Statistical Properties of Electromagnetic Environment in Wireless Networks, Intra-Network Electromagnetic Compatibility and Safety

Transcription

1 Sttisticl Properties of Electromgnetic Environment in Wireless Networs, Intr-Networ Electromgnetic Comptibility nd Sfety Vldimir Mordchev Belorussin Stte University of Informtics nd Rdioelectronics (BSUIR Electromgnetic Comptibility Lbortory 6, P.Brovi st., Mins 2213, Belrus E-mil: Abstrct Sttisticl properties of electromgnetic environment in wireless networs ffecting its intrnetwor electromgnetic comptibility nd sfety re studied. The nlysis is bsed on the stndrd propgtion chnnel model, Poisson model of rndom sptil distribution of trnsmitters, nd thresholdbsed model of the victim receptor behviour (rdio receiver or humn body. The distribution of dominnt interference level is derived nd nlysed under vrious networ nd system configurtions. The ggregte interference is dominted by the nerest trnsmitter one. The distribution of the unordered single-node interference is independent of the trnsmitters power nd their sptil density nd is the sme for homogeneous nd non-homogeneous networs. The outge probbility is used s mesure of not only the wireless lin qulity-of-service, but lso of environmentl riss induced by electromgnetic rdition. The mximum cceptble interference levels for relible lin performnce nd for low environmentl riss re surprisingly similr. 1. Introduction Electromgnetic comptibility (EMC nd sfety (EMS of wireless communiction networs hve been recently subject of extensive studies. Mutul interference mong severl lins (e.g. severl users operting t the sme time plces fundmentl limit to the networ performnce nd lso determines the level of electromgnetic environmentl riss to the popultion. The effect of interference in wireless networs t the physicl lyer hs been studied from severl perspectives [1]-[5]. A typicl sttisticl model of interference in networ includes model of sptil loction of the nodes, propgtion pth loss lw (which includes the verge pth loss nd, possibly, lrge-scle (shdowing nd smll-scle (multipth Sergey Loy School of Informtion Technology nd Engineering University of Ottw 161 Louis Psteur, Ottw, Cnd, K1N 6N5 E-mil: sergey.loy@ieee.org fding nd threshold-bsed receiver performnce model. The most populr choice for the model of the node sptil distribution is Poisson point process on plne [1]-[5]. Bsed on this model nd ignoring the effect of fding, Sous [2] hs obtined the chrcteristic function (CF of the ggregte (totl interference t the receiver, which cn be trnsformed into closed-from probbility density function (PDF in some specil cses, nd, bsed on it, the error rtes for direct sequence nd frequency hopping systems. While using the LePge series representtion, Ilow nd Htzinos [3] hve developed generic technique to obtin the CF of ggregte interference from Poisson point process on plne (2-D nd in volume (3-D, which cn be used to incorporte the effects of Ryleigh nd log-norml fding in strightforwrd wy. Relying on homogeneous Poisson point process on plne, Weber et l [4] hve chrcterized the trnsmission cpcity of the networ subject to the outge probbility constrint vi lower nd upper bounds. In recent wor, Weber et l [5] use the sme pproch to chrcterize the networ trnsmission cpcity when the receivers re ble to suppress some powerful interferers. A common feture of ll these wors is the use of ggregte interference (either lone or in the form of signl-to-interference-plus-noise rtio, nd common lesson is tht it is very difficult to del with: while the CF of ggregte interference cn be obtined in closed form, the PDF or CDF re vilble only in few specil cses. This limits significntly the mount of insight tht cn be extrcted from such model, especilly if no pproximtions or bounds re used. To overcome this difficulty, we dopt different pproch: insted of relying on the ggregte interference power s performnce indictor, we use the power of the dominting interfering signl [7]-[11]. While this is clerly n pproximtion, closed-form performnce evlution becomes fesible nd significnt insight cn be extrcted from such model.

