INVITED PAPER A Mathematical Theory of Network Interference and Its Applications A unifying framework is developed to characterize the aggregate interference in wireless networks, and several applications are presented.
By Moe Z. Win, Fellow IEEE, Pedro C. Pinto, Student Member IEEE, and Lawrence A. Shepp
ABSTRACT | In this paper, we introduce a mathematical I. INTRODUCTION framework for the characterization of network interference in In a wireless network composed of many spatially wireless systems. We consider a network in which the scattered nodes, communication is constrained by various interferers are scattered according to a spatial Poisson process impairments such as the wireless propagation effects, and are operating asynchronously in a wireless environment network interference, and thermal noise. The effects subject to path loss, shadowing, and multipath fading. We start introduced by propagation in the wireless channel include by determining the statistical distribution of the aggregate the attenuation of radiated signals with distance (path network interference. We then investigate four applications of loss), the blocking of signals caused by large obstacles the proposed model: 1) interference in cognitive radio net- (shadowing), and the reception of multiple copies of the works; 2) interference in wireless packet networks; 3) spectrum same transmitted signal (multipath fading). The network of the aggregate radio-frequency emission of wireless net- interference is due to accumulation of signals radiated by works; and 4) coexistence between ultrawideband and nar- other transmitters, which undesirably affect receiver nodes rowband systems. Our framework accounts for all the essential in the network. The thermal noise is introduced by the physical parameters that affect network interference, such as receiver electronics and is usually modeled as additive the wireless propagation effects, the transmission technology, white Gaussian noise (AWGN). the spatial density of interferers, and the transmitted power of the The modeling of network interference is an important interferers. problem, with numerous applications to the analysis and design of communication systems, the development of KEYWORDS | Aggregate network interference; coexistence; interference mitigation techniques, and the control of cognitive radio; emission spectrum; packet networks; spatial electromagnetic emissions, among many others. In partic- Poisson process; stable laws; ultrawideband systems; wireless ular, interference modeling has been receiving increased networks interest in the context of ultrawideband (UWB) technol- ogies, which use extremely large bandwidths and typically operate as an underlay system with other existing narrowband (NB) technologies, such as GSM, GPS, Wi-Fi, and WiMAX. The most common approach is to
Manuscript received June 16, 2008; revised October 3, 2008. Current version published model the interference by a Gaussian random process March 18, 2009. This work was supported by the Portuguese Science and Technology [1]–[6]. This is appropriate, for example, when the inter- Foundation under Grant SFRH-BD-17388-2004, the Charles Stark Draper Laboratory Robust Distributed Sensor Networks Program, the Office of Naval Research under ference is the accumulation of a large number of inde- Presidential Early Career Award for Scientists and Engineers (PECASE) pendent signals, where no term dominates the sum, and N00014-09-1-0435, and the National Science Foundation under Grant ANI-0335256. M. Z. Win and P. C. Pinto are with the Laboratory for Information and Decision thus the central limit theorem (CLT) applies [7]. The Systems, Massachusetts Institute of Technology, Cambridge, MA 02139 USA Gaussian process has many well-studied properties and (e-mail: [email protected]; [email protected]). L. A. Shepp is with the Department of Statistics, RutgersVThe State University, often leads to analytically tractable results. However, there Piscataway, NJ 08854 USA (e-mail: [email protected]). are several scenarios where the CLT does not apply, e.g., Digital Object Identifier: 10.1109/JPROC.2008.2008764 when the number of interferers is large but there are
0018-9219/$25.00 2009 IEEE Vol. 97, No. 2, February 2009 | Proceedings of the IEEE 205 Win et al.: A Mathematical Theory of Network Interference and Its Applications
dominant interferers [8]–[11]. In particular, it is known that effects (e.g., Nakagami-m fading and log-normal shadow- the CLT gives a very poor approximation for modeling the ing). The main contributions of the paper are as follows. multiple access interference in time-hopping UWB systems • Distribution of the network interference: We provide [12], [13]. In many cases, the probability density function a complete probabilistic characterization of the (pdf) of the interference exhibits a heavier tail than what is aggregate network interference generated by the predicted by the Gaussian model. Several mathematical nodes in a wireless network. Our analysis reveals models for impulsive interference have been proposed, the innate connection between the distribution of including the Class A model [14], [15], the Gaussian-mixture the network interference and various important model [16], [17], and the stable model [18], [19]. system parameters, such as the path loss exponent, A model for network interference should capture the the transmitted power, and the spatial density of essential physical parameters that affect interference, the interferers. namely: 1) the spatial distribution of the interferers • Interference in cognitive radio networks: We char- scattered in the network; 2) the transmission character- acterize the statistical distribution of the network istics of the interferers, such as modulation, power, and interference generated by the secondary users synchronization; and 3) the propagation characteristics of when they are allowed to transmit in the same the medium, such as path loss, shadowing, and multipath frequency band of the primary users according to a fading. The spatial location of the interferers can be spectrum-sensing protocol. We analyze the effect modeled either deterministically or stochastically. Deter- of the wireless propagation characteristics on such ministic models include square, triangular, and hexagonal distribution. lattices in the two-dimensional plane [20]–[23], which are • Interference in wireless packet networks: We obtain applicable when the location of the nodes in the network is expressions for the throughput of a link, consid- exactly known or is constrained to a regular structure. ering that a packet is successfully received if the However, in many scenarios, only a statistical description signal-to-interference-plus-noise ratio (SINR) ex- of the location of the nodes is available, and thus a sto- ceeds some threshold. We analyze the effect of the chastic spatial model is more suitable. In particular, when propagation characteristics and the packet traffic the terminal positions are unknown to the network de- on the throughput. signer a priori, we may as well treat them as completely • Spectrum of the aggregate network emission: We derive random according to a homogeneous Poisson point process the power spectral density (PSD) of the aggregate [24].1 The Poisson process has maximum entropy among all radio-frequency (RF) emission of a network. Then, homogeneous processes [25] and corresponds to a simple we put forth the concept of spectral outage and useful model for the location of nodes in a network. probability (SOP) and present some of its applica- The application of the spatial Poisson process to tions, including the establishment of spectral regula- wireless networks has received increased attention in the tions and the design of covert military networks. literature. In the context of the Poisson model, the issues • Coexistence between UWB and NB systems: We con- of wireless connectivity and coverage were considered in sider a wireless network composed of both UWB and [26]–[32], while the packet throughput was analyzed in NB nodes. We first determine the statistical distribu- [33]–[38]. The error probability and link capacity in the tion of the aggregate UWB interference at any loca- presence of a Poisson field of interferers were investigated tion in the two-dimensional plane. Then, we in [39]–[43]. The current literature on network interfer- characterize the error probability of a given NB vic- ence is largely constrained to some particular combination tim link subject to the aggregate UWB interference. of propagation model, spatial location of nodes, transmis- Our framework accounts for all the essential physical sion technology, or multiple-access scheme; and the exist- parameters that affect network interference and provides ing results are not easily generalizable if some of these fundamental insights that may be of value to the network system parameters are changed. Furthermore, none of the designer. In addition, this framework can be applied to mentioned studies attempts a unifying characterization various other areas, including noncoherent UWB systems that incorporates various metrics, such as outage probabi- [44], transmitted reference UWB systems [45], and lity, packet throughput, power spectral density, and error physical-layer security [46]. probability. This paper is organized as follows. Section II describes In this paper, we present a unifying framework for the the system model. Section III derives the representation characterization of network interference. We consider that and distribution of the aggregate interference. Section IV the interferers are scattered according to a spatial Poisson characterizes the interference due to secondary users in process and are operating in a wireless environment sub- cognitive radio networks. Section V analyzes the packet ject to path loss as well as arbitrary fading and shadowing throughput of a link subject to network interference. Section VI characterizes the spectrum of the network 1The spatial Poisson process is a natural choice in such situations because, given that a node is inside a region R, the pdf of its position is interference. Section VII characterizes the error probabil- conditionally uniform over R. ity of NB communication in the presence of UWB network
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interference. Section VIII concludes this paper and summarizes important findings.
II. SYSTEM MODEL
A. Spatial Distribution of the Nodes We model the spatial distribution of the nodes according to a homogeneous Poisson point process in the two- dimensional infinite plane. The probability of n nodes being inside a region R (not necessarily connected) depends only on the total area AR of the region and is given by [24]
n ð ARÞ IP fn in Rg ¼ e AR ; n 0 n! Fig. 1. Poisson field model for the spatial distribution of nodes. Without loss of generality, we assume the origin of the coordinate where is the (constant) spatial density of interfering system coincides with the probe receiver. nodes, in nodes per unit area. We define the interfering nodes to be the terminals that are transmitting within the fre- quency band of interest, during the time interval of interest, fZkg are independent random variables (RVs), which and hence are effectively contributing to the total interfer- account for propagation effects such as multipath fading ence. Then, irrespective of the network topology (e.g., point- and shadowing. The term 1=R2b accounts for the far-field to-point or broadcast) or multiple-access technique (e.g., path loss with distance R, where the amplitude loss time or frequency hopping), the above model depends only exponent b is environment-dependent and can approxi- on the density of interfering nodes.2 mately range from 0.8 (e.g., hallways inside buildings) to 4 TheproposedspatialmodelisdepictedinFig.1.For (e.g., dense urban environments), with b ¼ 1correspond- analytical purposes, we assume there is a probe link ing to free-space propagation [48].5 The proposed model is composed of two nodes: the probe receiver, located at the general enough to account for various propagation origin of the two-dimensional plane (without loss of scenarios, including the following. generality), and the probe transmitter (node i ¼ 0), 1) Path loss only: Z1 ¼ 1. 2 deterministically located at a distance r0 from the origin. 2) Path loss and Nakagami-mfading:Z1 ¼ ,where All other nodes ði ¼ 1 ...1Þ are interfering nodes, whose 2 Gðm; 1=mÞ.6 1 2 G random distances to the origin are denoted by fRigi¼1, 3) Path loss and log-normal shadowing: Z1 ¼ e , 7 2 G where R1 R2 . where G Nð0; 1Þ. The term e has a log- normal distribution, where is the shadowing B. Wireless Propagation Characteristics coefficient.8 In many cases of wireless systems design and analysis, 4) Path loss, Nakagami-m fading, and log-normal 2 2 it is sufficient to consider the power relationship between shadowing: Z1 ¼ with Gðm; 1=mÞ,and 2 G the transmitter and receiver to account for the propagation Z2 ¼ e with G Nð0; 1Þ. characteristics of the environment. Specifically, we For cases where the evolution of signals over time is of consider that the power Prx received at a distance R from interest, we consider the waveform relationship between a transmitter is given by
5 2b Q As we shall see in the rest of this paper, the simple 1=R dependence allows us to obtain valuable insights in numerous wireless Ptx k Zk scenarios. However, care must be taken for extremely dense networks, Prx ¼ (1) R2b since the close proximity of nodes to the origin would invalidate this far- field dependence. 6We use Gðx; Þ to denote a gamma distribution with mean x and 2 where Ptx is the average power measured 1 m away from variance x . 7 2 the transmitter3; b is the amplitude loss exponent4;and We use Nð ; Þ to denote a Gaussian distribution with mean and variance 2. 8This model for combined path loss and log-normal shadowing can be 2Time and frequency hopping can be easily accommodated in this expressed in logarithmic form [48]–[50], such that the channel loss in model, using the splitting property of Poisson processes [47, Section 6] to decibels is given by LdB ¼ k0 þ k1 log10 r þ dBG, where G Nð0; 1Þ. obtain the effective density of nodes that contribute to the interference. The environment-dependent parameters ðk0; k1; dBÞ can be related to 3 Unless otherwise stated, we will simply refer to Ptx as the ðb; Þ as follows: k0 ¼ 0, k1 ¼ 20b,and dB ¼ð20= ln 10Þ .The Btransmitted power.[ parameter dB is the standard deviation of the channel loss in dB (or, 4Note that the amplitude loss exponent is b, while the corresponding equivalently, of the received SNR in decibels) and typically ranges from power loss exponent is 2b. 6 to 12.
