,. ,

?

JOURNAL O F RESEARCH of the Nationa l Bureau of Standards-D. Ra dio Propagation Vo!' 66D, No. 3, May- June 1962

Statistical Distribution of the Amplitude and Phase of a Multiply Scattered Field

Petr Beckmann

(Received D ecember 26, 1961)

Contribution from the Institute of Radio Engineering and Electronics, Czechoslovak Academy of Sciences, Prague

The probability dist ribu Lion of the ampli tude and phase o f the sum of a large num ber of random two-di mensional vectors is derived under t he fo llowing general condit ions : Both t he ampli tudes a nd th e phases of the component vectors a re random, t he di sLribu tions b eing a rbitrary within the vali dit.\7 of the Central Lim it Theorem ; in part icular, the d is­t r ibutions of t he individual veetors need not be identical , the amplitude and phase of each component vec tor need not be independent a nd the di tribu tions Ileed not be symmel ri cal. T he disLributions formerly derived by R ayleigh , nice, H oy t, a nd Heckmanll are shown lo be special cases of this di sLribut ion .

1. Introduction

The c1ecLromagneLic field scattered by randomly distributed scn,tLerer necessarily always consist of the individually scattered wave which mutually interfere to form the resulting total field. Vve may thus write

n E = reiO = ~Ajei4>J

j = l (1)

where l' is the amplitude and B t he phase of the resulting field fUld A j exp (i<p j) arc the elementary component waves. The amplitudes A } and pha es <Pi> and hence also l' and 0, arc random quantities.

The sum (1) may be r epresented in the compl ex plane as the stUn of random veeLors (fi g. 1) ; the

FIGURE 1. Random vector sum in the complex plane.

231

problem is also idenLicnl with the "random-walk problem" in math emalical laLisLics, for our task: is to determine the pl"Ob~lbility distribu lions of 1"

and 0 if the distributions of t.ho A j and <P j are known. Tb e problem in its mosL elementary form , when

the A } a1"e constan t and Lbe <P j arc all distributed uniformly over an interval of lengLh 27r, ,va olvcd by Rayleigh [1896) and l eH cls to the well-known Rayleigh distribution

21' 2 p (1') = -e- r I n. (2)

n

OLhel' special cases or 1,he distribu lion of E were derived by Rice [1944, 1945) and by Hoyt [1947).

A 1110re general olu tion, namely for all A } COJl ­

stant and arbitrary symmetrical distributions of the phases was derived by Beckmann [1959). The Rayleigh , Rice, and HoyL distribution arc included in this distribution as special cases.

The purpose of the present paper i to derive the distributions of l' and B in t he general ca e when both the A j and the <PJ arc r andom and arbitrarily distributed. The di tributions of the individual A } or rP} need not be identical, A } Hnd rP j may be correlated and the distributions of tJ lO rP} need not be symmetrical. Thc only restriction is that the distributions of the quantities A j cos rP j and A} sin <p j satisfy the condiLions of the Central Limit Theorem [Gnedenko and Kolmogorov, 1949). In physical practice the e conditions are pracLi­cally always satisfied provided that n, the number of interfering waves, is large and that these waves ar e statistically independent.!

Our problem, then, is to determine the probability distribution p(r) and the (more rarely required) distribution p(B) when the two-dimensional distri­butions wj(A, rP) are given.

1 Generalization for dependent variables is also possible [Bernstein, 1944J.

2. General Solution

We introduce the quantitif's

x = r cos 8; y = r sin 8 (3)

and denot(' the mean value by angular brackets ( ) and the variance by the symbol D . Then

(x)=a= ttJ -"'", J -"'", w j(A,<t»A cos <t> dAd<t>,

(y)=,6= t1 J -"'", J -"'", wj(A,<t»A sin <t> dAd<t>.

