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IEEE TRANSACTIONS ON SIGNAL PROCESSING 1

Co-Channel Interference Modeling and Analysis ina Poisson Field of Interferers in Wireless

CommunicationsXueshi Yang and Athina P. Petropulu

Abstract— The paper considers interference in a wireless com-munication network, caused by users that share the same propaga-tion medium. Under the assumption that the interfering users arespatially Poisson distributed, and under a power-law propagationloss function, it has been shown in the past that the interference in-stantaneous amplitude at the receiver is α-stable distributed. Pastwork has not considered the second-order statistics of the interfer-ence, and has relied on the assumption that interference samplesare independent. In this paper, we provide analytic expressionsfor the interference second-order statistics and show that, depend-ing on the properties of the users’ holding times, the interferencecan be correlated. We provide conditions under which the inter-ference becomes m-dependent, φ-mixing or long-range dependent.Finally, we present some implications of our theoretical findings onsignal detection.

Index Terms—Interference modeling, wireless communications,signal detection, non-Gaussian, alpha-stable distributions, long-range dependence.

I. INTRODUCTION

IN WIRELESS communication networks, signal reception isoften corrupted by interference from other sources, or users,

that share the same propagation medium. Knowledge of thestatistics of interference is important in achieving optimum sig-nal detection and estimation.

Existing models for interference can be divided into twogroups: empirical models and statistical-physical models. Em-pirical models, e.g. the hyperbolic distribution and the K-distribution [1], fit a mathematical model to the practically mea-sured data, regardless of their physical generation mechanism.On the other hand, statistical-physical models are groundedupon the physical noise generation process. Such models in-clude the Class A noise, proposed by Middleton [2], and the α-stable model initially proposed by Furutsu and Ishida [3], andlater advanced by Giordano [4], Sousa [5], Nikias [6], Ilow [7]et al. A common feature in these interference models is the rateof decay of their density function, which is much slower thanthat of the Gaussian. Such noise is often referred to as impul-sive noise.

Xueshi Yang was with ECE Department, Drexel University, Philadelphia,PA 19104. He is now with Department of Electrical Engineering, PrincetonUniversity, Princeton, NJ 08544. (e-mail: [email protected])

Athina P. Petropulu is with ECE Department, Drexel University, Philadel-phia, PA 19104. (e-mail: [email protected])

Part of this work was presented on IEEE - EURASIP Workshop on NonlinearSignal and Image Processing, Baltimore, Maryland, June, 2001 and 11th IEEEWorkshop on Statistical Signal Processing, Orchid Country Club, Singapore,August, 2001.

Impulsive noise has been observed in several indoor (cf. [8],[9]) and outdoor [2], [10] wireless communication environ-ments. In [11], measurements of interference in mobile com-munication channel suggest that in some frequency ranges, im-pulsive noise dominates over thermal noise. Impulsive noiseattains large values (outliers) more frequently than Gaussiannoise. Such noise behavior has significant consequences in op-timum receiver design [6], [12]. Moreover, as optimum signaldetection relies on complete knowledge of the noise instanta-neous and second-order statistics [13], it is important to studythe spatial and/or temporal dependence structure of the noise aswell as its instantaneous statistics. In the last decade, many ef-forts have been devoted in this direction (see for example [14],[15], [16], [17]). In [15], the authors consider a physical sta-tistical noise model originated from antenna observations thatare spatially dependent. It is shown [15] that the resulted inter-ference can be characterized by a correlated multivariate ClassA noise model. For mathematical simplicity, the temporal de-pendence structure has been traditionally modeled by movingaverage (MA) [18], or Markov models [19]. However, it is notclear whether these models have assumed a temporal depen-dence structure that is consistent with the physical generationmechanism of the noise. Thus, use of these models in receiverdesign may lead to schemes that function poorly in practice.

In this paper, we consider statistical-physical modeling forco-channel interference. In particular, we are interested in thetemporal dependence structure of the interference. We adoptand extend the statistical-physical model investigated in [5],[6], [7]. The model considers a receiver, surrounded by in-terfering sources. The receiver picks up the superposition ofall the pulses that originate from the interferers, after they haveexperienced power loss that is a power-law function of the dis-tance traveled. We assume a communication network with ba-sic waveform period T (time slot). We here focus on the in-terference sampled at rate 1/T . As assumed in [5], [6], [7],at any time slot, the set of interferers forms a Poisson field inspace. Assuming that from slot to slot these sets of interfererscorrespond to independent point processes, it was shown in [5],[6], [7] that the sampled interference constitutes an independentidentically distributed (i.i.d.) α-stable process. However, the in-dependence assumption is often violated in a practical system.To see why this would be the case, consider an interferer whostarts interfering at some slot n, and remains active for a ran-dom number of slots. That interferer will still be active at timeslots n + i, i ∈ N

+, with certain probability, and will be one ofthe interferers at n + i, i ∈ N

+. A direct consequence of this is

2 IEEE TRANSACTIONS ON SIGNAL PROCESSING

that, location-wise, the interferers at time slots n and n + i arenot independent of each other. Independence can only be validif the interferer remains active for at most one time slot.

In this paper, we assume that the interferer’s holding time,or session life, is a random variable with some known distribu-tion. We obtain the first- and second-order characteristic func-tion of the sampled interference and show that, under certainassumptions, the interference becomes jointly α-stable. In thelatter case, we give conditions for the interference to be m-dependent, φ-mixing or long-range dependent. In particular,if the session life is heavy-tail distributed, the interference con-stitutes a long-range dependent process in a generalized sense(defined in Sec.II-C). It should be noted that, in this paper, weonly consider modeling of co-channel interference, which doesnot include thermal noise which is also present in a practicalcommunication systems.

This paper is organized as follows. Section II provides therelevant mathematical background. Section III details the in-terference model. The statistics of the interference are derivedin Section IV, while the dependence structure of the interfer-ence is studied in Section V. Numerical simulations are pre-sented in Section VI. Section VII discusses some implicationsof long-range dependence in signal detection. Finally, SectionVIII contains some concluding remarks.

II. MATHEMATICAL BACKGROUND

A. Heavy-tail Distributions and α-Stable Distributions

A random variable X is regularly-varying with index α, if

P (|X| ≥ x) ∼ L(x)

xαas x → ∞. (1)

Here, L(x) is slowly varying function, i.e., for all positive x,limτ→∞ L(τx)/L(τ) = 1 (typically such slowly varying func-tions are constant functions or ratios of two polynomials withidentical degree). The variable X is said to be heavy-tailed withinfinite variance if it is regularly varying with index 0 < α < 2.In those cases, the variance of X is infinite (if α < 1 the meanand moments of order greater than or equal to α are infinite).

A particular class of heavy-tail distributions with infinitevariance is the α-stable distribution. The α-stable distributionis a generalization of the Gaussian distribution. It is often clas-sified as non-Gaussian, although it reduces to the Gaussian casewhen α = 2 (to be defined). The difference between α-stableand Gaussian densities is that the tails of the former are heavierthan those of the latter. Due to lack of closed form expressionfor their probability density functions, α-stable distributions aremore conveniently characterized by their characteristic func-tions.

A vector X = (X1, X2, ..., Xd)T is α-stable in R

d, if andonly if there exists a finite measure Γ on the unit sphere Sd ofR

d and a vector u in Rd, such that the characteristic function,

Φ(w) = E exp{jwTX} is given by [20]:

Φ(w) = exp{−∫

Sd

|(w, s)|α(1 − jsign((w, s)) tan

πα

2

)

·Γ(ds) + j(w,u)}, (2)

for 0 < α ≤ 2, α 6= 1; If α = 1, tan πα2 is replaced by

− 2π ln |(w, s)|. The parameter α is the characteristic exponent.In the case when d = 1, S1 consists of two points {−1}

and {1}, and the spectral measure Γ is concentrated on them.Correspondingly, the distribution becomes a univariate α-stabledistribution, and for α 6= 1, its characteristic function equals:

Φ(ω) = exp {−|ω|α [(Γ({1}) + Γ({−1}))−jsign(ω)(Γ({1}) − Γ({−1})) tan

πα

2

]+ jµω

}

4= exp

{−σα|ω|α(1 − jβsign(ω) tan

πα

2) + jµω

},

where σ is the scale parameter, β is the skewness parameter,and µ is the location parameter. In short, such a univariate α-stable random variable will be denoted by X ∼ Sα(σ, β, µ).If β = 0, the distribution is symmetric about µ, and is termedsymmetric α-stable, or simply SαS.

