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Nonlinear Dyn (2014) 75:7–16DOI 10.1007/s11071-013-1044-z
O R I G I NA L PA P E R
BER performance enhancement for secure wireless opticalcommunication systems based on chaotic MIMO techniques
Lwaa Faisal Abdulameer · Jokhakar D. Jignesh ·U. Sripati · Muralidhar Kulkarni
Received: 30 April 2013 / Accepted: 17 August 2013 / Published online: 21 September 2013© Springer Science+Business Media Dordrecht 2013
Abstract There has been a growing interest in the useof chaotic techniques for enabling secure communi-cation in recent years. This need has been motivatedby the emergence of a number of wireless serviceswhich require the channel to provide low bit errorrates (BER) along with information security. The aimof such activity is to steal or distort the informationbeing conveyed. Optical Wireless Systems (basicallyFree Space Optic Systems, FSO) are no exception tothis trend. Thus, there is an urgent necessity to de-sign techniques that can secure privileged informationagainst unauthorized eavesdroppers while simultane-ously protecting information against channel-inducedperturbations and errors. Conventional cryptographictechniques are not designed for protecting informationintegrity when data is being transferred over a harsh
L.F. Abdulameer (B) · J.D. Jignesh · U. Sripati ·M. KulkarniElectronics & Communication Engineering Department,National Institute of Technology, Karnataka,Mangalore 575025, Indiae-mail: [email protected]
J.D. Jigneshe-mail: [email protected]
U. Sripatie-mail: [email protected]
M. Kulkarnie-mail: [email protected]
L.F. AbdulameerUniversity of Baghdad, Baghdad, Iraq
communication medium. Hence, a separate channelcoding protocol is often necessary to achieve this goal.Our work indicates that the use of a suitable ChaoticShift Keying (CSK) map combined with an appropri-ate Space-Time Code (STC) can allow both require-ments to be met. In this paper, we have concentratedon investigating the error rate performance of chaotic-wireless optical communication links operating overatmospheric channel, where the turbulence-inducedfading is described by the Gamma–Gamma and log–normal distributions. The main aim of the paper is toassess the feasibility of employing Space-Time Codedchaotic communications over Multiple Input MultipleOutput (MIMO) communication channels. Our simu-lations indicate that the combination of the STC andtent map provides the best BER performance in addi-tion to security when compared to the choice of othermaps.
Keywords Chaotic techniques · MIMO · STBC ·FSO
1 Introduction
Recently, there has been growing interest in the useof chaotic techniques for enabling secure communi-cation [1, 2]. It has been demonstrated that even one-dimensional discrete chaotic system is able to providea high level of security [3]. It is well known that main-taining information security on wireless channels is
8 L.F. Abdulameer et al.
a challenging task. This is because with suitable re-ceivers, anybody can intercept information from wire-less transmission in the local area. In addition, it isdifficult to discover such interceptions. So, security ofwireless transmission is very important. Chaotic sig-nals can be used in secure communications due to theirwideband property and sensitive dependence on initialconditions. The use of Chaos Shift Keying (CSK) pro-vides strong resistance against interception and cap-ture for long periods of time. So, it is said that wire-less communication method based on chaotic systemis robust and secure [4, 5]. Various methods for chaos-based secure transmission of private information sig-nals have been proposed by several authors; see [6–9].In these schemes, a chaos generator is used to gener-ate CSK sequences, where different sequences can begenerated using the same generator but with differentinitial conditions [10]. Over the past decade, the prob-lem of synchronization of chaotic systems and theirpotential application in securing communication hasreceived a lot of attention. In [11] Carroll and Pecorahave proposed a method to synchronize two identicalchaotic systems. In this paper, we assume that our sys-tems can correctly achieve the synchronization as pro-posed in [12]. We have concentrated on investigatingthe error rate performance of chaotic free space op-tics (FSO) links operating over atmospheric channel,where the turbulence-induced fading is described bythe Gamma–Gamma distribution. Atmospheric turbu-lence induced fading is one of the main impairmentsaffecting FSO. In recent years, Gamma–Gamma dis-tributed fading has become accepted as the dominantfading model for FSO links because of its excellentagreement with measurement data for a wide range ofturbulence conditions [13]. This mathematical modelcan accurately describe channel behavior for strong,moderate and weak turbulence conditions. Highly se-cure communication links with an optimum bit-errorrate (BER) performance are required to protect in-formation integrity against channel-induced impair-ments and criminal activity directed at wireless op-tical communication systems [10]. Many researchershave studied the BER performance of the chaos-basedcommunication system for the Single Input SingleOutput (SISO) channels [14]. It is now well estab-lished that exploitation of channel diversity via theuse of Multiple Input Multiple Output (MIMO) tech-niques channel utilizing multiple antennas is an op-timal method to combat fading in wireless optical
