ON THE K-FACTOR DISTRIBUTION AND DIFFRACTION FADING FOR SOUTHERN AFRICA

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1 7 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vl.97() June 006 ON THE K-FACTOR DISTRIBUTION AND DIFFRACTION AND DIFFRACTION FADING FOR SOUTHERN AFRICA T.J. Afull and P.K. Odedina Schl f Electical, Electnic and Cmpute Engineeing, Univesity f Kwazulu-Natal (WestvilleCampus), Duban 4000, Suth Afica Abstact: In pevius pesentatins, we have discussed mdels f detemining the pbability density functin (pdf) f the effective eath adius fact f Btswana as a means f pedicting k- fact vaiatin. These mdels ae based n the adisnde measuement data btained ve thee yeas in Btswana. In this pesentatin, we cmpae the k values btained via the cuve-fitting mdel f Btswana and Duban, Suth Afica. We cmpae the median k-fact values and k values exceeded 99.9% f the time. Tw design examples shw that the type f teain plays a maj le in diffactin fading esulting fm k-fact vaiatin. While designing with k=4/3 nly esults in lsses f db and 5 db f Duban and Btswana links, espectively, multiple bstuctins wuld esult in highe fades. Meve, the gentle teain in Btswana esults in lwe lsses that wuld have been the case if the k-fact f Duban had been.. Als we bseve that 99.9% link availability wuld be hade t attain in Duban than in Btswana, due t the highe k-fact vaiability and hillie teain in Duban. Key wds: The k-fact distibutin, k-type fading f Suthen Afica.. INTRODUCTION The ppagatin f electmagnetic waves aund the eath is influenced by the ppeties f the eath and the atmsphee [-5]. The eath is an inhmgeneus bdy whse electmagnetic ppeties vay cnsideably as we g fm ne pint t anthe. Sea wate is highly cnducting wheeas deset sands ae dielectic, having vitually ze cnductivity but dissipating enegy by vitue f plaizatin. The atmsphee ve the eath is a dynamic medium, its ppeties vaying with tempeatue, pessue, and humidity. Accding t ITU-R Recmmendatin P.530 [], the ppagatin lss n a teestial line-f-sight path elative t fee space lss is the sum f diffeent cntibutins, including the fllwing: attenuatin due t atmspheic gases; diffactin fading due t bstuctin patial bstuctin f the path; fading due t multipath; and attenuatin due t pecipitatin. Each f these cntibutins has its wn chaacteistics as a functin f fequency, path length, and gegaphical lcatin. The nmal cnditin f the atmsphee is a tempeatue and pessue gadient with which is assciated a gadient f efactin index and cnsequently a efacted wave [,3,6]. The vaying index f efactin esults in the bending f ppagating adi waves, with the sense f the gadient being such that the ays ae bent twad the eath. Changes in tempeatue and mistue cntent esult in changes in the gadient f the efactin index, n. Cnditins may esult in an incease in index with height and a bending f ays away fm the eath. The signal intensity at the eceive then fades makedly. Cnvesely, metelgical cnditins can esult in a steepe gadient such that ays ae bent twads the eath me stngly. Unde such cnditins, the atmsphee fms a duct waveguide that esticts the wave ppagatin ve the eath s suface. Abnmally lage anges beynd line f sight ae then btained, with the effects being me pnunced in the VHF t micwave egins [, 6-7]. The k-fact fading aises due t the abve-mentined vaiatins in atmspheic efactive cnditins, which cause changes in the k-fact fm its median value. When the atmsphee is sufficiently sub-efactive (lw k-fact values), the ay path will be bent in such a way that the eath appeas t bstuct the diect path between tansmitte and eceive, giving ise t the kind f fading called diffactin fading. This fading is the fact that detemines the antenna heights []. The k-fact statistics f a single pint can be detemined fm measuements pedictins f the efactive index gadient in the fist 00 m f the atmsphee []. These gadients need t be aveaged in de t btain the effective value f k f the path length in questin. Diffactin lss will depend n teain and vegetatin. The diffactin lss ve aveage teain can be appximated f lsses geate than 5 db by the fmula []: h G d ( db) = () e whee h is the height diffeence (m) between the mst Cpyight 004 IEEE: An ealie vesin f this pape was fist published in AFRICON '04, 5-7 Septembe 004, Gabne, Btswana.

