FACTA UNIVERSITATIS 

Series: Electronics and Energetics Vol. 34, No 1, March 2021, pp. 89-104 

https://doi.org/10.2298/FUEE2101089P 

© 2021 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Original scientific paper 

GPU-SUPPORTED SIMULATION FOR ABEP AND QOS 

ANALYSIS OF A COMBINED MACRO DIVERSITY SYSTEM 

IN A GAMMA-SHADOWED K-µ FADING CHANNEL 

Nenad Petrović1, Selena Vasić2, Dejan Milić1,  

Suad Suljović1, Samir Koničanin1 

1Faculty of Electronic Engineering, University of Niš, Serbia 
2Faculty of Information Technologies, Metropolitan University, Belgrade, Serbia 

Abstract. In this paper we have analyzed macro-diversity (MD) system with one macro 

SC diversity (MD SC) receiver and two micro MRC (mD MRC) receivers over 

correlated Gamma-shadowed k-µ fading channel. The average bit error probability 

(ABEP) is calculated using the moment generating function (MGF) approach for 

BDPSK and BPSK modulations. Graphical representation of the results illustrates the 

effects of different parameters of the system on its performance as well as the 

improvements due to the benefits of a combined micro and macro diversity. The 

obtained analytical expressions are used for the GPU-enabled mobile network 

modeling, planning and simulation environment to determine the value of Quality of 

Service (QoS) parameter. Finally, linear optimization is proposed as an approach to 

improve the QoS parameter of the fading-affected system observed in this paper. 

Key words: Gamma shadowing, k-µ fading, MGF, ABEP, GPGPU, SDR 

1. INTRODUCTION 

In cellular communications, due to specific nature of transmission channels, signal power 

level at the receiver is affected by macroscopic and microscopic degrading effects [1]. 

Macroscopic effects are usually shadowed fading effects from buildings, foliage, and other 

objects. Microscopic fading is a result of multipath characteristics. In urban areas, there is no 

direct line-of-sight (LoS) along the transmission line. The channel characteristics between 

transmitter and receiver will change instantaneously as the mobile user moves in the 

environment with not-stationary surrounding objects that additionally affect the transmission. 

Therefore, a multipath fading, a shadowed fading, or a mixture of these two can characterize 

 
Received July 22, 2020; received in revised form November 4, 2020 

Corresponding author: Nenad Petrović 
Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, 18000 Niš, Serbia  

E-mail: nenad.petrovic@elfak.ni.ac.rs 



90 N. PETROVIĆ, S. VASIĆ, D. MILIĆ, S. KONIČANIN, S. SULJOVIĆ 

the transmission channel [2]. In overcrowded downtown environments where foliage and 

urban shadowing is found, both multipath and shadowed fading need to be considered while 

analyzing system performances. Long-scale fading is manifested in variations of the signal 

average strength at the receiver. Small-scale fading emerges in the form of rapid 

fluctuations of the amplitude and phase of a signal. In a homogeneous channel short-term 

fading is successfully described by Rayleigh, Nakagami-m, Rice or Hoyt distributions. In a 

real system, the surfaces are spatially correlated, so the channel is non-homogeneous [3]. 

Distributions that model non-homogeneous fading channels are generalized n-distribution 

and generalized q-distribution. They are valid in both cases-when LoS (line-of-sight) 

component is present or absent and are more appropriate in modeling practical fading 

channels. Parameterized distributions such as κ-μ and η-μ distributions are used to model 

this type of non-homogeneous channels which show functional similarities with 

generalized-n and generalized-q distributions, respectively. The η-μ distribution is used to 

model the non-homogeneous small-scale fading in the absence of LoS component [4]. 

Distribution κ-μ is generalized distribution used to model small-scale fading in line-of-sight 

(LoS) non-homogeneous fading channel. For μ=1, κ-μ distribution becomes Rice 

distribution, for κ→0 Nakagami-m distribution, for μ=1, κ→0 Rayleigh distribution, and for 

μ=0.5, κ→0 one-sided Gaussian distribution [5]. Applying diversity techniques eliminates 

negative effects of fading [6].  

