BSTJ 50: 10. December 1971: Statistical Behavior of a Fading Signal. (Lin, S.H.)

21 MB·English

Statistical Behavior of a Fading Signal

By $.H.TIN

(ofan eae Jly22, 01)

A general analyse ofthe aan! Peharar ofthe enebspe of «Jang
Ce" Ge prowl sn thie paper, The eatiwics nal the
‘PU 2) thot he amplitude FC wil fois below a epee
sigue D; the exposed numer (Lt of foes of V0) btn Te amit
fine; ond the average duration 10) of fate bio J The easel forthe
‘Sinaloa annstaatector plas a sande neering air whic
Sepreants tw rele! of ot she rosie extraneous signals and nae
ha ioral ee ees th hon areal obaered peeve
shipe obtained Gn deep fais af wondiersitysigntas FQN 1) = Le
NC) = Land IL} = ba Th tonraton sere ave applioats 6a its
‘dau of lading prablnes. Phe anaiain inate th frais wc of Fey,
Nabugave, Norton, Pole, Manel, cud Shark os eset exes

A gonoral eaulyie of tn statistial butane oft mvs PG

ffs fading gral be owen in this poper. Ute prizepal interests ane
the probability, PV), thet tho amplioude PG will fade below a
specied sigusl level E* the eqpoeted sumber, N(L}, of fdas pee unit
time below the sported evel Js and the wvenage darting, FL}, of
tases blow Z, Thee sttistes are ll Eanes ofthe signa level

"The theory presented neren has been developed to cunplesnent an
texte tho empinal ruts developed by my aolleguos at Bel Tele.
hone Laboratories!“ from their axtasive experimental experience,
Published dats of other warkors have also boen consdest.

‘The previous theoretitl works on the slefistes of x. fing signal
often assume a compen Claueslan ore! forthe fading sign, "Phe
thooretical eupsort Zor tie aeeumotion i (ant, by the eontral Tit
theorem, the real end the magina’s pris of Uae sur uf Tae or
of independent iaerfring sis ill be noprorinatals Gaursia,

Aen the Heguaak PLE Loomer be wpeted arn of Sang ee
vi Ral teal pte Fe tee
8212 com mene crore wecunw ToORNAL, pees TL

Fr troposphere rin Hinks, this model seems to be sttsfanary
However, for Eneofsicht radio inks, th results of shor. puis
cxpertnen? and the arylcofarrival nesrurements"* indiete thas
the tuber of interfering wii ia usually firy small. Thy tesine
theory an indiates tit for typi! ineo-sght radio inks the mumbo
bf paths contibuting to vnltpath propagation ix unlly wo be Tare.
Furtheraere, the theoretic! reall of the complex Gousion moll
Aly mot agro wal wits the experimeatal data on tho statisties of feling
Signals of Iime-oFsight rio lnks, pve for certain overwater paths

In this paper, we do sot impose the restrictive casumprions of the
omples Causa noel, Iather, wo szaply mel Ube fading signal
Ve'tas a constant venir pean interfering rtdom veelar Ly

Vet atthe m1 tet i®, 0
shore Ry @, a nd @ ane tae ample, olises, wal pax, and imeginary
pcr repectively of the interfring vortor. The interfering vetoes
seri iy the fiz: probity deity funetion fe 6) and represents
the resultant of all the oacived extras signals echons, rays, and
fie, The waulyeeapplion for td @ iter dependint or independout;
Punitvoly of toauaonmiy dairboted; « and eicber Gaussian oF
fuonChausan, ‘hus, the resuly of dia anys may be applied to a
‘nie cae o ding problems.

"This saper tats fhe piblem in thee perts: The fate were
vith the amplisadsdistbution nf V."Thesoeoza eonsidrs he number
Ut fades A(L) ord the avenage fade duration 1). The fal seeion
inwergates seven) nyeial topics including m-distibutions
fistribations, the Resligh ditribntion, Live dtrbution, lognormal
Uisteibation and the mam af unit randoms veetor.

