Performance Analysis of Bfsk Multi-Hop Communication Systems Over K-μ Fading Channel Using Generalized Gaussian-Finite-Mixture Technique

International Journal of Wireless & Mobile Networks (IJWMN) Vol. 7, No. 6, December 2015
DOI : 10.5121/ijwmn.2015.7604 37
PERFORMANCE ANALYSIS OF BFSK MULTI-HOP
COMMUNICATION SYSTEMS OVER Κ- FADING
CHANNEL USING GENERALIZED GAUSSIAN-
FINITE-MIXTURE TECHNIQUE
Murad A.AAlmekhlafi1
, Shafea M. Al-yousofi1
, Mohammed M. Alkhawlani2
1
Department of Electrical Engineering – Division Electronic Communications.
Sana’a University- Faculty of Engineering, Sana’a, Yemen
2
Department of Electronics Engineering, University of Science and Technology, Sana'a,
Yemen
ABSTRACT
Multi-hop communication systems gained popularity in wireless communications; they can be used to
extend the coverage of the network and reduce the transmitted power. The transmission of data from the
source node to the destination node in multi-hop communications undergoes through intermediate relay
nodes. In this paper, we study the performance of multi-hop communication systems, in terms of average bit
error rate (BER) with Binary frequency shift keying assuming the κ-µ fading channel model. Due to the
difficulty in finding the probability density function (PDF) of the end-to-end signal to noise ratio (SNR) and
hence for the performance metrics, we use Gaussian Mixture (GM) approximation technique to
approximate the PDF of the end to end SNR assuming the κ-µ fading models as weighted sums of Gaussian
distributions. Numerical results are provided for the BER of binary frequency shift keying (BFSK) of
amplify and forward (AF) multi-hop communication systems assuming different values for the fading
parameters (ߢ, ߟ) and for different number of hops. Numerical results are validated by comparing them
with simulation results.
KEYWORDS
Multi-hop;κ-µ ; gaussian mixture ;BER ; BFSK
1. INTRODUCTION
In wireless communication networks, there is a high degradation in the signal quality because of
severe attenuation due to fading. Therefore, several techniques have been used in the literature
such as the well-known space diversity technique, maximum ratio combining (MRC), equal gain
combining (EGC), selection combining (SC) and switching combining (SWC). Cooperative
diversity is also considered as another type of space diversity since it provide relay nodes as a
virtual antennas to assist the transmission between the source node and the destination node.
Moreover, using cooperative communication with diversity is the good solution to combat the
effects of fading and enhance the communication system performance in terms of outage
probability, capacity and bit error rate. Cooperative communications, is a promising technique in
utilization use of the spectrum.This technique has an important advantage, expanding the
coverage of the network without the need for additional power at the source furthermore,

expanding battery life of the communication devices.
of the wireless channel to make nodes help each other
classified into two types, the non
iswillingness to share power between all nodes, and thus, savings of overall network resources
[2]. The performance of communicatio
relaysystem as it allows nodes to send information through relay
wireless communications face a fading problem, in which the signals decay as they travel for long
distances and different paths until they arrive to the receiver
of dual hop presented in [5]. The source node sends information, the destination node receives the
information and the relay node receives the information from source node and
information to the destination node, while promoting and developing the communication between
the source and the destination [6
2. GAUSSIAN MIXTUE TECHNIQUE
GM is a technique used to estimate the
GM Estimation, we assumed that the PDF
(‫;ݔ‬ ߠ݅). The parameters of each PDF will be estimated using
distribution, where (‫ܩ‬ << N). Mathematically, the PDF of a random variable
represented by ݂‫̃ݔ‬ (‫)ݔ‬ [8,9,10].
݂௫෤(‫)ݔ‬
Where ‫݅ݓ‬ is the weight coefficient of the
representedby vectorߠ݅. The term G represents the number of PDFs which must be determined in
advanced.
Fig.1 shows the algorithm that we use in our work.
nternational Journal of Wireless & Mobile Networks (IJWMN) Vol. 7, No. 6, December 2015
ing battery life of the communication devices.The ideas comes from the broadcast nature
of the wireless channel to make nodes help each other [1]. In addition, cooperative systems are
into two types, the non-regenerative and regenerative. The goal behind this technique
. The performance of communication system and transmission capacity are improved by the
relaysystem as it allows nodes to send information through relay [3]. As commonly known,
distances and different paths until they arrive to the receiver [4]. The cooperative communications
. The source node sends information, the destination node receives the
information and the relay node receives the information from source node and
[6,7].
ECHNIQUE
is a technique used to estimate the PDF of random variables from their statistical samples. In
GM Estimation, we assumed that the PDF ݂‫̃ݔ‬ (‫)ݔ‬ can be estimated by a weighted sum of
. The parameters of each PDF will be estimated using N samples taken from the
). Mathematically, the PDF of a random variable ‫̃ݔ‬ be estimated or
≃ ෍ ‫ݓ‬௜
ீ
௜ୀଵ
ߔ௜(‫;ݔ‬ ߠ௜)
is the weight coefficient of the ݅‫ݐ‬ℎ term and the mean and variance parameters
The term G represents the number of PDFs which must be determined in
shows the algorithm that we use in our work.
Fig. 1: GM procedural algorithm.
38
The ideas comes from the broadcast nature
. In addition, cooperative systems are
regenerative and regenerative. The goal behind this technique
n system and transmission capacity are improved by the
. As commonly known,
he cooperative communications
. The source node sends information, the destination node receives the
information and the relay node receives the information from source node and retransmits
of random variables from their statistical samples. In
can be estimated by a weighted sum of ‫-ܩ‬PDFs
samples taken from the
be estimated or
(1)
variance parameters
The term G represents the number of PDFs which must be determined in