Diversity combining improves receiver performance by processing numerically 

statistically independent copies of the same signal carrying information across multiple 

fading channels and by efficiently combining the two or more signals. The idea behind 
this concept is that it is highly unlikely that deep fading will simultaneously occur in all 

the diversity channels, thus reducing error probability. The minimum antenna spacing 

for a mobile unit is at least half a wavelength [7]. Macro-diversity (MD) reception 
reduces both long-term and short-term fading effects on performance of cellular mobile 

radio systems [8]. MD reduces the shadowing effect, and micro-diversity (mD) systems 
decrease rapid fading. MRC and EGC combining techniques, followed by SC and SSC 

techniques reduce the impact of fading and increase channel capacity [9]. MRC and EGC 
combiners are complex to implement in practice; hence, the use of SC combining, 

especially in cases with minimal correlation between channels, has proved as more 

efficient. The SC receiver choses the branch with the highest signal-to-noise ratio (SNR). 

For MRC harvesters, if the noise power is equal at all inputs the square of the output 

signal is equal to the sum of the squares of the input signals [10]. 

In this paper, a combined MD system with MD SC receiver and two mD MRC receivers 

in a Gamma-shadowed k-µ multipath fading channel is analyzed. Using the calculated 

expression for the moment-generating function (MGF) of the signals at the output of the 

mD MRC receivers, MGF of the signal at the output of MD SC receiver is determined. In 

the following, the obtained expression for MGF can be used to evaluate the first-order 

statistics such as outage probability (OP) and bit error probability (BEP) of the observed 

MD system. The average bit error probability (ABEP) for non-coherent binary differential 

phase-shift keying (BDPSK) and non-coherent binary phase-shift keying (BPSK) is studied 

using the MGF-based approach. To the best authors’ knowledge deriving ABEP using 

MGF method for the combined MD system with MD SC receiver and two mD MRC 

receivers in a Gamma large-scale fading and κ-μ small-scale fading in a simulated 



 GPU-Supported Simulation for ABEP of Macro diversity System in Gamma Shadowed κ-μ Fading Channel 91 

environment is not analyzed in open technical literature. Numerical results are obtained using 

software Mathematica and Origin and they illustrate the proposed analytical expressions and 

impact of the parameters of the system on the statistical characteristics observed. Moreover, 

the derived expressions were integrated in GPU-enabled mobile network modeling, planning 

and simulation environment.  

The paper is organized in the following way: In Section II, system ABEP is calculated for 

BFSK and BDPSK modulations schemes of the observed system using MGF approach. In 

Section III, numerical results and the corresponding analysis of the system performance are 

presented. GPU-enabled modelling, simulation and network planning environment for the 

system undergoing conditions described in this paper are given in Section IV. The conclusion 

is presented in Section V. 

2. AVERAGE BIT ERROR PROBABILITY OVER MOMENT GENERATING FUNCTION 

In this section macro diversity system (MD) with MD SC receiver and two mD MRC 

receivers in a channel affected by Gamma long-term and k-µ short-term fading is 

investigated. The block-diagram of a macro-diversity model under consideration is shown 

in Fig. 1. The envelopes of the useful signals at the inputs of the first and the second L-

branch mD are denoted by: r11,…,r1L, where r21,…,r2L are the second mD inputs, and x1 and 

x2, as the outputs of mDs. The envelope of the useful signal at the output of MD SC receiver 

is denoted by r. 

 

 
 

Fig. 1 Block-diagram of the combined MD system 

 

In mobile radio channels with LoS (Line-of-Sight) components the envelopes of the 

useful signals rij, affected by a small-scale fading, follow the κ-μ distribution [11]:  

2
1

( 1)
2

11

K 2

2 1 ( 1)
( ) e 2

e

i
ij

i

ij

K
r

ij i i i
r ij ij

i i

i

r K K K
p r I r

K










+
+

−


−−

  + +
=         

                (1) 



92 N. PETROVIĆ, S. VASIĆ, D. MILIĆ, S. KONIČANIN, S. SULJOVIĆ 

In the above expression I0 (x) denotes modified Bessel function of the first kind and of 

zeroth order, 
2
iji x=  are the powers of the useful signals, xij  0. Parameter μ describes 

the number of multipath clusters. The smaller values of μ cause the increase in fading 

severity. The parameter K describes the ratio of the power of a dominant component and 

the power of the scattered components. 