B. Codifference

α-stable distributions are known for their lack of momentsof order greater than or equal to α. In particular, for α < 2,the second-order statistics do not exist. In such case, the role ofcovariance is played by the covariation or the codifference [20].

The codifference of two jointly SαS, 0 < α ≤ 2, randomvariables x1 and x2 equals:

Rx1,x2= σα

x1+ σα

x2− σα

x1−x2(3)

where σx is the scale parameter of the SαS variable x.A quantity that is closely related to the codifference

Rx(t+τ),x(t) is [20]:

I(ρ1, ρ2; τ) = − ln E{ej(ρ1x(t+τ)+ρ2x(t))}+ ln E{ejρ1x(t+τ)} + ln E{ejρ2x(t)}.(4)

This quantity is referred to as generalized codifference [21]. Itreduces to the codifference for the case of jointly SαS pro-cesses, i.e.

Rx(t+τ),x(t) = −I(1,−1; τ). (5)

I(ρ1, ρ2; τ) is defined for any stationary heavy-tailed randomprocess.

C. Long-Range Dependence

A second-order process x(t) is called a (wide-sense) station-ary with long memory, or long-range dependence, if its auto-correlation function, ρ(τ), is finite and satisfies [22]:

limτ→∞

ρ(τ)/τβ−1 = c (6)

for some positive constant c and β ∈ (0, 1). From (6), it canbe seen that a long-memory process is characterized by an au-tocorrelation that decays hyperbolically, as the lag τ increases.This is in contrast with the exponential decay correspondingto short memory processes, e.g. auto-regressive moving aver-age(ARMA) processes.

The following generalization of the concept of long memoryprocess can be useful for processes which lack autocorrelation.

YANG AND PETROPULU: CO-CHANNEL INTERFERENCE MODELING AND ANALYSIS 3

Definition 1: [21], [23] Let x(t) be a stationary process. Wesay that x(t) is a long-memory process in a generalized sense,if I(1,−1; τ), as defined in (4), satisfies

limτ→∞

−I(1,−1; τ)/τβ−1 = c (7)

where c is some real positive constant and 0 < β < 1.

III. THE INTERFERENCE MODEL

In the sequel, we present the interference model, which is anextension to the model described in [5], [6], [7].

Consider a wireless communication scenario without powercontrol, where a receiver receives the signal of interest in thepresence of other interfering signals. For the sake of simplicity,we will assume that the users, which are the potential interfer-ers, and the receiver are all on the same plane and concentratedin a disk Rb of radius b. The modeling is performed first for afinite b, and at the final step, the limit for b → ∞ is taken. Thethree-dimensional scenario is a straightforward extension of thetwo-dimensional one. The receiver is placed at the origin of thecoordinate system, and the users are distributed within the diskaccording to a two dimensional Poisson point process.

Let us define the term emerging interferers at time intervalm, to describe the interfering sources whose contribution ar-rives for the first time at the receiver in the beginning of timeinterval m. The interferers that emerged at any time intervalare located according to a Poisson point process in the space(Poisson field) with density λ. 1 It is of course reasonable toassume that the interferers that emerged at two different timeslots are independent, or more precisely, correspond two inde-pendent Poisson point processes.

One issue that the model of [5], [6], [7] does not take into ac-count is that, at time n, in addition to interferers that emerge atn, there could be interferers that emerged at some slot m < n,and still stay active at n. It is understood that the latter groupwould exist if the holding times of the users were longer thanone time slot T . The combination of these two groups wouldmake the interferers at slots m and n dependent (location-wise).We will assume that a user, once started transmission, continu-ously emits pulses for a duration of L time slots, where L is arandom variable with known distribution.

At time n, the signal transmitted from the i-th interferinguser, i.e., pi(t) propagates through the transmission mediumand the receiver filters, and as a result gets attenuated and dis-torted. For simplicity, we assume that distortion and attenu-ation can be separated. Let us first consider only the filter-ing effect. For short time intervals, the propagation channeland the receiver can be represented by a time-invariant filterwith impulse response h(t). Due to filtering only, the contribu-tion of the i-th interfering source at the receiver is of the formxi(t) = pi(t) ∗h(t), where the asterisk denotes convolution. Inwireless communications, the power attenuation increases log-arithmically with the distance ri between the transmitter and

1λ may be a function of time and the locations of the unit area/volume, whichforms a non-homogeneous Poisson point process. A non-homogeneous Poissonprocess can be mapped to a homogeneous one through transformations, cf. [5],[7]. In this paper, we only consider the homogeneous case, i.e. λ is a constant.

the receiver (cf. [2]). The power loss function can be expressedin terms of signal amplitude loss function a(ri), i.e.,

a(ri) =1

rγ/2i

, (8)

where γ is the path loss exponent; γ is a function of the antennaheight and the signal propagation environment. It may varyfrom slightly less than 2, for hallways within buildings, to largerthan 5, in dense urban environments and hard partitioned officebuildings ([11]). Thus, the total signal at the receiver is:

x(t) = s(t) +∑

i∈N

a(ri)xi(t), (9)

where s(t) is the signal of interest, and N denotes the set ofinterferers at time t. Note that the transmitting power of user ihas been implicitly incorporated into xi(t).

The receiver consists of a signal demodulator followed by thedetector. A correlation demodulator decomposes the receivedsignal into an K-dimensional vector. The signal is expandedinto a series of orthonormal basis functions {gk(t), 0 < t ≤T, k = 1, ...,K}. Let Zk(n) be the projection of x(t) ontogk(·) at time slot n, i.e.,

Zk(n) =

∫ T

0

x((n − 1)T + t)gk(t)dt. (10)

It holds

Zk(n) = Sk(n) +∑

i∈N

a(ri)Xi,k(n) (11)

4= Sk(n) + Yk(n) (12)

where Xi,k(n) and Sk(n) are, respectively, the result of thecorrelations of xi(t) and s(t) with the basis functions gk(·),and Yk(n) represents interference.

The Xi,k(n)’s are assumed to be spatially independent (e.g.,Xi,k(n) is independent of Xj,k(n) for i 6= j). We shall fo-cus on the statistics of Yk(·) for a particular dimension k. Fornotational convenience, we drop the subscript k, thus denotingYk(n) and Xi,k(n) by Y (n) and Xi(n) respectively. At time n,Y (n) is the sum of a random number of i.i.d. random variables,which are contributions of active interferers.

IV. STATISTICAL ANALYSIS OF THE RESULTEDINTERFERENCE

A. Instantaneous Statistics

Now let us consider the instantaneous interference at time n.In order to calculate the instantaneous statistics, the number andlocations of the active interferers at any given time interval needto be specified.

Proposition 1: Assume that the mean of the interferers’ ses-sion life is finite, denoted by µ, and the density of the emerginginterferers at each time slot is λ. As the sampling time n tendsto infinity, the active interferers becomes Poisson distributed inthe space, with asymptotic density λµ.

Proof: See Appendix A.


