Average Symbol Error Probability in the Presence of ... - IEEE Xplore

IEEE ICC 2012 - Communications Theory

Average Symbol Error Probability in the Presence of Network Interference and Noise Cristina Merola(1) , Alessandro Guidotti(2) , Marco Di Renzo(3) , Fortunato Santucci(1) , Giovanni E. Corazza(2) (1)

University of L’Aquila, Dept. of Electrical and Information Engineering, Center of Excellence DEWS Via G. Gronchi 18, Nucleo Industriale di Pile, 67100 L’Aquila, Italy (2) University of Bologna, Dept. of Electronics, Computer Engineering and Systems (DEIS), Bologna, Italy Viale Risorgimento 2, 40136 Bologna, Italy (3) L2S, UMR 8506 CNRS – SUPELEC – Univ Paris–Sud Laboratory of Signals and Systems (L2S), French National Center for Scientific Research (CNRS) ´ ´ ´ Ecole Sup´erieure d’Electricit´ e (SUPELEC), University of Paris–Sud XI (UPS) 3 rue Joliot–Curie, 91192 Gif–sur–Yvette (Paris), France E–Mail: [email protected], {a.guidotti, giovanni.corazza}@unibo.it, [email protected]

Abstract— In this paper, we introduce a new framework to compute the Average Symbol Error Probability (ASEP) of an intended wireless communication system subject to network interference and noise. The interfering nodes are assumed to be randomly distributed in the 2D Euclidean plane according to a homogeneous Poisson point process. Our framework is applicable to performance prediction and optimization of, e.g., emerging heterogeneous cellular and cognitive radio networks. More specifically, we move from and generalize the semi– analytical framework recently introduced by Pinto and Win [1], and develop a new mathematical model which offers a simple single–integral expression of the ASEP under very general channel and interference conditions. The framework is exact, avoids Monte Carlo methods for its computation, and is applicable to asynchronous and synchronous scenarios. Our numerical examples show that both setups have almost the same performance, and that the ASEP in the presence of synchronous interference is a very tight upper–bound of the ASEP in the presence of asynchronous interference. This is a relevant result, as we show in this paper that in the former case all parameters of interest can be computed in closed–form. Our analytical derivation is substantiated through extensive Monte Carlo simulations.

I. I NTRODUCTION Accurate performance prediction of emerging heterogeneous wireless communication systems is instrumental to circumvent time–consuming computer simulations, to avoid expensive field test campaigns, as well as to shed lights on the dependance on many design parameters, to facilitate system optimization via a suitable design choice for some given practical implementation constraints, and, ultimately, to inspire optimal and innovative algorithms and designs. The derivation of tractable and accurate frameworks for performance prediction and system optimization is becoming even more important with the advent of new technologies and designs in the wireless communication industry. Two notable examples are heterogeneous femtocell–overlaid cellular networks [2] and cognitive radio networks [3]. These networks are characterized by a large number of nodes, e.g., femto and macro base stations in the first case, and secondary and primary users in the second case, which coexist and contend to have access to the wireless medium. Unlike traditional wireless networks, these systems are characterized by the inherent unplanned, irregular, and random locations of the nodes, whose positions may vary widely over a very large (ideally infinite) area. In this context, system performance and optimization depend critically on the spatial configurations of the nodes [4], and the computation of new “average” perfor-

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mance metrics, which take into account the random nature of these positions, are instrumental for the system designer and the network planner. The analysis of such systems typically requires extensive, complex, and time–consuming system– level simulations to average over the spatial distributions of the network nodes. Thus, new and advanced mathematical and statistical tools seem to be required to explicitly and accurately modeling the random distribution of these nodes, as well as to avoiding lengthy and seldom insightful numerical simulations. Luckily, recent research advances in this field have shown that stochastic geometry and Poisson point processes theory can be instrumental and essential tools to develop tractable and compact analytical frameworks, which can completely avoid Monte Carlo simulations [5]–[11]. In this depicted context, the analysis of the error performance of an intended (probe) link in the presence of interference generated by many randomly distributed nodes is receiving significant attention in current scientific literature. Notable contributions in this area are [1], [9], and [12]– [18]. However, these works have the following limitations. In [12] and [13], only binary modulation is considered and channel fading is neglected. In [14], the authors propose a very interesting framework, which, however, assumes that network interference is conditional Gaussian, which might not be the case for many scenarios of interest [1]. In [15] and [16], optimal combiners in the presence of network interference are derived, but no performance analysis is conducted. In [17] and [18], only binary modulation is considered and background noise is neglected. In [1] and [9], the authors propose a very general and elegant framework that can be applied to many cases of interest. However, the framework is semi– analytical and Monte Carlo methods are needed to remove the conditioning over network interference, even though extensive system–level simulations are completely avoided. In this paper, we move from the semi–analytical framework introduced in [1], and provide a new, exact, and single–integral expression of the Average Symbol Error Probability (ASEP) of an intended link, which is subject to receiver noise and aggregate interference generated by network nodes distributed according to a homogeneous Poisson point process. Furthermore, we study and compare the performance of asynchronous and synchronous network scenarios, and show that the latter setup is a very tight upper–bound of the former. This is a relevant result, as for synchronous systems we show that all parameters of interest can be computed in closed–form.

