pulse distortion and system penalties occur.
Different from the fiber chromatic dispersion(CD), PMD is a stochastic
quantity influenced by the external conditions such as temperature or fiber
vibration, et.al, which makes PMD particularly difficult to manage or
compensate. In a specific system, dissipation due to PMD is determined by the
theoretical structure model and physical factors (infrastructure,
environmental factor, et.al), and different systems have different modulation
techniques as well as PMD compensation techniques.

Many researches in recent years are concentrated on PMD tolerance which based
on different modulation modes and coding schemes[5, 6]. In [5], the article
proposed a 20Gb/s high-speed optical fiber transmission systems with Non
Return to Zero (NRZ) and Return to Zero (RZ) code and the differences of the
PMD-induced fiber channel is discussed by numerical simulation. In [6],
compared with on-off keying(OOK), it is demonstrated that differential phase
shift keying(DPSK) signal has large first-order PMD tolerate ability in a
40Gb/s optical fiber transmission system. Other researches are focused on the
PMD compensation techniques[7, 8, 9], and the compensation of system PMD is
demonstrated experimentally by electronical and optical compensators. Adaptive
optical PMD compensation, which based on feedback, courts a balance between
speed and accuracy. Recently, investigations pay more attention to the devised
electronic PMD compensation schemes, such as iterative decoding techniques,
maximum likelihood sequence estimation (MLSE), and transversal digital
filtering[10, 11, 12]. In [10], PMD equalizers based on constant modulus
algorithm (CMA) is presented in coherent optical polarization-division-
multiplexed (PDM) QPSK systems taking account of PMD effect.

Recently, OFDM has been recommended as an effective PMD-resilient modulation
format for high speed optical fiber transmission systems[13, 14, 15]. OFDM
employs multi-carrier transmission of orthogonal, and has lower data rate
subcarriers, therefore it simplifies PMD equalizer structure, and achieves
high spectral efficiency in frequency-selective channels via the fast Fourier
transform(FFT)[16]. In [15], the possibility of PMD compensation in fiber-
optic communication systems with direct detection using a simple channel
estimation technique and low-density parity-check (LDPC)-coded OFDM is
demonstrated. In [16], it has presented that, if it is not necessary for RF
guard bands, OFDM capacitates high speed transmission with PMD tolerance which
is at least twice greater than that of uncompensated OOK at a given bit rate,
while in systems with RF guard bands, a PMD tolerance trade-off proportional
to guard band size was shown, where in guard band and constellation sizes may
be viewed as design parameters. In [17], it has shown that, without requiring
any feedback, OFDM can mitigate pulse distortion caused by all-order PMD in
long-haul optic fiber communication systems. Like the OFDM, FBMC is another
multi-carrier technology with higher spectral efficiency and has been
perceived as an alternative to OFDM in recent years. While compared with OFDM,
FBMC has a larger PMD tolerance because of its large stop-band attenuation and
the frequency selective fading channel. Due to the advanced digital modulation
technique, the FBMC technique is quite fit for the high speed optical fiber
transmission systems. As a result, in optical FBMC communication system, more
research should be done on the impairment caused by the PMD.

In this paper, we focus on the system power penalty due to first-order PMD in
multi-carrier optical transmission system which use OFDM/QAM and FBMC/OQAM
modulation format respectively, by comparing it with single carrier QPSK
modulation, the theoretical model of power penalty in multi-carrier optical
system impacted by first-order PMD is proposed, and then numerical simulation
verification are given. The rest of the paper is organized as follow. In
Section 2, A brief introduction to the basic theory of OFDM/QAM and FBMC/OQAM
modulation has been given. And in Section 3, the PMD principles, as well as
the derivation of power penalty in optical OFDM/QAM and FBMC/OQAM is proposed.
In Section 4, the derivation results verified by the simulation results is
proposed and finally the conclusion is given in Section 5.

