RADIO SCIENCE Journal of Research NBS/USNC-URSI Vo!' 68D, No.9, September 1964

Rayleigh Distribution and Its Generalizations

Petr Beckmann 1

Department of Ele ctrical Engineering, University of Color ado, Boulder, C olo.

(R eceived D eccmber 7, ] 963)

Th e Rayleigh d istribution is the distribution of thc sum of a large number of coplanar (or t imc) vectors \\'ith random. amplitudes alld Ull ifO rlllly distributed phases. As such, it is the lim iting case of distributions a ~soci at('d II ith 111 0re gl'IlPral vpctor surn s t hat arise in practical problems. S uch cases arC' til(' folloll' ing: (a) 'I'll(' p hasC' distributions of the vector tC'rms are not unifo rm, e.g., in the case of scatt l' ring fro m rough surface'S; (b) One or more vector tcrms predominaLP, their nwan sq uare value' not I>l'illg neglig;ble compared to th e m ean square valup of Uw SU Ill , c.g., in tJle ea~e of sigl als propagated in eil i (,'. ll1<'teor-scatter, and atmosplwrie noise; (0) The numiJer of \'('ctO I' t('rn ' j~ small, ('.g., in radar ret urns from several clo",(' targets; (d) '1'110 Jlumi)('r of vector lel'm o is ilsl'H randolll, e.g., in atmosp heric t urbuh'nce, mekor-scatter and atm ~ pl'(' .. ic noise'. The ('('suI ting distributions for t hese cases and their deviations from the R ayleigh distribution will be co nsidered.

1. Introduction

In m any pl'Oblems arisin g in radio waye p ropaga­ Tho present paper considers (1) under more Lion LllC roslil Lant field is rormed by Lhe stlper­ gell eral co ncli Lions. J n principle the problem can posi Lion or interference of a number o[ clelllcntclry a l ways be sol ved by resol "ing (1) into its roctangular waves: cO lllponen Ls (i.o ., r eal and imaginary parts) x and y,

J/ Ee iO = L:: Ejei~j, (1) j=J X= E.I cos 8= 1.:'Ej cos Rayleigh distribution satisfy t he Central Limit Theorem , i. e. , whether or not x and yare normally distributed as n-?oo . ?E Each of these two cases again includes several p ( E) = ~sJ e-E 2 1 (2) s, further possibilities. L et D(x) denote the variance of x; t hen the di sLri­ where s=< 11;2)= nEJ and the brad ets < ) denote t he bution of x will tend to a normal distri bution as luean value; or n-?oo provided t hat the Xj a re mutuall y independent and that lim D (x;) = 0 for all j 1 On leave of absence from the Institute of Badio Engin eerin g and Electron ics. (6) Czechoslovak Academy of ScicJlccs , Praf,!; lIc. 11--) 00 D (x) , 927 with a similar statement for y. (This statement (14) usually suffices for engineering purposes; for a more rigorous enunciation of the Central Limit Theorem and Lindebol'g conditions d. Gnedenko and Kol­ Thus a Rayleigh vector is a UDP \'ec tol' whose x mogoro\' [1954].) Condition (6) essentially means and y components are di stributed normally with that nono of the terms Xj must predominate in the (x)=(y)=O and D (x) = D (y) = s/2. From this it fol­ resulting sum x. Howeyer, if the X j are themselves lows that the sum of any number of Rayleigh vectors normally distributed, t hen x will, of course, be also is itself a Rayleigh vectol'. normally distributed even if (6) does not hold. A R ayleigh distribution will thus be found when­ In most (but not all) applications the

In a number of applications the phases

probability theory one then :finds the quantities

But for j~k , and the integration (5) yields (9)

Substitu ti ng (9) in (8) we find the importan t relation (10) where 1m is the modi:rred Bessel function of order valid for UDP vectors regardless of the value of n m and Eo = l , E",=2 for m~O. Details of the pro­ or the distributions of the E j (possibly all different). cedure and Clll'Yes of (15) will be found in [Beckmann, 1962a] . The general distribution (15) simpli fies in certain 2. Rayleigh Distribution special cases. Ifa= O, but sl ~ s, theD (15) l'educes to If the terms in (1) are UD P Yectol's, then from (4)

(X) = (y)= O (11) a distribution derived directly by Hoyt [1947]. On the other hand, if SI= sl = 4 s, but a~ O, then (15) reduces to the N akagami-Rice di stribution

