NTIA-CR-78-1 

Statistical-Physical Models 

of Man-Made and 

Natural Radio Noise 

PART Ill. FIRST-ORDER 

PROBABILITY MODELS OF 

THE INST ANT ANEOUS AMPLITUDE 

OF CLASS B INTERFERENCE 

contractor reports 

U.S. DEPARTMENT OF COMMERCE • National Telecommunications and Information Administration 
NTIA-CR-78-1 

Statistical-Physical Models 

of Man-Made and 

Natural Radio Noise 

PART Ill. FIRST-ORDER 

PROBABILITY MODELS OF 

THE INST ANT ANEOUS AMPLITUDE 

OF CLASS B INTERFERENCE 

D. Middleton 

U.S. DEPARTMENT OF COMMERCE 

Juanita M. Kreps, Secretary 

Henry Geller. Assistant Secretary-Designate 

for Communications and Information 

June 1978 

PREFACE 

This Report is the thi rd ( i . e. Part III) in a series of ongoi ng studi es 

[Mi ddl eton, 1 974, 1 976] of the general el ectromagneti c (EM ) i nterference 

envi ronment ari sing from man-made and natural EM noise sources , and is al so 

part of the conti nui ng analyti cal and experimental effort whose general aims 

are [Spaul ding and Mi ddl eton, 1 975, 1 978] : 

(1 ). to p rovide q uanti tati ve , statis ti cal descri pti ons of man-made 

and n atu ral el ectromagneti c i nterference ( as in this seri es ) ; 

(2) . to i ndi cate and to gui de experi ment , not only to obtain pertinent data for urban and other EM envi ronments , but al so to 

generate standard procedures and techniques fo r assessing such 

envi ronments ; 

(3 ). to determi ne and predi ct system performance in these general 

el ectromagnti c mi lieux , and to obtain and eval uate optima l 

system structures therein, for 

( a ) . the general purposes of s pectrum management ; 

( b ) . the estab l i s hment of appropri ate data bases thereto ; and 

( c ) . the analysis and eva luation of l arge-scale te l ecorr

ni cati on systems. 

Wi th the aid of (1 ) a nd (2) one can p redi ct the i nterference characteri sti cs 

in sel ected re gi ons of the el ectroma gneti c s pectrum , and with th e resul ts of 

(3 ), rati onal cri teria of perfo rmance ca n be devel oped to predi ct the successful or unsuccess ful operati on of tel ecorr

uni cati on l in ks and systems in 

vari ous cl asses of i nterference . Thus , the combinati on of the results of 

(1 ) -(3 ) prov i de speci fi c, quanti tati ve p rocedures for spectra l management , 

and a rel i able techni cal base for the choi ce and i mpl ementati on of pol i cy 

deci s i ons thereto . 

The man-made EM envi ro nment, and most natu ra l EM noise sources as wel l, 

are bas i cally 11 i mpul sive 1 1 , in the sense that the emi tted waveforms have a 

highly structu red character , with sign i fi cant probabiliti es of l a rge 

iii 
i nterference l evel s. Thi s is noti ceably di fferent from the usual norma l 

( gaussi an) noise processes i nh erent in transmi tting and recei ving el ements . 

This h ighly structured character of the i n terference can drasti cal ly degra de performance of conventi onal sys tems , whi ch a re op timi zed , i . e. 

desi gned to operate most effecti ve ly , against the cus tomari ly assumed 

norma l background no ise processes. The present Report is devoted to the 

prob l ems of ( 1 ), (2) above , namely , to provi de adequate stati s ti cal physi cal 

model s, veri fi ed by experiment , of these genera l "impul sive" , highly nongaussi an i nterference processes , whi ch consti tute a princi pal corpus of the 

interference envi ronment, and whi ch are required in the s uccess ful pursuit of 

(3 ), as wel l. The princi pal new resul ts here a re: 

( i ). Canoni cal ana lyti cal models fo r the fi rst-order statisti cs 

of the i ns tantaneous ampl i tude and i ts magni tude for Class 8 

noise* ; 

(ii ). Procedures fo r obtaini ng the (canonical ) model parameters , 

from the APD1s ( = exceedance probabil ities, P

1

(t>£0), etc . ), 

the ca l cul ati on of moments , and probabil ity densi ty functions 

( pdf's) , and a vari ety of other rel ated stati s ti cs; [see the 

Tabl e of Con tents] . 

Fina l ly, we emphasize, again, that it is the quanti ta tive i nterpl ay 

between the expe ri mental ly establ ished,** ana lyti ca l mo del -buildi ng fo r the 

el ectroma gneti c envi ronment, and the eval uati on of system performance 

therein, whi ch provi des essenti al tool s fo r predi cti on and performance , for 

the deve l opmen t of adequate , appropri ate data bases , procedu res for effecti ve 

s tan dardi zati ons , and sp ectrum assessment, requi red for the effecti ve 

mana gement of the spectral-use envi ronment. 

* Class A and Cl ass B noi se are distinguished , qual i tati vely, by having 

i nput bandwi dths whi ch a re respecti ve ly narrower and broader than that 

of the (linear ) fro nt-end s tages of the typ i cal ( narrow- band) recei ver 

in use . Mo re p reci se definitions are devel oped in the text fo llowi ng. 

** Experi mental co rroboration has been achi eved on the basi s of envel ope 

data for both the Cl ass A and B interference proces5es [cf. Mi ddleton, 

1976 , Secti on 2.4]. An equival ent corroborat i on for the corresponding 

ampl i tude data is accordi ngly i nferred . 

iv 
Table of Conten ts 

Abs tract : 

PART I: INTRODUCT I ON, RESULTS AN D CONCLUS I ONS 

1 . I ntroducti on 

2. Resul ts and Canel usi ons 

Acknowl edgements 

PART II. ANALYS IS 

3 

5 

11 

3. Introd uction to the Analysis 12 

4. I ntegral Forms : P1

( X�X

0

), w

1

(X) , Class A , B ,C I nterference 12 

5. Probab ility Di stri but i ons P

1

( X�X

0

)

B+C ' 0 < a < 2: 23 

5 . 1 Eva l uations of P

l - B- I ,II 

5 . 2 The Composi te Approxi mati on 

5 . 2 . 1 Method 1 . 

5.2 . 2 Method 2 

5.3 Hal l - type Mode ls 

6. Proba bility Densiti es: w

1

(z) 8

: 

7. Remarks on Moments and Parameters : 

Appendi x A. The Approxi mating PD's . 

Gl os sary of Pri nci pal Symbol s. 

References. 

v 

25 

32 

35 

39 

42 

48 

54 

58 

62 

65 
F igure No . 

5. 1 

5. 2 

5. 3 

5. 4 

5. 5 

5. 6 

5. 7 

LIST OF F IGURES 

A A 

The amp l itude d is tr ibution , Pro b( \z\�\z

0

\) I

, 

c a l culated for Clas s B interferenc e for a = 1.0, 

for var ious value s of the Ind ex A , from 

a 

Eqs . ( 5. l l b), ( 5. 1 5) . 

A A 

The amp l itude d is tr ibut io n , Pro b( \z\>\z \) I - , 

0 

calculated for Clas s B int erference for 

A

a 

= 1.0, for var ious va lues of a, from 

Eqs . ( 5. l l b), ( 5.1 5) . 

The amplitude d istribut ion , Pro b( \z\>\z - \ 

0 

( �z

0B

) ) II' calculated for Class B interferen c e 

for AB = 1.0, a= 1.0, for var ious values of 

rB from E q . ( 5. 1 7 ) . 

The amplitude d is tr ibut ion , Pro b( \z\>\z - \> 0 -

( z0B) ) II' calculated

-

�or Clas s B int erfer enc e 

for a= 1.0, rB = 10 , for var ious values of 

AB, from E q . ( 5. 1 7 ) . 

S chema of P

l

-

B

' Eq. ( 5. 27b), o btained by jo in ing 

the t wo appro x imat ing forms ( 5. l l b), ( 5. 1 7 ) , a c ­

cord ing to the procedures ( 5. 2 2 ) , Me thod 1, or 

( 5. 2 6 ) , Me thod 2. 

The ( complete) amp l itude d istribut ion P1( \z\� 

\z0\) B, E q . ( 5. 27b), c a l culated for Class B in ­

terference for var ious A , g iven a, from 

a 

Eqs . ( 5. l l b) , ( 5. 1 7 ) , ac c ording to the pro -

cedu res of ( 5. 2 6 ) . 

The (complete) amp l itude d istribut ion P1( \z\� 

\z

0

\) B

, E q . ( 5. 27b) , calculated for Class B in ­

t erferenc e for var ious a, g iven A , from 

a 

Eqs . ( 5. l l b), ( 5. 1 7) accord ing to the procedures 

of ( 5. 2 6 ) . 

vi 

30 

31 

33 

34 

38 

43 

44 
F igure No . 

6.1 

6 . 2 

A - 1 

LIST OF F IGURES 

(con t inued) 

The (complete) pdf w

1

( z)

B

, (6. 3 ) , O<z<00 of the 

instantaneous amp l itude for Class B int erfer ­

e nc e , calculated from Eqs . (6. ?a ) , (6. 8a) , for 

var iou s A

a, g iven a, [ S ee F ig. 5. 6 for the assoc iated PD and parameter values .] 

The (complete) pdf w

1

( z)

B

, (6.3 ) , O<z<00, of the 

ins tantaneous amplitude for C lass B interfere nc e , calcu l ated from Eqs . ( 6 . 7a) , ( 6 . 8a) , for 

var ious a, g iven A

a. [ S ee F ig. 5. 7 for the a s ­

s oc iated PD and p ar ameter v a lue s .] 

S chema of t he pdf's , w

1

( z)

B- I II' ap pr o pr iate 

' 

to the regions B- I, II. 

v ii 

52 

53 

61 
STATIST I CAL-PHYSICAL MODELS OF MAN- MADE AND NATURAL RAD IO NO I SE 

Pa rt III. Fi rst- Order Probabi l i ty Model s of the I nstantaneous 

Ampl i tude of Cl ass B I nterference* 

by 

David Mi ddl eton** 

ABSTRACT 

This Report is the thi rd (

11Part III11 ) in a co ntinui ng seri es devoted 

to th e devel opme nt of analytical ly tractab le, stati s ti ca l -physi cal model s 

of man-made a nd natural el ectromagneti c i nterference . Here , the fi rs t-order 

s tati sti ca l probab ility densi ti es (pd f's) and the associ ated exceedance probabi liti es ( PD1s , or APD's ) are obta ined for the i nstantaneous ampl i tudes 

(X), a nd instantaneous magni tudes , I XI, of Cl ass B noi se. These a re needed 

not only for experi men tal s tudi es but, al so , pa rti cul arly for the analysis 

and eva luati on of the performance of optimum and s ubop timum recei vers in 

Class B i nterfe rence e nvi ronments . 

As in the earl i er studi es of the envel ope stati s ti cs of Cl ass B noi se 

[Mi ddl eton, 1 976] , a two-functi on approxi mati on is needed for the cha racteri stic functi on and hence for the correspondi ng pdf's and PD's . Two methods 

of determi ni ng the six ( basic) parameters whi ch descri be these fi rst-order 

stati sti cs and thus j oi ni ng the approximate forms ( pdf's and PD's ) are 

outl i ned . Met hod l is approx i mate , was used earl i er [Mi dd l eton, 1 976 , 1 977] , 

and has the adva ntage of s omewhat greater compu tati onal simpl i ci ty, wi th the 

d isadvantage , however , of yi elding too l ow va l ues of the PD at l ow values 

of the argument (X), when the gaussian component is smal l. Method 2 is 

11 exact11 , and somewhat mo re compl ex computati onal ly. The joining process 

i nvol ved in both methods has been essenti all y descri bed earl i er [Mi ddl eton, 

* This is the third in a continui ng seri es of Repo rts devel oping the fi rsto rder (and l ater , vari ous higher-order) stati sti cs of natural and manmade rad io noi se. Earl i er studi es in this seri es a re: Pa rt I- OT Repo rt 

74- 36 , Apri l , 1 974 ; Pa rt I I-OT Repo rt 76-86 , Apri l, 1 976 . 

** The a uthor is under co ntract wi th the U.S. Department of Commerce , 

Nati onal Tel ecommunicati ons and I nformati on Admi nistrati on, Insti tute for 

Tel ecommuni cation Sci ences , Boul der, Col orado 80303. 
1976 , 1977] bu t is devel oped further here . The basic parameters are , in 

any case, the s ame as those derived for the envel ope stati sti cs . The 

excel l ent agreement wi th experi ment observed fo r the envel ope data accord ingly appl i es here , as wel l . 

2 
STAT IST I CAL-PHYSI CAL MODELS OF MAN-MADE AND NATURAL RAD IO NOISE 

Pa rt III. Fi rst- Order Probab ility Model s of the Instantaneous 

Ampl i tude of Cl ass B Interference* 

by 

Dav id Mi ddl eton** 

PART I: I NTRODUCT ION , RESULTS AND CONCLUSI ONS 

1. I ntroducti on: 

Agai n, as in earl i er studi es ( for exampl e, [Mi ddl eton , l 972a , 1 972b , 

1 973 , 1 974 , 1 976 , 1 977]), our central probl em here is to construct analyti cal ly tractabl e model s of man-made and natural radio noise, based on the 

pertinent s tatis ti cal-physics and possessi ng a canoni cal s tructure , i nvari ant 

of a ny parti cul ar source mechanisms . This is done for three princ i pal 

reas ons : 

( i ). To provi de real i sti c, q uanti tati ve descripti ons of man-made 

and natura l electroma gneti c ( EM) i n terference envi ronmen ts ; 

(ii ). To s peci fy and guide experiments fo r measuring s uch i nterference 

envi ronments ; and , 

(iii ). To determi ne the structure of optimal communi cation systems and 

to eval uate and compare thei r performan ce wi th that of specifi ed 

s ubopti mum systems, when cperating in these genera l classes 

of EM i nterferen ce . 

These three tasks , in turn, are central features of any adequate program of 

spectrum management (fo r exampl e, [Mi ddl eton, 1 975a] ) . 

* Thi s is the thi rd in a continuing series of Reports devel opi ng the fi rstorder ( and l ater , vari ous h i gher-order) stati s ti cs of natural and man-made 

rad io noi s e. Earl i er studi es in this seri es a re: Part I - OT Report 74-36 , 

Apri l , 1 974 ; Part I I- OT Report 76-86 , Apri l , 1 976 . 