For convenience, we will set the time origin for the systemconsidered to be −∞ from now on. Therefore, at any time n,the set of active interferers form a Poisson field in the spacewith density λµ. Now we can use related results in [5], [7],where, by imposing the condition that Xi(n) is symmetricallydistributed, it is shown the resulted interference is SαS dis-tributed (cf. Eqs. (13)-(15) in [5]). Hence, we can conclude thefollowing Theorem.

Theorem 1: The instantaneous interference, observed at anygiven time interval at the receiver, is SαS distributed with char-acteristic exponent α = 4/γ, and scale parameter

σα = −λµπ

∫ ∞

0

x−αdΨ(x) (13)

were Ψ(x) is the common characteristic function of{Xi(·), i = 1, 2, ...}.

B. Joint Statistics

We next study the joint statistics of interference samples ob-tained at different time slots m and n. We assume n − m =τ > 0.

Let us denote the interference at m and n by Y (m) and Y (n)respectively. It holds:

Y (m) =∑

i∈Nm

a(ri)Xi(m), (14)

Y (n) =∑

i∈Nn

a(ri)Xi(n), (15)

where Nm and Nn represent the set of interferers that are ac-tive at time m and n, respectively. The following propositionprovides the joint characteristic function of Y (m) and Y (n),

Φm,n(ω1, ω2)4= E {exp[jω1Y (m) + jω2Y (n)]}.

Proposition 2: Let the session life L have finite mean andwe denote by FL(k) its survival distribution function. Then,the joint characteristic function of Y (m) and Y (n) (n − m =τ > 0) equals

Φm,n(ω1, ω2) = exp {−σαH1(τ) (|ω1|α + |ω2|α)

+H2(τ)Θm,n(ω1, ω2)} , (16)

where

α = 4/γ, (17)

σ =

(−λπ

∫ ∞

0

x−αdΨ(x)

)1/α

, (18)

H1(τ) =

n−m∑

l=1

FL(l), (19)

H2(τ) =∞∑

l=n−m+1

FL(l), (20)

Θm,n(ω1, ω2) = limb→∞

λπb2[

∫ b

0

Ψm,n(a(r)ω1, a(r)ω2)

·2rb2

dr − 1]. (21)

Here, Ψ(x) is the characteristic function of Xi(·), andΨm,n(ω1, ω2) is the second-order characteristic function ofXi(·), i.e.,

Ψm,n(ω1, ω2) = E[ejω1Xi(m)+jω2Xi(n)]. (22)Proof: see Appendix B.

C. Remarks on Proposition 2

1) Set ω2 = 0, we obtain the first order characteristic func-tion of the interference process, i.e.,

Φ(ω1) = e−σα∑

∞l=1

FL(l)|ω1|α

. (23)

Recognizing that∑∞

l=1 FL(l) = µ, which is the mean ofthe session life, we get a result consistent with Theorem1.

2) In the special case

FL(l) =

{1 l = 10 l > 1

(24)

we have H2(τ) = 0 for τ = 1, 2..., and H1(τ) = 1.Hence,

ln Φm,n(ω1, ω2) = −σα (|ω1|α + |ω2|α) . (25)

Equation (25) implies that the interference samples ob-tained at different time slots are independent and jointlyα-stable distributed. Indeed, this is consistent with thefindings in [5] and [7].

3) If H2(τ) tends to zero as τ tends to infinity, and H1(τ)approaches the mean of session life, µ. The joint charac-teristic function may be simplified as

limτ→∞

Φm,n(ω1, ω2) = e−σαµ(|ω1|α+|ω2|

α). (26)

(26) implies that when the distance between two samplesbecomes asymptotically large, they are becoming inde-pendently α-stable distributed (see also Sec.V-B).

4) The loss function in (8) is a far field approximation. As rapproaches zero, the power of the signal becomes infinite.Hence, it is conventional to use a truncated form

a(r) = min(r−γ/2, r0), (27)

for some constant r0 > 0. However, such truncationleads to an intractable analytical solution. In many cases,for typical values of r0, (8) may serve as a good approxi-mation [5].

D. Special Cases – Jointly α-Stable Interference

It is interesting to note that, although the sampled interfer-ence is marginally α-stable, in general, it is not jointly α-stable.The latter can be verified by checking the characteristic func-tion of (16) for general distributed Xi(m). However, for somespecial cases of Xi(m), the interference does become jointlyα−stable. We next consider two such cases.

1) Xi(m) does not vary with m.


Such a case would arise if the i−th interferer remained atsome constant level, Bi, throughout its session life. Thismay be a reasonable assumption for such interferers asdischarging particles, or automobile ignition.Let Bi be uniformly distributed between [−1/2, 1/2]. Asbefore, let us also assume that the i-th interferer emergesat Γi, and its session life is Li. The contribution from thisinterferer can be written as

Xi(m) = Bi1[Γi≤m<Γi+Li], (28)

where m is integer number, and 1[·] is the indicator func-tion. Then, it can be shown (see Appendix C) that

ln Φm,n(ω1, ω2)

= −σα[H1(τ)|ω1|α + H2(τ)|ω1 + ω2|α+H1(τ)|ω2|α], (29)

where α, H1(τ), H2(τ) are defined as (17), (19), (20),and

σ =

(−λπ

∫ ∞

0

x−αdΨB(x)

)1/α

(30)

where ΨB(x) denotes the characteristic function of Bi.Equation (29) implies that the interference is jointly α-stable (see Appendix C).

2) Bernoulli distributed Xi(m)In wireless communications, particularly in spread spec-trum networks, most of the interference is due to co-channel users who are transmitting data sequences dur-ing their session life. A more realistic model would beto assume that the symbols are either 1 or -1 with equalprobability, and independent from slot to slot.In that case, the contribution from interferer i is

Xi(m) = B(m)1[Γi≤m<Γi+Li], (31)

where B(m) is i.i.d. Bernoulli distributed for differentm, taking 1 or -1 with equal probability of 1/2. Then, itcan be shown that (see Appendix D)

ln Φm,n(ω1, ω2)

= −σαH1(τ)(|ω1|α + |ω2|α)

−σα H2(τ)

2(|ω1 + ω2|α + |ω1 − ω2|α), (32)

where α, H1(τ), H2(τ) are defined as before, and

σ =1

2

(λπ3/2Γ(1 − α/2)

Γ(1/2 + α/2)

) 1α

. (33)

Again, (32) implies that the interference at m and n arejointly α-stable.

V. DEPENDENCE STRUCTURE OF CO-CHANNELINTERFERENCE

Eq. (16) implies that the dependence structure of the co-channel interference is determined by the session life distribu-tion P [L ≥ k], k = 1, 2.... In this section, we shall give con-ditions under which the resulted interference is m-dependent,

φ-mixing and long-range dependent respectively. For con-venience, we will focus on the case of Bernoulli distributedXi(m) (see Eq.(32)) for which the interference is jointly α-stable distributed, although it is not difficult to verify that the re-sults presented also hold for generally distributed Xi(m), pro-vided that {Xi(m)}∞m=−∞ has finite second-order statistics forall i such that (21) is well defined.

A. m-dependent

A stationary random sequence {Y (i), i = 1, 2, ...} is saidto be m-dependent if there is a nonnegative integer m such thatthe sequences {Y (i), i = 1, ..., a} and {Y (i), i = b, ...,∞} arestatistically independent for all integers b > a ≥ 1 satisfyingb−a > m. For the interference model on hand, it is straightfor-ward to show that if the maximum possible value of the sessionlife of users is m − 1, the interference becomes m-dependent.It lies in the fact that if P [L ≥ m] = 0, the interferers at timeinstances separated more than m are independent, so is the in-terference.

