Download Statistical Estimation of Error Probability in a Digital Wireless

2014 UKSim-AMSS 16th International Conference on Computer Modelling and Simulation
Statistical Estimation of Error Probability in a Digital Wireless Communication
Network
Clement Temaneh-Nyah
Department of Electronics and Computer Engineering, University of Namibia
Ongwediva, Namibia
[email protected], [email protected]
Abstract —This paper proposes a Monte - Carlo based
statistical simulation technique for estimating the error
probability for different modulation schemes of a log-normal
Raleigh fading channel in a digital wireless communication
network. The generalised expressions which are found in the
literature involve not only tedious mathematical analysis and
lengthy computational time but also need to be re-derived
for any change in modulation type. The proposed method
allows some random parameters like user’s random
positions to be accounted for without any mathematical
challenge. There is excellent agreement between error
probabilities computed through the proposed technique and
that obtained in the literature.
where
s(t ) is the transmitted signal, n(t ) is the
Gaussian noise and φ (t ) is the phase of the received
signal. Because of the complexity in accounting for the
fluctuation of the received signal phase φ (t ) , in [1], it is
assumed that the phase is constant within the signal
interval [0, T], an assumption which is valid for the digital
modulation type: FSK, QPSK and QAM.
Different statistical distributions are used to estimate
the received signal level under fading conditions. In [2],
the multipath fading environment is generally modelled as
Raleigh or Rician distributions, while in [3], it is shown
that experimental data agrees with the m-Nakagami –
distribution. In [4], the authors give a unified theoretical
analytical approach to BER estimation in a CDMA
system for the Nakagami-m fading channel. However, in
the above approach, it is assumed that, the channel
attenuation is deterministic and that the error is
mainly due to multiple access interference (MAI) plus
noise, an assumption which might not always be true
in a real mobile channel. A number of authors have
obtained expressions for BER under specified conditions
and assumptions. The authors of [5] for example, gives a
generalized closed form expression for the computation of
average bit error probability over the Rayleigh fading
channel for any digital modulation technique. The
drawback with this approach includes: the tedious and
lengthy mathematical expressions involved which need to
be re-derived for any change in the modulation type. The
computation involved in this process can be lengthy.
In [6], the authors proposed a close form
expression for the average BER by statistically
averaging the conditional BER in the case of different
independent paths. However, with this approach, it is
difficult to obtain a simple expression even when the
paths have the same distribution but with different
parameters. A difficult case is when the probability
distribution functions (PDFs) of the different signal and
interference paths belongs to different families of
distributions. The authors of [7] presented an exact and
closed-form expression for the average BER of coherent
INTRODUCTION
Communication networks are designed to provide
service to their users with acceptable quality level. This
means that network traffic should be analyzed and
properly controlled so that the desired Quality of Service
requirements of applications and Services are
achieved. The diversity of applications characteristics,
including current network technologies demands specific
modeling and design tools.
Moreover, different
techniques including analytical modeling, simulation, and
measurement and monitoring are required in order to
support design and dimensioning of communication
networks and services.
One of the key parameters characterising the quality of
digital radio communication channel is the bit error rate
(BER) which is influenced by the signal to noise ratio.
The radio communication channel between stationary
base stations (BS) and mobile stations (MS) are
characterised by impairments such as noise, fast and slow
fading of the received signal due to a number of factors
including: multiple users’ random movements’, multipath
fading, inter-symbol interference (ISI), e.t.c. The received
signal in the signal interval [0, T] for a Gaussian channel
can be expressed as [1]:
r (t ) = α (t )e − jφ (t ) s (t ) + n(t ) .
978-1-4799-4923-6/14 $31.00 © 2014 IEEE
DOI 10.1109/UKSim.2014.16
is the attenuation of received signal
amplitude,
Keywords - Digital Radio Communication, Bit Error Rate,
Error Probability, Fading Radio Communication Channel,
Monte-Carlo Method.
I.
α (t )
(1)
216
TABLE I
BER EXPRESSION FOR VARIOUS MODULATION TYPES
binary phase-shift keying (BPSK) using maximal ratio
combining (MRC) in the presence of multiple cochannel
interfering users for in the case of Rayleigh fading. The
expression was obtained with the assumption that the
branch gains of the desired user signal and interfering
signals experience correlated Rayleigh fading. Even
though the results are valid for arbitrary correlation
models, number of interfering users and receiver
antennae, but will not be valid for a different fading
characteristic.