2 Vl.97() June 006 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 73 Height h, agl α (a) θ pans f Btswana, and f Duban n the Indian Ocean shes. As epted befe, the Btswana data cves thee yeas ( ) while the Duban data spans nly ne yea due t unavailability f such detailed data f lnge duatin (003/04). Then we use the k-fact values s btained t discuss the level f diffactin fading (diffactin lss/gain) when the value k=4/3 is used in link design instead f the actual values btained f the tw egins. Als nte that while it is imptant t aveage the k-values ve the design path t btain the effective k-fact ve the link, the enmus amunt f data equied cmpels us t assume the pint k-fact values as the effective values.. LINK DESIGN ISSUES d s In detemining the k-fact, the elatinship between the adius f cuvatue f the ay, ρ, and the change f efactivity, N, with height, h, is key. Cnside an atmsphee f which the index f efactin, n, deceases linealy with height [6-7,5]: ρ d θ θ h ( ) n = n β h () ( b ) Fig.. Refactin f adi waves in the atmsphee at height h a.g.l. [7, 5] whee n is the index f efactin at the eath s suface (h=0), and β is a pptinality cnstant. Such a ay ppagating in the atmsphee is shwn in Fig. [7,5]. A ay stats at an angle α with espect t the hizn at the eath suface level, and makes an angle θ at height h abve gund level. Fm Snell s law: significant path blckage and the path tajecty, and e is the adius f the fist Fesnel zne ellipsid [,6-7]. Detemining the eginal value f k is theefe key t detemining the diffactin lss f adi links. n cs α = n ( βh)csθ cs θ β secα = sin θ dθ dh (3) In ealie wk, Bake and Palme have discussed mdels f k-fact distibutin f Suthen Afica [8-]. Thei ange was tuncated in the mdel at k= because f a lack f data belw this value. Afull et al have als discussed k-fact mdels f Btswana [-4]. They have used cuve-fitting mdels as well as kenel mdels, t help detemine the k-fact pdf distibutin f Btswana. In thei cmpaisn f the tw mdels, they have cncluded that the biweight kenel gives the best estimate f k-fact distibutin f an ptimum smthing paamete f 0.5. By cmpaing the cuve-fitting and kenel methds, they have fund that thee is little diffeence between the median k value (appximately.) and the k-value exceeded 99.9% f the time (0.6) f Btswana [4]. In this pape, pesentatin, we biefly we biefly exple exple the cuve-fitting the cuve-fitting mdels f mdels f(k), f as fist f(k), pesented as fist pesented in [3]. Then in [3]. we cmpae Then we k- fact cmpae estimates k-fact f estimates Btswana f Btswana and Duban and Duban in Suth in Afica. Suth Afica. The mdels The mdels ae ae based based n n the the adisnde measuement data btained f Maun, in the Kgalagadi Hweve, fm gemety f Fig.(b), with K=/ρ, the invese f the adius f cuvatue, we have: ρdθ sinθ = dh (4) K = = β cs θ secα ρ F small launch angles, θ and α ae small; hence, fm (4), K -β. Fm (), we have: dn dh = n β dn K = = β n dh (5)