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This paper is organized as follows. In Section II, the system model is introduced. In Section III, the new single–integral framework to compute the ASEP is presented. In Section IV, numerical simulations are shown to validate the analytical derivation. Finally, Section V concludes this paper. II. S YSTEM M ODEL AND P ROBLEM S TATEMENT Similar to [1], we consider a 2D network deployment where the nodes are distributed according to a homogeneous Poisson point process of density λ. In this network, we focus our attention on studying the ASEP of a “typical” intended (probe) link between a pair of transmitter, TX0 , and receiver, RX0 , nodes. Without loss of generality, we assume that the intended receiver is located at the origin of the 2D Euclidean plane, and that the transmitter is located at a fixed distance R0 from the origin. On the other hand, the network nodes of the homogeneous Poisson point process act as interferers, and they are assumed to transmit in the same band of the intended link. According to the properties of Poisson point process, these interfering nodes are located at random positions, which are conditionally uniform distributed over the 2D Euclidean plane [7]. The random distances from the origin are denoted by Ri {\TX } {\TX } for i ∈ ΦTX 0 , where ΦTX 0 is the homogenous Poisson point process of all the transmit nodes in the 2D Euclidean plane except the transmitter, TX0 , of the intended link. A. Notation and Definitions The following notation and definitions are used throughout the paper: i) intended transmitter and interfering nodes use a M –ary Phase Shift Keying (M–PSK) modulation scheme. The symbol period is denoted by T , which is the same for all the nodes in the network. The generic complex M–PSK symbols emitted by TX0 and the i–th interfering node are denoted by s0 = a0√exp (jθ0 ) and si = ai exp (jθi ), respectively, where j = −1 is the imaginary unit. Our framework can be generalized to other linear modulation schemes, such as M –ary Quadrature Amplitude Modulation (M–QAM), but, due to space limitations, this is not considered in this paper; ii) intended transmitter and interfering nodes use a constant average transmit energy per symbol, which is denoted by E0 and EI , respectively. In particular, all interfering nodes use the same transmit energy; iii) the interfering nodes are assumed to transmit independently and asynchronously. In detail, the transmission delay, at the intended receiver, of the i–th interfering node located at distance Ri is denoted by Di , which is assumed to be a uniform Random Variable (RV) over the interval [0, T ). The RVs Di are independent {\TX } for i ∈ ΦTX 0 . On the other hand, and without loss of generality, it is assumed that the transmission delay of the intended transmitter is equal to zero. In fact, we assume that TX0 and RX0 are perfectly synchronized in time, phase, and frequency, and that optimum coherent decoding can be used; and iv) the additive noise at the input of the intended receiver, RX0 , is assumed to be white complex circularly symmetric Gaussian with power spectral density N0 /2 per dimension. As far as the channel model is concerned, we have the following assumptions: i) fast–fading has a Rayleigh distribution in all the wireless links. More specifically, the complex fading gain of intended and i–th interfering link are denoted by {\TX } α0 = |α0 | exp (jφ0 ) and αi = |αi | exp (jφi ) for i ∈ ΦTX 0 , respectively, where |α0 | and |αi | are Rayleigh distributed with

