Probability distributions relevant to radiowave propagation modelling

Transcription

1 Recoendaion ITU-R P.57-3 (9/3) Probabiliy disribuions relevan o radiowave propagaion odelling P Series Radiowave propagaion

2 ii Rec. ITU-R P.57-3 Foreword The role of he Radiocounicaion Secor is o ensure he raional, equiable, efficien and econoical use of he radio-frequency specru by all radiocounicaion services, including saellie services, and carry ou sudies wihou lii of frequency range on he basis of which Recoendaions are adoped. The regulaory and policy funcions of he Radiocounicaion Secor are perfored by World and Regional Radiocounicaion Conferences and Radiocounicaion Asseblies suppored by Sudy Groups. Policy on Inellecual Propery Righ (IPR) ITU-R policy on IPR is described in he Coon Paen Policy for ITU-T/ITU-R/ISO/IEC referenced in Anne of Resoluion ITU-R. Fors o be used for he subission of paen saeens and licensing declaraions by paen holders are available fro hp:// where he Guidelines for Ipleenaion of he Coon Paen Policy for ITU-T/ITU-R/ISO/IEC and he ITU-R paen inforaion daabase can also be found. Series of ITU-R Recoendaions (Also available online a hp:// Series BO BR BS BT F M P RA RS S SA SF SM SNG TF V Tile Saellie delivery Recording for producion, archival and play-ou; fil for elevision Broadcasing service (sound) Broadcasing service (elevision) Fied service Mobile, radiodeerinaion, aaeur and relaed saellie services Radiowave propagaion Radio asronoy Reoe sensing syses Fied-saellie service Space applicaions and eeorology Frequency sharing and coordinaion beween fied-saellie and fied service syses Specru anageen Saellie news gahering Tie signals and frequency sandards eissions Vocabulary and relaed subjecs Noe: This ITU-R Recoendaion was approved in English under he procedure deailed in Resoluion ITU-R. Elecronic Publicaion Geneva, 3 ITU 3 All righs reserved. No par of his publicaion ay be reproduced, by any eans whasoever, wihou wrien perission of ITU.

3 Rec. ITU-R P.57-3 RECOMMENDATION ITU-R P.57-3 Probabiliy disribuions relevan o radiowave propagaion odelling ( ) Scope This Recoendaion describes he various probabiliy disribuions relevan o radiowave propagaion odelling and predicions. The ITU Radiocounicaion Assebly, considering a) ha he propagaion of radio waves is ainly associaed wih a rando ediu which akes i necessary o analyse propagaion phenoena by eans of saisical ehods; b) ha, in os cases, i is possible o describe saisfacorily he variaions in ie and space of propagaion paraeers by known saisical disribuions; c) ha i is herefore iporan o know he fundaenal properies of he probabiliy disribuions os coonly used in saisical propagaion sudies, recoends ha he saisical inforaion relevan o propagaion odelling provided in Anne should be used in he planning of radiocounicaion services and he predicion of syse perforance paraeers; ha he sep-by-sep procedure provided in Anne should be used o approiae a copleenary cuulaive disribuion by a log-noral copleenary cuulaive disribuion. Anne Probabiliy disribuions relevan o radiowave propagaion odelling Inroducion Eperience has shown ha inforaion on he ean values of he signals received is no sufficien o characerize he perforance of radiocounicaion syses. The variaions in ie, space and frequency also have o be aken ino consideraion. The dynaic behaviour of boh waned signals and inerference plays a decisive role in he analysis of syse reliabiliy and in he choice of syse paraeers such as odulaion ype. I is essenial o know he een and rapidiy of signal flucuaions in order o be able o specify such paraeers as ype of odulaion, ransi power, proecion raio agains inerference, diversiy easures, coding ehod, ec.

