Non-Gaussian MAI Modeling to the Performanceof TH-BPSK/PPM UWB Communication Systems

Ehab M. ShaheenDepartment of Electronic Warfare

Military Technical CollegeCairo, Egypt

efareed2007@yahoo.com

Abstract—This paper investigates the impact of multiple ac-cess interference (MAI) on the performance of ultra-wideband(UWB) communication systems. Since the exact characteristicfunction for the MAI is not known and the Gaussian distribu-tion is not considered an accurate model for approximating theprobability density function (PDF) of the MAI in UWB systems.To this end; we approximate the PDF of the MAI by theLaplacian, Generalized Gaussian and the Symmetric α-Stabledistributions. It has been shown that a Gaussian PDF is not agood approximation even when there are a moderately largenumber of interferers in the UWB system and the GeneralizedGaussian distribution gives a much better approximation thanboth the Laplacian and Gaussian distributions. In this paper,it has been observed that the symmetric α-stable distributiongives the best approximation to the PDF of the MAI. Thebit error rate (BER) performance of the time hopping UWBcommunication systems in the presence of MAI using thesenon-Gaussian modeling approaches is numerically investigated,validated with simulation results, and compared with thecommon Gaussian approximation.

Keywords; Ultra wideband - Multiple Access Interfer-ence - Probability Density Function Approximation.

I. INTRODUCTION

Ultra-Wide Band (UWB) technology has risen dramati-cally, since the Federal Communication Commission (FCC)authorized UWB for the unlicensed use of short-distancecommunication in early 2002 [1]. It can be used in highspeed, low power and short range wireless communicationapplications. Its extremely larger bandwidth (range of 3.1-10.6 GHz) provides high multiple access capability androbustness to multi-path conditions.

In order to evaluate the performance of time hopping (TH)UWB communication systems, it is important to estimateits ability to work in multi user environment. Even if UWBcommunication systems are designed for indoor applications,it seems to be still important to guarantee the coexistence ofa large number of transmitters in the covered area.

Since it is not possible to find an exact closed-form ex-pression for the characteristic function and, correspondingly,the PDF of the MAI, throughout the literature the MAI hastypically been approximated by a Gaussian random variable,based on a Central Limit Theorem (C.L.T.) argument.

Time hopping pulse position modulation (TH-PPM) UWBsystem performance was previously analyzed in many liter-atures, using Gaussian approximation to statistically modelthe MAI. But the results in [2] show that the Gaussian

approximation is accurate only at very low signal to noiseratio (SNR) values; due to the domination of thermal noise atlow SNR values. In [3], it has been shown that a generalizedGaussian distribution approximation (GGA) can lead tosuperior receiver designs and performances compared to theperformance of the conventional matched filter receiver forthe TH-PPM UWB system [4].

The main contribution of this paper is to evaluate an exactor close-to-exact BER performance of two UWB communi-cation systems; TH-PPM and TH-binary phase shift keying(BPSK) under the impact of MAI. The PDF modeling ofthe MAI is empirically approximated by the symmetric α-stable distribution; where it will be proved that it gives abetter approximation to the PDF of the MAI than all otherdistributions.

The paper is organized as follows. The UWB systemmodel is presented in section II. Section III presents ananalysis to the receiver decision statistic. The PDF of theMAI for practical UWB pulses is investigated in sectionIV. Section V presents a comparison between the BERperformances of the two TH-UWB communication systemsusing the non-Gaussian approximations and the commonGaussian approximation. Numerical and simulation resultsare depicted in section VI. Finally, section VII draws theconclusions.

II. SYSTEM MODEL

For a matched filter reception, the kth transmitted UWBuser can be written in the form of a TH-PPM as

SkPPM (t) =√Eb

∞∑j=−∞

p(t− jTf − ckjTc − δdkbj/Nsc) (1)

and in the form of TH-BPSK as

SkBPSK(t) =√Eb

∞∑j=−∞

dkbj/Nsc · p(t− jTf − ckjTc) (2)

where p(t) is the shape of the transmitted pulse with pulsewidth Tm, dj , is the transmitted jth binary data bit andcomposed of equally likely bits, and Eb is the bit energy.Ns is the number of pulses transmitted per bit, Tc is thehop width and ckj is the TH code of the kth user, ckj ∈{0, 1, . . . , Nh − 1}, such that an additional time shift of“cjTc” is introduced when the jth pulse is transmitted. Tf is

978-1-4577-1379-8/12/$26.00 ©2012 IEEE 916

the frame duration, satisfying Tf = NhTc, the bit durationcan be represented as Tb = NsTf , and δ is the modulationindex with binary PPM (the time shift added to a pulse withan optimal value of 20% of a pulse width).