2 Furthermore, since the ggregte interference is dominted by the most powerful interferer in the region of low outge probbility (i.e. the prcticlly-importnt region, both models give roughly the sme results (see [11] for detils. This observtion is lso consistent with the corresponding results in [4][5], when the nerfield region contins only one interferer. Thus, in the frmewor of [4][5], our results represent the (tight lower bound on the outge probbility. Using this model, we study the power distribution of the ensemble of interferences nd the dominnt interferer in vrious scenrios, which is further used to obtin compct closed-form expressions for the outge probbility of given receptor (or, equivlently, of the lin of given user in the 1-D, 2-D nd 3-D Poisson field of interferers, for both uniform nd non-uniform verge node densities nd for vrious vlues of the verge pth loss exponent. Comprison to the corresponding results in [2] (obtined in terms of the error rtes indictes tht the dominnt contribution to the error rte is due to the outge events cused by the closest (i.e. dominnt interferer, which increses with the verge node density. The proposed method is flexible enough to include the cse when given number of strongest interferers re suppressed. The outge probbility is shown to scle down exponentilly in this number. Contrry to [5], we do not rely in this cse on the simplifying ssumption of cnceling ll interferers in the dis with the given verge number of interferers; neither we ssume tht only interferers more powerful thn the required signl re cncelled (the lst ssumption ffects significntly the result, i.e. our nlysis of interference cncelltion is exct. The proposed method cn lso be used to include the effect of fding. We rgue tht Ryleigh fding hs negligible effect on the distribution of dominnt interferer s power nd the effect of log-norml fding (shdowing is to shift the distribution by constnt non-negligible fctor [11]. Our nlysis culmintes in the formultion of the outge probbility-networ density trdeoff: for given verge density of the nodes, the outge probbility is lower bounded or, equivlently, for given outge probbility, the verge density of the nodes is upper bounded. This trdeoff is result of the interply between rndom geometry of node loctions, the propgtion pth loss nd the distortion effects t the victim receiver. Our nlysis is bsed on the frmewor originlly developed in [7]-[11]. We rgue tht the outge probbility, which is trditionlly used s mesure of qulity-of-service in wireless systems nd networs, lso mesures the environmentl riss to the popultion induced by electromgnetic rdition of wireless devices. The pper is orgnized s follows. In Section 2, we introduce the system nd networ model. In Section 3, the distribution of interference levels nd of the dynmic rnge (dominnt interference-to-noise rtio is given for this model. Bsed on this, the node density outge probbility trdeoff is presented in Section 4. The mximum cceptble interference levels for high qulity-of-service wireless networ performnce nd for low electromgnetic environmentl riss to the popultion re shown to be surprisingly similr in Section Networ nd System Model As n interference model of wireless networ t the physicl lyer, we consider number of point-lie trnsmitters (Tx nd receptors (Rx tht re rndomly locted over certin limited region of spce S m, which cn be one ( m = 1, two ( m = 2,or three ( m = 3 -dimensionl (1-D, 2-D or 3-D. This cn model loction of the nodes over highwy or street cnyon (1-D, residentil re (2-D, or downtown re with number of high-rise buildings (3-D. In our nlysis, we consider single (rndomly-chosen receiver (or some other receptor which is susceptible to electromgnetic fields generted by trnsmitters nd number of trnsmitters tht generte interference to this receiver. We ssume tht the sptil distribution of the trnsmitters (nodes hs the following properties: (i for ny two non-overlpping regions of spce S nd S b, the probbility of ny number of trnsmitters flling into S is independent of how mny trnsmitters fll into S b, i.e. non-overlpping regions of spce re sttisticlly independent; (ii for infinitesimlly smll region of spce ds, the probbility P( = 1, ds of single trnsmitter ( = 1 flling into ds is P ( = 1, ds = ρds, where ρ is the verge sptil density of trnsmitters (which cn be function of position. The probbility of more thn one trnsmitter flling into ds is negligible, P( > 1, ds << P ( = 1, ds s ds. Under these ssumptions, the probbility of exctly trnsmitters flling into the region S is given by Poisson distribution, (, N P S = e N /!, N = ρds (1 where N is the verge number of trnsmitters flling into the region S. If the density is constnt, then N = ρ S. Possible scenrios to which the ssumptions bove pply, with certin degree of pproximtion, re sensor networ with rndomly-locted noncooperting sensors; networ(s of mobile phones from the sme or different providers (in the sme re; networ of multi-stndrd wireless devices shring the sme resources (e.g. common or djcent bnds of frequencies or n d-hoc networ. S