4 Rec. ITU-R P.57-3 For he descripion of counicaion syse perforance i is ofen sufficien o observe he ie series of signal flucuaion and characerize hese flucuaions as a sochasic process. Modelling of signal flucuaions for he purpose of predicing radio syse perforance, however, requires also knowledge of he echaniss of ineracion of radio waves wih he aosphere (neural aosphere and he ionosphere). The coposiion and physical sae of he aosphere is highly variable in space and ie. Wave ineracion odelling, herefore, requires eensive use of saisical ehods o characerize various physical paraeers describing he aosphere as well as elecrical paraeers defining signal behaviour and he ineracion processes via which hese paraeers are relaed. In he following, soe general inforaion is given on he os iporan probabiliy disribuions. This ay provide a coon background o he saisical ehods for propagaion predicion used in he Recoendaions of he Radiocounicaion Sudy Groups. Probabiliy disribuions Sochasic processes are generally described eiher by a probabiliy densiy funcion or by a cuulaive disribuion funcion. The probabiliy densiy funcion, here denoed by p() for he variable, is such ha he probabiliy of aking a value in he infiniesial inerval o + d is p() d. The cuulaive disribuion funcion, denoed by F(), gives he probabiliy ha he variable akes a value less han, i.e. he funcions are relaed as follows: or: where d p ( ) = d F( ) = c [ F( )] p( ) c is he lowes lii of he values which can ake. The following disribuions are he os iporan: noral or Gaussian disribuion; log-noral disribuion; Rayleigh disribuion; cobined log-noral and Rayleigh disribuion; Nakagai-Rice disribuion (Nakagai n-disribuion); gaa disribuion and eponenial disribuion; Nakagai -disribuion; Pearson χ disribuion. d 3 Noral disribuion This disribuion is applied o a coninuous variable of any sign. The probabiliy densiy is of he ype: p() = e T () ()

5 Rec. ITU-R P T() being a non-negaive second degree polynoial. If as paraeers we use he ean,, and he sandard deviaion,, hen p() is wrien in he usual way: hence: wih: F( ) = π p ( ) = ep () π ep erf ( z) = π z e d = d + erf (3) (4) The solid lines in Fig. represen he funcions p() and F() wih equal o zero and equal o uniy. The cuulaive noral disribuion F() is generally abulaed in a shor for for he sae condiions. Table gives he correspondence beween and F() for a nuber of round values of or F(). TABLE F() F() For he purpose of pracical calculaions, F() can be represened by approiae funcions, for eaple he following which is valid for posiive wih a relaive error less han.8 3 : ep ( / ) F ( ) = π (5) A noral disribuion is ainly encounered when values of he quaniy considered resul fro he addiive effec of nuerous rando causes, each of he of relaively sligh iporance.

6 4 Rec. ITU-R P.57-3 In propagaion os of he physical quaniies involved (power, volage, fading ie, ec.) are essenially posiive quaniies and canno herefore be represened direcly by a noral disribuion. On he oher hand his disribuion is used in wo iporan cases: o represen he flucuaions of a quaniy around is ean value (scinillaion); o represen he logarih of a quaniy. We hen obain he log-noral disribuion which is sudied laer. Diagras in which one of he coordinaes is a so-called noral coordinae are available coercially, i.e. he graduaion is such ha a noral disribuion is represened by a sraigh line. These diagras are very frequenly used even for he represenaion of non-noral disribuions. 4 Log-noral disribuion This is he disribuion of a posiive variable whose logarih has a noral disribuion. I is possible herefore o wrie direcly he probabiliy densiy and he cuulaive densiy: ln p ( ) = ep (6) π ln ln F( ) = = + π ep d erf (7) However, in hese relaions and are he ean and he sandard deviaion no of he variable bu of he logarih of his variable. The log-noral disribuion is very ofen found in connecion wih propagaion, ainly for quaniies associaed eiher wih a power or field-srengh level or a ie. Power or field-srengh levels are generally only epressed in decibels so ha soeies reference is ade o a log-noral disribuion siply as a noral disribuion. This usage is no recoended. In he case of ie (for eaple fading duraions), he log-noral disribuion is always used eplicily because he naural variable is he second or he inue and no heir logarih. Since he reciprocal of a variable wih a log-noral disribuion also has a log-noral disribuion, his disribuion is soeies found in he case of raes (reciprocals of ie). For eaple, i is used o represen rainfall rae disribuions. In coparison wih a noral disribuion, i can be considered ha a log-noral disribuion eans ha he nuerical values of he variable are he resul of he acion of nuerous causes of sligh individual iporance which are uliplicaive. When considered in nuerical ers, a log-noral disribuion is ereely asyerical, unlike he noral disribuion. In paricular, he ean value, he edian value and he os probable value (ofen called he ode) are no idenical (see he dashed lines in Fig. ). The characerisic quaniies of he nuerical variable are: os probable value: ep ( ); edian value: ep (); ean value: ep + ;