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the transmitter and receiver in terms of their equivalent a given realization of P.Asweshallsee,this low-pass (ELP) representations. Specifically, the ELP naturally leads to the derivation of outage metrics received signal can be written as9 [51]–[54] such as interference outage probability, spectral outage probability, and error outage prob- ability, which in the described scenario are more Q pffiffiffiffiffi Z Z meaningful than the corresponding P-averaged YðtÞ¼ k k hðt; ÞXðt Þd (2) Rb metrics. 2) Fast-varying P:Inthiscase,theinterferershavea short session lifetime, where each node periodi- cally becomes active, transmits a burst of symbols, where the kernel hðt; Þ is the time-varying ELP impulse and then turns off (dynamic scenario). Then, the response of the multipath channel, and XðtÞ is the ELP set of interfering nodes (i.e., nodes that are trans- transmitted signal. In this case, the RVs fZ g account for k mitting and contributing to the interference) the slow-varying propagation effects (typically, slow- changes often, and hence P experiences a varia- varying log-normal shadowing), while hðt; Þ implicitly tion analogous to that of a fast block fading model. accounts for the multipath fading. A canonical example In this scenario, it is insightful to average the for hðt; Þ is the tapped-delay line model given by interference analysis over P,whichnaturally leads to the derivation of the average metrics. X ÀÁ The framework proposed in this paper enables the analysis |2 fc qðtÞ hðt; Þ¼ hqðtÞe qðtÞ (3) of both scenarios. In the rest of this paper, we illustrate q several applications of our framework for the case of slow- varying P only. Some examples of the analysis for fast- varying P can be found in [55] and [56]. where fc is the carrier frequency; hqðtÞ and qðtÞ are, respectively, the time-varying amplitudes and delays associated with the qth multipath; and ðtÞ denotes the III. INTERFERENCE REPRESENTATION Dirac-delta function. AND DISTRIBUTION For the purpose of this paper, the power relation in (1) In this section, we consider a general model for interfer- is sufficient for characterizing interference in cognitive ence, which will be used to investigate several applications radio networks in Section IV, and for analyzing interfer- in wireless networks throughout Sections V–VII. Specif- ence in packet wireless networks in Section V. The time- ically, let Y ¼½Y ; ...; Y T be a real random vector varying representation in (2) is useful for determining the 1 Nd ðRVÞ of arbitrary dimension Nd, representing the aggre- spectrum of the aggregate network emission in Section VI, gate interference at the probe receiver, located in the and for characterizing coexistence between UWB and NB origin of the two-dimensional plane (see Fig. 1). For exam- systems in Section VII. ple, the RVYcan correspond to a projection of the aggregate interference process YðtÞ onto some set of basis functions C. Mobility and Session Lifetime of the Interferers Nd 1 f iðtÞgi¼1. We can express the aggregate interference Y as Typically, the time variation of the distances fRigi¼1 of the interferers is highly coupled with that of the shadowing 1 X1 fGigi¼1 affecting those nodes. This is because the shadow- Qi ing is itself associated with the movement of the nodes Y ¼ 1AðQi; RiÞ (4) Ri near large blocking objects. Thus, we introduce the nota- i¼1 B 1 tion P to succinctly denote the distances fRigi¼1 and 1 [ shadowing fGigi¼1 of the interferers. We consider the dis- where tance Ri and the shadowing Gi associated with each interferer to be approximately constant over at least the 1; ðq; rÞ2A duration of a symbol, i.e., RiðtÞ Ri and GiðtÞ Gi.Fur- 1Aðq; rÞ¼ (5) thermore, we can consider two scenarios that differ in the 0; otherwise speed of variation of P over larger time windows (e.g., the session lifetime of communication). T and Q ¼½Q ; ; ...; Q ; represents an arbitrary random P i i 1 i Nd 1) Slow-varying :Inthiscase,theinterferershavea quantity associated with interferer i. The purpose of the term long session lifetime during which P is approxi- Qi is to accommodate various propagation effects such as mately constant (quasi-static scenario). Thus, it is multipath fading and shadowing. The indicator function insightful to condition the interference analysis on 1Aðq; rÞ allows for the selection of the nodes that contribute to the aggregate interference–the active users–based on some 9 Boldface letters are used to denote complex quantities and vectors. condition relating Qi and Ri. The proposed model is general
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enough to accommodate various choices of the active set A, where QðwÞ is the characteristic function of Qi.Forthe including the following. case of A¼fðq; rÞ : ðjqj2=r Þ G P g,weobtain 1) If A¼fðq; rÞ : r 2Ig, then (4) represents the aggregate interference resulting from all the nodes ZZ |w q I inside a region described by .Forexample,if Yðw; AÞ¼exp 2 1 e r fQðqÞdqrdr (8) I¼½u; vÞ, then (4) represents the aggregate Q interference resulting from the nodes inside the G annulus u r v. 2 G 2 2b where Q¼fq : jqj P r g. 2) If A¼fðq; rÞ : ðjqj =r Þ G Pthg, then (4) repre- sents the aggregate interference resulting only from 2 2b B. Distribution of the the nodes for which the received power jQij =Ri at 10 Aggregate Interference Amplitude the origin is below some threshold P . th Using the general result in Section III-A, we now The distribution of the aggregate interference Y plays an characterize the distribution of the aggregate inter- important role in the design and analysis of wireless net- ference amplitude generated by all the nodes in the works, such as in determining the probabilities of inter- plane. The following corollary provides such statistical ference outage, spectral outage, and error outage. characterization.
A. Interference From the Active Set 1 Corollary 3.1 (Symmetric Stable Distribution): Let fRigi¼1 1 denote the sequence of distances between the origin and Theorem 3.1: Let fR g denotethesequenceof i i¼1 random points of a two-dimensional Poisson process with distances between the origin and the random points of a 1 spatial density .LetfQigi¼1 be a sequence of spherically two-dimensional Poisson process with spatial density . 11 T 1 symmetric (SS) RVs Qi ¼½Qi;1; ...; Qi;Nd , i.i.d. in i, Let fQigi¼1 be a sequence of Nd-dimensional RVs T independent of the sequence fR g.LetY denote the Q ¼½Q ; ...; Q , independent identically distributed i i i;1 i;Nd aggregate interference at the origin generated by the (i.i.d.) in i,andindependentofthesequencefR g.Let i nodes scattered in the infinite plane, such that YðAÞ denote the aggregate interference generated by the nodes from the active set A,suchthat X1 Q Y ¼ i X1 Rb Qi i¼1 i YðAÞ ¼ 1AðQi; RiÞ i¼1 Ri
for b > 1. Then, its characteristic function YðwÞ¼ IE fe|w Yg is given by where >1. Then, its characteristic function |w YðAÞ Yðw; AÞ ¼ IE fe g is given by YðwÞ¼expðÞ jwj (9) ZZ |w q 1Aðq;rÞ Yðw; AÞ ¼ exp 2 1 e r fQðqÞdqrdr where (6) 2 ¼ (10) where fQðqÞ is the pdf of Qi. b no Proof: This follows immediately from Campbell’s 1 2=b ¼ C IE jQi;nj (11) theorem [24]. Ì 2=b Using the above theorem with A¼fðq; rÞ : r 2Ig, we obtain and C is defined as Z hi w 1 ; 6¼ 1 Yðw; AÞ ¼ exp 2 1 Q rdr (7) ð2 Þ cosð =2Þ I r C ¼ 2 (12) ; ¼ 1
with ð Þ denoting the gamma function.
10 11 As we shall see in Section IV, this is useful for characterizing A RVXis said to be spherically symmetric if its pdf fXðxÞ depends interference in cognitive radio networks. only on jxj.
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Proof: Using (7) with ¼ b,andI¼½0; 1Þ,weget C. Distribution of the Aggregate Interference Power Z hi In Section III-B, we considered the distribution of the 1 w w exp : aggregate interference amplitude.Wenowfocusonthe Yð Þ¼ 2 1 Q b rdr 0 r aggregate interference power generated by all the nodes scattered in the plane, where each interferer contributes 2b If the RV Q has an SS pdf, its characteristic function is also with the term Pi=Ri .TheRVPi represents an arbitrary SS, i.e., QðwÞ¼ 0ðjwjÞ for some 0ð Þ. As a result quantity associated with interferer i and can incorporate variouspropagationeffectssuchasmultipathfadingor Z hi shadowing, as described in (1). The following corollary 1 w w exp provides such statistical characterization. Yð Þ¼ 2 1 0 b rdr 0 r 1 Corollary 3.2 (Skewed Stable Distribution): Let fRigi¼1 which, using the change of variable jwjr b ¼ t, can be denote the sequence of distances between the origin and the random points of a two-dimensional Poisson process rewritten as 1 with spatial density .LetfPigi¼1 be a sequence of i.i.d. Z real nonnegative RVs and independent of the sequence 1 = 2 1 0ðtÞ fR g.LetI denote the aggregate interference power at the ðwÞ¼exp jwj2 b : i Y 2=bþ1 dt b 0 t origin generated by all the nodes scattered in the infinite plane, such that Appendix I shows that X1 Pi Z I ¼ 1 2b 1 ðtÞ C 1 ÈÉ i¼1 Ri 0 IE þ1 dt ¼ jQi;nj (13) 0 t for b > 1. Then, its characteristic function IðwÞ¼ IE fe|wIg is given by where C is defined in (12), and thus we conclude that the Y characteristic function of has the simple form given by hi (9), with parameters and given by (10) and (11), ðwÞ¼exp jwj 1 | signðwÞ tan (15) respectively. This completes the proof. g I 2 Random variables with characteristic function of the form of ðwÞ in (9) belong to the class of symmetric stable Y where RVs. Stable laws are a direct generalization of Gaussian distributions and include other densities with heavier 1 (algebraic) tails. They share many properties with ¼ (16) Gaussian distributions, namely, the stability property and b the generalized central limit theorem [7], [57]. Equations ¼ 1(17) 12 no (9)–(11) can be succinctly expressed as 1 1=b ¼ C1=bIE Pi (18) no 2 1 2=b Y SNd Y ¼ ; Y ¼ 0; Y ¼ C2=bIE jQi;nj b and C is defined in (12). (14) Proof: Using (7) with Nd ¼ 1, ¼ 2b, I¼½0; 1Þ, we get and this notation will be used throughout this paper. Z hi 1 w 12 Sð ; ; Þ exp We use to denote the distribution of a real stable RV with IðwÞ¼ 2 1 P 2b rdr characteristic exponent 2ð0; 2 , skewness 2½ 1; 1 , and dispersion 0 r 2½0; 1Þ. The corresponding characteristic function is
ÀÁÂÃÀÁ where PðwÞ is the characteristic function of Pi.Usingthe 2b expÀÁ jwjÂÃ1 | signðwÞ tan 2 ; 6¼ 1 ðwÞ¼ðVARIABLEERROR 2 unrecognisedsyntaxÞ change of variable jwjr ¼ t, this can be rewritten as exp jwj 1 þ | signðwÞ ln jwj ; ¼ 1. ÈÉ! We use S ð ; ¼ 0; Þ to denote the distribution of an Nd-dimensional Z Nd 1 |signðwÞPit SS stable RV with characteristic exponent and dispersion , and 1=b 1 1 IE e ðwÞ¼exp jwj dt : whose characteristic function is ðwÞ¼expð jwj Þ. Note that each of I 1=bþ1 b 0 t the Nd individual components of the SS stable RV is Sð ; ¼ 0; Þ.
