OT REPORT 76-86 

STATISTICAL -PHYSICAL MODELS 

OF MAN-MADE AND NATURAL 

RADIO NOISE 

PART II: 

FIRST ORDER PROBABILITY MODELS 

OF THE ENVELOPE AND PHASE 

D. MIDDLETON 

U.S. DEPARTMENT OF COMMERCE 

Elliot L. Richardson. Secretary 

Betsy Ancker-Johnson. Ph.D. 

Assistant Secretary for Science and Technology 

OF FICE OF TE LECOMM UNICATIONS 

Jo hn M. Rich ardson , Acti ng Dire c1or 

April 1976 

For sale by the Super intendent of Documents, U.S. Government Printing Office, Wash ington , D.C. 20402 


interference levels. This is noticeably different from the usual normal 

(gaussian) noise processes inherent in transmitting and receiving elements. 

This highly structured character of the interference can drastically degrade performance of conventional systems, which are optimized, i.e. 

designed to operate most effectively, against the customarily assumed 

normal background noise processes. The present Report is devoted to the 

prob~ems ·of (1), (2) above, namely, to provide adequate statistical physical 

models, verified by experiment, of these general 11 impulsive 11

, highly non-

. gaussian interference processes, which constitute a principal corpus of the 

interference environment, and which are required in the successful pursuit 

of (3), as well . The principal new results here are: 

(i). Canonical analytical models, experimentally corroberated, for the 

first-order statistics of the envelope and phase of Class A and 

Class B noise*;_ 

(ii). Procedures for estimating the (canonical) model parameters, 

calculation of moments, APD 1 s (= exceedance probabilities, 

P

1(DE0 ), e_tc . ) and probability density functions (pdf's), and 

a variety ·of other pertinent statistics; [see the Table of 

Contents]. 

~inally, we emphasize, again, that it is the quantitative interplay 

between the experimentally established, analytical model-building for the 

electromagnetic environment, and the evaluation of system performance 

therein, which provides essential tools for prediction and performance, for 

the development of adequate , appropriate data bases, procedures for effective standardizations, and spectrum assessment, required for the effective 

management of the spectral-use environment. 

* Class A and C1ass B noise are distinguished, qualitatively, by having 

input bandwidths which are respectively narrower and broader than that 

of the (linear) front-end stages of the typical (narrow-band) receiver 

in use. More precise definitions are developed in the text following. 

iv 
TABLE OF CONTENTS 

Abstract 

PART I: INTRODUCTION, RESULTS AND CONCLUSIONS 

1. INTRODUCTION 

1. 1 Classification of EM Interference 

1.2 Earlier Work 

1.3 New Results 

1.4 Organization of the Report 

2. RESULTS AND CONCLUSIONS 

2.1 Phase Statistics 

2.2 Envelope Statistics: The APO and pdf for Class A 

Interference 

2.3 Envelope Statistics: The APO and pdf for Class B 

Interference 

2.4 Comparisons with Experiment 

2.5 Remarks on the Estimation of Model Parameters 

2.6 Some Additional Results 

2.7 General Corrunents; Next Steps 

208 Acknowledqments 

PART II: ANALYSIS 

1. INTRODUCTION TO THE ANALYSJS 

2. FORMULATION : THE CHARACTERISTIC FUNCTION 

2.1 The Basic Statistical Model A 

2. 2 First Reduction of the C.F. F

1: The Narrow-band 

Receiver Condition 

2.3 Interference Classes A,B, and C: the Role of 

Input and Receiver Bandwidths. 

2.3.1 Some Extensions 

2.4 Large Impulsive Indexes · 

2.5 Second Reduction of the C.F. F,: the Role of 

Source OistributiGn and Propagation Law 

V 

1 

3 

4 

8 

9 

10 

12 

12 

13 

16 

20 

30 

32 

34 

37 

38 

39 

39 

44 

51 

56 

58 

61 
2.5.1 . Propagation Law 

2.5.2 Source Distributions 

2.5.3 Role of Input Signal Duration, Ts 

2.6 The C.F. for Class A Interference 

2.7 The C.F. for Class B Interference 

A. The Approximating C.F. 1s for (O<a<2) 

3. PROBABILITY DISTRIBUTIONS: P

1(£>£0 )A,B'(O<a<2) : 

3.1 Class A Interference 

3. 2 Class B Interference (O<a<2) 

A. The Composite Approximation 

B. Remarks on Hall-type Models 

4. PROBABILITY DENSITIES: w1(t:,)A,B' (O<a<2) : 

4.1 Class A Interference 

4.2 Class B Interference 

5. MOMENTS 

5.1 Existence and Direct Calculation (Approximate Forms) 

5.2 Class A and B Moments: Exact Forms (Even 

Moments Only) 

A. Exact Class A Even Noments 

B. Exact Class B Even Moments 

5.3 Remarks 

6. DETERMINATION OF THE BASIC FIRST-ORDER PARAMETERS 

A. Class A Interference 

B. