A Novel Design Architecture of Secure Communication System with Reduced-Order Linear Receiver

International Journal of Trend in
International Open Access Journal
ISSN No: 2456
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A Novel Design Architecture
Reduced
Professor, Department of Electrical Engineering
ABSTRACT
In this paper, a new concept about secure
communication system is introduced and a novel
secure communication design with reduced
linear receiver is developed to guarantee the global
exponential stability of the resulting error signals.
Besides, the guaranteed exponential convergence rate
of the proposed secure communication system can be
correctly calculated. Finally, some numerical
simulations are given to demonstrate the feasibility
and effectiveness of the obtained results.
Key Words: Chaotic system, secure c
system, reduced-order linear receiver
1. INTRODUCTION
As we know, because chaotic system
sensitive to initial value, the output behaves like a
random signal. Frequently, chaos in many dynamic
systems is an origin of the generation of oscillation
and an origin of instability. Several kinds of chaotic
systems have been widely applied in various
applications such as secure communication,
slave chaotic systems, image encryption
systems, chemical reactions, system identification
and ecological systems; see, for instance
the references therein.
In recent years, numerous secure communications
have been extensively explored; see, for example, [
12] and the references therein. Generally speaking
secure communication is composed of transm
receiver and reduced-order linear receiver
merits of low price and easy implementation.
Therefore, searching a lower-dimensional
order linear receiver for the secure chaotic
communication system constitutes an important area
for practical control design.
International Journal of Trend in Scientific Research and Development (IJTSRD)
International Open Access Journal | www.ijtsrd.com
ISSN No: 2456 - 6470 | Volume - 3 | Issue – 1 | Nov
www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018
A Novel Design Architecture of Secure Communication System
Reduced-Order Linear Receiver
Yeong-Jeu Sun
Electrical Engineering, I-Shou University, Kaohsiung
communication system is introduced and a novel
secure communication design with reduced-order
linear receiver is developed to guarantee the global
exponential stability of the resulting error signals.
guaranteed exponential convergence rate
of the proposed secure communication system can be
correctly calculated. Finally, some numerical
simulations are given to demonstrate the feasibility
and effectiveness of the obtained results.
Chaotic system, secure communication
chaotic system is highly
the output behaves like a
, chaos in many dynamic
systems is an origin of the generation of oscillation
everal kinds of chaotic
have been widely applied in various
applications such as secure communication, master-
image encryption, biological
system identification,
instance, [1-3] and
, numerous secure communications
have been extensively explored; see, for example, [4-
erally speaking, a
secure communication is composed of transmitter and
order linear receiver has the
s of low price and easy implementation.
dimensional reduced-
secure chaotic
constitutes an important area
In this paper, we will propose a new
communication system and a
communication system with
receiver will be developed
resulting error signals can converge to zero in some
exponential convergence rate
guaranteed exponential convergence rate
proposed chaotic secure communication system
be accurately estimated. Finally,
simulations are proposed to exhibit
feasibility of the main results.
This paper is organized as follows. The problem
formulation and main results
2. Several numerical simulations are
3 to illustrate the main result. Finally,
remarks are drawn in Section
paper, n
ℜ denotes the n-dimensional Euclidean space,
xxx T
⋅=: denotes the Euclidean norm of the
vector x, and a denotes the
number a.
2. PROBLEM FORMULATION AND MAIN
RESULTS
In this paper, we develop the following
communication system with simple
linear receiver and its block diagram is shown in
Figure 1.
Transmitter:
( ) ( ) ( ),3211 txtxatx =&
( ) ( ) ( ),23122 txatxatx +=&
( ) ( ) ( ),21543 txtxaatx +=& (1c)
( ) ( ) ( ),2716 txatxaty +=
( ) ( ) ( ) .0,1 ≥∀+= ttmtxCt mmφ
Research and Development (IJTSRD)
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1 | Nov – Dec 2018
Dec 2018 Page: 1154
Secure Communication System with
Kaohsiung, Taiwan
we will propose a new idea about secure
a novel design of secure
communication system with reduced-order linear
to guarantee that the
onverge to zero in some
convergence rate. Meanwhile, the
guaranteed exponential convergence rate of the
proposed chaotic secure communication system can
Finally, some numerical
exhibit the capability and
This paper is organized as follows. The problem
are presented in Section
simulations are given in Section
to illustrate the main result. Finally, conclusion
in Section 4. Throughout this
dimensional Euclidean space,
the Euclidean norm of the column
he absolute value of a real
ROBLEM FORMULATION AND MAIN
the following new secure
communication system with simple reduced-order
and its block diagram is shown in
(1a)
(1b)
(1d)
(1e)

