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Proceedings, 5th Chaotic Modeling and Simulation International 
Conference, 12 – 15 June 2012, Athens Greece 
Performance Analysis of Time Hopping Ultra
Wide Band System Using Chaotic vs.
Conventional system
Anis Naanaa and Safya Belghith
Sys’comlaboratory, National EngineeringSchoolof Tunis (ENIT),Tunisia.
(E-mail: anis
naanaa@yahoo.com, safya.belghith@enit.rnu.tn)
Abstract. This paper proposes to use chaotic modulation and coding for Time-
HoppingUltra Wide Band (TH-UWB) system with multi-path channel.
We compare the performances of chaotic systems using the Skew tent map
against those of a conventional systems, as the Gold code, and demonstrate that
chaotic system enhances the system performances when compared with the con-
ventional systemin terms of Bit Error Rate(BER).
We report simulation results clearly showing that the chaotic system outperforms
the conventional system.
Keywords: Ultra wide band systems, Time-Hopping, Multi-path, Chaotic sys-
tem, Bit error rate.
1 Introduction
Ultra-wideBand (UWB) systems is a spread-spectrum technique, that em-
ploys pulses oftemporal extension ofless than onenano second[1].
ThesucessofUWBsystemsforshort-rangewirelesscommunications[1,2]
is dueto thefactthattheypotentiallycombinereducedcomplexitywithlow
powerconsumption, lowprobability-of-intercept(LPI)andimmunitytomul-
tipathfading. In2004, theIEEE802.15.4a group presented acomprehensive
study of the UWB channel over the frequency range 2-10 GHz for indoor
residential, indooroce,industrial, outdoorandopenoutdoorenvironments
[3].
In this workweareconcerned with theindoorresidential environmentchan-
nel.
In TimeHopping format (TH-UWB), TH codes areused as a multiple user
diversity andPulsePosition Modulation (PPM) asdata [1,2].
In wireless communicationsystemwheremorethanoneusersharethesame
channel, the interference between users represents an additional source of
noise. this may degrade the performanceof the system. Thus the choice of
the modulation type, the multipleaccess techniques and the codes allowing
multiple access iscrucial forthedetermination of system performance.
Dierent works have dealt with the statistical characteristics of the Multi-
User Interference (MUI). In many works, the MUI has been modeled by a
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Proceedings, 5th Chaotic Modeling and Simulation International 
Conference, 12 – 15 June 2012, Athens Greece 
randomGaussian process [1,2,4],othersworks tackled with theoptimization
of the performance by code selection [5], and thus no optimization with re-
spect to the TH codes has been done. In themost detailed analysis [5], the
authors considered the asynchronous case, multi-channel propagation such
as IEEE 802.15.3a channel. They derived a criterion to nd optimal codes
that minimizes the variance of MUI. But, the optimization is done for only
onereference user, and so even if thecodessatised thegiven criterion, it is
notguaranteedthattheperformanceoftheother users isoptimizedor even
improved. In [6] we introduced a criterion namedAverageCollision Number
(ACN) that minimize the MUI variance. It has been shown that sequences
having smallerACN allow better BER. In anotherwork [7] we showed that,
ACNcriterion isunsuitable in somecases for selecting codes. Hence an im-
proved criterion called AverageofSquared Collision Number(ASCN).
Basedonthiscriterionwestudyinthepresentpaper how much chaoticityof
thechaoticcodesaectstheperformanceoftheconsideredTH-UWBsystem.
To validate our criterion, the performance in terms of BER is computed by
simulating theTH-UWBsystemwithline-of-sight(LOS)multi-path channel
in a residential environmentIEEE 802.15.4a.
This paperis organized as follows. Section 2 gives a detailed description
of UWB signal generations; after introducing the TH-UWB-PPM system
model, we give the format of the receiver signal. In Section 3, we dene
the dierent sequences considered in this work. For chaotic sequences, the
ASCNiscomputedversus bifurcation parameter andcomparedto Lyapunov
exponent.
In Section 4, we dene the new chaotic modulation and we validateour
methodbyreportingsimulationresultsshowingtheadvantageofusingASCN
andtherelevance of chaoticsequences. Finally weconclude in section5.
2 UWB signal generations
In this section, we begin bydescribing theTH-UWB systemmodel and the
expression of the receivedsignal inan asynchronous TH-UWB systemusing
thePPMmodulation. ThenwecomputethevarianceoftheMUIinthecase
of TH-codesequences when a correlating receiver is used.