##  2 Optical OFDM/QAM and FBMC/OQAM System Model

OFDM/QAM and FBMC/OQAM are kinds of multi-carrier modulation technique that
can modulate and demodulate signals in frequency-domain by Inverse Fast
Fourier Transform/Fast Fourier Transform (IFFT/FFT). OFDM technique can reduce
the effects of dispersion and ISI efficiently and now is considered to be an
effective solution to high speed optical communication in the future. High
speed optical OFDM transmission will also be influenced by PMD effect as it is
in single carrier modulation. Optical OFDM transmission systems have better
anti-PMD effect ability compared with single carrier system because of the
OFDM principles that separate a high speed data stream into several orthogonal
low speed stream.

As we known that OFDM/QAM (i.e. CP-OFDM) has been widely used and considered
as the core technique solution for next generation wireless communication, and
FBMC/OQAM (i.e. OFDM/IOTA) is an alternative approach according to 3GPP
protocols. Compared with traditional OFDM/QAM based on cyclic prefix (CP),
FBMC/OQAM without CP can achieve greater spectral efficiency, furthermore,
FBMC/OQAM has better performance in wireless channel via choosing well time-
frequency localized pulse shaping prototype filters[18]. Applying OFDM
techniques into high-capacity and high-speed optical fiber communication
systems is a major research direction[19, 20], and it will achieve a high
flexibility and capacity in dynamic resource allocation and user access by
combining with new technologies like PON, et.al[21]. The basic principles of
these two OFDM techniques are introduced bellow:

###  2.1 OFDM/QAM System Model

High speed information bit stream with bite rate $R_{b}=1/T_{b}$ is modulated
in baseband using M-QAM modulation with symbol duration
$T_{s}=T_{b}\log_{2}M$, and then divided in to $N$ parallel symbol streams
which are filtered by a pulse shape function $g_{n,k}(t)$, the time-domain
OFDM/QAM signal can be written in the following analytic form[18]

$s_{\textrm{QAM}}(t)=\sum_{k=1}^{+\infty}\sum_{n=1}^{N}a_{n,k}g_{n,k}(t)$ (1)

where

$g_{n,k}(t)=e^{j2\pi nFt}g(t-kT)$ (2)

$F$ denotes the inter-carrier frequency spacing and $T$ is the OFDM symbol
duration. $a_{n,k}$ represents the QAM baseband modulation output data on the
$n$th subcarrier at time index $k$. In a OFDM/QAM system,
$F=1/NT_{s}=\nu_{0}$, $T=\tau_{0}$ and in order to satisfy the orthogonality,
$\tau_{0}\nu_{0}=1$, and the prototype function is defined as

$g(t)=\left\\{\begin{array}[]{lc}1/\sqrt{\tau_{0}},&0\leq t<\tau_{0}\\\
0,&\textrm{elsewhere}\end{array}\right.$ (3)

###  2.2 FBMC/OQAM System Model

Under the same initial conditions as OFDM/QAM, the time domain FBMC/OQAM
signal can be expressed as Eq.(4)[18, 22]

$s_{\textrm{OQAM}}(t)=\sum_{k=1}^{+\infty}\sum_{n=1}^{N}a_{n,k}g_{n,k}(t)$ (4)

where

$g_{n,k}(t)=e^{j2\pi n\nu_{0}t}g(t-k\tau_{0})\times
e^{j(n+k)\pi/2},\>\nu_{0}\tau_{0}=1/2$ (5)

$g(t)$ is the prototype pulse shaping function that can be different from
rectangular window, for example, Extended Gaussian Function (EGF) and
Isotropic Orthogonal Transform Algorithm (IOTA) Function, et.al. Unlike the
original OFDM/QAM, FBMC/OQAM employs a modified inner product by taking a real
component to maintain the orthogonality among the synthesis and analysis
basis, as show in bellow

$\mathcal{R}\left\\{g_{n,k}^{*}\times
g_{n^{\prime},k^{\prime}}\right\\}=\left\\{\begin{array}[]{lc}1,&(n,k)=(n^{\prime},k^{\prime})\\\
0,&(n,k)\neq(n^{\prime},k^{\prime})\end{array}\right.$ (6)

The purpose of pulse shaping in FBMC/OQAM is to find an efficient transmitter
and a corresponding receiver waveform for the current channel condition[23,
24], a well time-frequency localized waveform should satisfy

$\frac{\tau_{0}}{\Delta t}=\frac{\nu_{0}}{\Delta f}$ (7)

where $\Delta t$ and $\Delta f$ is the RMS delay spread and Doppler spread,
respectively.

