Prediction of the Optimal Threshold Value in DF
Relay Selection Schemes Based on Artificial Neural
Networks
1

Ferdi KARA, 1Hakan KAYA, 2Okan ERKAYMAZ, 1Ertan ÖZTÜRK
Department of Electrical Electronics Engineering, 2Department of Computer Engineering
Bulent Ecevit University,
Zonguldak, Turkey
{f.kara@beun.edu.tr, hakan.kaya@beun.edu.tr, okan.erkaymaz@beun.edu.tr, eozturk@beun.edu.tr}
1

Abstract—In wireless communications, the cooperative
communication (CC) technology promises performance gains
compared to traditional Single-Input Single Output (SISO)
techniques. Therefore, the CC technique is one of the nominees
for 5G networks. In the Decode-and-Forward (DF) relaying
scheme which is one of the CC techniques, determination of the
threshold value at the relay has a key role for the system
performance and power usage. In this paper, we propose
prediction of the optimal threshold values for the best relay
selection scheme in cooperative communications, based on
Artificial Neural Networks (ANNs) for the first time in literature.
The average link qualities and number of relays have been used
as inputs in the prediction of optimal threshold values using
Artificial Neural Networks (ANNs): Multi-Layer Perceptron
(MLP) and Radial Basis Function (RBF) networks. The MLP
network has better performance from the RBF network on the
prediction of optimal threshold value when the same number of
neurons is used at the hidden layer for both networks. Besides,
the optimal threshold values obtained using ANNs are verified by
the optimal threshold values obtained numerically using the
closed form expression derived for the system. The results show
that the optimal threshold values obtained by ANNs on the best
relay selection scheme provide a minimum Bit-Error-Rate (BER)
because of the reduction of the probability that error
propagation may occur. Also, for the same BER performance
goal, prediction of optimal threshold values provides 2dB less
power usage, which is great gain in terms of green
communication.
Keywords—relay selection, cooperative, ANNs, MLP, RBF,
optimal threshold

I.

INTRODUCTION

The performances of wireless communication techniques
are mostly evaluated according to bit-error rate (BER) and
outage probability. These two criteria’s depend on the wireless
channel properties such as interference, fading, shadowing and
path-loss etc[1]. Researchers have been studying on several
techniques in order to increase the performance. One of these
techniques for improving the performance is diversity in which
the copies of data are transmitted over independent dimensions
such as time, frequency and space (antennas). In spatial
diversity, the copies of data are transmitted/received by
multiple antennas. Therefore, the spatial diversity is mostly
called as Multiple Input Multiple Output (MIMO) system [2],
This work has been supported by The Scientific and Technological
Research Council of Turkey (TUBITAK) with the name 2211A PhD
Scholarship Program.

which is key technology in 4G standards i,e LTE and LTEAdvanced [3]. However, in mobile communications multiple
antennas cannot be easily adapted due to physical limitations.
To overcome these limitations the relaying systems are
proposed in literature. In the relaying systems, the other users called relays- in the environment forward a processed version
of the received data from source to the destination. The
relaying systems are seen the nominees for the 5G and beyond
technologies. This relaying system is called cooperative
communication (CC) and the obtain diversity is called
cooperative diversity or virtual spatial diversity [4]. In
literature, different relaying protocols such as amplify-andforward (AF) and decode-and-forward (DF) are used according
to the employed processing at the relay. In AF protocol, the
received data is amplified by a relay (to equalize the effect of
the channel fade between the source and the relay) then the
relay retransmits the amplified version of data to a destination
[5]. On the other hand, in DF protocol, a relay first decodes the
received data, then re-encodes and forwards to a destination. In
the DF protocol, the relay decides whether it transmits or not
according to the source-relay link quality. If the source-relay
link quality is less than a threshold value on the relay, relay
remains silent. Otherwise the relay transmits [6]. However, it is
still possible that the relay may decode incorrectly even if a
threshold value is greater than the source-relay link quality and
decode the data from the source and then forward this incorrect
data to the destination erroneously. This phenomenon is called
error propagation problem. The probability of this incorrect
detection at the relay depends on the threshold value [7].
In a CC system, the source could transmit the signal to the
destination not only through the one relay but also through
multi relays (all available relays) to increase diversity order
gain .Instead of using all available relays, only one relay, which
is the best one among the multi relays to obtain best
performance, can be selected. The selected relay has the
highest link qualities in terms of the transmission path and
called best relay. Hence this scheme is called relay selection
[8].The use of only the best relay also provides the full
diversity order with an efficient use of the bandwidth.
In the DF protocol and relay selection scheme, the
threshold value has great effect on the performance of
cooperative communication. To minimize BER, an optimal
threshold values has to be obtained. This optimal threshold
value depends on the number of relays and the link qualities