In the considered environment with a dominant LoS component, after the transformation 

xij=rij
2, xij≥0; i=1,2, j=1,2..L., the PDF of k-µ random variable xi can be written as [12]:  

1 1
( 1)

2 2

1

1 ( 1)
( ) e 2

e

i i
i i

i
i

i ii i

L L
K

x
i i i i i i

x i i L iL K
i i i

L K x K K x
p x I L

L K L

 







+ −
+

−


−

    + +
 =             

    (2) 

Using well-known transformation [13; 17.7.1.1] the modified Bessel function of the 

first kind can be written as:  

 
2

2
0

( ) .
2 ! ( 1)

v k

v v k
k

x
I x

k v k

 +

+
=

=
 + +

   (3) 

where (m) is the Gamma function. Using the expression (3) we can obtain the PDF of 

the SNR at the output of the mD MRC combiner: 

( 1)
2 1

0

( ) 1
( ) e

! ( )

i i i
i i i

i

i

K i Li i L i Lx L K
i i i

x i i
i ii

LK x K
p x

i i L

   



+ + + + −− −


=

 +
 =  

 +  
                      (4) 

Parameter μ can take only integer values as it represents the argument in the Gamma 

function. Moreover, the moment-generating function (MGF) of a random variable xij > 0 

is given by [14]:  

2

00

( ) 11
( ) d e ( )

! ( 1)e

i
ii

i i

i i i i

i Li Li
x s x s i i

x ij x i L K
i i ii

LK K
M s e y p x

i K s









+  +
−

=

 +
= = =  

+ +  
    (5) 

In the observed system, there are i=2 branches of MD SC receiver, and powers of the 

corresponding useful signals are represented by the random variables Ω1 and Ω2.  

In Gamma-shadowed channels, these random variables follow the correlated Gamma 

distribution [15], [16]:  

 
1 2 0

1 2

1( )/( (1 ))
1 2

1 2 2 2 2
00

( )e
( , )

( ) ! ( ) (1 )

i i c

i c i c
i

p
c i i c






 + −−  +  −

  + +
=

 
  = 

  +  −
   (6) 

In the last expression, c is the order of the Gamma distribution, Ω0 denotes the average 

value of the signal powers Ω1 and Ω2 (
0 1 2

 =  =  ) and ρ is the correlation coefficient. MD 

SC receiver selects the signal from the micro diversity MRC receiver that has the largest 

average power of the signal envelope. Hence, the moment generating function of the signal at 

the output of the macro diversity SC receiver will be given by [17], [18]:  



 GPU-Supported Simulation for ABEP of Macro diversity System in Gamma Shadowed κ-μ Fading Channel 93 

1

1 11 2 2 1 2

0 0

( ) d d ( ) ( )x xM s M s p



 =     +   

2

2 12 1 2 1 2

0 0

d d ( ) ( )xM s p



 +     =   

1

1 11 2 2 1 2

0 0

2 d d ( ) ( ).xM s p



 =                                            (7) 

By substituting expressions (5) and (6) in the expression (7), we obtain the MGF at 

the output of the MD SC receiver:  

2 21 2

1 1

1 2

2

2 2 2
1 2 1 00 0

( ) ( 1)2
( )

! ! ( ) (1 )( )e

i i

i i

i L i Li i
i i i

x L K i c i c
i i

LK K
M s

i i i cc

 



 



  + +

+ +
= =

+
= 

 +  −
  

 

1
1 01

2 0 1

2

/ (1 )1
/ (1 ) 11

1 2 2

0 0

e
d d e

( ( 1) ) i

i c
i c

i L
i i iK s







 −  −+ −
−  − + −

+


   

+ + 
   (8) 

The integral I1 in the above expression is equal:   

1

2 01 1/( (1 ))1 1
1 2 2 0 1

0
0

d e ( (1 )) , .
(1 )

i c i c
I i c


 





−  −+ − +  
=   =  − + 

 − 
                (9) 

Here, γ(α, x) is lower incomplete gamma function [19; 6.5.2],  that can be developed by 

using the following form of the series [20]:   

 

0

1
( , ) e .