B. φ-mixing

A more general dependence structure of importance is φ-mixing. φ-mixing defines for a stationary sequence of randomvariables say {Y (i), i = 1, ...,∞} satisfying the following:For a ≤ b, define Mb

a = σ{Y (a), Y (a + 1), · · · , Y (b)}, theσ-algebra generated by the indicated random variables. Then{Y (i), i = 1, ...,∞} is φ-mixing if there exists a nonneg-ative sequence {φi}∞i=1 with φi → 0 such that for each k,1 ≤ k < ∞ and for each i ≥ 1, E1 ∈ Mk

1 , E2 ∈ M∞k+i,

together imply |P (E1 ∩ E2) − P (E1)P (E2)| ≤ φiP (E1). Ifwe also have |P (E1 ∩ E2) − P (E1)P (E2)| ≤ φiP (E2), wesay that the process is symmetrically φ-mixing. Intuitively, in aφ-mixing process, the distant future is virtually independent ofthe past and the present. If Y (i) is φ-mixing, then ejY (i) is alsoφ-mixing ([24], p.182). The following provides the necessarycondition for the interference to be φ-mixing.

Proposition 3: A necessary condition for {Y (i), i =1, ...,∞} to be φ-mixing is:

H2(τ) → 0 as τ → ∞, (34)

where H2(τ) was defined in (20).Proof: Assuming that Y (i) is φ-mixing, it implies ([24],

Lemma 1)

|E{ejY (m)ejY (n)} − E{ejY (m)}E{ejY (n)}| ≤ 2φ1/2n−m (35)

From (32), the left side of (35) is

|Φm,n(1, 1) − Φm,n(1, 0)Φm,n(0, 1)|= e−2σα(H1(τ)+H2(τ))|eσαH2(τ)(2−2α−1) − 1|∼ e−2σαµ2σαH2(τ)|2 − 2α−1| (36)

and the result follows. �

Note that for most distributions, H2(τ) tends to zero as τ →∞, so that the condition is automatically satisfied.


In applications, of particular interest is the class of φ-mixingfor which ∑

n

φ1/2n < ∞. (37)

Under this condition, samples of the φ-mixing noise (or theirnonlinear transformations) comply with the standard centrallimit theorem (CLT). Such a property is useful when asymp-totic convergence problem is considered. For example, in [25],to apply the asymptotic relative efficiency as a criterion for sig-nal detection, (37) is imposed upon the φ-mixing noise consid-ered.

Now, by (36), a necessary condition of (37) is given by∑

τ

H2(τ) < ∞, (38)

which can be satisfied by most distributions, e.g. the exponen-tial family.

C. Long-range dependent

In this section, we study the asymptotic dependence struc-ture of the interference when the session life is heavy-tail dis-tributed. In this case, condition (38) is not met. However, aswe shall see, it forms another interesting class of dependencestructures.

The motivation behind considering heavy-tail distributed ses-sion life is that such a distribution can well characterize manycurrent and future communication systems. For example, inspread spectrum packet radio networks, multiple access termi-nals utilize the same frequency channel. The signals receivedat the receiver consist of superposition of the signals from allthe users in the network. Assuming that multi-user detec-tion and power-control are not implemented, the interferencefrom other users, or otherwise referred to as self-interference,can be characterized by our interference model. As more andmore wireless users are equipped with internet-enabled cellphones, their resource-request holding times (session life) aredistributed with much fatter tail than that of voice only networkusers (cf. [26]). The session life distributions in future wire-less systems are expected to be similar to the ones in currentwireline networks, for which extensive statistical analysis ofhigh-definition network traffic measurement has shown that theholding times of data network users are heavy-tailed distributed[23], [27]. In particular, they can be modelled by Pareto distri-butions [23], [27].

For a discrete-time communication system, we here assumethat the session life is Zipf distributed (a discrete version of thePareto distribution). A random variable X has a Zipf distribu-tion [28] if

P{X ≥ k} = [1 + (k − k0

σ)]−α, k = k0, k0 + 1, k0 + 2...

(39)where k0 is an integer denoting the location parameter, σ > 0denotes the scale parameter, and α > 0, is the tail index. In thispaper, for simplicity, we set σ = k0 = 1, and α > 1, whichimplies that E{X} = ζ(α), where ζ(·) is the Riemann Zetafunction. We shall denote the tail index of the session life byαL to avoid confusion with the α defined in (17).

Since the interference is marginally heavy-tail distributed,conventional tools such as covariance that measures the depen-dence structure are not applicable. Instead, we use the codiffer-ence (see Eq.(4)) to explore the dependence structure.

Proposition 4: If the session life of the interferers are Zipfdistributed, with tail index 1 < αL < 2, and Xi(m) are i.i.d.Bernoulli random variables taking possible values 1 and -1 withequal probability 1/2, then, the resulted interference is long-range dependent in the generalized sense, i.e.,

limτ→∞

−I(1,−1; τ)

τ−(αL−1)=

(2 − 2α−1)σα

αL − 1(40)

where

τ : time lag between time intervals;αL : tail index of the session life distribution;σ : as defined in (33).

(41)

Proof: See Appendix E.

VI. NUMERICAL SIMULATION RESULTS

In this section, we simulate a wireless network linkwith Poisson distributed interferers and Bernoulli distributedXi(m)’s, as described in Sec.IV-D. Our goal is to show that thesimulated interference is consistent with our theoretical find-ings, i.e., jointly α-stable distributed with long-range depen-dence in the generalized sense when users’ holding time isheavy-tail distributed.

The wireless communication network link was subjected tointerferers which are spatially Poisson distributed over a planewith density λ = 2/π. The path loss was power-law withγ = 4. Once the interferers become active, they stay in theactive state for random time durations, which in our simula-tions are Zipf [28] distributed with k0 = σ = 1 and αL = 1.4.Xi(m) was taken to be i.i.d. Bernoulli distributed, taking values±1 with equal probabilities. Then, according to (33), σ = π,and the instantaneous interference is SαS distributed with scaleparameter σµ1/α, where µ is the mean of the session life. Notethat µ = ζ(1.4) ' 3.1.

One segment of the simulated interference process is shownin Fig. 1(a) (for comparison purposes, we also simulate an-other trace with γ = 2.2 corresponding to α = 1.82, as shownin Fig. 1(b). Both traces have been normalized with respectto their empirical standard deviation). As expected, it exhibitsstrong impulsiveness (impulsiveness becomes weaker as γ de-creases, as evidenced by Fig. 1(b)). We employ the methodof sample fractiles ([6] Chap. 5.3) to estimate the various pa-rameters for the first trace (Fig. 1(a)). Assuming the data is α-stable distributed, we find that α = 1.0044, and scale parameterσ = 9.34, which are very close to their theoretical value 4

γ = 1and 3.1π, respectively. A more rigorous statistic method to testwhether the experimented data is indeed SαS distributed is theQQ-plot, which compares the quantiles of experimented datato those of the ideally SαS distributed (α = 1) data. Shouldthe experimental data be indeed SαS distributed, the QQ-plotwould be linear. We synthesized ideally SαS distributed noisewith the same parameters (α and σ) and plotted the quantilesplot versus that of the interference (length of 5000 points) in


Fig. 2. The figure clearly demonstrates that the instantaneousinterference can be modeled well by the SαS distribution.

As shown in Sec.V-C, when the session life is heavy-tail dis-tributed, the interference exhibits long-range dependence in thegeneralized sense. We estimated the codifference of the inter-ference yk according to [29]:

τ(n) = − ln | 1

K

K∑

k=1

eis(yk+n−yk)| + ln | 1

K

K∑

k=1

eisyk+n |

+ ln | 1

K

K∑

k=1

e−isyk | (42)

where K is the data length and s is some small multiplicativeconstant. 40 Monte Carlo simulations were run for the param-eters described above. Each run was based on 5, 000 pointsand s = 0.1. In Fig.3(a), we overlap the estimated codiffer-ences in a log-log scale. The linear trend is clearly seen in thegraph. In Fig.3(b), we also plotted the mean (solid line) and thestandard deviation (dotted line) of the estimated codifference.A least squares line was fitted to each simulation run, and themean slope of the fitted lines is found to be −0.3829. Note thataccording to Proposition 4, the theoretical value is −0.4. Theestimated value is in good agreement with the theory, indicatingthat the interference is long-range dependent in the generalizedsense.