** The author is under contract wi th the U.S. Department of Commerce , 

Nati onal Tel ecommuni cati ons a nd I nformati on Admin istrati on, I nstitute for 

Tel ecommuni cati on Sc i ences , Boul der, Col orado 80303 . 

3 
Accordingly, our aim, as in prev ious s tudi es in this seri es [cf . 

Mi ddl eton, Part I, 1 974 ; Part II, 1 976] is to p rovi de anal yti cal model s, whi ch: 

( 1 ). combi ne the appropri ate physi cal and statis ti ca l descri pti ons 

of general EM i nterference envi ronmen ts ; 

(2) . are anal yti cal ly manageabl e; 

(3) . possess canonical properties, i . e . , a re not speci al i zed to i ndiv i dual noi se mechani sms , source dis tributi ons , and emission 

waveforms , propagati on l aws , etc . ; and , mos t i mportant , 

( 4) . are experimen ta l ly veri fyabl e and predi cti ve . 

In addi tion, the basic parameters of s uch stati sti cal -physi cal model s must 

be measurab le quanti ti es with a speci fi ed physi cal structure and i nterpretati on. 

To achi eve the above is cl early a non-tri vial- problem, as we have 

noted befo re [Mi ddl eton, l 972a , 1 974 , 1 976] , chiefl y because of the i nherent 

h i ghly nongaussi an nature of these random processes , a characteris ti c whi ch 

at once i nsures compl ex descri pti ons and ensuing diffi cul ties for the 

analysis of sys tem performance . That these diffi cul ti es can be effecti vely 

overcome for mo del-bui l di ng and experimenta l veri fi cati on has been demonstrated by recent efforts [Mi ddl eton, 1 976 , 1 977] . For rece i ver desi gn and 

performance, using these mo del s, we cite the recent work of Spaul ding and 

Mi ddl eton [l 975 , 1 977] , whi ch establishes the appl i cabi l i ty of the canoni cal 

model s to recei ver eval uati on in these EM envi ronments . 

Speci fi cal ly, earl ier work here has focused on ampl i tude and enve l ope 

stati s ti cs: we note 

( i ). Mi ddl eton, [1 974] : Fi rst-order statis ti cs of the i nstantaneous 

ampl i tude (X), Class A noi se ; 

(ii ). Mi ddl eton, [1 976] : First-order stati stics of th e enve l ope (E) 

and ph ase (�) , Cl ass A and Class B no ise; 

The p resent effort (Part III, 1977) is devoted to the corres pond i ng statis ti cs 

of the i nstantaneous ampl i tude (X), fo r Cl ass B i nterference , whi le Part IV 

(in preparati on) extends the above to the genera l Class C ( = Cl ass A+ Class B) 

4 
envi ronme nts.* Still earl i er, i mportant rel ated studies have been carri ed 

out ( essenti al ly for what is now cal l ed Class B no ise) by Furutsu and 

Ishi da [1 960] , and Giordano [1 970 , 1 972] and Haber [1 972] , whi ch , however , 

a re not canoni cal and which focus primari ly on atmospheri c noise . The 

pri nci pal new resul ts obta i ned here are the probaoi lity distri buti ons ( PD's) 

and probabi lity densities ( pdf's ) , moments , and characteristic functi ons 

(c. f. 's ) for the Cl ass B i nstantaneous ampl i tudes (cf. analytical deta i ls 

in Pa rt II ff. ). The analysis and approach here is s tructured di rectly 

upon that of Pa rt II [Mi ddl eton , 1 976] , wi th appropri ate modi fi cati ons . 

Thus , the present Report is organi zed as fol l ows : Sect i on 2 conta i ns a 

concise acco unt of the p ri nci pa l resul ts; Secti ons 3-7 comprise the body 

of Part II here , vi z . , the anal yt ical deta i ls. These are : Section 3 , an 

i ntroducti on to the analysis ; Section 4, i n tegral forms for the PD , 

P

1

( X.:X0) and pdf, w

1

(X) , for Cl ass A, B, and C i nterference ; Secti on 5, 

the PD's , P1

( X.:X0

) for C l ass B noi se , wi th particular attenti on to an 

(earl ier) app roxi mate method ( Method 1) of obtai ning the di stri bution 

paramete rs , and an exact method ( Method 2 ) , whi ch is pertinent for al l Class 

B mo del s� whi l e Secti on 6 gi ves the p df1s, w

1

(x) 8

. Section 7 comp l etes 

the Report wi th remarks on mome nts and mode l parameters . 

2. Resul ts and Concl usi ons : 

Let us now s ummari ze the princi pal resul ts of thi s study in a concise 

fashi on. The analyti c detai ls and rep resentati ve ca l cul ati ons ( and Fi gures ) 

for the PD's and pdf's a re deve l oped and presented in Pa rt II fol l owi ng . 

Th e main anal ytic results a re: 

Characteri s ti c functi ons (c. f. 's ): 

Aa

l la 2A2 2

A ; 

• 

- b

1 aA

8

a A -�oG

a A 2 • . 

F

1

( 1aA)

B-I::

 e (=F1

(1aA)

B-I )

' jz

0

j, jzj jz

08

j ,

, ( Eq • ( 5. 4a ) 

j 

(2. la) 

* See Section 1 .1 (Part I), Mi ddl eton [1 976] , or Mi ddl eton [1 977] for 

detai l ed defi ni ti ons of Cl ass A, B, and C noi se ( as seen by the recei ver) . 

Essential ly, Cl ass A sources are characteri zed by having emi ssi on spectra 

narrower than the recei ver bandpass , while Class B sources are spectral ly 

broader than the rece i ver bandpass. 

· 

5 
A -

I - l /2 where a = {n28

(l+r8

) , cf . (5. 2 ) . Here z, z

0

, z08 a re normal i zed 

ampl i tu des , e.g . 

The corresponding PD 's are 

p

l ( Z2:_Zo

) = p

l ( Z2:_Z

o

)

B- I

. 

· {-z <Z <Z } ) (2.2a) 

oB o oB 

B-I I: {z <- z 8

,z >z B

} 0 0 0 0 

(2.16) 

(2. lc) 

P

1

( i z i 2:. i z

0

i ) = P

1

( i z i 2:. i z

o

i )

B- I: O<iz

0

i.::_z

08

(>0) (2 2b ) 

---------

B- r r 

I 

z

o

I >z

oB

. 

) •. �J 

Wi th 1

r

1 and e respecti vel y the usual confl uent hypergeometri c functi on and 

error i ntegra l , we have from (5. 11 ) fo r the P

l - B-I: 

6 

(-l )

n 

(

na.+l

)

An 

n! r 2 a. 

(2.3a) 

(2.3b) 
For the P

l -B-I I ' we have from (5.1 9): 

Here, specifically, we have 

(2.4b) 

r 1+4-a 

" a a " "' 2 B 2-a A

a 

= A/2 GB

; z0B = z0,

B

N /2 2 GB

; GB

= 4(l+rB) , Eq. (5. 20p) ; 

(I z0B I >O , cf. p. 37 • ) 

(2;�B

=) ��B = (m/AB

+r8)/(l+r8) ; AB

= (t�)AB ' cf. (5.7c) ; 

[A , cf. Eq. (5. l lc) 

a 

a , cf. Eq. (4. 24b)]. 

The associ ated pdf1s , w1

(z)B ' are from (6. 7) and (6. 8) : 

(-�oB

<�<�o

B

)

' ; = zNI

/2/2 GB } �

o

B = 2

o

B

N I

/ 212 GB 

(2.6a) 

7 

(2.5) 
I 

I 

w

l (

I z I ) B- I

' II = 2wl ( z) B-I

' II ' z > 0. 

(z < -z

o

B ' z > z

o

B 

(2. 5b) 

(2. 5c) 

Figures 5. 1, 5. 2; 5.3, 5.4 are respectively analogous to Figs. 3. 3, 

3.4, and 3. 1, 3.2 [Middleton, 1 976] and are cal culated: Figs. 5. 1, 

5.2 from Eq. (2. 3) above [or Eq. (5.11 ) ] ; Fi gs. 5.3, 5.4 from Eq. (2. 4) 

above [or Eqs. (5.17), (5. 1 9)]. The composite PD and pdf, e.g. 

P

1

( / z / �/ z0 / ) = P

l

(z�z

o

)B-I = P

1

( /

�/ �/

�

o

/ )

1 ' (0.:

/z/ , / z0 / < / z08 / ) ; I 

and 

= 

w

l (z)B 

= �

l (�)

I 

p 1 ( I z I �1 z

o I ) B-II ' 

A 

l

� o .:

 I z I < os l 

= w

l (z)B-II , I z I > l 20s l 

I z

o /�I z

oB I (>O) ) 

(2. 7a) 

} (2. 7b) 

are il l ustrated, respectivel y, in Figs. 5.6, 5. 7 [analogous to Figs. 3.6, 

3. 7, Middleton [1 976]] , and in Figs. 6.3, 6. 2, which are l i kewise analogous 

to Fi gs. 4.3, 4.4 [Middleton,1 976]. The composite PD's exhibit the characteristic 11bend-over11 at the smal ler probabilities (and larger arguments , 

/ z0

/, etc. ). Fig. (5. 5) ill ustrates the composite, approximating procedure 

schematical l y, cf. Sec. (5. 2)ff. For smal l / z0 / the gaussian character of 

the PD's is exhibited, while for the l arger val ues (and correspondingl y 

smal l values of P1) the strong departures from gauss appear. Simil ar observations are noted for the pdf' s, Figs. 6. 1, 6. 2. 

8 
The global and generic parameters for these ampl i tude di s tributions 

are p recisely the same as for the envelope dis tributions studi ed earl i er 

[Mi ddl eton, 1 976 , Secs . 2.2, 2.3, 2 . 5], in these Cl ass B cases . We revi ew 

them concise ly, for conveni ence , here : 

( 1 ). A

B = the Impul sive Index , for the "Cl ass A-form" of approx imation 

( l zl>z0B

), used in the overal l Class B model . The Impul sive 

I ndex is a measure of the nongaussi an nature of the noi se : the 

smal l er A

B

' the more "i mpul sive" the i nterference ; 

(2) . rB : o�/ Q2B = rat io of i ntens ity of the i ndependent gaussian componen t, 

a

G

, of the i nput i nterference (incl udi ng "front-end " rece i ver 

noise) to the i ntensity, Q

2B , of the impul sive component , in 

the l arge-amp l i tude approximation pdf ; 

(3) . Q2B = the i ntensi ty of the above-menti oned i mpul sive ( "Class A- form" ) 

component; 

A 

( 4 ) . 

A 

= 

2r(l- a/2) A /( B

OB )

a) a 2

3af2r(l +a/2 ) G

a B \ /2Q (l +r 1) B 2B B 

( 5). a 

( 5.l l c), an "effective" Impul sive Index , propo rtional to the 

I mpu l sive I ndex A

B

' which depends on the generic parameter a; 

= (

2- μ

) f

; (3- μ

) 1 = spatial densi ty-propagati on parameter, cf. 

y sur y vo 

( 4 . 24h ) ; re l ati ng the exponents μ

, y of the range power-l aw 

dependenci es of source density and propagation l aw, respectively, 

cf. (4. 24f ,g ) ; 

(6). N

I = scal i ng factor which i ns ures that P

l -B-I

' w

l -B-I are properly 

joi ned to p

l -B-I I

' w

l -B-I I

; 

(7). z

0B = z

0B

/h28

(l +r8) : the (norma l i zed ) 11bend-over11 poi nt, at whi ch 

the two approx i mating fo rms of pdf ( and PD , for Jz l � 0) are 

joi ned , according to the proced ures di scussed in Sec. (5.2) ff. 

This is an empi ri cal ly determined poi nt , cf. Fig. 5.5. 

9 
As before , in the case of Class B envelope di stributions , we have the fol ­

l owing (same ) sets of global and generi c model parameters : 

Class B: Gl obal : 

Generic: 

A 

A

a

(or A

a

) , a ,A8

,rB,�2

8

, NI 

2 A A2 A

B

,a ,crG

'<

B�B

) ,(B

oB)'N

I } (2.8 ) 

It is i mportant to stress the fact that (except for z

08

) al l model paramete rs 

are physical ly structu red and hence a re canoni cal in form : they are not ad 

hoc quanti ti es. [We remark, also, that N

I is not ful l y depende nt on the other 

generic parameters and is i ndependent of z

08

. Thus , NI may be rega rded as 

generic.] We include two methods , one approxi mate and one exact [cf. Secs . 

5.2-1 , 2] for obtaini ng the model parameters . 

Fi nal ly, a l though we have no readi ly obtai nabl e di rect experimental 

comparisons vi s-a-vi s theory for the i nstantaneous ampl i tude, z, or i ts magni tude , iz l, we have i nfe rentia l ly, demons trated excel l ent agreement between 

theory and experiment on the basis of the much mo re eas i ly obtainabl e 

envelope data [cf. Figs . (2. 1 ) - (2.8) , Mi ddl eton, [1 976]]. Si nce the model s 

are basi cal ly the same , we can accordingly regard the experi mental agreements , 

canonical ly achi eved over a vari ety of disparate (Cl ass B) sources , as 

confi rmi ng the adequacy of our model , al so canonical l y. Because Class B 

mo del s contain the a-parameter they are al so capabl e of provi di ng mo re 

s tructural i nfo rmati on regardi ng thes e i nterference mechani sms , viz . , average 

source densities (in space ) for given power l aws of propagation. [This poi nt 

is discussed more ful ly, p.82, Mi ddl eton [1 976]. ] Again , Hal l - type model s 

are deriveabl e from the B-I cl ass of approxi mati ons [iz

0

l<lz08i) , when the 

addi tive gaussian component is negl i gi ble, cf . Section (5.3) below. Accordingly, wi th this Report we have accompl i s hed I tem 1 of 11Next Steps" 

(cf. Sec. 2 . 7-I, Mi ddl eton, [ ( 1976), and wi th current efforts ( FY ' 78) we 

are underway on Item 2: fi rst-order stati stics of Class C i nterference , for 

various combi nat i ons of the Class A and Class B components which can comprise 

a typi cal EM i nterference envi ronment. These resul ts , in turn , are to 

be employed in the eva l uation of the performance of optimum and subopti mum 

receivers in s uch envi ronmen ts , a long the l ines a l ready developed by Spaul d i ng 

and Mi ddl eton [1975, 1 977]. 