(1 )

j
a x

jj

x
x x

a
 




−

=

=
+

   (10) 

where (a)n denotes the Pochhammer symbol. Using (10), we can evaluate the integral I1 

as follows:  

 
1 31 0

3

33

/( (1 ))
1

1
1 1 00

e

( 1) ( (1 ))

i i c

i
ii

I
i c i c





 + +−  −

=


=

+ + +  −
  (11) 

After substituting back, the expression (11) into the expression (8) for the MGF at the 

output of the MD SC receiver, we can rewrite the expression for the MGF as:  

1 2

31 2 3
1 2 1 1 10 0 0

( )2
( )

! ! ( )( )( 1)( )e i i

i i
i i

x L K
ii i i

L K
M s

i i i c i c i cc


 
  

= = =

= 
 + + + +

  

 
1 3 1 0

1 3 1 3 2

2 2 1 2 / (1 )
1

12 2 2
0 0

e1
d

(1 ) (1 / ( ( 1)) ) i

i i c

i i c i i c i L
i i is K




 

 + + − −   −

+ + + + +


 
 − + + 

  (12) 



94 N. PETROVIĆ, S. VASIĆ, D. MILIĆ, S. KONIČANIN, S. SULJOVIĆ 

Using the following identity [21; 3.383]:  

1

0

e
d ( ) , 1 ,

(1 )

q px
q

v

x p
x a q q q v

aax

 − −
−  

=   + − 
+  

                                    (13) 

where ψ (s, d, t) denotes the confluent hyper-geometric function, the last part of the 

expression (12) labeled as integral I2, can take the form:  

1 31 3 1 0

2

2 22 2 1 2 / (1 )
1

2 1

0

( 1)e
d

(1 / ( ( 1)) ) i

i i ci i c
i i

i L
i i i

K
I

ss K









 + ++ + − −   −

+

+  
=  =  

+ +   
  

1 3 1 3 1 3 2
0

2 ( 1)
(2 2 ) 2 2 , 2 2 1 ,

(1 )

i i
i

K
i i c i i c i i c i L

s






 +
 + +  + + + + + − − 

 − 
          (14) 

Substituting back the expression for the integral I2 into expression (12), we obtain the 

MGF at the output of the MD SC receiver:  

1 2 31 2

1 3

1 2 3

2 2

2 2
1 2 1 00 0 0

( )2
( )

( )e ! ! ( )i i

i i i ci i
i i

x L K i i c
i i i

LK
M s

c i i i c


 
   + + +

+ +
= = =

= 
  + 

  

  
1 3

1 3

3

2 2

1 3

2
1 1

(2 2 ) 1

(1 ) ( )( 1)

i i c

i

i i c
i

i i c K

si c i c

+ +

+ +

 + + + 
  

− + + +  
                             

1 3 1 3 2
0

2 ( 1)
2 2 , 2 2 1 ,

(1 )

i i
i

K
i i c i i c i L

s






 +
 + + + + + − − 

 − 
                      (15) 

One of the performance measures of the wireless communication system is the 

average symbol error probability (ASEP). If there are two bits per symbol, this property 

is equivalent to the average bit error probability (ABEP) [8]. 

In the following, using the obtained expression for MGF, the ABEP of non-coherent 

BFSK and BDPSK modulation signals can be directly calculated as [6]:  

 
BDPSK.for   (1),0.5M=)(P

  BFSK;for   (0.5),0.5M=)(P

x0be

x0be




  (16) 

Graphical representation of the ABEP for the range of different values of parameters 

from expression (16) for BFSK and BDPSK modulation is given in Figures 2, 3, 4 and 5. 

3. NUMERICAL AND GRAPHICAL RESULTS 

Figures 2 and 3 show the ABEP of the MD SC receiver output in terms of the parameter 

Ω0 (the average value of the signal powers Ω1 and Ω2) for several different values of fading-

severity parameters, correlation coefficient ρ and different number of branches (L) on the 

input of mD MRC receivers for BFSK modulation.   



 GPU-Supported Simulation for ABEP of Macro diversity System in Gamma Shadowed κ-μ Fading Channel 95 

0 5 10 15 20

1E-3

0,01

0,1

1

=1, =0.2, L=2

A
B

E
P

[





[dB]

 c=1, k=1

 c=1.5, k=1

 c=2, k=1

 c=2.5, k=1

 c=3, k=1

 c=1, k=1.5

 c=1, k=2

 c=1, k=2.5

 c=1, k=3

 
Fig. 2 ABEP in terms of Ω0 for BFSK modulation and different values of c and K  

0 5 10 15 20

1E-3

0,01

0,1

1

c=1, k=1

A
B

E
P

[





[dB]

 =1, =0.2, L=2

 =1.5, =0.2, L=2 

 =2, =0.2, L=2 

 =2.5, =0.2, L=2 

 =3, =0.2, L=2

 =1, =0.4, L=2

 =1, =0.6, L=2

 =1, =0.8, L=2 

 =1, =0.2, L=3

 =1, =0.2, L=4

 
Fig. 3 ABEP in terms of Ω0 for BFSK modulation and different values of µ, ρ and L  