3 Consider now given trnsmitter-receptor (trnsmitter-receiver pir. The power t the Rx ntenn output P r coming from the trnsmitter is given by the stndrd lin budget eqution [6], P = PG G g (2 r t t r where P t is the Tx power, Gt, G r re the Tx nd Rx ntenn gins, nd g is the propgtion pth gin (=1/pth loss, g = ggl gs, where g is the verge propgtion pth gin, nd gl, g s re the contributions of lrge-scle (shdowing nd smll-scle (multipth fding, which cn be modeled s independent lognorml nd Ryleigh (Rice rndom vribles, respectively [6]. The widely-ccepted model for g is g = R, where is the pth loss exponent, nd is constnt independent of R [6]. In the trditionl lin-budget nlysis of point-to-point lin, it is deterministic constnt. However, in our networ-level model g becomes rndom vrible since the Tx-Rx distnce R is rndom (due to rndom loction of the nodes nd it is this rndom vrible tht represents new type of fding, which we term networ-scle fding, since it exhibits itself on the scle of the whole re occupied by the networ. Since g does not depend on the locl propgtion environment round the Tx or Rx ends tht ffect gl, g s but only on the globl configurtion of the Tx-Rx propgtion pth (including the distnce R, of which gl, g s re independent [6], the networ-scle fding in this model is independent of the lrge-scle nd smll-scle ones, which is ultimtely due to different physicl mechnisms generting them. The distribution functions of g in vrious scenrios hve been given in [8]-[1]. 3. PDF of Interference Levels nd the Interference to Noise Rtio We consider fixed-position receptor nd number of rndomly locted interfering trnsmitters (interferers, e.g. mobile units of other users of the sme power P t (following the frmewor in [7]-[1], this cn lso be generlized to the cse of unequl Tx powers. Only the networ-scle fding is ten into ccount in this section, ssuming tht gl = gs = 1 (this ssumption is relxed in section 4. For simplicity, we lso ssume tht the Tx nd Rx ntenns re isotropic (this ssumption is relxed below, nd consider the interfering signls t the receiver input. The sttistics of trnsmitters loction is given by (1. Trnsmitter i produces the verge power Pi = Pt g ( Ri t the receiver input, nd we consider only the signls exceeding the Rx noise level P, Pi P. We define the interference-to-noise rtio (INR d in the ensemble of the interfering signls vi the most powerful (t the Rx input signl (It cn be shown tht, in the smll outge region, the totl interference power (i.e. coming from ll trnsmitters is dominted by the contribution of the most powerful signl, i.e. the single events dominte the outge probbility [11], d = P / P (3 1 where, without loss of generlity, we index the trnsmitters in the order of decresing Rx power, P 1 P 2... PN. The most powerful signl is coming from the trnsmitter locted t the minimum distnce r 1, P 1 = Pt g ( r1. The cumultive distribution function (CDF of the minimum distnce cn be esily found [7]-[9], 1 ( 1 exp ( N ( V F r =, N ( V = ρdv, (4 V where N ( V is the verge number of trnsmitters in the bll V ( r of rdius r. The corresponding PDF cn be found by differentition, f N 1( r = e ρ dv (5 V ( r where V ( r is sphere of rdius r nd the integrl in (5 is over this sphere. The probbility tht the INR exceeds vlue D is Pr{ d > D} = Pr { r1 < r( D } = F1 ( r( D, where r( D is such tht P ( r ( D = P D, so tht the CDF of d is F ( D = 1 Pr d > D = exp( N( D (6 d { } where N( D is the verge number of trnsmitters in 1/ the bll V ( r( D of the rdius r( D = ( P t / P D. The corresponding PDF cn be obtined by differentition, N ( D r( D e f ( D = ρdv d D (7 V ( r( D When the verge sptil density of trnsmitters is constnt, ρ = const, (6, (7 simplify to [7]-[1], m / P t Fd ( D = exp cmρ = exp N D P D m m / ( mx 1 m f exp{ mx / d D = N D N D } m / { mx } where c 1 = 2, c 2 = π nd c 3 = 4 π / 3, N mx = cmrmxρ is the verge number of trnsmitters in the bll of rdius R mx, which we term potentil interference zone, nd R mx is such tht P ( Rmx = P, i.e. trnsmitter t the boundry of the potentil interference zone produces signl t the receiver exctly t the noise level; trnsmitters locted outside of this zone produce weer signls, which re neglected in the nlysis. Note tht (8 gives the distribution of the INR s simple explicit function of the system nd geometricl prmeters, nd ultimtely depends on N mx, m, only. m, (8