between source-relay, relay-destination and source-destination
[7]. The computation of optimal threshold value changing relay
numbers and link qualities by analytically is hardly possible.
In this paper, for the prediction of optimal threshold values
by using MLP and RBF networks, the number of relays and the
average link qualities of source-relay-destination paths are used
as inputs after normalization.
The optimal threshold values on the best relay selection
scheme are determined for different scenarios by numerically
minimizing the closed form BER equation similar to given in
[6]. Different types of scenarios having different relay numbers
and different link qualities have been used to test the proposed
techniques. The outputs of ANNs are perfect match with the
numerical results.
This paper organized into six sections. In section II, we
have briefly explained the best relay selection in DF network
models. Also the numerical results for optimal threshold values
are given in this section. Section 3, has an overview of
literature in determination of threshold values on DF networks.
Section 4 deals with the description of ANNs as proposed
techniques. Application of ANNs for the prediction of optimal
threshold values in DF networks and results are given in
Section 5. In section 6, some results are discussed, and the
paper is concluded.
II. SYSTEM MODEL
In the relay selection scheme, one relay is selected among
multi-relays in order to utilize full cooperative diversity [8].
The relay selection is performed by considering source-relaydestination cascaded link qualities of all available relays. Relay
selection is done in two steps: First, the decoding set (C) of
relays which are called reliable relays, is determined by
considering source-relay link qualities of all available relays. If
the received SNR from source at the relay is less than a
threshold value, the relay cannot decode the source message
correctly and it does not transmit. If the received SNR at the
relay is greater than the threshold, this relay is added into the
decoding set. It is also assumed that the relays belong to the
decoding set may detect the signals from source erroneously
and forward incorrect data to the destination, causing error
propagation that can reduce the performance of the system [9].
Second, the relay selection is completed at the destination by

taking the relay having the highest relay-destination link
quality from the decoding set. After determining the best relay,
the destination informs all relays about which relay is selected
to transmit as the best relay through a reverse broadcast
channel then other unselected relays turn to be idle [10]. The
destination combines the data received from source and best
relay by using Maximum Ratio Combining (MRC). The best
relay selection scheme is given Fig 1.
For the best relay selection scheme, the BER of system,
which is given in the (1), is obtained by using the equations
given in [5] and [11] under the condition that the destination
uses MRC and the modulation is Binary Phase Shift Keying
(BPSK). The BER of systems depends on the threshold value
(𝛾𝑡ℎ ), the average link SNRs (𝛾𝑆𝑅 , 𝛾𝑅𝐷 , 𝛾𝑅𝐷 ) and the number of
relays (M).
The definition of erfc(.) function used in (1) is 𝑒𝑟𝑓𝑐(𝑥) =
𝜋⁄2
𝑥2
∫ 𝑒𝑥𝑝⁡(− 𝑠𝑖𝑛2 (𝜃))𝑑𝜃 .
𝜋 0
2

𝑜𝑝𝑡

The optimal threshold (𝛾𝑡ℎ ) is the value which minimizing
the BER of the system.
𝑜𝑝𝑡

𝛾𝑡ℎ = 𝑎𝑟𝑔, 𝑚𝑖𝑛{𝛾𝑡ℎ , 𝐵𝐸𝑅}

(2)