10 
Acknowl edgments : 

The author wishes to thank Dr. A.O. Spaulding of the National Tel ecommunications and Information Admin i strati on ( ITS-Boul der) for hi s constructive comments and cri ticisms , and for hi s supervisi on of the numeri cal calculations for the Figures . 

11 
PAR T II. ANALYSIS 

3. I ntroducti on to th e Anal ys i s: 

Part II of this Report , as in Ref. 11 earl i er, is devoted to the derivation of the fi rst-orde r i nstantaneous ampl i tude s tatis ti cs w1

( X) , P1

( X_:X0

), 

and associated moments <I X!

s

), primari ly for our stati sti cal -physi cal model 

of Class B i n terference . The analysi s para l l ells much of Sections 2-7 

( Part II) of Ref. 1 1 , starting with the basic poi sson model which describes 

these broad classes of i nterference . We are abl e here to take advantage of 

our previous results fo r the envelope statis ti cs for Class A, B, 

, 

and C noi se general ly ,* and Class B in particular, usi ng the methods developed 

in Sec. 2 (Part II), Ref . 1 1 , to obtain fi rst the needed characteristic functions and their approximating forms . Our pri ncipal attention is then di rected 

to the Cl ass B case . [ For a full account of the physi cal reasoning underlying 

the approxi mations , see Sec. 2 ( I I), Ref. 1 1 .] 

The material of Part II here is organi zed as fol l ows : in Sect i on 4 

be l ow we obta in the general ( integral ) express ions for the desi red PD 's and 

pdf's of the i nstantaneous ampl i tudes , X, for Class A, B, and C i nterference .* 

Then , in Section 5 we special i ze these resul ts to Class B noise and eval uate 

the various integral forms i nvol ved , for P1(X_:X0)8. Correspond i ng resul ts 

for the pdf, w

1

(x)8

, are derived· in Secti on 6. Sect i on 7 presents an evaluation of the general fi rs t-order moments (!Xis), s _:

 O; and offers rema rks on 

the global and generic parameters of the model , al ong the l ines of Sec. 

6 (II) , Ref. 1 1 . Appendi x A concludes the analysis wi th the derivation of 

the 11joi ni ng11 condi tions on the PD 1 s. 

4. I ntegral Forms : P

1

(X_�:_x0

) �1(X) , Cl ass A, B, C I nterference : 

Our s tarting poi nt fo r th e present anal ysis is the basic poi sson model 

[Eq. (2. 1), Ref. 1 1] for the fi rst-order characteristic function of the i nstantaneous ampl i tude , X, of the recei ved i nterference process ( at the i nput 

to the IF stage of the typi cal narrow-band rece i ver), viz: 

* Class C model s are to be treated in detai l in a subsequent Report in this 

seri es (e. g . , Part I V, 1 978). 

12 
Here e is an epoch indicating, vis- a-vis the receiver 1 s time t, when a typi cal 

s ource may emi t. [ The basic model here , as before , consi s ts of an i n finite 

number of poten tial ly i nterferi ng sources in a source domain , A, and while 

the bas i c wave forms emi tted al l have the same form , U, thei r scale, du ration, 

frequency , etc. may be ran domly distributed . Th e fundamental postulate of 

this basic inte rference mo del is that : the l ocat i ons in space , and emi ssion 

epochs , in time, are i ndependent , and hence poi sson distributed , cf. ( 4 . 1 ) ] 

The \ = (\,e, ¢) are coordi nates , or a vector magni tude , appropriate to the ""' 

geometry of the source fi el d, in A, and of the receiver, wi th .sl,\ (= d\d¢) for 

a s urface di s tribution , an d vv 

d\ (= d\ded¢) fo r a vol ume element . The quanti ty 

p(1,€) is the 11process densi ty 1 1 of this point space- time poi sson i nterference 

process, e.g. , p .:_O (

'V probabi l i ty densi ty ) . The average <>e denotes a 

s tati s ti cal average o ver various random pa rameters which may be perti nen t to 

the particular model , e.g. , dopp l er, signal dura tion, ampl i tude , etc. Final ly, 

U(t ;\, ... ) is the typical waveform� in the receiver , fol lowing the ( l inear) vv 

aperture x RF x IF s tages (i.e. , ARI stages ) of our narrow-band receiver , 

and X(t) is the resul tant recei ver p rocess, given by 

X(t) = [ U ( tJ �) dN( �) 

Z=Axe 

(4. 2) 

,, vv "" 

where the {dN} form a (usual ly zero-mean) poi sson process (in lX�), i . e . , in 

( \xe) , such that (4. 1) is the fi rs t-order cha racteri stic function. 

"" .,., 

The desi red PD 1s and pdf1s (probabi l i ty distributions and probabili ty 

densi ty functions ) are fo rmal ly 

(4.3 ) 

We can rewri te the express ion for P1 in the fol l owing form , more conveni ent 

for eva luation: 

*A further gen era l i zation is possi .bl e when the waveform i tsel f (U) is regarded as random : the averages ( >e 1

, become< )

e 1 ,U' cf. ( 4 . 1 ), etc. 

V\I CV 

13 
Usi ng w1

(X) , (4.3) , in this gives 

x 

J

oo d f 1 

p

l (Xl

>X�X

o

) = F

l (i�) x 2; -oo x 0 

00 

-i�X

0 -i�x

1 

_ J ( . ) [

e -e J d� 

- F

l 1� 

-00 

X i� 2TI" ' 

(4. 3a) 

(4.4) 

and since X(t) is an i nstantaneous ampl i tude of a narrow-band process , 

X i s a zero mean ((X) = 0) , s ymmetrical process about X = 0, e.g., w

1

( X) 

= w

1

(-X) and F

1

(i�) X = F

1

(-i � ) X

' i . e . , the c.f. is l i kewi se even i n �­

Equati on (4. 4) accordi ngly can be rewri tten as 

(4.4a) 

(4.46 ) 

so that 

(4. 5) 

since ( heuristi cally) 

= � cS(�- 0) (a 1 ways) , (4.5a) 

and J�_cS(y-O)F(y)dy = F(O) . [Note , as expected , that if X

0 

= 0, P

1 (X�O ) 

= 1/2, while X

0 

+ -00 , P1

(X�-00) = 1/2- (-1/2) = 1 , from (4.5) , al so as requi red.] 

Similarly, for the mo re special case P

1 (IX l �IX0

j) , we have di rectly 

here [w1

( x) = w

1

(-X)]: 

14 
F

1 (i�) X

d� . 

(4. 5b) 

Fi na l ly, fo r these symmetrical distributions , we obtain from (4. 3) 

(4.6) 

Both (4. 5 ) , (4. 5b) , and (4. 6) wi ll be employed here in the subsequent analys is. 

Our principal task, accordi ngly , is to apply (4. 1 ) to these relations , to obtain analytical ly tractabl e expressions for the various PD 1 s and pdf 1 s of X , 

as in our earl i er studies [Mi ddl eton, 1 974 , 1 976]. 

Our next s tep is to use (2. 1 0) , Ref. 11 for the basic waveform , U, in the 

recei ver viz: 

where B

0 (� 0) is an enve lope (cf. (4 .lOa ) ) and 'l' is a phase , wi th the form 

(4 . 7a) 

Here �

S

' ¢

r

, ¢

R are respecti vely the typi cal source phase, and the phase 

angl es of the (complex) beam patterns of the source ( T) and recei ver (R) , 

[cf . Sec. 2 . 5, (I I ), Ref . 1 1 ]. The quanti ty ed (=μ

d

-1 ) is the s um of the 

re l ative doppl ers between a source and the receiver , and is al ways smal l 

O(l o

-5 or l ess ) in our app l ications , viz. e

d = 2v/c = 0 ( 1 0

-6

) for v=l 05 mph, 

so that B0 is i ndependent of e

d

. I ns ert�ng (4. 7) into (4. 1 ) and applying the 

na rrow-band condi tions, e . g., fol lowing the procedure on pp. 44 , 45, Ref. 11 

( or pp. 24 , 25, Ref. 9 ), we obtain for the c.f. 

(4.8) 

15 
which is sti ll essential ly exact . (Here J0 is a Bessel function of the 

fi rst-kind , order zero, as before .) 

Our next step is to fol l ow the analysis [pp. 46-50 , Ref. 11], making 

the fol l owing not very restrictive assumptions : 

( i ). average no. of emi ssions per unit domain (dA) and per i n terval 

de in ( O ,T) a re i ndependent [(2. 25 ) I I , Ref. 11]; 

(ii ). " local stationari ty": no changes in average source numbers 

and emi ss ion propert i es during the observation period ( O ,T) , 

and the emission epoch probabi lity ( densi ty) w

1

( €) is uni form 

in al l i nterval s ; 

( i ii). i deal i zed 11s teady-state 11 condi tion ( T-+«>) hol ds . 

The resul t is that (4.8 ) is now reduced to the basic form 

(4.9) 

where speci fical ly, 

( 4. 1 Oa) 

A0 = (peak) ampl i tude of the rece i ved envelope (at output of the IF stage) 

(4. l Ob) 

e =a l i mi ti ng vol tage setti ng (in suitabl e dimensi ons ), at whi ch the Oy 

recei ver wi ll respond to a test signal , above recei ver noise, at 

the output of the IF [cf. (2.34 ) , II, Ref. 11]; 

( 4. 1 Oc) 

u

0

(z) = normal i zed e�vel ope wave- form at IF output ; (z= t/Ts

); 

= 0, z > Ts

/T

s

' z < O ; (4. l Od) 

16 
T

s = duration of typical signal from an i nterfering s ource 

= mean durat i on of typi cal s ignal from an i nterfering 

source 

e1 = oth er random parameters of i nterference wavefo rm (other JV 

(4. l Oe) 

( 4. l Of ) 

( 4. l Og) 

g(A) = a geometrical factor, which describes the propagation l aw, 

"" 

from sou rce to recei ver (in thei r mutual-far fi el ds); (4. l Oh) 

a:.T, t2R=(compl ex) beam patterns of typi cal source and the receiver . 

( 4. 1 Qi ) 

[Note that (4. 9) is forma l ly i dentical to exp [Eq. (2.38 ) , Mi ddl eton , 1 976] , 

wi th � replaci ng r therin.] 

The quanti ty A is the Impul sive Index ( as defined in (2.37 ) , (2. 38 ) II 

00 

( Mi ddl eton , 1 976) ). As noted in our earl i er s tudi es [Mi ddl eton , 1 972b, 1 973, 

1 974 , 1 976] , the I mpul sive Index is a measure of the temporal overlap or 

densi ty , at any i ns tant , of the s uperposed i nterference wavefo rms at the recei ve r 's IF output . It is a key parameter of the i n terference model because 

it cri tical ly i nfl uences the form of the pdf1s and PD1 s of the i n terference , 

as observed at the o ut put of the i nit i al ( l i near) stages of our typical 

narrow-band recei ver . Then , for smal l val ues of A the stati stics of the 00 

resu l tant o uput wave a re domi nated by the overlappi ng of comparatively few , 

dete rmi ni stic wavefo rms , of di fferent ampl i tudes and durations. The resul ting 

interference has then an 11i mpul si ve11 , structured character . For i ncreasi ngly 

l a rge val ues of A this resul tant approaches a normal (or gaussian) process 00 

as one woul d expect from the Central Li mit Theorem (CL T) [Mi ddl eton, 1 960, Sec. 

7 . 7], cf. Eq. 4. 1 9 fol l owing. 

We a re now ready to use the p rocedu res and resul ts of Sections 2.3, 

2 . 4 (II) , [Mi ddl eton , 1 976] to obtain expressi ons for F1

( i �)

X when the 

i n terference bel ongs to Class A, B, or C types . Si nce r (therei n ) -+ �( here ) , 

we have at o nce , for the general Cl ass C case : 

17 
-

z ( =T IT ) <oo 

F

l

( i �)

C = exp A

A � 

o s s 

[J (�

B

oA)-l ]dz)z A e• 

l �o 

0 

o

' u'V'v-1 

= 

where the i mpul sive Indexes A

A

, A8 are speci fical ly 

and 

(4. 1 1 ) 

( 4. l la) 

(4. 1 2) 

in whi ch A

A

= f A

P(1)�A = av. no. of emi tting sources in A. Note that 

B

0A 1 B

08

, cf. ( 4.lOa ) , through the waveform u0

(z) , e.g. u

0

(z)8

f0 (0 ..: z < 00) , 

whereas u

0

( z)

A = 0 outsi de (0, z0 < 00) , cf. Fig. 2. l ( II), Mi ddl eton , 1976] 

p. 53. 

In general , there is al ways an accompanying gaussi an background , whi ch 

arises from a number of mechani sms : 

( i ). as sys tem noi se in the recei ver ; 

(ii ). as external i nterference , whi ch is the resul tant of many 

. ._ . i ndependent sources , none of which is domi nati ng v1s-a-v1s 

the others (so that the CL T appl i es ) ; 

(iii ). as a mi xtu re of ( i ) and ( i i ) . (4.13) 

This component of the i nterference is al so i ndependent of the 1 1i mpul siven 

component. Accordingly, we may write for the c.f . of the (sum of) these 

components: 

18 
for P (= poisson) and G (=gauss ), where specifically 

2 2 2 

crG = crE + crR ' 

(4. 1 4) 

(4. 1 5) 

where cr�, cr� are respectively the variances of the external and the receiver 

(gauss) components . 

Applying (4. 1 4 ) , (4. 1 5) to (4. 1 1 ), and noting that for Class A interference (AA

>> AB) ; for Class B interference (AB>> A

A

) ' while for the 

general Class C 11mixture11 , (AA"' AB), we have specifically 

(4. 1 6) 

Class A (Amplitude) Interference+Gauss: 

(4. 1 7) 

Class B (Amplitude) Interference+Gauss : 

(4. 1 8) 

It is Eq. (4. 1 8) with which we shall be primarily considered here. 

Parallelling the analysis of Sec. 2.4( I I), [Middleton, 1 976] we readily 

find that for large Impulsive Indexes the c.f. 's are asymptotically gaussian, 

e.g. 