International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456
Receiver:
( ) ( ) ( ),
1
6
2
6
7
1 ty
a
tz
a
a
tz +−= (2a)
( ) ( ) ( ),
6
2
23
6
72
2 ty
a
a
tza
a
aa
tz +







−−=& (2b)
( ) ( ) ( ) ,0,2 ≥∀−= ttzCttm mmφ (2c)
where ( ) ( ) ( )[ ] 2
21: ℜ∈=
T
txtxtx is the partial
of transmitter, ( ) ℜ∈ty is the output of transmitter
( ) ( ) ( )[ ] 2
21: ℜ∈=
T
tztztz is the state vector
( ) 1
1
×
ℜ∈ q
tm is the information vector,
( ) 1
2
×
ℜ∈ q
tm is the signal recovered from
Nq∈ . It is noted that the chaotic Sprott B
the special case of the system (1) with
153321 =−=−=== aaaaa . In the sequel, we adopt the
same parameters of the chaotic Sprott B
076 >aa . Apparently, a good secure communication
system means that we can recover the message
in the receiver system; i.e., the error vector
( ) ( ) ( )tmtmte 12: −= can converge to zero in some sense.
Before presenting the main result, let us introduce a
definition which will be used in the main theorem
Definition 1: The system (1) with (2) is called secure
communication system with exponential convergence
type if there are positive numbers k and
( ) ( ) ( ) ( ) 0,exp: 12 ≥∀−≤−= ttktmtmte α .
In this case, the positive number α
exponential convergence rate.
Now we present the main results
communication system of (1) with (2).
Theorem 1: The system (1) with (2) is a secure
type. Besides, the guaranteed
convergence rate is given by
6
7
1
a
a
+=α .
Proof. Define
( ) [ ] [ ] 2
221121 ℜ∈−−==
TT
zxzxwwtw .(3)
Thus, from (1)-(3), one has
( ) ( ) ( )tztxtw 222
&&& −=
( ) ( ) ( )
( )ty
a
a
tza
a
aa
txatxa
6
2
123
6
72
2312
−








−++=
in Scientific Research and Development (IJTSRD) ISSN: 2456
partial state vector
of transmitter,
is the state vector of receiver,
is the information vector, 2×
ℜ∈ q
mC , and
from ( )tm1 , with
haotic Sprott B system is
the special case of the system (1) with
. In the sequel, we adopt the
chaotic Sprott B system with
, a good secure communication
system means that we can recover the message ( )tm1
in the receiver system; i.e., the error vector
can converge to zero in some sense.
let us introduce a
which will be used in the main theorem.
The system (1) with (2) is called secure
and α such that
is called the
s for secure
tem (1) with (2) is a secure
exponential
( ) ( )
( ) ( )[ ]txatxa
a
a
a
a
aa
txatxa
2716
6
2
3
6
72
2312
+−




−++=
( )
( ) ( )[ ]tztxa
a
aa
a
aa
txa
a
aa
223
6
72
6
72
23
6
72
−







−−=




−+







+−=
( )
( )twa
a
aa
a
aa
txa
a
aa
23
6
72
6
72
23
6
72








−−=




−+







+−=
( ).1 2
6
7
tw
a
a






+−=
This implies that
( ) ( )
















+−⋅= t
a
a
wtw 1exp0
6
7
22 .
From (1)-(4), it is easy to see that
( ) ( ) ( )tztxtw 111 −=
( ) ( )
( ) ( )





+−−






−=
ty
a
tz
a
a
tx
a
a
ty
a
6
2
6
7
2
6
7
6
1
1
( ) ( )[ ]
( )tw
a
a
tztx
a
a
2
6
7
22
6
7
−=
−−=
( )
