96 N. PETROVIĆ, S. VASIĆ, D. MILIĆ, S. KONIČANIN, S. SULJOVIĆ 

0 5 10 15 20

1E-3

0,01

0,1

1

=1, =0.2, L=2

A
B

E
P

[





[dB]

 c=1, k=1

 c=1.5, k=1

 c=2, k=1

 c=25, k=1

 c=3, k=1

 c=1, k=1.5

 c=1, k=2 

 c=1, k=2.5 

 c=1, k=3

 
Fig. 4 ABEP in terms of Ω0 for BDPSK modulation and different values of c and K  

0 5 10 15 20

1E-3

0,01

0,1

1

c=1, k=1

A
B

E
P

[





[dB]

 =1, =0.2, L=2 

 =1.5, =0.2, L=2

 =2, =0.2, L=2 

 =2.5, =0.2, L=2

 =3, =0.2, L=2 

 =1, =0.4, L=2 

 =1, =0.6, L=2 

 =1, =0.8, L=2 

 =1, =0.2, L=3

 =1, =0.2, L=4

  

Fig. 5 ABEP in terms of Ω0 for BDPSK modulation and different values of µ, ρ and L  

 
The figures 2, 3, 4, 5 for both BFSK and BDPSK modulations show that, as 

parameters µ, c, k and L are increasing ABEP function is decreasing and the system has 
better performance and stability. The increase in correlation coefficient ρ influences the 
increase in ABEP and the system becomes unstable. It can be observed that the higher 
values of Ω0 lead to a lower error and better performance.  

ABEP is significantly reduced with increasing L-input in micro MRC, and the system 
performance will improve. On the other hand, larger ABEP results in a degradation of system 
performance. Comparing the graphics results we can conclude that the better stability of the 
system and a smaller ABEP are achieved for the BDPSK than BFSK modulation.  



 GPU-Supported Simulation for ABEP of Macro diversity System in Gamma Shadowed κ-μ Fading Channel 97 

To further analyze the ABEP performances, error related to truncation of infinite 

series will be considered.   

Tables 1 and 2 illustrate the number of terms in the series that need to be added in 

order to assure the accuracy of the expression (16) to 5 decimal places, for the given 

values of BFSK parameters, as given in Figures 2 and 3.  

Table 1 Number of terms needed in the infinite series expansion in the expression (16) 

for the ABEP for the BFSK with accuracy to 5 decimal places: µ=1, L=2, ρ=0.2, 

while changing parameters c and K (Fig.2). 

 Ω0=0 dB Ω0=5 dB Ω0=10 dB 

c=1, K=1 11 10 7 

c=1.5, K=1 12 9 7 

c=2, K=1 13 11 7 

c=2.5, K=1 13 10 7 

c=3, K=1 14 11 8 

c=1, K=1.5 13 11 9 

c=1, K=2 13 12 11 

c=1, K= 2.5 15 14 12 

c =1, K=3 18 15 13 

For three different values of Ω0 [dB] and different values of parameters c and K, more 

terms are needed in the expansion and convergence will occur at a slower pace. 

Table 2 Number of terms needed in the infinite series expansion in the expression (16) 

for the ABEP for the BFSK with accuracy to 5 decimal places, where c=K=1 

and parameters µ, ρ and L are varied (Fig.3). 

 Ω0=0 dB Ω0=5 dB Ω0=10 dB 

µ=1, ρ=0.2, L=2  11 10 7 

µ=1.5, ρ=0.2, L=2  12 11 10 

µ=2, ρ=0.2, L=2  14 13 10 

µ=2.5, ρ=0.2, L=2  15 14 13 

µ=3, ρ=0.2, L=2  18 15 13 

µ=1, ρ=0.4, L=2  13 10 8 

µ=1, ρ=0.6, L=2  15 13 10 

µ=1, ρ=0.8, L=2  11 20 16 

µ=1, ρ=0.2, L=3   11 10 8 

µ=1, ρ=0.2, L=4   13 11 10 

As the values of the parameters µ, ρ and L increase, the number of terms in the 

expression needed for the accuracy to 5 decimal places increases too, and convergence is 

obtained at a slower pace. 