##  3 Power Penalty duo to First-order PMD

Currently, OFDM is proposed to be a promising modulation technique for high-
speed optical transmission systems, owing to high tolerance to CD and PMD.
However, PMD still degrades the performance of the high-speed optical
transmisson systems, due to lacking of mature compensation techniques.
Subsequent bit error and system power penalty analysis seeks to assess in
order to evaluate the PMD tolerance of high speed fiber OFDM/QAM and FBMC/OQAM
transmission system.

In single mode fiber (SMF) transmission, optical signals are composed by two
orthogonally polarized $HE_{11}$ mode. If the SMF is ideal, the two polarized
mode have the same refractive index and transmitting speed, so there won’t be
any PMD as it shown in Fig. 1. However, in practical fiber, it’s impossible to
achieve identical refractive index, thus there will be a different delay
between the two polarized mode and causes the DGD, as shown in Fig. 1, this
phenomenon is called PMD. In long haul and high speed optical fiber
communication system, pulse broadening caused by PMD effect can lead to
serious ISI, which will degrade the system performance, and that is why PMD
has been considered as a key factor after CD and fiber attenuation.

Figure 1: Time-domain behavior of PMD in a short birefringent fiber. Figure 2:
System model and corresponding lowpass equivalent.

Fig. 2 illustrate the block diagram of basic optical fiber transmission system
and it’s frequency-domain equivalent, the input electronic signal after
electro-optical modulation (EOM) is set as $i_{in}(t)$, the envelope of the
input signal is $\tilde{x}(t)=\alpha i_{in}(t)$ and $\alpha$ is a
proportionality coefficient. The Jones vector of the resulting electronic
field at the fiber input is given by $\tilde{E}_{in}(t)$ as[16]

$\tilde{E}_{in}(t)=\tilde{E}_{in}(t)\hat{e}_{in}=R[\sqrt{\tilde{x}(t)}e^{j\omega_{0}t}]\hat{e}_{in}$
(8)

where $\hat{e}_{in}=[\hat{e}_{1},\hat{e}_{2}]^{T}$ is the polarization state
Jones vector, whose entries denote the two orthogonal Principle State of
Polarizations (PSPs) at the fiber input. Under the First-order PMD
approximation, PMD vector
$\vec{\Omega}=\vec{\Omega}_{0}=\Delta\tau\hat{e}_{1}$, $\hat{e}_{1}$ is the
unit vector in fast PSP. The Fourier transform of fiber output signal
$\hat{E}_{out}(t)$ is given by[16]

$\hat{E}_{out}(\omega)=\textbf{F}(\omega)\hat{E}_{in}(\omega)=\textbf{RU}(\omega)\textbf{R}^{-1}\hat{E}_{in}(\omega)$
(9)

where $\hat{E}_{in}(\omega)=\mathcal{F}(\tilde{E}_{in}(t))$,
$\textbf{F}(\omega)=\textbf{RU}(\omega)\textbf{R}^{-1}$ denotes the fiber
Jones matrix. R denotes the random, frequency-independent rotation matrix and
$\textbf{U}(\omega)$ denotes the time delay matrix caused by first-order PMD
which can be expressed as [25]:

$\textbf{U}(\omega)=\left[\begin{array}[]{cc}e^{j\omega\Delta\tau/2}&0\\\
0&e^{-j\omega\Delta\tau/2}\end{array}\right]$ (10)

where $\Delta\tau$ is the differential group delay (DGD). And the explicit
expression for R is given by[25]