+−⋅−= t
a
a
a
wa
1exp
0
6
7
6
27
Hence, from (3)-(5), it results
( ) ( ) ( )
( )022
6
2
7
2
6
2
2
2
1
w
a
aa
twtwtw
⋅
+
≤
+=
.0,1exp
6
7
≥∀
















+−⋅ tt
a
a
Thus, it can be readily obtained that
( ) ( ) ( )
( ) ( ) ( ) (
( )twC
xCttzCt
tmtmte
m
mmmm
⋅≤
+−−=
−=
φφ
12
( )
,0,1exp
0
6
7
22
6
2
7
2
6
≥∀














+−⋅
⋅⋅
+
≤
tt
a
a
Cw
a
aa
m
in view of (1), (2), and (6). This
in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
Dec 2018 Page: 1155
( )tz23 



( )tza 23 



−
( )tza 23 



−
(4)
), it is easy to see that
. (5)
(6)
t can be readily obtained that
( )t
This completes the proof.

Remark 1: It should be emphasized that
receiver of (2) is linear and with lower dimens
than that of the transmitter. Consequently
proposed receiver of (2) has the superiorities
price and easy implementation by electronic circuit
3. NUMERICAL SIMULATIONS
Consider the novel secure communication system of
(1)-(2) with 176 == aa and [ ]11 −=mC . By Theorem 1,
the synchronization of signals ( )tm1 and
proposed secure communication (1)
achieved with guaranteed convergence
The real message ( )tm1 , the recovered message
and the error signal are depicted in Figure 2
respectively, which clearly indicates
message ( )tm1 is recovered after 3 seconds.
4. CONCLUSION
communication system has been introduced and a
novel secure communication design with
order linear receiver has been developed to guarantee
the global exponential stability of the resulting error
signals. Meanwhile, the guaranteed exponential
convergence rate of the proposed secure
communication system can be correctly calculated.
Finally, some numerical simulations have been
offered to show the feasibility and effectiveness of the
obtained results.
ACKNOWLEDGEMENT
The author thanks the Ministry of Science and
Technology of Republic of China for supporting
work under grants MOST 106-2221
MOST 106-2813-C-214-025-E, and MOST
E-214-030. Besides, the author is grateful to
Professor Jer-Guang Hsieh for the useful
( ) ( )daSystem 11 −
channelchannel
( )tz
( )ty( )tx
Transmitter
( )tmφ+
+
( )tm2
+
−
( )tm1
mC
mC
Systems (2a)-(2b)
Figure 1: Secure-communication scheme
information vector and ( )tm2 is the recovered vector)
It should be emphasized that the proposed
lower dimensions
Consequently, the
superiorities of low
by electronic circuit.
novel secure communication system of
. By Theorem 1,
and ( )tm2 for the
proposed secure communication (1)-(2) can be
rate of 2=α .
message ( )tm2 ,
ure 2-Figure 4,
that the real
seconds.
communication system has been introduced and a
cation design with reduced-
has been developed to guarantee
the global exponential stability of the resulting error
signals. Meanwhile, the guaranteed exponential
convergence rate of the proposed secure
ectly calculated.
Finally, some numerical simulations have been
offered to show the feasibility and effectiveness of the
Ministry of Science and
of Republic of China for supporting this
2221-E-214-007,
, and MOST 107-2221-
grateful to Chair
for the useful comments.
Transmitter
Receiver
scheme ( ( )tm1 is the
is the recovered vector).
Figure 2: Real message of
transmitter of (1)
Figure 3: Recoverd message of
receiver of (2)
Figure 4: Error signal of
0 5 10 15 20 25
3.5
4
4.5
5
5.5
6
6.5
t (sec)
m1(t)
0 5 10 15 20 25
3.5
4
4.5
5
5.5
6
6.5
t (sec)
m2(t)
0 5
0
0.05
0.1
0.15
0.2
0.25
t (sec)
m2(t)-m1(t)
Dec 2018 Page: 1156
Real message of ( )tm1 described in the
transmitter of (1).
Recoverd message of ( )tm2 described in the
receiver of (2).
Error signal of ( ) ( )tmtm 12 − .
30 35 40 45 50
t (sec)
30 35 40 45 50
t (sec)
10 15
t (sec)

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