3 74 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vl.97() June 006 whee h is the height abve gund level. The efactive index, n, is elated t the elative pemittivity, ε and efactivity, N, as fllws [,3]: ( ) 6 n = ε ; N = n 0 (6) Then, with ε defined as the elative pemittivity f the tpsphee at height h, we have: dn dh dε = ε dh dn dε K = = = ρ n dh n ε dh dε K ε dh Theefe the adius f cuvatue f the ay path, ρ, is given by [6-7]: (7) the aea fm a neaby weathe bueau. If N is knwn then the suface efactivity can be btained as [5]: N s = N e h s () whee h s is the altitude abve sea level (in km) f the LOS adi site. If the k-fact is geate than (k>), the ay beam is bent twad the eath, which essentially allws us t shten adi-link twes. If k<, the eath bulge is effectively inceased, and the path is shtened the twe height must be inceased. Many adi links efe t the standad atmsphee with k=4/3; and theefe pfile chats have been develped f k=4/3. Hweve, cae shuld be taken when engineeing adi links that the k=4/3 they may nly be used f gss planning; in eality, it is imptant t detemine the aveage and effective k-facts f adequate path engineeing. The actual eath bulge (EC) t be added t the intevening teain and vegetatin placed between teminals A and B is: dε ε ρ = = = K ε dh d dh ( ε ) / (8) 0.078dd EC( m) = () k The adius f cuvatue f the path, ρ, being a functin f the change f dielectic cnstant with height h, vaies fm hu t hu, day t day, and seasn t seasn. That is why the ITU-R gives plts f efactivity and efactivity gadient f the mnths f Febuay, May, August, and Nvembe. The k-fact is detemined fm [-, 6-7]: k = = a / ρ d + a ε / dh (9) whee d is the distance between the antennas at teminals A and B; d and d ae the distances between the intevening pint and the antennas. 3. CURVE-FITTING MODEL REVISITED This sectin biefly discusses the simple cuve-fitting mdel as pesented in [3-4]. Equatins (8) and (9) ae used in [3-4] t calculate k as a functin f height. Fm the esults in [], we cnclude that the distibutin f k is bell-shaped, centeed almst symmetically abut a mean value, µ k. We theefe ppse the fllwing appximate pdf, f*(k): whee, a=6375 km is the actual eath adius; then we calculate the effective eath adius, ρ, fm the fmula [,3]: a ρ = N ( e ) (0) s f α ( k µ ( k) = Ae ) (3) * k The paamete A is the value f f*(k) at the mean, µ k ; while the paamete α detemines the gith f f*(k). The task is theefe t detemine A, µ k, and α. The fllwing cnditins have t be satisfied [3]: whee Ns is the suface efactivity. This is the efactivity at the altitude f the LOS micwave sight that was selected, the aveage efactivity f the path. The sea-level efactivity, N, can ften be btained f f * ( k) dk = α ( k µ k ) Ae dk = (4)

4 Vl.97() June 006 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 75 Table : Values f mean, A, and ISE f Btswana and Duban Peid Best estimates fm Btswana measuement Best estimates fm Duban measuement µ k A ISE µ k A ISE Feb (500m) May (500m) Aug (500m) Nv (500m) All Y (500m) All Y (00m) Table : Calculated value f k exceeded 99.9% f the time, Btswana & Duban Peid Btswana Duban Febuay (500m a.g.l.) May (500m a.g.l.) August (500m a.g.l.) Nvembe (500m a.g.l.) All Yea (500m a.g.l.) All Yea (00m a.g.l.) 0.6 A = f ) ( k (7) k=µ k Nw detemine the cespnding α t fulfill the cnditins (4) and (5). Calculate the ISE fm (6). Then, with mean cnstant at µ k, vay A and α t satisfy cnditins (4) and (5). F each cmbinatin (A, α), detemine the ISE. Fm the pcedue in (e) abve, identify the cmbinatin (µ k, A m, α m ) such that ISE is a minimum. Finally, with (A m, α m ) fixed, vay the mean, µ k until the lwest ISE is achieved. This gives the ptimum cmbinatin (A m, α m, µ km ). Then the best estimate f the pdf becmes: f α ( ) ( ) m A k µ k = m e (8) * km Nte that stpping at (8) abve may ften nt pvide a glbal minimum f the ISE. Theefe futhe efinement f bth A m and µ is necessay in de t btain the equied glbal minimum. It is due t this futhe efinement that the values f µ shwn hee diffe slightly fm thse shwn in ealie epts [3,4]. α { } ( k µ k kae ) k E dk = µ = k (5) 4. RESULTS OF THE CURVE-FITTING MODEL It is easy t shw that if f*(k) defined in (3) is a tue pdf, then nce it satisfies (4), then (5) is autmatically satisfied. In de t ascetain f*(k) as the best appximatin f f(k), we need t ptimize a cst functin, elated t the e, ε(k). The cst functin we chse is the integal f the squae e (ISE): ε ( k) = ISE = [ f ( k) f * ( k) ] [ f ( k) f *( k) ] dk (6) Thus the minimum value f the squae f the e in (6) will detemine the ptimum chice f f*(k) paametes A, µ k, and α. We have theefe ppsed a simple algithm f the deteminatin f the paametes A, µ k, and α, which give us the minimum e in (6). The algithm uns as fllws: As a fist estimate, chse µ k fm the plt f f(k). Next, chse A such that: Using the feging pcedue, we wee able t btain the ptimum f*(k) (which gives the best estimate f f (k)) f the mnths f Febuay, May, August, and Nvembe. We als detemined the veall f*(k) f the whle yea. Initially, we esticted u esults t data within the height ange m a.g.l. These ae shwn in Figues t 6. Table gives the best estimates f µ k, A, and ISE f the mnths f Febuay, May, August, and Nvembe f the height ange 0-500m a.g.l., as well as all-yea values f the anges 0-500m and 0-00m, f bth Btswana and Duban. Table gives the values f k exceeded 99.9% f the time f bth Btswana and Duban. Nte that bth the mean and effective values f k in the tw tables ae detemined fm the appximate distibutin plts, given in Figues t 6 as k vaies. Theefe the nmal cuvefitting distibutins f k-fact f Btswana and Duban ae as fllws [3]: Btswana, 00 m agl:. ( k. ) f ( k) =. 5e Btswana, 500 m agl:. ( k. ) f ( k) = 33. e Duban, 500 m agl:. ( k. ) f ( k) = 75. e 9 66