$\textbf{R}=\left[\begin{array}[]{cc}r_{1}&-r_{2}^{*}\\\
r_{2}&r_{1}^{*}\end{array}\right]$ (11)

where

$\begin{split}r_{1}=&\cos{\theta}\cos{\phi}-j\sin{\theta}\sin{\phi}\\\
r_{2}=&\sin{\theta}\cos{\phi}+j\cos{\theta}\sin{\phi}\end{split}$ (12)

where $\theta$,$\phi$ are independent random variables,representing the fast
PSP azimuth and ellipticity angle respectively.

And then we can get[16]

$\hat{E}_{out}(t)=c_{1}E_{in}(t+\Delta\tau/2)\hat{e}_{1}+c_{2}E_{in}(t-\Delta\tau/2)\hat{e}_{2}$
(13)

where $c_{1},c_{2}$ depend on the rotation matrix R, and can be expressed by

$\begin{split}c_{1}=&|\cos{\varphi}|\\\ c_{2}=&|\sin{\varphi}|\end{split}$
(14)

where $2\varphi$ denotes the angle between the fast PSP and the signal
polarization state in Stokes space.After photoelectric detection (PD) and
post-detection filtering, the electronic signal[16]

$\begin{split}i_{out}(t)&=\rho|\hat{E}_{out}(t)|^{2}\\\
&=\rho\left[|c_{1}\sqrt{\tilde{x}(t+\Delta\tau/2)}|^{2}+|c_{2}\sqrt{\tilde{x}(t-\Delta\tau/2)}|^{2}\right]\\\
&=\rho\alpha\left[\gamma
i_{in}(t+\Delta\tau/2)+(1-\gamma)i_{in}(t-\Delta\tau/2)\right]\end{split}$
(15)

where $\rho$ is photoelectric detector sensitivity and $\gamma=|c_{1}|^{2}$ is
the PSP power splitting ratio.

In Eq.(15), it is obvious that first-order PMD causes pulse broadening and
results in adjacent pulses overlap, which will finally cause the Power Penalty
at the receiver. System Power Penalty due to PMD effect is defined as the
difference of receiver sensitivity (in dB) between two conditions, with or
without PMD effect. Considering first-order PMD approximation, the input
optical signal is only divided into two orthogonal polarized mode and causes
one DGD $\Delta\tau$, and leads to power penalty[26].

Under this assumption, we assume that the received signal $i_{out}(t)$ can be
determined by input signal $i_{in}(t)$ after fiber transmission via Eq.(15)

$i_{out}(t)=\gamma i_{in}(t+\Delta\tau/2)+(1-\gamma)i_{in}(t-\Delta\tau/2)$
(16)

The Root-Mean-Square (RMS) pulse width $\delta_{2}$ of the output signal is
given by[26]

$\delta_{2}^{2}=\frac{\int_{-\infty}^{+\infty}t^{2}i_{out}(t)dt}{\int_{-\infty}^{+\infty}i_{out}(t)dt}-[\frac{\int_{-\infty}^{+\infty}ti_{out}(t)dt}{\int_{-\infty}^{+\infty}i_{out}(t)dt}]^{2}$
(17)

Assuming that $i_{in}(t)$ is symmetrical about $t=0$, therefore
$i_{in}(t+\Delta\tau/2)$ and $i_{in}(t-\Delta\tau/2)$ are symmetrical about
$t=0$ too, so Eq.(17) can be expressed as

$\delta_{2}^{2}=\frac{\int_{-\infty}^{+\infty}t^{2}i_{in}(t)dt}{\int_{-\infty}^{+\infty}i_{in}(t)dt}+\Delta\tau^{2}\gamma(1-\gamma)$
(18)

When $\Delta\tau=0$ (without PMD), RMS pulse width $\delta_{1}$ of the input
signal is given by