    2 2 parameter1 Ω0 = Eα0 |α0 | = 1 and ΩI = Eαi |αi | = {\TX }

1 for i ∈ ΦTX 0 , respectively; and φ0 and φi are the channel phases, which are uniform distributed over the interval [0, 2π). We emphasize that our framework can be extended to more general fading distributions, such as Nakagami–m fading [19]. However, due to space limitations, in this paper we omit the analysis of this scenario; ii) shadowing has a Log–Normal distribution in all the wireless links. More specifically, the shadowing gain of the i–th interfering link is denoted by Si = exp (σI Gi ), where σI is the shadowing {\TX } standard deviation (in Neper2 ), and Gi for i ∈ ΦTX 0 are independent standard Gaussian RVs with zero mean and unit standard deviation. On the other hand, we assume that slow power control is used on the intended link [10], and, hence, that the shadowing gain S0 is a fixed value; and iii) the path–loss follows a typical exponential–decaying law in all the wireless links [6]. In particular, the transmitted power of intended and i–th interfering link decays with  the transmission distance according to the functions l0 = κ0 R0b0  {\TX } and li = κI RibI for i ∈ ΦTX 0 , respectively, where κ0 and κI are environmental–dependent constants, and b0 > 1 and bI > 1 denote the path–loss exponents. Finally, it is worth emphasizing that in our channel model all interfering nodes are independent and identically distributed (i.i.d.) with common fast–fading (ΩI ), shadowing (σI ), and path–loss (κI and bI ) parameters. Accordingly, the combined complex channel gain at RX0 of intended transmitter and i–th interfering node is {\TX } h0 = α0 S0 l0 and hi = αi Si li for i ∈ ΦTX 0 , respectively. B. Meaning of “Average” when Computing the ASEP With the many random variables introduced so far, we feel important to provide some clarifications about the physical meaning of “average” when computing the ASEP. With ASEP we denote the SEP that is averaged over fading channel statistics (fast–fading and shadowing) and random positions, {\TX } Ri for i ∈ ΦTX 0 , of all the interfering nodes. On the other hand, as far as the intended link is considered, the average is computed only over fast–fading, while shadowing (S0 ) and distance (R0 ) are, as mentioned in Section II-A, fixed. This “average” performance metric is useful for network planning and optimization in many cases of interest. Two examples are as follows. 1) The interferers have a short session lifetime compared to the duration of the communication of the intended link, as explained in [9]. In this case, each interfering node periodically becomes active, transmits a burst of symbols, and then turns off. Thus, the set of interfering nodes, i.e., fading/shadowing and spatial positions, changes so often that the interference process can be considered to be ergodic, and averages over fading, shadowing, and space (positions) are meaningful. 2) In many scenarios of interest, e.g., in ad hoc networks [5], [6], and femtocell–overlaid networks [4], [10], the positions of the interfering nodes are unknown to the network designer a priori. In such a context, we are interested in the performance of an “average” network configuration, i.e., by averaging over all possible network configurations, which implies averaging over fading, shadowing, and random spatial positions of the interferers. X {·} denotes the expectation computed over RV X. 2 Let σ (dB) be the shadowing standard deviation in dB, 1E

have σ = (ln (10)/20) σ (dB) .

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then in Neper we

 XI = exp

2σI2 b2I





      ai D i 2/bI  1 Di Γ 1+ cos (θi + φi ) + a ¯i 1 − Eai ,θi ,¯ai ,θ¯i ,φi ,Di  cos θ¯i + φi  bI T T



(2)





 +∞  (b) κ20 E0 Ω0 S02 √ = 1− p t−1/2 J1 2 pt MηA (t) dt 2b0  2 BEI σI + N0 R0 0      +∞  (c) Ψ INR √ 0 = 1− p t MB Ψ0 σI2 t dt t−1/2 J1 2 pt exp − SNR SNR 0        +∞  (d) Ψ0 INR 1/bI 1/bI √ −1/2 2 = 1− p dt t exp − Ψ0 σI t J1 2 pt exp − t SNR SNR 0 (a)

MηB (p) = EηB {exp (−pηB )} = EB

1 M −1 π− ASEP = M π

 T (t, ξ) =



−p

exp

M −1 π M



+∞

−1/2

t

0

0

sin2 (π/M ) sin2 (ξ)



+∞

0

√

     1/bI Ψ0 2 INR 1/bI t exp − Ψ0 σI exp − t T (t, ξ) dtdξ SNR SNR

  z exp (−z) J1

2

sin2 (π/M ) √ √ t z sin2 (ξ)

(6)

 dz (7)

    (a) √ sin2 (π/M ) sin2 (π/M ) sin2 (π/M ) (b) sin2 (π/M ) √ = t = F t exp − t t 2; 2; − 1 1 sin2 (ξ) sin2 (ξ) sin2 (ξ) sin2 (ξ)

C. Problem Statement In [1], by exploiting Slivnyak and Probability Generating Functional (PGFL) theorems from stochastic geometry [7] along with the decomposition property of alpha stable RVs [20], the authors have shown that the ASEP, for M–PSK modulation and Rayleigh fading, can be written as follows: ASEP =

1 π

where: i) ηB =





M −1 π M

0



EB

κ20 E0 Ω0 S02

1+



sin2

(π/M ) ηB sin2 (ξ)

R02b0 σB2



σB2

=

BEI σI2

N0 ; iii) B ∼   ˜ γ˜ denotes a real skewed stable distribution where S α ˜ , β, with characteristic exponent α ˜ ∈ (0, 2], skewness β˜ ∈ [−1, 1], and dispersion γ˜ ∈ [0, +∞) [20]; and iv) σI2 =  bI −1 , where, for Rayleigh fading, XI is given 4κ2I ΩI πλXI C2/b I in (2) on top of this page, and C2/bI is as follows: C2/bI =

