Narrowband Communication in a Poisson Field ofUltrawideband Interferers

(Invited Paper)

Pedro C. Pinto, Chia-Chin Chong, Andrea Giorgetti, Marco Chiani, and Moe Z. Win

Abstract— This paper puts forth a mathematical model for nar-rowband (NB) communication subject to ultrawideband (UWB)network interference and additive white Gaussian noise (AWGN).We introduce a framework where the UWB interferers arespatially scattered according to a Poisson field, and are operatingasynchronously in a wireless environment subject to path loss,shadowing, and fast fading. Under this scenario, we determinethe statistical distribution of the aggregate network interferenceat the output of a linear NB receiver, located anywhere in the two-dimensional plane. Then, we provide an exact expression for theerror probability of the NB link, which is valid for any linearmodulation scheme (e.g., M -ary phase shift keying or M -aryquadrature amplitude modulation). Our work generalizes theconventional analysis of coherent NB detection in the presenceof AWGN and fast fading, allowing the traditional results to beextended to include the effect of UWB network interference.

Index Terms— Ultrawideband systems, spectral coexistence,aggregate network interference, Poisson field, stable laws.

I. INTRODUCTION

There has been an emerging interest in transmission sys-tems with large bandwidth for both commercial and mili-tary applications. For example, ultrawideband (UWB) systemscommunicate with time-hopping (TH) or direct sequence (DS)spread-spectrum (SS) signals using a train of extremely shortpulses, thereby spreading the energy of the signal very thinlyover several GHz [1]–[5]. However, the use of large transmis-sion bandwidths introduces new challenges. The successfuldeployment of UWB systems requires that they coexist witha variety of signals over the already populated frequencybands. In particular, UWB devices must not cause harmfulinterference to existing narrowband (NB) systems (e.g., GPS,cellular, WLAN, and public safety).

The issue of coexistence in heterogeneous networks withboth UWB and NB nodes has received considerable attentionlately [6]–[9]. However, the effect of UWB interference on NB

P. C. Pinto and M. Z. Win are with LIDS, Massachusetts Institute ofTechnology, Cambridge, MA 02139, USA, e-mail: ppinto@mit.edu,moewin@mit.edu.

Chia-Chin Chong is with DoCoMo Communications Labs USA, Palo Alto,CA 94304, USA, e-mail: cchong@docomolabs-usa.com.

Andrea Giorgetti and Marco Chiani are with the DEIS, WiLAB,University of Bologna, 47023 Cesena ITALY, e-mail: {agiorgetti,mchiani}@deis.unibo.it.

This research was supported in part by DoCoMo Communications LabsUSA; the Portuguese Science and Technology Foundation under grant SFRH-BD-17388-2004; the University of Bologna (Progetto Internazionalizzazioned’Ateneo) and the Ministero dell’Istruzione, dell’Universita e della RicercaScientifica (MIUR); the Institute of Advanced Study Natural Science &Technology Fellowship; the Office of Naval Research Young InvestigatorAward N00014-03-1-0489; the National Science Foundation under Grant ANI-0335256; and the Charles Stark Draper Endowment.

systems has only been partially addressed in the literature.In [10], the bit-error probability (BEP) is analyzed for thecase of a single UWB pulse interfering with a binary phaseshift keying (BPSK) NB system, in an additive white Gaus-sian noise (AWGN) channel. In [11], a semi-analytical BEPexpression is derived for the case of one SS signal interferingwith a NB-BPSK system, also in an AWGN channel. Using ashot noise perspective, [12] analyzes the combined energy ofmultiple UWB signals at the output of a square-law receiver,without taking into account the error performance. In sum-mary, the existing literature ignores the channel propagationeffects (i.e., path loss, shadowing and fading), as well as thespatial distribution of the terminals whenever multiple UWBinterferers are considered.