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Appendix II shows that according to an homogeneous Poisson process sec,with density sec. When a primary transmission occurs, the Z ÈÉ handshake procedure between the primary transmitter and 1 1 IE e| signðwÞPit dt receiver would trigger the primary receiver to transmit a t þ1 0 hi beacon with power Ppri, thus indicating its presence to the ÈÉC 1 secondary nodes. When a secondary user senses such ¼ IE P 1 | signðwÞ tan (19) i 2 beacon, it is not allowed to transmit, in order to reduce interference caused to the primary link. However, due to the wireless propagation effects such as multipath fading where C isdefinedin(12),andthusweconcludethatthe and shadowing, the secondary users may miss detecting characteristic function of I has the simple form given by this beacon, and may still transmit and interfere with the (15), with parameters , ,and given by (16)–(18). This primary link. The probability p ðrÞ that a secondary user completes the proof. g d detects the beacon transmitted by the primary receiver at a Random variables with characteristic function of the distance r is given by form IðwÞ in (15) belong to the class of skewed stable RVs. Using the notation introduced before, (15)–(18) can be Q succinctly expressed as Ppri k Zk pdðrÞ¼IP P (21) fZkg r2b no 1 1 1=b I S ¼ ; ¼ 1; ¼ C1=bIE Pi : (20) b where Ppri is the average transmit power of the beacon; P denotes the threshold for beacon detection (e.g., related to the detection sensitivity of the secondary In the remainder of this paper, we will show how the users); and the other parameters have the same meaning theory developed in this section is useful to characterize as in (1). interference in wireless networks, enabling a wide variety In what follows, we characterize the distribution of the of applications. network interference generated by all the secondary nodes that are not able to detect the primary beacon, and thus IV. INTERFERENCE IN COGNITIVE contribute to a performance degradation of the primary RADIO NETWORKS link. With the increasing need for higher data rates, the elec- A. Distribution of the Network Interference tromagnetic spectrum has become a scarce resource. Such Due to Secondary Users shortage is in part due to the current allocation policies, The power I of the network interference generated which allocate spectral bands for exclusive use of a single sec by the secondary nodes can be written as entity within a geographical area. Studies show that the licensed spectrum is mostly underused across time and geographical regions [58]. One solution that enables a more X1 PsecZi efficient use of the spectrum is cognitive radio, a new para- Isec ¼ 1Aðq; rÞ R2b digm in which a wireless terminalVthe secondary userVcan i¼1 i dynamically access unused licensed spectrum, while avoid- Q ing interference with its ownerVthe primary user [59]. One where Zi ¼ k Zi;k accounts for the random propagation example of such cognitive behavior is spectrum sensing, i.e., effects associated with node i and the ability to actively monitor the occupation of spectrum bands. It is of critical importance to develop models for cognitive radio networks, which not only quantify their P z A¼ ðz; rÞ : pri G P : communication performance but also provide guidelines for r2b the design of practical cognitive algorithms. In this section, we consider an application of the proposed framework to the characterization and control of Using (8) with Nd ¼ 1, ¼ 2b,andQ ¼ PsecZ, we can the aggregate interference in cognitive radio networks. write Specifically, we wish to quantify the performance of the primary users, when the secondary users are allowed to ! Z Z 2b 1 P r transmit in the same band, according to a spectrum- Ppri |wPsecz r2b sensing protocol. We consider the spatial model depicted Isec ðwÞ¼exp 2 sec 1 e fZðzÞrdzdr in Fig. 1, where a primary link operates in the presence of 0 0 secondary users, which are scattered in the plane (22)
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where fZðzÞ is the pdf of Zi. Alternatively, we can obtain an where equivalent expression for (22) by changing the order of integration as P r2bm 1 ¼ : Z Z !Ppri 1 1 |wPsecz ðwÞ¼exp 2 ÀÁ 1 e r2b f ðzÞrdrdz : Isec sec P z 1=2b Z 0 pri P For the particular case of Rayleigh fading ðm ¼ 1Þ, we obtain
Although in general Isec ðwÞ cannot be determined in closed form, it can be used to numerically evaluate [60], P r2b [61] various performance metrics of interest, such as the pdðrÞ¼exp : interference outage probability, given by Ppri
Pout ¼ IP fIsec > I g¼1 FI ðI Þ 3) Path loss and log-normal shadowing: In this case, (21) sec 2 G 2b reduces to pdðrÞ¼IP Gfe P r =Pprig, where G Nð0; 1Þ. Using the Gaussian Q-function, we where FIsec ð Þ is the cumulative distribution function (cdf) obtain of the RV Isec. 2b B. Effect of the Propagation Characteristics on pd 1 P r pdðrÞ¼Q ln : The ability of a secondary user to detect the primary 2 Ppri beacon is essential in controlling the interference gener- ated to the primary link. Therefore, it is important to understand the behavior of the detection probability pd for 4) Path loss, Nakagami-m fading, and log-normal various propagation conditions, including the four differ- shadowing: In this case, (21) reduces to 2 2 G 2b ent scenarios described in Section II-B. pdðrÞ¼IP ;Gf e P r =Pprig.Condition- 1) Path loss only: In this case, a secondary node can ing on G, using the cdf of a gamma RV, and detect the primary beacon if it is located inside a then averaging over G,weobtain 1=2b circle of radius ðPpri=P Þ around the primary receiver, also known as the exclusion or no-talk 1 P r2bm radius. Thus, (21) reduces to IE ; : pdðrÞ¼1 G inc m 2 G ðmÞ Pprie ( 1 Ppri 2b 1; 0 r pdðrÞ¼ P In [63], we show that if we consider m to be an 0; otherwise. integer and also approximate the moment gener- ating function of the log-normal RV e 2 G by a Gauss–Hermite series, p ðrÞ simplifies to 2) Path loss and Nakagami-mfading:In this case, (21) d 2 2b reduces to pdðrÞ¼IP f P r =Pprig,where ! 2 ; = Xm 1 XNp Gðm 1 mÞ. Using the cdf of a gamma RV, we ! k 2 ðm 1Þ 1ffiffiffi 2e obtain pdðrÞ 1 1 p Hx (24) n ! ðmÞ k¼0 n¼1 k 1 P r2bm pdðrÞ¼1 inc m; (23) where ðmÞ Ppri
R ffiffi x a 1 t P r2bm p where incða; xÞ¼ t e dt is the lower in- 2 2 xn 0 2 ¼ e complete gamma function. For integer m, we can Ppri express incða; xÞ in closed form [62], so that (23) simplifies to and xn and Hxn are, respectively, the zeros and the ! weights of the Np-order Hermite polynomial. m 1 X k 1 Both xn and Hxn are tabulated in [62, Table 25.10] ðm 1Þ! 1 e pdðrÞ¼1 1 for various polynomial orders N . Typically, ðmÞ k! p k¼0 setting Np ¼ 12 ensures that the approximation
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the case in which only path loss is present, the Rayleigh fading improves the performance of the primary link, while the log-normal shadowing degrades such performance.
V. INTERFERENCE IN WIRELESS PACKET NETWORKS A wireless network is typically composed of many spatially scattered nodes, which compete for shared network resources, such as the electromagnetic spectrum. A traditional measure of how much traffic can be delivered by such a network is the packet throughput. In a wireless environment, the throughput is constrained by various impairments that affect communication between nodes, such as wireless propagation effects, network interference, and thermal noise. Here we develop a framework that Fig. 2. Interference outage probability Pout versus the normalized quantifies the impact of all these impairments on the detection threshold P =Ppri for various wireless propagation 2 packet throughput, incorporating other important network characteristics (I =Psec ¼ 1, sec ¼ 0:1m , b ¼ 2, dB ¼ 10). parameters, such as the spatial distribution of nodes and their transmission characteristics [63], [65]. The proposed spatial model is depicted in Fig. 1. Our in (24) is extremely accurate [64]. For the goal is to determine the throughput of the probe link particular case of Rayleigh fading ðm ¼ 1Þ, (24) subject to the effect of all the interfering nodes in the simplifies to network. We consider that the power Prx received at a distance R from a transmitter is given by (1) and carry out the analysis in terms of general propagation effects fZkg. XNp 2b pffiffi 1 P r Interfering nodes are considered to have the same p ðrÞ pffiffiffi H exp e2 2 xn : d xn V n¼1 Ppri transmit power PI a plausible constraint when power control is too complex to implement, for example, in decentralized ad hoc networks. For generality, however, we allow the probe node to employ an arbitrary power P0, C. Numerical Results not necessarily equal to that of the interfering nodes. Fig. 2 illustrates the dependence of the interference We analyze the case of half-duplex transmission, where outage probability Pout on the beacon detection threshold each device transmits and receives at different time P for various wireless propagation characteristics. Note intervals, since full-duplexing capabilities are rare in that both the multipath fading and shadowing have two typical low-cost applications. Nevertheless, the results opposing effects in Pout:ononehand,ifasecondaryuser presented in this paper can be easily modified to account experiences a severely faded channel, then it misses the for the full-duplex case. We further consider the scenario primary beacon detection and still transmits, thus where all nodes transmit with the same traffic pattern. In increasing the Pout of the primary link; on the other particular, we examine three types of traffic, as depicted in hand, due to the reciprocity of the wireless channel, the Fig. 3. interference contribution of such secondary user will be 1) Slotted-synchronous traffic: Similar to the slotted severely attenuated, thus decreasing Pout.Fig.2shows ALOHA protocol [66], the nodes are synchro- that,fortheconsideredexample,whencomparedto nized and transmit in slots of duration Lp
Fig. 3. Three types of packet traffic, as observed by the node at the origin. (a) Slotted-synchronous transmission. (b) Slotted-asynchronous transmission. (c) Exponential-interarrival transmission.