2.1 TH-UWB System model
Inthispaper,weassumepulsepositionmodulationforthetransmittedbinary
symbols in both the UWB techniques. Second derivative of the Gaussian
pulsewas used as basicUWB pulseshape. Thepulsewaveform, w(t)canbe
expressed as,
w(t)=(1 4(
t
)
2
)exp( 2(
t
)
2
)
(1)
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Proceedings, 5th Chaotic Modeling and Simulation International 
Conference, 12 – 15 June 2012, Athens Greece 
Thepulsewaveformw(t) is assumed to benon-zero onlyduring theinterval
0 t T
c
,T
c
is thechip duration and  is the pulse duration. Thesecond
derivative of Gaussian pulse is shown in Figure 1.
−1
0
1
2
3
4
5
6
7
8
x 10−9
−0.5
0
0.5
1
Time [s]
Amplitude
Fig.1. Thesecond derivativeof Gaussian pulse
Atypical expression of theTH-UWBtransmitted signal fora user j is given
byequation 2.
s
(j)
(t)=
X1
k= 1
N
f
1
X
l=0
w(t kT
s
lT
f
ec
(j)
l
T
c
d
(j)
k
)
(2)
WhereT
s
istheperiodofonebit. EverybitisconveyedbyN
f
frames. Each
framehasa duration ofT
f
and isdivided into N
c
timeslots. Each timeslot
has a duration of T
c
. ec
(j)
l
is the TH code sequence assigned to the user j,
whereec
(j)
l
2f0; 1; :::;N
c
1g. The location ofeach pulsein eachframe is
dened by the code ec
(j)
l
. d
(j)
k
2f0;1g is the binary transmitted symbol at
timek byuserj,  is thetimeshiftassociatedwith binaryPPM, the pulses
corresponding tobit 1aresent secondslaterthan thepulsescorresponding
to bit 0. N = N
c
N
f
presents the total processing gain of the system. The
signals corresponding to bits 1 and 0 aredepicted in Figure 2.
In this work, weuse theIEEE 802.15.4a UWB channel model [3] in a resi-
dential area. According to this model theimpulse responseis [3,8],
h
(j)
(t)=
MX 1
m=0
RX 1
r=0
(j)
r;m
(t T
(j)
m
(j)
r;m
)
(3)
where
r;m
is thetap weight of the r-th ray (path)in the m-th cluster, T
m
is thearrival timeofthe m-thclusterand 
r;m
is thearrival timeofthe r-th
rayin the m-thcluster. The distribution of theclusterarrival timesisgiven
bya Poissonprocess andthedistribution oftherayarrival times isgivenby
amixedPoissonprocess[3]. Thechannel model whichisused in thepaperis
for LOS scenarios in residential environments as shown in Figure3, referred
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Proceedings, 5th Chaotic Modeling and Simulation International 
Conference, 12 – 15 June 2012, Athens Greece 
0
0.5
1
1.5
2
2.5
3
x 10−8
−0.5
0
0.5
1
AMPLITUDE
(a)
Time [s]
0
0.5
1
1.5
2
2.5
3
x 10−8
−0.5
0
0.5
1
Time [s]
AMPLITUDE
(b)
Ts
delta
Tf
Tf
Tc
Fig.2. A TH-UWB signal with binary PPM modulation where N
f
=4, N
c
=8
and the THcodesequenceis f0121g. (a): Bit 1, (b): Bit 0.
0
50
100
150
200
250
300
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
Time [ns]
Impulse response
Fig.3. Impulse response channelmodel 1inresidential environments.
to asCM1 [3].
IfN
u
isthenumberofactiveuserstransmittingasynchronously; thereceived
signal is
r(t)=
N
u
X
j=1
M 1
X
m=0
RX 1
r=0
(j)
r;m
s
(j)
(t T
(j)
m
(j)
r;m
)+n(t)
(4)
2.2 Receiver Signal Processing
Theoutputof thecorrelation receiver of the i
th
user at timeh is givenby:
s
(i)
h
=
N
f
1
X
p=0
Z
hT
s
+pT
f
+
e
c
(i)
p
T
c
+T
c
+
(i)
0;0
+T
(i)
0
hT
s
+pT
f
+
e
c
(i)
p
T
c
+
(i)
0;0
+T
(i)
0
r(t)v(t hT
s
pT
f
ec
(i)
p
T
c
(i)
0;0
T
(i)
0
)dt
(5)
wherev(t)isthereceiver’s templatesignal denedbyv(t)=w(t+) w(t).