In this paper, we derive an exact expression (without resort-ing to Gaussian approximations) for the error performance ofa NB system subject to multiple UWB interferers and AWGN.We introduce a framework where the UWB interferers are spa-tially scattered according to an infinite Poisson field, and areoperating asynchronously in a wireless environment subject topath loss, shadowing, and fast fading. Under this scenario, wedetermine the statistical distribution of the aggregate networkinterference at the output of a linear NB receiver, locatedanywhere in the two-dimensional plane. Then, we provideexpressions for the error performance of the NB link, whichare valid for any linear modulation scheme, such as M -aryphase shift keying (M -PSK) or M -ary quadrature amplitudemodulation (M -QAM). Lastly, we quantify these metrics as afunction of various important system parameters, such as thesignal-to-noise ratio (SNR), interference-to-noise ratio (INR),path loss exponent, and spatial density of the interferers. Ouranalysis shows how the aggregate UWB interference affectsthe performance of the NB system, thereby providing insightsthat may be of value to the network designer.

The paper is organized as follows. Section II describesthe system model. Section III derives the representation anddistribution of the aggregate UWB interference. Section IVanalyzes the error performance of the NB link subject to UWBinterference and noise. Section V provides a simple case studywith DS-BPAM interferers. Section VI concludes the paperand summarizes important findings.

II. SYSTEM MODEL

A. Spatial Distribution of the Nodes

The spatial distribution of the UWB interferers is modeledaccording to a homogeneous Poisson point process in the two-

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r0UWB interferers

NB transmitter

R1

R3

R2

NB receiver

Fig. 1. Poisson field model for the spatial distribution of nodes. Withoutloss of generality, we assume the origin of the coordinate system coincideswith the NB receiver.

dimensional infinite plane. Typically, the terminal positions areunknown to the network designer a priori, so they may betreated as completely random according to a spatial Poissonprocess. Then, the probability P{k in R} of k nodes beinginside a region R (not necessarily connected) depends onlyon the total area A of the region, and is given by

P{k in R} =(λA)k

k!e−λA, k ≥ 0,

where λ is the (constant) spatial density of the UWB interfer-ing nodes, in nodes per unit area. We define the interferingnodes to be the set of terminals which are transmitting withinthe frequency band of interest, during the time interval ofinterest (e.g., one symbol period), and hence are effectivelycontributing to the total interference. Thus, irrespective ofthe network topology (e.g., point-to-point mode or broadcastmode) or the session lifetime of each interferer, the proposedmodel depends only on the effective density λ of interferingnodes.1

The proposed spatial model is depicted in Fig. 1. Here, weassume there is a NB link composed of two nodes: one receivernode, located at the origin of the two-dimensional plane, andone transmitter node (node i = 0), deterministically located ata distance r0 from the origin. All the other nodes (i = 1 . . .∞)are UWB interfering nodes, whose random distances to theorigin are denoted by {Ri}∞i=1, where R1 ≤ R2 ≤ . . .. Ourgoal is then to determine the effect of the UWB interferingnodes on the NB link.

B. Transmission Characteristics of the Nodes

We consider the case where the UWB interferers transmitasynchronously and independently, using the same power PU.This is a plausible constraint when power control is too

1For example, if the interfering nodes are idle for a fraction of the time,then the splitting property of Poisson processes [13] can be used to obtainthe effective density of nodes that contribute to the interference.

complex to implement, and is applicable in decentralized ad-hoc networks, cellular systems, WLANs, and WPANs. Then,the signal si(t) transmitted by the ith UWB interferer can bedescribed as

si(t) =∑n

ai,npi(t− nTU − bi,n∆−Di), (1)

where pi(t) is the ith interferer symbol waveform,2 normalizedto have unit energy; TU is the symbol period; {ai,n}+∞

n=−∞and {bi,n}+∞

n=−∞ are, respectively, independent identicallydistributed (i.i.d.) pulse amplitude modulation (PAM) andpulse position modulation (PPM) sequences, not necessar-ily binary; ∆ is the modulation index associated with thePPM; and Di ∼ U(0, TU)3 is a random delay modeling theasynchronism between interferers. Since the UWB interferersoperate independently, all the random variables (r.v.’s) in (1)are independent for different i.

The NB transmitter employs a linear modulation scheme,such as M -PSK or M -QAM. To derive the error probabilityof the NB link, we only need to analyze a single NB symbol.Without loss of generality, we consider symbol 0 and writethe corresponding transmitted signal for all t as

sN(t) = c0√

2g(t) cos(2πfct+ θ0), (2)

where g(t) is a unit energy pulse-shaping waveform satisfyingthe Nyquist criterion; c0 and θ0 are the amplitude and phaseassociated with symbol 0, respectively; and fc is the carrierfrequency of the NB signal. Furthermore, we consider ascenario where the NB receiver performs demodulation of thedesired signal using a conventional linear detector. Typically,parameters such as the spatial density of interferers and thepropagation characteristics of the medium (e.g., shadowingand path loss parameters) are unknown to the receiver. Thislack of information about the interference, together withconstraints on receiver complexity, justify the use of a simplelinear detector, which is optimal when only AWGN is present.