 1− 2/π

2 bI

 Γ 2−

2 bI





cos

π bI

−1

III. S INGLE –I NTEGRAL E XPRESSION OF THE ASEP

if

bI = 2

if

bI = 2

By using [21, Eq. (6)], the integral in (1) can be written as:

(1)

+   S α ˜ = 1/bI , β˜ = 1, γ˜ = cosbI [π/(2bI )] , ; ii)

expression of (1) can be obtained for the system model under analysis. Furthermore, we provide a closed–form expression of {\TX } (2) for the synchronous scenario, i.e., Di = 0 for i ∈ ΦTX 0 , which is shown in Section IV to be useful to derive a tight upper–bound of the ASEP for the asynchronous scenario.

−1

dξ

(3)

+∞ with Γ (x) = 0 tz−1 exp (t) dt being the Gamma function; and Eai ,θi ,¯ai ,θ¯i ,φi ,Di {·} being the expectation computed over information symbols, channel phases, and propagation delays of the interferers. It is  worth noticing that si = ai exp (jθi ) and s¯i = a ¯i exp j θ¯i denote a pair of consecutive symbols transmitted by the same interfering user. Two symbols have to be considered because of the assumption of asynchronous communication system. Also, since the interfering signals are i.i.d., then Eai ,θi ,¯ai ,θ¯i ,φi ,Di {·} is the same for each interferer. The result summarized in (1) is highly remarkable as it avoids the need of pure network simulations. However, this framework is semi–analytical as the expectation EB {·} must be computed by using Monte Carlo methods. Furthermore, Eai ,θi ,¯ai ,θ¯i ,φi ,Di {·} in (2) must be computed numerically too. The main objective of this paper is to provide a simplified expression of (1) by completely avoiding Monte Carlo simulations. We show that a simple and exact single–integral

(5)

ASEP =

1 π



M −1 π M

0

 0

+∞

 MZξ (z; ξ) dz dξ

(4)

where MZξ (p; ξ) = EZξ {exp (−pZξ )} is the Moment (MGF) of RV Zξ = 1 +  2 Generating

2 Function  sin (π/M ) sin (ξ) η . Thus, MZξ (p; ξ) = B

by definition,   exp (−p) MηB psin2 (π/M ) sin2 (ξ) with MηB (p) = EηB {exp (−pηB )}. MηB (·) can be re–written, after some algebra, as shown (a)

in (5) on top of this page, where: i) = follows from the (b) definition of RV ηB in (1); ii) = follows from [22, Theorem 1] by introducing the RV ηA = 1/ηB , with MηA (p) = EηA {exp (−pηA )} being the MGF of ηA and Jν (·) being the Bessel function of the first kind and order ν [23, Ch. (c) 9]; iii) = follows from of ηA , where the  the definition  2b0 2 2 κ0 Ω0 S0 , SNR = E0 /N0 , and constants Ψ0 = R0 INR = EI /N0 have been introduced. In particular, SNR and INR are the symbol–energy–to–noise–spectral–density–ratio of intended and interfering links, respectively. Furthermore, (d) MB (p) = EB {exp (−pB)} is the MGF of RV B; and iv) = follows from [20], where it is shown that the MGF of the real skewed stable RV B is available in closed–form and is equal  to MB (p) = exp −p1/bI . Thus, by substituting (5) in (4), the ASEP can be re–written as shown in (6) on top of this page, where T (·, ·) is given in (a)

(7) on top of this page. In particular, in (7): = follows from [24, Eq. (6.631)] with 1 F1 (·; ·; ·) being the Kummer confluent (b)

hypergeometric function [23, Ch. 13]; and ii) = follows from the identity 1 F1 (2; 2; ±x) = exp (±x).

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ASEP =

ASEP =

sin (π/M ) M −1 π− √ M 2 π

Eθ¯i ,φi

 0

1 M −1 π− M π

+∞



+∞ 0

      Ψ0 INR 1/bI 1/bI t exp − Ψ0 σI2 exp − t Q (t) dt SNR SNR

      π  π   √ 1 INR 1/bI 1/bI  Ψ0 √ exp − + sin2 t exp − Ψ0 σI2 dt (10) t t cos 1 + erf SNR M SNR M t

   −1  +∞   +∞      1 d 1/b MΥI (x)  ξ (1/bI −1) J0 2 ξt dξ dt ΥI I = − Γ bI dx 0 0 x=t    −1  +∞   (a) 1 d −1/b I = − Γ 1− MΥI (x)  t dt bI dx 0 x=t