communications [15]. Motivated by these considera-tions, we have investigated the feasibility of employ-ing chaotic techniques to enhance information securityin MIMO channels by implementing space-time cod-ing schemes combined with CSK modulation. Space-Time Bock Codes (STBCs) have attached much atten-tion in radio-wireless communications, while space-time codes provide both space and time diversity bycoding over multiple apertures (antennas) and severaltime slots. A very simple and effective scheme referredto as STBC was introduced by Alamouti [16, 17]. Inthe proposed systems STBC is used to overcome theinfluence of the turbulence-induced fading. We showthat the proposed systems can achieve a better BERthan the BER of FSO system with Single Input Sin-gle Output (SISO), especially with the chaotic se-quences generated electrically. Although end devicesmay use cryptographic theories to protect the integrityof data, it is still critical that the network assures thatthe FSO links also secure at the physical layer. Be-cause of the non-periodic nature of chaotic signals,it is certainly the case that after passing through thechaotic modulator, the transmitted bit energy variesfrom one bit to another. However, most papers com-pute the BER performance by considering the bit en-ergy as constant. This approximation, which is widelyknown as the Gaussian Approximation (GA), suffersfrom a low precision when the spreading factor is low.Another approach integrates the BER expression fora given chaotic map over all possible spreading se-quences of a given spreading factor [14]. This lattermethod, when compared with the BER computationunder the Gaussian assumption, gives more accurateresults but suffers from high computational complex-ity. Another accurate computation is the exact BERperformance for coherent and non-coherent chaos-based communication systems. The idea of this ap-proach, is to first compute the probability density func-tion (PDF) of the bit energy, and then use the com-puted PDF to compute the BER expression [18–20].Recently, a MIMO system has been proposed for CSKsystem.
2 Chaotic map generators
These maps are chosen because of the simplicity interms of generating chaotic sequences. Here we willillustrate the dynamics of these maps.
BER performance enhancement for secure wireless optical communication systems based on chaotic 9
2.1 Tent map
The tent map is defined as follows [21]:
xn+1 ={
2xn mod 1, xn < 12
1 − (2xn mod 1), xn ≥ 12
2.2 Logistic map [22]
xn+1 = p × xn × [1 − xn]
2.3 Bernoulli map
xn+1 = 2xn mod 1
2.4 Chebyshev map
Its dynamics are described by [14]:
xn+1 = 1 − 2x2n
In the above dynamical equations, xn and xn+1 rep-resent the current and next symbols of the chaotic se-quence, respectively. In the logistic map, p representsthe control parameter.
3 Chaotic shift keying (CSK)
Consider a binary intensity modulation (IM) wheresymbols ‘0’ and ‘1’ are represented by x0 and x1.A description of chaotic shift keying (CSK) techniquemethod that can be employed in microchip lasers foreffecting secure communications as given in [23]. TheCSK modulated data symbols are from the set {−1,1}.After IM, the data symbols sent to Optical AlamoutiBlock are from the set {0, I }.s0(t) = 0, 0 ≤ t ≤ βTc
s1(t) = I, 0 ≤ t ≤ βTc
(1)
Consider the data information symbols (sl = ±1) withperiod Ts spread by a sequence of chaotic samples (orchips). The time interval of each chip is equal to Tc
(xk = x(kTC)). I is the constant intensity of the laserat the transmitter for symbol 1. The emitted signal atthe output of the transmitter is:
u(t) =∞∑l=0
β−1∑k=0
slxlβ+k (2)
where the spreading factor β is equal to the number ofchaotic samples in symbol duration (β = Ts
Tc). For the
AWGN channel, the received signal is
r(t) = u(t) + n(t) (3)
where n(t) is the AWGN with zero mean and powerspectral density equal to N0/2. In order to demod-ulate the transmitted bits, the received signal is firstdespread by the local chaotic sequence, and then in-tegrated over symbol duration Ts . Finally, the trans-mitted bits are estimated by computing the sign of thedecision variable at the output of the correlator.
