This is an open access article under the CC BY license: 

 

 
 

 

   

Al-Khwarizmi 

Engineering 

Journal 

 
Al-Khwarizmi Engineering Journal, Vol. 18, No. 4, December, (2022) 

P. P. 32- 44 

 

 

 

BER Performance Improvement of Dual Chaotic Maps Based on 

STBC Communication System 

 
Lwaa Faisal Abdulameer *            U. Sripati**             Muralidhar Kulkarni*** 

*Department of Information and Communication/ Al-khwarizmi College of Engineering/ University of Baghdad/ Iraq 

**,***Electronics and Communication Engineering Department/ NITK. Mangalore/ India 

*Email: lwaa@kecbu.uobaghdad.edu.iq 
sripati_achary@yahoo.co.in :Email**                                                                   

mkuldc@gmail.comEmail: ***                                                                 

 
(2022 October 4 2022; Accepted May 15 Received) 

https://doi.org/10.22153/kej.2022.10.001 
 

 
 

Abstract  
 

Sensitive information of any multimedia must be encrypted before transmission. The dual chaotic algorithm is a good 

option to encrypt sensitive information by using different parameters and different initial conditions for two chaotic maps. 

A dual chaotic framework creates a complex chaotic trajectory to prevent the illegal use of information from 

eavesdroppers. Limited precisions of a single chaotic map cause a degradation in the dynamical behavior of the 

communication system. To overcome this degradation issue in, a novel form of dual chaos map algorithm is analyzed. To 

maintain the stability of the dynamical system, the Lyapunov Exponent (LE) is determined for the single and dual maps. 

In this paper, the LE of the single and dual maps have been computed numerically. Increasing the dynamical behavior of 

the system by using more complex chaotic maps leads to inferiority in the overall system performance. So, in this work, 

the BER performance for the dual and single chaotic maps by exploiting the benefits of a hybrid Chaos Shift Keying-

Multiple-Input-Multiple-Output (CSK-MIMO) communication system has been investigated. The results show that the 
dual tent map has more randomness, whereas the single logistic map has the least randomness. As well as the CSK-MIMO 

gives an outstanding BER performance when it compared with the SISO system which helps in reducing the system’s 

inferiority.   

 

Keywords: Chaotic technique, dual, Lyapunov Exponent, STBC. 

 

 

1. Introduction  
 

A chaotic waveform is characterized as being 

deterministic, non-periodic, and very sensitive to 
any change in initial conditions, i.e., initializing the 

system with a slight difference in initial conditions 

generates a new sequence completely different 

from the original signal [1]. So, to encrypt sensitive 
data, a chaotic technique can be utilized to produce 

a key stream with high randomness. In any case, a 

single chaotic map is attackable to be descended by 
chaotic reconstruction since it has a simple 

structure and weak control boundaries. Dual 

dimensional chaos has a complicated structure and 

more parameters [2,3].  
To adjust these two factors, a dual chaotic 

system can be considered [4]. In this paper, dual-

chaotic systems that consist of two one-
dimensional tent and logistic maps have been 

designed. 

The chaotic attractor can be characterized by 
measuring the degree of sensitivity to initial 

conditions. Lyapunov Exponents (LEs) describe 

the increase of very small perturbations in different 

directions of the state space on a logarithmic scale, 
when at least one LE is positive, the attractor under 

mailto:lwaa@kecbu.uobaghdad.edu.iq
mailto:sripati_achary@yahoo.co.in
mailto:mkuldc@gmail.com
https://doi.org/10.22153/kej.2022.10.001


Lwaa Faisal Abdulameer                              Al-Khwarizmi Engineering Journal, Vol. 18, No. 4, P.P. 32- 44 (2022) 

 

33 

realization characterize by sensitive dependence on 
initial conditions property and this is termed 

chaotic [5].  

Chaos is utilized in many research areas, such 
as engineering, mathematics, chemistry, and 

biology. In communication engineering, chaos 

introduces important aspects in security as well as 
multiuser applications in cellular communications. 

Therefore, chaotic communications is a promising 

technique in the next wireless generation.  

A compound of two 1-D chaotic maps suggests 
by I. Aouissaoui et al.  [6] for information security. 

Which achieved a logistic-sine map based on a tent 

chaotic map as well as a cubic-tent map but for 
SISO channel. The analysis of LE for the proposed 

maps has been achieved. The bifurcation diagram 

was plotted to measure the chaotic range but the 
authors did not calculate the BER for the proposed 

system. 

J. Pedro et al. [7] propose a compound of two 1-

D chaotic maps; logistic and tent maps for the deep 
zoom technique. They found that the random 

quality of chaos improved with PRNG when the 

chaotic parameter and a are increased for SISO 
communication systems. But also they did not 

compute the BER to ensure the system 

performance is not affected by the changing in 

chaotic behavior.  M. Lawnik et al. [8] suggest M-
map chaotic system for cryptography applications. 