7 Rec. ITU-R P roo ean square value: ep ( + ); sandard deviaion: ep + ep ( ). FIGURE Noral and log noral disribuions.9.8 Probabiliy Mode Median Mean p ( ), noral F (), noral p ( ), log-noral F ( ), log-noral P.57-5 Rayleigh disribuion The Rayleigh disribuion applies o a posiive coninuous variable. I is linked wih he noral disribuion as follows. Given a wo-diensional noral disribuion wih wo independen variables y and z of ean zero and he sae sandard deviaion, he rando variable = y + z (8) has a Rayleigh disribuion. The os probable value of is equal o. The Rayleigh disribuion represens he disribuion of he lengh of a vecor which is he su of a large nuber of vecors of siilar apliudes whose phases have a unifor disribuion. The probabiliy densiy and he cuulaive disribuion are given by: p ( ) = ep (9) F ( ) = ep () Figure gives eaples of hese funcions p() and F() for hree differen values of b.

8 6 Rec. ITU-R P FIGURE Rayleigh disribuion p() is shown as solid lines and F() as dashed lines for hree differen values of b: blue b = ; red b = ; green b = 4.8 Probabiliy P.57- The characerisic values of he variable are as follows: os probable value: edian value: ; ln.833b; ean value: π.886b; roo ean square value: b; sandard deviaion: π 4 π 4.463b. The Rayleigh disribuion is ofen only used near he origin, i.e. for low values of. In his case we have: () () This epression can be inerpreed as follows: he probabiliy ha he rando variable X will have a value of less han is proporional o he square of his value. If he variable in quesion is a volage, is square represens he power of he signal. In oher words, on a decibel scale he power decreases by db for each decade of probabiliy. This propery is ofen used o find ou wheher a received level has a Rayleigh disribuion a leas asypoically. I should be noed, however, ha oher disribuions can have he sae behaviour.

9 Rec. ITU-R P In paricular he Rayleigh disribuion occurs for scaer fro independen, randoly-locaed scaerers for which no scaering coponen doinaes. Foonoe: b =. 6 Cobined log-noral and Rayleigh disribuion In soe cases he disribuion of a rando variable can be regarded as he resulan of a cobinaion of wo disribuions, i.e. a log-noral disribuion for long-er variaions and a Rayleigh disribuion for shor-er variaions. The disribuion of insananeous values is obained by considering a Rayleigh variable whose ean (or ean square) value is iself a rando variable having a log-noral disribuion. If and are used o designae he ean and he sandard deviaion of he noral disribuion associaed wih he log-noral disribuion, he following disribuion is obained: F( ) = π + u u ep e du () In his forula he sandard deviaion is epressed in nepers. If ' is used o designae is value in decibels, we have: =.5 ' (3) Figure 3 shows a graph of his disribuion for a nuber of values of he sandard deviaion, he value of being aken o be equal o zero. The characerisic values of he variable are as follows: os probable value of : ln ; edian value of : ln ( ); ean value of : ln ( ); roo ean square value of : ; sandard deviaion of : ln ( ). The disribuion occurs ainly in propagaion via inhoogeneiies of he ediu when he characerisics of he laer have non-negligible long-er variaions, as for eaple in he case of ropospheric scaer.

10 8 Rec. ITU-R P.57-3 FIGURE 3 Cobined log-noral and Rayleigh disribuions (wih sandard deviaion of he log-noral disribuion as paraeer) Apliude (db) db Percenage probabiliy ha ordinae will be eceeded, ( F( )) (%). P Nakagai-Rice disribuion (Nakagai n-disribuion) (See Noe ) NOTE No o be confused wih he Nakagai -disribuion. The Nakagai-Rice disribuion is also derived fro he noral disribuion and i generalizes he Rayleigh disribuion. I ay be considered as he disribuion of he lengh of a vecor which is he su of a fied vecor and of a vecor whose lengh has a Rayleigh disribuion. Alernaively, given a wo-diensional noral disribuion wih wo independen variables and y and wih he sae sandard deviaion, he lengh of a vecor joining a poin in he disribuion o a fied poin differen fro he cenre of he disribuion will have a Nakagai-Rice disribuion.