39
In this technique, we have several steps:
We guess the initial parameters and weighting coefficients. Once the initial parameters and
weighting coefficients are estimated, they can be updated by GM update equations, which is
derived from the following equation [10].
෍ ߬௜௝
(௞)
ே
௝ୀଵ
߲
߲ߦ௜
݈‫݃݋‬ ൥෍ ‫ݓ‬௜
ீ
௜ୀଵ
ߔ௜(‫ݕ‬௝;ߠ௜)൩ = 0 (2)
whereߦ௜ can be weighting coefficient ‫ݓ‬௜ or, which represent the mean and variance parameters,
߬௜௝
(௞)
is the ݇‫ݐ‬ℎ estimated posteriori probability that the sample point ‫ݕ‬௝ belongs to the ݅‫ݐ‬ℎ
weighted PDF, which determined by the following equation:
߬௜௝
(௞)
=
‫ݓ‬௜
(௞)
ߔ௜൫‫ݕ‬௝;ߠ௜
(௞)
൯
݂௒൫‫ݕ‬௝൯
, ݅ = 1,2, … … … . , ‫ܩ‬ , ݆ = 1,2, … . , ܰ (3)
The updated equations which are used to estimate‫ݓ‬௜ ,ߤ௜ and ߪ௜
ଶ
respectively [10]:
‫ݓ‬௜
(௞ାଵ)
=
∑ ߬௜௝
(௞)ே
௝ୀଵ
ܰ
ߤ௜
(௞ାଵ)
=
∑ ߬௜௝
(௞ାଵ)
‫ݕ‬௝
ே
௝ୀଵ
∑ ߬௜௝
(௞)ே
௝ୀଵ
ߪ௜
ଶ(௞ାଵ)
=
∑ ߬௜௝
(௞)
൫‫ݕ‬௝ − ߤ௜
(௞)
൯
ଶே
௝ୀଵ
∑ ߬௜௝
(௞)ே
௝ୀଵ
(4)
we can expressed these PDFs mathematically as the sum of the weighted Gaussian PDFs using
GM with EM technique as follows [10].
݂௡෤మ(݊ଶ) ≃ ෍
‫ݓ‬௡,௜
√2ߨߪ௡,௜
ே೙
௜ୀଵ
exp ൭−
൫݊ଶ
− ߤ௡,௜൯
ଶ
2ߪ௡,௜
ଶ ൱ (5)
We summaries the GM algorithm in this work as follows:
1. We use the weighting coefficients, mean and variance values to find the sum of the
weighted Gaussian PDFs of κ-µ fading models. In addition, we compare the exact analytical PDF
with GM PDF of κ-µ.
2. Study the BER performance metric of multi-hop κ-µ fading models hence, we compare
the simulation BER with GM BER of κ-
3. SYSTEM MODEL
Multi-hop wireless communication is a very promising technique used to reduce the power of the
transmitter and to increase the coverage by dividing the link between the source node (S) and the
destination node (D) into several hops hence, extending the battery life. In multi-hop technique,
the transmission of data from source to the destination undergoes through intermediate relay
nodes (ܴ௜). Each link between successive nodes undergoes a certain fading model (ܽ௜) that
might be same for all nodes [11]. In this study, it’s important to obtain some performance
measures for the multi-hop system, such as the bit error rate. In order to do this, it’s important to
PDF of the overall SNR at the link between (ܵ → ‫)ܦ‬. However, it is not easy to find the PDF
unless we use certain technique to approximate this PDF, such as the Gaussian Mixture
technique.