In Tables 3 and 4 the results for the similar analysis are presented for the BDPSK 

modulation scheme. The number of terms in the series expansion in the expression (16) 

needed to achieve the accuracy to 5 decimal places for the given parameter values, are 

presented all together with the results given in Figures 4 and 5. 



98 N. PETROVIĆ, S. VASIĆ, D. MILIĆ, S. KONIČANIN, S. SULJOVIĆ 

Table 3 Number of terms in the infinite series expansion of the expression (16) for the 

ABEP for BDPSK at the accuracy of 5 decimal places, where µ=1, L=2, ρ=0.2, 

and various values of parameters c and K (Fig.4). 

 Ω0=0 dB Ω0=5 dB Ω0=10 dB 

c=1, K=1 11 8 7 

c=1.5, K=1 11 8 5 

c=2, K=1 12 9 5 

c=2.5, K=1 12 9 5 

c=3, K=1 12 9 5 

c=1, K=1.5 12 9 8 

c=1, K=2 13 12 10 

c=1, K=2.5 14 13 12 

c=1, K=3 17 15 13 

Again, for three different values of the average power Ω0 [dB] as the parameters c and K 

increase, the number of terms needed for the series convergence increases, so convergence is 

slower.  

 

Table 4 Number of terms in the infinite series expansion of the expression (16) for the 

ABEP for BDPSK at the accuracy of 5 decimal places, where c=K=1 and 

parameters µ, ρ and L take various values (Fig.3). 
 

 Ω0=0 dB Ω0=5 dB Ω0=10 dB 

µ=1, ρ=0.2, L=2 11 8 7 

µ=1.5, ρ=0.2, L=2 12 9 9 

µ=2, ρ=0.2, L=2 13 11 10 

µ=2.5, ρ=0.2, L=2 14 13 12 

µ=3, ρ=0.2, L=2 17 15 12 

µ=1, ρ=0.4, L=2 12 9 7 

µ=1, ρ=0.6, L=2 13 11 7 

µ=1, ρ=0.8, L=2 23 17 12 

µ=1, ρ=0.2, L=3 10 9 7 

µ=1, ρ=0.2, L=4 11 10 9 

Table 4 shows that for higher values of the parameters µ, ρ and L, more terms should 

be added in the expression to achieve the accuracy up to and including the fifth decimal, 

so convergence becomes slower. Also, the higher values of the correlation coefficient 

require more terms. 

4. GPU-ENABLED MODELLING, SIMULATION AND NETWORK PLANNING ENVIRONMENT 

4.1. Framework overview 

Although many of the network simulators conceptualize mobile propagation models up 

to some point, often they involve time-consuming and inefficient computational schemes 

resulting in difference between simulated and realistic scenarios. To assure better simulation 

results and optimal computation algorithms, in this section, we present the framework for 

simulated transmission in a gamma-shadowed kappa-mu fading environment analyzed in 



 GPU-Supported Simulation for ABEP of Macro diversity System in Gamma Shadowed κ-μ Fading Channel 99 

the previous sections. The idea behind the efficiency of the approach is the implementation 

of the GPGPU computational units [22], [23]. The software framework has several phases. 

In the first phase, user defines mobile network within the graphical environment running in 

a web browser. Various modelling aspects related to infrastructure, terrain, channel, and 

service consumers within smart cities, have been considered earlier [24], [25]. The Quality 

of Service (QoS) parameter values are calculated with the reference to the ABEP values. 

These values and the user-defined model are taken as the input of a linear optimization 

mechanism. The process of linear optimization is executed, giving the optimal base station 

configuration and deployment as output, which is further translated to target Software 

Defined Radio (SDR) commands. Fig. 6 depicts the modeling and simulation environment 

used for network planning. 

 

Fig. 6 Overview of GPU-enabled mobile network modeling, planning and simulation 

environment: 1-Creation of user-defined smart city mobile network model diagram 

2-Mobile network model 3-Calculated QoS values 4-SDR commands 

4.2. GPU-enabled fading calculation 

Traditionally, graphics processing units (GPU) perform computations only for computer 

graphics. The goal is to implement general-purpose GPU (GPGPU) programming to 

accelerate the calculations in applications that are not graphic. In [22] and [23], GPGPU 

has been considered as an effective approach towards performance improvement when it 

comes to simulations of fading effect. A pseudo-code depicting the structure of CUDA 

kernel is given in Fig. 7.  
 