(4)

(5)

D eno ting th E' individual terms of the sums (4) and (5) by 2

(6)

,6 j= JJwj(A ,<t»Asin4>dAd<t>, (7)

we have

D {x } =8[ = tt [Jf Wj (A,<t»A2 cos2 <t> dAd<t> - a7], (8)

D {y } =82= "tt [J J w/ A ,<t»A2 sin2 <t> dAd<t>- ,6J]. (9)

The covariance of x and y is, after elementary manipulations,

cov (.e, y) =~ t1 J Wl fl ,<t» A2 sin 2<t> dAd<t>

+ ~ tt' J}Vi(A,<t»A cos <t> dAd<t>

J J w/A,4»A sin <t> dAd<t>- a,6 , (10)

where the apostrophe at the second sum indicates that the terms with i=j are to be omitted.

We now temporarily assume that the distribution of <t> is symmetrical about its mpan value zero. Then the integrals in (5), (7), and (1 0) vanish owing to the factor sin <t> or sin 24> in the integrands, so that

,6j=,6=cov (x, y) = 0. (11)

Now according to the Central Limit Th eorem the distribution of the sum of many independent random variables tends under certain conditions that in practical a pplications are almost invariably satisfi ed, to the normal (Gaussian) distribution .3

2 To save space, we henceforward omit the integration limits - co, co; unless other wise indicated, all integrals in this paper are thereIore to be taken Irom -00 to 00,

3 For an exact statement of the Central L imit Theorem and the weakest (Linde­herg) conditions of its vqlidity, sec [Gnedenko and Kolmogorov, 1949, Ch.5, Sec, 26), or [Richter, 1956, Ch. VII, Sec, 4] , An unnecessarily, st,:ong, coudition fo r the validity of the Ceotral Llm ,t Theorem IS e.g, that the distributIOns 01 tbe ter ms be iden'tical an d their variance exist (Bernoulli condit ion).

Since by (11) x and yare uncorrclated (and h ence, as normal random variables, independen t) , the two­dimensional probabili ty density of x and y is

(12)

Transformin g to polar coordinates as l.ll (3), we 'find the req uil'ed distribution in the form

r- sm- 8 ? • ? ]

282 d8,

(13)

which after elementary manipulations may b e 'VTitten as

p(r)= Se - T12T e-Pcos28+Qco sO de , (14)

where S, T , P, and Q are functions of r, a , 81, and 82.

The in tegral in (14) may be expressed as a series of modified Bessel functions of the first kind (1m) as follows [Beckmann and Schmelovsky, 1958]:

" , ,

l

12"- e- P cos20+Q coSOde= 27re- P12i;o (- l )me,Jm(f)lzm(Q) I (15)

where

(16)

so that (13) may be expressE'd as follows:

(17)

B efore normalizing the distribution (15), we calcu­late the mean square of r, which, apart from a con­stant factor (1 / 12071" in MKS units), equals the mean scattered power. From (3) and the formula for the variance of a random quantity

'1

and from the independence of x and y it follows that

(18)

The RMS value of the scattered field equals the square root of this expression.

It is easily SE'en from (12) that the resulting vector r exp (i8) is the sum of a constant vector ~ directed along the x-axis and a random vector B (fig. 2); the x and y components of this random (Hoyt) vector are normally distributed with mean values zero and unequal variances 81 and 82'

I

232

> I

I I

\ /

y

FIGURE 2. Components of the Tesultant vector r and its equi-pTobability wn·es.

vVe -now normalize in a ':col'dance with (18) by using Ithe ratio of the RMS values of the constant and random components

B = a (19) .y81+ 8z

and introducing the "normalized ampliLude"

(20)

and the "asyml1wtry factor"

(21)

.x

FIGU RE 3. TransfoTmation of coordinares whqn I\'(</» is not an even func tion.

FIG lJ RI, 4. Scattering by Tough 0 1' turbulent layers in the troposphel·e.

1,2-r-r-r~-T~~~~~r-r-r-~~~~~.-,

p{fl

! 8=0

~O~--------~--------~---------r----~

Using this normalization,4 we obtain the rcquircd distribution by substituting (19) to (21) in (17) in the generalfor111 ~8~--++~~~-?L-------+---------~----~

( )_ K 2+ 1 , [ _ I+ K 2(B2+ 1+K 2 2)J P p - K p exp 2 2K2 P

The distribution (22) is determined through the parameters Band K according to (19) to (2 1), which in turn are determined through a, 81, and 82 according to (4), (8), and (9) .