(4 . 1 9) 

19 
I 

I 

I 

I 

wh ere 

( 4. l 9a) 

The s peci fic structu re of the "correction" (or Edgeworth seri es) terms in 

(4 . 19) fol lows at once from (2.53 ) , (2. 54 ) , [Mi ddl eton , 1976] . The associated pdf and P. O. for (4. 19) are obtained di rectly from (4. 3)- (4.6) and are 

the fami l iar forms 

-x2 / 2cr

2 

e o 

�2ncr� 

(4.20) 

2 

where e(z) = ( 2//;) J�e

-t dt is the fami liar ( tabulated ) error i ntegral . For 

Class A, Cl ass B noi s e these re l ations simpl i fy at once on setti ng 

Class A: 

Class B: 

( 4. 21) 

There remai ns the fi na 1 reduction of the c. f. 1 s to gi ve us the desired 

tractable analytic forms , sui table for i nversion , to obtain the pdf1s and 

P. D. 1s i nvolved. At thi s poi nt we take advanta ge of the fo rmal i denti ty 

of our results (4.11) , (4. 16)- (4. 18 ) , wi th the express ions in Secs . ( 2.3) 

- (2 . 5 ) II, [Mi ddl eton , 1976], where now r + �. The same physi cal and analyt ical arguments for the reducti on [Sec. 2 . 5 ( II) therein] apply here , so that 

we can wri te down at once th e va rious desi red (approximate ) forms for the 

20 
c . f. 1 s. These a re: 

(4 . 22) 

wi th correcti on terms gi ve n speci fical ly by (2. 78 )II , [Mi ddl eton, 1 976] , 

r + s. The various pdf1s , PD1 s, moments , etc. for the i nstantaneous ampl i tude 

X have been determined and discussed in detai l in 11Part I11 [Mi ddleton, 

1 974] for this Cl ass A i nterference , and so wi ll not.be consi dered further 

here . 

On the oth er hand , fo r the ( fi rs t-order) s tatis ti cs of the Cl ass B 

noise process, X, we must consi der the dual characteri stic functions * 

(4.23a ) 

( O<a<2) 

F 1 ( i s ) B+G-I I (4. 23b) 

where the subscript -I i ndicates the c.f . appropriate to the range (02IXI, IX

0

1 

.2.I X

8

1) of ampl i tude val ues , whi le th e s ubscri pt -II de notes the c.f . associated 

with the ampl i tude range (I X

8

1.:

IX

0

1.:

I XI), cf. (3. l Oa,b) , [Mi -ddl eton, 1 976 ] . 

The p reci se defi ni ti on of the boundary poi nt, IX8

1, wi ll be given presently, 

in Sec. 5. The pa rameters of (4 . 23a,b) are 

b r(l-a/2) ;Ga )/"Aay = 

r(l -a/2) 

la -

2

a- l r

( l +a

/2) '\ oB ·max 2

a

/2-

l r

( l +a

/2) 

A Cl 

<(

B

oB )) ( >O) ; 

12 (4 . 24a) 

* Note that we have s+lsl here , s ince J0(sB0s) = J0(lslB0s), and since the 

c.f. must be an even function of s, in view of the symmetry of the pdf 

about X = 0. 

21 
r' 

b = (

4- a

) /G

2 );-;._ 2 

= (

4-a)(B

2 );2 ( >O) 2a 2-a \ oB max 2- a oB (4. 24b) 

(4. 24c) 

(4. 24d ) 

The key analytic di fferences between (4. 23a) and (2.90)II , [Mi dd l eton, 1 976] 

is the term Is l a

: the absol ute val ue of Isl is requi red in this approxi mation, 

to conform to the symmetry condition impos ed on F1(is)

x , cf . (4.4) et seq . , 

whi ch is in turn demanded by the narrow-ban d nature of the i nstantaneous 

ampl i tude X, which is symmetrical about(X) = X = 0. [ From the purely analytic vi ewpoint, the exact expressions (4. 11), (4. 1 7 ) , (4. 1 8) a re even i n s: 

hence the above approxi mate forms (4. 22) , (4 . 23) must be likewise .] 

Fi nal ly, we must speci fy the remaining pa rameters : \

max'Y' a. Thes e are: 

\ = l i mi ting range ( fc), depending on e

0 [cf . ( 4 .lOc ) ], of the max Y 

receiver ; (4. 24e ) 

y = exponent of the propagati on l aw: g ( �) = [gs

( ¢ ,gv

(e, ¢) ]/ ( 4�c\) Y, 

μ 

cf. (4. l Oh) above , and [ (2 .61 ), Mi ddl eton , 1 976] , e.g. 

g "" l /).Y, O < y ; 

= expone�t of the densi ty l aw wi th range ( ""c\) , e.g. 

a

S , V

( \) = l /\

μ

, 0 < μ

; cf. [2.63 ) , Mi ddl eton , 1 976] ; 

- 2- μ 1 . 3- μ 1 

= -Y- surface' -Y- vol ume 

(4. 24f) 

(4. 24g) 

(4. 24h) 

spatial densi ty-propagation pa rameter ; [In genera l , see Sec. 2 . 7, 

and Eq. (2.82) , Mi ddl eton, 1 976] . ( For our pres ent app l i cations , 

the range of val ues of a i s : 0 < a < 2 . ) (4 . 24i ) 

g

s ,v = angular geometrical factors in the propagation l aw, usual ly taken 

to be uni ty; cf . discussi on, Sec. 2 . 5.l [Mi ddl eton , 1 976] . 

(4.24j ) 

22 
As we have noted earl i er [p. 82, Mi ddl eton , 1 976] , it is i mportant to 

distinguish the q ual i tati ve di fferences between Class A and Class B i n terference , in thei r ampl i tude stati stics (as wel l as earl i er, in thei r envel ope properties): 

( i ). Unl ike Class A i nterference , th e ( fi rst-order) statis ti cs of 

Cl a ss B noi se are clea rly sensi ti ve to the comb ined effects 

of source- di s tribution l aw ( μ) and propagat i on l aw (y), through 

the generi c parameter a, cf. (4 . 24h ) . 

(ii ). Consequently , the sensi ti vi ty to a is a rece i ver bandwi dth 

p henomenon , which can be (in principl e) removed by suitably 

broadeni ng receiver bandwi dth 6fR vis-�-vis 6fN' or hei ghtened 

by suitably narrowing 6fR vis-a-vis 6fN" In fact , Class B operation is often more desi reabl e from a meRsu rement viewpoint, as 

it i ncreases the number of descriptive parameters of the i nterference source from 3 [Class A, cf. (4 . 22)] to 6 [Cl ass B, 

cf . (4 . 23) ], and thus provi des the poten tial for a more detai l ed 

des cription of s ource s tructu re and characteri s tics , via (4. 24 ) . 

( i ii). The Cla ss A and B noi se parameters are the same (as we woul d 

expect on refl ection) for both the ampl i tude ( here) and envelope 

s tati s ti cs [Mi ddleton, 1 976] : (4. 1 7 ) , (4 . 22) and (4. 1 8 ) , (4. 23) 

here . 

5. Probabi lity Di stributions P1

(X�X0

)

B+ G

;O < a < 2: 

We proceed here to use the desi red approxi mations (4. 23a,b) to the characteris ti c funct i on F(i 1

�)

B+G in (4.5, 5b ) , to obtain the corresponding (approxi ­

ma te ) fo rms for P1

(X�X0

)

B+G · 

Firs t, we remark that just as in the earl i er analysis fo r the envelope 

s tatistics of Cl a ss B noi se the approxi mati ng c.f . 's here [cf. (4 . 23a ,b)] 

yi eld pdf ' s (and PD's ) which a re not properly s cal ed . In fact , as 

we s hal l note belo\'J, the c . f. Fl -I

' (4.23a) , yi elds a pdf w

1 (X)B-T , 

23 
whi ch requi res a change of scale for the argument X (or z = X/(X2

)

1

1

2 

= 

X/lri2

8(l+rB ) and.·. X0 (or :?'.0

) in the associated PD. Moreover, if w

1

( x)8_

I 

is used over the whole range of X: ( -00 < X < 00) , then (x2)

I + 00• [On the 

other hand, the c.f. F

l-II ' (4.23b) , yields a pdf, w

1

(X)B- II ' which gives a 

finite second moment over ( -00,00 ) , e.g. , (z

2

) r r = 4G�(tl ), cf. Eq. (7 .4). below, 

but a moment which is not sui tabl y normal ized until we divide by 4G�. ] 

with 

Wi th thi s in mind l et us next introduce the following normal izations : 

. 

. . 

z

o = X/h2

B

(l+r8) z

oB = X

oB

//ri

2B

(l+r8) (>O) 

( 5. 1) 

(5. l a) 

(5. lb) 

A (as before, cf. (3. 2a) , [Middleton, 1 976]). 

factor 

Letting a be a normal izing 

(5.2) 

anal ogous to a = {2ri2

8(l+rB)}-l/2 ( = 2

-l/2a) , cf. (3.3) , [Middleton , 1 976] , 

we can write 

A 

z = ax 

A A 

z0 = aX0 · z - ax cf (4 23) ' oB - B ' . . ' (5.3) 

where A, z, z0 are now dimensionless quantities. Then the approximating 

c. f. 1s (4. 23) become 

-b A a

a

1A l

a

-�

cr

2a

2

A

2

/2 · e la B G 

24 

- F

l ( i aA )B- I ; I : (I z 1, I z

o I< I z

oB I

), 

(5.4a) 
A2 2 

. • 

-AB 2A2 2 -b2aa A /2 

F1(1�)B- II = e exp[-cr

G

a \ /2+ABe ] _ 

II : I z I , I z0 I> I z

oB I · 

(5.4b) 

The associated PD' s are accordingl y from (4. 5) and (4. 5b) , cf. Appendix A: 

:. P 1 ( I z I .=:.I z0 I ) 

(5. 5b) 

for these symmetrical distributions , subject to the 11joining11 condition 

P1(Jzl�Jz0l

= l z0B l )B-I = P1

(JzJ�l

z08J=Jz081)B-II ' cf. (5.22a) , (5.26a) . 

5.1 Eval uations of Pl-B-I, II : 

To faci l itate the integration we repl ace sin \z0 by its equival ent 

Bessel-function form [Watson, 1944] viz: 

. 

z0

\ 

sin ZOA = 2� Jl/2

(ZO

A), (5.6) 

so that (5. 5a,b) , with the help of 5.4a,b) , become expl icitl y 

z 00 -b A a

aJ\J

a

-licr

2a

2

\

2

/2 pl (z�

z

o

)

B- I '.

 � -� 2

� ( A -l/2J

l/2(z

o

\)e la B G d\ 

Jo

zs 

(5. 7a) 
wi th : 

( 5. 7c) 

Al so, we have ( from ( 5. 5b ) ) 

( 5.8b) 

wi th ( 5. 7c ) , and where we have used (4 . 24b ) , ( 5. 1 )-(5. 3) in defining the 

A2 A 

parameters c

mB' A

B

. 

Our principal task now is to evaluate ( 5.6)-( 5. 8 ). For this, we need 

the Hankel i ntegral resul t [cf. (A. 1 -1 9) , Mi ddl eton, 1 960 , for exampl e] 

Re ( v+ μ ) > 0; I a rg b I < 7f I 4 . 

(5.9) 

Our fi rst step in dealing with the B- I fo rm (for 0.: !XI .: !XBI) para l l el s 

the earl ier analysis to obtai n the analogous envel ope statis ti cs: 

P

1

( E. > l0)

B-I = P

1

(l> f:

0

), Eqs. ( 3.lla ,b ) , [Mi ddl e ton , 1 976] . We fi rst 

replace z

0 by z

0

N

1 in (5. 7a), (5.8a), where N1

( >0) is a scal i ng factor, 

which helps to i nsure that (x

2)

B =l. Next , we define a parameter G

B by 

, or 

I 4-a 

2 1 r B

+ 2-a G

B = 4 ( 1 + r 1 ) ' 

B 

26 

( 5. 1 0) 
(which is the same as the G

8

, (3. 1 2b), [Mi ddl eton, 1 976] ). The simpl es t 

useful form of resul t is obtained by expanding the exponent containi ng 

/A/a and then usi ng (5. 9 ) . We obtain after a l i ttl e manipulation 

where 

and where 

(5. l la) . 

(5. llb) 

A 

)

n (

na+l 2 / z I 00 ( -1 r ) 2

An na+ 1 3 " 2 � 1 - -f I n! A

a 1 1

(

2

; 2 ;- /z

0

/ ) , 

n=O 

(5. l l d) 

He re A

a 

is aga i n the Effective Class B I mpul sive I ndex, which i s. unchanged 

from that quanti ty defi ned in (3. 1 2a ,b) , [Mi ddl eton , 1 976] , and is proportional 

A 

to the I mpulsive Index, A8, here . Li ke A8

, A

a 

al so depends on the spatial ly 

sensi tive pa ramete r a, and on the rel ative gauss component rB' cf. ( 5. la ) . 

27 
1 

Wi th the help of Kurnmer 's transformation [cf. Mi ddl eton, 1 960 , Eq. 

A. 1-1 7] we see that when z -+ +oo, 0 -

( 5. l 2a) 

so that the l ead i ng term in the seri es in (5. lla,b) become respectively 

Z + +oo. 0 - ( 5. 1 2b) 

Accordingly , we see , as expected , that P1 takes the proper l i mi ts as z

0 + ±_oo,O: 

= 0 ( 5. 1 3) 

= 1/2 

A 

Si mi l arly , we have at once P

1

(1z l.:

)8_I =l, P

1

( lzl>00) = 0, al so, as expected. 

This shows that these expressi ons for the PD gi ve the correct limi ti ng forms , 

and are now scal ed (N1) to permi t proper "joining" to P

B-I I' so as to give 

(z2

)=1 (or (x

2

) = Q

28+0� = Q

2

8

( l +r8), as no ted at the beginni ng of this 

section. 

A 

Fo r l arge val ues of z

0 we us e, as before , [cf. ( 3. 1 4 ) , [Mi ddl eton, 1 976] ) 

the asymptotic expans i on 

( .. ) _l'_(§_l_ -a

[ ci(a-s+l) + a(a+l)(a-s+l)(a-s+2) + ] l

F

l a ,S,-x .'.

m-aTX l + l!x 2 · · · ' 

to get fo r ( 5.lla ) , ( 5.ll b)*: 

2!x ( 5. 1 4) 

* For ( 5.llb) , in (5. 1 5) we repl ace z

0 by 1�0

J and mul tply by 2. 