In our scenarios, it is assumed that the average source-relay,
relay-destination and source-destination link SNRs are
𝜀
2 ). 𝜀𝑏
2
𝛾̅𝑆𝑅 = 𝑬(ℎ𝑆𝑅
𝛾̅𝑅𝐷 = 𝑬(ℎ𝑅𝐷
). 𝑏⁄𝑁 ,
𝛾̅𝑆𝐷 =
⁄𝑁 ,
0
0
𝜀𝑏
2
𝑬(ℎ𝑆𝐷 ). ⁄𝑁 respectively. E(.) is the statistical average (or
0
expectation) operator. hSD, hSR and hRD are the channel
coefficients and they are modeled as Rayleigh flat fading
2
2
2
channels with variance of 𝜎𝑆𝐷
, 𝜎𝑆𝑅
and 𝜎𝑅𝐷
, respectively. In
𝑜𝑝𝑡
Fig. 2, the optimal threshold value (𝛾𝑡ℎ ) numerically obtained
𝜀
minimizing BER as a function of 𝑏⁄𝑁 is showed for different
0
number of relays (M) on the symmetric network which has
2
2
2
𝜎𝑆𝐷
= 𝜎𝑆𝑅
= 𝜎𝑅𝐷
= 1.
𝑜𝑝𝑡

In Fig. 3, the optimal threshold value (𝛾𝑡ℎ ) is showed on
2
2
2
another network in which 𝜎𝑆𝑅
= 𝜎𝑅𝐷
= 10 and 𝜎𝑆𝐷
= 1. It is
mostly possible scenario if the relay stands between source and
destination.

𝑀

𝑀−𝑖

𝑖

𝛾
𝛾𝑡ℎ
𝛾
𝐵𝐸𝑅 = (1 − exp (− 𝑡ℎ⁄ )) × 0.5⁡𝑒𝑟𝑓𝑐 (√𝛾𝑆𝐷 ⁡) + ∑𝑀
⁄𝛾 ) (1 − exp (− 𝑡ℎ⁄𝛾 ))
𝑖=1 ⌈exp (−
𝛾𝑆𝑅
𝑆𝑅
𝑆𝑅
{0.5 (𝑒𝑟𝑓𝑐(√𝛾𝑡ℎ ⁡) − exp (
𝑒𝑟𝑓𝑐 (√𝛾𝑡ℎ (1 +

1
𝛾𝑆𝑅

[(

𝛾𝑅𝐷

𝛾𝑅𝐷 +𝛾𝑆𝐷

)×

𝛾𝑆𝑅
1
𝛾𝑆𝑅
𝛾𝑡ℎ
𝛾
⁄𝛾 ) √1+𝛾 𝑒𝑟𝑓𝑐 (√𝛾𝑡ℎ (1 + 𝛾 )⁡))} + {1 − 0.5 (𝑒𝑟𝑓𝑐(√𝛾𝑡ℎ ⁡) − exp ( 𝑡ℎ⁄𝛾 ) √1+𝛾 ×
𝑆𝑅
𝑆𝑅
𝑆𝑅
𝑆𝑅
𝑆𝑅

)⁡))} ⁡ × 0.5 {⁡𝑀 ∑𝑀−1
𝑘=0

(−1)𝑘 𝑀−1
( 𝑘 )
𝑘+1

× (1 −

𝛾𝑅𝐷 ⁄𝑘+1
𝛾𝑅𝐷 ⁄𝑘+1−𝛾𝑆𝐷

𝛾

√1+𝛾𝑅𝐷

⁄𝑘+1

𝑅𝐷 ⁄𝑘+1

+

𝛾𝑆𝐷
𝛾𝑅𝐷 ⁄𝑘+1−𝛾𝑆𝐷

𝛾

√1+𝛾𝑆𝐷 )}]⌉ (1)
𝑆𝐷

Relay 1

Yes

Reliable

No
İdle

.
.
.