The probability densities pep ;B,K ) accordi.ng to (22) for various values of Band K are shown in figLlTes 5 to 9.

The complement of the distribution function, i.e.,

P { p>z;B ,K } = 1-i~(p;B,K)dp (23)

• This normalization differs [rom that used by Beckmanu, [1959 and 1960J.

233

0,' -I-3-------+-~\\---+-----t_-____1

~2'~W------~~----~--+---------_r----~

2 3 -f

FIGU RE 5. P Tobability densities p (p) fOT B = O.

-(0 p{f)

K2 =IO

t 8=05 5 I

0,8 3

2

0,6

0," -+-.t+-----+----\-\\-- --+------+---i M----- 3 -1----\\\

0,2'-+~-----+---~.+-----+--;

2 3

- f FIGURE 6. P robability densities p (p) f or B = 0.5.

p{f) 8= 1

! ~o~-----+~r-~-~----~--~

o,8~-----U~~~~~-----~-_;

O, 6-1-----IIA---'>,~~___'___1-----~-_;

0," -+---HIH--+-;-;

2 3 -f'

FIGURE 7. P robability densities p (p) f or B = 1.

234

1,2

p{fl

14,0

0,8

0,6 -t---_t_----ff_t_----1~---"_+----___1

O'''-t---_t_--~~-t---~mr_+----___1

0,2 -t------t--f-t;rF--t-----t---;;----""i--ti'l:\--------j

2 3 -- f "

FIG URE 8. P robability densities p (p) for B = 2. ~2~,-~~~~-,~~~~~~,-,-~~

p(f)

t4PI-r----1-----++-~---r_-~

O'64------r---~_r--¥~-_t_--~

o"-r----_+-~*ff-_+--~~_+--_;

2 3_ f FIG URE 9. P robability densities p (p) for B = 3.

, \

'r

I 1

:;\

for yal'10US value of 13 and K arc ploUed on proba­bilily paper In fi gure 10 lo 14. 5

, The curves in figs. 5 to 14 " 'ere calculated by direct ullmer' ieal integration of (13) by plulched carcl machine un der the gllidan·~e of Mr. H. Vieh.

• .. ...---, 99.8

~ 995 ~ '9 ,-.....

N .. A

~ 95 a. 90

1 I/O

'" '0 50 W

30

20

"- " "\ '\ '\

~\ '" '" I~ \ '" ~"'-\\ "-

'\"\

/ 8 - 0

8 ",05

"- K2' 1 " I" " "". "" 8'2

1""'- ""'- K "" X "" "- " "- "- 8 = 25

" " "-. " 1/

" " " " " " ," " "-. " "-.

"''' '" "- "- I'-. "-8'3

"X " "'>--"" ""'''''' 10 / ,,"- y"",- :""'- "- "-

I

Cl5

a2 al

Q2

8~f \, /

8 "".5 K. "" " " " "-.

"" " "" " "" '" '" "-. "-. "-.

" " " " '" '" FIGURE 10. Distribution curves oj P ! p> z} Jo r 1\:2= L

,-.....99 -r~~~~~~,-----------

N 98 A

0 95

a. '0 80

70

50

10

I -t-------i- - --,:--&-r-05'--t-------i--­

-- K'-2---

D.2-O.I-+....,...--;-r-1-r....,...-..,..-r-1-r-r-..,.;>.~~--.'>_~h~--~

a' - - 2

FIG U RE 11. Dist1ibntion curves oj P {p > z} fo r K 2= 2.

• 99. 9 •. 6

;It. 99.