11 Rec. ITU-R P If a is used o designae he lengh of he fied vecor and he os probable lengh of he Rayleigh vecor, he probabiliy densiy is given by: + a a p ( ) = ep I (4) where I is a odified Bessel funcion of he firs kind and of zero order. This disribuion depends on wo paraeers bu for he purposes of propagaion probles i is necessary o choose a relaion beween he apliude a of he fied vecor and he roo ean square apliude of he rando vecor. This relaion depends on he applicaion envisaged. The wo ain applicaions are as follows: a) Power in he fied vecor is consan, bu he oal power in fied and rando coponens varies For sudies of he influence of a ray refleced by a rough surface, or for a consideraion of ulipah coponens in addiion o a fied coponen, he ean power is given by ( a + ). The disribuion is ofen defined in ers of a paraeer K: a K = log db (5) which is he raio of he powers in he fied vecor and he rando coponen. b) Toal power in he fied and rando coponens is consan, bu boh coponens vary For he purpose of sudying ulipah propagaion hrough he aosphere, i can be considered ha he su of he power carried by he fied vecor and he ean power carried by he rando vecor is consan since he power carried by he rando vecor originaes fro ha of he fied vecor. If he oal power is aken o be uniy, one hen has: = a + (6) and he fracion of he oal power carried by he rando vecor is hen equal o. If X is used o designae he insananeous apliude of he resulan vecor and a nuerical value of his apliude, we find ha he probabiliy of having an insananeous level greaer han is given by: a νa Prob (X > ) = F() = ep ep ( ) I dν / ν ν (7) Figure 4 shows his disribuion for differen values of he fracion of power carried by he rando vecor.

12 Rec. ITU-R P.57-3 FIGURE 4 Nakagai-Rice disribuion for a consan oal power (wih he fracion of power carried by he rando vecor as paraeer) Apliude (db) Percenage probabiliy ha ordinae will be eceeded, ( F( )) (%). P.57-4 For he purpose of pracical applicaions use has been ade of a decibel scale for he apliudes, and for he probabiliies, a scale such ha a Rayleigh disribuion is represened by a sraigh line. I will be seen ha for values of he fracion of power in he rando vecor above abou.5, he curves approach a lii corresponding o a Rayleigh disribuion. This is because in his case he fied vecor has an apliude of he sae order of agniude as ha of he rando vecor and i is pracically indisinguishable fro i. On he oher hand for sall values of his fracion i can be shown ha he disribuion of he apliude ends owards a noral disribuion.

13 Rec. ITU-R P.57-3 While he apliude has a Nakagai-Rice disribuion, he probabiliy densiy funcion of he phase is: a cos θ a cos cos π a θ ( ) a θ p θ = + e + erf π e (8) where: erf ( ) = e d (9) π 8 Gaa disribuion and eponenial disribuion Unlike he previous disribuions which derive fro a Gaussian disribuion, he gaa disribuion is derived fro he eponenial disribuion of which i is a generalizaion. I is applied o a posiive and non-liied variable. The probabiliy densiy is: ν α ν p( ) = e Γ( ν) α () where Γ is he Euler funcion of second order. This disribuion depends on wo paraeers α and ν. However α is only a scale paraeer of variable. Characerisic values of he variable are: ν ean value: α roo ean square value: ν ( + ν) α ν sandard deviaion: α The inegral epressing he cuulaive disribuion canno be evaluaed in closed for ecep for inegral values of ν. On he oher hand he following epansions are possible: Series approiaion for << : F( ) = e Γ( ν + ) Asypoic approiaion for >> : α ( α ) ν α ( α ) ν + ( ν + ) ( ν + ) () F( ) = e Γ( ν) α ( α ) ν ν ( ν ) ( ν ) α ( α ) ()

14 Rec. ITU-R P.57-3 For ν equal o uniy we find an eponenial disribuion. For ineger ν he asypoic epansion has a finie nuber of ers and gives he gaa disribuion in an eplici for. In propagaion he useful values of ν are very low values of he order of o 4. For ν in he viciniy of zero, we have: ~ ν ~ ν Γ( ν) Γ( ν + ) (3) I is possible herefore o wrie for ν sall and α no oo sall: e F( ) ~ ν d (4) For pracical calculaions i is possible o find an approiaion o he above inegral, for eaple he following: α α e F( ) ~ ν (5).68 + α +.8 log α which is valid for ν <. and α >.3. The cuulaive disribuion of he copleenary gaa funcion for sall values of ν is shown in Fig. 5. I can be seen ha he probabiliy of he variable being significanly greaer han zero is always sall. In paricular his eplains he use of he gaa disribuion o represen rainfall raes since he oal percenage of rainfall ie is generally of he order of o %. 9 Nakagai -disribuion (see Noe ) NOTE In his secion denoes a paraeer of he Nakagai -disribuion; i is no a ean value as in previous secions of his Anne. This disribuion is applied o a non-liied posiive variable. The probabiliy densiy is equal o: Ω p( ) = e (6) Γ( ) Ω Ω is a scale paraeer equal o he ean value of. = Ω (7) This disribuion has various relaionships wih he previous disribuions: if a variable has a Nakagai -disribuion, he square of his variable has a gaa disribuion; for = we obain a Rayleigh disribuion; for = / we obain he one-sided noral disribuion.