C. Propagation Characteristics of the Medium

To account for the path loss affecting both the NB and UWBnodes, we assume a k/rν median signal amplitude decay withdistance r, for some given constant k. The amplitude lossexponent ν is environment-dependent, and can approximatelyrange from 1 (e.g., hallways inside buildings) to 4 (e.g., denseurban environments) [14], [15]. The use of such a decay lawalso ensures that interferers located far away have negligiblecontribution to the total interference observed at the NBreceiver, thus making the infinite-plane model reasonable. Forgenerality, we assume different path loss parameters for theNB signal (kN, νN) and the UWB signals (kU, νU).

To capture the shadowing affecting both NB and UWBnodes, we use a log-normal model where the probability

2Note that pi(t) itself may be composed of many monocycles, anddepending on its choice, can be used to represent both DS or TH spreadspectrum signals [4].

3We use U(a, b) to denote a real uniform distribution in the interval [a, b].

388

density function (p.d.f.) of the received signal strength S isgiven by

fS(s) =1

sσ√

2πexp

[− 1

2σ2ln2

(s

µ

)], s ≥ 0,

where µ = k/rν is the median of S, and σ is an environment-dependent parameter [15], [16]. The shadowing is responsiblefor random fluctuations in the signal level around the determin-istic path loss k/rν . A useful fact is that a log-normal r.v. Swith parameters µ and σ can be expressed as S = µeσG, whereG ∼ N (0, 1).4 For generality, we assume different shadowingparameters for the NB signal (σN) and the UWB signals (σU).

To account for the fading affecting the UWB interferers,we consider a fast, frequency-selective multipath channel withimpulse response

hi(t) =L∑l=1

hi,lδ(t− ti,l),

where {hi,l}Ll=1 and {ti,l}Ll=1 are, respectively, the amplitudesand delays (with arbitrary statistics) describing the L pathswhich affect the ith UWB interferer; and δ(t) denotes theDirac-delta function. In addition, we normalize the powerdelay profile of the channel such that

∑Ll=1 E{h2

i,l} = 1,where E{·} denotes the expectation operator. Using thismodel, the overall channel impulse response between the ithUWB interferer and the NB receiver becomes

hi(t) =kU

RνUi

eσUGihi(t). (3)

To account for the fading affecting the NB link, we considera fast, frequency-flat Rayleigh channel. Specifically, the chan-nel introduces in the received NB signal a Rayleigh-distributedamplitude factor α0 (normalized so that E{α2

0} = 1), as wellas a uniform phase φ0. We assume the NB receiver perfectlyestimates φ0, thus ensuring that coherent demodulation of thedesired signal is possible (for this reason, we can set φ0 = 0without loss of generality). Under this model, the overallimpulse response of the NB channel becomes

hN(t) =kN

rνN0

α0eσNG0δ(t). (4)

In this paper, we assume the shadowing and the fading tobe independent for different nodes (both NB and UWB), andapproximately constant during at least one symbol interval.

III. INTERFERENCE REPRESENTATION AND DISTRIBUTION

A. Complex Baseband Representation of the Interference

Under the system model described in Section II, the aggre-gate signal z(t) at the NB receiver can be written as

z(t) = d(t) + y(t) + n(t),

4We use N (0, σ2) to denote a real, zero-mean, Gaussian distributionwith variance σ2 , and Nc(0, σ2) to denote a circularly symmetric (CS)complex Gaussian distribution, where the real and imaginary parts are i.i.d.N (0, σ2/2).

where d(t) = sN(t) ∗ hN(t) is the desired signal from theNB transmitter corresponding to symbol 0, with ∗ denotingthe convolution operator; y(t) =