VII. IMPLICATIONS OF IMPULSIVE AND LRDINTERFERENCE

The dependence structure of the interference should be takeninto account in signal detection and estimation. In particular,applying signal detection algorithms optimized for i.i.d noiseto scenarios where noise is highly correlated, can yield unex-pected degradation of receiver performance. This point is high-lighted in the following example. We consider the problem ofbinary signal detection in non-i.i.d noise, but for detection, weoverlook the noise dependence structure and treat it as if it werewhite. The consequence of ignoring the dependence is studiedvia the bit-error-rate (BER).

Let us assume that the transmitter is sending binary signals,s0 = 0 or s1 = 1 with equal probability and the transmissionis corrupted by the noise n. For deciding between the two hy-potheses, let us adopt the Cauchy receiver developed in [30]. Ithas been shown in [30] that the Cauchy receiver performs ro-bustly in the presence of i.i.d. noise modeled as α-stable for awide range of α, despite the fact that Cauchy noise only con-stitutes a special case of α-stable noise (α = 1). Given theobservation {x(k), k = 1, 2, ...}, the test statistic is:

Λ(k) =γ2 + [x(k) − s0(k)]2

γ2 + [x(k) − s1(k)]2, k = 1, 2, ... (43)

where γ is the dispersion of the interference (γ = σα). IfΛ(k) > 1, we decide that s1(·) has been transmitted at timek, otherwise, we decide in favor of s0(·).

Two different noise processes are simulated based on the pro-posed model, both having identical marginal distributions. Inthe first case, we set the session life to be Zipf distributed with

αL = 1.5, which should lead to a long-range dependent inter-ference. In the second case we set the session life to be equal to1, which should result in an i.i.d. interference. Other parame-ters are selected such that the dispersion of the interference arethe same for the two situations. Denoting the density of the in-terferers in first scenario by λ1, and in second scenario by λ2,we note that λ2 = λ1µ, where µ is the mean the Zipf distribu-tion.

We performed 40 Monte-Carlo simulations, with data lengthof 5000 each, corresponding to various values of λ. For 40Monte-Carlo simulations, the corresponding mean BER alongwith the standard deviation (normalized by the mean BER) areshown in Figs.4(a) and (b) respectively. Diamond-marked linesrepresent the LRD case; star-marked lines are for the i.i.d. case.We observe that although the mean BER are close to each otherin the two cases, there is a large discrepancy between the stan-dard deviation of the BER. The standard deviation of the BERin the LRD case is significantly larger than in the i.i.d case. Thisobservation implies that the highly correlated interference candegrade the performance of Cauchy receiver. This is of particu-lar importance in practice, where finite data length is available,and where, instability of the receiver may lead to erroneous per-formance.

Figs. 5(a)(b) illustrate the case when αL=1.2, vis-a-vis thei.i.d. case. λ’s are chosen such that the mean BERs are similarto the previous example. It is interesting to see that, in this case,as smaller αL implies stronger dependence, the discrepancy be-tween the BER standard deviations further increases.

A heuristic explanation for this phenomenon lies in the obser-vation that LRD time series exhibit strong low frequency com-ponent. There exist long periods where the maximal level tendsto stay high, and also there exist long periods where the se-quence stays in low levels [22]. Therefore, in a short segmentof the series, one often observes cycles and trends, althoughthe process is stationary. Thus, for signal detection in LRD in-terference based on finite data length, the probability of erroroscillates in a wider range than in an i.i.d noise environment.

It would be of interest to study analytically the dependenceof the BER statistics on the noise LRD index (αL − 1, see Eq.(40)). This will be subject of future investigation.

VIII. CONCLUSIONS

In this paper, we investigated the statistics of the interferenceresulted from a Poisson field of interferers. Key assumptionswere that individual interferers have certain random sessionlife, whose distribution is a priori known, and the signal propa-gation attenuation is power-law. We obtained the instantaneousand second order distributions of the interference. We showedthat although the interference process is marginally SαS, it is ingeneral not jointly α-stable distributed, except in some specialcases, such as the interferers sending BPSK signals, or constantamplitude signals. We provided conditions under which the in-terference becomes m-dependent, φ-mixing or long-range de-pendent.

The dependence structure of the interference must be takeninto account to attain optimum signal detection. Some pre-liminary results shown in this paper indicate that, the perfor-mance of traditional detectors may deteriorate significantly un-


der long-range dependent noise. Signal detection in the pres-ence of impulsive and long-range dependent noise is still anopen problem and is currently under investigation.

APPENDIX APROOF OF PROPOSITION 1

Proof: Let C(n) denote the number of active interferers atthe n-th time interval. It holds

C(n) =∑

k

1[Γk≤n<Γk+Lk], (44)

where Γk is the time when the sources emerged, and Lk’s cor-respond to their session life (in multiples of T ). 1[.] is the indi-cator function.

Let FL(·) be the survival function of the session life, i.e.,

FL(k)4= P [L ≥ k], k = 1, 2, ... (45)

Since the number of new interferers at each time interval isa Poisson random variable, C(n) is Poisson distributed withparameter λ

∑nk=0 FL(n − k + 1). Letting n tend to ∞,

or alternatively, taking the time origin to be −∞, the corre-sponding counting quantity C(n) asymptotically is λµ, whereµ =

∑∞k=1 FL(k).

At any given time interval, by noting that a combined Poissonpoint process is still Poisson, the active interferers are spatiallyPoisson distributed in the space. �

APPENDIX BPROOF OF PROPOSITION 2

Proof: The joint characteristic function of the total interfer-ence at time m and n is:

Φm,n(ω1, ω2) = E{exp[jω1Y (m) + jω2Y (n)]}

= E

{exp

[jω1

∑

i∈Nm

a(ri)Xi(m)

+jω2

∑

i∈Nn

a(ri)Xi(n)

]}(46)

Nm represents the set of interferers which are active at timem. According to their emerging time, this set can be furtherpartitioned as:

Nm =m⋃

t=−∞

Nm,t (47)

where the set Nm,t contains the interferers that are active at mand emerged at time slot t. We should note here that the inter-ferers emerged after time m will not contribute to the interfer-ence received at m. Thus the summations of the form

∑i∈Nm

can be replaced by the double summation∑m

t=−∞

∑i∈Nm,t

.By assumption, the interferers in Nm,t, t = 1, 2... are inde-

pendent of each other for different t. The same independence

conclusion applies to interferers in Nn,t, t = 1, 2.... Thus, thejoint characteristic function becomes:

Φm,n(ω1, ω2)

= E

exp

m∑

t=−∞

jω1

∑

i∈Nm,t

a(ri)Xi(m)

+jω2

∑

i∈Nn,t

a(ri)Xi(n)

+

n∑

t=m+1

jω2

∑

i∈Nn,t

a(ri)Xi(n)

=m∏

t=−∞

E

exp

jω1

∑

i∈Nm,t

a(ri)Xi(m)

+jω2

∑

i∈Nn,t

a(ri)Xi(n)

·n∏

t=m+1

E

exp

jω2

∑

i∈Nn,t

a(ri)Xi(n)

(48)

=

m∏

t=−∞

I1(t) ·n∏

t=m+1

I2(t) (49)

where the definitions of I1(t) and I2(t) can be easily deducedby (48).

We divide the calculation I1(t) and I2(t) into cases for t <m, t = m, m < t < n and t = n respectively.

I) t < mFor interferers started at time t < m, they can be classified

into 3 groups according to their active/inactive states in m andn:

1) inactive at both m and n, i.e. their session life are shorterthan m − t + 1. Let K ′′

t denote this set of interferers;2) active at m but not n (m−t+1 ≤ session life < n−t+1)

(denoted by K ′t);

3) active at both m and n ( session life ≥ n−t+1). (denotedby Kt.)