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seconds.13 A node transmits in a given slot with Using (1), the desired signal power S can be written as probability q. The transmissions are independent for different slots and for different nodes. Q 2) Slotted-asynchronous traffic: The nodes transmit in P0 k Z0;k S ¼ 2b (26) slots of duration Lp seconds, which are not r0 synchronized with other nodes’ time slots. A node transmits in a given slot with probability q. where the subscript 0 refers to the probe link. Similarly, The transmissions are independent for different the aggregate interference power I can be written as slots or for different nodes. 3) Exponential-interarrivals traffic: The nodes trans- Q mit packets of duration Lp seconds. The idle time X1 PI i k Zi;k between packets is exponentially distributed with I ¼ 2b (27) 14 i¼1 Ri mean 1= p. In what follows, we analyze the throughput of the probe link from an SINR perspective. In such approach, a node where PI is the transmitted power associated with each can hear the transmissions from all the nodes in the two- interferer and i 2½0; 1 is the (random) duty-cycle factor dimensional plane.15 A packet is successfully received if associated with interferer i.Asweshallsee,theRV i the SINR exceeds some threshold. Therefore, we start with accounts for the different traffic patterns of nodes and is the statistical characterization of the SINR and then use equal to the fraction of the duration Lp during which those results to analyze the throughput. interferer i is effectively transmitting. Note that since S and I depend on the random node positions and random A. Signal-to-Interference-Plus-Noise Ratio propagation effects, they can be seen as RVs whose value is Typically, the distances fRig and propagation effects different for each realization of those random quantities. fZ g associated with node i are slowly varying and i;k The distribution of the RV I canQ be determined from remain approximately constant during the packet dura- Corollary 3.2, setting Pi ¼ PI i k Zi;k.Then,itfollows tion Lp. In this quasi-static scenario, it is insightful to that I has a skewed stable distribution given by define the SINR conditioned on a given realization of those RVs. As we shall see, this naturally leads to the derivation of an SINR outage probability, which in turn 1 I S ¼ ; ¼ 1; determines the throughput. We start by formally defining b ! the SINR. noY no 1 1=b 1=b 1=b ¼ C1=bPI IE i IE Zi;k (28) Definition 5.1 (SINR): The SINR associated with the k node at the origin is defined as
where b 9 1andC is defined in (12). As we shall see, the S probe link throughput depends on the traffic pattern of SINR ¼ (25) 1=b I þ N the nodes only through IE f i g in (28).
B. Probe Link Throughput where S is the power of the desired signal received from We now use the results developed in Section V-A to the probe node, I is the aggregate interference power characterize the throughput of the probe link, subject to received from all other nodes in the network, and N is the the aggregate network interference. We start by defining (constant) noise power. Both S and I depend on a given the concept of throughput. realization of fRig, i ¼ 1 ...1,andfZi;kg, i ¼ 0 ...1. Definition 5.2: The throughput T of a link is the proba- bility that a packet is successfully received during an in- 13 By convention, we define these types of traffic with respect to the terval equal to the packet duration Lp. For a packet to be receiver clock. In the typical case where the propagation delays with successfully received, the SINR of the link must exceed respect to the packet length can be ignored, all nodes in the plane observe exactly the same packet arrival process. some threshold. 14This is equivalent to each node using a M=D=1=1 queue for packet Using the definition above, we can write the through- transmission, characterized by a Poisson arrival process with rate p,a put T as constant service time Lp, a single server, and a buffer of one packet. 15This contrasts with a connectivity-based framework [63], where a node can only hear the transmissions from a finite number of nodes (called audible interferers) whose received power exceeds some threshold. T¼IP fprobe transmitsgIP freceiver silentgIP fno outageg: In such approach, for a node to successfully receive the desired packet, it must not collide with any other packet from the audible interferers. (29)
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The first probability term, which we denote by pT, where the distribution of I in (28) reduces to depends on the type of packet traffic. The second term, whichwedenotebypS,alsodependsonthetypeof no packet traffic and corresponds to the probability that the 1 1 1=b 1=b I S ¼ ; ¼ 1; ¼ C1=bPI IE i : node at the origin is silent (i.e., does not transmit) b during the transmission of the packet by the probe node. This second term is necessary because the nodes are half- Note that the characteristic function of I was also duplex, so they cannot transmit and receive simulta- obtained in [35] using the influence function neously. The third term is simply IP fSINR g,where method, for the case of path loss and slotted- is a predetermined threshold that ensures reliable synchronous traffic only. packet communication over the probe link. Therefore, 2) Path loss and Nakagami-mfading:In this case, (31) the throughput of a wireless packet network can be reduces to written as
T¼pTpSIP fSINR g: (30) IP fSINR g 2b 1 r0 ðI þ NÞm ¼ 1 IE I inc m; : ðmÞ P0 Using (25), (26), and the law of total probability with respect to RVs fZ0;kg and I, we can write For integer m, this can be expressed in closed form ()() [63] as Y r2b IP fSINR g¼ IE IP Z 0 ðI þ NÞ I I fZ0;kg 0;k P0 Xm 1 Xk j k ðm 1Þ! ð Þ IP fSINR g¼1 1 3 (31) ! ðmÞ k¼0 j¼0 j ! k j 3N j ð 3NÞ e d IðsÞ or, alternatively (34) ðk jÞ! dsj s¼ 3 Q P Z IP fSINR g¼IE F 0 k 0;k N (32) fZ0;kg I 2b where r0
2b where F ð Þ is the cdf of the stable RV I,whose r0 m I 3 ¼ distribution is given in (28). As we shall see, both forms P0 are useful depending on the considered propagation characteristics. Equations (30)–(32) are general and valid and for a variety of propagation conditions as well as traffic patterns. As we will see in the next sections, the pro- no 0 ÀÁ 1 pagation characteristics determine only IP fSINR g, 1 1=b 1 1=b C PI m þ IE i while the traffic pattern determines p , p ,and @ 1=b b ÀÁ 1=bA T S IðsÞ¼exp s 1=b IP fSINR g. m ðmÞ cos 2b
C. Effect of the Propagation Characteristics on T We now determine the effect of four different propa- for s 0. For the particular case of Rayleigh gation scenarios described in Section II-B on the through- fading ðm ¼ 1Þ,weobtain put. Recall that the propagation characteristics affect the throughput T only through IP fSINR g in (30), and IP fSINR g so we now derive such probability for these specific scenarios. r2b N ¼ exp 0 1) Path loss only: In this case, the expectation in (32) P 2 0 ÀÁno 3 disappears and we have 1 1 1=b IE 2b 1=b C1=b 1 þ b i P r exp4 ÀÁ I 0 5: cos P P 2b 0 IP SINR 0 f g¼FI 2b N (33) r0 (35)
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3) Path loss and log-normal shadowing: In this case, and (31) reduces to 0 ÀÁno1 1 1=b 2 2=b2 1 1=b C PI e m þ IE i 2b @ 1=b ÀÁb 1=bA 1 r0 ðI þ NÞ IðsÞ¼exp s IP fSINR g¼IE I Q ln (36) m1=b ðmÞ cos 2 P0 2b
where Qð Þ denotes the Gaussian Q-function, and for s 0. For the particular case of Rayleigh the distribution of I in (28) reduces to fading ðm ¼ 1Þ,weobtain
no 1 1 1=b 2 2=b2 1=b IP fSINR g I S ¼ ; ¼1; ¼ C1=bPI e IE i : 8 0 ÀÁno b < 1 2 2=b2 1 1=b 2b C e 1þ IE i r0 N @ 1=b ÀÁb ¼ IE G exp 0: 2 G0 P0e cos 2b 19 4) Path loss, Nakagami-m fading, and log-normal 2b 1=b = PIr A 0 : (39) shadowing: In this case, (31) reduces to 2 G0 ; P0e
1 IP fSINR g¼1 D. Effect of the Traffic Pattern on T ðmÞWe now investigate the effect of three different types of 2b traffic pattern on the throughput. Recall that the traffic r0 ðI þ NÞm IE G ;I inc m; (37) 0 2 G0 pattern affects the throughput T through pT, pS, and P0e IP fSINR g in (30). The type of packet traffic deter- mines the statistics of the duty-cycle factor i, and in parti- cular IE f 1=bg in (28), which in turn affects IP fSINR g. where the distribution of I in (28) reduces to i 1) Slotted-synchronous traffic: In this case, pT ¼ q and pS ¼ 1 q. The duty-cycle factor i is a binary RV taking the value zero or one, and we can show [63] 1 1=b I S ¼ ; ¼ 1; that IE f i g¼q. b ÀÁ 2) Slotted-asynchronous traffic: In this case, pT ¼ q no 1 2 2 2 m þ and pS ¼ð1 qÞ . The duty-cycle factor i is ¼ C 1 P1=be2 =b IE 1=b b : 1=b I i m1=b ðmÞ zero, one, or a continuous RV uniformly distrib- uted over the interval [0,1], and we can show [63] 1=b 2 that IE f i g¼q þ 2qð1 qÞb=ðb þ 1Þ. 3) Exponential-interarrivals traffic: Considering that For integer m, this can be expressed in closed form L 1, then p L and p ¼ e 2Lp p .The [63] as p p T p p S duty-cycle factor i is either zero or a uniform RV in the interval [0,1], and we can show [63] that 1=b IE f g¼ð1 e 2Lp p Þb=ðb þ 1Þ. ðm 1Þ! i IP fSINR g¼1 ðmÞ () !E. Discussion Xm 1 Xk j k j 4N j Using the results derived in this section, we can obtain ð 4Þ ð 4NÞ e d IðsÞ 1 IE G0 j! ðk jÞ! dsj insights into the behavior of the throughput as a function k¼0 j¼0 s¼ 4 of important network parameters, such as the type of pro- (38) pagation characteristics and traffic pattern. In particular, the throughput in the slotted-synchronous and slotted- asynchronous cases can be relatedasfollows.Considering where that b > 1, we can easily show that q q2 þ 2qð1 qÞb= ðb þ 1Þ, with equality iff q ¼ 0orq ¼ 1. Therefore, 1=b IE f i g is smaller (or, equivalently, IP fSINR g is larger) for the slotted-synchronous case than for the r2b m 0 slotted-asynchronous case, regardless of the specific propa- 4 ¼ 2 G 2 P0e 0 gation conditions. Furthermore, since qð1 qÞ qð1 qÞ ,
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we conclude that the throughput T given in (30) is higher for slotted-synchronous traffic than for slotted- asynchronous traffic. The reason for the higher throughput performance in the synchronous case is that a packet can potentially overlap with only one packet transmitted by another node, while in the asynchronous case, it can collide with any of the two packets in adjacent time slots. We can also analyze how the throughput of the probe link depends on the interfering nodes, which are characterized by their spatial density and the transmitted power PI. In all the expressions for IP fSINR g,we can make the parameters and PI appear explicitly by noting that if I Sð ; ; Þ,theneI ¼ 1= I Sð ; ; 1Þ is a normalized version of I with unit dispersion. Thus, we can for example rewrite (32) as
IP fSINR g Fig. 4. Throughput T versus the transmission probability q,for 8 0 Q 19 various types of packet traffic (path loss and Rayleigh fading, > P0 Z0;k > 2 < k = P0=N ¼ PI=N ¼ 10, ¼ 1, ¼ 1m , b ¼ 2, r0 ¼ 1m). B r2b N C ¼ IE F @ hino0 noA fZ0;kg> ~I Q b > : 1 1=b 1=b ; PI C1=bIE i k IE Zi;k cause interference to other systems operating in over- (40) lapping frequency bands. To prevent interference, many commercial networks operate under restrictions which e where I Sð ¼ 1=b; ¼ 1; ¼ 1Þ only depends on the often take the form of spectral masks, imposed by a regu- amplitude loss exponent b.Furthermore,sinceFI~ð Þ is latory agency such as the Federal Communications monotonically increasing with respect to its argument, we Commission (FCC). In military applications, on the other conclude that IP fSINR g and therefore the through- hand, the goal is ensure that the presence of the deployed put T is monotonically decreasing with and PI.In network is not detected by the enemy. If, for example, a > particular, since b 1,thethroughputismoresensitiveto sensor network is to be deployed in enemy territory, then an increase in the spatial density of the interferers than to the characterization of the aggregate network emission is an increase in their transmitter power.Thisanalysisisvalid essential for the design of a covert system. for any wireless propagation characteristics and traffic In this section, we introduce a framework for spectral pattern. characterization of the aggregate RF emission of a wireless F. Numerical Results network [67], [68]. We first characterize the PSD of the Fig. 4 shows the dependence of the packet throughput on the transmission probability for various types of packet traffic. We observe that the throughput is higher for slotted-synchronous traffic than for slotted-asynchronous traffic, as demonstrated in Section V-E. Fig. 5 plots the throughput versus the spatial density of interferers for various wireless propagation effects. We observe that the throughput decreases monotonically with the spatial den- sity of the interferers, as also shown in Section V-E.