They measured the LE and plotted the bifurcation 

diagram. They showed that the minimum value for 
LE for the proposed analysis was 2 and the 

maximum value was 12.  

A dual map is suggested to resolve the problem 
of dynamical degradation. A significant advantage 

in the dynamical systems analysis is the stability of 

the system, which can be satisfied by using the LE.   

In this paper, the LEs for dual tent and logistic 
maps have been derived.   This paper realizes the 

feasibility of using MIMO to improve the 

performance of the communication system due to 
chaotic degradation. We have studied the 

performance of CSK with the Alamouti scheme for 

these dual chaotic maps under the AWGN channel. 
The use of these schemes allows us to enhance 

security without degrading the BER performance. 

An exact method to compute the BER has been 

utilized and analyzed. This method considers the 
aperiodicity nature of the chaotic signals [9]. The 

simulation results refer to the combination of 

STBC and a single chaotic map can degrade the 
BER when compared to a dual chaotic map in cost 

of reducing the information protection. 

 

 
 

2. Iteration of Chaotic Map 
2.1 Tent Map Iteration 
 

       Consider the tent map trajectory 𝑥: [0,1] is 
expressed as [10] 

  xn+1 = {

xn

a
                         , 0 ≤ xn < 𝑎

1−xn

1−a
                    , a ≤ xn < 1

           

                                                                        …(1)                                                    

The dynamical system trajectory 𝑥(𝑥) is obtained 
by iterating the chaotic map, i.e.,   

𝑥(𝑘) = 𝑓𝑘 [𝑥(0)] =               𝑓({… 𝑓[𝑥(0)] … }), 
k=0,1,2,……                                                  …(2) 

                                                                          

2.2 Logistic Map Iteration 

 
The non-linear paradigm of the logistic map is 

given in equation (3) [11] 

  xn+1 = μxn(1 − xn),         μ ∈ [0,4], xn ∈ (0,1)                     
                                                                        …(3)                                                                                 

     The sequence {𝑥1, 𝑥2, … 𝑥𝑛}  is generated by 
initializing the initial value 𝑥0 ∈ (0,1) depending 
on the value of a parameter 𝜇 . A new sequence 
generates when a small change in the initial value 

occurs and changes in the value𝜇. The chaos state 
behavior occurs with the increase of 𝜇 values and 
achieving a chaos state ultimately. When 𝜇 >
3.75, movement from the sequence {𝑥𝑛} appears 
chaotic behavior.  
 

 

3. Dual Chaotic Iteration  
        

As mentioned above, a single chaotic system is 

attackable to be attacked by chaotic reconstruction 
since it has a simple structure and weak control 

boundaries. A 2-D chaotic map has more 

parameters and a complex form, but it has a more 

complex load.  To balance these two factors, a dual 
chaotic map may be utilized. A dual-maps that 

consists of two 1-D maps has been implemented.  

 

3.1 Dual-Logistic Map 

 

Chaos may be appeared by LOGISTIC 1 and 
LOGISTIC 2 such as, 

LOGISTIC 1:           xn+1 = μxn(1 − xn),         μ ∈
[3.757,4], xn ∈ (0,1)                                       … (4) 
LOG2:                   xn+1 = ηxn(1 − xn),         η ∈
[3.57,4], xn ∈ (0,1)                                        … (5) 
where 𝜇 and η represent the control parameters. In 
this case, 𝑥𝑛+1  represents the output sequence of 
LOGISTIC 1, while 𝑦𝑛+1  represents the output 
sequence of LOGISTIC 2 considering 𝑦0 = 𝑥𝑛+1. 
So, {𝑦𝑛} is the last output of those two sequences. 



Lwaa Faisal Abdulameer                              Al-Khwarizmi Engineering Journal, Vol. 18, No. 4, P.P. 32- 44 (2022) 

 

34 

It is produced by composing the sequences of 
LOGISTIC 1 and LOGISTIC 2 [4,12]. 

 

3.2 Dual-Tent Map   
 

TENT MAP 1 and TENT MAP 2 are represented 

as follows, 
TENT MAP 1:               

  xn+1 = {

xn

a
                         , 0 ≤ xn < 𝑎

1−xn

1−a
                    , a ≤ xn < 1

       … (6)                 

                                                                 
TENT MAP 2:              

 yn+1 = {

yn

b
                         , 0 ≤ xn < 𝑏

1−yn

1−b
                    , b ≤ xn < 1

       …(7) 

                     

where 𝑎  and 𝑏  are the control parameter, 𝑥0  and 
𝑦0 are initial values, 𝑥𝑛+1  is the output sequence of 
TENT MAP 1, and 𝑦𝑛+1 is the output sequence of 
TENT MAP 2 considering 𝑦0 = 𝑥𝑛+1. 
       So, {𝑦𝑛}  is the last output of these two 
sequences. It is produced by composing the two 

sequences of TENT MAP 1 and TENT MAP 2 
leading to increasing the complexity of chaotic 

behavior.  