We then have, Nm,t = K ′t

⋃Kt and Nn,t = {Kt}. Therefore,

for t < m,

I1(t) = E

exp

jω1

∑

i∈K′t

+∑

i∈Kt

a(ri)Xi(m)

+jω2

∑

i∈Kt

a(ri)Xi(n)

]}

= E

[exp{j

∑

i∈Kt

a(ri)(ω1Xi(m) + ω2Xi(n))

+jω1

∑

i∈K′t

a(ri)Xi(m)}

. (50)

To compute (50), the sum is first taken for all sources re-stricted in a disk centered at the receiver, Rb, which has a radius


of b. Assuming there are total k sources that started emission attime t, due to the Poisson assumption their locations are inde-pendent and uniform distributed on the disk Rb. Let us use kt,k′

t and k′′t to denote the number of elements in sets set Kt, K ′

t

and K ′′t respectively. From Eq. (50), we get:

I1(t)

= limb→∞

∞∑

k=0

P{k sources started in Rb at t}

·E[exp{j

∑

i∈Kt


+jω1

∑

i∈K′t

a(ri)Xi(m)}|kt + k′t + k′′

t = k

= limb→∞

∞∑

k=0

P{k in Rb at t}

·E[∏

i∈Kt

E[eja(ri)(ω1Xi(m)+ω2Xi(n))

]

·∏

i∈K′t

E[ejω1a(ri)Xi(m)

]|kt + k′

t + k′′t = k

= limb→∞

∞∑

k=0

P{k in Rb at t}

E

[∏

i∈Kt

Ψm,n(a(ri)ω1, a(ri)ω2)

·∏

i∈K′t

Ψ(a(ri)ω1)|kt + k′t + k′′

t = k

, (51)

where Ψ(ω1) and Ψm,n(ω1, ω2) are the first and second ordercharacteristic function of Xi(n).

Since the survival function of the session life is FL(·), theprobability that an interferer that emerged at time t will remainactive until n is

P1(n, t) = FL(n − t + 1) (52)

The probability that an interferer that emerged at t will surviveuntil m but will die out at n is

P2(m,n, t) = FL(m − t + 1) − FL(n − t + 1), (53)

while the probability that an interferer will be inactive at m is

P3(m, t) = 1 − FL(m − t + 1). (54)

If there are k interferers beginning their emission at t, the prob-ability that l of them are active until time n, and p of them areactive at m but not n is

P{kt = l, k′t = p}

=

(kl

)(k − lp

)P1(n, t)lP2(m,n, t)pP3(m, t)(k−l−p)

=k!

l!p!(k − l − p)!P1(n, t)lP2(m,n, t)pP3(m, t)k−l−p (55)

where l + p ≤ k.

Combining (55) and(51) we get

I1(t)

= limb→∞

∞∑

k=0

e−λπb2(λπb2)k

k!

k∑

l=0

k−l∑

p=0

P{kt = l, k′t = p}

·∏

i∈Kt

E [Ψm,n(a(ri)ω1, a(ri)ω2)]∏

i∈K′t

E [Ψ(a(ri)ω1)]

= limb→∞

∞∑

k=0

e−λπb2(λπb2)k

k!

k∑

l=0

k−l∑

p=0

P{kt = l, k′t = p}

·[∫ b

0

Ψm,n(a(ri)ω1, a(ri)ω2)fr(r)dr

]l

·[∫ b

0

Ψ(a(ri)ω1)fr(r)dr

]p

= limb→∞

e−λπb2∞∑

k=0

k∑

l=0

k−l∑

p=0

1

l!p!(k − l − p)!U lV p

·P3(m, t)k−l−p(λπb2)k, (56)

where U = P1(n, t)∫ b

0Ψm,n(a(r)ω1, a(r)ω2)fr(r)dr and

V = P2(m,n, t)∫ b

0Ψ(a(r)ω1)fr(r)dr. Then, from(56) we

get:

I1(t)

= limb→∞

e−λπb2∞∑

k=0

k∑

l=0

k−l∑

p=0

1

l!p!(k − l − p)!

·[P3(m, t)λπb2

]k[

U

P3(m, t)

]l [V

P3(m, t)

]p

= limb→∞

e−λπb2∞∑

k=0

k∑

l=0

[P3(m, t)λπb2

]k[

U

P3(m, t)

]l

·[1 +

V

P3(m, t)

]k−l

/(l!(k − l)!)

= limb→∞

e−λπb2∞∑

k=0

[P3(m, t)λπb2

]k

·[1 +

U

P3(m, t)+

V

P3(m, t)

]k

/k!

= limb→∞

e−λπb2eλπb2(P3(m,t)+U+V ) (57)


The logarithm of (50) equals:

ln I1(t)

= limb→∞

−λπb2 + λπb2(P3(m, t) + U + V )

= limb→∞

λπb2[U + V − P1(n, t) − P2(m,n, t)]

= limb→∞

λπb2

[P1(n, t)[

∫ b

0

Ψm,n(a(r)ω1, a(r)ω2)fr(r)

−1] + P2(m,n, t)[

∫ b

0

Ψ(a(r)ω1)fr(r) − 1]

]

= limb→∞

λπb2

[P1(n, t)[

∫ b

0

Ψm,n(a(r)ω1, a(r)ω2)2r

b2

−1] + P2(m,n, t)[

∫ b

0

Ψ(a(r)ω1)2r

b2− 1]

](58)

Proceeding in the manner of [5], the second term in (58) canbe simplified as

limb→∞

λπb2P2(m,n, t)

[∫ b

0

Ψ(a(r)ω1)

b2br2 − 1

]

= −σαP2(m,n, t)|ω1|α (59)

where

σα = −λπ

∫ ∞

0

x−αdΨ(x) (60)

Denoting

Θm,n(ω1, ω2)

= limb→∞

λπb2

[∫ b

0

Ψm,n(a(r)ω1, a(r)ω2)2r

b2dr − 1

],

we finally get:m−1∏

t=−∞

I1(t) =

m−1∏

t=−∞

exp{P1(n, t)Θm,n(ω1, ω2)

−σαP2(m,n, t)|ω1|α}

= exp

{m−1∑

t=−∞

P1(n, t)Θm,n(ω1, ω2)

−m−1∑

t=−∞

σαP2(m,n, t)|ω1|α]

}. (61)

Let τ = n − m, then,m−1∑

t=−∞

P1(n, t) =

m−1∑

t=−∞

FL(n − t + 1) =

∞∑

l=τ+2

FL(l), (62)

andm−1∑

t=−∞

P2(m,n, t)

=

m−1∑

t=−∞

FL(m − t + 1) −m−1∑

t=−∞

FL(n − t + 1)

=

τ+1∑

l=2

FL(l) (63)

Equations (61) (62) (63) define the contributions from interfer-ers which emerged before time m.

II) t = m

Interferers that emerged at time m need to be treated dif-ferently, since they always contribute to the interference at m,while not necessarily at n. Nevertheless, these interferers canbe grouped as either active or inactive at n. Hence, for t = m,we have

Nm,m = Nn,m

⋃N n,m (64)

where N n,m represents the interferers started at m, but died outbefore n. We will use kn,m and kn,m to enumerate Nn,m andNn,m respectively.