28 
00 ( l )n+l An

r(na+l ) 

P

A ( ) \ 

-

a 2

/z

A /.

-na

[l+ (na+l )na 

1 z.:_z

o '\, l 

n=l !TI n'r(2-na

) o Ll.z

A

2 . 2 . 0 

A 

(O<<)z

o < z

oB ( = z

oB

N

I

/2/2 GB

) . 

(5. 1 5) 

�fo note agai n that thi s asymptotic series (5.1 5)* appl ies here onl y for 

A A 

"l arge11 z0, smal l er than some (large) val ue of z0B

. For l arger val ues of 

z0 (>z0B) we must use the second form of the PD, (5.7b) , (5.8b) , as 

expl ai ned earl ier. Thi s is done immediatel y below. Figures 5.1 , 5.2, 

based on (':;. l l b) , (5.15), shows typical PD1 s for P1, provided /z0 / < /z0B

/. 

For P

l -B-I I ' cf. (5. 7b) , (5.8b) , we use (5. 9) di rectl y, to obtain 

m 

1 -A

B 00 A

B A 

P

1 (z.:_z0)8_

II � -

2 {1 -e L -1 e[z //2 c B

]} , (region I I , cf. (5. 5a) ) 

m=O m. o m 

(5. 1 6) 

[ �

�B

(=2;

�B

) = (� +r8

)/ (l+r8

) ' 

AB 

(5. 16a) 

l e(x) = £_ -t

2 

cit , 

/TI o 

( 5. l 6b) 

cf. (5. 7c) and Eq. 3. 1 6a) , [Middleton, 1 976] . Similarl y, for (5.8b) we 

have at once 

Note that (5. 1 6) , (5.17) have the proper behaviour as z0 + ±_00, 0, viz: 

l im P (z>z ) = 

z ++00 1 - o B- II 

0 -

1/2- ( 1/2 or -1/2) " 0, 1 ·� 

P

l (z.:_O)

B-II = 1/2 j 

as was the case above for P

l-B- I

, cf. ( 5 .13) . 

* See footnote on preceedi ng page. 

29 

(5. 18) 
F i gure 5.1 

0 

<N 

60 

40 

20 

0 

Class B Interference 

a = 1.0 

" 

Aa = 1 

-20 

10

-

6 

'-'- 

10-4 I0-3 0.01 0.1 0.2 0.4 0.6 0.8 0.9 0.95 

A A 

0.98 0.99 

The amplitude distri bution, Pro b( lzl>lz - j)1, cal ­

0 

culated fo r Clas s B i nt e rfe renc e f o r a = 1.0, fo r 

A 

va rious value s of the Ind e x A

a, f rom Eqs . (5.l l b), 

(5. 1 5 ) . 

30 
Figure 5.2 

-

0 

CN 

60 

a= 0.2 

40 

20 

0 

Class B Interfe rence 

/\ 

Aa= 1.0 

0.8 0.9 0.95 0.98 0.99 

" " 

The amplitude distribution, Prob(Jzl�!z

0J)I

, calculated for Class B interference for A = 1.0, for 

a 

various values of a, from Eqs. (5.llb), (5.15). 

31 
I 

Final ly, when z0 is l arge, we use the asymptotic rel�tion [Middleton, 

1 960, Eq. (7. 5)] 

2 

e-y 1 4 e(y) -v 1 - - [1- - + O(y

-

)] - /;y 2y2 

to write (5.16) as 

1 

V2TI 

-A A m 2 A

B 2 oo c A -z /2c e l B

e o mB

[l-O(z�

2

)], 2o m=O 

( 5. 1 Sa) 

(_5. l 9b) 

when z; >> 1. [For (5. 1 7) we simply repl ace z0 by Jz01 (and z by l z J) and 

multiply the right member of (5.1 9b) by 2. ] Figures 5.3, 5.4 illustrate 

Pl -B-II ' Eq. (5. 17). 

5.2 The Composite Aproximation : 

As we have al ready remarked, the problem with the approximating resul ts 

for P

1

_8 in the present Class B model , cf. , (5. 1 1 ), (5.15) for P

1

_8_

1

, (5. 1 6) 

for P

l -B-II ' is that these forms , stemmi ng as they do from approximate 

characteri stic functions (4. 23a,b) , are not necessarily properly scaled in 

that each approxima,ti ng form P

1

_

1

,P1

_

II does not yield the correct mean-square 

val ues of (z� B = 1, or (X2

)8 = ·n2

8(l+r8L cf. (7. 5) . The approximations 

P

l

-I

' (5.11), in fact, and the associated pdf's, w

1

(z)8_

1

, Eq . (6. 7) , al though 

suitably scaled [e. g. z+z, etc. , cf. remarks (5.9)ff] , in accordance with 

the "joini ng" procedures outl ined below, cf. (5. 22)-(5. 26) , do not possess 

fnni:te mean squares on (-00 ,00 ) , or (0,oo ) , cf. (7. 3) ff. The approximations 

P

l-B-II ' (5. 16), (5. 1 7), for Type-A forms , and their pdf' s, w

l -B-II ' Eq. (6.8) , 

although also proper PD's (and pdf's) of (5. 18) , likewise do not yield the 

exact mean-square (z

2

)8 = l, but rather a finite (z

2

)8_

11 � l, cf. (5. 20a) 

below. 

Accordingly, we must devi se suitable methods of combi ning, or 11joining1 1 , 

the B-I , II approximati ons, so as preserve the proper characteristics 

[(5. 13) , (5. 18)] of the PD's, (and associated pdf's) ; provi de the correct 

second moment, (7. 7b) ; determine the various (six) B-model parameters , and 

thus effectvvely specify an acceptable approximation to P

1

_8 , and w

1

_8 , over 

32 
0 

N 

Figure 5.3 

20 

0 

-20 

-40 

C l ass B I nterfe rence 

A8 = 1 .0 

a= 1.0 

0.8 0.9 0.95 0.98 0.99 

The amplitude distribution, Prob(J zl�Jz0 J (�z0B) ) 1 1, 

calculated for Class B interference for A

B = 1.0, 

a = 1.0, for various values of rB from E q . ( 5 . 1 7 ) . 

33 
.J ' 

l 

I 

0 

N 

0 

-

20 

- 40 

Class B Interference 

r� = 10-

4 

a = 1.0 

A8= 0.I 

-

60

I o·

L-L- 

6 10·4 10-3 0.01 0.1 0.2 0.4 0.6 0.8 0.9 0.95 0.98 0.99 

Figure 5.4 The amplitude distribution , Prob ( l z l -> l z I 0 -> 

(z0B

))1 1 , calculated

-

�or Class B interference 

for a= 1.0, rB = 10 , for various values of 

A

B

, from Eq� ( 5 . 1 7 ) . 

34 
the en ti re range of t he arguments (z,z0

). Two pri ncipal methods for accompl i shing this are outl i ned be l ow: Method No. 1 is a somewhat more simpl e, 

but approxi mate one , described earl i er in Mi ddl eton [1 976, Sec. 3.2-A; 

1 977, Sec. 3.2. l); Method No. 2 is preci se, but i nvol ves more elaborate 

calcul at i ons. The former employs a suitably 11normal i zed 11 pdf , (wl-B-II )

norm 

for the B-l� region, to determine (z

2

/

B

=l , over the enti re range ( -00 , 00 ) , or 

(0 ,00), of the random variabl es in question, and can thus yi el d errors of 

somewhat too s mall val ues of P0 for l arge (z,z0), when rB is smal l. The l atter 

method however , yiel ds the correct val ues. The greater comp l exi ty of the 

l atter l i es in t he expl i cit eva l uation of (z�B properly using the approxi ­

mating pdf's for the appropriate regi ons B-l , Il, cf. ( 5.23) bel ow. 

5.2. l Method No . 1: 

Let us o utl i ne Method No. 1 fi rst , fol l owi ng the author 's earl i er 

analysis [Mi ddl eton, 1 976, 1 977] . Therefore , since the p recise mean square 

is (z

2

)

B � 1, cf. (7.5 ) , and since in this approach we now use w

l -b-ll 

over a l l z to establ i sh (z

2

)

B = 1, this means a 11 renormal i zation1 1 of w

l -B-I I

' 

whi ch is accompl i shed as fol l ows : 

(A ). Cal cu late (z

2

; 11 on ( -00 , 00 ) according to (7.4 ) , usi ng wl -B-l l ' 

Eq . (6.8), e.g. 

A Am -A 00 A

m 

A 

I 

2 -

-

B � __.§_ �

2 = e B \ _§_I (m/A +r') ( l +r ' )-l 

\2 )

11 - e 

m O ml mB m O m. B B B ( 5.20 ) 

(

4-a)+r' 2-a B 4G

2 ) ) 

. A 

(

2-a = ) ( ) l +r' = B (tl , cf. (5.10 , wi th A

B = 4

_

a 

A

B

. 5.20a 

B 

The requ i red normal i zation factor is thus Ni = ( 4G� )

- 1

, e.g. , 

1 2 (wl -B- l l )

norm = 

4G

2 ' w

l -B- ll = Nl lw

l

(z)B- ll" 

B 

(5.20b) 

Usi ng ( 6 .8), applying Nir di rectly to (5. 16), (5. 1 7) gives us now 

35 
( 5. 2la) 

where the domai ns of appl i cabi l i ty ( for z

0 vs z

08

) are as i ndicated in (5.5a ,b) 

above . [ lhese ampl i tude re l ations , (5.2la,b ) , are the analogues of (3. 1 7 ) , 

[Mi ddl eton, 1 976] , for the enve l ope .] 

(B) . The case of w

1

_8

;.

I

' Eq. (6.7) , requi res a di fferent approach , since 

(z

2

)

I on ( -00 , 00 ) , or (0,00) is i nfini te ( O<a<2 ) , cf. Sec. 7.3. Instead of 

normal i zi ng with respect to the second moment , [as is done in ( i ) above , which 

is, of course , not possibl e here] , we must scale z

0 (and z) according to 

(5.lld ) , e.g. , z0

+z� = Ni

z�

, wi th z� = X/212 G

8

1�2

8

(l +ri3) , so that z0 

= z

0

N

1

/212 G

8

. The rational e for this is 

1 ). The obs ervation that the approximating P

1

_8 I (and w

1

_8

_1

) must 

have the same val ues in the gauss region ( z;<<l ), where P

1

>(0. 9) , 

or more , as does the preci se PD , r1

_8 (and pdf w

1

_8

), derived from 

the exact (but i ntractabl e) c.f . [ (2. 89) , r+� , in Mi ddl eton [ 1 976] ] . 

Hence we have (5. 11) above . 

• •• 2 ) . The sca l i ng factor N

I is to be determi ned by fi tt i ng the two 

approximate PD 1 s, P

1

_1

, P

1

_1 1

, together by the procedure outl ined 

be l ow, which is based on the canonical properti es of the Class B 

model general ly. 

The 11joini ng procedu re11 for P

l -I , II ( and therefo re for w

l -I , I I ) now 

di rectly paral l els that stated i.n [Mi ddl eton , Sec. 3.2B, 1976; Sec. 3.2. l , 1 977] , 

and uses (5. ll), (5.20) speci fical ly : we set 

( i ) . in the gauss regi on (e .g. l z0

l rv0 or smal l ) . (5,22a) 

Equal i ty of the two approxi mations in thi s region is requi red, since both represent the same 

( smal l - ) ampl i tude behaviour , characteri s tic of 

a l l PD 1 s here , and , of course , s ince both are approxi mations to the same , s ingl e, exact PD ; 

36 
(ii ). 

( i ii). 

(iv). 

( v ) . 

in the gauss region : for the same reasons , 

both approx imating PD 's must have the same 

( constant ) slopes , ( P1

> 0 . 9) ; 

at +z

08

, i . e. at the 11bendover 1 1 poi nt (cf. 

Fig. 5.5) , z08

(>0) , where the two approxi ­

mations a re joined together . Thi s cond i ­

tion is requi red for (5.5) to hol d, as shown 

in Appendix A, cf. Eq . (A. l -6 ) , et seq. I t 

a l so i nsures that the PD here remai ns con ­

tinuous . The poi nt z08 is empi rica l ly determi ned from the data , i . e. from the experimental APO , or exceedance probabi l i ty curve 

dP P

l

(Z_'.:_Z

o

)

exp· 

I I ) the ( fi ni te) slopes of the PD 's must 

0 .:!:.

20B also be equal at the bend-over point(s ) 

+z

08

). This i ns ures that the associated 

pdf ' s a re continuous at (+z

08

), and that 

d

2

P the PD 's are smooth here . 

1

2

I I ) (fO) : 

dzo .:!:_z

oB 

(5.22b) 

(5.22c) 

(5. 22d ) 

(5. 22e ) 

this is requi red by the conditi on that the pdf ' s 

be not only conti nuous but " smooth", i . e. have 

a common tangent , at .:!:_z08. This, in turn , is 

requi red by the fact that the exact pdf is not 

only continuous but smooth , as wel l . 

(vi ). z

08

(>0) : this is the point of i n fl ect i on ( d2

P

1

/dz�8=0 ) (5. 22f) 

of the exact P

1

, and is determined as s uch 

(usual ly by i.nspecti on of the P

l

-data ) , wi th 

P

1

+Pl -expt ' cf . Fig. 5 . 5. This val ue of z

0 

(=+z 8) cannot be determined analytical ly from -o 

either approx imati ng form , P

l -I ,

I I " 

The five re l ations , ( i ) - (v ) , with the s i xth, (z

2

)1 1=1 , e.g. , (x

2

)

II = 

2rt28

( l +r.8) = \x

2

)

expt cf. ( i ), Eqs . (5 . 20 ) - (5.2l b) , a re suffici ent to 

37 
0 

N 

40 

2os 

20 

0 

-20 

Figur e 5.5 

�-Gauss-- 

0.8 0.9 0.95 0.98 0.99 

S chema of P

l

-

B

' E q . (5. 2 7 b ) , ob tained by j o in ing 

t he two approx imat ing forms (5. l lb ) , ( 5 . 17) , a c ­

cording to t he proc edur es ( 5 . 22) , Me thod 1, or 

( 5 . 2 6 ) , Method 2. 

38 
determine the six global parameters (A8,Q2

8 ,rg,a ,A

a

,N

I

)

' as long as z08 is 

empirical ly available. 