15 Rec. ITU-R P The Nakagai -disribuion and he Nakagai-Rice disribuion can hus be regarded as wo differen generalizaions of he Rayleigh disribuion. I should be noed ha for very low signal levels, he slope of he Nakagai -disribuion ends owards a value which depends on he paraeer, unlike he Nakagai-Rice disribuion for which he lii slope is always he sae ( db per decade of probabiliy). The cuulaive Nakagai -disribuion for various values of paraeer is shown in Fig. 6. FIGURE 5 Gaa disribuion (α =, ν.) ν =. Variable Percenage probabiliy ha ordinae will be eceeded, ( F( )) (%) P.57-5

16 4 Rec. ITU-R P.57-3 FIGURE 6 Nakagai- disribuion ( = ) Percenage probabiliy ha ordinae will be eceeded, ( F ( ) ) (%) = 5 3 Apliude (db) Probabiliy ha ordinae will no be eceeded, F( ) P.57-6

17 Rec. ITU-R P Pearson χ disribuion The probabiliy densiy is given by he equaion: χ ν p ( χ ) = e ( χ ) (8) ν ν Γ χ is a non-liied posiive variable and he paraeer ν, a posiive ineger, is ered he nuber of degrees of freedo of he disribuion. Γ represens he Euler funcion of second order. Depending on he pariy of ν, one has ν ν ν even: Γ =! (9) ν odd: ν ν ν Γ =... π (3) The cuulaive disribuion is given by: The ean and sandard deviaion are given by: χ ν F( χ ) = e d ν (3) ν Γ = ν (3) = ν (33) An essenial propery of he χ disribuion is ha, if n variables i have Gaussian disribuions of ean i and sandard deviaion i, he variable: n i i (34) = l i i has a χ disribuion of n degrees of freedo. In paricular, he square of a sall Gaussian variable has a χ disribuion of one degree of freedo. If several independen variables have χ disribuions, heir su also has a χ disribuion wih a nuber of degrees of freedo equal o he su of he degrees of freedo of each variable.

18 6 Rec. ITU-R P.57-3 The χ disribuion is no fundaenally differen fro he gaa disribuion. Conversion fro one o he oher can be effeced by he equaions: χ = α (35) ν = n (36) Siilarly, conversion fro χ disribuion o he Nakagai- disribuion can be achieved by: χ = (37) Ω ν = (38) The χ disribuion is used in saisical ess o deerine wheher a se of eperienal values of a quaniy (rainfall rae, aenuaion, ec.) can be odelled by a given saisical disribuion. Figure 7 gives a graphic represenaion of he disribuion for a nuber of values of ν.

19 Rec. ITU-R P FIGURE 7 χ disribuion 3 ν = ν = 5 ν = ν = ν= 8 ν = 6 ν = 5 ν = 4 ν = 3 ν = ν = 3 χ Percenage probabiliy ha ordinae will no be eceeded, F( χ) % P.57-7

20 8 Rec. ITU-R P.57-3 Anne Sep-by-sep procedure o approiae a copleenary cuulaive disribuion by a log-noral copleenary cuulaive disribuion Background The log-noral cuulaive disribuion is defined as: + = π = ln erf d ln ep ) ( F (39) or equivalenly: F d ep ) ( ln π = (4) Siilarly, he log-noral copleenary cuulaive disribuion is defined as: = = π = ln erf ln erfc d ln ep ) ( G (4) or equivalenly: = π = Q G ln d ep ) ( ln (4) where (). Q is he noral copleenary cuulaive probabiliy inegral. The paraeers and can be esiaed fro a se of n pairs (G i, i ) as described in he following paragraph.

21 Rec. ITU-R P Procedure Esiae he wo log-noral paraeers and as follows: Sep : Consruc he se of n pairs (G i, i ) where G i is he probabiliy ha i is eceeded. Sep : Transfor he se of n pairs fro (G i, i ) o (Z i, ln i ) where: i ( G ) = erf ( G ) Z = erfc i i or, equivalenly, Z = Q ( ) i G i Sep 3: Deerine he variables and by perforing a leas squares fi o he linear funcion: ln i =Zi + as follows: = n n Z ln i i i i= i= i= n n n Zi Zi i= i= n n n ln Z n i = i= i= n Z i ln i