For t = m, we get

I1(m) = E

exp{j

∑

i∈Nn,m


+jω1

∑

i∈Nn,m

a(ri)Xi(m)}

. (65)

For any interferer starting at time m, it remains active until nwith probability Pm1 = FL(n−m+1), while it ends emissionbefore n with probability Pm2 = 1 − Pm1. Following similarreasoning as before, we may proceed to calculate (65) as

I1(m)

= limb→∞

∞∑

k=0

P{k in Rb at m}

·E

∏

i∈Nn,m

E[eja(ri)(ω1Xi(m)+ω2Xi(n)

]

·∏

i∈Nn,m

E[ejω1a(ri)Xi(m)

]|kn,m + kn,m = k

= limb→∞

∞∑

k=0

e−λπb2(λπb2)k

k!

k∑

l=0

k!

l!(k − l)!P l

m1Pk−lm2

·[∫ b

0

Ψm,n(a(r)ω1, a(r)ω2)fr(r)dr

]l

·[∫ b

0

Ψ(a(r)ω1)fr(r)dr

]k−l

= exp{Pm1Θm,n(ω1, ω2) − σαPm2|ω1|α}, (66)

where σ as defined in (60).III) m < t < n

So far, we are done with the computation for interferersemerged at or before time m. Next, we are continuing withcalculation of the second product term in equation (49). Theseinterferers emerged after time m, contributing interference onlyat time n. Since interferers emerged between m and n may dieout before n, we need to consider them separately from the in-terferers emerged at n.


Reasoning as before, for m < t < n,

lnn−1∏

t=m+1

I2(t)

=n−1∑

t=m+1

ln limb→∞

∞∑

k=0

P{k started in Rb at t}

·E[ejω2

∑i∈Nn,t

a(ri)Xi(n)|k in Rb

]

=

n−1∑

t=m+1

ln limb→∞

∞∑

k=0

e−λπb2(λπb2)k

k!

k∑

l=0

k!

l!(k − l)!

·P1(n, t)l(1 − P1(n, t))k−l

[∫ b

0

Ψ(a(r)ω2)fr(r)

]l

=

n−1∑

t=m+1

limb→∞

λπb2P1(n, t)

(∫ b

0

Ψ(a(r)ω2)fr(r) − 1

)

= −σαn−1∑

t=m+1

P1(n, t)|ω2|α, (67)

where σ as defined before in (60).Note that

n−1∑

t=m+1

P1(n, t) =

τ∑

l=2

FL(l), (68)

where τ = n − m.IV) t = nFinally, following similar steps, we can show that, for t = n,

ln I2(t) = −σα|ω2|α. (69)

Plugging Eqs.(61)-(67) and (69) into (49), Eqs.(16)-(21) fol-low. �

APPENDIX CPROOF OF EQ.(29)—CONSTANT CASE

We here prove Eq. (29).Proof: We need to calculate Θm,n(ω1, ω2). Eq. (28) gives us

Ψm,n(ω1, ω2) = E[ejω1Xi(m)+jω2Xi(n)]

= E[ej(ω1+ω2)Bi ]

= Ψ(a(r)(ω1 + ω2)). (70)

Since,

Θm,n(ω1, ω2)

= limb→∞

λπb2[

∫ b

0

Ψ(a(r)(ω1 + ω2))

b2dr2 − 1]

= −σα|ω1 + ω2|α, (71)

plugging (71) into (16), we obtain (29).Next, we show that Y (m) and Y (n) are jointly α-stable.

To see this, noting that Y (m) and Y (n) are symmetricallyα-stable, we only need to show that the linear combination

a1Y (n)+a2Y (m) is symmetrically α-stable for any real num-ber a1, a2 (Theorem 2.1.5 [20]). The log-characteristic functionof the random variable a1Y (n) + a2Y (m) equals:

ln Φ(ω) = ln E{exp(jω(a1Y (n) + a2Y (m)))}= ln Φm,n(ωa1, ωa2)

= −σα[H1(τ)|a1|α + H2(τ)|a1 + a2|α+H1(τ)|a2|α]|ω|α (72)

which indeed is symmetrically α-stable.�

APPENDIX DPROOF OF EQ.(32)—BERNOULLI CASE

The characteristic function of a Bernoulli random variable,which takes 1 or -1 with equal probability 1/2, is cosω. Hence,

Ψm,n(ω1, ω2) = E[ejω1Xi(m)+jω2Xi(n)]

= cosω1 cos ω2. (73)

In the equation above, we have used the fact that Xi(m) is in-dependent of Xi(n) for m 6= n.

Therefore, we have

Θm,n(ω1, ω2)

= limb→∞

λπb2[

∫ b

0

Ψm,n(a(r)ω1, a(r)ω2)

b2dr2 − 1]

= limb→∞

λπb2[

∫ b

0

cos(r−γ/2ω1) cos(r−γ/2ω2)2r

b2dr − 1]

= λπ

∫ ∞

0

(1

2cos(ω1 + ω2)r

−γ/2

+1

2cos(ω1 − ω2)r

−γ/2 − 1

)2rdr

=λπ

2

∫ ∞

0

(Ψ(a(r)(ω1 + ω2)) − 1) dr2

+λπ

2

∫ ∞

0

(Ψ(a(r)(ω1 − ω2)) − 1) dr2

= −σα

2(|ω1 + ω2|α + |ω1 − ω2|α) (74)

where α = 4/γ and σ is defined as in (60), which can be calcu-lated as:

σ =

(−λπ

∫ ∞

0

x−αdΨ(x)

) 1α

=

(−λπ

∫ ∞

0

x−αd cos(x)

) 1α

. (75)

The above integral exists as long as 0 < α < 2 [31]. Using theintegral identity

∫ ∞

0

x−α sin xdx =

√π2−αΓ(1 − 1/2α)

Γ(1/2 + 1/2α), (76)

we have

σ =1

2

(λπ3/2Γ(1 − α/2)

Γ(1/2 + α/2)

) 1α

. (77)

Now, (32) is readily verified.


APPENDIX EPROOF OF PROPOSITION 4

Proof: The codifference of the interference separated by τcan be calculated as

−I(1,−1; τ)

= ln E{ej(x(t+τ)−x(t))} − ln E{ejx(t+τ)}− ln E{e−jx(t)}

= ln Φt,t+τ (−1, 1) − ln Φt,t+τ (0, 1) − ln Φt,t+τ (−1, 0)

= (2 − 2α−1)σαH2(τ) (78)

Plugging (39), we get

−I(1,−1; τ) = (2 − 2α−1)σα∞∑

l=τ+1

l−αL . (79)

Note that∞∑

l=τ+1

l−αL ≤∫ ∞

τ+1

(t − 1)−αLdt =τ−αL+1

αL − 1(80)

and∞∑

l=τ+1

l−αL ≥∫ ∞

τ+1

t−αLdt =(τ + 1)−αL+1

αL − 1. (81)

Since the upper bound (80) and lower bound (81) converge asτ → ∞, we conclude that

limτ→∞

−I(1,−1; τ)

τ−αL+1=

(2 − 2α−1)σα

αL − 1(82)

where σ is as given as in (33). Note that, for α ∈ (0, 2), 2 −2α−1 and σ are positive. Hence, for αL > 1, (2−2α−1)σα

αL−1 ispositive, and the interference process is long-range dependentin the generalized sense. �

ACKNOWLEDGEMENT

We would like to thank Dr. Steve Lowen at Harvard MedicalSchool for the interesting discussions on Proposition 4. We alsothank the anonymous reviewers for the careful reading of earlierversions of this paper.

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[9] T. K. Blankenship, D. M. Krizman, and T. S. Rappaport, “Measurementsand simulation of radio frequency impulsive noise in hospitals and clin-ics,” in Proc. 47th IEEE Veh. Technol. Conf., vol. 3, pp. 1942-1946,Phoenix, AZ, May 1997.

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[15] K. F. McDonald and R. S. Blum, “A statistical and physical mechanisms-based interference and noise model for array observations,” IEEE Trans.Sig. Proc., vol. 48, no. 7, pp. 2044-2056, July 2000.

[16] D. Middleton, “Threshold detection in correlated non-Gaussian noisefields,” IEEE Trans. Inform. Theory, vol. 41, no. 4, July 1995.