Fig. 7 GPU-powered CUDA kernel pseudo-code for QoS impact calculation 

In this paper, we adopt NVIDIA CUDA for GPU-enabled calculation of ABEP that is 

further used for QoS estimation of the observed system. The pseudo-code shows how 

calculations are distributed among different GPU computation/processing subunits to 

__global__ void calc_impact(Location* l, float* qi) 

{

    int k = threadIdx.x + blockIdx.x * blockDim.x;

    while (k < N) 

    {

        qi[k] = MGF(l->o[k])/qnf;

  k += blockDim.x * gridDim.x;   

    }

}



100 N. PETROVIĆ, S. VASIĆ, D. MILIĆ, S. KONIČANIN, S. SULJOVIĆ 

achieve the speed-up. The input of this algorithm is a set of locations (denoted as *l) 

within the map, while the output is a set of QoS impact coefficients determined for each 

location (denoted as *qi). To each location, Ω0 value is assigned (denoted as o) and used 

as input of MGF function (denoted as MGF) to calculate the value of ABEP. 

Furthermore, qnf is used for normalization of ABEP to a real number from range [0, 1] in 

order to express the weight of fading effect expressed by ABEP value among different 

parameters that might affect quality of service, as discussed in [25]. This value can be set 

by user and is determined empirically. Moreover, the obtained result is further used for 

calculation of the next QoS value at the time point t+1 at location L:  

(t 1, L) (t) ( 1,  )QoS QoS qi t L+ =  +                                  (17) 

To assure the computational pace, Intel i7 7700-HQ quad-core CPU machine at 2.8 

GHz and GTX 1050 GPU with 2 GB VRAM, 1TB HDD and 16GB DDR4 RAM is used. 

According to the results, the GPU-enabled calculation of QoS based on ABEP for this 

type of fading is around 42 times faster than calculation done on CPU using the tool from 

[24]. Moreover, the performance improvement using the GPGPU approach for level-

crossing rate (LCR) of the SC receiver output over α-κ-µ multipath fading channels [25] 

was 39 times faster while performance evaluation of MD wireless communication system 

within Weibull multipath fading channel [26] was 47 times faster. 

4.3. Linear optimization and QoS  

One of the main objectives in designing and implementing wireless networks is to 

provide the best possible quality of service (QoS). In this section, we overview the linear 

optimization approach to improve the QoS in a real-life scenario of the transmission over 

Gamma long-term and k-µ short-term fading channel employing a combined MD system 

with MD SC receiver and two mD MRC receivers. In every fading-affected environment, 

QoS requirements can be recognized in terms of data rates, average bit error rates, delay 

limits, outage probabilities and so forth. The linear optimization method presented in the 

following is aimed to assure the best possible data rates and lowest error rates, as well as 

minimum possible outage probability of the system proposed in the previous sections. 

The analysis is based on a mathematical model with the requirements expressed by linear 

relationships [27].  

We define the base station BaseStationb, energy consumption ecb and an estimated 

capacity, that is the highest number of connected users cb. Also, we define the location 

Locationl for each base station and encounter the cost of energy distribution dcl as well as 

an estimated demand dl of service by the customers for the given location covered by the 

allocated base station. We will contemplate a level of QoS drop qd[b,l] for each pair 

(BaseStationb, Locationl). This parameter depends not only on the specific characteristics 

of the channel (in our scenario these would be specific fading conditions) but also the 

design and properties of the base station itself. The parameter is represented by the ratio 

between the values of maximum possible and estimated QoS for the observed part of the 

channel: 

0 max

0

[ , ]
[ , ]

[ , ]
estimate

Q S b l
qd b l

Q S b l
=                                                (18) 



 GPU-Supported Simulation for ABEP of Macro diversity System in Gamma Shadowed κ-μ Fading Channel101 

where x[b,l] represents the decision variable. It can take only two possible values: 1 if 

BaseStationb is at the Locationl, while it takes 0 in the other case. Linear optimization 

allows for allocating the base stations to optimal locations. This will have positive effects 

on the QoS parameters such as the minimization of QoS drop, together with reduction of 

costs related to energy distribution and consumption:  

, 

[ , ] [ , ] [ ] [ , ]
b BaseStation l Location

minimize qd b l dc b l ec b x b l
 

                     (19) 

There are several conditions that must be fulfilled to successfully complete the 

optimization. Primarily, a single base station has to be assigned to each location:   

[ , ] 1,  
b BaseStation

x b l l Location


=                                               (20) 

Moreover, each base station is assigned to at most one location at the moment:  

[ , ] 1,  
l Location

x b l b BaseStation


                                        (21) 

Each base station assigned to a specific location has to assure the needed capacity to 

meet the service demand for the corresponding location:  

[ ] [ , ] [ ],
b BaseStation

c b x b l d l l Location


                                 (22) 

To implement the linear optimization program we have used the AMPL1 optimization 

supporting system while the optimization process itself has been completed using IBM 

CPLEX2 optimizer and its option of simplex method-based solver.    