Furthermore, by substituting (7) in (6), the ASEP simplifies as shown in (8) on top of this page, where Q (·) is equal to:   sin2 (π/M ) sin2 (π/M ) exp − t dξ 2 2 sin (ξ) sin (ξ) 0  π  π  (a) 1 π (9) = sin exp −t sin2 2 t M M  π  π  π  √ 1 π sin + t cos exp −t sin2 erf 2 t M M M 

Q (t) =

(8)

M −1 π M

  √ x where erf (x) = (2/ π) 0 exp −t2 dt is the error function,

that appear in it still need some expectations to be computed numerically. More specifically, XI in (2) is not available in closed–form. The aim of this section is to provide a closed– form expression of (2) for a network setup where all the {\TX } transmitters are synchronous, i.e., Di = 0 for i ∈ ΦTX 0 . In Section IV, we will show that the synchronous scenario turns out to be a tight upper–bound for the asynchronous scenario. {\TX } If Di = 0 for i ∈ ΦTX 0 , since for M–PSK modulation {\TX } ¯i = 1 for i ∈ ΦTX 0 , then (2) simplifies as: we have ai = a     2/b   2σI2 1 Γ 1 + XI = exp Eθ¯i ,φi cos θ¯i + φi  I b2I bI       2σI2 (a) 1 1/b = exp Γ 1+ Eθ¯i ,φi ΥI I 2 bI bI 

(a)

and = is obtained from:  2 solution of the  i) the closed–form 2 t sin (ξ) dξ = indefinite integral  μ2 sin2 (ξ) exp √−μ  √ −2 πμt−1/2 exp −μ2 erf μ cot (ξ) t ; and ii) the notable limit lim {erf (ε cot (x))} = 1 for any constant ε > 0. x→0 Finally, by substituting (9) in (8), the final single–integral expression of the ASEP in (10) on top of this page can be obtained. We can notice that, in spite of the complexity of the problem at hand, the ASEP in (10) is very simple with a computational complexity that is similar to many conventional modulation schemes without network interference [19]. For example, the single–integral in (10) can be efficiently computed by using conventional quadrature rule methods [23, Ch. 25], e.g., the Gauss–Chebyshev Quadrature (GCQ) integration rule [25, Eq. (9)]. Once again, we emphasize that, unlike [1], neither Monte Carlo methods nor the generation of alpha stable RVs are needed to compute (10). As a sanity check, we have verified that when INR → 0 ⇒ EI → 0, the framework reduces to well–known formulas without interference [19]. Furthermore, for high SNR, simple closed–form and asymptotically–tight approximations can be found [26]. Due to space limitations, these details are here omitted. Finally, we would like to emphasize that the main design parameters of the system are clearly shown in (10), such as the symbol transmit energy of intended and interfering links in SNR and INR, respectively, the fading/shadowing/path–loss parameters of intended and interfering links in Ψ0 and σI2 , respectively, the modulation order M , and the path–loss exponent of the interfering links bI . Thus, the obtained framework is not only simple to compute and completely avoids Monte Carlo methods, but it also allows us to readily understand the system performance under a variety of practical scenarios. To the best of the authors knowledge, this framework is new and is not available in the open technical literature. A. Closed–Form Expression of XI for Synchronous Systems

(14)

(11)

  2 (a) where in = we have introduced the RV ΥI = cos θ¯i + φi .   1/b By using [21, Eq. (6)], Eθ¯i ,φi ΥI I , can be written as: Eθ¯i ,φi

    1/b 1/b Υ I I = E ΥI Υ I I   −1  +∞ 1 ξ (1/bI −1) M1/ΥI (ξ) dξ = Γ bI 0

(12)

where M1/ΥI (·) is the MGF of RV 1/ΥI . This latter MGF, can be computed using [22, Theorem 1], as follows:  +∞  √ M1/ΥI (p) = 1 − p t−1/2 J1 2 pt MΥI (t) dt 0   +∞  d  (a) =− MΥI (x)  J0 2 pt dt dx 0 x=t

(13)

(a)

where: i) MΥI (·) is the MGF of RV ΥI ; and ii) = follows by using indefinite integral −1/2integration √ by parts and the notable √ t J1 (2 pt) dt = −p−1/2 [2J0 ( pt) − 1]. By substituting (13) in (12), we can obtain (14) on top of (a) this page, where = is obtained by first expressing the Bessel function J0 (·) in terms of the Meijer–G function [27, Eq. (8.4.19.1)], and then computing the Mellin transform of the Meijer–G function with the help of [27, Eq. (2.24.2.1)]. To solve the integral in (14), a closed–form expression of MΥI (·) is needed. By definition, we have:    2  exp −p cos θ¯i + φi   2π p p   (b)  (a) 1 = I0 exp −p cos2 θ¯i + ξ dξ = exp − 2π 0 2 2 (15)