 

 

4. Lyapunov Exponent 
 

Consider |𝛿𝑛| = 𝑓
𝑛 (𝑥0 + 𝛿0) − 𝑓𝑛 (𝑥0)  being 

the 𝑛 iteration number of the chaotic sequence, 𝑥0 
denotes the initial condition, and 𝛿0 be the small 
random number. However, LE formula can be 
given as, 

𝜆 =
1

𝑛
𝑙𝑛 |

𝛿𝑛

𝛿0
| =

1

𝑛
𝑙𝑛|(𝑓𝑛)́(𝑥0)|                        … (8)   

       By expanding the term within algorithm, the 
approximation value of the LE can be given as,  

𝜆 =
1

𝑛
∑ 𝑙𝑛|�́�(𝑥𝑖 )|

𝑛−1
𝑖=0                                           …(9) 

If the equation (10) have limit when 𝑛 → ∞, then 
the LE for the orbit that begins in 𝑥0 is                                             

𝜆 = lim
𝑛→∞

{
1

𝑛
∑ 𝑙𝑛|�́�(𝑥𝑖 )|

𝑛−1
𝑖=0 }                              …(10) 

       Note that λ depends on 𝑥0 . For points and 
cycles stable, λ is negative and for chaotic 
attractors, λ is positive [13]. 

4.1 LE of Single Logistic [14] 

        According to equations (3 and 10),                

𝜆 = lim
𝑛→∞

{
1

𝑛
∑ ln(𝜇) − ln (2𝜇𝑥𝑖 )

𝑛−1
𝑖=0 }             … (11) 

4.2 LE of Single Tent [15-17] 

       According to Equs. (3) and (10), 

𝜆 = ∫ 𝑙𝑛 |
1

𝑎
| 𝑑𝑥 + ∫ |

1

1−𝑎
| 𝑑𝑥

1

𝑎

𝑎

0
                      … (12) 

 

 

 

4.3 LE of Dual Logistic 

LEs of a 𝑥𝑛 sequence denoted by, 

𝜆 = lim
𝑛→∞

1

2𝑛
ln [𝐷𝑛

𝑇 𝐷𝑛 ]                                    … (13) 

whenever 𝐷𝑛  is the limit exists. Uppercase T 
represents the transpose.  

      From Equs. (10, 11 and 13), a dual logistic map 

can be expressed as, 

Let ,  xn+1 = μxn(1 − xn), 𝑥𝑛+1́ = 𝜇 − 2𝜇𝑥𝑛 
For Dual;  

𝜆 = lim
𝑛→∞

1

2𝑛
∑ ln {[𝜇 − 2𝜇𝑥𝑛]

𝑇 [𝜇 − 2𝜇𝑥𝑛] −
𝑛−1
𝑖=1

𝑑} = lim
𝑛→∞

1

2𝑛
∑ ln {𝜇(1 − 2𝑥𝑛 )]

𝑇 [𝜇(1 −𝑛−1𝑖=1

2𝑥𝑛)] − 𝑑}                                                   … (14) 
where 𝑑 is the change in initial value. 
 

4.4 LE of Dual Tent 
       

According to equations (10,12 and 13), a dual 

tent map can be derived as, 

Let,   xn+1 = {

xn

a
                         , 0 ≤ xn < 𝑎

1−xn

1−a
                    , a ≤ xn < 1

  

For dual; 

𝜆 = lim
𝑛→∞

1

2𝑛
{[∫ 𝑙𝑛 |

1

𝑎
| 𝑑𝑥 +

𝑎

0

∫ |
1

1−𝑎
| 𝑑𝑥

1

𝑎
]𝑇 [∫ 𝑙𝑛 |

1

𝑎
| 𝑑𝑥 + ∫ |

1

1−𝑎
| 𝑑𝑥

1

𝑎

𝑎

0
] − 𝑑}  

                                                                         

                                                                      …(15)  

 

    

5. CSK-MIMO Communication System 
 

Let us consider x(t)  as the signal that is 
generated by any chaotic map and defined in the 

previous section. In CSK, the chaotic signal will 

occur If a ′ + 1′ is transmitted and an inverted copy 
of the chaotic signal will appear If ′ − 1′  is 
transmitted. Therefore,  the transmitted signal can 

be denoted as [18-20,24],         

s(t) =

     {
x(t),    if  ′+1' is transmitted                
−x(t),                    otherwise                

             

    …(16) 

                                             

       Let us consider the time interval of the data 

symbols (sl = ±1)  denoted by Ts , then the chip 
interval of the chaotic sequence will be Tc(xk =
x(kTC)) . However, The transmitted signal is 
expressed as, 

  u(t) = ∑ ∑ slxlβ+k
β−1
k=0

∞
l=0                               …(17) 

where 𝛽 represents the spreading factor (β =
Ts

Tc
).  