The task of estimating the signal level in a fading
environment requires some special statistical methods for
example the Monte – Carlo method. In [8], a Monte
Carlo based methodology for estimating the signal and
interference powers while accounting for the different
random processes is presented, however, the signal to
interference ratio which is related to BER is not
estimated. In this paper, we perform a Monte-Carlo base
statistical estimation of BER as a function of the signal to
noise ratio. The estimation is based on the channel’s
statistical model in which the slow (lognormal) and fast
(Raleigh) fading including the random movements of the
mobile users are taken into account. By using this method,
there is no need to obtain the PDF of signal to interference
ratio through the complicated analytical expression. This
methodology can therefore, simplify such analysis. In the
next section, we present the expressions for the bit energy
to noise ratio as a function of bit error rate, for different
types of frequently used modulation techniques and the
mathematical model for estimating the signal power.
Section III presents the fading and shadowing models.
The stages involved and the algorithm for a MonteCarlo based estimation of error probability are
presented in section IV. In section V, the estimate of the
error probability by simulation based on the proposed
methodology is presented. A qualitative comparison of
the results and a recommendation concludes the paper in
Section VI.
II.
γb
where
)
1
exp(− γ b )
2
§ 3k
≤ 4Q¨¨
.γ b
© M −1
1
§ γ ·
exp¨ − b ¸
2
© 2 ¹
DBPSK
M-QAM
BFSK,
For coherent
detection
·
¸
¸
¹
In table 1 above,
Q(x ) =
∞
§t2
exp
³ ¨¨© 2
2π x
1
·
¸¸dt ,
¹
x ≥ 0.
(3)
The signals level Prx at the receiver is estimated using the
mathematical model of the propagation medium as
follows:
Prx = Ptx + Gtx + G rx − η tx − η rx − L .
(4)
where Prx is the signal power at the receiver (dBm), Ptx is
the transmitter output power (dBm), Gtx , G rx are the
transmitter and receiver antenna gains respectively (dB),
η tx ,η rx are the transmitter and receiver feeder loss
respectively (dB), L is the propagation loss (dB). In order
to test the results of this work we use the fourth order
propagation law [9, 10]:
­
°L
°°
®L
°L
°
°¯
) at the receiver
according to [1], can be expressed in the form:
γ b = α 2 . Eb N 0 .
(
Q 2γ b
BPSK
CALCULATION OF BIT ENERGY TO NOISE
RATIO
Bit energy to noise ratio (
Expression for BER
Modulation type
2
r =0
r ≤ r0
r > r0
§
·
λ2
¸¸ − 10
= 10 log ¨¨
© 8π H rx H tx ¹
.
= L r = r − 20 log (r r0 )
0
= L r = r − 38 .4 log (r r0 )
(5)
0
where r0 = 4 H rx H tx λ , H tx , H rx are the transmitter and
receiver antenna heights respectively in meters.
(2)
Eb is the mean bit energy and N 0 is the noise
III.
spectral density.
The relationship between the bit error rate and the bit
energy to noise ratio for different types of frequently used
modulation techniques (table I) in digital communication
systems are given in [1].
ACCOUNTING FOR THE EFFECTS OF SLOW
AND FAST FADING
A random process
α dB (t ) describing
the fading in a
mobile radio communication channel can be expressed as
a sum of independent random processes [11]:
α dB (t ) = x dB (t ) + y dB (t ) .
217
(6)
where,
x dB (t ) characterises the lognormal distribution
3.
with probability density function:
1
f xdB =
2π σ xdB
(
)
2
(7)
4.
σ 2y
§ − y dB
. exp¨
¨ 2σ y2
dB
©
dB
·
¸,
¸
¹
(8)
Calculate the value of
(10
p dBm / 10
i
× 10 Ri / 10
6.
Calculate the bit error rate
7.
8.
table 1.