In Table 5, an overview of the results obtained in optimization for different model 

sizes is given. The first model represents the number of different configurations of the 

involved base stations (nBS). The second column is the number of considered locations 

(nL). These two values determine the size of the model. The third column represents the 

time needed for optimization processes for the given model size.  

In the last column, a percentage of the cost reduction obtained in each scenario is 

given. According to the results in the Table 5, we can observe that the cost reduction 

depends on the specific instance model while the time spent on optimization increases 

with the increase of the model size. 

Table 5  Results of the optimization for different sizes of the network model 

Number  

of base stations  

[nBS] 

Number  

of locations  

[nL] 

Optimization  

 

[s] 

Cost  

reduction  

[%] 

5 4 0.05 77 

10 6 0.08 54 

15 8 0.14 83 

 
1 https://ampl.com/  
2 https://www.ibm.com/analytics/cplex-optimizer   

https://ampl.com/
https://www.ibm.com/analytics/cplex-optimizer


102 N. PETROVIĆ, S. VASIĆ, D. MILIĆ, S. KONIČANIN, S. SULJOVIĆ 

5. CONCLUSION 

We have considered the combined MD system with one macro-diversity SC receiver 

and two micro-diversity MRC receivers in a Gamma long-term and κ-μ short-term fading 

channel has been studied. Micro-diversity combines signal envelopes from multiple L 

antennas at the base stations. In this way multiple effects of κ-μ fading are reduced. On 

the other hand, MD combines signals with antennas at two or more base stations which 

helps mitigating the effects of a long-term fading. We have derived the closed-form 

expressions for the moment-generating function of the signal at the output of the system 

for correlated composite non-homogenous fading channel for two modulation schemes: 

BFSK and BDPSK.    

Using the obtained expressions, analytical expressions for ABEP for both modulation 

schemes are evaluated. The ABEP is improved with an increase of the number of 

antennas L. The increase in correlation coefficient ρ weakens the system performance. 

The correlation coefficient ρ has higher influence on ABEP for higher values of Gamma 

long-term fading severity parameters. When ρ is one, the lowest value of the signal 

occurs simultaneously resulting in MD-reception becoming mD-reception.  

The ABEP function is smaller and the system performance is better when the number of 

branches in mD is greater than L=2. In order to estimate the rate at which the convergence 

of the expressions for ABEP developed in the infinite series occurs, the analysis has been 

conducted where the needed number of terms of the series is determined for the rounding 

accuracy of 5 decimal places. It is observed that the series converges at a high rate, and in 

general, 10-15 terms need to be taken to achieve the expected accuracy. The increase of the 

values of parameters µ, ρ and L affects the convergence rate as more terms should be 

encountered to assure the stated accuracy.  

In the final section of the work, we have proposed the implementation of GPU-

powered calculations that significantly speed up the determination of QoS parameters 

based on ABEP function. This is of the utmost importance for the QoS performances in 

real-time wireless systems dealing with data transmission such as real-time video. The 

usage of GPGPU in a simulated model of a Gamma long-term and κ-μ short-term fading 

environment dramatically improves the response. Within the analysis for the improvement in 

QoS parameter of the fading-affected system observed in this paper the linear optimization is 

proposed.   

The idea behind the linear optimization model is built on the optimal base station 

configuration and deployment scheme that can be further translated to target Software 

Defined Radio (SDR) commands. The implementation of command generation mechanisms 

for specific SDR hardware solutions is outside the scope of this paper and will be covered 

by our future research. Moreover, we would like to include the channel capacity calculation 

techniques [28] to dynamically determine the capacity parameter in linear optimization 

model.  

Acknowledgement: This paper has been supported by the Ministry of Education, Science and 

Technological Development of the Republic of Serbia.  



 GPU-Supported Simulation for ABEP of Macro diversity System in Gamma Shadowed κ-μ Fading Channel103 

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