THE α-µ DISTRIBUTION: A GENERAL FADING DISTRIBUTION. Michel Daoud Yacoub

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1 TH - DISTRIBUTION: A GNRAL FADING DISTRIBUTION Mchel Doud Ycoub Unvest of Cmns, DCOM/FC/UNICAMP, C.P. 60, , Cmns, SP, BRAZIL, mchel@decom.fee.uncm.b Abstct - Ths e esents genel fdng dstbuton the - Dstbuton tht ncludes the Ngm-m nd the Webull s secl cses. One-Sded Gussn, Rlegh, nd Negtve xonentl dstbutons e lso secl cses of the - Dstbuton. Kewods Fdng dstbuton, Ngm, Webull, Rlegh, Negtve xonentl, One-Sded Gussn. I. INTRODUCTION A get numbe of dstbutons exst tht well descbe the sttstcs of the moble do sgnl. The long tem sgnl vton s well chctezed b the Lognoml dstbuton whees the shot tem sgnl vton s descbed b sevel othe dstbutons such s Rlegh, Rce, Ngm-m, nd Webull, though to the ltte, ognll deved fo elblt stud uoses, lttle ttenton hs been d. It s genell cceted tht the th stength t n del s chctezed b the shot tem dstbutons ove stl dmenson of few hunded wvelengths, nd b the Lognoml dstbuton ove es whose dmenson s much lge []. Among the shot tem fdng dstbutons, the Ngm-m dstbuton hs been gven secl ttenton fo ts ese of mnulton nd wde nge of lcblt. Although, n genel, t hs been found tht the fdng sttstcs of the moble do chnnel m well be chctezed b the Ngm-m, stutons e esl found fo whch othe dstbutons such s Rce nd even Webull eld bette esults [3, 4]. Moe motntl, stutons e encounteed fo whch no dstbutons seem to dequtel ft exementl dt, though one o nothe m eld modete fttng. Some eseches [4] even queston the use of the Ngm-m dstbuton becuse ts tl does not seem to eld good fttng to exementl dt, bette fttng beng found ound the men o medn. The well-nown fdng dstbutons hve been deved ssumng homogeneous dffuse sctteng feld, esultng fom ndoml dstbuted ont scttees. The ssumton of homogeneous dffuse sctteng feld s cetnl n oxmton becuse the sufces e stll coelted chctezng non-homogeneous envonment []. Moe ecentl [5, 6], two new fdng dstbutons hve been oosed tht nclude o closel oxmte the most common fdng dstbutons. The η- Dstbuton [5] ncludes the Hot nd the Ngm-m dstbutons s secl cses. The κ- Dstbuton [6] ncludes the Rce nd the Ngm-m dstbutons s secl cses. Theefoe, n both fdng dstbutons, the One-Sded Gussn nd the Rlegh dstbutons lso consttute secl cses nd the Lognoml dstbuton m be well-oxmted b these dstbutons. Ths e esents nothe genel fdng dstbuton the - Dstbuton tht ncludes the Ngm-m nd the Webull s secl cses. Theefoe, One-Sded Gussn, Rlegh, nd Negtve xonentl dstbutons e lso secl cses of the - Dstbuton. II. TH - DISTRIBUTION The - Dstbuton s genel fdng dstbuton tht cn be used to eesent the smll- scle vton of the fdng sgnl. Fo fdng sgnl wth enveloe, n bt mete > 0 ˆ,, nd -oot men vlue ( ) the - obblt denst functon ˆ ( ) ex ˆ of s wtten s whee s the nvese of the nomlzed vnce of,.e. ( ) ( ) ( ) () () z nd ( z) t ( t) ex dt s the Gmm functon. 0 The th moment ( ) s obtned s ˆ ( + ) Fo nomlzed enveloe ( ) denst functon ( ) s obtned s ( ) (3) ˆ, the obblt ( ) ( ) ex (4) /0/$ I PIMRC 00