4 When ( 1 most powerful signls, which re coming from ( 1 closest trnsmitters, do not crete ny interference (i.e. due to frequency, time or code seprtion in the multiple ccess scheme, or due to ny other form of seprtion or filtering, the CDF nd PDF of the distnce r to the most powerful interfering signl of order cn be found in similr wy. The CDF of the INR d in this cse is given by N ( D 1 i F ( D = e N( D / i! (9 d i= In the cse of constnt verge density ρ = const, the CDF nd PDF of the INR simplify to [7]-[1], m { } 1 mx 1 N Fd ( D = exp N D, / mx m / i= i! D m mx 1 / m { mx } m N fd ( D = D exp N D ( 1! i (1 which re lso explicit functions of N mx, m,. On the other hnd, the PDF of interference power P coming from single, rndomly-selected node locted in the potentil interference zone is, for ρ = const, [7]- [1]: mp m/ f ( P =, P ( m / P + (11 P Using this, (1 cn be derived s well by nlysing the mx/min rtio for n ensemble of interfering signls, ech hving the PDF in (11. Note tht this PDF does not hve 1 st nd 2 nd moments for some importnt vlues of m nd (becuse of its long til, e.g. m = 2, = 2 (plnr distribution nd free-spce propgtion. Another interesting property of (11 is tht it is independent of the Tx power nd node density nd depends on only three bsic prmeters ( m,, P, so tht nodes with different powers nd densities s well s their combintions (e.g. non-homogeneous networ will induce the sme distribution. The sme pplies to the cse of rndom Tx power (including fding chnnels. Unfortuntely, this conclusion does not extend to (1 (see [11] for detiled nlysis of the impct of fding in this cse. 4. Outge Probbility-Node Density Trdeoff Powerful interfering signls cn result in significnt performnce degrdtion due to liner nd nonliner distortion effects in the receiver when they exceed certin limit, which we chrcterize here vi the receiver distortion-free dynmic rnge (i.e. the mximum cceptble interference-to-nose rtio Ddf = Pmx / P, where P mx is the mximum interfering signl power t the receiver tht does not cuse significnt performnce degrdtion. If d > Ddf, there is significnt performnce degrdtion nd the receiver is considered to be in outge, which corresponds to one or more trnsmitters flling into the ctive interference zone (i.e. the bll of rdius r( D df ; the signl power coming from trnsmitters t tht zone exceeds P mx, whose probbility is { } P = Pr d > D = 1 F ( D (12 out df d df For given P out, one cn find the required distortion-free dynmic rnge ( outge dynmic rnge D D = F 1 (1 P (13 df d out We note tht, in generl, D df is decresing function of P out, i.e. low outge probbility clls for high distortion-free dynmic rnge. For simplicity of nottions, we further drop the subscript nd denote the spurious-free dynmic rnge by D. All interfering signls re ctive (=1: We consider first the cse of = 1, i.e. ll interfering signls re ctive. The outge probbility cn be evluted using (6 nd (12. From prcticl perspective, we re interested in the rnge of smll outge probbilities P out << 1, i.e. high-relibility nd communictions. When this is the cse, Fd ( D 1 nd using N McLuren series expnsion e 1 N, (12 simplifies to P out N = ρdv (14 V ( r( D which further simplifies, in the cse of m / out N mx D df ρ = const, to P (15 Note tht, in this cse, the outge probbility P out scles linerly with the verge number N mx of nodes in the potentil interference zone, nd it effectively behves s if the number of nodes were fixed (not rndom nd equl to N mx. Bsed on this, we conclude tht the single-order events (i.e. when only one signl in the ensemble of interfering signls exceeds the threshold P mx re dominnt contributor to the outge. This immeditely suggests wy to reduce significntly the outge probbility by eliminting (e.g. by filtering the dominnt interferer in the ensemble. Using (15, the required spurious-free dynmic rnge of the receiver cn be found for given outge probbility, / m D ( N mxp / out. Note tht higher vlues of nd lower vlues for m cll for higher dynmic rnge. Intuitively, this cn be explined by the fct tht when the trnsmitter moves from the boundry of the potentil interference zone (i.e. R = Rmx, P ( R = P closer to the receiver ( R << Rmx, the power grows much fster when is lrger, so tht closely-locted trnsmitters produce much more interference (compred to those locted close to the boundry when is lrge, which, combined with the uniform sptil density of the trnsmitters, explins the observed behvior. The effect