MΥI (p) = EΥI

(a)

Even though, as mentioned in Section III, (10) needs no Monte Carlo methods for its computation, some parameters

{\TX } where: i) = follows because RV φi for i ∈ ΦTX 0 is (b)

uniform distributed in [0, 2π); and ii) = can be obtained from

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Eθ¯i ,φi

  −1  +∞ −1  +∞             t t 1 t t 1 1/b t−1/bI exp − t−1/bI exp − ΥI I = 2Γ 1 − I0 dt − 2Γ 1 − I1 dt bI 2 2 bI 2 2 0 0  −1      −1       √ √ (a) 1 1 1 1 1 1 1 1 = 2 πΓ Γ − Γ 2− − Γ 1+ Γ − 2 πΓ 1 − bI bI 2 bI bI bI bI 2

0

(16)

0

10

10

1

1

10

10

2

2

ASEP

10

ASEP

10

3

3

10

10 INR = 20dB INR = 40dB INR = 60dB INR = 80dB INR = 100dB

4

10

5

10

0

10

INR = 20dB INR = 40dB INR = 60dB INR = 80dB INR = 100dB

4

10

20

30

40

50

5

10

1 0 1

SNR [dB]

0

10

20

30

40

50

SNR [dB]

Fig. 1. ASEP of M–PSK modulation for the asynchronous system setup. Solid lines show the analytical model and markers show Monte Carlo simulations. Setup: i) M = 4; ii) R0 = 1, κ0 = 1, G0 = 1, S0 = exp (σ0 G0 ); (dB) (dB) iii) b0 = bI = 4; iv) σ0 = σI = 3dB; and v) λ = 10−3 .

[28, Eq. (14)] with Iν (·) being the modified Bessel function of the first kind and order ν [23, Ch. 9]. Thus, by substituting (15) in (14), and computing the (a) derivative, we obtain (16) on top of this page, where = is obtained by solving both integrals with the help of the Meijer– G function and the Mellin transform theorem [27]. More specifically, each integral can be solved by first expressing the Bessel function I0 (·) in terms of the Meijer–G function [27, Eq. (8.4.22.3)], and then computing the Mellin transform of the Meijer–G function with the help of [27, Eq. (2.24.2.1)]. Finally, by substituting (16) in (11) we obtain a very simple closed–form expression of XI for the synchronous scenario. IV. N UMERICAL AND S IMULATION R ESULTS In this section, we compare our analytical framework with Monte Carlo simulations. As far as Monte Carlo simulations are concerned, we have simulated a real network with nodes randomly distributed in the 2D Euclidean plane according to a homogeneous Poisson point process (see Section II). We have implemented the whole communication system with modulator, channel, and demodulator. In particular, the same signal model as in [1, Eq. (1)] has been accurately implemented, and no a priori model has been assumed for the network interference. Of course, even though in our system model the interference is distributed in the whole 2D Euclidean plane, in practice only a finite, even though very large, area can be simulated. The radius of this area has been determined by using the approach proposed in [6] and [29]. With the simulation parameters used in our study, which are summarized in the caption of each figure, we have found that a radius equal to 40 is sufficient to avoid truncation problems due to the finite simulation area. In Figs. 1–4, we study the accuracy of our framework for two system setups.

Fig. 2. ASEP of M–PSK modulation for the synchronous system setup. Solid lines show the analytical model and markers show Monte Carlo simulations. Setup: i) M = 4; ii) R0 = 1, κ0 = 1, G0 = 1, S0 = exp (σ0 G0 ); iii) (dB) (dB) b0 = bI = 4; iv) σ0 = σI = 3dB; and v) λ = 10−3 .

• In Fig. 1 and Fig. 2, we consider a scenario where the interfering nodes have a constant transmit energy, i.e., INR is fixed, while the intended user increases its transmit energy for better performance, i.e., SNR increases. Overall, we can observe that the ASEP gets better when SNR increases, while it gets worse when INR increases, as expected. The interesting result is that no error–floor can be observed for this case study. As far as the accuracy of the proposed analytical derivation is concerned, we can see that simulation and framework closely overlap. Furthermore, by comparing the results for the asynchronous scenario in Fig. 1, where XI in (2) is computed numerically, with the results for the synchronous scenario in Fig. 2, where XI is computed using the closed–form formula in (11) and (16), we can observe that the ASEP is almost the same. In fact, the difference between the curves is negligible. However, a close inspection reveals that the curves in Fig. 2 are a tight upper–bound of the curves in Fig. 1. • In Fig. 3 and Fig. 4, we consider a scenario where the interfering nodes have the same transmit energy as the intended user, i.e., INR = SNR. Overall, we can observe that the ASEP gets better when SNR increases, while it gets worse when the density, λ, of the interfering nodes increases, as expected. However, unlike the first case study analyzed in Fig. 1 and Fig. 2, we can clearly observe an error–floor when SNR increases. The reason is that in Fig. 3 and Fig. 4 the interfering nodes increase their transmit energy together with the intended user. Thus, the system tends to be heavily interference limited. Similar to Fig. 1 and Fig. 2, our analytical model is very accurate, and there is only a negligible difference between synchronous and asynchronous scenarios. Thus, the closed– form expression of XI in (11) and (16) can efficiently be used to get tight upper–bound estimates for the asynchronous scenario too.