2 whee s gven b The (5) V ( ) th moment ( ) ( ) ( + ) s obtned s (6) ( ) Defnng w s he-owe, b smle tnsfomton of vbles the obblt denst functon w of w s found s w w ( w) ex w w ( ) Fo nomlzed he-owe ω denst functon ( ω ) of ω s obtned s ( ) (7) w w, the obblt ω ( ω ) ex( ω) (8) The obblt dstbuton functon ( x) P fo n vte x of the - te cn be found n closed-fom fomul. P fo the enveloe s gven b In tcul, P (, ˆ ) ( ) z whee ( z ) t ex( t) Gmm functon. (9), dt s the ncomlete 0 III. PHYSICAL MODL The fdng model fo the - Dstbuton consdes sgnl comosed of clustes of multth wves ogtng n non-homogeneous envonment. Wthn n one cluste, the hses of the sctteed wves e ndom nd hve sml del tmes wth del-tme seds of dffeent clustes beng eltvel lge. The clustes of multth wves e ssumed to hve the sctteed wves wth dentcl owes. The esultng enveloe s obtned s non-lne functon of the modulus of the sum of the multth comonents. The non-lnet s mnfested n tems of owe mete > 0, such tht the esultng sgnl ntenst s obtned not sml s the modulus of the sum of the multth comonents, but s ths modulus to cetn gven owe. IV. DRIVATION OF TH - DISTRIBUTION Assume tht t cetn gven ont the eceved sgnl encomsses n bt numbe n of multth comonents, nd the ogton envonment s such tht the esultng sgnl s obseved s non-lne functon of the modulus of the sum of these comonents. Suose tht such non-lnet s n the fom of owe so tht the esultng enveloe s obseved s the modulus of the sum of the multth comonents to the owe of, > 0. Hence, fo the - Dstbuton the enveloe cn be wtten s functon of the n-hse nd qudtue elements of the multth comonents so tht whee x nd ( + ) n x (0) e mutull ndeendent Gussn x nd ocesses, wth 0 ( x ) ( ) σ. Defnng stndd ocedue to fnd tht ( ) n B tnsfomton of vbles It cn be shown tht ex n ( σ ) ( n) σ n, we follow the ex n σ n σ ( n + ) ( σ ) ( n) () () (3) Theefoe ( ) σ n ( ) ( σ ) ( n + )n nomlzed vnce of nd. Hence, the nvese of the, defned s, s obtned s ( ) ( ) ( ) n (4) Note tht the vble n s totll exessed n tems of hscl mete, nmel the nomlzed vnce of. Note lso tht whees ths hscl mete s of contnuous ntue, n s of dscete ntue. It s lusble to esume tht f such mete s to be obtned b feld mesuements, fgues detng fom the exct n wll cetnl occu. Sevel esons exst fo ths. One of them,