Repeat step 2 – 6 a large number ( N ) times
Calculate the mean error probability as
¦ BER
i
i =1
∞
Compute
(9)
(
γb
.
and
p(γ b ) is the probability distribution of
2.
parameters
p dBm
and
σ p [8]
Compute mean BER
Fig. 1 Major steps for BER estimation
V.
BER ESTIMATION RESULTS AND
DISCUSSION
Using the typical network parameters shown in Table
II, the Monte-Carlo simulation based on the proposed
methodology was performed.
TABLE II
PARAMETERS USED IN THE SIMULATION
p dBm (dBm)
at the receiver according to (4).
Generate the random instantaneous signal level
p dBmi according to the normal distribution with
)
Statistical Processing
expression for BER from (4) may be mathematically
rather complicated or awkward to handle therefore in this
case, we implore the use of simulation based on the
Monte-Carlo method which bypasses the problem of the
analytical awkwardness or complications encountered in
the estimation of BER according to (6). The major steps
in the Monte-Carlo simulation are shown in Fig. 1.
A linear congruent generator (RNG) [12] is used for
the generation of a series of pseudo random numbers. The
calculation of the number of samples was performed in
accordance with the Kolmogorov – Smirnov criteria [13].
The exact proposed procedure for a Monte-Carlo based
estimation of error probability is given below:
Calculate the local mean of the signal
bi
Criteria for required number of samples
The process of obtaining the final analytical
1.
{γ }
BERi = f γ bi , mod ulation , i = 1,2,3,...
where P (γ b ) is the BER without considering the random
γb
(10)
{α i } from corresponding distribution
RNG [0 1]
−∞
effects of
.
N
The error probability (BER) for a particular
modulation type, with the statistics of α accounted for
can be estimated by the following average [1]:
³ P(γ b ) p(γ b )dγ b .
BERi corresponding to
N
MONTE-CARLO BER ESTIMATION
BER =
2
5.
and defines the fast fading of the received signal α (t ) .
IV.
)
α i 2 as
according to (5)
Calculate the instantaneous value of the bit energy to
noise ratio γ bi according to (2) for a given Eb N 0
with probability density function:
y dB
σ R = p dBm 1.253 (expressed in
dB) [11] (characterising the fast fading of the
received signal)
·
¸,
¸
¹
and defines the slowly varying envelop of the received
signal α (t ) , y dB (t ) characterises the Raleigh distribution
f ydB =
Ri according to the Raleigh distribution
with parameter
§− x −x
dB
dB
. exp¨
¨
2σ x2dB
©
2
Generate
Parameter
MS transmit power, dBm
BS(MS) antenna height, m
BS(MS) antenna gain, dB
BS(MS) feeder loss, dB
Standard deviation, dB
(characterising the
slowly varying envelop of the received signal)
218
GSM-900
33
30(1.5)
18(0)
5(0)
10
Fig. 2 shows the results of BER as a function of bit energy
to noise ratio E b N 0 for different modulation types for
has been proposed. The result shows excellent
correlation with those in the literature.
These results can be used in the algorithm for the
statistical estimation of electromagnetic compatibility of a
wireless communication network.
the case of uniform distribution of users in a circle of
radius Rmax. The simulation was ran with N = 5000
samples.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
Fig. 2 BER vs.
Eb N 0
for the case of uniform distribution of the
[7]
mobile stations.
From Fig. 2, as expected, the BER is monotonically
decreasing with increase in the bit energy to noise ratio,
and 16QAM gives the smallest BER than all the other
modulation schemes considered for the same bit energy to
noise ration. Qualitatively, the estimated results obtained
as shown in Fig. 2 is in agreement with that obtained
from experimental data for the multipath urban
channel provided in [14]. If the user’s distribution
changes from being uniform, then this algorithm can be
applied with the new users’ distribution without any
difficulties to obtain the BER, otherwise solving this
problem analytically with the closed form expression
BER can be a very complicated task. A quantitative
comparison of the BER value obtained by the proposed
methodology and the closed form expression can be
carried out provided that the network parameters used in
both cases are the same.
VI.
[8]
[9]
[10]
[11]
[12]
[13]
CONCLUSION
[14]
In this paper, a Monte-Carlo based simulation
technique enabling the estimation of the value of the
error probability for a digital wireless communication
system which takes into consideration, the effects of
slow and fast fading and also the users’ random position
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