Then using (10), condi tion (6) becomes 2 peE)= 2~ exp [ _ a : E] 10 (2 ~E), (17) (13) a distribution derived by Rice [1944 and 1945] 2 and further analyzed, e.g., by Norton et al. [1955], and Zuhrt [1957]. The distribution (17) is obtained If (13) is satisfied and n is large, x and y can be when a constaDt yector (E 1= a, normal distribution with mean value zero and the same variance. The integration Z '-rhe distribution w as origina11 y derived h y N akagami in 1940. A summary and bibliography or t he ",-ark on this and related top ics by :-Jakagam; and other (5) then leads to the R ayleigh distribution (2) with Japanese scientists will be found in [~a k aga,m i , 1960]. 928 1_ R ayleigh nctor , for the x a nd y co mponents of th e for one (or m ore) j. If more than one of such waves sum will then obviously be di stributed normally are presen t, we m ay sum them by standard methods wit h (x)=a, (y)=O, D (x) = D (Y) = 8/2 (where 8 is (conITolutions, characteristic functions) and regard the mean square value of the R ayleigh vector) thi s p ar ti,)'l sum as on.e wave. Since assumption (21 ) just as assumed in deriving (17). lL should be noted will hold for all other j , the rema ining terms will add that (J 7) will equ ally well h old for E [ exp (ict>l) a up to a R ayleigh ITector, so that the problem reduces UDP yector with constant a mplitude E[ = a; t his to :finding the distri bution of t he sum of a UDP may be shown by measuring the p hases from ct>l as Yectol' witb random ampliLude El and a Rayleigh a reference phase: the distribu tions of ct>; = ct> j- ct>l yector. will for j ~ 11'em ain uniform ,),s before, whereas ct>; = 0, This problem ma,y be sollred directly from first thus reducing to t he same co nditions under whicb principles by (4) to (5) or more quickly by random­ (1 7) was d er i,~ e d . izing a= E[ in (17) a nd using Lh e theorem of total The Rayleig h distribution, as may easily be probability: if the density of a is w(a), then (17) verified, is the limiting di stribution of (15), and of gilres t he density p(E la), so t ltat Li tO required total its special cases (16) and (1 7), for a= O; 8[ = S2 = S/2. prob ability density is If the Cen tral Limit Theorem may be applied to (4), so th at x and y will be distributed n ormally, 2E !c '" ( a2+ E 2) ( 2aE) then in the m os t gen er al case (corresponding to peE) = - w(a) exp - --. - 10 - da. SO . 1"l"" S 8 asynlln etricnl p hilse distributions) we hal'e foul' p arametcrs: (22)

(18) It m ay be ve ri fied fr om (22) t hat p eE ) will approach ,l R ayleigh dis tribu b on for (a2)<< s, as wn,s The integration (5) t hen leads to to be expected. The co mpleme nt of the di stri­ buLion fUll cLion of (22) is

P (E> R )= fR'" p (E )dE= i '" w(a)j (R , a)da (23)

cos [2m ( a r c t a n ~)] (19) where the order of in. tegration J1<"lS been reversed with

2 where j(R , a) = s2 JrR'" E exp [ - a- +s-E 2] 10 ( -8-2aE) dE., (24)

(20) Now if H is large (Ii > >s/a) , the Bessel function in (24) m ay be replaced by its asymptotic expression : The distribution (19) was found by N ak aga,mi a saddle-point integration then leads to [1960]; it is the 1110sL general distribution for the case when the Central Limit Theorem is applicable to j(R , a) = 21 [ l -erf (R-is- a) ] . (25) (4); for {3 = 0 a nd hence R = O, i t reduces to (15). The R ayleigh distribution is again obtained from (19) for a= {3 = O; s[=s2=s/2. Now (25) cha nges i tsn)'lue from < 0.01 to > 0.99 If the Central Limit Theorem is not applicable to near the p oint a= R wi thin a n inter val !J. a= 2.3{SH, (4), this may b e for one of the following reasons: (a) tending to zero below and to unity ab ove tha t condition (6) is not satisfied (this will be considered interval; for R > > -Js we m ay t herefore well approx­ in sees. 4 a nd 5) ; (b) the number of terms n in (1) is ima te (25) by not large (sec. 6) or random (sec. 7). o for a< R 4. Dominant Terms j(R , a) ~ { (26) 1 for a> R. F or UDP vectors, which we shall henceforth as­ sume, (6) reduces to (1 3). If the number of inter­ Substituting this v alue in (23) we find fering wayes n is large, but finite, we may replace (13) by n (for R > > {S) . (E ;) < < ~ (E ;) for any j. (21 ) j~ l (27)