~9'

--;::; 98

5

~ 95

--""'- 90 a.

t 80

70

60 50 '0 30

20

10

I

as a 2 a I

'\ '\ '\ "-f\~ "\

'\

\ \ \ \ 1\\ \ \ \ \\ \ \

"" --"'\ "- '\ '\ '\ 1,\ "\

"\ / "\ "\

8 ' 0 ,\. '\ '\ -V\. '" " 8~O5 , ""'~

/"-,. '" 8 : ' "" / 8 ' .s

a.

K" 3 -

"- 8 '2

Y f>( ""

'\ '\ 8 · 2.5

'\ '\ / 1"\ "\

"\ "\ "\

1\ " " 8'3

"" "'- "'- "< f'\-A '" \

~"" 1"'- "'- 1\ ~" " " '\ \. ~ " " " " ~" " " I\. "

- - 2

FIGURl~ 12. Distribution curves ofP{p > zl for 1\:2= 3.

,

235

As lllay be seen .from figures 10 to 14, the distr'i­button becoJll cs practically normal for B "2:.3, K:;' 3, as may also be shown theoretically (cf. appendix) . Th e mcan value and variance in th;s case is given by

(24)

From (18), (9), IUld (20) we have

(25)

Hence

p{-!-->z;B,K ~ =p~--P-->z ;B,K \.. I RM S .) \.. PRM S )

J' Z, ll+B' = 1- 0 ]i (p;13,K )dp· (26)

.. H

*' 99. ~ .. ---;::; 98

~ 95

-..--' ' 0 a.

60

70

60

50 (Q

, 6

5

30

10

5

2

,

'\ '\

'\

\ \ ~\ \ 1\\ \

\"-"\

/ '\ 8'0

8:05

I

a o. o. I I

02

\ '\ \

'\ '\

\ \ 1\ \

\ \ \ \

,'\ '\

'\ '\ ,", '\ '\

""-" " ~~ ~

8 - 1

B · 1.5

1,\ K2' 5 -'\

\ 8 - 2

\ \ / '\

"- \ 8 -2.5

'\ \/ '\ '\ ' '\ '\ V '\

"- \ "- 8 :J

'\ '\ ""

'\ ~ "- \ "-~ '" '" '" '" ............ " " " ""''' '' '' " " ~'t '" " '" I I I

___ Z 5

FI GU RE 13. Distribution curves of P {p > z I for 1\:2 = 5.

99.9 -)9.9

(/. 9 9.5 ~ g;

---;::;-- 98

~ 95

- '0 a. 60

70

50 50

<0

30

10

10

I

as OJ 0 .1

\ \

\

\ , \ \ 1\ \ \

\ \

'""\ 8 =0

8'cii

OJ

\ \ \ \

\

1\ \ 1\ \ \ \ \ \ \ \ \ ~

\ / '\ \ v

'\ \ \ '\

L,\ '\ \

'*''' \ \

~ I "'- \ / ~ A "-

8 =4

~ '" "" "\ / ....... " " 8 : 1.5 ~ ~

K2·/0 -

8 =2

8 "' 25 /

)( -\

\ \ ;!=J

\ '\

"'- "'-" "-. " 1"\

""'-" "-. " , - - 2

FIG1iRE 1 -1. Distribution curves of P{p> zl for K' -o= 10.

Figures 15 to 19 show thi.s r elation plotted on Rayleigh paper (on which the Rayleigh distribution appears as a straight line with 450 slope).

T h e less often r equired statistical distribution of the phase is founi from (13) by integrating over r from 0 t o 00 instead of over 8 from 0 to 211". After a somewhat tedious calculation one obtains

]J (0)

where

G- BK I I + K 2

- -V 2(K 2 cos2 8+ sin2 8) II ncl

vVe now ret mn to th e case when the distribut ions of the phases 1> j are not symmetrical abou t zero . The general formulas (4) to (9) are still valid, but n either {3 in (5) nor the covariance cov (x,y) in (10) will vanish, so that x and y are no longer independ ent and om derivation breaks down from (11) onward. In this case we proceed as follows.