∑∞i=1 si(t) ∗ hi(t) is the

aggregate network interference; and n(t) is the AWGN withtwo-sided power spectral density N0/2, and independent ofy(t). By performing the indicated convolutions, we can furtherexpress the desired signal as

d(t) =kNα0e

σNG0

rνN0

c0√

2g(t) cos(2πfct+ θ0),

and the aggregate interference as

y(t) =∞∑i=1

∑n

kUeσUGi

RνUi

ai,nvi(t− nTU − bi,n∆−Di),

where vi(t) = pi(t) ∗ hi(t).The NB receiver demodulates the aggregate signal z(t)

using a conventional linear detector. This can be achievedby projecting z(t) onto the orthonormal set {ψ1(t) =√

2g(t) cos(2πfct), ψ2(t) = −√2g(t) sin(2πfct)}. Defin-ing the in-phase and quadrature (IQ) components Zk =∫ +∞−∞ z(t)ψk(t)dt, k = 1, 2, we can write

Z1 =kNα0e

σNG0

rνN0

c0 cos(θ0) + Y1 +N1 (5)

Z2 =kNα0e

σNG0

rνN0

c0 sin(θ0) + Y2 +N2, (6)

where N1 and N2 are N (0, N0/2) and mutually independent.Furthermore, Y1 and Y2 can be expressed as

Yk =∫ +∞

−∞y(t)ψk(t)dt =

∞∑i=1

kUeσUGiX

(k)i

RνUi

, k = 1, 2,

(7)where

X(k)i =

∑n

ai,n

∫ +∞

−∞vi(t− Di,n)ψk(t)dt, (8)

with

Di,n = nTU + bi,n∆ +Di. (9)

Using the Parseval relation and the fact that Pi(f) = F{pi(t)}and Hi(f) = F{hi(t)} are approximately constant over thefrequencies of the NB signal,5 (8) reduces after some algebrato

X(1)i =

∑n

ai,n√

2|Pi(fc)||Hi(fc)|g(Di,n) cos(φi,n) (10)

X(2)i =

∑n

ai,n√

2|Pi(fc)||Hi(fc)|g(Di,n) sin(φi,n), (11)

where

φi,n = arg{Pi(fc)}+ arg{Hi(fc)} − 2πfcDi,n. (12)

5We use F{·} to denote the Fourier transform operator.

389

Using complex baseband notation, (5)-(12) can be conciselywritten as6

Z =kNα0e

σNG0

rνN0

c0ejθ0 + Y + N (13)

Y =∞∑i=1

kUeσUGiXi

RνUi

, (14)

where

Xi =∑n

ai,n√

2|Pi(fc)||Hi(fc)|g(Di,n)ejφi,n , (15)

and the distribution of N is given by

N ∼ Nc(0, N0). (16)

Note that range of the summation of n in (15) depends onthe duration of the shaping pulse g(t) relative to TU. In effect,in the usual case where g(t) decreases to 0 as t→ ±∞, asn and thus Di,n grow, the r.v.’s g(Di,n) become increasinglysmall, and so the sum in n can be truncated.

B. Interference Distribution

The distribution of the aggregate interference Y is im-portant in the evaluation of the error probability of theNB link. To derive such distribution, we first analyze theproperties of Xi in (15). Typically, it is accurate to con-sider that arg{Hi(fc)} ∼ U(0, 2π), and independent ofthe magnitude |Hi(fc)|. Then, we can rewrite (15) asXi = Xie

j arg{Hi(fc)}, where arg{Hi(fc)} is independent ofXi; thus, Xi is circularly symmetric (CS). Furthermore, sincedifferent interferers i transmit asynchronously and indepen-dently, the r.v.’s Xi are i.i.d. for different i.

Sums of the form of (14) – where the r.v.’s {Ri}∞i=1

correspond to distances in a spatial Poisson process and the{Xi}∞i=1 are CS and i.i.d. – belong to the class of symmetricstable distributions [17].7 The p.d.f. of a real symmetric stabler.v. is plotted in Fig. 2 for various α. It can thus be shown [18]that Y has the CS complex stable distribution given in (17) atthe bottom of the page, where 0 < αY < 2 (or equivalently,νU > 1), and Cx is given by

Cx =

{1−x

Γ(2−x) cos(πx/2) , x �= 1,2π , x = 1.

(18)

6Boldface letters are used to denote complex quantities; for example,Z = Z1 + jZ2.