$\begin{split}\delta_{1}^{2}&=\frac{\int_{-\infty}^{+\infty}t^{2}i_{in}(t)dt}{\int_{-\infty}^{+\infty}i_{in}(t)dt}-[\frac{\int_{-\infty}^{+\infty}ti_{in}(t)dt}{\int_{-\infty}^{+\infty}i_{in}(t)dt}]^{2}\\\
&=\frac{\int_{-\infty}^{+\infty}t^{2}i_{in}(t)dt}{\int_{-\infty}^{+\infty}i_{in}(t)dt}\end{split}$
(19)

Furthermore, we can observe that[26]

$\delta_{2}^{2}=\delta_{1}^{2}+\Delta\tau^{2}\gamma(1-\gamma)$ (20)

Irrespective of the polarization dependent loss (PDL), power penalty due to
PMD effect $\epsilon$ can be represented as

$\epsilon(dB)=10lg\frac{\delta_{2}}{\delta_{1}}=5lg(1+\frac{\Delta\tau^{2}\gamma(1-\gamma)}{\delta_{1}^{2}})$
(21)

where $\Delta\tau$ is very small in a general way, so

$\epsilon(dB)\approx 5\frac{\Delta\tau^{2}\gamma(1-\gamma)}{\delta_{1}^{2}}$
(22)

For general system, the RMS pulse width $\delta_{1}$ of input signal is
proportional to bit interval $T_{b}$, power penalty $\epsilon$ can be
represented as[26]

$\epsilon=\frac{A}{T_{b}^{2}}\Delta\tau^{2}\gamma(1-\gamma)$ (23)

where $A$ is a coefficient concerned with pulse shape, modulation format and
receiver characters, et.al. $T_{b}$ is bit interval and $\Delta\tau$ is the
instant DGD. The PSP power splitting ratio $0<\gamma<1$. For a fixed DGD,
power penalty is maximized when $\gamma=0.5$, so it should be considered as a
requirement while designing a system to ensure performance. In the follow
simulations, $\gamma$ is fixed to be 0.5.

In original single carrier optical fiber communication systems, power penalty
due to PMD effect $\epsilon$ (in dB) can be seen in Eq.(23) , and when it
comes to the multi-carrier optical fiber transmission, power penalty due to
PMD can be deduced by the following content.

Assuming an OFDM/QAM system with symbol duration $\tau_{0}=T$, sub-carrier
number $N$ and inter-carrier frequency spacing $\nu_{0}=F=1/T$. $\Delta\tau$
represents the DGD caused by first-order PMD. The baseband modulation scheme
is SC-QPSK and the OFDM signal bandwidth $BW=N\times F$, bit rate
$R_{b}=N\times F\times\log_{2}M$ (for QPSK, 4QAM, $M=4$ and for 16QAM, $M=16$,
et.al) and bit duration $T_{b}=1/R_{b}$.

For FBMC/OQAM, inter-carrier frequency spacing $\nu_{0}=F=1/T$ and
$\tau_{0}=T/2$. Firstly, we can get the power penalty $\epsilon_{n}$ in $n$th
sub-carrier via its angular frequency $\omega_{n}$

$\omega_{n}=2\pi\nu_{0}n$ (24)

For OFDM/QAM and FBMC/OQAM, the cycle of the $n$th sub-carrier
$T_{n}=1/\nu_{0}n=T/n$ and put it into Eq.(23)

$\epsilon_{n}=\frac{A}{T_{n}^{2}}\Delta\tau^{2}\gamma(1-\gamma)=\frac{A\gamma(1-\gamma)n^{2}\Delta\tau^{2}}{T^{2}}$
(25)

Assuming that transmitting power of each sub-carrier is $P_{0}$ without PMD
can satisfy the requirement of receiver PMD while the $n$th sub-carrier
transmitting power is $P_{n}$ with first-order PMD to achieve the same
performance. According to the definition of power penalty we can define the
power penalty of the $n$th sub-carrier as