Direct Link

Source

Relay
Selection

Destination

Relay M
Yes

Reliable

No
İdle

Figure 1. Best Relay Selection Scheme
Figure 1
Figure 2

𝜀𝑏
⁄𝑁 on the symmetric
0
= 1 for M=4

Figure 2. The optimal threshold values versus
network

2
𝜎𝑆𝐷

=

2
𝜎𝑆𝑅

=

2
𝜎𝑅𝐷

III. RELATED WORKS
The optimal threshold value in DF schemes could have
been obtained by analytically just for a single relay i,e M=1
over Rayleigh fading channels [7] and over Nakagami-m
fading channels [12] by using optimum decision rule similar to
given (2). Similarly in [13] , the authors proposed a method for
the optimal threshold value and optimal power allocation when
single relay is used. In [7] , the authors have obtained the
optimal values numerically by minimizing BER values as
described Section II for the multiple relay scenarios. However,
this method could not be used at the relay because of the
computational burden. In [14], the authors used the MATLAB

𝜀𝑏
2
=
⁄𝑁 on the network 𝜎𝑆𝑅
0
= 1 for M=4

Figure 3. The optimal threshold values versus
2
𝜎𝑅𝐷

= 10 and

2
𝜎𝑆𝐷

Optimization Toolbox for the determination of the threshold
values.
IV. OVERVIEW OF ARTIFICIAL NEURAL NETWORKS
The Artificial Neural Networks (ANNs) have been studied
by researchers since 1970s. Artificial Neural Networks
(ANNs) have become commonly used tool to generate proper
outputs for given inputs in the absence of a formula between
inputs and outputs. An ANN model is formed by input(s) with
weight(s), activation function(s), biases an output(s) [15].The
weights and biases are changed by the time an output is
generated with a tolerated error for given inputs. Thanks to

their computational speed, ability to handle complex functions
and great efficiency even if full information is absent for the
problem, ANNs have become one of the most preferred
methods to solve non-linear engineering problems. ANNs are
mainly used for classification, function approximation,
clustering and regression [16]. ANNs have been also used in
wireless communications for the purpose of channel
estimations, channel equalizations etc [17] [18] [19]. In the
last years, researchers have started to use the ANNs for the
network solutions as relay selection methodology[3].
ANNs have different types which are called according to
network connections and the activation functions used. MultiLayer Perceptron (MLP) and Radial Basis Function (RBF)
networks are two of the most popular ANNs.
A. Multi-Layer Perceptron (MLP) Network
Multi-Layer Perceptron (MLP) network which is proper for
the linear and non-linear applications is the most used ANN
network. MLP network consists of an input layer, one or more
hidden layers and an output layer. A MLP network with one
hidden layer is shown Fig. 4.

output layer activation function, hidden-output layer weights
and output layer biases, respectively. In this work “tangent
sigmoid” and “pure linear” functions are used as a hidden layer
activation and output layer activation function.
B. Radial Basis Function (RBF) Network
Radial Basis Function (RBF) network also consists of three
layers: input layer, hidden layer and output layer like MLPs.
However, RBF networks are trained faster than MLPs. The
RBF networks could be defined as an application of neural
networks for function approximation on the multi dimension
space. RBF networks use Gauss function as hidden layer
activation function. The output of a RBF network is:
𝑦 = ∑𝑛𝑘=1 (𝑒

−‖

𝑥𝑗 −𝜇𝑗
‖
2𝜎2

) 𝑤𝑜𝑘 + 𝑏𝑜𝑘.

(4)

In (4) 𝑥𝑗 is the input and 𝑦 is output of network. 𝜇𝑗
𝜎 2 are neurons center points and spread, respectively. The
output of the hidden layer is transferred to output after
multiplied by hidden-output layer weights (𝑤𝑜𝑘 ) and summed
with output biases (𝑏𝑜𝑘 ). In the RBF network the input-hidden
layer weights are all 1 [20].
V. PREDICTION OF OPTIMAL THRESHOLD VALUES
The BER values for the selection relaying scheme given in
(1) are calculated for various threshold values on the different
scenarios having changing number of relays and link qualities
as described Part II. Then, the threshold value which minimizes
the BER is assumed the optimal. The data set explained above
how to obtain is divided into two parts for training and testing.
The training data has 12500 samples and the testing data has
3125 samples. After normalized, the training data is applied to
ANNs: MLP and RBF networks, in the same way.