Appendix A is ist symbole and their definition

Tm spite of gaast uration in fading environment end test
tiwltions, the experimental dau" on PV 2 D)y

Wir), an {1 of mast nondiversiey* Fading egnals obey the
follwing three prevailing powcr Ins of deep fades

opti Pita Shae
dese tons steristes saa

psn t @
NUL) © de? Gaver 8
y= 4! a

he hootrtical anatase shows taat 2 the probability density
funetion fl, 2) of ths resulta ntrfering weet, lee" = + i
isa anooth fusetion which i nether aingalor not zaro at he
ep Tide point (a= —1, 8 = M), then the atatstios PLP = 1),
NUL}, and 1) of deep fel Fallow the three provi power
lows (2, (Bh an A}. Tho exily satiod emdition, =>
(1,0) > 0, iesuiclent toeblsin thee Tunesonal relationships

(G8) The set of power laws (2, (8, and (1D apwly for R gud @ ther
independent or depen, # ther wfirny ne noniforey
Aitrbote, Cor,» sid wiles Gans natant, her
Jndependet or depesdent ws In ws fe, 8) bx sat

(is) LG, 8) nsingut (@ = 1, 4= 0), den the henry penis
that for sl

ms
wing
4b

te aod

hs exceptions betaian (6) eostent se 4 = 122 has eon
observe experiment em gerle'n uvarcoter rai line with

rte fading, In thie cass. the sorutant intarering weotor
fontsina the tong wsternalevied my ey dominant eompo-
next, Therefor, sho proboblity dorwty fret fla, 8)
ingulasity at the positon of the dominant component vootae*

vay eh ot ®
men | oo

The composite fading siguale of divericy combining eystome
te the set of power aw (), (9 and (10) Por everiend eaco

snl ly fy a abla spe sei Ua 19) al
8214 ae meLL ave MHCMTOIN, JoLRNAL, puER LOT

Unt, the pace onder of diverse. Jv our
cxperiente, the nondives' Galing geal of a alativey short”
ra Tink wish pat Tongth ISAT rake hae also shawn tho
terion behiariondeseibes 8), 19}, ad (10)

(8) Thr thencetical rests (8), 7), snd (10) indionta thet tho power
wr, ib} Tor the wverage fale dration ie more universe
than thove of PAF 2 £) end NUL). This prediction ares wit
‘sallaleesporimental dat,

(os) In genera, the lationship betwoen f(a, 8) and the amplitude
distribution PCV 1) 9 nos unique, As an example, this non
‘umiquenes ehows Us spying the Tasleigs dsrbution for
tine anpltads of a ding ign os nt neeeseily empl that
tr ond # ary Caueinn, nor does it aeceazuily imply a large
Thumbor of ineersering sighs

Ac aH eormsing foe, Ay aswnge ath Ina of Knog allo
i ro
ns sae ae 15

Part 1. Amplitude Distribution

tne ae
1 tno nl Seay. a

Pome Soe Repetto Amide Date oo
VI, Pesvng Sates Ly of Fate sas
eA Bonin Canoe om Teanga am
EX, Orsi Ned 26

In the study of fling syste dim tn msipath interiernee, tae
exporinenta da on the eusnlative atapitute distribution, PV 2 23,
ff op fle an Ten plotted oa a graph papor wher hw fede
Fe exproorl inl atid the paobabsity of Fale we exremnd tm Lgeale
fie sown in Fig, 1, 'The eonatnses fala Tange anna of experi
smontal dase je thst the cnulative nuplitade distibsion
St mos overt fading signals in the doo Fade cog ea wall be
represented by a straight fine its «preva inverse lope of 10-41
Ie seeade nf penbabibity

"The mation hich deseibes this typical trbatin

a Big Lt

WW 24) =I, for By 2 EEO ay

‘where V isthe envelope voltae of the random fading igmal normaliced
ties nonfaded ign level, Zs wy apie signal evel, «isa param
tier depending on fuding cnvirommont, atid 4, i the upper bowl of
‘gna lovelbolow whieh the xesght cline epeveonasion of (0 = ZL) on
Fig. 1 is valid
3216 ur mma avenune sursion. JOURNAL, BECHER AL

ra
NS

4

4

4

ig Come apd ston ane sins

‘The erin! nent (11} mewn thal the amplitude distributions of
nos: rovaivensty lading gras ibe te folowing square nw of deep