3 obbl the most menngful one, s tht, lthough the model oosed hee s genel, t s n fct n oxmte soluton to the so-clled ndom hse oblem, s e oxmte soluton to the ndom hse oblem ll the othe well-nown fdng models. The lmtton of the model cn be mde less stngent b defnng s the el extenson of n. Non-ntege vlues of the mete ccount fo ) non-zeo coelton mong the clustes of multth comonents, b) non-zeo coelton between the n-hse nd qudtue comonents, c) non-gussnt of the n-hse nd qudtue comonents of the fdng sgnl, mong othes. (We note tht n devton of the Ngm-m model [7], the mete n, whch descbes the numbe of comonent sgnls [7], theefoe dscete, s lso wtten n tems of the Ngm contnuous mete m s m n.) Fom the defnton of the - fdng sgnl, we fnd tht, the condton coesondng to the exstence of one of the Gussn elements onl. Now elcng n b nd usng ( ) ˆ, the obblt denst functon of the - sgnl s obtned s ˆ ( ) ex ˆ V. TH - DISTRIBUTION AND TH OTHR FADING DISTRIBUTIONS (5) The - Dstbuton s genel fdng dstbuton tht ncludes the Webull nd the Ngm-m dstbutons s secl cses. Webull ncludes the Rlegh nd the Negtve xonentl dstbutons whees Ngm-m ncludes the Rlegh nd the One-Sded Gussn dstbutons. The Lognoml dstbuton m lso be well oxmted b the - Dstbuton. A. Webull, Rlegh, nd Negtve xonentl The Webull dstbuton cn be obtned fom the - Dstbuton b settng. In such cse β ex( β ) (6) whee β ˆ. Fom the Webull dstbuton, b settng, the Rlegh dstbuton s obtned s ˆ ex (7) γ γ whee γ. Stll fom the Webull dstbuton, the Negtve xonentl dstbuton s obtned b settng, n whch cse whee δ ˆ. δ ( δ ) ex (8) B. Ngm-m, Rlegh, nd One-Sded Gussn The Ngm-m dstbuton cn be obtned fom the - Dstbuton b settng. In such cse ex Ω Ω Ω ˆ ( ) (9) whee. Fom the Ngm-m dstbuton, b settng, the Rlegh dstbuton s obtned s ex (0) γ γ ˆ whee γ. Stll fom the Ngm-m dstbuton, the One-Sded Gussn dstbuton s obtned b settng, n whch cse πˆ ˆ ex VI. STIMATORS FOR TH - DISTRIBUTION () The - Dstbuton s chctezed b the nd metes s well s the -oot men vlue ˆ of the enveloe. The detemnton of ethe o ˆ nvolves the evous nowledge of the mete, s cn be seen fom the defnton of nd ˆ. In ode to estmte such metes, we defne mesuble mete such tht ( ) ( ) ( ) () whee > 0 s n bt mete. Note tht genelzes the defnton of nd educes to t fo,.e.. Usng the defnton of the moments s gven evousl, the mesuble mete cn be wtten s ( + ) ( + ) ( ) ( + ) (3) Fo gven, chosen btl, the mesuble mete s gven s functon of two metes, nmel nd. Theefoe, n ode to obtn these two metes, t