VI. SPECTRUM OF THE AGGREGATE NETWORK EMISSION The spectral occupancy and composition of the aggregate RF emission generated by a network is an important con- sideration in the design of wireless systems. In particular, it is often beneficial to know the spectral properties of the aggregate RF emission generated by all the spatially scat- tered nodes in the network. This is useful in commercial Fig. 5. Throughput T versus the interferer spatial density ,for applications, for example, where communication designers various wireless propagation characteristics (slotted-synchronous must ensure that the RF emission of the network does not traffic, P0=N ¼ PI=N ¼ 10, ¼ 1, q ¼ 0:5, b ¼ 2, r0 ¼ 1m, dB ¼ 10).
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16 SXðfÞ¼F t!f fRXð tÞg. We define the PSD of the 17 process XiðtÞ as SXðfÞ¼F t!f fRXð tÞg. Since dif- ferent nodes operate independently, the processes XiðtÞ are also independent for different i, but the underlying second-order statistics are the same (i.e., the autocorrela- tion function and the PSD of XiðtÞ do not depend on i). In terms of the multipath channel, we consider a wide- sense stationary uncorrelated scattering (WSSUS) channel Fig. 6. Channel model for spectral analysis. [69]–[73], so that the autocorrelation function of hiðt; Þ can be expressed as aggregate emission process YðtÞ, measured by the probe ÈÉ receiver in Fig. 1. The spectral characteristics of YðtÞ can Rh ðt1; t2; 1; 2Þ¼IE h ðt1; 1Þhiðt2; 2Þ be inferred from the knowledge of its PSD. Then, we put i i ; forth the concept of spectral outage probability and pre- ¼ Phð t 2Þ ð 2 1Þ sent some of its applications, including the establishment of spectral regulations and the design of covert military for some function Phð t; Þ. Such channel can be repre- networks. As described in Section II-C, we consider the sented in the form of a dense tapped delay line, as a scenario of slow-varying P, where the analysis is first con- continuum of uncorrelated, randomly scintillating scat- ditioned on P to derive a spectral outage probability. Other terers having WSS statistics. The functions hiðt; Þ are fast-varying propagation effects, such as multipath fading considered to be independent for different nodes i,butthe due to local scattering, are averaged out in the analysis. underlying second-order statistics are the same (i.e., the autocorrelation function of hiðt; Þ does not depend on i). A. Power Spectral Density of the Aggregate WSSUS channels are an important class of practical Network Emission channels, which simultaneously exhibit wide-sense statio- The aggregate network emission at the probe receiver narity in the time variable t and uncorrelated scattering in Y can be characterized by the ELP random process ðtÞ, the delay variable . defined as We now wish to derive the PSD of the aggregate RF emission YðtÞ of the network, and with that purpose we X1 introduce the following theorem. YðtÞ¼ YiðtÞ (41) i¼1 Theorem 6.1 (WSS and WSSUS Channels): Let hðt; Þ denotethetime-varyingELPimpulseresponseofamulti- path channel, whose autocorrelation function is given by where YiðtÞ is the received process originated from node i. Rhðt1; t2; 1; 2Þ.LetuðtÞ denote the ELP WSS process that In the typical case of path loss, log-normal shadowing, and is applied as input to the channel and zðtÞ denote the multipath fading, the signal YiðtÞ in (2) reduces to corresponding output process of the channel. 1) If the channel hðt; Þ is WSS, i.e., Rhðt1; t2; 1; Z 2Þ¼Rhð t; 1; 2Þ, then the output zðtÞ is WSS e Gi and its PSD is given by18 YiðtÞ¼ b hiðt; ÞXiðt Þd (42) Ri ZZ hi f |2 fð 1 2Þ SzðfÞ¼ Psð ; 1; 2Þj ¼f SuðfÞe d 1d 2 where XiðtÞ is the ELP transmitted signal and hiðt; Þ is time-varying ELP impulse response of the multipath (43) channel associated with node i.Thesystemmodel describedby(42)isdepictedinFig.6.Sinceinthis where P ð ; ; Þ¼F fR ð t; ; Þg and section we are only interested in the aggregate emission of s 1 2 t! h 1 2 S ðfÞ is the PSD of uðtÞ. the network, we can ignore the existence of the probe link u depicted in Fig. 1. In what follows, we carry out the 16 analysis in ELP, although it can be trivially translated to As we will show in the case study of Section VI-C, if XiðtÞ is a train of pulses with a uniformly distributed random delay (which models the passband frequencies. asynchronism between emitting nodes), then it is a WSS process. 17 In the remainder of this section, we consider that We use F x!yf g to denote the Fourier transform operator, where x and y represent the independent variables in the original and transformed the transmitted signal XiðtÞ is a wide-sense stationary domains, respectively. x (WSS) process, such that its autocorrelation function has 18We use to denote the convolution operation with respect to the form RXi ðt1; t2Þ¼IE fXi ðt1ÞXiðt2Þg ¼ RXð tÞ,where variable x.
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2) If the channel hðt; Þ is WSSUS, i.e., Rhðt1; t2; 1; 2Þ¼Phð t; 2Þ ð 2 1Þ for some function Phð t; Þ, then the output zðtÞ is WSS and its PSD is given by
f SzðfÞ¼Dhð Þj ¼f SuðfÞ (44)
R where Dhð Þ¼ Psð ; Þd is the Doppler power spectrum of the channel hðt; Þ,andPsð ; Þ¼ F t! fPhð t; Þg is the scattering function of the channel hðt; Þ. Proof: See [55]. g In the specific context of (42), the theorem implies that YiðtÞ is WSS and thus the aggregate network emis- Fig. 7. PDF of A for different amplitude loss exponents b and interferer sion YðtÞ is also WSS. Furthermore, the PSD of YiðtÞ is given by densities ð dB ¼ 10Þ. Stable laws are a direct generalization of Gaussian distributions and include other densities with heavier (algebraic) tails. e2 Gi SYi ðfÞ¼ 2b ½ DhðfÞ SXðfÞ (45) Ri 1 1 depends on P (i.e., fRigi¼1 and fGigi¼1), it can be seen as an RV whose value is different for each realization of P. D ð Þ where h f is the Doppler power spectrum of the time- The distribution of the A can be determined from varying multipath channel h ðt; Þ,andS ðfÞ is the PSD i X Corollary 3.2, setting P ¼ e2 Gi . Then, it follows that A X i of the transmitted signal iðtÞ. Because the processes has a skewed stable distribution given by YiðtÞ associated with different emitting nodes i are statistically independent when conditioned on P, we can write 1 2 2 A S ¼ ; ¼ 1; ¼ C 1 e2 =b (49) A b A A 1=b X1
SYðfÞ¼ SYi ðfÞ: (46) i¼1 where b > 1andCx is defined in (12). This distribution is plotted in Fig. 7 for different values of b and .
Combining (45) and (46), we obtain the desired condi- B. Spectral Outage Probability tional PSD of the aggregate network emission YðtÞ as In the proposed quasi-static scenario, the PSD of the aggregate network emission SYðf; PÞ is a function of the random node positions and shadowing P.Then,withsome SYðf; PÞ ¼ A ½ DhðfÞ SXðfÞ (47) probability, P is such that the spectrum of the aggregate emission is too high in some frequency band of interest, where A is defined as thus causing an outage in that frequency band. This leads to the concept of spectral outage probability, which we denote by Ps ðfÞ and generally define as [67], [68] X1 2 G out e i A ¼ 2b : (48) i¼1 Ri s 9 PoutðfÞ¼IP PfgSYðf; PÞ mðfÞ (50) Note that in (47), we explicitly indicated the conditioning of SY on P.SinceSYðf; PÞ depends on P, it can be where SYðf; PÞ is the random PSD of the aggregate viewed, for a fixed f, as an RV whose value is different for network emission YðtÞ and mðfÞ is some spectral mask each realization of P.19 Lastly, note that since A in (48) determining the outage (or detection) threshold at the receiver. The SOP is a frequency-dependent quantity and, 19 SYðf; PÞ is in fact a random process whose sample paths evolve in in the case of slow-varying P, is a more insightful metric frequency instead of time. For each realization P¼P , we obtain a sample 0 than the PSD averaged over P. Note that this definition is path SYðf; P0Þ that is a function of f; for a fixed frequency f ¼ f0, SYðf0; PÞ is a RV. applicable in general to any emission model: the spectral
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s outage probability PoutðfÞ represents the probability that 2) Application to Covert Military Networks: In military the PSD of the aggregate network emission, measured at applications, the goal is to ensure that the presence of the an arbitrary location in the plane and at a particular deployed network is not detected by the enemy. If, for frequency f, exceeds some predetermined mask. example, a surveillance network is to be deployed in s For the signal model considered in this section, PoutðfÞ enemy territory, then the characterization of its aggregate can be derived by substituting (47) into the general emission is essential for the design of a covert network definition of SOP in (50), leading to with low probability of detection. In such application, the function mðfÞ in (50) can be interpreted as the frequency- dependent mask, which determines the detection threshold s mðfÞ (not the outage threshold as before) [75]. In other words, if PoutðfÞ¼IP A > DhðfÞ SXðfÞ the aggregate spectral density SYðf; PÞ measured at a mðfÞ given location exceeds the mask mðfÞ,thenthepresenceof ¼ 1 FA (51) DhðfÞ SXðfÞ the deployed network may be detected by the enemy.