$\epsilon_{n}=10\log\frac{P_{n}}{P_{0}}$ (26)

then the total power penalty of the multi-carrier system is given by

$\epsilon=10\log\frac{\sum_{n=1}^{N}P_{n}}{NP_{0}}$ (27)

put Eq.(26) into Eq.(27)

$\epsilon=10\log\left(\frac{1}{N}\sum_{n=1}^{N}10^{\frac{\epsilon_{n}}{10}}\right)$
(28)

Equation Eq.(27) is the conclusion that the theoretical expression of power
penalty due to first-order PMD in OFDM/QAM and FBMC/OQAM system and in the
next section, numerical simulation will be given to prove its correctness.

##  4 Numerical Simulation

In this section,focused on studying the power penalty due to first-order PMD
in SC-QPSK, OFDM/QAM and FBMC/OQAM system respectively. Specially, in OFDM/QAM
and FBMC/OQAM simulation, we fixed inter-carrier frequency spacing
$\nu_{0}=100$MHz and PSP power splitting ratio $\gamma=0.5$, sub-carrier
number $N=64$ and $N=128$ respectively. The prototype pulse shaping function
in FBMC/OQAM is set to be Square Root Raised Cosine (SRRC) filter with the
length of $L=4N$. Obviously, in order to compare with the multi-carrier
condition on the PMD tolerance problem, we set the SC-QPSK modulation with the
same transmission bit rate as the OFDM/QAM and FBMC/OQAM. For example, SC-QPSK
modulation bit rate $R_{b}=2N\nu_{0}=25.6$Gb/s for $N=128$ and
$R_{b}=2N\nu_{0}=12.8$Gb/s for $N=64$. Other factors (like FEC et.al) remain
unchanged when comparing OFDM/QAM with FBMC/OQAM.

The Bit Error Rate (BER) vs Signal to Noise Ratio (SNR) simulation results of
the OFDM/QAM system with $N=128$ and inter-carrier frequency spacing
$\nu_{0}=100$MHz is shown in Fig. 3, DGD with 0, 0.2, 0.4, 0.8 and 1 times of
the bit duration $T_{b}$. caused by first-order PMD is separately simulated.
It’s clearly shown in the figure that system BER can reach $10^{-9}$ when
$E_{b}/N_{0}$ is about 6.8dB without PMD(DGD$=0$) and with the growth of DGD,
we have to increase the transmitter power to improve the channel SNR in order
to maintain the system BER performance in $10^{-9}$. When
DGD$=0.4\times/2N\nu_{0}=15.6ps$, thus 0.4 times of bit duration, the
$E_{b}/N_{0}$ is 7.7dB at the point that BER is $10^{-9}$, and the power
penalty due to PMD in this situation $\epsilon=7.8-7.1=0.7(dB)$. Comparing all
the simulation results, We can find that the power penalty due to PMD effect
is growing more faster when DGD is getting bigger, which means the signal
distortion is getting more worse.

Figure 3: Simulation BER versus $E_{b}/E_{o}$ results for several
$\Delta\tau/T_{b}$ values under OFDM/QAM. Figure 4: Simulational BER versus
$E_{b}/E_{o}$ results for several $\Delta\tau/T_{b}$ values under FBMC/OQAM.
Figure 5: Simulation BER versus $E_{b}/E_{o}$ results for several
$\Delta\tau/T_{b}$ values under SC-QPSk.

The BER performance of the FBMC/OQAM modulation scheme is show in Fig. 4. The
simulation condition is the same with that of the OFDM/QAM system Without the
influence of PMD effect, the system BER reaches $10^{-9}$ at
$E_{b}/N_{0}=$7.1dB, and when DGD is 0.4 times of bit duration ,$E_{b}/N_{0}$
should be 7.8dB to achieve the same BER performance. The power penalty
$\epsilon=7.8-7.1=0.7$dB at that moment. Same with the OFDM/QAM the power
penalty in the FBMC/OQAM caused by the first-order PMD is growing faster when
DGD is getting bigger and leads to more serious signal distortion.