From thepreviousequations andafter variable changes, weobtain
s
(i)
h
=T
U
(i)+T
ISI
(i)+T
I
(i)+T
N
(i)
(6)
with T
U
is the useful signal, T
ISI
is inter-symbol interference signal, T
I
is
theMUI andT
N
is thetermcorresponding to thenoise. Formoredetails see
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Proceedings, 5th Chaotic Modeling and Simulation International 
Conference, 12 – 15 June 2012, Athens Greece 
[6,7].
In [6], it hasbeen shown that,
T
I
(i)=E
w
N
u
X
j=1;j6=i
(j)
(2d
(j)
h
1)cn(i;j)
(7)
whereE
w
is theamplitudewhich controlsthetransmitted power, 
(j)
isthe
tap weight of the user j, d
(j)
h
is the binary sequence, cn(i;j) is the number
of collision between codes ec(i) and ec(j). ec(j) can becomputed bytaking into
account the developed Time-Hopping Codes (DTHC) corresponding to TH
codes as follows, fora given codeec(j), the DTHC is a binarycodeof length
N
c
N
f
and is dened by
c
(j)
r
=
1if r=ec
(j)
l
+lN
c
; 0lN
f
1
0otherwise:
(8)
where r= 0:::;N
c
N
f
1. The relation between TH sequence ec
(j)
l
and the
developed codec
(j)
r
isillustrated inFigure4.
Fig.4. Correspondencebetween DTHC and THC.
cn(i;j)=
N
f
N
c
1
X
l=0
c
(i)
l
c
(j)
l
(9)
TheAverageCollision NumberACN ofthesequenceset(ec
(j)
), j =1;:::N
u
is therefore denedby [6]:
ACN =
1
N
u
(N
u
1)
N
u
X
i=1
N
u
X
j=1;j6=i
cn(i;j)
(10)
In [6] we have dened the ACN criterion, and we have showed that the
experimentalresultsvalidatetherelevanceoftheACNasan ’o-line’perfor-
mance evaluation criterion for codes sequences. These results motivated us
to usethe ACNas a tool to predict the performance of codesequences.
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Proceedings, 5th Chaotic Modeling and Simulation International 
Conference, 12 – 15 June 2012, Athens Greece 
However, we found intuitively that this criterion may in some cases be un-
suitableforcodeselection [7]. Toremedytothisdrawback, wedeneda new
criterioncalledAverageofSquaredCollisionNumberASCNwhich isdened
as:
ASCN =
1
N
u
(N
u
1)
N
u
X
i=1
N
u
X
j=1;j6=i
cn
2
(i;j)
(11)
This is motivated by the observation that when the collisions are regrouped
on few positions the performance aresignicantly degraded.
3 Generation of sequence
Chaotic sequences have some properties that motivate researchers to use
them in various applications: determinism, long term unpredictability and
high sensitivityto initial conditions. Especially, chaoticsequencesgenerated
by onedimensional non linear transformation have been used in cryptogra-
phy, watermarking, spectrum spreading systems [9].
Webegin bydening chaoticand Goldsequences thatwill beconsidered
inthiswork. ThenweshowtheASCNforchaoticsequenceversustheirbifur-
cation parameter. Next, we analyse how chaoticity measured by Lyapunov
exponent is correlated withtheASCN.
3.1 Sequences generated by Skew tent map
Chaoticsequencesaregenerated bytheSkew tent map(STM)dened by:
x
n+1
=
x
n
r
;
0x
n
r
1 x
n
1 r
; r<x
n
1
(12)
TheSTM exhibitschaoticbehaviorfor everyvalueofthebifurcationparam-
eterr2 [01].
Figure 5, show the Lyapunov exponent and ASCN versus the bifurcation
parameter r for STM chaotic sequences. We can see that the curve of the
ASCNfollow theone ofLyapunov exponent and that thegreater the expo-
nent is the smaller theASCN. Forr=0:5, theSTM have a best ASCNand
Lyapuonvexponent. Weshowed numericallythat theASCN of a quantized
chaotic sequence depends on thechaoticity of these sequences measured by
theirLyapunov exponent.
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Proceedings, 5th Chaotic Modeling and Simulation International 
Conference, 12 – 15 June 2012, Athens Greece 
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
r
Lyapunov exponent
0
0.2
0.4
0.6
0.8
1
0.5
1
1.5
2
2.5
r
ASCN
Fig.5. Lyapunovexponent and ASCNfor skew tent map.