C. Numerical Results where FAð Þ is the cdf of the stable RV A, whose distri- We now present a case study to quantify the spectral butionisgivenin(49). densities and outage probabilities derived in the previous In what follows, we present two possible applications section. We also illustrate their dependence on the various of the SOP. parameters involved, such as the transmitted pulse shape, spectral mask, transmitted power, and spatial density of 1) Application to Spectral Regulations: The concept of SOP the emitting nodes. For all numerical examples, we con- can provide a radically different way to establish spectral sider that the emitting nodes employ a two-dimensional regulations. Current regulations and standards (e.g., FCC modulation (e.g., M-PSK or M-QAM), such that trans- Part 15 or IEEE 802.11) impose a spectral mask on the PSD mitted signal XiðtÞ can be written for all t as at the transmitter, and the type of mask often depends on the environment in which the devices are operated (e.g., Xþ1 indoor or outdoor). The purpose of this mask is to limit RF XiðtÞ¼ ai;ngðt nT DiÞ (52) emissions generated by a terminal and to protect other n¼ 1 services that operate in dedicated bands (e.g., GPS, public safety, and cellular systems). However, the transmitted þ1 where the sequence fai;ngn¼ 1 represents the stream of PSD is usually not representative of the aggregate PSD at complex symbols transmitted by node i,assumedtobe the victim receiver due to the random propagation effects i.i.d. in n and zero mean, for simplicity; gðtÞ is a real, (multipath fading and shadowing) and the random baseband, unit-energy shaping pulse, defined for all values position of the emitting nodes. Thus, spectral regulations of t; T is the symbol period; and Di Uð0; TÞ is a random that are based only on the transmitted PSD do not delay representing the asynchronism between different necessarily protect a victim receiver against interference. emitting nodes. The type of constellation employed by the The approach proposed here is radically different, in emitting nodes is captured by the statistics of the symbols the sense that the spectral mask is defined at the victim 20 fai;ng. Note that the process XiðtÞ in (52) is WSS, as receiver, not at the transmitter [74]. In effect, the mask 21 required by Theorem 6.1. The PSD of XiðtÞ is then given mðfÞ introduced in (50) represents the outage threshold by [78]–[80] with respect to the accumulated PSD at the receiver,notthe individual PSD at the transmitter (this follows from the 2 fact that SYðf; PÞ is measured at an arbitrary location in SXðfÞ¼PtxjjGðfÞ (53) the plane, where a probe receiver could be located). ; Therefore, the received aggregate spectrum SYðf PÞ and 2 s where Ptx ¼ IE fjai;nj g=T is the power transmitted by the corresponding PoutðfÞ can be used to characterize and control the network’s RF emissions more effectively, since each emitting node and GðfÞ¼FfgðtÞg. they not only consider the aggregate effect of all emitting In terms of the multipath channel, we consider for nodes at an arbitrary receiver location but also incorporate simplicity that hðt; Þ is time-invariant such that it does the random propagation effects and random node posi- 20 | i;n tions. Furthermore, the use of different masks for indoor Note that each complex symbol ai;n ¼ ai;ne can be represented in the in-phase/quadrature (IQ) plane by a constellation point with or outdoor environments is no longer necessary, since the amplitude ai;n and phase i;n. environment is already accounted for in our model by 21This can be shown in the following way: first, if we deterministically e parameters such as the amplitude loss exponent b,the set Di to zero in (52), the resulting process XiðtÞ is wide-sense cyclostationary (WSCS) with period T [76], [77]; then, since spatial density of the emitting nodes, and the shadowing e XiðtÞ¼Xiðt DiÞ, where Di Uð0; TÞ and is independent of everything coefficient . else, it follows that XiðtÞ is WSS.
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Fig. 8. Effect of the transmitted baseband pulse shape gðtÞ on the Fig. 9. Effect of the spectral mask shape mðfÞ on the outage probability s 9 2 s 9 P ðfÞ (square gðtÞ, Ptx ¼ 10 dBm, T ¼ 10 s, ¼ 0:1m , b ¼ 2, PSD and the outage probability PoutðfÞ (Ptx ¼ 10 dBm, T ¼ 10 s, out 2 10). (a) Plot of various spectral masks m f , which define the ¼ 0:1m , b ¼ 2, dB ¼ 10, RRC pulse with rolloff factor 0.5). dB ¼ ð Þ (a) PSD of the individual transmitted signal versus frequency outage threshold at the receiver (top curves). Also shown is the PSD (bottom curves) for various pulse shapes gðtÞ. The pulses are of the individual transmitted signal versus frequency (bottom curve). s normalized so that the transmitted signals have the same (b) Spectral outage probability PoutðfÞ versus frequency for the various masks m f shown in (a). power Ptx. The piecewise-constant spectral mask mðfÞ ð Þ (top curve) determines the outage threshold at the receiver. s (b) Spectral outage probability PoutðfÞ versus frequency for the piecewise-constant mask mðfÞ shownin(a). Fig. 8 shows that for a fixed spectral mask mðfÞ,the SOP can be highly dependent on the pulse shape gðtÞ,such 22 not introduce any Doppler shifts, i.e., Dhð Þ¼ ð Þ. as square, Hanning, or root raised-cosine (RRC) pulse. In s Substituting the expressions for SXðfÞ and Dhð Þ in (51), fact, PoutðfÞ is a nonlinear function of jGðfÞj,wherethe we obtain the SOP as nonlinearity is determined in part by the cdf FAð Þ of the stable RV A,asshownin(54).Thus,theSOPcanbeused ! as a criterion for designing the pulse shape: for example, s mðfÞ we may wish to determine the baseband pulse gðtÞ and PoutðfÞ¼1 FA 2 : (54) s PtxjjGðfÞ transmitted power Ptx such that maxf PoutðfÞ p ,where p is some target outage probability which must be satisfied at all frequencies. 22 For typical node speeds or channel fluctuations, the frequencies of Fig. 9 shows that for a fixed pulse shape gðtÞ, Ps ðfÞ the Doppler shifts are on the order of few kilohertz [73]. As a out can significantly depend on the spectral mask mðfÞ (e.g., consequence, when the considered XiðtÞ is UWB, Dhð Þ can be well approximated by a Dirac-delta function. piecewise-linear, Gaussian, or constant mask). Since
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s PoutðfÞ accounts for both GðfÞ and mðfÞ,itquantifiesthe nth transmitted symbol of the ith UWB interferer; TU is 24 compatibility of the transmitted pulse shape with the the symbol period; and Di Uð0; TUÞ is a random spectral restrictions imposed through mðfÞ. delay modeling the asynchronism between nodes. The signal transmitted by the NB probe transmitter can be written as VII. COEXISTENCE BETWEEN UWB AND NB SYSTEMS ffiffiffiffiffiffiffiffi X1 ÀÁ The increasing proliferation of heterogeneous communica- p N N sNðtÞ¼ 2EN an gðt nTNÞ cos 2 fNt þ n (56) tion devices sharing the same frequency bands makes mutual n¼ 1 interference a key issue, in order to ascertain whether their coexistence is possible. For example, UWB signals, which occupy extremely large bandwidths, usually operate as an where EN is the average transmitted symbol energy; gðtÞ is a unit-energy pulse-shaping waveform satisfying the underlay system with other existing, licensed and unli- N N j n censed, NB radio systems [81]–[85]. Because of their Nyquist criterion; an e is the nth transmitted NB symbol, N N N characteristics, UWB systems are considered among key belonging to a constellation C ¼fs1 ; ...; sMg,and N 2 technologies in the context of cognitive radio [86]–[88]. As a satisfying IE fjan j g¼1; and fN is the carrier frequency result, the deployment of UWB systems requires that they of the NB signal. coexist and contend with a variety of interfering signals. To account for the propagation characteristics, we Thus, they must be designed to account for two fundamental consider that the signal transmitted by the ith UWB aspects: 1) UWB devices must not cause harmful interfer- interferer experiences a frequency-selective multipath ence to licensed wireless services and existing NB systems channel with impulse response (e.g., GPS, GSM, UMTS, 3G, Bluetooth, and WLAN), and 2) UWB devices must be robust and able to operate in the eU kU presence of interference caused by both NB systems and h ðtÞ¼ e UGi h ðtÞ (57) i bU i other UWB-based nodes. Ri The framework proposed in this paper enables the characterization of coexistence in heterogeneous wireless where networks, and captures all the essential physical param- eters that impact UWB and NB network interference. In this section, we consider an application of the framework XL to the scenario of NB communication in the presence of hiðtÞ¼ hi;q ðt i;qÞ (58) UWB interferers [56], [89]. We first determine the sta- q¼1 tistical distribution of the aggregate UWB interference, which naturally leads to the characterization of the cor- L L with fhi;qgq¼1 and f i;qgq¼1 representing, respectively, the responding error probability. The application of the amplitudes and delays (with arbitrary statistics) of the L proposed framework to the case of UWB communication paths. In addition,P we normalize the PDP of the channel in the presence of NB interferers is considered in [56], [90]. L 2 25 such that q¼1 IE fhi;qg¼1. We assume the shadowing The proposed spatial model is depicted in Fig. 1, where and fading to be independent for different interferers i, the interferers are UWB transmitters scattered in the two- and approximately constant during at least one symbol dimensional plane, according to a spatial Poisson process interval. with density U. In terms of transmission characteristics of On the other hand, the signal transmitted by the NB nodes, we consider the case where the UWB interferers node experiences a frequency-flat multipath channel with operate asynchronously and independently, using the same U power PU.Thesignalsi ðtÞ transmitted by the ith UWB interferer can be described as 24We use Uða; bÞ to denote a real uniform distribution in the interval ½a; b . 25 ffiffiffiffiffiffi X1 For strict consistency with the linear time-varying (LTV) descrip- U p U tion in (3),P the impulse response in (58) would be written as si ðtÞ¼ EU ai;nwiðt nTU DiÞ (55) L hiðt; Þ¼ q¼1 hi;qðtÞ ð i;qðtÞÞ. For a slow-varying channel, n¼ 1 i;qðtÞ i;q, i;qðtÞ i;q, and hiðt; Þ does not depend on t, which corresponds to a linear time-invariant (LTI) system. In such case, we proceed as is usually done in the literature by dropping the variable t and where EU is the average transmitted symbol energy; wiðtÞ is replacing with t. Note that such LTI impulse response is applicable to the unit-energy symbol waveform23; aU 2f 1; 1g is the error probability analysis, since the interval of interest for symbol i;n detection is typically much smaller than the coherence time of the channel. On the other hand, an LTI impulse response is generally not 23 Note that wiðtÞ itself may be composed of many monocycles, and applicable to the spectral analysis of Section VI, since the interval of depending on its choice, can be used to represent both direct-sequence interest is much larger than the coherence time of the channel, and (DS) or time-hopping (TH) spread spectrum signals [84]. therefore an LTV representation is needed.