5 of m cn be explined in similr wy. To vlidte the ccurcy of pproximtion in (14, nd lso the expressions for the dynmic rnge PDF nd CDF in the previous section, extensive Monte-Crlo (MC simultions hve been underten. Fig. 1 shows some of the representtive results. Note good greement between the nlyticl results (including the pproximtions nd the MC simultions. It cn be lso observed tht the tils of the distributions decy much slower for the = 4 cse, which indictes higher probbility of high-power interference in tht cse nd, consequently, requires higher spurious-free dynmic rnge of the receiver, in complete greement with the predictions of the nlysis. Note lso tht the outge probbility evluted vi the totl interference power coincides with tht evluted vi the mximum interferer power, t the smll outge region (this result hs been rigorously proved in [11]. Consider now scenrio where the ctul outge probbility hs not to exceed given vlue P out for the receiver with given distortion-free dynmic rnge D. Using (8 nd (12, the verge number of trnsmitters in the ctive interference zone (bll of rdius r( D cn be upper bounded s N ln(1 P out. Using the expression for N, one obtins bsic trdeoff reltionship between the networ density nd the outge probbility, N = ρdv ln(1 P V ( r( D out (16 i.e. for given outge probbility, the networ density is upper bounded or, equivlently, for given networ density, the outge probbility is lower bounded. In the cse of uniform density ρ = const nd smll outge probbility, P out << 1, this gives n explicit trdeoff reltionship between the mximum distortionfree interference power t the receiver P mx, the trnsmitter power P t nd the verge node density for distortion-free receiver opertion, ( 1 m out mx / t m / ρ c P P P (17 or, equivlently, n upper bound on the verge density of nodes in the networ. As intuitively expected, higher P out, Pmx, nd lower Pt, m llow for higher networ density. The effect of is intuitively explined by the fct tht higher results in lrger pth loss or, equivlently, in smller distnce t the sme pth loss, so tht the trnsmitters cn be locted more densely without incresing interference level. The effect of the other prmeters cn be explined in similr wy. ( 1 strongest interfering signls re inctive: We now ssume tht ( 1 strongest interfering signls re eliminted vi some mens (e.g. by filtering or resource lloction. In this cse, (9, (1 pply nd (14 generlizes to 1 1 out =!! m / ( mx P N N D (18 1 which cn be expressed s Pout = P! out,1 P out,1, where P out,1 is the outge probbility for = 1 (see (14. In the smll outge region, P out,1 << 1 nd Pout << P out,1, i.e. there is significnt beneficil effect of removing ( 1 strongest interferers, which scles exponentilly with. Further comprison to the corresponding result in [5] shows tht the ssumption there of cncelling ll interferers, which exceed the required signl nd re in the dis with the given verge number of interferers, ffects significntly the result (no exponentil scling. It should lso be noted tht, contrry to the = 1 cse, P out in (18 is super-liner in N mx : doubling N mx increses P out by the fctor 2 > 2, i.e. P out is more sensitive to N in this cse. CCDF (outge prob mx nu= INR (dynmic rnge, db Fig. 1. The CCDF of d = P 1 / P nd dtot = Ptot / P (lso the outge probbility evluted from Monte-Crlo (MC simultions for m = 2, = 2 & 4, 1 5 P = 1, P t = 1, ρ = 1 ; nlytic CCDF of d (derived from (8 nd its pproximtion in (15 re lso shown. mx. power (MC totl power (MC nlytic pproximte nu=4 In similr wy, the node density-outge probbility trdeoff cn be formulted. In the for smll outge probbility region P out << 1, it cn be expressed s ( 1/ N = ρdv! P V ( r( D out (19 Compring (19 to (16, one cn clerly see the beneficil effect of removing ( 1 most powerful interferers on the outge probbility-networ density trdeoff, since (! 1/ Pout >> P out in the smll outge regime, so tht higher node density is llowed t the sme outge probbility. In the cse of uniform density, (19 reduces to (! ( / 1 1/ m / m out mx t ρ c P P P (2 which is generliztion of (17 to 1. Impct of Ryleigh nd log-norml fding: Following the sme pproch s in [3], it cn be shown tht the impct of Ryleigh nd log-norml fding on