7We use S(α, β, γ) to denote a real stable distribution with characteris-tic exponent α ∈ (0, 2], skewness β ∈ [−1, 1], dispersion γ ∈ [0,∞), andlocation µ = 0. The corresponding characteristic function is

φ(ω) =

(exp

ˆ−γ|ω|α `1 − jβ sign(ω) tan πα

2

´˜, α �= 1,

expˆ−γ|ω| `

1 + j 2πβ sign(ω) ln |ω|´˜

, α = 1.

We use Sc(α, β, γ) to denote CS complex stable distribution, where the realand imaginary parts are i.i.d S(α, β, γ).

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

x

fX(x)

α=0.5α=1α=1.5α=2

Fig. 2. Real stable p.d.f.’s for varying characteristic exponents α (β = 0,γ = 1, µ = 0). Stable laws are a direct generalization of Gaussian distribu-tions, and include other densities with heavier (algebraic) tails. For α = 2,the stable distribution reduces to a Gaussian.

IV. ERROR PROBABILITY

In previous section, we determined the statistical distribu-tion of the aggregate interference at the output of a conven-tional linear receiver. We now use this to directly characterizeof the error probability of the NB link, when subject to bothnetwork interference and AWGN.

To derive the average error probability, we use the decom-position property of stable r.v.’s [17], which allows Y in (17)to be decomposed as

Y =√BG, (19)

where B and G are independent r.v.’s, and

B ∼ S(αB =

1νU, βB = 1, γB = cos

(π

2νU

))(20)

G ∼ Nc(0, 2VG), VG = 2γνUY , (21)

with γY given in (17). Conditioning on the r.v. B, we thenuse (16) and (19) to rewrite the aggregate received signal Zin (13) as

Z =kNα0e

σNG0

rνN0

c0ejθ0 + N,

where

N =√BG + N

|B∼ Nc(0, 2BVG +N0). (22)

Thus, our framework has reduced the analysis to a Gaussianproblem, where the combined noise N is a Gaussian r.v.Note that this result was derived without resorting to any

Y ∼ Sc

(αY =

2νU, βY = 0, γY = λπC−1

2/νUk

2/νU

U e2σ2U/ν

2U E{|X(k)

i |2/νU})

(17)

390

approximations – we merely used the decomposition propertyof stable r.v.’s.

The corresponding error probability8 Pe(G0) can be foundby taking the well-known error probability expressions forcoherent detection of linear modulations in the presence ofAWGN and fast fading [19], then using 2BVG +N0 insteadof N0 for the total noise variance, and lastly averaging over ther.v. B. Note that this substitution is valid for any linear modu-lation, allowing the traditional results to be extended to includethe effect of network interference. For the general case wherethe NB transmitter employs an arbitrary signal constellationin the IQ-plane, the resulting symbol error probability Pe(G0)is given in (23) at the bottom of the page, where

η =k2

Ne2σNG0EN

r2νN0 (2BVG +N0)

; (24)

M is the constellation size of the NB system; {pk}Mk=1 arethe symbol probabilities; Bk, φk,l, wk,l, and ψk,l are theparameters that describe the geometry of the constellation em-ployed in the NB link (see Fig. 3); EN = E{c20} is the averagetransmitted symbol energy of the NB link; B is distributedaccording to (20); and VG is given in (21). When the NBtransmitter employs M -PSK and M -QAM modulations withequiprobable symbols, (23) is equivalent to9

PMPSKe (G0) = I

(M − 1M

π, sin2( πM

))(25)

and

PMQAMe (G0) = 4

(1− 1√

M

)· I(π

2,

32(M − 1)

)

− 4(

1− 1√M

)2

· I(π

4,

32(M − 1)

), (26)

where the integral I(x, g) is given by

I(x, g) =1π

∫ x

0

EB

{(1 +

g

sin2 θη

)−1}dθ. (27)

V. CASE STUDY: DS-BPAM INTERFERENCE

We now particularize the general analysis developed in theprevious sections, using a simple case study with DS-BPAMinterferers. In this example, the signal si(t) transmitted by theith UWB interferer in (1) becomes

si(t) =∑n

ai,npi(t− nNsTf −Di),

8The notation Pe(X, Y ) is used as a shorthand for P{error|X,Y }.9For M -QAM, we implicitly assume a square signal constellation with

M = 2k points (k even).