The difference of PMD tolerance between single carrier and multi carrier
system can be seen from Fig. 5 which demonstrate the BER performance of a
25.6Gb/s SC-QPSK signal with different DGD due to first-order PMD . Same as
above, $E_{b}/N_{0}$ is 7.2dB and 10.1dB when DGD is 0 and $0.4T_{b}$
respectively, and derivatives the power penalty is about 2.9dB. By observing
Fig. 3, Fig. 4 and Fig. 5, we can find that both OFDM/QAM and FBMC/OQAM has
better anti-PMD ability than SC-QPSK.

Figure 6: Power penalty under SC-QPSK,OFDM/QAM and FBMC/OQAM for N=128. Figure
7: Power penalty under SC-QPSK,OFDM/QAM and FBMC/OQAM for N=64.

Fig. 6 presents the power penalty with different normalized mean DGD with the
SC-QPSK, OFDM/QAM and FBMC/OQAM modulation at the same transmission bit rate
of 25.6Gb/s, and the sub-carrier number $N=128$. Correctness can be verified
by the simulation results that the SC-QPSK power penalty matches well with the
theoretical curve which is given by Eq.(23). Theoretical curves of OFDM/QAM
and FBMC/OQAM power penalty due to first-order PMD which are given by our
derivation in Eq.(28) are also shown in this figure , and it’s clear to see
that the simulation results are consistent with theoretical values pretty
well. The coefficient $A$ is set as 68, 64 and 60 in SC-QPSK, OFDM/QAM and
FBMC/OQAM respectively as its value is dependent on multi-factors like
modulation schemes, pulse shaping techniques and receive modes, et.al, that
is, $A$ is a different value for different systems.

When power penalty is 1dB for SC-QPSK, OFDM/QAM and FBMC/OQAM systems,
$\Delta\tau/T_{b}$ is about 0.24, 0.41 and 0.45 in Fig. 6, and OFDM/QAM and
FBMC/OQAM have about twice more PMD tolerance than SC-QPSK from this view
point and show better anti-PMD abilities. FBMC/OQAM shows better performance
than OFDM/QAM at the same time because of its out of band attenuation. For
example, the power penalty of OFDM/QAM and FBMC/OQAM is 0.9dB and 0.7dB
respectively when $\Delta\tau/T_{b}=0.4$, thus the latter one is 0.2dB less
than the former. And when $\Delta\tau/T_{b}=0.6$, the gap between these two
schemes is increased to about 1dB. We can deduce a conclusion that FBMC/OQAM
will achieve better anti-PMD ability with the growth of PMD effect.

Fig. 7 presents the same situation with sub-carrier number $N=64$, and the
same conclusion can be deduced from this figure compared with Fig. 6.
comparing these two figure we can find that the difference between OFDM/QAM
and FBMC/OQAM is turning smaller with a lower number of sub carrier which give
us the conclusion that FBMC/OQAM shows better anti-PMD performance in power
penalty than OFDM/QAM with larger sub-carrier number $N$.

##  5 Conclusion

In this paper, we discussed the System Power Penalty due to first-order PMD
effect in multi-carrier optical communication system, especially two
modulation schemes OFDM/QAM and FBMC/OQAM. Theoretical derivation of the
multi-carrier condition were given by comparing with original single carrier
situation at the first, and then confirmed its validity via the numerical
simulation. Through the simulation results we can find that the power penalty
due to first-order PMD in OFDM/QAM and FBMC/OQAM systems is about a half
smaller than that of the single carrier SC-QPSK system at the same
transmitting bit rate, and with the growth of sub-carrier number and bit rate,
the latter one can achieve better PMD resistance ability than the the former.

## Acknowledgments

This research is supported by the Fundamental Research Funds for the Central
Universities (No.FRF-TP-09-015A), and also supported by the National Natural
Science Foundation of P.R.China (No.61272507).

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