Then, the signal received at the receiver side is 

denoted as, 

r(t)=u(t) + n(t)                                           …(18) 



Lwaa Faisal Abdulameer                              Al-Khwarizmi Engineering Journal, Vol. 18, No. 4, P.P. 32- 44 (2022) 

 

35 

where n(t) represents the AWGN with zero mean 
and variance  𝜎𝑛

2 . At the receiver, the received 

signal is first despreaded by generating an exact 

replica of the chaotic signal that is generated at the 
transmitter, after that the symbols  are integrated 

over symbol duration  Ts . at last, the sign of the 
decision variable will be computed [9,21], such as, 

   Dsl = sign (slTc ∑ (xlβ+k)
2 + wl)

β−1
k=0

=
 

sign(slEb
(l)

+ wl)                                               …(19)  

Where sign(. )  represents the sign operator, 

 E
b
(l)

denotes lth  bit energy and wl is the AWGN 
computed after despreading processing.  

2×2 Space Time Block Code (STBC) 
       Fig. 1 shows the block diagram of CSK-

MIMO transmitter with two antennas and two 

receive antenna. 
 

              

 

 
 

Fig. 1  𝟐 × 𝟐 MIMO-CSK communication scheme. 
 
The designed symbols for STBC denoted by,  

S = [
s1 −s2

∗

s2 s1
∗ ]                                        … (20) 

 

Table. 1, 

The transmitted signa 

Time interval 𝐬𝟏(𝐭)𝐟𝐫𝐨𝐦  first transmitter 𝐬𝟐(𝐭)𝐟𝐫𝐨𝐦 second transmitter 

[0, βTc ] s1xk s2xk 
[βTc , 2βTc ] −s2

∗ xk+β s1
∗ xk+β 

 
 
Table. 2, 

The signal received by the first receiving antenna 

Time Received signal on 𝐟𝐢𝐫𝐬𝐭 𝐫𝐞𝐜𝐞𝐢𝐯𝐢𝐧𝐠 𝐚𝐧𝐭𝐞𝐧𝐧𝐚 
[0, βTc ] h11s1xk + h21s2xk + nk

1  

[βTc , 2βTc ] −h11s2
∗ xk+β + h21s1

∗ xk+β + nk
2 

 
 

 

Noise 

Noise 
Chaotic generator 

෍ ⬚

𝛽

 

෍ ⬚

𝛽

 

STBC 

decoder 

βTc 

s1,́ s2́ 

x(t) 

h11 

h21 

h12 

h22 

𝑇𝑥1 

𝑇𝑥2 

Data 

Alamouti 

encoder 
Chaotic generator 

s1, s2 

Tx1 

Tx2 



Lwaa Faisal Abdulameer                              Al-Khwarizmi Engineering Journal, Vol. 18, No. 4, P.P. 32- 44 (2022) 

 

36 

 

Table. 3, 

The signal received by the second receiving antenna 

Time Received signal on second receiving antenna 

[0, βTc ] h12s1xk + h22s2xk + nk
1  

[βTc , 2βTc ] −h12s2
∗ xk+β + h22s1

∗ xk+β + nk
2 

 The t
th
 bit energy is denoted by E

b
(l)

= ∑ x
k
(l)2β

k=1
. 

 
Table. 4, 

The equivalent baseband of the symbol at 𝑹𝒙𝟏 
Time The equivalent baseband of the symbol at 𝑹𝒙𝟏 
[0, βTc ] Y11 = Eb(h11s1 + h21s2) + N11 
[βTc , 2βTc ] Y21 = Eb(−h11s2

∗ + h21s1
∗ ) + N21 

Where, N11 = ∑ nk
1β

K=1 xk  and 𝑁21 = ∑ 𝑛𝑘
2𝛽

𝐾=1 𝑥𝑘+𝛽  represent AWGN components.  
 
Table .5, 

The equivalent baseband of the symbol at 𝑹𝒙𝟐 
Time The equivalent baseband of the symbol at 𝑹𝒙𝟐 
[0, βTc ] Y12 = Eb(h12s1 + h22s2) + N12 
[βTc , 2βTc ] Y22 = Eb(−h12s2

∗ + h22s1
∗ ) + N22 

 

 

 N12 = N11  and 𝑁22 = 𝑁21  represent AWGN 
components and h11, h21 , h12  and h22  are the 
channel coefficients for 2 × 2 scheme. The above 
formulas are derived in [6]. 
The received vector is expressed as,  

  Y = Eb HS + N                                         … (21)             
       The data are estimated by multiplying the 

vector 𝑌 by the conjugate transpose of H  (equation 
22) and then computed by the decision variable 

(equation 23); 

(
Ds1
Ds2

) = H*Y                                                 …(22) 

        The estimated bits are computed from the sign 

of the decision variables, 

  s1́ = sign Ds1;   s2́ = sign Ds2                       …(23) 
 

5.1 Performance Analysis  
        

Let us consider the channel coefficient is 

constant for 2×2 STBC under the assumption of 
AWGN. The BER formula of the proposed scheme 

is expressed in (24) [21,22,24].  