6 the distributions bove is shift by constnt fctor. In the cse of Ryleigh fding, the constnt is close to 1 nd, thus, cn be neglected so tht the distributions re roughly not ffected. In the cse of log-norml fding, the constnt is not negligible. The intuition behind this result is tht the distributions in (12, (15, (18 re much more hevily-tiled (slowly-decying thn the Ryleigh distribution so tht outge events in the combined distribution re mostly cused by nerby interferers without deep Ryleigh fdes nd the combined distribution is roughly the sme s the originl one (without fding. On the other hnd, the log-norml distribution is lso hevily-tiled, so it cnnot be neglected (see [11] for detiled nlysis of the fding effects. 5. Outge Probbility: Mesure of Induced Electromgnetic Environmentl Riss As it ws mentioned before, environmentl riss induced by EME in wireless networs re determined by the level of dominnt interference t the receptor lloction. The threshold vlues П E1 -П E4 of these levels in terms of the power flux density (PFD of n electromgnetic field nd used s n electromgnetic sfety criteri, re given below [1]. Tble 1 PFD, Description µw/cm 2,1 (П E1 1, (П E2 2, (П E3 1, (П E4 Preliminry preventive Mximum Permissible Level (MPL for «totl common electromgnetic irrditions from ll high-frequency equipment with very low pulsing modultion» (recommended. Equl or close to the MPL ccepted in some countries/regions. Highest level of intensity of n electromgnetic bcground tht is sve for the popultion. Accepted erlier in the USSR up to 1984 s the MPL for the popultion The MPL ccepted in Moscow nd Pris for plces of round-the-cloc sty of people The MPL for the popultion, ccepted in Russi, Belrus nd lso in number of the Europen nd Asin countries The level П b [W/m 2 ] of the receiver desensitiztion/blocing, determined t the ntenn loction, exceeds the receiver sensitivity П min [W/m 2 ] by the blocing dynmic rnge D dfb : Π b = DdfbΠ min. (2 A comprison between the level of rdio receiver desensitiztion/blocing nd the levels in the electromgnetic sfety criteri, for typicl vlues of receiver sensitivity П min nd the receiver dynmic rnge D dfb, re given in Tble 2. Tble 2. П min, П b, µw/cm 2 W/m 2 D dfb =7dB D dfb =8dB D dfb =9dB П E П E1 П E2, П E3 1-1 П E1 П E2, П E3 П E4 Clerly, the levels of out-of-bnd interference cusing desensitiztion for highly-liner receivers (high D dfb roughly equl to the MPL levels required from the ecologicl point of view, which is n indiction of close similrity of EMC nd EMS problems in wireless communictions. 6. References [1] E.S. Sous, J.A. Silvester, Optimum Trnsmission Rnges in Direct-Sequence Spred-Spectrum Multihop Pcet Rdio Networ, IEEE JSAC, v. 8, N. 5, pp , June 199. [2] E.S. Sous, Performnce of Spred Spectrum Pcet Rdio Networ Lin in Poisson Field of Interferers, IEEE Trns. Inform. Theory, v.38, No.6, pp , Nov [3] J. Ilow, D. Htzinos, Anlytic Alph-Stble Noise Modeling in Poisson Field of Interferers or Sctterers, IEEE Trns. Signl Processing, v.46, No.6, pp , June [4] S. P. Weber et l, Trnsmission Cpcity of Wireless Ad Hoc Networs With Outge Constrints, IEEE Trns. Inform. Theory, v.51, No.12, pp , Dec. 25. [5] S. P. Weber et l, Trnsmission Cpcity of Wireless Ad Hoc Networs With Successive Interference Cncelltion, IEEE Trns. Inform. Theory, v.53, No.8, pp , Aug. 27. [6] G. L. Stuber, Principles of Mobile Communiction (2nd Ed., Kluwer, Boston, 21. [7] V.I.Mordchev, "Sttisticl Chrcteristics of Dynmic Rnge of Indvertent Disturbnces with Spce-Scttered Groupings of Their Sources", in Proc. of the 9-th Wroclw Symp. on EMC, June 1988, pp (in Russin [8] V.I. Mordchev, Typicl Models of Electromgnetic Environments for Sptilly-Scttered Rdio Trnsmitters, in Proc. of the 1-th Wroclw Symp. on EMC, June 199, pp (in Russin [9] V.Mordchev. Mthemticl Models for Rdiosignls Dynmic Rnge Prediction in Spce-Scttered Mobile Rdiocommuniction Networs, IEEE VTC Fll, Boston, Sept , 2. [1] V.Mordchev. System Ecology of Cellulr Communictions, Belrus Stte University Publishers, 29, 319 p. (in Russin. [11] V.Mordchev, S.Loy. On Node Density Outge Probbility Trdeoff in Wireless Networs, IEEE Journl on Selected Ares in Communictions (Specil Issue on Stochstic Geometry nd Rndom Grphs for the Anlysis nd Design of Wireless Networs, v.27, No.7, p , Sept. 29.