PERFORMANSE BPSK SISTEMA U PRISUSTVU LASERSKOG FAZNOG [UMA
BPSK SYSTEM PERFORMANCES IN THE PRESENCE OF THE LASER PHASE NOISE
Mihajlo Stefanovi}, Dragan Dra~a, Daniela Milovi}, Elektronski fakultet u Ni{u,
]emal Doli}anin, Elektrotehni~ki fakultet u Pri{tini
Sadr`aj –
Modulacija pomo}u vi{e nosioca pokazuje znatnu
osetljivost na fazni {um oscilatora. Iz ovog razloga veoma je
va`no odrediti uticaj faznog {uma na performanse sistema. U
ovom
radu
predstavljamo
ta~an
metod
za
odre|ivanje
verovatno}e gre{ke BPSK modulacione {eme. Rezultati su
bazirani
na
numeri~kom
re{avanju
Fokker-Planck-ove
diferencijalne jedna~ine i prikazani su grafi~ki.
Abstract – Multicarrier modulation exhibits a significant sensitivity to the phase noise of
the oscillator used for frequency down-conversion at the receiver. For this reason, it is
important to evaluate the influence of the phase noise on the system performance. In this
paper we present an accurate method to determine the error probability of binary phaseshift keying (BPSK) modulation scheme. The results in this paper are based on numerical
solution of Fokker-Planck differential equation and are shown graphically.
zt   A cos 0 t   t   x(t ) cos  0 t  y(t ) cos  0 t
1. INTRODUCTION
Orthogonal frequency-division multiplexing (OFDM) for
the transmission of asynchronous transfer mode (ATM) data
packets over indoor radio channels has been suggested in
many papers [2],[3]. The main advantage of OFDM is to
split the available signal bandwidth into a large number of
independent narrow-band subchannels. In this paper we
considered the effect of Wiener phase noise in OFDM
systems. Some analysis of the effect of phase noise has been
presented in [3]. In [3] authors do not derive exact error
probability formulas because they approximate the
contribution of the phase noise with an additional noise
component. This approach is accurate enough in cases when
the linewidth-symbol period product is sufficiently small and
the signal-to-noise ratio large such that the error rate
degradation is mainly determined by the intercarrier
interference. Instead, the analysis presented in [2] takes into
account the exact statistic of the phase noise to evaluate the
error probability. This statistic is obtained by using Taylor
expansion model. Taylor expansion method slightly overestimates the influence of laser phase noise, as it has been
shown[2]. Since the method requires the use of inverse
Laplace transform on highly oscillatory function, it could be a
problem to obtain the reliable results. In order to obtain the
more precise results, we have used the Fokker-Planck
approach, or more precisely, the direct numerical solving of
the Fokker-Planck equation.
2. ERROR PROBABILITY
Let consider BPSK modulation scheme in the presence of
laser phase noise. The signal at the receiver input is
(1)
where (t) is laser phase noise process and the second term in
(1) presents narrow-band Gussian noise process. Signal z(t) is
then multiplied by 2 cos  0 t in order to remove highfrequency components and then sent to the integrate and
dump filter (I&D) with integration time T. At the integrator
output, the signal component related to laser phase noise is

1 T j (t )
e
dt   e j (t ) dt  r ( )e j ( )   ( )

T 0
o
(2)
The phase noise (t) is assumed to be a continious-path
Brownian motion with zero mean and variance 2 (  is
laser linewidth). As it can be seen, phase noise is continious
and nonstationary Markov process. Considering stochastic
nature of laser phase noise, it is rather difficult to obtain the
probability density function of  ( ) . For this reason, we
have used the exact approach that leads to the Fokker-Planck
partial differential equation. The solution of Fokker-Planck
equation is joined pdf of random variable  ( ) . In polar
coordinates, FP equation can be expressed as
p
p sin  p D  2 p
  cos