  𝑃𝑒 (𝐸𝑏
(𝑙)

) =
1

2
𝑒𝑟𝑓𝑐(√

(ℎ11
2 +ℎ21

2 +ℎ12
2 +ℎ22

2 )𝐸
𝑏
(𝑙)

𝑁0
)                     

Now,                                                                                   

𝐸𝑥𝑖 [𝑄{𝐸(𝑥)/√𝑉𝑎𝑟(𝑥)}]  
where  

𝑄(𝑥) = ∫
1

√2𝜋
exp (−𝑢2 /2

+∞

𝑥
)                                                                                                        

𝐵𝐸𝑅 = ෍ ℎ𝑛,𝑝
2

2

𝑛,𝑝=1

෍ 𝐸𝑥𝑖
𝛽−1

𝑙=0
(෍

𝑥𝑖
2

𝑁0

𝛽−1

𝑖 =0

)2

= ෍ 𝐸𝑥𝑖
𝛽−1

𝑙=0
(෍

𝑥𝑖
2

𝑁0

𝛽−1

𝑖 =0

)
1
2  (ℎ11

2

+ ℎ21
2 + ℎ12

2 + ℎ22
2 )𝐸𝑏

(𝑙)
 

 ∑ 𝑥𝑖
2𝛽−1

𝑖=0
 =E

b
(l)

 is the chaotic symbol energy, and 

𝐸𝑥𝑖 = ∫ p (Eb
(l)

) dE
b
(l)∞

0
,  

       The BER formula for 2×2 STBC is formulated 
as, 

BER =

∫
1

2

∞

0
erfc (√

(h11
2 +h12

2 +h21
2 +h22

2 )E
b
(l)

N0
) p (E

b
(l)

) dE
b
(l)

     

                                                              … (24) 

Where p (E
b
(l)

) is the PDF of the E
b
(l)

. The BER 

formula resulted from integration given in equation 

(24). To compute the integral in equation (24), the 

distributed energy must be considered. Because of 
the PDF consider to be intractable, the method to 

compute the BER is evaluating the histogram of 

E
b
(l)

 followed by a numerical integration 

[18,19,18]. Fig. (2) presents the histogram of the 

E
b
(l)

, for 50000 samples obtained by simulation and 

𝛽=4. 



Lwaa Faisal Abdulameer                              Al-Khwarizmi Engineering Journal, Vol. 18, No. 4, P.P. 32- 44 (2022) 

 

37 

 
Fig. 2 𝐄

𝐛
(𝐥)

 histogram.      

 
 

Applying numerical integration, the BER formula 
is expressed as, 

BER =

∑
1

2
erfc (√

(h11
2 +h12

2 +h21
2 +h22

2 )E
b
(l)

N0
) p(E

b
(l)

)ml=1                    

                                                                     … (25) 

       In the above equations, 𝑚 denotes the number 

of histogram classes and p (E
b
(l)

)is the probability 

of the energy in interval centered on E
b
(l)

. 

 

 

6. Results and Discussions 
        

Fig.3 shows the single logistic map providing 

the chaotic behavior beyond 𝜇 = 3.57. This map 
has more randomness 𝜇 = 3.8 and LE equals 0.5. 
Fig.4 presents the single tent map. It is observed 

that this map has chaotic properties within the 
range (0,1) of a control parameter (a). It has more 

randomness when the value of (a) is greater than 

0.5 and LE equals 0.85. A dual logistic is provided 

in Fig.5. It has more randomness when  𝜇 > 2.35 
and 𝜇  < 4. More randomness appeared at 𝜇 = 4 
and LE equals 0.81. 

       Fig.6 presents the dual tent which has chaos 

properties in the range (0,1) of 𝑎 . It has more 
randomness when 𝑎 = 0.5 and LE at equals 1.2. 
We observed from Fig. 3 and Fig. 4 that when a 

very small change in initial values say 10
-16

, a new 
chaotic sequence is generated completely different 

from the current state. Therefore, with increasing 

the time of iteration more than 20 times, the 
difference exponentially increases. For the dual 

logistic map in Fig. 5, this difference appears with 

increasing the iteration time more than 40 times, 
after this value, the difference exponentially 

increases. 

       In Fig.6, the dual tent map has the difference 
between two sequences is very sensitive to a very 

small change in initial conditions. However, this 

map shows better sensitivity to initial conditions. 

Whereas, the single tent map provides the best 
sensitivity than dual single and logistic maps 

according to results obtained from LE.       

Table (6) summarizes the values of LEs for 
presented chaotic maps 

 
Table 6, 

values of LE 

Chaotic map LE 

Single logistic  0.5 

Single tent  .0.85 

Dual logistic  0.81 

Dual tent  1.2 

 

 

       From the above table, it is observed that the 
dual tent has the most randomness whereas the 

single logistic map has the least randomness. 