[17] X. Yang and A. P. Petropulu, “Interference modeling in radio communi-cation networks,” Trends in Wireless Indoor Networks, Wiley Encyclope-dia of Telecommunications, J. Proakis Eds, John Wiley and Sons, NewYork, 2002.

[18] A. M. Maras, “Locally optimum detection in moving average non-Gaussian noise,” IEEE Trans. Commun., vol. 36, no. 8, pp. 907-912,Aug. 1988.

[19] A. M. Maras, “Locally optimum Bayes detection in ergotic Markovnoise,” IEEE Trans. Inform. Theory, vol. 40, no. 1, pp. 41-55, Jan. 1994.

[20] G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random pro-cesses: Stochastic Models with Infinite Variance, New York: Chapmanand Hall, 1994.

[21] A.P. Petropulu, J-C. Pesquet and X. Yang, “Power-law shot noise andrelationship to long-memory processes,” IEEE Trans. on Sig. Proc.,Vol.48, No.7, July 2000.

[22] J. Beran, Statistics for Long-Memory Processes, Chapman & Hall, NewYork, 1994.

[23] X. Yang and A.P. Petropulu, “The extended alternating fractal renewalprocess for modeling traffic in high-speed communication networks”,IEEE Trans. Sig. Proc., Vol.49, No.7, July, 2001.

[24] P. Billingsley, Convergence of Probability Measures, New York: Wiley,1968.

[25] D.R. Halverson and G. L. Wise, “Discrete-Time Detection in φ-MixingNoise,” IEEE Trans. Inform. Theory, Vol. IT-26, No.2, pp.189-198, Mar.1980.

[26] T. Kunz, T. Barry, X. Zhou et al, “WAP traffic: description and com-parison to WWW traffic,” Proc. of 3rd ACM international Workshopon Modeling, Analysis and Simulation of Wireless and Mobile Systems,Boston, USA, Aug. 2000.

[27] W. Willinger, M.S. Taqqu, R. Sherman, and D.V. Wilson, “Self-similarity through high-variability: statistical analysis of Ethernet LANtraffic at the source level,” IEEE/ACM Trans. Networking, Vol.5, No.1,Feb. 1997.

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[30] G.A. Tsihrintzis and C.L. Nikias, “Performance of optimum and subop-timum receivers in the presence of impulsive noise modeled as an alpha-stable process”, IEEE Trans. Comm, Vo.43, No.2/3/4, Feb/March/April,1995.

[31] I.S. Gradshteyn, I.M. Ryzhik, A. Jeffrey, Eds., Table of Integrals, Seriesand Products, New York: Academic Press, 1994.


Xueshi Yang received B.Sc. degree from the Elec-tronic Engineering Department, Tsinghua University,Beijing, China, in 1998 and the Ph.D. degree in elec-trical engineering from Drexel University, Philadel-phia, PA, in 2001.

During September 1999 - April 2000, he was a visiting researcher at Labora-toire des Signaux et Systemes, CNRS-Universit Paris Sud, SUPELEC, France.Since 2002, he has been with the Electrical Engineering Department, Prince-ton University, Princeton, NJ, where he is a Postdoctoral Research Associate.His research interests are in the areas of non-Gaussian signal processing forcommunications, fractional-order statistics and communication system model-ing and analysis.

Athina P. Petropulu received the Diploma in Electri-cal Engineering from the National Technical Univer-sity of Athens, Greece in 1986, the M.Sc. degree inElectrical and Computer Engineering in 1988 and thePh.D. degree in Electrical and Computer Engineeringin 1991, both from Northeastern University, Boston,MA.

In 1992, she joined the Department of Electrical and Computer Engineeringat Drexel University where she is now a Professor. During the academic year1999/2000 she was an Associate Professor at LSS, CNRS-Universit Paris Sud,cole Suprieure d’Electrcit in France. Dr. Petropulu’s research interests spanthe area of statistical signal processing, communications, higher-order statis-tics, fractional-order statistics and ultrasound imaging. She is the co-authorof the textbook entitled, “Higher-Order Spectra Analysis: A Nonlinear Sig-nal Processing Framework,” (Englewood Cliffs, NJ: Prentice-Hall, Inc., 1993).She is the recipient of the 1995 Presidential Faculty Fellow Award. She hasserved as an associate editor for the IEEE Transactions on Signal Processingand the IEEE Signal Processing Letters. She is a member of the IEEE Con-ference Board and the Technical Committee on Signal Processing Theory andMethods.


LIST OF FIGURES

1 Interference presented in a communication link in a Poisson field of interferers: (a) α = 1; (b) α = 1.82. . . . . . . 152 QQ-plot of simulated interference and ideally SαS distributed random variable with the same parameter. α = 1,

which is the Cauchy distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 (a) Log-log plot of the codifference of the interference in 40 Monte Carlo simulations. (b) Mean (solid line) and

standard deviation (dotted lines)of the codifference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Mean and standard deviation (normalized by mean) of bit-error-rate of Cauchy receiver in the presence of long-range

dependent (αL = 1.5) and i.i.d α-stable noise, for 40 Monte Carlo simulations. Diamond-marked lines represent thelong-range dependent case, while star-marked lines denote the i.i.d. case. . . . . . . . . . . . . . . . . . . . . . . . 18

5 Mean and standard deviation of bit-error-rate of Cauchy receiver in the presence of long-range dependent (αL = 1.2)and i.i.d α-stable noise, for 40 Monte Carlo simulations. Diamond-marked lines represent the long-range dependentcase, while star-marked lines denote the i.i.d. case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19


0 500 1000 1500 2000−2500

−2000

−1500

−1000

−500

0

500

1000

1500

2000

2500α=1.82

Time

Inte

rfere

nce

Leve

l

0 500 1000 1500 2000−2500

−2000

−1500

−1000

−500

0

500

1000

1500

2000

2500α=1

Time

Inte

rfere

nce

leve

l

Fig. 1. Interference presented in a communication link in a Poisson field of interferers: (a) α = 1; (b) α = 1.82.


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 104

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 104

Ideally α−stable variables Quantile

Inte

rfere

nce

Qua

ntile

Fig. 2. QQ-plot of simulated interference and ideally SαS distributed random variable with the same parameter. α = 1, which is the Cauchy distribution.


0 0.5 1 1.5 2 2.5 3−6

−5

−4

−3

−2

−1

0

log10(Lag)

log1

0(C

odiff

eren

ce)

(a) Overlapped Codiffference

0 0.5 1 1.5 2 2.5 3−2.5

−2

−1.5

−1

−0.5

0

log10(Lag)

log1

0(C

odiff

eren

ce)

(a) Mean and Standard Deviation of Codifference

Fig. 3. (a) Log-log plot of the codifference of the interference in 40 Monte Carlo simulations. (b) Mean (solid line) and standard deviation (dotted lines)of thecodifference.


0 2 4 6 8 10 12 14 16 18 200

0.01

0.02

0.03

0.04

0.05

0.06

BER mean, αL=1.5

λ

Pe

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

BER std., αL=1.5

λ

Std

(Pe)/M

ean(

Pe)

Fig. 4. Mean and standard deviation (normalized by mean) of bit-error-rate of Cauchy receiver in the presence of long-range dependent (αL = 1.5) and i.i.dα-stable noise, for 40 Monte Carlo simulations. Diamond-marked lines represent the long-range dependent case, while star-marked lines denote the i.i.d. case.


0 1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

BER mean, αL=1.2

λ

Pe

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

BER std., αL=1.2

λ

Std

(Pe)/M

ean(

Pe)

Fig. 5. Mean and standard deviation of bit-error-rate of Cauchy receiver in the presence of long-range dependent (αL = 1.2) and i.i.d α-stable noise, for 40Monte Carlo simulations. Diamond-marked lines represent the long-range dependent case, while star-marked lines denote the i.i.d. case.

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