4 suffces to estmte two mesuble metes so tht sstem of two equtons nd two unnowns e set u. Moe secfcll, f nd e these two mesuble metes, then b solvng the sstem of equtons ( + ) ( + ) ( ) ( + ) (4) ( + ) ( + ) ( ) ( + ) (5) the metes nd e detemned. Unfotuntel, these two metes cnnot be wtten exlctl n tems of nd. Theefoe, the equed soluton fo the sstem of tnscendentl equtons s gven bove must be ovded b mens of tetve o numecl methods. As n llustton, f nd e btl chosen to be equl to nd, esectvel, then nd e obtned s the nomlzed vnces of nd, esectvel. These, o n othe nomlzed vnce of, s gven b (3), cn be dectl obtned though feld mesuements. Theefoe, n set of two equtons cn be set u n ode to detemne nd metes. Then the -oot men vlue ˆ cn be estmted nd the comlete - Dstbuton s detemned. In the cse of, then m, m beng the so-clled Ngm mete. VII. SAMPL SHAPS OF TH - DISTRIBUTION The followng bsc fetues of the - obblt denst functon cn be obseved dectl fom (4). Fo <, ( ) tends to nfnt s oches zeo nd ( ) deceses monotoncll wth the ncese of. Fo, ( ) hs non-zeo nd non-nfnt vlue t the ogn nd deceses monotoncll to zeo wth the ncese of. Fo >, ( ) s nl t the ogn; t nceses wth the ncese of to ech mxmum, then decesng s nceses. Fgue lots ( ) fo 7 4 nd vng. Fgue lots ( ) vesus vesus fo 4 7 nd vng. Aentl, the effects of seem comble but, n fct, the nd on shes of the cuves v substntll f the e obseved moe closel. We obseve tht the - Dstbuton hs one mete moe thn the Ngm-m dstbuton. In tcul, s led onted out, m, beng functon of both nd. Theefoe, t s ossble to fnd n nfnte numbe of combntons of the metes - ledng to the sme mete m. Fgues 3 nd 4 show the vous shes of the - denst nd dstbuton functons, esectvel, fo the sme Ngm mete m Fgues 5 nd 6 m.5, esectvel. In ll the cuves, the mete hs chosen s 0. 5, 0. 75,,. 5,, 5,0, 50, nd 00, nd hs been clculted fom the elton m fo the esectve. The effects of e such tht s t nceses the obblt dstbuton cuves tend to occu the lowe otons of the fme. We obseve tht n enomous vet of shes cn be found fo the sme Ngm mete m. Theefoe, the - Dstbuton cn be used to bette djust to feld dt. VIII. CONCLUSIONS Ths e esents genel fdng dstbuton the - Dstbuton tht ncludes the Ngm-m nd the Webull s secl cses. Theefoe, One-Sded Gussn, Rlegh, nd Negtve xonentl dstbutons e lso secl cses of the - Dstbuton. It s notewoth tht, lthough the - Dstbuton hs one mete moe thn Ngm-m o Webull dstbutons, no ddtonl dffcult s osed b the ncese n the numbe of metes. The flexblt the - Dstbuton conves s outstndng nd endes t suted to bette djust to feld dt. The fdng model descbed hee llows one to develo hghe ode sttstcs (enveloe devtve, etc.) of the - Dstbuton, esech cuentl unde couse. RFRNCS [] G. L. Tun, Intoducton to sed-sectum ntmultth technques nd the lcton to ubn dgtl do, Poc. I, vol. 68, no. 3, , Mch 980. [] W. R. Bun nd U. Desch, A hscl moble do chnnel model, I Tns. Veh. Technol. vol. 40, no., , M 99. [3] J. D. Psons, The Moble Rdo Chnnel, nd dton, John Wle & Sons, Chcheste,, 000. [4] S. Sten, Fdng chnnel ssues n sstem engneeng, I J. Selected Aes n Commun., vol. 5, no., , Feb [5] M. D. Ycoub, The η- dstbuton: A genel fdng dstbuton, I Vehcul Technolog Confeence Fll 000, Boston, 000. [6] M. D. Ycoub, The κ- dstbuton: A genel fdng dstbuton, I Vehcul Technolog Confeence Fll 00, Atlntc Ct, 00. [7] M. Ngm, The m-dstbuton - A genel fomul of ntenst dstbuton of d fdng, n Sttstcl Methods n Rdo Wve Pogton, W. C. Hoffmn, d. lmsfod, NY: Pegmon, 960.

5 ,0,5 / 4/7 5/8 3/4.0 (Webull) Ngm Webull (),0 P() 0-0, ,5,0,5,0,5 3, log() Fgue. The vous shes of the - denst functon fo 7 4. Fgue 4. The vous shes of the - dstbuton functon fo the sme Ngm mete m 0. 5.,0,4 (),5,0.5 7/ (Ngm) (),,0 0,8 0,6 Ngm Webull 0,5 0,4 0, 0,5,0,5,0,5 3,0 0,5,0,5,0,5 3,0 Fgue. The vous shes of the - denst functon fo 4 7. Fgue 5. The vous shes of the - denst functon fo the sme Ngm mete m. 5., ,8 Ngm Webull 0 - Webull Ngm 0,6 () 0,4 P() 0-0, 0-3 0,5,0,5,0,5 3, log() Fgue 3. The vous shes of the - denst functon fo the sme Ngm mete m Fgue 6. The vous shes of the - dstbuton functon fo the sme Ngm mete m. 5.

Lecture 5 Single factor design and analysis

Lecture 5 Single factor design and analysis Lectue 5 Sngle fcto desgn nd nlss Completel ndomzed desgn (CRD Completel ndomzed desgn In the desgn of expements, completel ndomzed desgns e fo studng the effects of one pm fcto wthout the need to tke