       In Fig.7, the BER is degraded for single map 

when a 2 × 2  is considered for the system. Also, 
for a dual tent, when the system is compensated 

with  2 × 2 STBC, the  SNR will gain 0.5 dB as 
compared to 2 × 2  single logistic for the same 
BER under the AWGN assumption.  It is noted that 

there is a tradeoff between the randomosity and 

BER performance. So, the BER performance can 

be achieved by implementing 2 × 2   single tent 
map at the cost of reducing randomosity. 

       From Figs. 3,4, 5, and 6, we observed that 

chaotic maps give highly distinct sequences as 

1.85 1.855 1.86 1.865 1.87 1.875
0

0.5

1

1.5

2

2.5
x 10

4

Bit energy

n
0
. 

o
f 

s
a
m

p
le

s



Lwaa Faisal Abdulameer                              Al-Khwarizmi Engineering Journal, Vol. 18, No. 4, P.P. 32- 44 (2022) 

 

38 

compared to the PN sequence generator. The 
degree of distinctness is more for the dual chaotic 

map as compared to the single chaotic map. 

       Now, let us suppose that 𝑥𝑛  and 𝑦𝑛  are two-
sequence with length 𝑁. The auto-correlation and 
the cross-correlation are defined in equations (26 

and 27) respectively [4]. 

𝑅𝑥 (𝑗) =
1

𝑁
∑ 𝑥𝑖 𝑥𝑖+𝑗

𝑁
𝑖=1                                   …(26) 

𝑅𝑥𝑦 (𝑗) =
1

𝑁
∑ 𝑥𝑖 𝑦𝑖+𝑗

𝑁
𝑖=1                                   …(27) 

where 𝑁  is the sequence’s length and 𝑗  is the 
distance between two chaotic elements.  

Fig.8 shows the auto-correlation of dual tent. The 

auto-correlation is maximum at zero time shift. Fig. 
9 presents that the single tent map provides a cross 

correlation function near zero (maximum value 

0.0085) when compared to dual tent (highest value 
0.028). However, the two sequences of single tent 

satisfy the orthogonality property more than dual 

tent map 

 

 

 
Fig. 3. The difference between two single logistic and its LE. 

0 10 20 30 40 50 60 70 80 90 100
-4

-2

0

2

4

6

8
x 10

-14

Iterations

D
if

fe
re

n
c
e
 b

e
tw

e
e
n

 t
w

o
 s

e
q

u
e
n

c
e
s

 

 

single logistic map



Lwaa Faisal Abdulameer                              Al-Khwarizmi Engineering Journal, Vol. 18, No. 4, P.P. 32- 44 (2022) 

 

39 

 

 
Fig. 4. a) The difference between two single tent and its LE. 

 

 
 

 

0 10 20 30 40 50 60 70 80 90 100
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Iterations

D
if

fe
re

n
c
e
 b

e
tw

e
e
n

 t
w

o
 s

e
q

u
e
n

c
e
s

 

 

Single tent map

0 10 20 30 40 50 60 70 80 90 100
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Iterations

D
if

fe
re

n
c
e
 b

e
tw

e
e
n

 t
w

o
 s

e
q

u
e
n

c
e
s

 

 

dual logistic logistic



Lwaa Faisal Abdulameer                              Al-Khwarizmi Engineering Journal, Vol. 18, No. 4, P.P. 32- 44 (2022) 

 

40 

 
Fig. 5. The difference between two dual logistic and its L.E  

 

 
Fig. 6. The difference between two dual tent and its LE. 

0 10 20 30 40 50 60 70 80 90 100
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Iterations

D
if

fe
re

n
c
e
 b

e
tw

e
e
n

 t
w

o
 s

e
q

u
e
n

c
e
s

 

 

dual tent maps



Lwaa Faisal Abdulameer                              Al-Khwarizmi Engineering Journal, Vol. 18, No. 4, P.P. 32- 44 (2022) 

 

41 

 
Fig. 7. BER for single and dual maps. 

 
Fig. 8. auto-correlation of dual tent.  



Lwaa Faisal Abdulameer                              Al-Khwarizmi Engineering Journal, Vol. 18, No. 4, P.P. 32- 44 (2022) 

 

42 

 

 
Fig. 9. cross-correlation of single and dual tent. 

 

 

6. Conclusions 
        

The data encryption based on the dual chaotic 

presented in this work has excellent randomness. 
The LEs for the single and dual maps have been 

computed numerically. In this work, the exact BER 

for dual and single chaotic maps modulated CSK 

2 × 2 STBC under AWGN has been investigated. 
The final results show that the randomness of the dual 

tent map is better than the single tent map, single 

logistic map, and dual logistic map. As well as the 

2 × 2  CSK-MIMO gives a better BER 
performance as compared with the SISO system. 