' Ve calculate the covariance cov (x,y) according to (1 0) and determine the correlat ion coefficient

G cov (X, lI)

"';8182 (28)

' Ve now in trod uce n ew coordinate axes x' and y' which are tmned t hrough an angle 1>0 with respect to t h e original axes X, y (fig. 3) . The angle 1>0 is so chosen that th e quantities x' = 1' cos 0' and y ' = 1'

sin 8' are uncorrelated , where

8' = 8- 1>r. (29)

For x ' and y ' to be u.neorrclated (and h ence, as normal random variables, independent), it is suffi­cien t that the two-dimensional distribution liV(x' ,Y' ) h av e an axis of symmetry parallel to on e of th e coor­dinate axes. Since the cmves liV(x ,Y) = const arc concentric ellipses with cen t er X= a, y = {3, it is th er e­fore sufficien t t o choose the x' and y' axes parallel to th e axes of the ellipses. The r equired angle 1>0 th en follows from [Hristow 1961 , p . 125]

tan 2¢0 2G~S:S;

8 [- 82 (30)

In th is n ew coordinate system we th en proceed as b efore ; the only d ifference is that {3' does not vanish now. Inst ead of (13) we therefore have

(31)

3.0

t,f

/'D 0.9

Q8

0,7

46

o,J

0,2

." -.1

~

,, 1 ,

,

- ,

B =

~

l' I R IC E §

.~ of ;- " ; < .. r:x

< 0

oS'

B = 0

..

~

- ..:::,* ~. ~ ....... ~'W> .... ~ .... ~ ""~""~"" ~ "' li "' :t ... ~ ... '<I .... ~ i ... ... ... ~;:... ..... ~

p{ :JIM'; >.} (Il

2

1

o

F I GURE 15. Di stribu tion curves of Plr/ rnMs> zl f or K 2= 1 plotted on Rayleigh paper .

J O

2,0

t, f

I,D

0.. as 0,7

4' ' ,M o,s ,

, o,J

0,2

1

~

~

-

B = 0

2

2

B .0 ..:".

.:::,..

""

F IGURE 16, Distribution curves of P lr jrHM s > zl f or K 2= 2 plotted on R ayleigh pap er,

236

) ,

J.O

~o

1,5

(0

~9

aD 0.7

Q6 '. J> 0.5 , 0.>

o.J

-,'

f-1

b:,2

f-

-

a" 0

)

<Xl

J

I , 2

1

B " 0

"-,;!

FIGU RE 17, Distribllti on curves oj P lr / I'JlMs > zl for J(2= 3 ploUed on Rayleigh pap er.

J.O

;:: B " .0

~o - 1

1,5 --'>. 2

(0 00 .9 .41

41

46 a.» 45 1.+'-

, , I- . ::.i:t"

0.> a .0

.... 4J

." 0.'

FIG URE 18. Distribu tion curves of Plr/ rRMs > zl f 01' I{2 = 5 ploUed on Rayleigh paper.

JO

. ~~" 20 ~ . _0

~I-i\~ , .

~ 5 t-r-; I- )

1.0 I-- I-?" 09

ae 0.7

0.6 ' .J> 0,5

'" 0.>

." o.J

o. 'f

0.'

---f-~1Tl-

, , , ' ;ttP - --' -t-=+::; - --

cUe

~. , . I J ._- - -_ .. -- 1-0- I----

1-- ·~~j~-W~ .~~~-~

n ,"

.1=1;0'

-

J

8 I • o

--

p{_r_ > .l (f)

rRMS ~

FI Gum;; 19. D ist1'ibu tion curves of PI r/ rRMR > zl f0 1' r z= 10, p lotted on Na ylei !J /! pap el ,

where the meaning or (J' , c/ and {3 ' is evidenL ['rom fi.gul'e 3. This integral may agai.n be evaluated as an infi.nite series of Bessel fun ctions (d . appendix), bu L for practical purposes it is usually m ore simple t o perform the num eri.cal integration 0[' (3 1) directly.