Fates

PU EMH, bg bEO

1

inanite ofthe gpestvorintensut inding easizoument wil test condition.
Fioever, shege are anne vxcotional eas. "Tho mest orofouns excep
ign ocean oh gettin overeoe asdio Links wit severe fading. In
these inetance, she priuliy of fades, Po) 1, deren very
lop 28 the sigaal Level 1 desmaes ana in chaeuteriaed by wn inves
Floj of 0 ad per dovede of babsity a shown in Fig. 1, plying
power law

ry sneh as)
in the deep nde regi.

"Another kind of exeontion oevury on etrinin endo Hinks wits relatively
litte multipath ling. The probablity of fudes, POV =}, doereses
‘very rapidly it, wl is eharatarzed by gn inverwe soge of 5 aR
Tr decade of probability’ as shown im ig. 1, This kind of distribution
Tollows the poner hw

anew as
nis stone snes ites nz

in the despfode region. Tor exsmple, at 4 GHz operating frequency,
the average path langth of line-of-sight radi links i show 27 mle.
"The behavior (14) has been ebserved to occur on a velatively shor path,
swith path length of 15.87 re.

"The those emplitive distributions previously derived were bused

yr Gaussian model nd pedir a enue law depend

) sn the deep fae reginn. For example he Naku
Aistsbasions! which ised Hoyt distibation, "Rie disoutiny
fond Tavieih distribution oe special cases, all exe equaee aw in the
‘leopetude ego. The exploit expression of Neha denn ea
be found in Eoustion (.0-25) of Rafeones 18, Figure 1 alo inludes
ayleigh dietribution (ached lize or comparison with the experimental
ats

"The smell number of interfering signs in line-of-sight radio Hinks
suggosta tat tho assumption of 4 comolex Gaussian model may be
Aju Dao of bie win chores thi et etenmine te
‘woke wt tesa er which mane e813) 3 abt,
tad the sondtion for which the exeopional cage auch ae 13) oF (14)
srl oneur

In eunmery:
19) Thu Uoretien! ose for te fading algal, 4, 18 & constant
nit vorlue plug reauand Satta veto
Vet 1 te” 1 te ti
‘he romltant inteoring vestor Be" — we + 22) wish joint
probability densi'y funeion fo Bb, repose dhe stn of all
‘the soceved extrseous signal, oehoes, cage, and mise
(8) An infinite fade (ce, V"~ 0} occurs weenevera— Tandg = 0
‘At thie point A —" Laud ¢-~ x. Therefore, the behavior of
‘a, 9) npae che inne fade point (x ~ —1, 3 ~ 0) i clecely
Ielated to tho pover law of amschtude eistsbation 1" 2 2)
in te donate rion
Fur snos radio Tinks, the intefeing sgla and ine my be
eres so thatthe jal probability Ueailyfeneton
1, 8) of the resultant interfering vector ie @ emooth fanetion

‘na the infinite fade point (2 = —1, 8 ~ M). The analysis
hows thal if flo, 8) is a ssusth fuolion his bs neither
singular nor row at fa = 1, = 0} then POP g 1) 12 for

fll [The simple soaiion > f(t, 0) > 0 i easly
tied by moet radio i Pe vay of Ussuace La doe
2218 up waa. vere new JouRNAL, pueRM TL

hot maine that fla, be 8 nom density funtion, Therefore,
the tuber of interiring signals docs not hase tobe lane.

(60) The anulyrisshowa tht if fl, 8) i not smooth, bu is singular
wt (a= 1,8 = 0), then for sna,

PED ei 1d eh

[A physnal example for this exw is the overwork
whore the walesteectad ray is almost as stable us the diet
sy. Tie roullant ineceving vector in this ease contains the
‘aterrfcel may which i "uot very random." In other word,
th Joins probable density furetiu f(a, 6) et « Wh peak
‘et tie portion of the dominant, stable component, and ma be
‘onaderod ingle thal in

(0) Tf fle, 2) is nro oi welisiy ral the ifite fd point
fan 1,@ ~ 0), then the analysis chorea that for ssa £