Now if one (or more) of the interfering waves is H ence for R > > -Js the distribu tion of a r andom powerful, so t h at its p ower is n ot n egligible when vector plus a R ayleigh vector will approach that of co mpared to the total p ower , (2 1) will be violated the random vector alone. This effect m ay be

731- SGG--G4----2 929 observed in several cases in radio wave propagation. _ Nj t ~- l -Nt .

o (3 1) One of these IS the field strength of VHF and UHF wj(t j)-(j_l)! e J in cities and other built-up areas, where the total signal may be due to a direct wave (attenuated at The probability density of B = exp (-tJia) is then random as it is transmitted through walls and other found from (3 1) by a simple transformation: building materials) onto which large numbers of re(lected waves are superimposed. The resultin o' N jaj (ln B j) j-l amplitude (which is constant in time, but (32) randor~ (j-1) !Bi N +1 when measured at different places) is then dis­ tributed as in (22). A survey conducted at various If the density of E p is }.. (Ep) , then the distribution parts of the city of Prague showed that in most of E j is found from (3 0) as t hat of a product of two areas w(a), the distribution of the attenuated random variables: direct wave, is lognormal; the same r esult need not necessarily hold for cities differing in character from the above (brick or concrete houses fiye to six stories high, streets relatively naIT~w and not forming a regular pattern). By analyzing the resulting distribution the propagation mechanism Substituting (33) in (28) will in general lead to may thus be investigated (separation of reflections great computational difficulties, which may be and attenu ation). o,rercome by the following approximation. From (33) we find 5. Converging Variances (34) In at least two cases met in propagation t heory, meteoric fo rward-scatter and noise due to atmos­ and pherics, t he signals arrive at random intervals of '" time with a random aJl1plitude which then decays (E 2) = ~ (E ;). (35) exponentially: The signals are mutually indepen'd­ j=l eDt and thelr p hase makes them UDP ,"ectors. Sin ce an exponentially decaying signal never vanishes This se ~'~es will ?onverge and hence violate (13); completely, there is an infinity of residual signals however, 1£ t!le se.nes converge~ quic~dy (Na< < 1), present at any time; but sin ce the power at any time t hen (21) WIll stIll hold for J ~ 1; If t his tertn is is finite, the in:6.nite series of signals must converge excluded, the rest of the series Illay Lhus be approxi­ a~d the de~ominator of (13) will not tend to in:6.ni ty mate~ by a Rayleigh vector. Physically this m eans wIth n (thIS can also be shown mathematically). thaL I~ the time constant of decay is sufficiently H.en~e (13) Will not b p satisfied for any j, the C'entrnl short for the SIgnal to decay to a low value before LImit Theorem cannot be used, and t he res11ltin o' the next signal. arrives (average inter','al is l iN), ampli tude distribution cannot be a pure Rayleio'h then the total slgnal at any time will be dominated distribution. b by.the last signal (or possibly the last few signals), T o solve the problem rigorously one therefore wJlllst the remnants of all previous sio'ncLls will has to .return to :first principles, e.g., by findin g x combi ne to fOrln a low-power Ravleio'h ve~tor. '1'1 d . b and y 111 (4) through their characteristic functions lUS un er t h e~e circumstances this case may be reduced approXImately to the one in section 4' the required d!stribution is thus given by (22) with X (v) = X (v). = IIro --1 £Z" Gl-l. ro GlE .w(E ) i,E · cos '" x y . (2 ) j 'f' i J . J j e J gll'en by (33) aLld found )=1 7r • 0 0 w(a)=w(Et) s=(EZ)-(E D from (34) cwd (3 5) . The di~Lribution }..(E,,) is given by the physical (28) nature of the problem. For atm ospherics, }.. (E p) lllay ? e sh?wn to be lognormal [Beckmann, 1964]; 111 Spl te of the several approximations in \~ol ved wh~r e E j is the ampli tude of the jth decayi ng signal, tbe agreement of t he d istribu tion as deri ved abov~ wInch dep e ~ld s on Lwo random quant ities: the Li me t j elapsed slll ce the signal aLLained its peak ulluE', and Lhe Ill eclsur ed distribution is ve ry good as shown and that peak,oalue E p: by flg:lre 1. An analysis of the disLribution permits ~he efleet or propagation conditions to be separated (29) from Llmt or ligllLni ng acti viLy in the total random atmospheric noise. .0' wit~ a the Li me COllsttUlI of decay. Since the number of SIgnals per uniL Li me is Poisson-dis tributed (about an average N ), tl ha,s an exponentiftl distribution 6. Small Number of Independent Components (30) . If t he number n of independent interfering waves apd the di str i bu~ion of tj is gi , ~ e n by aj-fold cOllyolu­ IS small (e.g., the total rftdar signal returned from a tlOn of (30), whICh leads Lo small number of independent targets in the same 930 40 we find an ex pression which on integration over E yields (Levi n, 1960 , pp . 184- 187]' ~ ) 0 P (E~1 S > R )=e- R2 [1+ 4~ (:i-2) R4+ .. .J. ~ (38) 20 For n ---'> oo t his lea\'es a pure R a.yleigh dis tribution; 1\ for fmite n, t he Rayleigh distribution .will be a good 10 approximation if the second ter.m JI1. the . square bracket will be small compared WIt h umLy ; I. e., the required criterion is E ~ w 0 w 1\ (39) > 0 ro moment Lo mOll1 en t; t he IlU m bel' of eff ecti I'e refle c­ ~ 0.... -40 tO I'S in cities or irregular terrai n ch'1I1 ges fr om loca­ tion to 10caLion, etc. Thus n itself will be random; if its distribu tion (discrete for inlegers only) is -50 p en), t hen the di stribuLion of x alld ?J in (4) will be qa:xJI qOl 0,1 1 5 10 20 JO 40 50 60 70 BO 90 95 98 99 PERCENTAGE OF TIME ORDINA T E IS EXCEEDED '" n = FIG U RE 1.- Ampliltbde-pTobabi li ty distTibution oj atmospheric px(x) "L,P(n) "L, E j cos 4> j. (40) radio noise . n~ O j~ l Circles: values m easured by C richlow ci HI. [19601. on 13.3 kc/s at Boulder, Colo., October 6, 1958. F ull C"rI'e: dist ri bu tio n corn puted as in (33) t lll"o ll gh N ow if p en ) assumes appreciable I'alues oll ly for (35) fo r u ~1.67, Nc~ O . Ol. Broken curve ill terpolate(i. (Cf. Beckmann [1964J.) laro'e n tlte terms of t he n-sum will be norm al, hence x ~ ll be !lonnaJ and E will be Rayleigh-di stribu ted. area), t he Central Limit 'J'heorem cannot be appli ed A more detail ed ill ves Li gaLio ll [Beckmanll , ]962b] and li ttle can be said about t he di stributions of x shows t hat t he cl istriblltion of E will closely H.pproach and y in (4) beyond the statement that they have a Rayleigh distribu tion as the cond i tion to be determined from t he w;(E j ) by con volu tions or characteristic fun ctions. However , we :may ask the opposite question: how large must n be in (1 ) (41) for the Rayleigh distribution to be a good approxima­ p eE) tion for in engineering practice? The answer lS more nearly satis:Ci.ed; it is also shown that for will ob\Tiously depend on the di stributions wj(Ej), a o'iven distribution pen) t lte deviation of p eE) which we here assume all identical and equal to fro~ a Rayleigh distribution will always be greater w(Ej ) . In that case one may derive the formula for large E than for small E. [Watson, 1944; Levin, 1960] As a rule n is Poisson-distribu ted abou t its mean value (n), i.e., peE) = E i OO (i'" w(Ej) J o(uEj) dEj)nuJo(Eu)du. P (n) = (n;n e(-n) . (42) (36) n.