Figure 4. Multi-Layer Perceptron Network with one hidden layer

The computations are located on the neurons at the each
layer and the information is transferred forward from node to
node via weighted connections. The weights on the connections
and the biases values for each neuron are adjusted according to
desired outputs via backpropagation algorithms [20]. There are
several learning algorithms in the literature. In this work, one
of the mostly used algorithm Levenberg-Marquardt back
propagation is preferred. MLP is chosen in this work due to its
simple structure for prediction problems and its efficiency in
learning large data sets. The output of MLP with one hidden
layer is:
𝑛
𝑦 = 𝑓𝑜 {∑𝑚
𝑘=1 𝑓ℎ (∑𝑗=1 𝑤ℎ𝑗 𝑥𝑗 + 𝑏ℎ𝑗 )𝑤𝑜𝑘 + 𝑏𝑜𝑘 }.

(3)

In (3) 𝑥𝑗 ⁡is input(s) and 𝑦 is the output. 𝑓𝑜 ,⁡⁡𝑤ℎ𝑗 ,⁡𝑏ℎ𝑗 ⁡ are
hidden layer activation function, input-hidden layer weights
and hidden layer biases, respectively. Likewise 𝑓ℎ , 𝑤𝑜𝑘 , 𝑏𝑜𝑘 are

During the training of MLP, the network was trained many
times for different numbers of hidden layer neurons, and the
performance criteria –Mean Squared Error (MSE) - was noted.
The more neurons in the hidden layer, the less MSE is obtained
on training. However, the changing of MSE after 12 neurons is
very slow. In addition to that, considering the complexity to
implement of MLP, the number of hidden layer neurons is
chosen 12. The training MSEs of MLP according to different
number of hidden neurons are given Table 1.
TABLE I.

THE TRAIN MSES OF MLP ACCORDING TO NUMBER OF
HIDDEN LAYER NEURONS

Number of Hidden Layer
Neurons
4
6
8
10
12
14
16
18

MSEs
2.62 E-04
8.49 E-05
1.45 E-05
1.16 E-05
8.59 E-06
2.53 E-06
1.58 E-06
1.11 E-06

The RBF network was trained with the same number of
neurons to compare with MLP network properly. The spread of
the network on RBF has effect on training MSE. However,

there is not a way to calculate the spread coefficient. Therefore,
the RBF network was retrained for different spread coefficients
until the minimum MSE is obtained. The spread coefficient is
chosen 0.8 for RBF network.
After training of two networks, they were tested with
dataset not used for training. For the 10 different samples
randomly selected from the test data, the outputs of ANNs are
obtained. Three statistical performance criteria’s –Mean
Squared Error (MSE), Mean Absolute Error (MAE) and
coefficient of determination (R2)- are calculated to compare the
ANNs outputs with the numerical optimal threshold values.
The results are given Table 2.
TABLE II.
THE COMPARISON OF NUMERICAL RESULTS WITH ANNS
OUTPUTS AND THE STATISTICAL PERFORMANCE CRITERIA'S OF ANNS FOR 10
SAMPLES RANDOMLY SELECTED
.
S.
No
1
2
3
4
5
6
7
8
9
10

Num.
Values

Inputs
𝝈𝟐𝑺𝑹
1
10
1
3.25
3.25
10
7.75
3.25
1
7.75

𝝈𝟐𝑹𝑫
5.5
3.25
3.25
5.5
1
10
3.25
5.5
7.75
3.25

𝝈𝟐𝑺𝑫
10
10
7.75
7.75
10
10
3.25
3.25
5.5
7.75

M
2
8
4
6
2
6
4
8
8
8

𝜀𝑏
⁄𝑁
0
9
8
0
0
3
13
20
10
18
16

𝒐𝒑𝒕

ANN Outputs

𝜸𝒕𝒉

MLP

RBF

0.1100
0.3432
0.3171
0.0753
0.0750
0.4438
0.4291
0.2839
0.4107
0.6127

0.1094
0.3371
0.3222
0.0786
0.0739
0.4446
0.4303
0.2805
0.4090
0.6125
9.225
E-06
0.0024
0.9997

0.989
0.3377
0.3056
0.0791
0.0652
0.4387
0.4190
0.2861
0.4540
0.6196
2.4516
E-04
0.0109
0.9914

MSE
MAE
R2

On the two network scenarios described in Part II, the
ANNs outputs are calculated like given in (3) and (4). In Fig. 5,
2
2
2
on the symmetric network -𝜎𝑆𝐷
⁡ = 𝜎𝑆𝑅
= 𝜎𝑅𝐷
= 1- the optimal
values numerically obtained and the ANNs outputs are given
according to 𝜀𝑏 ⁄𝑁0 values for M=4. In Fig. 6, the same
computations are given for M=6 on the other network 2
2
2
𝜎𝑆𝑅
= 𝜎𝑅𝐷
= 10 and 𝜎𝑆𝐷
= 1-.