ψ1,2

ψ1,3φ1,2

φ1,4

ψ1,4

s3

s2

s4

s1φ1,3

Fig. 3. Typical decision region associated with symbol s1. In general,

for a constellation with signal points sk = |sk|ejξk and ζk = |sk|2E{|sk|2} ,

k = 1 . . .M , four parameters are required to compute the error probability:φk,l and ψk,l are the angles that describe the decision region correspondingto sk (as depicted); Bk is the set consisting of the indexes for the signal pointsthat share a decision boundary with sk (in the example, B1 = {2, 3, 4}); andwk,l = ζk + ζl − 2

√ζkζl cos(ξk − ξl).

where the unit-energy waveform pi(t) for each bit is given by

pi(t) =Ns−1∑k=0

di,kw(t − kTf).

In these equations, Ns is the number of monocycles requiredto transmit a single information bit ai,n ∈ {−

√EU,√EU},

where EU is the bit energy; w(t) is the transmitted monocycleshape, with energy 1/Ns; Tf is the monocycle repetition time(frame length), and is related to the bit duration by TU = NsTf;and {di,k}Ns−1

k=0 is the spreading sequence.For such a DS-BPAM system, |Pi(fc)| in (15) can be easily

derived as

|Pi(fc)| = |W (fc)|∣∣∣∣∣Ns−1∑k=0

di,kej2πfckTf

∣∣∣∣∣ , (28)

where W (f) = F{w(t)}.Different monocycle shapes are considered to satisfy FCC

masks with the maximum transmitted power. Among these,the 5th derivative of a Gaussian monocycle is chosen in [20],so the received pulse can be modeled as the 6th derivative. Inthis case, we can write W (fc) in (28) as

W (fc) =8π3

3√

1155Nsτ13/2w f6

c e−π

2 f2c τ

2w , (29)

where τw is the monocycle duration parameter. Both (28) and(29) can be used in (15) and (17) when determining the BEPof the DS-BPAM system.

Pe(G0) =M∑k=1

pk∑l∈Bk

12π

∫ φk,l

0

EB

{(1 +

wk,l

4 sin2(θ + ψk,l)η

)−1}dθ (23)

391

0 5 10 15 20 25 3010

−4

10−3

10−2

10−1

100

SNR (dB)

Pe(G

0)

λ=0.5

λ=0.05

λ=0

Fig. 4. Pe(G0) versus the normalized SNR of the NB link, for various spatialdensities λ of the UWB interferers, in m−2 (NB link: BPSK, square g(t)of duration TN, 1/TN = 1 Mb/s, fc = 1 GHz, G0 = 0, r0 = 1 m, σN = 1,νN = 2 ; UWB interferers: DS-BPAM, 1/TU = 1.25 Mb/s, INR = −10 dB,Tf = 50 ns, Ns = 16 pulses/bit, τw = 0.192 ns, σU = 1, νU = 2, Nakagami-m fading with m = 3 and L = 1).

Fig. 4 quantifies the error performance of a NB-BPSKlink, subject to a Poisson field of DS-BPAM interferers andAWGN. For this purpose, we define the normalized SNR ofthe NB link as SNR = k2

NEN/N0, and the normalized INRas INR = k2

UEU/N0. The plot of Pe(G0) presented here isof semi-analytical nature. Specifically, we resort to a hybridmethod where we employ the analytical results given in(23)-(24), but perform a Monte Carlo simulation of the stabler.v. B according to [21]. Nevertheless, we emphasize that theexpressions derived in this paper completely eliminate the needfor bit-level simulation of the system, in order to obtain itserror performance.

VI. SUMMARY

This paper puts forth a mathematical model for NB com-munication subject to both UWB network interference andAWGN, where the spatial distribution of the nodes is cap-tured by a Poisson field in the two-dimensional plane. Weconsider the case of asynchronous UWB interferers, in arealistic wireless environment subject to path loss, log-normalshadowing and fast fading. Under this scenario, we determinethe statistical distribution of the aggregate UWB interferenceat the output of a conventional linear receiver, which leadsdirectly to an exact characterization of the error performance,without resorting to Gaussian approximations. The resultingsemi-analytical expressions are valid for any linear modulationscheme, and capture all the essential physical parameters thataffect network interference.

ACKNOWLEDGEMENTS

The authors would like to thank L. A. Shepp, L. J. Greenstein,J. H. Winters, and G. J. Foschini for their helpful suggestions.

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