Dsl = sign
(slTc
β−1∑k=0
(xlβ+k)2 + wl
)
= sign(slE
(l)b
) + wl (4)
where sign(.) is the sign operator, E(l)b is the bit energy
of the lth bit and wl is the noise after despreading andintegration.
4 Fading channel models
4.1 Log–normal fading channel [24, 25]
If a random variable X is said to be log-normally dis-tributed, then the random variable Y = lnx is Gaus-sian distributed. The log–normal distribution is used tomodel continuous random quantities when the distri-bution is believed to be skewed, such as certain incomeand lifetime variables. The PDF of the log–normal dis-tribution is given by
f (x) =⎧⎨⎩ 1
xσ√
2πe−[ (ln(x)−μ2)
2σ2 ], x > 0
0, x ≤ 0(5)
4.2 Gamma–Gamma fading channel [13, 26]
The atmospheric turbulence causes irradiance fluctu-ation known as scintillation, which is mainly causedby temperature variations in the atmosphere. Differ-ent models have been proposed to describe the atmo-sphere turbulence by varying degrees of strength. Fora wide range of turbulence conditions (weak to strong)
10 L.F. Abdulameer et al.
the fading gain Imn in FSO systems can be modeled bya Gamma–Gamma distribution.
fI (Imn) = 2(αβc)α+βc
2
Γ (α)Γ (βc)I
α+βc2 −1Kα−βc(2
√αβcImn)
(6)
where parameters α > 0 and βc > 0 are linked tothe so-called scintillation index SI � 1
α+ 1
βc+ 1
(αβc),
where α and βc can be adjusted to achieve good agree-ment between fI (Imn) and the measurement data. Al-ternatively, assuming spherical wave propagation, α
and βc can be directly linked to physical parametersvia
α =[
exp
(0.49x2
(1 + 0.18d2 + 0.56x12/5)7/6
)− 1
]−1
(7)
βc =[
exp
(0.51x2(1 + 0.69x12/5)−5/6
(1 + 0.9d2 + 0.62d2x12/5)5/6
)− 1
]−1
(8)
where x2 = 0.5C2nK7/6L7/6, d = (KD2/4L)1/2, and
K = 2π/λ. Here λ, D, C2n , and L are the wavelength
in the meter, the diameter of the receiver aperture inmeters, the index of the refraction structure parameter,and the link distance in meters, respectively.
5 Chaotic-MIMO using Alamouti space time code
Let
si(t) = −sj (t) + I, i �= j, i, j ∈ {0,1} (9)
To continue the analysis, we define the complement ofsignal ai by al to represent the opposite binary state ofthe signal ai (i.e., if ai = s0 then al = s1 and if a = s1
then al = s0). Applying this definition to (9) results inthe following relationship:
a(t) = −a(t) + I, a(t) ∈ {s0, s1} (10)
Noting that a(t) = 0, I the relationship in (10) en-sures that a(t) is non-negative. The primary prop-erty of a space-time code A(a1, . . . , am) = A(x) witha = (a1, . . . , am)T is
AT (a)A(a) = IM‖a‖2 (11)
where (∗)T denotes the transpose operation, Ik is theM × M unit matrix, M is the number of transmit-ter antennas and ‖a‖2 = (a2
1 + a22 + · · · + a2
m). How-ever, a coding scheme satisfying (11) cannot be im-plemented in IM/DD system because transmitted IMsignal must be non-negative at all times.
The above coding scheme shows that certain trans-mitter outputs must be negated to get the orthogonal-ity. Therefore the above coding scheme cannot be usedfor free-space IM/DD systems.
To overcome the above problem, we use the follow-ing STBC matrix:(|A|(a, a)
)i,j
=⎧⎨⎩
(A(a))i,jif (A(a1 = 1, . . . , am = 1))i,j ≥ 0
−(A(a))i,j otherwise(12)
This definition ensures that the transmitted symbolsare always non-negative for an IM/DD system.
Then,
a = AT (h)JK × (JKA(h)x + n
)(13)
Using the relation JK × JK = IK , we obtain
a = ‖h‖2x + AT (h)JKn (14)
where K represents the transmission time slots, JK
is K × K diagonal matrix, x = (x1, . . . , xm)T , n =(n, . . . , n)T are Gaussian noise, which may includecontribution from thermal and/or shot noise and h =(h1, . . . , hm)T is the quasi-static channel response fortransmitters i = 1,2, . . . ,M , respectively [17, 27–29].