III. ANALYSIS OF RECEIVER DECISION STATISTIC

The received UWB signal in the presence of Nu activeasynchronous users transmitting on an additive white Gaus-sian noise (AWGN) channel can be written as

r(t) =

Nu∑k=1

ak · Sk(t− τk) + n(t) (3)

where Sk(t) can be the TH-PPM or TH-BPSK signal,ak and τk represent the attenuation and the correspond-ing asynchronous delay of the kth user of the channel,where τk is uniformly distributed on [0,Tb]. n(t) is additivewhite Gaussian noise with two-sided power spectral density“No/2”.

A correlator is used to detect a single desired user at thereceiver. Without loss of generality, we will further assumethat the desired user is the first user and that c1j = 0 for allvalues of j. The decision statistic of a correlation receivercan be written as

Z =

Ns−1∑j=0

∫ (j+1)Tf

jTf

r(t) · v(t− τ1 − jTf )dt (4)

where v(t) is the template waveform, which for a TH-PPMsignal has the form

v(t) = p(t)− p(t− δ) (5)

Define the correlation of the template, v(t) with a time-shifted pulse, p(t) as

R̆(τ) =

∫ ∞−∞

p(t− τ) · v(t)dt (6)

and from equations (5) and (6), we obtain

R̆(τ) = R(τ)−R(τ − δ) (7)

where R(τ) is the UWB pulse auto-correlation function, then

Z = S + I +N (8)

N is a Gaussian random variable (RV) with zero mean andvariance, σ2

N = NoR̆(0), S = ±a1√EbR̆(0) is the desired

signal component.Note that, R̆(0) = (1 − %), where % is the correlation

coefficient between the two pulses p(t) and p(t−δ), for bits0 and 1 respectively, % can be written as

% =

∫ ∞−∞

p(t)p(t− δ)dt, % ∈ [−1, 1] (9)

The value of % ranges between [-1,1], where % = -1corresponds to the antipodal modulation, % = 0 correspondsto the orthogonal modulation, and % ∈ (0,1], % ∈ [-1,0)corresponds to modulation schemes that are inferior andsuperior to orthogonal modulation.

I is the total MAI due to all Nu − 1 interfering signals,which can be written as

I =√Eb

Nu∑k=2

Ns−1∑j=0

∫ (j+1)Tf

jTf

akSk(t−τk)·v(t−τ1−jTf )dt

(10)We model the difference of time shifts for user asynchronismas [5]

τk − τ1 = jkTf + βk (11)

in which jk is an integer value, and βk is the fractional partwhich is modeled by a RV uniformly distributed on [-Tf /2,Tf /2]. Then for TH-PPM, I can be rewritten as

I =√Eb

Nu∑k=2

Ns−1∑j=0

∫ (j+1)Tf

jTf

akp(x− βk − ckjTc−

δdkbj/Nsc) · v(x)dx(12)

which can be rewritten as [5]

I =√Eb

Nu∑k=2

ak

[ λk−1∑j=0

R̆(γk0,j) +

Ns−1∑j=λk

R̆(γk1,j)

](13)

whereγk0,j = βk + ckjTc + dkoδ (14)

γk1,j = βk + ckjTc + dk1δ (15)

where dko and dk1 represent the two adjacent bits of the kth

signal that overlap with the transmission time of d1o, and λkis uniform over [0,Ns-1], according to the given definitionof jk.

Similarly, for a TH-BPSK, I can be easily derivedas

I =√Eb

Nu∑m=2,m 6=k

Ns−1∑j=0

∫ (j+1)Tf

jTf

amdb (j−jk)

Nsc·

p(t− τk − jTf ) · p(t− τm − (j − jk)Tf − cm(j−jk)Tc)dt(16)

Assuming Tf >> Tm, which means that the durationof UWB pulses is effectively limited to one frame, thenequation (16) becomes

I =√Eb

Nu∑m=2,m 6=k

Ns−1∑j=0

amdb (j−jk)

Nsc ·R(βm − cm(j−jk)Tc)

(17)Without loss of generality, we can shift the time axis an

assume that the UWB pulse shape has non-zero values inthe range [-Tm/2, Tm/2], hence

I =√Eb

Nu−1∑k=2

Ns−1∑j=0

akbkR(γk) (18)

where bk is a binary RV which takes values -1, +1 withequal probability and γk is a RV uniformly distributed on[−Tf/2, Tf/2].