5.2-2: Method 2: 

As noted above, this is a precise version of (i )-(v) above, Eq. (5.22) , 

with the sixth , (approximate condition in Method 1) replaced by the 11exact1' 

rel ation (z

2

)8 = <z) B-region 1+<z�B-region II-1 , e.g. , (7. ?b) in (7.2a) . 

Now we use the proper, approximating PD's (and pdf' s), (5. 1 1 ), (5.16), (and 

(6.7) , (6.8) ) in (i )- (vi) above, cf. (5.22) . These rel ations here are 

then expl i citly: 

(5.23a) 

(5.23b) 

thi s last because of the syrrunetry of the pdf1 s, which also, cf. (6.8a) ', 

al l ows us to express (5. 23b) as 

(5.23c) 

in �erms of the pdf for i z l . Then , since w

1

(z)8_1dz = w1 (�)d� , where 

w

1

(z) is the "scaled" pdf f

�

r �

egion B-I , cf. (6.?a) , and z = z0N1

/212 G8, 

cf. (5. l ld) , we get, usi ng w1

(z) , (6.?a) , 

A 

2 oB 2 oB 8G8 A

2

A A A iz lz 2 

(

z fB-region I = 2 

0 

z w

l (z)B- I

dz = 2 

0 

( 

N

2

) z w

l (z)dz . 

I 

The desired condition using the correct mean square then reduces to 

39 

(5. 23d) 
I 

(5.23e) 

We can eval uate the second integral partial ly in closed form, using 

(6.8) , (6 .8a) 1, to get directly 

<z

2

) B . I I = (oo z

2

w

l ( I z I ) BI I.d I z I -reg1 on z 2

oB 

(5.24a) 

(5.24b) 

[We remark that these results necessari ly reduce to that of (i ), Method l, 

above, when we set z08-+0 in the cal culation of (z

2

)

I I ' cf. (5.20)- (5. 21 ) .] 

The first integral in (5. 23e) can be obtai ned formal ly on a term-wi se basis. 

We get 

where 

2 00 

� 

8GB 2 (

z 

l B-regi on I = N2 · 7T n�O I 

A 

nAn z (-l ) Aa na+l ( oB 2 na+l 2 

n ! (-2 ) 

n 

x 1 1

(

2-; 1/2;-x )dx , 

0 (5. 25) 

08 

2 na+l 2 � 2 t2

oB L

z (

na+l ) 

A

2H3 

0 

x l l

(

2-;1/2;-x )dx = 

t O t !(l/2)

t

(2t+3) 

(5.25b) 

Accordingly, (5. 24) and (5.25) in (5.23e) yield one rel ationship among the 

six global parameters of this Class B model . 

40 
The other five relations are obtained as above (5.22a-f) . We have 

expl i citly here, for Method 2: 

( i ) . 

(ii ) . 

(i i i ). 

(iv) . 

[P1- r

=P1 -IIJ l z j «l 

f 

0 . 

c . gauss region 

5. 22a) 

which becomes 

z 00 : l_ _ _.2. L (-l )n

r(

na+l )A

n /n! 2 n n=O 2 a 

m 

l � -AB oo A

B z �} = - 1 -e " - 0- - 2 l m' A n ' 

m=O · c mB 

1 (5.26a) 

for both z0 and j z0j, cf. (5. l l b) , (5. 1 7) . 

m [dPl-I 

-

dPl -I I J N

I 

dz

o 

- dz

o I z0 I «l G�l1f 

oo ( )

n -A oo A 

L �1 r(

n

�

+3)A

� = e 

B � ,

�

2 n=O n-0 m.

c

mB cf. (5. 22b) gauss region 

al so for both z0 and j z0

j 

[dP 1 -I 

= 

dP 1 - I I J 

'A r (

a+1

)

�-a 

a 2 oB [l+O(z

-2 ,a) J 12 r(l -a/2) oB 

where (5. T5) 

A ar (

a+l ) 

a 2 

and (5. 1 9b) are used. 

dz0 dz0 +z 

cf. (5. 22d) - oB /2r (l -a/2) 

-A oo 

rv -1- e 

B \' 

- 2 l 

z

0B m=O 

l5' 26d) 

again where (5. 1 5) qnd (5.l9b) are used 

41 
(v) . 

(vi ). 

2 

[

d P

1-t 

= 

dz

2 

0 

cf. (5.22e) , which insures that the pdf's 

at :!:_z08 are 11smooth11 , etc. [The expl i cit 

analytic form is obtained from (5. 1 5) and 

(5. 1 9b) by differentiation as before.] 

+z 8: the 11bend-over1 1 points again are empirical ly determi'ned and -o 

used in the above, cf. (5.22f) . 

We remark, also, that when the relations (i ii)-(v) are used, we may 

need the next set of "correction terms" in these various asymptotic developments. Since Class B model s i nvol ve six (ul timately) independent parameters , 

fixing any three in the above enables us tn determine the other three, from 

any three of the (i )- (v) , and (5.23) etc. Final ly, we note that the 11Class 

A1 1 form ( I I ) is coupled to the "Class 8 1 1 form ( I ) through the Class B 

parameter,a., and vice versa through the "Class A1 1 parameter, rs, appearing in 

G

8

, common to both approximations ! , II . In any case, we have, for ei.ther 

Method: 

(5.27 

(5.27b) 

Typical di stributions are shown in Figs . 5.6, 5.7. 

5.3 Hal l-Type Models: 

As in the envelope cases treated earl ier [cf. , Sec. 3.28, [Middleton, 

1 976]], we can also obtain a Hal l model form [Hal l, 1 966] from the P

1_1 

approximation used in the gauss and i ntermediate region 0 .: \ z0\ .: \ z0g l· 

Thi s is achieved provi ded (1 ), we omit the gaussian contribution in the 

c. f. (5.7a ) , or (5.8a) , e.g. , set lia� -rO therein , and (2) set z08-+<x>. 

The resulting expressi on for the PD is accordingly [cf. (5. 7a)] 

42 
0 

N 

60 

40 

20 

0 

Figure 5.6 

a= 1, N1 /2G8 = 1 

A = a 10-1 NI = 1.729 I 

As = 10-; f � = 5 X 10-3

, il2s = 199.5 

�Aa = 10-� NI = 1.723 

As = lff� f � = 1.6 X 10-� il2s = 70.8 

Aa= 10-� NI

--

= 1.53 

..-- As= 10-; r� = o.5, n2s = 2.01 

CT� = f� .Q28 

( B�a > = 2.U2alAa 

0.8 0.9 0.95 0.98 0.99 

The (complete) amplitude distribut i on P 1

C l z J � 

l z

0

l ) B , E q . (5. 2 7b ) , c a l cul at ed for C lass B in ­

terfer ence for vari ous Aa , g iven a , from Eqs . 

(5. l lb ) , (5. 17) , accord ing to the proc edures 

of (5. 26) . 

43 
I .I 

0 

N 

Figure 5.7 

60 

40 

20 

0 

a= 0.2, N1 = 1.45 

� f,' - -5 ll - 4 A s = 0.35, s - 2 x 10, .lL29 - 8.9 X IO 

z OB 

--

a = l.2, Nr = l .86 

-4 I 

-

2 

A s = 5X 10 , fs= l.6XIO , D,20=158.5 

a= 1.8, N1 = 1.63 

---A s = 10-� f� = 5.0, f228= 0.47 

a-i = r� f220 

( B�s) = 2 f228 I A8 

-20

I ff6 

'- 

10"10-3 0.01 0.1 0.2 0.4 0.6 0.8 0.9 0.95 0. 98 0.99 

The ( complete) ampli tude distr ibut i on P 1( \ z \ � 

\ z

0

\ ) B, E q . (5. 2 7b ) , calcu l ated for Class B int erference for var i ous a, g iven A , from Eqs . 

a 

(5. l lb ) , (5. 17) accord ing to the procedur es of 

(5. 2 6 ) . 

44 
(5. 28) 

We may proceed to eval uate (5.28) , fol lowi ng the approach used on pp. 

1 00-1 03 [Middleton, 1 976] . Let us start with the form convenient for smal l 

values of / z0/ (.:z08) and use the fol lowing transformations : 

ft _ fta _ -a/2 . B = b

1 A8

a (-A 2 , cf. (5. llc)) , 

a a a 

. A= (z /B )

l/a . dA = 

1-a/a

d /aBl/a 

• • a a ' � 1a 

and 

-z 

a 

e dz 

a 

Next, we use the Barnes-integral representation for J112

: 

* 

J ( * l/a

) _ 1 r(-s (

2

o

)

2s+l/2 (2s+l/2)/a � 

1/2 2o

2

a -

r r s+3/2 2 2

a 21Ti ' 

(5. 29a) 

(5. 29b) 

(5, 29c) 

(5.30) 

(5.31 ) 

cf. Eq. (13. 1 06) , [Middleton , 1 960] , when r is the contour (-00+c, i00+c) , 

with c(<O) so chosen that the i ntegral over z

a in (5.30) is convergent at 

z =O, e.g. 

a 

1 + 2s+l/2 1 

ioo 2a a 

-

-

z

a

z e d z = r (2s

a

+1 ), Re s > -1/2 

O a a 

Applying (5. 32) to (5. 30) , (5.31 ) then gives 

45 

-1/2 < c < 0. 

(5. 32) 
& r (- )r (

2s+l ) (z

*

)

2s+l/2 

( ) 1 1 s 

a o ds p Z> Z ru - - - --. 

, 1 o BI - 2 " � r (s+3/2)22s+l/2 2n1 (5. 32a) 

which is evaluated (at the (simple) poles s=O,l,2, ... ) to yield directly 

2n+l z

o 

2n+l 

A a 

a 

[Note that in the gauss region ( l z I << 1 ), Eqs . (5. 33a,b) exhi bit the 0 

A 

characteri stic gauss form (in the PD) ruz0

, cf. (5.11 ), as z0

,z0 -+ O.] 

(5.33b) 

(5.33c) 

For l arge val ues of z0(<z08-+m) (or smal l val ues of A

a

), we return to 

(5. 28) and use a Barnes integral representation for exp(-� \a ) , viz: 

a 

(5.34) 

to reexpress (5. 28) as 

P

l

(z�z

o

)

B-I I

Hal l ru l+ l_ r (-s) i i

-as �. y

as-l/2

J (y)dy , -2 r o 27f1 J0 1 /2 

7f (5.35) 

-1 < Re as < 0 • 

where we have employed the transformation z0\ = y and ( I ) -+ z0-+ �= here. 

Again , we use [Watson, ( 1 944) , p. 391 ] to eval uate the y-i ntegral : 

J (t)tμ

-v-ldt = r (μ/2)/r (v•μ/2+1 )2vμ

+l 

, Re μ

< Re(v+3/2) (5.36) J 0 \) 

with v = 1/2, μ =as+l here, and :. 0 < Re(as+l )<l (<2) , as requi red. Equation 

(5. 35) now becomes 

46 
( - ) ( as+l

)I *

i

-as 

1 - 1 [ r s r 2 z

o p

l ( z�z

o

)

B-I I Hal l 'V 2 + - 1 - liT r r ( l -as/ 2)a -as 

--

---------

ds 

2n

i 

(5. 37) 

whi ch s hows , as expected , th at 

l i m

p -+ O 

z � 1 0 

l im P -+ l Z -+-00 1 ' 

0 

(5.37a ) 

formal ly. Thu s , we can use the rel ations ( 5. 37) for l a rge val ues of l z0

1. 

Equati ons (5.33b , c ) , (5.37) cons ti tute the Hal l - type PD 's for Cl ass B i nterference . [As befo re , [Mi ddl eton , 1 976] , there are no Hal l - type model s for 

Cl ass A noi se. [ These resul ts are general i zati ons of the Hal l approach , based 

now on deta i l ed physi cal consi derati ons , i . e . , those l ead i ng to the resul ts 

of Sec . 4 above fo r the c.f . 's. ] 

Final ly , in the s pecial case a=l ( as in the envel ope cas e, cf . pp . 1 02-

1 03, [Mi ddl eton, 1 976] ) we can sum the seri es ( 5. 33c ) , or ( 5. 37 ) . Thus , we 

have ( from (5. 33c ) ), al l z0

, 

a=l : 

( ) n ( ) 2n+ 1 

1 /2 00 - 1 r 2n+ 1 z 

p ( z> z ) I 'V - - - I o 

1 - o B- I Hal l - 2 n n=O n! (3/ 2 )

n

2

n A

���l (5.38) 

1 /2 z 00 (-1 )

n

( l/2) /2 z 2n ( 1 ) 

- o I n ( o

) n ( 5. 38a ) 

- 2 - 1T A

a= l n=O (3/ 2)

n 

A

a=l � 

47 
1 2 1 /2 z

o 

= 2 {1- ; tan -

( A ) } . 

a 

Here the corrdspondi ng pdf is at once 

12 A

-l 

= 

dPl = l ( __ 

a=1-=-

) ' a=l . 

- dZ n 1+2z

2 I A: _ 0 Z +Z -.

 _ 1 0 

(5. 38b) 

(5. 39) 

It is at once evident that only the (l z i 8/-moment, -l <S<l, exi sts for thi s 

Hal l model , and consequently (z)-xo, as expected from our general analysis 

[cf. Sec. 7 ] . The PD, (5. 38b ) , and pdf, (5. 39 ) , correspond to the Hal l case 

B

Hal l

= 2 , for ampl itudes [cf. Spaulding, Miadleton, 1 976 , Eq . 2.33 et seq .] 

. 

Note that P

1

, (O<a.<2) , is monotonical ly decreasi ng as z0

++, and that 

1

1�� P

1

=(0,l ), wi th P

1

(z>O) = 1/2, as required of a proper PD. It is, how- zo�oo -

ever, an inappropriate approximate form when �oG is at al l comparable to 

(b1a.

A

B

)

l /a

. Al so, it is not appl icable for very large z0, since the true , 

(i .e. , physical ) pdf1s must fall off fast enough to guarantee the existence 

of a finite second moment [cf. comments in Sec . (2. 7A) , Middleton, 1 976] . 

In any case, Hal l -type PD1s and pdf1s are not possible for Class A interference, as noted earl ier. Moreover, because of the improper scal ing of 

these PD's, e.g. , Pl-B- I ' general ly, we must replace z, z0, by N!z, Niz0

, 

where the scal i ng factor Ni along with the four other parameters (A

a

,a'rB' 

a

2

8), are determined by the procedure outl ined in Section 2 (and in Section 

.. 