3. Special Cases

\Ve now consider some special cases or lhe distribu­tion (22) , In most cases the expression (22) will formally rf'rnain unchanged, but the values or a, 8l ,

and 82 will vary . L et the elementary scattel'erl waves be all 0[' th e

same kind , so that th e amplitudf' and phase distribu­tions are the same for each wave. W'e first assume the amplitudes constant and eq ual to unity . Let the probability density of the phases be symmetrical about zero and equal to w(c/». Then from (4), (8), and (9) we have

(32)

(33)

(34)

Table 1 gives the values of a , 81, and 82 as ealcu­l d from (32) to (34), and also of R, K, and p in

237

accordance with. (19) to (21 ) for the normal , uniform, and Simpson-distributions; the symbol sinc a = (sin a) / a is used in the table.

T ABLE 1.

w(q, ) Normal, (q, ) =0, Uniform from -a to +a Simpson·distributed sta ndard deviation q from - 2a t o +2a

a ne-1j2rr2 n s ine a 11, s ine2 a

8, %(1-e- I1Z )2 ~ (l +sinc2a - 2 sinc' a) ~ (1 +sine' 2a+sinc' a)

82 ~( l -e-'"') ~ (I-sine 2a ) ~ (l-sin ~' 2a )

B 2 e-(I'2 sinc2 a sinc4 a

n--11, I-si no:? a n ---

l-e-0"2 l-sinc~ a

K 2 u2 I -s ine 2a ] -sine:? 2a coth "2 "1 +s in c 2a-2 sinc:? a 1 +sinc' 2a+sinc' a

r' r 2 r ! p'

n(1-c"') ,, (I -s ine' a) n (l -sinc' a)

If the phases of the elementary waves are distrib­uted uniformly over the int erval (- 71',71'), we obtain for a=71' from table 1 a= O, sl=s2=n/2, or R = O, K = 1, p= r/ln; substituting these values in (22) we obtain, as was to be e:\ pected, the normalized R ayleigh distribution

(35)

The Ra~Tleigh distribution is also obtained if q:, is distri buted uniformly over any interval of length 2br (k= I ,2, ... ) or if the varianee of q:, is much greater than 71'2, in which case the distribution of q:, is (with some very unrealistic exceptions) arbitrary. IVe shall call a vector whose elementary components A j exp (iq:,) have their phases uniformly distribut ed over the interval (- 71',71') .a "R ayleigh vector."

The distribution of the amplit ude of the sum of a constant vector and a Rayleigh vector was fo und by Rice [1 944, 1945] and detailedly analyzed by Norton, Vogler , Mansfield, and Short [1955] and Zuhrt [1957]. Direc ting the cons tan t vector V along the x-axis, we immediately have a = V, Sj =sz= n/2, hen ce by (19) to (2 1) R = a/, In, q:, = r/, 'n, K = 1 and on sub­stitu ting these values in (22) we find the normalized Rice distribution

(36)

The distribution of a vector whose x and y compo­nents are distributed normally with mean values zero and unequal variances Sj and S2 was found by Hoyt [1947]. Here we have a = R = O; on substituting in (22) we obtain the normalized Hoyt distribution

I f is evident from (12) and figure 2 that the sum (1) may always be represented as the sum of a C011-

...., stant vec tor a and a H oyt vector H .

IVe next consie! er the case in which the amplitudes I

A j of the elementary waves are random and governed .( by the (same) probabili ty density wA(A). If A j and

q:, j are independent then it follows from the general formulas (4), (8), and (9) that

a = n (A ) f wq,(q:,) cos q:, clq:"

sl=n(AZ) J wq,(q:, ) cos2 q:,clq:,-~

s2=n(A 2) J~q,(q:,) sin2 q:,dq:,.