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IV. MAI PDF FOR PRACTICAL UWB PULSES

In this section, the six derivative Gaussian pulse is em-ployed, this pulse shape complies with the FCC emissionlimits and maximizes the available bandwidth provided toUWB systems. It can be written as

p(t) =

√640

231Nsτp

[1− 12π

(t

τp

)2

+ 16π2

(t

τp

)4

−

64

15π3

(t

τp

)6]

exp

[−2π

(t

τp

)2]

(19)where τp is the pulse shaping factor.

Since this pulse is even; its auto-correlation function can

be easily obtained as R(τ) = F.T−1{P (f)2

}, where F.T.

is the Fourier transform process and P (f) is the F.T. of theUWB pulse, p(t), which can be written as

P (f) =8π3

3√

1155Nsτ13/2p f6e−

π2 f

2τ2p (20)

UWB devices are envisioned to be deployed for indoor UWBapplications, thus it was reported in [8] that UWB devicesare subjected to impulsive (non-Gaussian) noise. To thisend, heavy tailed distributions that have a positive excesskurtosis are desirable. The PDF of the MAI can be computednumerically based on the inverse of R(τ ). The PDF of MAIfor 20 interferers has been found by means of simulationand is shown in figure (1).

Fig. 1. The normalized PDF of the MAI for the practical six derivativeGaussian TH-UWB pulse, for 20 interferers. The MAI model is approxi-mated by Gaussian, the Laplacian, the GGA, and the symmetric α-stableapproximations.

Assuming that the transmission powers of all the usersare equal. It is observed from figure (1) that the GaussianPDF is not a good approximation even when there is amoderately large number of interferers. Note that, the GGAmodel approximates the PDF of the MAI much better thanthe Laplacian and the Gaussian model, yet, the symmetricα-stable distribution is the best approximation to the PDFof the MAI.

V. BER PERFORMANCE ANALYSIS

In this section the BER performance of the TH-BPSK/PPM UWB communication system is analyzed. Start-ing from (8), the bit error probability can be written as

Pe = P{S + I +N < 0|d0 = 0

}= FY (−S) (21)

where FY (−S) is the cumulative distribution function of Y,Y = I + N.

Using the fact that the interferers and noise are indepen-dent, the CF of Y can be written as

φY (ω) = E{eiωY } = φI(ω) · φN (ω) (22)

The CF of the noise can be written as

φN (ω) = exp

(− σ2

Nω2

2

)(23)

Using the inversion theorem [7] and by evaluating φY (ω),a formula for the error probability can be obtained directly.By observing that Y is an even RV, the bit error probabilitycan be written as

Pe =1

2− 1

π

∫ ∞0

sin(ω)

ω· φI(ω

S

)· φN

(ω

S

)dω (24)

Now, in order to evaluate such BER probability, we willdevelop the CF of the MAI for four different distributions.

A. Gaussian Approximation

Assuming the C.L.T. is valid in this case, the MAI term Ican be approximated by a Gaussian distributed RV, and theCF of the RV I can be expressed as

φGAI (ω) = exp

(− σ2

Iω2

2

)(25)

where σ2I is the variance of the MAI.

For a perfect power control scenario, the signal to inter-ference ratio (SIR), can be written as [6]

SIR =(1− %)2 · γR

σ2M ·Rb · (Nu − 1)

(26)

where γR =NsTfTb

, γR ≤ 1, and σ2M is given by [6]

σ2M =

∫ Tf

0

(∫ 2Tm

0

p(t− τ) · v(t)dt

)2

dτ (27)

Note that, SNR =EbNo· R̆(0).

B. Laplacian and GGA

The Laplace distribution, which is usually adopted tomodel impulsive noises, is also used here to approximatethe MAI term I. In this case, the approximate PDF of I canbe expressed as f lapI (x) = 1

2q e−|x|/q , and the corresponding

CF can be written as

φlapI (ω) =1

1 + q2ω2(28)

Note that σ2I = 2q2.

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As presented in [4], the GGA gives much better approx-imation to the PDF of the MAI than the SGA and theLaplacian approximation.

The pdf of the GGA is [9]

fGGA(x;Sm, σ, p) =exp

(−∣∣ x−SmA(p,σ)

∣∣p)2Γ(1 + 1/p)A(p, σ)

(29)

where the function A(p, σ) =

[σ2Γ(1/p)

Γ(3/p)

] 12

is a scaling

factor. Γ(·) is the Gamma function, and p is the shapeparameter. Sm is the mean of the RV.