Fig.2: wireless communication system with K
We obtain some performance measures
outage probability. First, we considered the
(ܴ → ‫)ܦ‬with instantaneous SNR
generalize this procedure in multi
The PDF of κ-µ distribution is given by [1
݂ఊ(ߛ)
=
ߤ(1 ൅ ߢ)଴.ହሺఓାଵ)
ߛ଴.ହሺఓିଵ)
ߢ଴.ହሺఓିଵ) ݁‫݌ݔ‬ሺߤߢ)ߛ̅଴.ହሺఓାଵ)
݁‫݌ݔ‬
Where ߢ represents the ratio between the total power of the dominant components and the total power
of the scattered waves, ߤ is related to the Multipath clustering, the average signal to noise ratio is
exp ሺ. )is the exponential function.
We compared the exact PDF of
weighted Gaussian PDFs (with tolerance
of κ and µ, The weighted coefficients for estimated (GM) PDF of
the tables below:
These following figures are a part of our work which illustrate the agreement between the analytical
PDF and GM PDF at different values for
of κ or µ is increased with fixed another parameter, the degree of agreement between the analytical PDF
and GM PDF will increase.
Fig. 3: Comparison between exact and GM PDF of the
0 0.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
PDF
κκκκ=1
wireless communication system with K-hop
some performance measures for the multi-hop system, such as the bit error rate and
First, we considered the κ-µ fading model at the link (ܵ → ܴ) and at the link
with instantaneous SNR ߛଵ and ߛଶ respectively in the dual hop. In addition, we
multi-hop with many relays.
µ distribution is given by [12]:
݁‫݌ݔ‬ ൬െ
ߤሺ1 ൅ ߢ)ߛ
ߛ̅
൰ Ιఓିଵ ቌ2ߤඨ
ߢሺ1 ൅ ߢ)ߛ
ߛ̅
ቍ
represents the ratio between the total power of the dominant components and the total power
is related to the Multipath clustering, the average signal to noise ratio is
is the exponential function.
exact PDF of κ-µ fading model and GM PDF for these fading models using ten
(with tolerance ߝ ൌ 10ିଷ
ܽ݊݀ ܽܿܿ‫ ݕܿܽݎݑ‬ሺ|݁‫|ݎ݋ݎݎ‬ ൏ 1%)
The weighted coefficients for estimated (GM) PDF of κ-µ fading model
PDF at different values for κ and µ. From the figures bellow, we see that when the value
s increased with fixed another parameter, the degree of agreement between the analytical PDF
Comparison between exact and GM PDF of the κ-µ at κ=1 and various values of µ
1 1.5 2 2.5 3 3.5 4 4.5 5
X
PDF of κ-µ channel(105
samples)
GM PDF
Exact PDF
µµµµ=4
µµµµ=1( Rice,k=1)
µµµµ=0.7
µµµµ=0.5
40
system, such as the bit error rate and
and at the link
hop. In addition, we
(6)
represents the ratio between the total power of the dominant components and the total power
is related to the Multipath clustering, the average signal to noise ratio is ߛ̅ and
and GM PDF for these fading models using ten
))at different values
g model is summarized in
µ. From the figures bellow, we see that when the value
s increased with fixed another parameter, the degree of agreement between the analytical PDF
and various values of µ

41
Fig.3. illustrates a comparison between exact analytical PDF and estimated PDF using GM
technique of the κ-µ at κ=1 and various values of µ , it can be seen that the two PDFs matched
with each other at different values of µ, we are using 10ହ
sample.For this figure, special cases of
the κ-µ are taken,the weighting coefficients, means and variances of the ten components of GM
techniquefor κ=1, µ=1, are given in Table 1.
Table 1: GM parameters and weighting coefficients using for estimation κ-µ PDF for κ=1, µ=1.
i weight Mean Variance
1 0.0960 1.3193 0.0834
2 0.1049 1.0191 0.0654
3 0.0998 1.2452 0.0929
4 0.0634 1.5531 0.1411
5 0.1325 0.3860 0.0153
6 0.1213 0.6404 0.0211
7 0.0333 0.1681 0.0049
8 0.1252 0.7426 0.0353
9 0.1062 1.0046 0.0616
10 0.1172 0.9340 0.0503
4. AVERAGE BIT ERROR RATE (BER)
BER in wireless communication systems, is considered to be one of the main metrics to evaluate
the performance of several systems. In this paper, we study the BER performance of multi-hop
communication systems overκ-µfading model. Hence, other fading models such as Nakagami-m,
Rice, Rayleigh and one-sided Gaussian can be considered as special cases for the κ-µ fading
model. BER importance comes from its use to describe the functionality of a digital
communication system and modulation types. TheBER given by:
‫ܴܧܤ‬ ൌ න ‫݌‬௘ሺߛ)‫݌‬ఊሺߛ)
ஶ
଴
. ݀ߛ ሺ7)
In BFSK case,
‫݌‬௘ሺߛ) ൌ ܳ൫ඥߛ൯
So the equation of BER became
‫ܴܧܤ‬ ൌ න ܳ൫ඥߛ൯‫݌‬ఊሺߛ)
ஶ
଴
. ݀ߛ ሺ8)
In the dual-hop communication systems, the equivalent SNR ߛ௘௤ is given by:
ߛ௘௤ ൌ
1
2
ߤு ሺ9)
Where ߤு is the harmonic mean that is defined by [13]
ߤுሺܺଵ , ܺଶ) ൌ
2
ଵ
௑భ
൅
ଵ
௑మ
ൌ
2ܺଵܺଶ
ܺଵ ൅ ܺଶ
ሺ9)
Therefore, the equivalent end-to-end SNRߛ௘௤ for dual-hop is given by
ߛ௘௤ ൌ
ߛଵߛଶ
ߛଵ ൅ ߛଶ
ሺ10)
In addition, the equivalent end-to-end SNRߛ௘௤ for multi-hop is given by [14,15]
ߛ௘௤ ൌ ൥ෑ ൬1 ൅
1
ߛ௡
൰ െ 1
ே
௡ୀଵ
൩
ିଵ
ሺ11)