(38)

(39)

(40)

If, for example, the phases q:, are uniformly dis­tributed over an in terval of length 271', we obtain a = O,

S j =S2=~ n(A 2); substitu ting in (17) we find

( ) _~ _r2/n(A2) . p r - n (.I12) e (41)

Comparing this res ul t with (2) this will be recog­nized as a R ayleigh distribution, in which the num­ber of components n is multiplied by t he mean power 6

of eftch component. If a R ayleigh vector co nsists of components with

different (constant or random) amplitudes A » th en from (4), (8), and (9)

(42)

Subs ti tuting this in (17) and (2 7), cer tain proper­ties of a Rayleigh vector with components of uneq ual but constan t amplitudes postulated by Norton, Vogler, Mansfield, and Short [1955] are immediately

proved as correct (1. P {I'>z}= exp (-z2j± A]) ; )= 1 j

2. e distribu ted uniformls between ° and 271'; 3. r and e independent).

If the random amplitudes and phases of the ele­mentary waves are correlated, formulas (4), (8), and (9) may no t be simplified ; however, it is still true that the mean power of the random component equals the sum of the mean powers of the individual (identically distributed ) elementary components, for in this case we have from (8) and (9)

SI + s2=nJ A2 [J w (A,q:,)dq:, ] dA-~

=TI'JA2WA(A)clA-~=n(A2)-a2 n n

(43)

6 More precisely the mean square of the am plitude . This differs from the mean power by a constant factor F , which in MKS units eq uals 120 .. for propagation in free space. The distinction is immaterial for OUf present purposes an d will be d isregarded ; if the reader objects to this procedure, he may consider (1) to present n sinusoidal vol tages intt?rfering aCrOSS a one-ohm resistor , in wh ich case F equals un ity.

{

I

238

so that by (18)

(44)

Since n is by assumption large, the second term will practically equal a 2 , i.e., the power of the con­stan t component. Thus the power of the random component equals n(A2), or the sum of the mean powers or the individual components.

It is instructive to observe the transition from a purel~' coherent field (mean power equal to '(/,2) to a purely incoherent field (mean power equal to n). This depends on the phase distribution w(¢) ; if the phases are constant, i.e., D {¢ } = 0, then (r2 )=n2, whereas for phase distributions with large variances, i.e., for D {¢ } > > 71"2 the mean power (r2)=n. Thus for example from (18) and table 1 we find for nor­mally distributed pbases

(45)

yielding (r2)=n2 for 0"= 0, but (r2)=n for O"~ 00 (0"» 71"). Si mil arly, for uniformly disLribu Lcd phases we find

agai n yielding (1'2)=n2 for a= O, but r2= n for a~ oo (a> > 71"), and also for a= 2h.

4. Conclusion

The statistical distribution of the ampli tude and phase of a multiply cattered electromagneLic field is equal to the stati tical distribution of the sum of two-dim ensional vectors with random ampli tudes and phases. When these phases are distributed symmetrically, the ampli tude disLribution of the resulting vector is given by (17) or in the normalized form (22), or by the curves of figme 5 to 19; the phase dis tribution is given by (27). In the general case, which includes asymmetrica1 phase distribu­tions, the resulting distribution is given by the in­tegral (31). Various distribution laws of the am­plitudes and phases of the elementary vectors change the values of a, SI, and 82, but not the general form of the above formulas. The distributions derived by Rayleigh [1896], Rice [1944- 5], Hoyt [1947], and Beckmann [1959] are special cases of the above distribution.

The distribution derived here is met, among other cases, in the propagation of radio waves in irregnlar terrain and in tropospheric scatter propagation, since in both cases scattering from rough surfaces is involved. From the above derivation it is seen that the amplitude of a field consisting of very many elementary scattered waves is not necessarily Ray­leigh-distributed (as is often erroneously assumed), but that the Rayleigh distribution, even in its most general form, is met only if the phases of the in­dividual scattered waves are distributed uniformly over an interval of length 271" or in some equivalent way iodicated after equation (35). In practice this

239

will not be the case if, for example, the scattel'ers are distributed in space in such a way that the variance of path lengths between source and point of observation is smaller than one wavelength. Such a case is shown in figure 4, where a rough or turbulent layer is asslllled to be normally distributed about a mean level (N£) with variance (J" and the condition

(47)

holds. This condition is very often satisfied in practice, especially for the longer wavelengths A; experimental measmements of tropospheric propaga­tion beyond the horizon in the meter band [Beck­mann, 1960] have in fact shown distributions as in figmes 15 to 19 more of len than a pure Rayleigh distribution.