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of the system, and can efficiently be exploited for network planning and system optimization.

0

10

ACKNOWLEDGMENT

1

10

This work is supported, in part, by the research projects “GREENET” (PITN–GA–2010–264759), “WSN4QoL” (IAPP–GA–2011–286047), “HYCON2” (NoE–GA–2010–257462), and the Lifelong Learning Programme (LLP) – ERASMUS Placement.

2

ASEP

10

R EFERENCES 3

10

10

λ = 10

5

λ = 10

λ = 104

4

10

λ = 103 λ = 102 1

λ = 10

5

10

0

10

20

30

40

50

SNR [dB]

Fig. 3. ASEP of M–PSK modulation for the asynchronous system setup. Solid lines show the analytical model and markers show Monte Carlo simulations. Setup: i) M = 16; ii) R0 = 1, κ0 = 1, G0 = 1, S0 = exp (σ0 G0 ); (dB) (dB) iii) b0 = bI = 4; iv) σ0 = σI = 3dB; and v) SNR = INR.

0

10

1

10

2

ASEP

10

3

10

10

λ = 10

λ = 105 λ = 104

4

10

λ = 103 λ = 102 λ = 101

5

10

0

10

20

30

40

50

1 0 1

SNR [dB]

Fig. 4. ASEP of M–PSK modulation for the synchronous system setup. Solid lines show the analytical model and markers show Monte Carlo simulations. Setup: i) M = 16; ii) R0 = 1, κ0 = 1, G0 = 1, S0 = exp (σ0 G0 ); iii) (dB) (dB) b0 = bI = 4; iv) σ0 = σI = 3dB; and v) SNR = INR.

V. C ONCLUSION In this paper, we have proposed a new single–integral framework to compute the ASEP of M–PSK modulation in the presence of network interference produced by a large number of interferers, which are distributed according to a homogeneous Poisson point process. Unlike state–of–the–art frameworks, our formula requires no Monte Carlo methods to remove the conditioning over alpha stable interference. Furthermore, numerical examples have shown that the performance of an intended link in the presence of synchronous and asynchronous interference is almost the same. This result is very interesting because our framework can be further simplified, as closed–form expressions for all the parameters of interest are available for the synchronous scenario. Finally, the numerical study has shown that the performance of the intended user might change significantly in the presence of interferers with constant or non–constant transmit energy. In summary, our extensive numerical simulations have shown that the proposed framework can accurately predict the behavior