 

 

 

 

7. References   
 

[1] K. Daniela and E. Marcio, “On the Power 
Spectral Density of Chaotic Signals Generated 

by Skew Tent Maps,” IEEE International 

Symposium on Signals, Circuits and Systems”, 

Vol. 1, pp. 1-4, 2007. 
[2] A. Kathleen, S. Tim and Y. James, Chaos, An 

Introduction to Dynamical Systems. Springer 

1996. 
[3] F. Jiu and T. Chi, Reconstruction of Chaotic 

Signals with Applications to Chaos-Based 

Communications. Word Scientific, 2008.  
[4] C. Yi, Li Z. and W. Yifang, A Data Encryption 

Algorithm based on Dual Chaotic System: 

proceeding of the 2010 International 

Conference on Computer Application and 

https://ieeexplore.ieee.org/xpl/conhome/5602791/proceeding
https://ieeexplore.ieee.org/xpl/conhome/5602791/proceeding


Lwaa Faisal Abdulameer                              Al-Khwarizmi Engineering Journal, Vol. 18, No. 4, P.P. 32- 44 (2022) 

 

43 

System Modeling (ICCASM 2010), pp. V4-
431V4-435, 2010. 

[5] Parlitz U.: Lyapunov Exponents from Chau’s 
Circuit., Journal of Circuits, Systems and 
Computers, Vol. 3, No. 2, pp. 507-523, 1993. 

[6] I. Aouissaoui1, T. Bakir, A. Sakly, and S. 
Femmam, “Improved One-Dimensional 
Piecewise Chaotic Maps for Information 

Security”, “Journal of Communications”, Vol. 

17, No. 1 P.P. 1-6,, January 2022. 

[7] J. Pedro do V. Alvarenga, J. Machicao, and O. 
M. Bruno, “Chaotical PRNG based on 

composition of logistic and tent maps using 

deep-zoom”, “Chaos, solitons and fractals- 
Elsevier”, Vol. 161, 2022. 

[8] M. Lawnik and M. Berezowski, “New Chaotic 
System: M-Map and Its Application in Chaos-
Based Cryptography”, “Symmetry- MDPI”, 

Vol. 14, 2022. 

[9] G. Kaddoum, Mai V. and F. Gagnon, “On the 
performance of chaos shift keying in MIMO 
communications systems”. “IEEE WCNC-

PHY”, pp. 1432-1437 ,2011. 

[10] A. Mozsary, L. Azzinari, K. Krol and V. 
Porra, Theoretical connection between PN-

sequences and chaos makes simple FPGA 

pseudo-chaos sources possible: proceeding of 

the 8th IEEE International Conference on 
Electronics, Circuits and Systems, February 

2001. 

[11] A. Abir, A. Safwan, Q. Wang, V. Calin and B. 
Bassem, “Comparative study of 1-D chaotic 

generator for digital data encryption”, 

“IAENG International Journal of Computer 
Science”, 2008. 

[12] K. Ljupco, S. Janusz, A. Jose and T. Igor, 
“Discrete Chaos-I: Theory”, “IEEE 

Transaction of Circuits and Systems-I”, Vol 
53, No. 6, pp. 1300-1309, 2006. 

[13] E. Ibarra, R. Vazquez, M. Cruz and J. Rio, 
Numerical Calculation of the Lyapunov 
Exponent for the Logistic Map: proceeding of 

the 12th International Conference on 

Mathematical Methods in Electromagnetic 
Theory, pp. 409-411, 2008. 

[14] H. Martin and M. Yuri, “An Introduction to 
the Synchronization of Chaotic Systems: 

Coupled Skew Tent Maps”, “IEEE 
Transactions on Circuits and Systems-I: 

Fundamental Theory and Applications”, Vol. 

44, No. 10, pp. 856-866, 1997. 

[15] G. Kaddoum, P. Charge and D. Roviras, “A 
generalized methodology for bit-Error-rate 

reduction in correlation-based communication 

schemes using chaos”,” IEEE Communication 
Letters”, Vol. 13, No. 8. , 2009. 

[16] L. Abdulameer., D. Jokhakar, U. Sripati and 
M. Kulkarni: BER Performance Enhancement 
for Secure Wireless Communication Systems 

based on Chaotic MIMO Techniques., 

procedding of the International Conference on 

Communication and Electronics System 
Design, 2013. 

[17]  R. Sankpal, A. Vijaya, Image Encryption 
Using Chaotic Maps. A Survey, Lecture notes 
in networks and systems, Springer, p.p. 835-

844, 2014. 

[18] S. Das,  K. Mandal, and  M. Chakraborty, 
“LMMSE equalized DCSK communication 

system over a multipath fading channel with 

AWGN noise”, proceeding of the third 

International Conference on Computer, 
Communication, Control and Information 

Technology (C3IT), p.p 1-4, 2015. 

[19] V. Manikandan & Rengarajan Amirtharajan, 
“On dual encryption with RC6 and combined 

logistic tent map for grayscale and DICOM”,” 

Multimedia Tools and Applications”, P.P. 1-

30, 2021. 
[20] A. Imoize , A. Ibhaze , A. Atayero , and K. 

Kavitha, “Standard Propagation Channel 

Models for MIMO Communication Systems”, 
“Hindawi Wireless Communications and 

Mobile Computing”, PP. 1-36, 2021. 