5 . Appendix

To evaluate the integral (31), one may use a result derived by Chytil [1961], which after elementary modifications reads

( 2?r u(P,Q,R )= Jo exp (_ P2 cos2 e+ Q co e+ R sin e)cle

= 271"e-f -£, (- I )"'€71J ", (f.2) 12rn(,IQ2+R 2) m = O

cos [ 2m ( arctftn ~) ] (4 )

reducing to the formula (15) derived by Beckmann and Schmelovsky [1958] for R = O. For R = O and large Q(Q>P > > 1) one obtains by addle-point integration [Beckmann and Schmclov ky, 1958]

/271" P 00 (2P)'" u(P,Q,O) ~ -V Q eQ - ~o Am Q (49)

where A 1.3.5 ... (2m - I>. A = 1 (50)

m 2.4.6 . .. 2m ' 0 .

Using (49) to evaluate (14), we find after normal­izing by (19) to (21) for Q> 2P> > 1.

(51)

Now if

(52)

this expression will obviously be negligibly small for all values of p except in the neighborhood of p = B, where the exponential factor will dominate, tbe terms with p (l / 2J-m being either negligible or practically constant in this short interval. But the exponential is that of the normal distribution with mean value

(p)= B and variance D {p}= 1/ C1 + K2). hence for large B (52), the distribution of p beco~es normal. As may be seen from figures 10 to 14 in practice this is the case for B ?:.3, K2 <5. '

6 . References

Beckmann, P., The probability distribu t ion of the sum of n unit vectors with arbitrary phase distributions Acta TechDica CSAV (, 323- 335 (1959). '

Beckmann, P., A generalized Rayleigh distribution aDd its application to ~ropospheric propagation, Electromagnetic Wave PropagatlOn, pp. 445- 449, Academic Press London (1960). . ,

Bec kmann, P. , and Schmelovsky, K. H., Uber ein bei Schwun­derscheinungsuntersuchungen auftretendes Integral, Wiss. Z. HFE Ilmenau 4, 167-171 (1958).

Bernstein, S. N., Generalization of the central limi t theorem of probability t heory for sums of dependent random vari­abl.es (in Russian), Usp. fiz . nauk, No. 10, 55- 114 (1944).

Chyti l, B., D epolar ization by randomly spaced scatterers Prace URE - CSAV, Inst. R ep. No. 20 (1961). '

Gnedenko, B. V., and-A. N. Kolmogorov, Limit distributions for sums of random variables. Russian original: Go­stekhizdat, Moscow-Leningrad 1949. English translation: Wesley Pub!. Co., Cambridge, Mass. (1954) .

Hoyt, R . S., Probability functions for t he modulus and angle I of t he normal complex variate, Bell Syst. T ech. J . 26, ~ 318- 359 (1947). ,

Hristow, W . K., Grundlagen der Wahrscheinlichkeitsrech­nung, p. 125, Verlag Bauwesen, Berlin (1961).

Norton, K. A., L. E. Vogler, W. V. Mansfield and P. J . Short The probability distribution of the amplit~de of a constant vector plus a R ayleigh-distribu ted vector, Proc . LR.E. (3, 1354- 1361 (1955).

Rayleigh, Lord , The Theory of Sound, Sec. 42a, 3d ed., London (1896).

Rice, S. 0., Mathematical analysis of random noise, Bell Syst. .Tech. J. 23, 282- 332 (1944); 2(, 46- 156 (1945).

Richter, H ., Wahrscheinlichkeitsth eorie. Springer Verlag Berlin-Gottingen-Heidelberg (1956). '

Zuhrt, H ., Die SummeDhaufigkeitskurven cler exzentrischen Rayleigh-Verteilung und ihre Anwendung auf Ausbreitungs-messungen, AEU, 11, 478-484 (1957).

(Paper 66D3- 191)

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