3.2 Gold sequences
TheGold sequencebasedTHcodesaregenerated asshownin [7], wherewe
illustratehowisgeneratedasequencetakingvaluesinf0; 1;;N
c
1 =7g
andwith a lengthN
f
29.
In Figure 6, we represent the ASCN versus user number for N
c
=8; for
Gold sequences considered here as a reference and STM sequence dened
above; theASCNofchaoticsequenceareaveragedover100 realizations. For
STMweconsidered thebifurcationparameterthatgivesthebestASCN, i.e.
r=0:5.
Theresults show that STM, havea better ASCN than Gold sequences. We
can notice likewise that Gold sequences show better performancecompared
to the chaoticsequence whenN
u
<6, thisisbecauseoftheorthogonalityof
this sequences.
4
6
8
10
12
14
16
18
20
10−3
10−2
10−1
100
101
Users number
Average of Squared Collision Number
Gold
Skew tent map
Fig.6. ASCNversus user number for STMand Gold sequences. N
c
=8.
4 BER performance analysis
In oursimulation, thesecond derivativeof the Gaussian pulseis selected as
UWB pulse in a residential environment CM1 channel with the correlation
receiver. Thesimulationparameters arelistedin Table1. Forsimplicity, we
Proceedings, 5th Chaotic Modeling and Simulation International 
Conference, 12 – 15 June 2012, Athens Greece 
Table 1. Simulations parameters of TH-UWB system
Simulationparameters
acronym
Value
Chipduration
T
c
1ns
Pulse duration
0.2ns
Numberofchip
N
c
8
Numberofpath
L
10
Numberofframe
N
f
4
Samplingfrequency
F
s
8GHz
Numberofsampling
N
e
50
Signal toNoiseRatio
SNR
10dB
Numberofbitsforeach user
Nb
10
5
Factorforspread spectrum Gold
N
31
assume thatthenumberof paths Lis the sameforall users.
ToenhancestheperformanceoftheTH-UWBsystemweproposeachaotic
modulation combinedwith chaoticcode. Theidea isto splitthestatespace
of STM and to associate a symbol to each partition. We suppose that the
state space is represented by the interval I = [0 1], and we dene the two
sub-intervals I
0
=[0 0:5 [ and I
1
=[0:5 1 ], where isthepulse du-
ration. According to this consideration weassociatethe transmitted binary
sequence0 and 1 tothesub-intervals I
0
and I
1
,respectively. Thisprocessof
modulation is called Chaotic Position PulseModulation (CPPM). Asimilar
schemewas designed in [9].
ThesimulationresultsareshowninFigure7,wherewepresentedtheBER
ofthesystemversustheusernumberforGold,STM+PPMandSTM+CPPM
sequences. We can see that STM combined with CPPM modulation allow
the best performance. This is due to the random eect of chaotic systems
whichreducetheinterferencebetweenusers. Moreover,Gold sequence allow
the worst performance, excepted when N
u
<10. These results compared to
theresultsshownin Figure6, provethattheASCN is a suitablecriterion to
select TH-codes.
ForSTM,wexedthebifurcationparametersrto0:5. Thisvaluecorrespond
to the minimal oftheASCN (the maximal ofLyapunovexponent).
8
10
12
14
16
18
20
10−3
10−2
10−1
Number Users
BER
Gold
STM
STM+CPPM
Fig.7. BERperformanceevaluation of TH-UWB systems: Chaotic vs. Gold.
Proceedings, 5th Chaotic Modeling and Simulation International 
Conference, 12 – 15 June 2012, Athens Greece 
We can clearly see that the THC generated by the STM can get bet-
terperformancethanTHCgenerated byGold sequence. Thus, theproposed
STMchaoticsystemconsideredisnotonlyadvantageousintermsofsynchro-
nization, butitcanalsogenerateTH-codesthatoutperformtheconventional
system.
5 Conclusion
In this paper, weproposed a chaoticpulse position modulation and chaotic
timehopping spectrum spreading in UWB system. It has been proved that
thechaoticsequencesareappropriateforUWBsystemswithmultipleaccess.
TheperformancecomparaisonofTH-PPM-UWBsystemshowedthatchaotic
sequences has better performance in term of bit error rate than Gold se-
quences. We specically showed that the higher the Lyapunov exponent is
the loweris theASCN, and subsequently the bettertheperformances.
In addition, chaotic systems enhances the quality and the security of the
transmission.
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Documents you may be interested
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