5 76 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vl.97() June 006 Measued f(k), Duban measued pdf, f(k), Btswana 3.5 Best Estimate f*(k): A=., mean=., ISE=0.0984, Duban Best estimate, f*(k), Mean=.05, A=.5, ISE=0.3445, Btswana 3 Measued and Estimated pdf, f(k) and f*(k) K anges Fig.. Estimated k distibutin f Btswana and Duban, Febuay, m a.g.l. Measued pdf, f(k), Duban Measued pdf, f(k), Btswana 6 Best Estimate f*(k), A=.4, mean=.5, ISE=0.89, Duban Best estimate, f*(k), Mean=.06, A=5., ISE=0.0668, Btswana 5 Measued and Estimated k k anges Fig. 3. Estimated k distibutin f Btswana and Duban, May, m a.g.l. We bseve, fm this mdel, that the mean value f k f Btswana is.07 within 0-500m a.g.l ange, while it is.09 f 0-00m a.g.l. ange, thus slightly diffeent fm cespnding values btained in [3,4]. The value f k exceeded 99.9% f the time is 0.70 f the height ange m, while it is 0.6 f the height ange 0-00 m f Btswana. F Duban, the mean value f k is. ve the m ange while the value f k exceeded 99.9% f the time is 0.5. Nte that these values ae detemined fm the best estimate f f(k). Thus the median k-fact values f Duban ae much clse t the standad tempeate climate value f 4/3 than the Btswana k-fact values. We als nte that in bth egins, the highest median value takes place in the winte mnth f August, with µ k =. f Btswana and µ k =.7 f Duban. The same applies t the value f k exceeded 99.9% f the time.

6 Vl.97() June 006 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 77 Measued pdf, f(k), Duban Measued pdf, f(k), Btswana 6 Best Estimate f*(k), A=3.7, mean=.7, ISE=0.835, Duban Best estimate, f*(k), Mean=.0, A=5.7, ISE=0.796, Btswana 5 Measued and Estimated pdf, f(k) and f*(k) K anges Fig. 4. Estimated k distibutin f Btswana and Duban, August, m a.g.l. Measued pdf, f(k), Duban Measued pdf, f(k), Btswana 3 Best Estimate f*(k), A=.7, mean=., ISE= , Duban Best estimate, f*(k), Mean=.075, A=.6, ISE=0.795, Btswana.5 Measued and Estimated pdf, f(k) and f*(k) K anges Fig. 5. Estimated k distibutin f Btswana and Duban, Nvembe, m a.g.l. 5. K-FACTOR FADING The cncept f diffactin lss as a functin f path cleaance aund an bstuctin is explained by Fesnel znes. The Fesnel zne adius is given by [6,7]: whee n is an intege and is wavelength f the adi wave at any paticula tansmitting fequency, d is the distance fm the tansmitte t bstuctin pint and d is the distance fm bstuctin pint t the eceive. n = nλd d d + d (9) Assuming a knife-edge bstuctin made up by a pefect cnduct half plane, nmal t the diectin f ppagatin. In this ideal case, the bstacle may be defined by a single nn-dimensinal paamete v, given by [5]:

7 78 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vl.97() June 006 Measued pdf, f(k), Duban Measued pdf, f(k), Btswana Best Estimate f*(k), A=.75, mean=., ISE= , Duban Best estimate, f*(k), Mean=.07, A=3.3, ISE=0.75, Btswana Measued and Estimated pdf, f(k), f*(k) k-anges Fig. 6. Estimated k distibutin f Btswana and Duban, All Yea, m a.g.l. Fig.7: Path Pfile f SHF Radi Link between Shewd and Umlazi f k =.