42
Fig 4 : average bit error rate of the dual hop for fixed µ=0.5 and change κ using GM
Technique compared with simulation results
Fig.4 shows a comparison between the simulation results of the average bit error rate
performance employing BFSK modulation scheme and the approximated average bit error rate
using GM technique for dual-hop communication systems over κ-µ fading model with fixed
µ=0.5 and change κ=0.10,0.70, and 1.30. It is clear that the GM matched with simulation results
of average bit error rate. For this figure, the BER decrease by increase the value of κ.
Fig 5 : average bit error rate of the dual hop for fixedκ =0 and changeµ=0.5, 1, and 1.5
Using GM Technique compared with simulation results
0 5 10 15 20 25 30
10
-2
10
-1
10
0
. [dB]
BER
BER of BFSK κ-µ with fixed number of hops 2
Sim
GM
µµµµ=0.5
κκκκ=0.10
κκκκ=0.70
κκκκ=1.30
0 5 10 15 20 25 30
10
-4
10
-3
10
-2
10
-1
10
0
BER
. [dB]
BER of BFSK κ-µ with fixed number of hops 2
Sim
GM
µµµµ=1
µµµµ=1.5
µµµµ=0.5
κκκκ=0

43
Fig. 5 shows a comparison between the simulation results of the average bit error rate
using GM technique for dual-hop communication systems over κ-µ fading model with fixed κ=0
and change µ=0.5,1,1.5. It is clear that the GM matched with simulation results of average bit
error rate. For this figure, the BER decrease by increase the value of µ.
Fig 6 : Bit error rate of the multi-hop for κ=0 and µ =1 (Rayleigh) over κ-µ fading channel
Using GM technique compared with simulation results
Fig.6 shows a comparison between the simulation results of the average bit error rate
using GM technique for multi-hop communication systems over κ-µ fading model with κ=0 and
µ=1 (Rayleigh). It is clear that the GM matched with simulation results of average bit error rate.
For this figure, the BER increase by increase the number of hops.
5.CONCLUSION
In this paper, we studied the Performance Analysis of BFSK multi-hop communications systems
overthe κ-µfading model. In order to do this, we opted for the Gaussian Mixture technique, a
technique used to estimate the PDF of random variables from their statistical samples. In addition,
it provides us with a tool to study the performance metric, such as BER. In the BER case, we use
the BFSK modulation scheme and we observe improvement in BER by increasing parameters κ
and µ. On the other hand increase the number of hops can be the reason for the decline in
performance.This is because we use the non-regenerative (amplify-and-forward) relay type,
which is amplifies both signal and noise received from the first hop. The accumulated noises over
increasing number of hops lead to substantial performance loss.Increase in BER by increasing
number of hops is not a big problem we can solve it by using different diversity or coding
techniques. Moreover, it has been shown that the simulation results were identical to the GM
technique results at all special cases of κ-µ fading model. We found the simulation results almost
identical with GM technique result at all special cases of κ-µ fading model. Furthermore,these
results did not exist or have not been mentioned in literature earlier.
0 5 10 15 20 25 30
10
-3
10
-2
10
-1
10
0
. [dB]
BER
BFSK BER of multi-hops Rayleigh by using κ-µ channel(105
samples)
Sim
GM
Dual hop
4 hops
3 hops
κ=0
µ=1

44
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Performance Analysis of Bfsk Multi-Hop Communication Systems Over K-μ Fading Channel Using Generalized Gaussian-Finite-Mixture Technique

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