Expanding Jo (E ju) ill a series and integrating From (42) we find (n 2)=(n)+(n)2; hence b term by term, rearranging in ascending powers of (1 ln) and using t he 2d and 4th initial moments of r (n2) _ 1 w(EJ, i.e. , (n)2- (n) + 1 (43) I m2=(E;)= Sa'" E ;w(Ej)dEj ; which will approach unity as required by (41 ) for (n»>l. Thus if pen) is give n by (42), .a l ar~e j mean value is sufficient to make the RayleIgh dIs­ m4 =(E J)= i '" E'w(Ej) dE (37) tribu tion a good approximation for p (E). It should 931 be noted that for large (n), (42) is equivalent to a Crichlow, W. Q. , C. J. Roubique, A. D. Spaulding, and W NL Bee ry (J ltn.- Feb. 1960), Determination of t he ampli­ normal distribution with D (n) = (n) [Levin, 1960]. tude-probability distribution of atmospheri c radio noise For a general distribution P (n), however, a large from statistical moments, J. R es. N BS 6tD (Radio P rop.), m efl,n value (n) is not sufficient to guarantee No.1, 49- 56. t l~fl,t .peE) in (1) fl,nd (4) will tend to the R ayleigh Gnedenko, . B. V ., and A. N. Kolmogorov (1954), Limit cllstnbutlOns for SLIms of indepen dent random. variables dlstnbutlOn, but the en terion (4 1) must be sfl,tisfied. (Addison-Wesley Publishing Co., Cambridge, Mass.). Hoyt, R. S. (1947), Probabilitv functions for the modulus 8. Distribution of the Resulting Phase a nd angle of the normal complex variate, Bell System T ech. J . 26, 318- 359 . Levin, B. R. (1960), The theor y of random processes and its The distribution of B in (1) is found from the second application to radio engineering, 2d ed. (in Russian) relation in (5). For a sum of UDP vectors the Sovyetskoye Radio, Moscow. r~sul.tjng phase will. of course again be unifdrmly Nagakami, M. (1960), The m-distl'ibution-a general formula dlstnbuted over an mtervfl,l of leno-th 211"' for non­ ?f inter;sity distribut lon of rapid fading, Statistical Methods 111 R adIO Wave Propagation, ed. W. C. Hoffman (Pergamon uniform phase distributions of the ~ ' ector' terms we Preiis, Oxford). introduce the qUfl,ntities Norton, K. A ., L. E. Vogler, W. V. Mansfi eld, and P. J . Short (1955), The probability distribution of the amplitude T a ,- of a co nstant veetor plus a Raylcigh-distributed vector, Proc. IRE 53, 1354-1361. p= .JSI + S2; B = .JSt+sz; K= .y ~ (44) Rayleigh, Lord (1896), The Theory of Sound, sec. 42a, 2d .ed. (Macmillan, London), (R eprinted Dover, 1945) . RICe, S. O. (194.4), Mathemat ical fl1lalvsis of random noise and then obtain from (5) Bell System T ech . .J. 23, 282- 332. ~ , Rice, S. O. (1945), lVlathematical analysis of random noise I-{e- tB2(l+K2) 2 Bell System T ech. J . 240, 46- 156 . . , po(B) G 2 (K Z <) B+ . 2 e) [J + G,hre (1 + erf G)] Watson, G. N. (1944), A Treatise on the TheOl'v of Bessel 11" cos- SIn Functions (University Press, Cambridge). • (45) Znhrt, H . (l !J57) , Die Summenhiiufigkeitskurven del' exzcll t­ rischen Rayleigh-Vel'teilung und ihre AnwendllnO' auf where 3 Ansbreit.llngsmessungen, A. E. U. 11, 478- 484. '"