8 Vl.97() June 006 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS k=4/3 k=. k=0.9 k=0.6 Limit f 60% f B C Elevatin (m a.s.l.) A Distance fm Gabne (km) Fig. 8. Gabne-Mleplle Path Pfile: Antenna Heights Gabne-56m, Mleplle-30m, k=4/3, f=3 GHz d ν ± h λ d d = (0) whee h is the bstacle height abve (plus sign) belw (minus sign) the diect ay between the tansmitte and eceive antennas; d is the distance between the antennas; d and d ae the distances between the bstacle and the antennas; is the wavelength cespnding t the peating fequency f. Paamete v may als be expessed as a functin f the adius f the fist Fesnel ellipsid e at the bstacle. Fm equatins (9) and (0) we get: v = ± h () e Estimating the signal attenuatin caused by diffactin f adi waves ve hills and buildings is essential in pedicting the signal stength in a given sevice aea. The electic field stength E d f a knife-edge diffacted wave is given by [6,7]: E E d ( + j) = F( v) = ν exp( ( j j πt ) t ) / dt ) dt () Hee E is the fee space field stength in the absence f bth the gund and the knife edge, and F(v) is the cmplex Fesnel integal. The Fesnel integal, F(v) is a functin f the Fesnel-Kichff diffactin paamete v. In pactice, the additinal lss f () due t the bstuctin is ze when the ati f the cleaance t the fist Fesnel adius is appximately 0.6. The plain (flat) eath they and the knife-edge they ae the tw extemes; physical featues f eal teain lie in between these extemes. A typical design ule n any path f fequencies f 7 GHz and up is thus t allw cleaance f 0.6 e ve bstuctin at k = 0.7. The diffactin gain (lss) due t the pesence f a knifeedge bstuctin as cmpaed t the fee space electic field is given by [6.7]: G d ( db) = 0lg F( v) (3) As an example, a 48.8 km link is set up between Umlazi and Shewd in Kwazulu-Natal Pvince f Suth Afica. The path pfile diagam f the link is shwn in Fig. 7 f k-value f.. A design value f k =.33 is assumed, as ppsed t k =. which is the median k value f Duban. The link availability is examined using these diffeent values f k anging between 0.9 and.33. The esults in Table 3 shw that f the median k-value f., the diffactin lss (due t a knife-edge bstuctin 8 km fm Shewd) is baely nticeable at.8 db. On