C. α-Stable Approximations

Here we are proposing the symmetric α-stable distribution[10], where it gives the best approximation to the PDF ofthe MAI term, as presented in figure (1). Note that, anadaptation of the empirical PDF of the MAI as determinedby simulation is considered in order to construct an α-stablemodel for the MAI. The rationale for this adaptation is thatany estimate of the actual PDF of the MAI by simulation isan estimate of a locally averaged version of the actual PDF.

We can write E{fsimI (r)} = P actI (r−ε < I < r+ε)/2ε,where fsimI (r) is the pdf estimate by simulation and 2ε isthe length of a small segment in the r axis as depicted infigure (1). Thus, an accurate estimate of actual MAI pdf isobtained by simulation with a smaller value of ε.

The CF of the α-stable model can be expressed as

φasaI (ω;α, β, µ) = exp

[− β|ω|α + jωµ

], 0 ≤ α ≤ 2 (30)

where µ, β and α are the location parameter of the distribu-tion, the shaping parameter, and the characteristic exponentwhich determines the heaviness of the tail of the PDFrespectively.

Note that the Gaussian distribution is a special case ofthe alpha-stable distribution with α = 2 and a smaller valueof α represents an impulsive distribution. As far as ourknowledge, no optimal estimation algorithms are knownfor signals immersed in alpha-stable noise. However in[11], a sub-optimal estimation method which estimates thelocation parameter of an alpha-stable process is evaluated.To implement the decision rule of the detector, the value ofK2 is calculated according to

K2 = β2α

(α

2− α

)+ Ceσ

2n. (31)

where Ceσ2n accounts for the AWGN.

In order to estimate the parameters α and β, the empiricalcharacteristic function of the MAI from a single frame (Ii)is calculated over nω equally spaced sample points of ω.The CF of (Ii) can be written as

φIi(ω) ' exp(−β|ω|α) (32)

In order to estimate α and β, the natural logarithms will betaken twice on both sides to obtain

ln[− ln{φIi(ω)}] ' ln(β) + α ln(ω), ω > 0 (33)

if the left-side of the equation equals x1 +x2 ln(ω), then theestimates for β = ex1 , and α = x2.

VI. NUMERICAL AND SIMULATION RESULTS

In this section, numerical examples are presented to inves-tigate the performance of both TH-PPM/BPSK systems andvalidated with the aid of simulation. The number of activeasynchronous users Nu = 10 and 20 users, Ce is set at 40at Nu=10, and 80 at Nu = 20. A six derivative Gaussianreceived pulse will be used with values: τp = 0.192 ns, δ= 0.068 ns (for a TH-PPM system), Tf = 10 ns, and % =-0.824.

Figures (2) and (3) evaluate the BER performance of TH-PPM UWB system in the presence of MAI for differentnumber of active users Nu = 10 and 20 respectively. It can beseen that, the Gaussian approximation is in a good agreementwith the simulated MAI results only for small SNR values,say SNR < 5dB. Furthermore, the Gaussian approximationfails to predict the true BER of the TH-PPM UWB systemfor large SNR values. Thus, the Gaussian approximation isnot a reliable approximation. Also, it can be observed thatthe Laplacian approximation gives a better estimate than theGaussian approximation but still not accurately enough topredict the true error-rate floor of the TH-PPM UWB system.

It can be observed that the GGA approximation is in agood agrement with the simulated MAI BER results, yet,the symmetric α-stable is the best estimate to the PDF ofthe MAI.

Fig. 2. TH-PPM UWB performance in the presence of MAI, Nu=10.

Figures (4) and (5) evaluate the BER performance of TH-BPSK UWB system in the presence of MAI for differentnumber of active users Nu = 10 and 20 respectively. It canbe seen that, good estimated BER results can be achieved bythe GGA, yet the symmetric α-stable approximation givesthe best approximation to the PDF of the MAI.

VII. CONCLUSION

In this paper, the impact of MAI on the performance of aTH-PPM/BPSK UWB system is investigated and validated

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Fig. 3. TH-PPM UWB performance in the presence of MAI, Nu=20.

Fig. 4. TH-BPSK UWB performance in the presence of MAI, Nu=10.

Fig. 5. TH-BPSK UWB performance in the presence of MAI, Nu=20.

with the aid of simulation. It has been found that the Gaus-sian approximation underestimates the accurate BER of bothTH-BPSK/PPM systems for medium and large SNR valueseven when there is a moderately large number of interferersin the UWB system. Also, the Laplacian approximationunderestimates the accurate BER of both systems for largeSNR values and the GGA gives much better approxima-tion than the Gaussian and the Laplacian approximations.However, the symmetric α-stable approximation empiricallygives the best estimate for the BER performance for bothsystems.

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