Figure 6. The numerical optimal thresholds and ANNs’ outputs versus
2
2
2
𝜀𝑏 ⁄𝑁0 for M=6 on the network in which 𝜎𝑆𝑅
= 𝜎𝑅𝐷
= 10 and 𝜎𝑆𝐷
= 1.

VI. CONCLUSION
We have proposed the use of two different ANN models
(MLP and RBF) for the first time to obtain the optimal
threshold values for the best relay selection scheme in
cooperative communications systems. Different scenarios
having different link qualities and different number of relays
were used to test the validity of the proposed models.
Numerical results have shown that the optimal threshold value
increases as a function of relays number and of SNR which
represents all link qualities. The results have shown that two
ANN networks can predict the optimal values. However, the
MLP network outperforms the RBF network when using the
same number of hidden layer neurons.
By the prediction of the optimal threshold value at the relay
adaptively, BER value of the DF scheme remain minimum all
SNR values. Otherwise, small threshold values would have low
BER performance at high 𝜀𝑏 ⁄𝑁0 values because of the
propagation error; high threshold values would have low BER
performance at the low 𝜀𝑏⁄𝑁0 ⁡value. On the symmetric network
2
2
2
-𝜎𝑆𝐷
⁡ = 𝜎𝑆𝑅
= 𝜎𝑅𝐷
= 1- which has M=4 relays, the BER
performances of the system have obtained for different
constant threshold values and for the optimal threshold values
according
to
changing
𝜀𝑏
⁄𝑁 values.
0
TABLE III.
THE COMPARISON OF THE BER VALUES FOR ACHIVED BY
OPTIMAL THRESHOLD VALUES WITH THE ACHIVED BY CONSTANT THRESHOLD
VALUES ON TH SYMETRIC NEWROKS WITH 4 RELAYS
Threshold Values

3

0
7,32
E-02
0,130

5

0,144

10

0,147

𝜸𝒕𝒉 (MLP)

6,98
E-02
7,2
E-02

1

𝒐𝒑𝒕

Figure 5. The numerical optimal thresholds and ANNs’ outputs versus
2
2
2
𝜀𝑏 ⁄𝑁0 for M=4 on the symmetric network in which 𝜎𝑆𝐷
⁡ = 𝜎𝑆𝑅
= 𝜎𝑅𝐷
=1

𝒐𝒑𝒕

𝜸𝒕𝒉 (RBF)

4
1,54
E-02
2,8
E-02
4,96
E-02
7,31
E-02
1,48
E-02
1,51
E-03

8
4,74
E-03
2
E-03
4,67
E-03
1,58
E-02
1,86
E-03
1,85
E-03

𝜺𝒃
⁄𝑵 (dB)
𝟎
10
3
E-03
5,55
E-04
9,26
E-04
4,41
E-03
5,39
E-04
5,45
E-04

12
1,93
E-03
2,29
E-04
1,61
E-04
9
E-04
1,38
E-04
1,43
E-04

16
7,92
E-04
7,96
E-05
1,12
E-05
1,99
E-05
6,83
E-06
7,38
E-06

20
3,19
E-04
3,19
E-05
3,63
E-06
2,96
E-07
2,44
E-07
2,87
E-07

In summary, obtaining the threshold value adaptively with
the ANNs for average 𝜀𝑏 ⁄𝑁0 ⁡ values and number of relays
provides
minimum
BER
values
for
all
𝜀𝑏 ⁄𝑁0 values. The results also shown that, compared to constant
threshold values using optimal threshold values at the relay
gives us 2 dB less power opportunity for the same BER
performances. This is very promising result in terms of green
communication systems.
As a conclusion, this paper shows that by the time the
threshold value is obtained properly, the DF cooperation
communication systems are very promising for 5G standards
instead of the traditional MIMO techniques already exist in 4G
standards

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