9 80 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vl.97() June 006 the the hand, since typical link design is f 99.9% availability, we ntice that f this paticula link, a fade magin f 5 db wuld suffice f the mnth f Febuay, since the value f k exceeded 99.9% f the time wuld be 0.9. Yet this wuld be insufficient f the the times f the yea. Fm futh-de plynmial extaplatin (in de t ensue 99.9% link availability all yea) the cespnding value f k=0.5 wuld equie a fade magin exceeding 40 db f this path. A secnd example exples the 44-km Gabne- Mleplle SHF link peating at 3 GHz, als designed f 60% cleaance f the fist fesnel zne adius f k=4/3. In this case we nte, fm Fig.8, pssible bstuctins at pints A, B, and C placed at 8 km, 38 km, and 4 km fm Gabne. These esult in diffactin lsses f.6 db,.8 db, and.5 db, espectively, giving a ttal lss f 4.9 db at the median k=.. Hweve, f this same path, the ttal diffactin lss f k=0.6 (f link availability f 99.9%) is 5.9 db, fm k-type fading alne. Theefe, wing t the gentle, flatte teain in Btswana, a fade magin f 6 db wuld be adequate f k-type fading, even with the incect chice f median k-fact value. Table 3: Effect f the k-fact vaiatin n diffactin gain G (db) f Shewd-Umlazi Link k h v G d (db) Table 4: Effect f the k-fact vaiatin n diffactin gain G (db) f Gabne-Mleplle Link k Pint A Pint B Pint C Ttal h G h G h G G (db) CONCLUSIONS A cuve-fitting mdel f the pdf f the k-fact based n adisnde data cllected in Btswana and Suth Afica is discussed. In the Btswana case, it is bseved that at height spans f 0-500m and 0-00m a.g.l., the all-yea mean values f k, ae.07 and.09, espectively, f the cuve-fitting mdel. The value f k, exceeded 99.9% f the time is fund fm the deived analytical expessins t be 0.7 f the height span 0-500m a.g.l., while it is 0.6 f the lwe span 0-00m a.g.l. The cespnding values f Duban ae the mean value f., while the value exceeded 99.9% f the time is 0.5. Tw design examples shw that the type f teain plays a maj le in diffactin fading esulting fm k-fact vaiatin. While designing with k=4/3 esults in lsses f db and 5 db f Duban and Btswana links, espectively, multiple bstuctins wuld esult in highe fades. Meve, the gentle teain in Btswana esults in lwe lsses that wuld have been the case if the k-fact f Duban had been.. In additin, 99.9% link availability is hade t attain in Duban than in Btswana wing t the bade k-fact distibutin and the hillie teainf Duban. 7. ACKNOWLEDGMENT The auths ae indebted t the Depatment f Metelgical Sevices, Gabne, Btswana, f the adisnde data made available f the this study. In additin, the data fm the Suth Afican Weathe Bueau, btained thugh Tacy f Petia headquates, is gatefully acknwledged. 8. REFERENCES [] ITU Radicmmunicatin Study Gup III: The Radi Refactive Index: Its Fmula and Refactivity Data, ITU Radicmmunicatin Assembly, Dcument N. ITU-R P.453-7, July 000. [] ITU Radicmmunicatin Assembly Ppagatin data and pedictin methds equied f the design f teestial line-f-sight systems ITU-R P ( ). [3] ITU Radicmmunicatin Study Gup III: Effects f Tpspheic Refactin n Radiwave ppagatin, ITU Radicmmunicatins Assembly, Dcument N. ITU-R p ( ). [4] R. L. Olsen and Teje Tjelta Wldwide Techniques f Pedicting the Multipath Fading Distibutin n Teestial LOS Links: Backgund and Results f Tests IEEE Tansactins n Antennas and Ppagatin, Vl. 47, N. Januay 999. [5] R.L. Olsen Radiclimatlgical Mdeling f Ppagatin Effects in Clea-Ai and Pecipitatin cnditins: Recent Advances and Futue Diectins Pceeding f the Thid Reginal Wkshp n Radi Cmmunicatins in Afica Radi Afica 99, Gabne Btswana, Octbe 999. [6] R.L. Feeman: Radi System Design f Telecmmunicatins, Jhn Wiley, 997, ISBN [7] D.J. Angelaks and T.E. Evehat: Micwave Cmmunicatins, McGaw-Hill, 968, Chapte 6.

10 Vl.97() June 006 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 8 [8] D.C. Bake and A.J. Palme A Mdel f the Factin f time Availability f the Effective Eath Radius Fact f Cmmunicatins Planning in Suth Afica The Basic Mdel, Tansactins f SAIEE, Vl. 93, pp-7, 00. [9] D.C. Bake and A.J. Palme A Ppsed Empiical Mdel f the Effective Radius Fact f Telecmmunicatins use in Suth Afica, SAIEE Tansactins, Vl. 94, N., pp40-45, 00. [0] A.J. Palme and D.C. Bake, Pedicting the Lng- Tem Aveage f the Effective Eath Radius Fact f Suth Afica Using Gund-Based Obsevatins, Pceeding f IEEE AFRICON 004, Septembe 004, ISBN , pp [] A.J. Palme and D.C. Bake, Pedicting the Mnthly Cummulative Distibutin f the Effective Eath Radius Fact f Suth Afica, Pceeding f IEEE AFRICON 004, Septembe 004, ISBN , pp [] T.J.Afull, M.O. Adng, T.Mtsela and D.F Mltsi Estimates f Refactivity Gadient and K- fact Ranges f Btswana SAIEE Tansactins, Vl. 9, pp-6, 00. [3] T.J.Afull, M.O. Adng, T.Mtsela and D.F Mltsi Estimates f Refactivity Gadient and K- fact Ranges f Btswana SAIEE Tansactins, Vl. 9, pp-6, 00. [4] T.J. Afull, P.K. Odedina, Effective Eath Radius Fact Chaacteizatin f Line f Sight Paths in Btswana, Pceedings f IEEE AFRICON 004 Cnfeence, ISBN , pp7-3. [5] T.J. Afull, Effective Eath Radius Fact Measuement and Mdeling f Radi Link Design in Btswana, accepted f publicatin in SAIEE Tansactins, 005. [6] T. S. Rappapt: Wieless Cmmunicatins Pinciples and Pactice, Pentice Hall Inc. Uppe Saddle Rive, N.J , pp 90. [7] D.H. Hamshe: System Engineeing Handbk, McGaw-Hill, Inc. 967, pp

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