5.1 2 × 2 STBC
For 2 × 2 STBC,
A(a,a) =(
a1 a2
a2 a1
)(15)
Then(a1
a2
)= (
h211 + h2
21 + h212 + h2
22
)(a1
a2
)
+(
h11n1 + h21n2
h22n1 − h12n2
)(16)
where h = (h11, h21, h12, h22)T is the channel re-
sponse vector for two lasers (antennas).
BER performance enhancement for secure wireless optical communication systems based on chaotic 11
Fig. 1 Chaotic-MIMO(2 × 2), (a) transmitter withtwo antennas, (b) receiverwith two antennas
Fig. 2 2 × 1 optical STBCreceiver
The transmitter and receiver of 2 × 2 STBC is thesame as in Fig. 1(a); the receiver is shown in Fig. 2.
The design of the transmitted signal for time[0, βTc] from Tx1 and Tx2 is given by
U[0,βT c](t) = a1xk + a2xk (17)
The design of the transmitted signal for time [βTc,
2βTc] from Tx1 and Tx2 is given by
U[βT c,2βT c](t) = a2xk+β + a1xk+β (18)
The received signal on Rx1 for time [0, βTc] is givenby
r11(t) = h11a1xk + h21a2xk + n1k (19)
The received signal on Rx1 for time [βTc,2βTc] isgiven by
r21(t) = h11a2xk+β + h21a1xk+β + n2k (20)
The received signal on Rx2 for time [0, βTc] is givenby
r12(t) = h12a1xk + h22a2xk + n1k (21)
The received signal on Rx2 for time [βTc,2βTc] isgiven by
r22(t) = h12a2xk+β + h22a1xk+β + n2k (22)
The energy of a given bit l is E(l)b = ∑β
k=1 x(l)2k .
The equivalent baseband model of the receivedsymbol on the first antenna Rx1 for time [0, βTc] isgiven by
Y11 = Eb(h11a1 + h21a2) + N11 (23)
The equivalent baseband model of the received symbolon the first antenna Rx1 for time [βTc,2βTc] is given
12 L.F. Abdulameer et al.
by
Y21 = Eb(h11a2 + h21a1) + N21 (24)
where N11 = ∑β
K=1 n1kxk and N21 = ∑β
K=1 n2kxk+β .
The equivalent baseband model of the receivedsymbol on the second antenna Rx2 for time [0, βTc]is given by
Y12 = Eb(h12s1 + h22s2) + N12 (25)
The equivalent baseband model of the received sym-bol on the second antenna Rx2 for time [βTc,2βTc] isgiven by
Y22 = Eb
(h12s
∗2 + h22s
∗1
) + N22 (26)
where N12 = N11 and N22 = N21.The channel model can be written as
Y = EbHA + N (27)
The transmitted bits are estimated by multiplying thesignal Y by the conjugate transpose of the channel H :(
Ds1
Ds2
)= H ∗Y (28)
The estimated bits are computed from the sign of thedecision variables,
a′1 = signDa1; a′
2 = signDa2 (29)
5.2 2 × 1 STBC
For 2 × 1 STBC and from (15) we have(a1
a2
)= (
h21 + h2
2
)(a1
a2
)+
(h1n1 + h2n2
h2n1 − h1n2
)(30)
The transmitter of 2 × 1 STBC is the same as inFig. 1(a); the receiver is shown in Fig. 2.
The design of the transmitted signal of 2 × 1 Alam-outi is as in (17) and (18).
The received signal for time [0, βTc] is given by
r1(t) = h1a1xk + h2a2xk + n1k (31)
The received signal for time [βTc,2βTc] is given by
r2(t) = h1a2xk+β + h2a1xk+β + n2k (32)
The equivalent baseband model of the received symbolfor time [0, βTc] is given by
Y1 = Eb(h1a1 + h2a2) + N1 (33)
The equivalent baseband model of the received symbolfor time [βTc,2βTc] is given by
Y2 = Eb(−h1a2 + h2a1) + N2 (34)
where N1 and N2 represent the noise componentswhile h1 and h2 represent the channel gains.
The channel model, estimation and computation ofthe transmitted bits are similar to 2 × 2 STBC using(27), (28) and (29), respectively.