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pffiffiffi Y Y Y impulse response pffiffiffi 2gðtÞ sinð2 fNtÞg.Ifwelet ðtÞ¼ 1ðtÞþ| 2ðtÞ¼ 2gðtÞe |2 fNt, we can write
e kN NG0 Z hNðtÞ¼ e ðt Þ (59) þ1 bN 0 0 r0 Z ¼ zðtÞYðtÞdt 1 NG0 ffiffiffiffiffiffi kN 0e p N | N ¼ ENa e 0 þ Y þ N (60) where 0 is an amplitude factor with Nakagami-m dis- bN 0 2 r0 tribution (normalized so that IE f 0g¼1), and 0 is the time delay introduced by the channel. For generality, (57) and (59) can accommodate different path loss parameters where the distribution of N is given by28 for the UWB signals ðkU; bUÞ and the NB signal ðkN; bNÞ,as well as different shadowing parameters N and U. As described in Section II-C, we consider the scenario N Ncð0; N0Þ: (61) of slow-varying P, where the analysis is first conditioned R on P to derive an error outage probability.Otherfast- þ1 Y Furthermore, Y ¼ 1 yðtÞ ðtÞdt can be expressed as varying propagation effects, such as multipath fading due to local scattering, are averaged out in the analysis. X1 e UGi X Y ¼ i (62) bU A. Signals and Interference Representation i¼1 Ri Under the system model described in Section II, the aggregate signal zðtÞ at the NB victim receiver can be written as where
Xi ¼ Xi;1þþ||XX i;2 zðtÞ¼dðtÞþyðtÞþnðtÞ i i;1 i;2 Z X1 Z pffiffiffiffiffiffiffiffiffiffiffiffiffi X1 þ1þ1 ¼ k E aUU v ðt eDe YÞYðtÞdt: (63) ¼ kU EUU ai;i;nn viði t Di;ni;nÞ ðtÞdt: (63) ffiffiffiffiffiffiffiffi 1 p N N e nn¼ 1¼ 1 1 where dðtÞ¼½ 2ENa0 gðtÞ cosð2 fNt þ 0 Þ hNðtÞ is the desired signal from theP NB transmitter corresponding to 26 1 U eU symbol n ¼ 0 ; yðtÞ¼ i¼1 si ðtÞ hi ðtÞ is the aggregate Note that both HiðfÞ¼FfhiðtÞg and WiðfÞ¼FfwiðtÞg network interference; and nðtÞ is the AWGN with two- are Fourier transforms of UWB signals, and therefore are sided power spectral density N0=2, and independent of approximately constant over the frequencies of the NB yðtÞ. By performing the indicated convolutions, we can signal. As a result, we can show [89] that (63) reduces to further express the desired signal as27 pffiffiffiffiffiffiffiffi X1 e U e |2 fNDi;n Xi ¼kU 2EUWiðfNÞHiðfNÞ a gðDi;nÞe : k e NG0 pffiffiffiffiffiffiffiffi ÀÁ i;n N 0 N N n¼ 1 dðtÞ¼ 2ENa gðtÞ cos 2 fNt þ bN 0 0 r0 (64) and the aggregate interference as Note that effective range of the summation of n in (64) depends on the duration of the shaping pulse gðtÞ relative to TU. In effect, in the usual case where gðtÞ decreases to 0 as X1 X1 pffiffiffiffiffiffi UGi e kUe U e t ! 1,ther.v.’sgðDi;nÞ become increasingly small since yðtÞ¼ EU a v ðt D ; Þ bU i;n i i n e i¼1 Ri n¼ 1 Di;n grows as n increases, and so the sum in n can be truncated.
B. Distribution of the Aggregate UWB Interference e where viðtÞ¼wiðtÞ hiðtÞ,andDi;n ¼ nTU þ Di: The distribution of the aggregate UWB interference Y The NB receiver demodulates the aggregate signal zðtÞ plays an important role in the evaluation of the error using a conventional in-phase/quadrature (IQ) detector. probability of the victim link. Based on the Kullback- This can be achieved by projectingpffiffiffi zðtÞ onto the Leibler divergence [56], [91], it can be shown that the orthonormal set f 1ðtÞ¼ 2gðtÞ cosð2 fNtÞ; 2ðtÞ¼ P-conditioned distribution of the aggregate interference Y in (62) can be accurately approximated by a circularly 26To derive the error probability of the NB victim link, we only need to analyze a single NB symbol. 27 28 2 In what follows, we assume that the NB receiver can perfectly We use N cð0; Þ to denote a circularly symmetric (CS) complex estimate 0, and thus perfectly synchronize with the desired signal. As a Gaussian distribution, where the real and imaginary parts are i.i.d. 2 result, we can set 0 ¼ 0 in the remaining analysis. Nð0; =2Þ.
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symmetric (CS) complex Gaussian r.v. with the same of UWB network interference to a Gaussian problem, e variance as (62), i.e., where the combined noise N is Gaussian when condi- tioned on the location of the UWB interferers. ð ; PÞ jP The corresponding error probability Pe G0 can be Y N cð0; 2AVXÞ (65) found by taking the well-known error probability expres- sions for coherent detection of two-dimensional modula- where A is defined as tions in the presence of AWGN and fast fading [92]–[96], but using 2AVX þ N0 instead of N0 for the total noise variance. Note that this substitution is valid for any two- X1 2 G e U i dimensional modulation, allowing the traditional results to A ¼ (66) 2bU i¼1 Ri be extended to include the effect of UWB network inter- ference. For the case where the NB transmitter employs an
29 arbitrary signal constellation in the IQ-plane and the and fading has a Nakagami-m distribution, the conditional symbol error probability PeðG0; PÞ is given by VX ¼ VfXi;kg: (67) XM X 1 PeðG ; PÞ ¼ p 1 0 k 2 Note that since A in (66) depends on P (i.e., fRigi¼1 and k¼1 l2Bk 1 Z fGigi¼1), it can be seen as a r.v. whose value is different for k;l m wk;l each realization of P. The distribution of the A can be 1 þ 2 A d (71) 2 UGi 0 4m sin ð þ k;lÞ determined from Corollary 3.2, setting Pi ¼ e .Then, it follows that A has a skewed stable distribution given by where 1 1 2 2 =b2 A S ¼ ; ¼ 1; ¼ UC e U U A A A 1=bU 2 2 G bU k e N 0 E ¼ N N (72) A 2bN (68) r0 ð2AVX þ N0Þ
9 where bU 1, and C is defined in (12). is the received SINR averaged over the fast fading; M is the M NB constellation size; fpkg are the NB symbol proba- C. Error Probability k¼1 bilities; Bk, k;l, wk;l, and k;l are the parameters that de- In a quasi-static scenario of slow-varying P, it is insightful scribe the geometry of the NB constellation (see Fig. 10); A is to analyze the error probability conditioned on P, as well as defined in (66) and distributed according to (68); and VX is on the shadowing G0 of the NB victim link. We denote this 30 defined in (67). The result in (71) can be easily generalized conditional symbol error probability by PeðG0; PÞ. to the scenario where the victim link employs diversity To derive the conditional error probability, we techniques, using the methods described in [94]–[98]. employ the results of Section VII-B for the distribution When the NB transmitter employs M-PSK and M-QAM of the aggregate UWB interference Y. Specifically, using modulations with equiprobable symbols, (71) is equiva- (61) and (65), the received signal Z in (60) can be lent to31 rewritten as MPSK M1 NG0 ffiffiffiffiffiffi 2 kN 0e p N | N e Pe ðG0; PÞ ¼ IA ; sin (73) Z ¼ ENa e 0 þ N (69) M M bN 0 r0 and where MQAM 1ffiffiffiffi 3 e jP Pe ðG0;PÞ¼41 p IA ; N ¼ Y þ N N cð0; 2AV þ N Þ (70) 2 2ðM 1Þ X 0 M 2 1ffiffiffiffi 3 41 p I A ; (74) and A was defined in (66). Our framework has thus M 4 2ðM 1Þ reduced the analysis of NB communication in the presence
29We use Vf g to denote the variance operator. 31In this paper, we implicitly assume that M-QAM employs a square 30 n The notation PeðX; YÞ is used as a shorthand for IP ferrorjX; Yg. signal constellation with M ¼ 2 points (n even).
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In the case of slow-varying interferer positions, the error outage probability is a more meaningful metric than the error probability averaged over P, as discussed in Section II-C.
D. Numerical Results We now particularize the general analysis developed in the previous sections, using a simple case study of a BPSK NB victim link subject to the DS-BPAM UWB interferers. In this scenario, the signal siðtÞ transmitted by the ith interferer in (55) becomes
ffiffiffiffiffiffi X U p U si ðtÞ¼ EU ai;nwiðt nNsTf DiÞ n Fig. 10. Typical decision region associated with symbol s1. In general, j for a constellation C¼fs1; ...; sMg with signal points sk ¼jskje k and 2 2 with k ¼ jskj =IE fjskj g, k ¼ 1 ...M, four parameters are required to where the unit-energy waveform wiðtÞ for each bit is given by compute the error probability: k;l and k;l are the angles that describe the decision region corresponding to sk (as depicted); Bk is the set consisting of the indexes for the signal points that share a decision boundary with sk (in the example, B1 ¼f2; 3; 4g); and NXs 1 pffiffiffiffiffiffiffiffi wk;l ¼ k þ l 2 k lcosð k lÞ. wiðtÞ¼ ci;kpðt kTf Þ: k¼0 where the integral I ðx; gÞ is given by In these equations, Ns is the number of monocycles A U required to transmit a single information bit ai;n 2 f 1; 1g; pðtÞ is the transmitted monocycle shape, with Z 1 x g m energy 1=Ns; Tf is the monocycle repetition time (frame I ðx; gÞ¼ 1 þ d : (75) A 2 A length), and is related to the bit duration by TU ¼ NsTf ; 0 m sin Ns 1 and fci;kgk¼0 is the spreading sequence, with ci;k 2 f 1; 1g. For such a DS-BPAM system, WiðfNÞ in (64) can In the general expression given in (71) and (72), the be easily derived as network interference is accounted for by the term 2AVX, where A depends on the spatial distribution of the UWB NXs 1 interferers and propagation characteristics of the medium, j2 fNkTf WiðfNÞ¼PðfNÞ ci;ke while VX depends on the transmission characteristics of the k¼0 UWB interferers. Since 2AVX simply adds to N0,weconc- lude that the effect of the interference on the conditional error probability is simply an increase in the noise level, a where PðfÞ¼FfpðtÞg. Different monocycle shapes are fact which is intuitively satisfying. Furthermore, note that considered to satisfy FCC masks with the maximum trans- the modulation of the UWB interfering nodes only affects mitted power. Similarly to [99], we choose the 5th derivative the term VX, while the modulation of the NB link affects the of a Gaussian monocycle, so the received pulse can be type of error probability expression, leading to forms such modeled as the 6th derivative. In this case, we can write as (73) or (74). PðfNÞ as In our quasi-static model, the conditional error probability PeðG0; PÞ is seen to be a function of the slow- 3 varying interferer positions and shadowing (i.e., G0 and P). 8 2 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13=2 6 2 fN p ð ; PÞ PðfNÞ¼ p p fNe Since these quantities are random, Pe G0 is itself a r.v. 3 1155Ns Then, with some probability, G0 and P are such that the conditional error probability of the victim link is above some target p . The system is said to be in outage, and the where p is the monocycle duration parameter. error outage probability is Concerning the NB propagation characteristics, we consider frequency-flat Rayleigh fading (i.e., m ¼ 1). For the UWB propagation, we consider that the multipath e IP ; 9 Pout ¼ G0;PðÞPeðG0 PÞ p (76) fading channel in (58) is composed of L independent
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performance of NB communication subject to UWB network interference deteriorates as U or the INR increase, for a fixed SNR. This is expected because as the spatial density or transmitted energy of the UWB interferers increase, the aggregate interference at the NB victim receiver becomes stronger.