6C, [Middleton, 1 976] . [In the case of the Hal l model here where a=l, cf. 

(5. 38 ) , (5. 39) , there are only the four parameters (Ni ,A

a

,rB,a2

8) to be 

establ ished. ] 

6. Probabi l ity Densiti es: w1�_:_ 

The associated probabil ity densities (pdf' s) for the approximations 

w

1

(z)

I , II for these Cl ass B interference model s are now readtly obtained, 

48 
ei ther from 

dP I w 1 (z) = - -d zo z �z 0 

or by using ( the norma l i zed ) form of (4 . 6 ) , viz: 

( 6. 1 ) 

(6 . 2 ) 

for these symmetri c pdf 's [deri ved from (4. 6) wi th the help of (5.3) and the 

re l ati on w

1

(z) 8 = w

1 (X(z) ) / dX/dz/] . Here we have , in the approximate form : 

(6 . 3 ) 

----------

correspondi ng to the associ ated PD 's, (5.27) . For exampl e, ( 6 .2) becomes 

expl i ci tly. 

Let us use 

'V 1 f

"" = - cos 

1T 0-

(6.4a) 

(6.4) 

(6. 5) 

as with (5. 6) in (5. 5) ( (5. 7 ) , (5.8 )), to faci l i tate the i ntegration. The 

49 
I 

l 

resul t for (6 . 4 ) is expl i ci tly 

(6.6a) 

(6. 6b) 

Usi ng ( 5 .9) we find di rectly that,* in repl acing z by zN 1 in w

1

_

8 1

, we obtain 

expl i ci tly 

r----··-----------=-·-·----··--·--·-------- - ··- -···--·------ ---------·--·------ ------·-----·--·- ··--·-·---- - --� 

1 nAn I I A A 1 00 ( -l ) A

a na+ 1 na+ 1 A 2 : I w

1

(z) B 1 =w

1

(z)-v - I 1 (-2-) 1 1

(

2-;l/2;- z) , i { • - • 

- rr n=O n. • 

and 

(- z

0B<

z<z

0B

) ( z " zN 1

/ 212GB; 

�oB =. 2

oB

N

I

/ 2/2 G

B

; 

00 

I 

m=O 

2 A2 

A

m -z /2c

mB 

ii 

B e ---

B

- ' Z.2_-

ZoB , z.:

z

oB ' 

cf. (5. 7 ) ;A

B

= ( 2-a ) ( 4-a ) AB

. 

(6. ?a) 

(6. 7a ) 1 

(6.8a) 

(6. 8a ) 1 

(6 . 8b ) ' 

Wi th 1 a rge va 1 ues of J � I ( < z

08 ) , ( 6. ?a ) may be deve 1 oped in the asymptoti c 

* Here we go from z to z 1 =zN 1

; then from z 1 to �=z ' / 2/2G

8

, wi th a jacobi an 

=212 G

8

; thi s removes the factor ( 2/2 G

8

)

-1 in the eval uation of (6.6a). 

50 
series : 

,. ,. 00 (-l )n

A

n 

"-n _

1 (na.+l ) (�

a.

1 1 )+ ... 

w

1

(z)B- r

=w

1

(z)� I n! r(-n�/2) 2 a. [l+ "2 ] 

/if n=l 2z (6. 9) 

o << / � / � l �os

l . 

These forms , w

l -BI , I I ' are , like their corresponding PD1s, conti nuous at 

the 11bend-over11 point, +z08, cf. (5. 22) . Unlike the purely Cl ass A interference, when rB -+ 0 here there are no 11gaps-in-time1 1 [G8-+(4-a.)/4(2-a.)] : 

thi s is a consequence of the fact that even if the system and external gauss 

noise component vanishes effectively (cr�-+O) , there still remains an inherent 

gaussian contribution b

2a.

A8(>0) , cf. (4. 24d) , stemming from the interaction 

between the rel atively broad band incoming i nterference and the receiver1s 

i nitial (linear) stages , e.g. 4fN > 4fARI (cf. comments pp. 74, 82 

[Middleton, 1 976]). Figures 6. 1, 6. 2 show w1

(z)8 , Eq . (6. 3) , based on the 

composite approximation (6.7) , (6.8) . 

[We note, fi nal ly, that w1

(z) , (6.7) , is a proper pdf: w1

(z)2:_0* and 

1:

00

w

1

(�)dz = 1. We show here that this l atter condition is satisfied. Let 

I = b

l� I (b) ; I (b) = -1T

2 I 

00 � 

n=O 

(-l )

n An

r (

l+na.

)

J b 

F ( l+na.

·l/2·-"

2

)d,. n! a. 2 0 1 1 2 ' ' 2 z, 

(6. lOa) 

2 

(-l )n A

n

r (

l+na)

, (

b 

y

-1/2 F (

l+na.l. _

y)dy 1 00 

= - I 1T n=O n ! a 2 0 1 1 2 '2

' 

-

' 

( 6, l Ob} 

and then express 1F

1 in series form and integrate: 

2 (

na.+l ) )£ 2 

2 - Jb -1/2 na.+l . . _ 

00 2 i

(-l J

b i-1/2 K

n

(b ) = y 1

F

1 (

2

, 1/2,-y)dy - I (l/2) ii y dy 

0 i=O £ . 0 

A A 

oo (

n

2

+1

)

£

(-1 )

£

2b2i+l 

= 

i

i

o (1;2)

i

i! (2£+1) 

* Thi s fol lows from / F(iaA ) 1 / .. J, F1

-+0, / s /-ro, for the c.f. 

51 

( 6. 1 Oc) 
aJ 

N 

F igure 6.1 

a = l.O 

- 10 0 10 20 30 40 50 60 70 80 

z (dB) 

The (complete) pdf w1 ( z ) B

, ( 6 . 3 ) , O < z <00 of the instantaneous amp l i tude for C l ass B interference , 

calculated from Eqs . ( 6 . 7a) , ( 6 . 8a) , for var ious 

A

a

, g iven a , [See Fig . 5.6 for the assoc iated PD 

and p ar ame ter values .] 

52 
I 

I 0

-

9 L----

-----'----'- -'-- I _L__L___ _ _J___L_-'-_J___ _ _l_J'-_J_ _L_l.l_ 

- 20 -10 0 10 20 30 40 50 60 70 80 

F igure 6 . 2 

z (dB) 

The (complete) pdf w1 (z) B

, ( 6 . 3 ) , O < z <oo , of the 

instantan eous amp litude for Class B int erferenc e , 

calculated from Eqs . (6. 7a) , ( 6 . 8a) , for var ious 

a, g iven A. [See F i g . 5.7 for the associated PD 

a 

and parameter value s .] 

53 
I 

Si nce ( 2�+1 ) ( 1/2)i = ( 3/2)i , we have 

+ r(l/2) _g_ r (3/2 )+0(b- na

,n>l ) + 1, as b + 00• 

7f -

00 

· . I = l, or J_ �(�)d� = 1, as requi red ·. • 00 

-co 

( 6. 1 Od ) 

( 6. 1 Oe ) 

Clearly, w

1

_8

_11 is al so a proper pdf, wi th w

1

2.

o , (by i nspection) as di rect 

integrati on or (6.8a ) , (6.8a ) 1 s hows at once . 

7. Remarks on Moments and Parameters : 

Our discussi on here closely paral l ells the treatment of Sect i ons 5-7 

[Mi ddl eton, 1 976] , and so we wi ll content oursel ves wi th a seri es of s ummary 

remarks ; detai l ed eval uations may be found in the reference above . 

For the Cl ass B i nte rference under discussi on in this Report we may wri te 

for the general moments (lz/8), B�O , ( z

i

), i = 0,1 ,2, ... , usi ng (6.3) : 

s _:_ 0 ' ( 7. 1 ) 

where speci fical ly we employ (6. 7 ) , (6.8) above for w

l -B-I , I I . Simi larly, 

we have fo r the integral momen ts 

(7.2a) 
( /" > = 0 ' ,Q, = 0 dd . (7.2b) 

As expected the odd moments vanish , s ince w1(z) is symmetrical about z=O. 

Fu rthermo re , it is cl ear from the nature of w

1

(z)8_

1 1 , (6.8) , that al l 

( finite) ( amp l i tude) moments of Class B i n terfere nce ex ist, as requi red 

physical ly. [This was found to be the case for the envelope , as wel l 

[Mi ddl eton , 1 976, p. 1 13] .] 

Speci fic moments may be determined by di rect apl i cation of (6.7) , (6,8) 

to (7. 1), (7. 2). In fact, exact expressi ons for ( z2

t> , (7.2a ) , may be s imply 

found from the characteri stic function , as we note below , Eqs . (7.6) , 

so that we do not have to empl oy the approxi mate forms (6. 7 ) , (6.8) in (7.2) . 

[A similar set of exact relations appl i es for the envelope moments (c:.2t;8

, 

a l so . ] Of course , if we want ( l z l s>, then general ly we must use (6. 7 ) , (6. 8 ) , 

in (7. l). Let us examine a few s peci al cases of i nterest: 

( i). (I z I s \ � Here we have 

z B( «lO) A 

( 8) . ( 

o 8 2 ( -l ) . a J00 " S-a.-l " I z I I � 2j" z (w1 (z)8

_

1

dz + � - " z dz , 

0 !IT 2oB (7.3) 

"S a from (6.9) in (7. 1 ). Th e second i n tegral is O(z -

): accordingly , only 

moments of order S-a.<O or ( O.:) S<a. can exi st, based on w

1 (z)8_1

. Since 

( O< ) a.<2, this means that (z2>

1-?0

: w1(z)8_

1 does not s upport a fini te 

moment. (As noted earl i er [Mi ddl eton , 1 976, pp. 1 1 3-1 1 4] this is al so 

the case fo r (£2

)

1

, fo r Class B envel opes .) 

(ii ). (z2)

1 1 : Here we cal culate the second moment of z, based on w1

(z)8_

1 1 , 

(6 . 8) . Thus , we have 

55 
-- (�=�+rs ) = 4G2 

l + r 1 B , cf. ( 5 • 10 ) , ( 6 • Ba ) 1 , (7.4 ) 

2 whi ch demonstrates the proper normal ization factor (=l/4G8

) , cf. 

(5.20 ) for Method 1 . Accordingly, we have 

(7.5) 

Exact even moments for the i nstantaneous ampl i tudes of Cl ass B i nterference a re found at once from the resul ts of Sec. 5.2A,B , and 5.3 [Mi ddl eton , 

1 976] , when we note that 

(7.6) 

A 

wh ere F

1

( i aA j z�) is given by ( 4 . 1 8 ) when th e upper l i mi t on the i ntegral 

is repl aced by z� : see the comments of Sec. 5 . 2B [Mi ddl eton, 1 976] . The 

res ul ts are di rectly 

(z

0

)

8 = (as expected ) (7.7a) 

(z

2

)8 = (as expected ) (7.7b) 

56 
5 n6B + .!§_ n4B + 15 t : 2 3 ( 1 I )

3 2 2 (1 I )

2 ' e C . ' 

n2B +r

B n2B +rB 

whe re, as before , 

, cf. (5. la) ; and 

(7.7c) 

(7.7d) 

(7.8a) 

() =<fo00( ) dz)A0

, e

0y

, aRT 'A, cf. (4. 1 9a) (7.8b) 

cf. (5. 1 4a), [Mi ddl eton , 1 976] . 

The odd moments , (

z

2i

+ l

)8

, of course , vani s h , by vi rtue of the symmetry of 

w

1 (z)8

, or equi val ently, the even (i n �- ) character of the (exact , and approximate ) c.f . 's (4. 1 8 ) , (4. 23 ) . [We ca ll the reader 's attention , again, to the 

i mpo rtant condi tions in the p rocedures for eval uati ng these (exact) Cla ss B 

momen ts , and to the genera l canonical nature of the present analysis vi s-a-vis 

th e earl ier wo rk of Giordano and Haber [1 970, 1 972] , and Furutsu and Ishida 

[1 960] , discussed on pp. 117-1 1 9 of [Mi ddl eton , 1 976] . 

Final ly, we h ave , as before [Mi ddl eton , 1 976; Sec. 6] , the global and 

generic parameters of the Class B, which a re, as expected , the same as in 

the envelope cases , s i nce we are dealing wi th the same i nterference at the 

receiver i nput. These pa rameters may be obtained , as i ndi cated earl i er, 

from envel ope s tati s ti cs [Sec. 6B , Mi ddl eton, 1 976] , or by analogous forms 

usi ng the PD 's and moments derived above [Sect i ons 5 ,6] . These pa rameters 

are bri efly descri bed again in Section 2, preceeding. The practical condi tions for Class B vs . Class A i nteference model s (vis -a-vis the receiver , 

57 
"I 

of course) are j ust those al ready ci ted in Sect i on 7 [Mi ddl eton , 1 976] and 

wtl l no t be repeated here . 

APPEND IX A: The Ap

roxi mating PD 1s: 

In order to use (4. 5), (4. 5b) wi th the appropri ate approximating c.f . 1s, 

A 

F

1

(ia\ ) B-I,II ' we must ta ke care to emp l oy the correspondi ng pdf 1 s, w

l -B-I ,II 

* in the correct fashion . This is done as fol l ows , assisted by Fi gure A-1 , 

and remembering that w

I,I I are symmetri cal pdf1s (about z=O ) : 

We fi rs t d istinguish the fol l owing cases for the i nstantaneous 

ampl i tu de : 

A. - 1 . 

A. -2. 

A.-3. 

z < - z o oB 

-zoB < 2

o 

z B < z 0 0 

< 2

oB 

For the fi rst, A-1 ., we have at once , by definition of P

1 

P

l

( z>zol ) =�

-zoB 

wl l dz + �

z

oB 

wl

dz + �oo w! I

dz 

2

ol - zoB 2

oB 

which can be rewri tten di rectly as 

(A. l -1 ) 

(A. 1 -2 ) 

The fi nal term in (A. 1 - 3) is, wi th the ai d of the symmetry properti es of 

the pdf1s, 

* I t is assumed here that w8_ I II w

l - B-I II are proper pdf 1 s, i . e . , w

I I I..:

O 00 ' ' ' 

and J_00w

I,IIdz= l . This is es tab l is hed in the text , cf. Secti ons 5.1 , 6 ( end ) . 