r
r  2  2
(3)
with D=2 and initial condition
1
p(r ,  ,0)   (r )
r
The solution of FP equation possesses certain properties:
1. r  z  cannot be greater than , so that the solution is
confined within a disc of radius .
2. Since the phase process (t) has zero-mean, the solution
must be symmetric about the =0, = axis.
3. The solution p(r,,) must be non-negative everywhere.
4. p(r,,) must integrate to unity over the {r,} domain.
5. The solution has very steep front near the circle r=.
sin   D  2

r  2  2
Operator splitting method leads us to following equations:
 v
(t k  t  t k 1 )
  Lr v 
 
vr ,  ,  k   pr ,  ,  k 

L 
 w
 L w

 
wr ,  ,  k   vr ,  ,  k 1 

(t k  t  t k 1 )
w k 1  p k 1
Solving of these equations can be simplified so that first
equation:
v
v
  cos 


has an analytical solution:
vr ,  ,  0     vr   cos ,  ,  0 
Now, instead of using difference scheme, we can only
perform simple interpolation. Obtained results are shown on
Fig1.
r
p (r ,  )drd
(5)
1
1
Pe    
e
2 0 r  2 

( z  Ar cos ) 2
2 2
p(r ,  )drd dz
(6)


NUMERICAL RESULTS
Numerical results are obtained with following parameters:
Nr=100, N=20, N=100.
In an angular direction, relatively small number of points is
sufficient, but not less then N=20 on the interval  In a
radial direction we need large number of points, especially for
small values of D. It is shown that the best solution is
obtained if we use the same number of points for time and
radial step. In that manner interpolation will keep the
influence of the initial condition.
Error probability is obtained numerically and results are
shown on Fig.2.
-1
T=0.16
-2
-3
-4
0.0
4
-5
-6
-7
0.0
2
-8
-9
7.5
0.09
-10
0.011
5
5
10
15
20
snr[dB]
0.02
2.5
0.06
Figure 2. Error probability versus signal-to-noise ratio (snr)
for different ammount of laser phase noise
0
CONCLUSION
0.076
-2.5
e
2 
( z  Ar cos ) 2
2 2
Now, the BER of BPSK system is
Pe[dB]
We chose to develop the difference scheme and then use an
operator splitting method. Equation can be written as
p
 L r  p   L  p 

where L r and L are radial and angular operators:

Lr   cos 
r
p( z )   

1
0.04
-5
0.013
-7.5
-7.5
-5
-2.5
0
2.5
5
7.5
Figure 1. Contour plot of the joined probability density
function p(r) for D=8 and =8 (
The joined probability density function p(r) can be used
for evaluation of the error probability. The signal at the
output of the I&D filter is
z  Ar cos  x
(4)
As we estimated the statistic of laser phase noise, random
variable z is Gaussian variable and its pdf (when symbol –1
transmitted ) is
In this paper, BPSK system performances in the presence
of laser phase noise are obtained. Since method in [1]
requires the use of inverse Laplace transform on highly
oscillatory function, it could be a problem to obtain reliable
results. For more precise results, we have used the FokkerPlanck approach, or more precisely, the direct numerical
solution of the Fokker-Planck differential equation. The
analysis presented in this paper gives the exact statistic of the
laser phase noise to evaluate the error probability. BER is
then simply calculated and shown graphically. It is apparent
that performances strongly degrades for 
Fokker-Planck approach can be used for performance
evaluation of different modulation schemes such as DBPSK,
DQSK, QPSK.
LITERATURE
[1] Luciano Tomba, “On the effect of Wiener phase noise in
OFDM systems”, IEEE Transactions on Communications,
Vol.46, No.5, May 1998.
[2]
A. Chini, M. S. El-Tanany, S. A. Mahmoud,
"Transmission of high rate ATM packets over indoor radio
channels", IEEE Trans. J. Select. Areas Commun., vol.14, pp.
460-468, Apr.1996., AC-22, pp. 210-222, April 1977
[3]
H. Sari, G. Karam, I. Jeanclaude, “Transmission
techniques for digital terrestrial TV broadcasting”, IEEE
Trans. Commun. vol. pp. 479-486, 1985.