PY ED=E wrk

Apihysical example for tie ese 0 the short radio link where the
pte clffenees among the muah propagation are a
call, Thon the velo of ((—1, 0) fe unghigbly smal Breanze
the phase, ¢, of see reultany intereriag vector ia generally
nll, Anoubee camps for thie eaae ie the compeeite cgeal of
Aho output uf wiliyerity noahiningeystana whore the artiGeal
retire combining deve Forvez to areute a zero of f(a, B) a
@= Lest

‘Tap ronsved fnding tiga i¢ rmodsled as a constant voetor plus an
interfering runs vera ne liar in Fig. 2. Te latter reprints the
resulta uf Eve rind exlemneons signals, eeboos rays, and no,
‘The rcsived fading sgzal noreanzed to the maeitud of the caw
‘oul ens ie written

Heb RE, as,

shore Rn (at The nnnnataed magnitude and the phase ofthe
Intereringrudown veriae repmalvely: Pit and 3) are the normalised
aon nad mance saw

fan Tela amd a bo the real ent a he iginary ek te
comptes inleering random weer Ie,

Ride" = at j8t0 a
Fs $28 4,29, 9 nd ane all eal oncom vargbley. ee uma
zp power

Vise eems Relat iat as

Tho mfalve ne, 86 hetreea tho intanering vector and he consent
“porn en te 0 awe rales frm ls Ze i Bs (2 — 2a)
Iie say Tslager tate indctngsiile Uo lw rave epi al a
Dalny Ferguaney.

"A eomecicd erp of squesions (6) vo (18) snow ttt
step fares the, sac 7} neror wien W and 6 ae sear thointnts ade
foil (a; inthe (6) plave or eouivalontly whee x sad 3 are near
ho indi’ fade paint, (1,0), Hay 9 phan.

Tor Viney rio inks tie os ar ws the roviv agnal
Pete” i comers, culllaoa, etmorpberis diengoute, wel earth
Delge often way leo be replaced by © mathorntically uvalen
inesorng sgl Z2Gie""° wich wbow eombend wil Uy eoastane
‘oul give th wecived Bustuntn gel

‘rehome nods) deasibed beatin 16) ty (08) ines
mow some ‘ating mechanisms which miny éveus individuals ve
simultwreuusly on a ding envio,

‘Going ne st t=, thi oda tap cies eet whens
[veel armies verte roars knesuse the rhea deco
postion (16) 3 eopliecle to say arb Urry fling sig For ant
natin, a6 gears aston mse) repsvn ie average goal Jesel
"The usta of se rg cedeee lu Le caused by am eq eaend
estat intern wel Fe

Ter rao Tinke subj to unlepach intense, che ourer of
fngoming corpo raves estas sate ta to. rae the wlan
fntastenng vooten Rs" mut of mnee tan one gota s12 shold nel be
faresprered aa asacel physi! evo, The wait estineo 3 taut the
agnitads aad the time dey of x piysiel eo age not Fanetions of
parti fequsney ule os Ue agnirads and che ecuivalew sine
Geleys Pa vn on coule steteing water eonsetng of ure Uh
to gos ane Cala af speatingfeqgonr fan, are dienes
SRO es oa, svermMe HeemETGH souRSaty mecenED in

anton (18: cho tn Ube peobility Hat Lie dom signal
be faled bsvow a order gua level fa sual tothe probability that
ceond 3 fal-wthin Ue rebar neo

aie teen ass
in ef, pome aa shorn i Lig, 3. Let (3? bo tho joist probity

‘notion uf aad 8. Then PF £1) isthe Sateral of fy 2) over the
‘enue ep (19348,

pren- f

fleas} duds. 0)

‘he total hehas'or of Oh iteworny swelne, Ra? — a 8 ie
sorntinesceenbed tx the joint eobbiliy dona fonetin 9G @
ff the magnivade ead the ps @ ut the intr vote. A anlar
‘Sevivacon in vers of gn @ yell

vwan- [f° an

2

Mel ef she flowing analyse in ia toes of und 9. An equivalent
recule interme of H abd #8 given in Appendix

Veco es, ”

ig Hig ss motel fo, ons