6 Performance analysis
The main objective of this paper is to study the per-formance of the Chaotic-MIMO system under chan-nels perturbed by AWGN, Gamma–Gamma and log–normal fading channels. The channel gains for both2 × 2 STBC and 2 × 1 STBC are constant underthe AWGN assumption. The overall BER expres-sions of the Chaotic-MIMO system for 2 × 2 STBCand 2 × 1 STBC are given in (35) and (36), respec-tively:
BER =∫ ∞
0
1
2erfc
(√(h2
11 + h212 + h2
21 + h222)E
(l)b
N0
)
× p(E
(l)b
)dE
(l)b (35)
and
BER =∫ ∞
0
1
2erfc
(√(h2
1 + h22)E
(l)b
N0
)p(E
(l)b
)dE
(l)b
(36)
where p(E(l)b ) is the probability density function of the
energy E(l)b . The BER expression is the result of the
integral given in (35) and (36). To compute the inte-gral in (35) and (36), we must first have the bit energydistribution. Since the PDF seems to be intractable,the only way to evaluate the BER is to compute thehistogram of the bit energy followed by a numericalintegration. Figure 3 shows the histogram of the bitenergy, for spreading factor β = 4 and 50000 sam-ples.
BER performance enhancement for secure wireless optical communication systems based on chaotic 13
After numerical integration the BER expression for2 × 2 STBC is given by
BER =m∑
l=1
1
2erfc
(√(h2
11 + h212 + h2
21 + h222)E
(l)b
N0
)
× p(E
(l)b
)(37)
Equations (35)–(37) have been explained in [18–20].
Fig. 3 Histogram of the bit energy
For 2 × 1 STBC, the corresponding BER is givenby
BER =m∑
l=1
1
2erfc
(√(h2
1 + h22)E
(l)b
N0
)p(E
(l)b
)(38)
where m is the number of histogram classes andp(E
(l)b ) is the probability of having the energy in in-
terval centered on E(l)b .
7 Simulation results
In Fig. 4 we have compared the performance of differ-ent chaotic maps described in Sect. 2 under AWGN.The channel gain is constant and equal to 1, β = 4and Tc = 1. Our simulations indicate that the tentmap gives best performance when compared to theother chaotic maps. Hence, the use of the tent mapis preferred over other maps because it offers supe-rior BER performance in addition to security. In SISOsystems using BPSK, chaotic techniques give a degra-dation of 6 dB in BER performance. This degrada-tion is compensated by the use of MIMO systems.In Fig. 4 it can be seen that up to a BER of 10−5,chaos systems maintain the same BER performance
Fig. 4 BER of 2 × 2 STBC for various types of chaotic maps under AWGN
14 L.F. Abdulameer et al.
Fig. 5 BER of 2 × 2 STBC for various types of chaotic maps under Gamma–Gamma fading channel
Fig. 6 BER of 2 × 1 STBC for various types of chaotic maps under Gamma–Gamma fading channel
as BPSK without using chaos. Additionally, they pro-vide security against eavesdroppers. Therefore, thistechnique provides additional security while maintain-ing a BER of 10−5. Also, in Fig. 5 we have com-pared the BER performance of 2 × 2 STBC usingdifferent chaotic maps under Gamma–Gamma fadingchannel. It is observed in Fig. 6 that under Gamma–Gamma fading conditions, the tent map gives the best
BER performance as compared to other maps. In ad-dition from Figs. 5 and 6 it is clear that the 2 × 2STBC gives 2 dB gain in BER performance as com-pared to 2 × 1 STBC under Gamma–Gamma fad-ing conditions. In Fig. 7 it is observed that with theuse of the 2 × 2 STBC under Gamma–Gamma fad-ing conditions, the performance of the channel de-grades by only 1 dB as compared to AWGN chan-
BER performance enhancement for secure wireless optical communication systems based on chaotic 15
Fig. 7 BER performance compression between AWGN, Gamma–Gamma and log–normal fading channels for 2 × 2 STBC schemes
nels and 4 dB as compared to log–normal fadingchannel. Hence increasing the diversity along withthe use of MIMO communication system reduces thedegradation in a fading channel employing CSK. Suchcombined MIMO-CSK schemes can be gainfully em-ployed on channels where high levels of data integrityand security are simultaneously required. This im-provement is realized with only a marginal increasein computational complexity at the transmitter and re-ceiver.
8 Conclusions
The BER performance of 2×2 STBC and 2×1 STBCschemes combined with CSK for different chaoticmaps under different fading channel conditions havebeen computed and plotted. These schemes give thebenefit of providing additional security while main-taining BER performance at levels similar to that ob-tained by the use of a simple BPSK alone.
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