58 
(A. 1 -4) 

whi le the fi rst three terms reduce at once to 

(A .1-5) 

We next requi re th at P 1 _ 1 ,P1 _11 be equa l at J z0

J =z

08

: the respecti ve PD 's 

a re continuo us at the 11turning point" , z

08 (>0 ) . [Additi onal conditi ons for 

the composi te approximation usi ng the c.f. 's (or pdf's) fo r regi ons B-I,II are 

devel oped in Sec . ( 5. 2 ) .] Accordi ngly, we see from (A. 1 -3)-(A. l -5) that now 

s ubject to the 11joining 11 condi ti on 

cf. (A. 1 -5) et seq. 

We proceed in similar fashion for the cas es A.l- 2,3, to get 

(A .l-6) 

(A . 1 -7) 

(A . 1 -8) 

(A . 1 -9) 

A simi l ar procedure for the modul us , Jzl , ( shaded regi ons in Fi g. A . l) 

may be used to give us di rectly 

59 
P

1

( i z l > l z0

1) = P

1

( i z l�l z

0

1 ) 1 ' 0 .:

 i z0

i .:

 z

oB

' } = P 1 ( I z I � I z

o I ) Ir ' I z

o I >z

oB ( >O) 

(A. l-10 ) 

(A .1-11) 

(A . 1 -1 2 ) 

wi th the appropri ate doma i ns O.:

l z0

i <z

0B , l z

0

i >z

oB

' respecti ve ly . Hence 

follow Equati ons (5.5a,b) in the mai n text . (Simi l arl y, if we repl ace z 

by E, z0B by cB

' etc . , Eqs . (A. l -1 0-1 2) veri fy our earl i er resul ts , Eqs . 

(2 . 7 ) , [Mi ddl eton, 1 976 , 1 977] .) 

60 
O'I 

I-' 

Z o1 

Figure A- 1 

I 

rr_

I 

I 

- Z o s 

W1 (z ) s-r, rr 

I 

Z o2 0 

I 

II 

r - � 

1 

Z o

3 Zos Zo

4 z-+- + co 

Schema of the pdf ' s , w1 (z) B-I ,

I I ' appropriate to the regions BI ,

I I . 
A. A0 

A

A

, AB

,A ,A A

A 00 oo B , oo , 

A 

a 

b

l a ' b

2a ' b

2a+2 j a 

s 

C. c.f . 

D. o

1 

8 

E. E 

A 

F. F

1 , F1 

1 F

l 

Li.fN,Li.f ARI 

f 

G. G0 

g(J) 

r' B 

r (x) 

y 

H. 

Glossary of Pri nci pal Symbols 

= pea k ampl i tude of typi cal i nput s i gna l 

= Impul si ve I ndexes , ( Cl ass A,B i nterference ) 

= effecti ve Impul sive I ndex 

= norma l i zi ng factor 

=a pos teri ori probabi l i ty; here l - Distributi on=P1 

= combi ned apertu re-I F-IF rece i ver i nput stages 

= source , rece i ver beam patterns 

= spatial densi ty propagat i on parameter 

= generi c or typi cal envel ope of waveform from ARI 

recei ver s tage 

= wei ghted moments of the generic envel ope B

0 B 

= exponent of moment 

= characteristi c functi on 

= probabi l i ty dis tri buti on 

= del ta (singul ar ) functi on 

= i mpu l se epoch 

= no rma l i zed doppl er 

= characteri s ti c functi on 

= confl uent hypergeometri c functi on 

= no ise, recei ver bandwi dths 

= frequency 

= a basic waveform 

= geometrica l factor of recei ved waveform 

� rat io of ( i ntensity of ) gaussi an component to that 

of the "impul sive", or nongaussi an component 

= gamma functi on 

= exponent of propagation l aw, wi th range 

62 
K. s 

L. A 

A. 

A. IN 

N. n . b . 

O. rl2B 

p. p

l 

pdf 

'¥' � 

�

T '

�

R 

Q. 

R. r 

p 

= exponent of c ha racteri sti c functi on 

= i ncompl ete r- functi on 

= unit vector 

= Bessel funct i on, 1 st-ki nd , (0,1 order) . 

= j acobian 

= c.f . vari abl e 

= doma i n of i n tegrati on 

= argumen t of the c.f. 

= (A. ,6,�), coordinates 

= exponent of source dens i ty l aw wi th range 

= norma lized dopp l er 

= narrow-band 

= mean intensi ty of the nongaussian component 

= angul ar frequencies (w0= carri er angul ar fr . ) 

= APD or Exceedance Probabi l i ty 

= pro babi l i ty densi ty functi on 

= phase of narrow band wave 

= ape rture ph ase 

= c.f . variabl e 

= poi sson 11densi ty11 

s. 

A 2 2 2 

cr,crG,cr,�cr

G

, crA

, crR, crE = vari ances 

cr

s , v = so urce densi ty 

-

T . T

s

,Ts:B 

t , t

,

, t2 

e e' vv 

,.,.

 

= emission durati on 

= times 

= sets of waveform parameters 

63 
T. 8 

v. 

w. w

l

,wl 

x. x 

Y. 

= error function 

= basic waveforms out of ARI rece i ver stage 

= norma lized envel ope waveform at output of ARI 

s tages 

= probabi l i ty densi ty funct i on 

= i nstantaneous ampl i tude 

=a c.f . variabl e 

= (normal i zed ) i nstantaneous ampl i tude 

= a normal i zed time, al so , a normal i zed ampl i tude 

threshol d 

= (normal i zed ) 11bend- over1 1 point 

64 
REFERENCES 

l. Furutsu, K . , and T. Ishi da ( 1 960) , 110n the theory of ampl i tude di strib uti on of i mpul sive random noi se11 , J. App l . Phys . (J apan), Vol . 32 , 

pp . 1 206- 1 221 , July (1960 ). 

2. Gi ordano , A.A. ( 1 970) , 11Modeling of atmospheri c noise 1 1 , Ph. D . dissertati on, Dept. of El ect. Eng . , Univ. of Pennsyl vani a, Phi l adel phi a. 

3. Gi ordano , A.A. , and F. Haber (1972 ) , 11Model i ng of atmospheri c no i s e1 1 , 

Rad i o Sci ence , Vol . 7, No . 1 1 , pp . 1 01 1 - 1 023 . 

4. Ha l l , H.M . (1966 ) , 11A new model fo r 11i mpul sive 1 1 phenomena : appl i cation 

to atmospheric-noise communi cati ons channel s 1 1, Stanford Uni versi ty 

El ectroni cs Laboratori es Tech. Report Nos . 341 2-8 and 7050-7, SU-S EL66-052 . 

5. Mi ddl eton, D. ( 1 960 ) , An Introducti on to Statisti cal Communi cation 

Theory, McGraw Hi ll ( New York ) . 

6. Mi ddl eton, D. ( 1 972a ) , 11Statis ti cal -physi cal mode ls of u rban radi ono i se envi ronmen ts - Part I: Fo undat i ons 11 , I EEE Trans . Electromagneti c 

Comapt ., Vol . EMC-14, pp . 38-56 , May , 1 972 . 

7. Mi ddl eton, D. (1972b ) , 11Probabi lity model s of recei ved , scattered , 

and ambient fiel ds11 , i.n Proc. IEEE 1 972 Int 1 1. Con f. Engineering in 

the Ocea n En vi ronment , Newport , R. I . , Sept . 1 3- 1 6, 1 972 . 

8. Mi ddl eton, D. (1973 ) , 11Man-made no i se in urban envi ronments and 

trans portation sys tems : Model s and measurements 11 , I EEE Trans . Communicati ons , Vo l . COM-21 , No . 1 1 , pp . 1 232-1 241 , Nov . , 1 973 . 

9. Mi ddl eton, D. ( 1 974 ) , 11Statisti cal - physi cal model s of man-made radi ono i se, Part I: Fi rs t-order probab ility model s of the i nsta ntaneous 

ampl i tu de 1 1 , Offi ce of Te l ecommuni cati on s , Techni cal Report OT -74-36 , 

Apri l, 1 974 (U.S. Gov 1 t . Printing Offi ce , Washington , D.C. 20402 ). 

1 0 . Mi ddl eton, D. ( 1 975a ) , 11Perfo rmance of tel ecommuni cati on systems in 

the spectral -use envi ronment : I., Obj ecti ves and approaches for the 

general data base 11 , Executi ve Offi ce of the Presi dent, Offfi ce of 

Tel econmunicat i ons Pol i cy, Jan. 3 1 , 1 975 ( Res earch Paper) . 

11. Mi ddl eton, D . , ( 1 976 ) , 11Stati sti ca l physi cal model s of man-made and 

natural radi o noi se . Part II: Fi rs t-order probability models of the 

envel ope and phase . 

65 
I 

I 

1 2 . Mi ddl eton, D . , (1977 ) , 11 Statisti ca l - physi cal model s of el ectromagneti c 

i n terferen ce 11 , I EEE Trans . El ectromagneti c Compati bi l i ty, Vol . EMC- 1 9, 

No . 3, Aug . 1 977 , pp . 1 06-1 27. 

1 3 . Spaulding , A. O . , and D. Mi ddl eton ( 1 975 ) , 11 0ptimum recepti on in an 

impulsive i nterference envi ronment11 , Offi ce of Tel ecommuni cati ons , 

Technical Report OT-75-67 , J une , 1 975 (U.S. Govt . Pri nti ng Offi ce 

Washington, D.C. , 20402 ). 

1 4 . Spaulding, A. D . , and D. Mi ddl eton , 1 977 ) , 1 1 0ptimum recepti on in an 

impulsive i nterference envi ronment - Pa rt I, Coherent detection11 , 

pp . 91 0- 923 ; 11Part II, I ncoherent recepti on, pp . 924-934; I EEE Trans . 

on Comm . , Vol . COM-25, No . 9, Sep tember, 1 977. 

15. Watson , G.N. ( 1 944 ) , Theory of Bessel Functi ons , MacMi llan (New York ) , 

1 6 . Spa ul ding , A.D. a nd D. Mi ddleton ( 1 978 ) , 11 0ptfmum recepti on in a Cl ass· 

B i nterference envi ronment 11 (tent. -ti tl e; s chedul ed , Fal l , 1 978 , Qffice 

of Tel ecommuni cati ons . 

66 
F O RM OT-29 U .S. DE PA RTMENT OF COMMERCE 

( 3·73) OFF ICE OF T E L E C OMMUNI CATIO NS 

B I BLIOGRAPHIC DATA SH EET 

1. PUBLICATION OR R E PORT NO. 2. Gov 'r Access ion No. 3. Rec ipient 's Accession No. 

NTIA-CR- 78-1 

4. TITLE AND SUBTITLE 5. Pub l ication Date 

Stati stical -Physical Model s of Man-Made and Natural June 1978 

Rad io Noi se Part III : Fi rs t-Order P roba bi lity Model s of 6. Performing Orga nizar ion Code 

the I nstantaneous Ampl i tude of Cl ass B i nterference 

7. AUTHOR (S) 9. Project/Task/Work Unit No. 

Dr. Davi d Mi ddl eton 

8. PER FORMING ORGANIZATION NAME AND ADDRESS 

U.S. Deoartment of Commerce 

National Tel ecommun i cati ons and 1 0. Contract/Geant No. I n formati on Admi nistrat i on/ I TS Boul der Laboratori es OT-01 24 Bou l der. Col orado 80303 

11. Sponsoring Organization Name and Addre ss 1 2. Type of Report and Period Covered 

U.S. De1a rtment of Commerce 

Nati ona Tel ecommuni cati ons and 

i nformati on Admi nistration 1 3. 

�Jas hi ngton , D.C. 20005 

1 4. SUPPLEMENTARY NOTES 

Prepa red under contract by D. Mi ddl eton , 1 27 E. 9l st St . , New York , N.Y. l 0028 

1 5. A BSTRACT (A 2 00·word or less factual summary o f m o s t significan t informa tion. If documen t in cludes a significant 

bibliography ot literature survey, men tion it here. ) 

Th is Report is the thi rd ( 11 Pa rt II I 11 ) in a continu i ng seri es devoted to the 

devel opment of anal yti cal ly tracta bl e, statistica l -phys i cal model s of man -made 

and natura l el ectromagneti c i nterference. Here , the fi rst-order stati sti cal 

proba bi l i ty densi ti es ( pdf ' s) and the assoc i ated exceedance probabi liti es ( PD1s, 

or APD 1 s ) a re o btai ned for the i n stantaneous ampl i tudes ( x) ' and i n stantaneous 

ma gni tudes , I x I , of Class B noi se . These are needed not on ly for experi menta l 

studi es but , a l so, particul arly for the analysis and eval uati on of the performance 

of opti mum and suboptimum recei vers in Cl ass B i nterference envi ronments. 

As in the earl i er stud i es of the envel ope stati sti cs of Cl ass B no i se 

[Mi ddl eton , 1 976] ' a two-function approximation is needed for the characteri s ti c 

function and hence for the correspondi ng pdf ' s and PD 's. Two methods of determi ning the six ( b as i c ) parameters wh i ch descri be these fi rst-order statisti cs and 

thus joi nin g the approximate forms ( pdf's and P D 1 s ) are outl i ned . Method 1 is approximate , was used earl i er [Mi ddl eton , 1 976 , 1 977] ' and has the advantage of 

16. Key Words (A lphabetical order, s eparated by semicolons) 

Interference Model s; Man-Made Noise ; Natura 1 No i se; P robabl i s ti c Noi se Model s 

1 7. AVAILABILITY STATEMENT 1 8. Secur i ty Class (This report) 20. Numbet of pages 

[X] UNLIMITED. Uncl as sifi ed 74 

1 9. Se c ur ity Class (This page) 21. Pr ice: 

0 F OR OFFICIAL DISTRIB UTION . 

Unc 1 ass i fi ed 

L1 SC OMM- DC 297 1 6- P7 3 
somewhat greater computational simpl i ci ty, wi th the disadvantage , however, of 

yiel ding too l ow val ues of the PD at l ow va l ues of the argument (X) , when the 

gaussian component is smal l . Method 2 is 11 exact" , and somewhat more compl ex 

computati onally . The joining process i nvol ved in both methods has been essentially descri bed earl i er [Mi ddl eton, 1 976 , 1 977] but is devel oped further here . 

The basic parameters are , in any case , the same as those deri ved for the envel ope stati sti cs. The excel l ent agreement wi th experiment observed for the 

envel ope data accordingly a ppl i es here , as wel l. 

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