(46 ) Additional Related References

l' l1l'ntsll , K., and T. Ishida (July 1961), On the theory of 9. Reference amplitude distribution of impulsive random noise, J. App\. Phys. 32, No.7, 1206- 1221. Beckmann, P . (1962a), Statistical distribution of the a mpli­ Siddiqui, M. 1\1. (Mar·.- Apr. 1962), Some problems connected tude and phase of a multiply scattered fi eld, J . Res. NBS with Rayleigh distribution, J. Res. NBS 66D (lhdio 66D (Ra dio Prop.), No.3, 231-240. Prop.) , No. 2, 167- 174. Beckmann, P. (1962b), Deviations from the Ravleiah di s­ Wheelon, Albert D . (Sept.-Oc t. 1960), Amplitude distribut ion tributior:! for a_small and for a random number of ·i nt~"f ering for radio signals refl ected b y m eteor trails, .J . R es. NBS waves, URE- CSAV Inst. Rept. No. 2.5. 6tD (Radio Prop.), No.5, 449- 454. Beckmann, P. (1.964), The amplit ude-probabili ty distribut ion Wheelon, Albert D. (May- June 1962), Amplitude distribution of atmospheric noise, Radio Sci. J . Res. NBSjUSNC/UHSI for radio signals refl ected by meteor trails, II, J. Res. NBS 68D, No.6, 723-735. 66 D (R adio Prop.), No.3, 241- 247.

3 Note the factor cos 0 in (46), \I'hidl U11fortunatel y dropped out from the correspond inp; eq uation on p. 236 of Beckmann [1962aJ .. (Pfl,per 68D9-392)

932