VIII. CONCLUSION In this paper, we presented a unifying framework for the characterization of network interference in wireless systems, where the interferers are scattered according to a spatial Poisson process. We first derived the character- istic function of the aggregate interference generated by a general set of active nodes. We then showed that when all the nodes in the plane are active, the aggregate interfer- e Fig. 11. Pout versus the SNR of the NB link for various spatial densities ence amplitude follows a symmetric stable distribution, U of the UWB interferers (NB link: BPSK, square gðtÞ of duration while the aggregate interference power follows a skewed TN ¼ 1 s, fN ¼ 5010 GHz, bN ¼ 2, N;dB ¼ 3, frequency-flat Rayleigh stable distribution. These results are valid for arbitrary 2 fading, p ¼ 10 ,r0 ¼ 1m; UWB interferers: DS-BPAM, TU ¼ 0:8 s, DS k wireless propagation effects. p ¼ 0:192 ns, INR ¼ 20 dB, Ns ¼ 16, ck ¼ð 1Þ ,bU ¼ 2, U;dB ¼ 8, frequency-selective Nakagami-m fading, L ¼ 8, p ¼ 3, We further investigate four applications of the pro- m1 ¼ 3, m ¼ 4). posed model: 1) interference in cognitive radio networks; 2) interference in wireless packet networks; 3) spec- trum of the aggregate RF emission of a wireless network; and 4) interference between UWB and NB systems. Our Nakagami-distributed paths having random delays and an framework accounts for all the essential physical param- exponential power dispersion profile [100], [101] given by eters that affect network interference, such as the type of wireless propagation effects, the type of transmission tech-
1= p nology, the spatial density of interferers, and the transmit- e 1 q= p q ¼ e ; q ¼ 1 ...L (77) ted power of the interferers. While we have explored some 1 e L= p applications of the proposed framework, it is our hope that this paper will inspire other researchers to explore other where p is a decay constant that controls the multipath applications. h dispersion. We also consider different Nakagami param- eters for each path [101] according to APPENDIX I
ðq 1Þ= m DERIVATION OF (13) mq ¼ m1e ; q ¼ 1 ...L (78) Since Qi is SS, we can write the characteristic function of each coordinate Qi;n as 0ðtÞ¼ QðwÞjjwj¼t. where m controls the decay of the m-parameters. As a result The plots presented in this section are of semi-analytical nature. Specifically, we resort to a hybrid method where we employ the analytical expressions for Pe , and perform a out 0ðtÞ¼ Qi;n ðtÞ Monte Carlo simulation of the stable RV A according to [102]. ¼ IE fe|tQi;n g Nevertheless, we emphasize that the expressions derived in ÈÉÈÉ IE c o s | IE s i n this paper completely eliminate the need for bit-level ¼ ðtQi;nÞ þ |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}ðtQi;nÞ simulation, in order to obtain error performance results. ÈÉ ¼0 Fig. 11 quantifies the error performance of the BPSK ¼ IE c o s ðtQi;nÞ : NB link, subject to DS-BPAM UWB network interference and noise. For this purpose, we define the normalized SNR 2 of the NB link as SNR ¼ kNEN=N0,andthenormalized 2 32 Using the elementary integral [103, Eq. 3.823], we can write INR as INR ¼ kUEU=N0. We conclude that the error
32 Since EN corresponds to the average transmitted symbol energy of Z ÀÁ 2 the NB victim link, then k EN can be interpreted as the average symbol 1 N 1 cosðztÞ ð1 Þ cos 2 energy measured 1 m away from the NB transmitter. An analogous dt ¼ jzj (79) 2 t þ1 interpretation applies to kUEU. 0
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33 for any real constants z and 0 G G 2. For z ¼ Qi;n and for any real constants z and 0 G G 1. For z ¼ signðwÞPi taking expectations on both sides, we obtain and taking expectations on both sides, we obtain
Z ÀÁ Z ÈÉ 1 ÈÉ 1 |signðwÞP t 1 0ðtÞ ð1 Þ cos 2 1 IE e i dt ¼ IE jQi;nj : (80) dt t þ1 þ1 0 0 t ÀÁhi ÈÉ ð1 Þ cos ¼ IE P 2 1 | signðwÞ tan : Noting that ð1 Þ¼ ð2 Þ=ð1 Þ for 6¼ 1, we i 2 rewrite (80) as
Z Noting that ð1 Þ¼ ð2 Þ=ð1 Þ,werewrite 1 1 ðtÞ C 1 ÈÉ (81) as 0 IE þ1 dt ¼ jQi;nj 0 t Z ÈÉ 1 1 IE e| signðwÞPit dt where C is defined in (12). This is the result in (13), and t þ1 0 hi the derivation is complete. ÈÉC 1 ¼ IE P 1 | signðwÞ tan i 2 APPENDIX II DERIVATION OF (19) where C is defined in (12). This is the result in (19), and We start with the elementary integral [104, Eq. (4.11), the derivation is complete. p. 542]
Z 1 1 e|zt Acknowledgment dt þ1 The authors gratefully acknowledge helpful discus- 0 t hi sionswithN.C.Beaulieu,M.Chiani,G.J.Foschini, ð1 Þ ¼ jzj cos | signðzÞ sin L.J.Greenstein,R.A.Scholtz,andJ.H.Winters.They 2 2 would also like to thank A. Conti, D. Dardari, A. Giorgetti, 33Note that (79) is still valid for ¼ 1, provided that the right side is Y.Shen,W.Suwansantisuk,andA.Zanellafortheir interpreted in the limiting sense of ! 1. suggestions and careful readings of the manuscript.
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Vol. 97, No. 2, February 2009 | Proceedings of the IEEE 229 Win et al.: A Mathematical Theory of Network Interference and Its Applications
ABOUT THE AUTHORS Moe Z. Win (Fellow, IEEE) received both the Ph.D. Pedro C. Pinto (Student Member, IEEE) received in Electrical Engineering and M.S. in Applied the Licenciatura degree with highest honors in Mathematics as a Presidential Fellow at the Electrical and Computer Engineering from the University of Southern California (USC) in 1998. Oporto University, Portugal, in 2003. He received He received an M.S. in Electrical Engineering from the M.S. degree in Electrical Engineering and USC in 1989, and a B.S. (magna cum laude)in Computer Science from the Massachusetts Insti- Electrical Engineering from Texas A&M University tute of Technology (MIT) in 2006. Since 2004, he in 1987. has been with the MIT Laboratory for Information Dr. Win is an Associate Professor at the and Decision Systems (LIDS), where he is now a Massachusetts Institute of Technology (MIT). Prior Ph.D. candidate. His main research interests are in to joining MIT, he was at AT&T Research Laboratories for five years and at wireless communications and signal processing. He was the recipient of the Jet Propulsion Laboratory for seven years. His research encompasses the MIT Claude E. Shannon Fellowship in 2007, the Best Student Paper developing fundamental theory, designing algorithms, and conducting Award at the IEEE International Conference on Ultra-Wideband in 2006, experimentation for a broad range of real-world problems. His current and the Infineon Technologies Award in 2003. research topics include location-aware networks, time-varying channels, multiple antenna systems, ultra-wide bandwidth systems, optical trans- Lawrence A. Shepp received the B.S. degree in mission systems, and space communications systems. applied mathematics from the Polytechnic Insti- Professor Win is an IEEE Distinguished Lecturer and an elected tute of Brooklyn in 1958, and the M.A. and Ph.D. Fellow of the IEEE cited for Bcontributions to wideband wireless degreesinmathematicsfromPrincetonUniversity transmission.[ He was honored with the IEEE Eric E. Sumner Award in 1960 and 1961, respectively. (2006), an IEEE Technical Field Award for Bpioneering contributions to He joined the Mathematics Research Center of ultra-wide band communications science and technology.[ Together AT&T Bell Laboratories in 1962 and became a with students and colleagues, his papers have received several awards Distinguished Member of Technical Staff at Bell including the IEEE Communications Society’s Guglielmo Marconi Best Laboratories in 1986. He was a Professor of Paper Award (2008) and the IEEE Antennas and Propagation Society’s Statistics and Operations Research at Columbia Sergei A. Schelkunoff Transactions Prize Paper Award (2003). His other University from 1996 to 1997. In 1997, he joined the Department of recognitions include the Laurea Honoris Causa from the University of Statistics at Rutgers University, where he is Board of Governor’s Ferrara, Italy (2008), the Technical Recognition Award of the IEEE Professor. During his career at AT&T, he held various joint appointments: ComSoc Radio Communications Committee (2008), Wireless Educator of he was a Professor of Radiology at Columbia University from 1973 to the Year Award (2007), the Fulbright Foundation Senior Scholar 1996, a Mathematician in Radiology Service at the Columbia Presbyterian Lecturing and Research Fellowship (2004), the U.S. Presidential Early Hospital from 1974 to 1996, and a Professor of Statistics (1/4 time) at Career Award for Scientists and Engineers (2004), the AIAA Young Stanford University from 1978 to 1992. He was a member of the Scientific Aerospace Engineer of the Year (2004), and the Office of Naval Research Boards of American Science and Engineering Inc. from 1974 to 1975 and Young Investigator Award (2003). of Resonex Inc. from 1983 to 1984. Dr. Shepp spent extended visiting Professor Win has been actively involved in organizing and chairing a periods at many places including the Mittag-Leffler Institute, MIT, the number of international conferences. He served as the Technical University of New South Wales, the Australian National University, and Program Chair for the IEEE Wireless Communications and Networking Simon Bolivar University in Caracas. He was an exchange scholar at the Conference in 2009, the IEEE Conference on Ultra Wideband in 2006, Steklov Institute in Moscow for 6 months in 1966. He has a permanent the IEEE Communication Theory Symposia of ICC-2004 and Globecom- visiting position at University of Coral Gables and also at Stanford 2000, and the IEEE Conference on Ultra Wideband Systems and University. His current research interests include functional magnetic Technologies in 2002; Technical Program Vice-Chair for the IEEE resonance imaging, tomography, wireless telephony, cosmology, and the International Conference on Communications in 2002; and the Tutorial mathematics of finance. Chair for ICC-2009 and the IEEE Semiannual International Vehicular Prof. Shepp has made fundamental contributions to CAT scanning, Technology Conference in Fall 2001. He was the chair (2004–2006) and emission tomography, probability theory (random covering, Gaussian secretary (2002–2004) for the Radio Communications Committee of the processes, connectedness of random graphs), mathematical finance and IEEE Communications Society. Dr. Win is currently an Editor for IEEE economics. He is a Co-Editor for the Wiley International Journal of TRANSACTIONS ON WIRELESS COMMUNICATIONS.HeservedasAreaEditorfor Imaging Systems and Technology and was an Associate Editor for Journal Modulation and Signal Design (2003-2006), Editor for Wideband of Computer Assisted Tomography from 1977 to 1992. He was elected to Wireless and Diversity (2003–2006), and Editor for Equalization and the National Academy of Science, the Institute of Medicine, the Academy Diversity (1998–2003), all for the IEEE TRANSACTIONS ON COMMUNICATIONS. of Arts and Science, and a Fellow of the Institute of Mathematical He was Guest-Editor for the IEEE JOURNAL ON SELECTED AREAS IN Statistics. He was the winner of the William Lowell Putnam Intercollegiate COMMUNICATIONS (Special Issue on Ultra-Wideband Radio in Multiaccess Mathematics Competition in 1958, the Paul Levy Prize in 1966, and he Wireless Communications) in 2002. received an IEEE Distinguished Scientist Award in 1979.
230 Proceedings of the IEEE |Vol.97,No.2,February2009