[1] P. C. Pinto and M. Z. Win, “Communication in a Poisson field of interferers – Part I: Interference distribution and error probability”, IEEE Trans. Wireless Commun., vol. 9, no. 7, pp. 2176–2186, July 2010. [2] V. Chandrasekhar, J. G. Andrews, and A. Gatherer, “Femtocell networks: A survey”, IEEE Commun. Mag., vol. 46, no. 9, pp. 59–67, Sep. 2008. [3] A. Rabbachin, T. Q. S. Quek, S. Hyundong, M. Z. Win, “Cognitive network interference”, IEEE J. Sel. Areas Commun., vol. 29, pp. 480–493, Feb. 2011. [4] J. G. Andrews, R. K. Ganti, M. Haenggi, N. Jindal, and S. Weber, “A primer on spatial modeling and analysis in wireless networks”, IEEE Commun. Mag., vol. 48, no. 11, pp. 156–163, Nov. 2010. [5] J. Venkataraman, M. Haenggi, and O. Collins, “Shot noise models for outage and throughput analyses in wireless ad hoc networks”, IEEE Military Commun. Conf., pp. 1–7, Oct. 2006. [6] M. Haenggi and R. K. Ganti, Interference in Large Wireless Networks, Foundations and Trends in Networking, vol. 3, no. 2, pp 127–248, 2009. [7] F. Baccelli and B. Blaszczyszyn, Stochastic Geometry and Wireless Networks, Part I: Theory, Part II: Applications, Now Publishers, Sep. 2009. [8] M. Haenggi, J. G. Andrews, F. Baccelli, O. Dousse, and M. Franceschetti, “Stochastic geometry and random graphs for the analysis and design of wireless networks”, IIEEE J. Sel. Areas Commun., vol. 27, pp. 1029–1046, Sep. 2009. [9] M. Z. Win, P. C. Pinto, and L. A. Shepp, “A mathematical theory of network interference and its applications”, IEEE Proc., vol. 97, pp. 205–230, Feb. 2009. [10] J. G. Andrews, F. Baccelli, and R. K. Ganti, “A tractable approach to coverage and rate in cellular networks”, IEEE Trans. Commun., vol. 59, no. 11, pp. 3122–3134, Nov. 2011. [11] H. S. Dhillon, R. K. Ganti, F. Baccelli, and J. G. Andrews, “Modeling and analysis of K–tier downlink heterogeneous cellular networks”, IIEEE J. Sel. Areas Commun., (to appear), Dec. 2011. [Online]. http://arxiv.org/abs/1103.2177. [12] S. Ambike, J. Ilow, and D. Hatzinakos, “Detection for binary transmission in a mixture of gaussian noise and impulsive noise modeled as an alpha–stable process”, IEEE Signal Process. Lett., vol. 1, no. 3, pp. 55–57, Mar. 1994. [13] G. A. Tsihrintzis and C. L. Nikias, “Performance of optimum and suboptimum receivers in the presence of impulsive noise modeled as an alpha–stable process”, IEEE Trans. Commun., vol. 43, no. 2/3/4, pp. 904–914, Feb./Mar./Apr. 1995. [14] Y. Shobowale and K. A Hamdi, “A unified model for interference analysis in unlicensed frequency bands”, IEEE Trans. Wireless Commun., vol. 8, no. 8, pp. 4004–4013, Aug. 2009. [15] S. Niranjayan and N. C. Beaulieu, “The BER optimal linear rake receiver for signal detection in symmetric alpha–stable noise”, IEEE Trans. Commun., vol. 57, no. 12, pp. 3585–3588, Dec. 2009. [16] S. Niranjayan and N. C. Beaulieu, “BER optimal linear combiner for signal detection in symmetric alpha–stable noise: Small values of alpha”, IEEE Trans. Wireless Commun., vol. 9, no. 3, pp. 886–890, Mar. 2010. [17] A. Rajan and C. Tepedelenlioglu, “Diversity combining over Rayleigh fading channels with symmetric alpha stable noise”, IEEE Trans. Wireless Commun., vol. 9, no. 9, pp. 2968–2976, Sep. 2010. [18] J. Lee and C. Tepedelenlioglu, “Space–time coding over fading channels with stable noise”, IEEE Trans. Vehic. Technol., (to appear), Feb. 2011. [Online]. Available: http://arxiv.org/abs/1102.3392. [19] M. K. Simon and M.–S. Alouini, Digital Communication over Fading Channels, John Wiley & Sons, Inc., 1st ed., 2000. [20] G. Samorodnitsky and M. S. Taqqu, Stable Non Gaussian Random Processes Stochastic Models with Infinite Variance, Chapman and Hall/CRC, June 1994. [21] M. Di Renzo, F. Graziosi, and F. Santucci, “Further results on the approximation of log–normal power sum via Pearson type IV distribution: A general formula for log–moments computation”, IEEE Trans. Commun., vol. 57, no. 4, pp. 893–898, Apr. 2009. [22] M. Di Renzo, F. Graziosi, and F. Santucci, “A unified framework for performance analysis of CSI–assisted cooperative communications over fading channels”, IEEE Trans. Commun., vol. 57, no. 9, pp. 2552–2557, Sep. 2009. [23] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, June 1964. [24] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, 5th ed., Jan. 1994. [25] F. Yilmaz and M.–S. Alouini, “A unified MGF–based capacity analysis of diversity combiners over generalized fading channels”, IEEE Trans. Commun., (submitted), Dec. 2010. [Online]. http://arxiv.org/abs/1012.2596. [26] C. Merola, Network Interference Modeling and Performance Analysis of Next Generation Wireless Networks Through Stochastic Geometry, Master Thesis, University of L’Aquila, July 2011. [27] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series – Vol. 3: More Special Functions, ISBN 5–9221–0325–3, 2003. [28] M. Di Renzo and H. Haas, “A general framework for performance analysis of space shift keying (SSK) modulation for MISO correlated Nakagami–m fading channels”, IEEE Trans. Commun., vol. 58, pp. no. 9, pp. 2590–2603, Sep. 2010. [29] S. Weber and M. Kam, “Computational complexity of outage probability simulations in mobile ad–hoc networks”, IEEE Conf. Inform. Sciences and Systems, pp. 1–5, Mar. 2005.

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