[21] R. Dhia’a, “Design a Security Network 
System against Internet Worms”, “Al-

Khwarizmi Engineering Journal”, Vol. 8, No. 

3, PP 40 – 52, p.p. 1-13, 2012. 

[22] R. Yamazaki, Y. Shimada, and T. Ikeguchi, 
“Chaos MIMO system with efficient use of 

information of bits”, “NOLTA, IEICE”, P.P. 

1-15, 2020.  
[23] I. Kadhem, “A New Audio Steganography 

System Based on Auto-Key Generator”, “Al-

Khwarizmi Engineering Journal”, Vol. 8, No. 
1, p.p. 27 – 36, 2012. 

[24] L. Abdulameer, ANALYSIS AND DESIGN 
OF RELIABLE AND SECURE CHAOTIC 

COMMUNICATION SYSTEMS FOR 
OPTICAL AND WIRELESS LINKS, Dept. 

of Electronics and Communication Engg., 

NITK, 2014. 
 

 

 

 
 

https://ieeexplore.ieee.org/xpl/conhome/5602791/proceeding
https://ieeexplore.ieee.org/xpl/conhome/4569857/proceeding
https://ieeexplore.ieee.org/xpl/conhome/4569857/proceeding
https://ieeexplore.ieee.org/xpl/conhome/4569857/proceeding
http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=6754860&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D6754860
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http://ieeexplore.ieee.org/search/searchresult.jsp?searchWithin=p_Authors:.QT.Das,%20S..QT.&newsearch=true
http://ieeexplore.ieee.org/search/searchresult.jsp?searchWithin=p_Authors:.QT.Mandal,%20S.K..QT.&newsearch=true
http://ieeexplore.ieee.org/search/searchresult.jsp?searchWithin=p_Authors:.QT.Chakraborty,%20M..QT.&newsearch=true
http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=7055073
http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=7055073
http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=7055073
http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=7055073
http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=7055073


 (2022) 32-44، صفحة 4العدد، 18مجلة الخوارزمي الهندسية المجلد                                                 لواء فيصل عبد االمير

44 

 

 

 

 

 تحسين اداء منظومة االتصاالت الفوضوية الثنائية باستخدام تقنية الهوائيات المتعددة 

 
 لواء فيصل عبد األمير*        اودبي سريباتي**          مولدير كولكرني***

العراق /جامعة بغداد /كلية الهندسة الخوارزمي /قسم هندسة المعلومات واالتصاالت*  
 دالهن /قسم هندسة االلكترونيك واالتصاالت /المعهد الوطني التكنولوجي كارناتاكا**،***

  lwaa@kecbu.uobaghdad.edu.iq*البريد االلكتروني:
sripati_achary@yahoo.co.in:البريد االلكتروني** 

mkuldc@gmail.com:البريد االلكتروني*** 

 

 

 
 

 الخالصة

 
تعد من الخيارات الممتازة  الفوضويةان الحفاظ على سرية المعلومات عند انتقالها خالل الوسط من والى المستخدم تعد من االمور الضرورية جدا. االشارة 

 جدا فياالشارة ال تكرر نفسها بعد مدة من الزمن وان اي تغيير حتى لو كان صغيرا لتشفير االشارات وخصوصا انها تمتاز عن الخوارزميات االخرى بكون 

فوضوية من االحادية مما  أكثر. االشارة الفوضوية الثنائية تمتاز بكونها القيم االبتدائية سوف يولد اشارة جديدة مما يصعب على المتنصتين استراق االشارة

الفوضوية الثنائية وقد تم قياس  لإلشارةالفوضوية االحادية ونوعين مماثلين  لإلشاراتيولد اشارة معقدة جدا بالنسبة للمتنصت. في هذا البحث، تم تصميم نوعين 

(. ايضا تم في هذا البحث استخدام تقنية متعددة Lyapunov Exponent (LE)مية لكل نوع وتتمثل بـ )مدى فوضوية كل اشارة من خالل اشتقاق خوارز

)العالقة العكسية بين  الهوائيات لتحسين االداء العام لمنظومة االتصاالت لكون استخدام االشارات الفوضوية يؤدي الى تقليل االداء بسبب زيادة سرية المعلومة

فوضوية من بقية االنواع المستخدمة في البحث.  أكثرمما يعني انها  (LE)اعطت اعلى قيمة من الـ  Dual chaotic mapالنتائج بينت بأن  االنتروبي واالداء(.

 االخرى المستخدمة.  باألنواعاداء مقارنة  أحسنيقدم  single chaotic mapمع  2x2ايضا من خالل استخدام تقنية الهوائيات المتعددة تبين ان استخدام 

 

 
 

 

 

mailto:lwaa@kecbu.uobaghdad.edu.iq
mailto:sripati_achary@yahoo.co.in**البريد
mailto:sripati_achary@yahoo.co.in**البريد
mailto:mkuldc@gmail.com***البريد
mailto:mkuldc@gmail.com***البريد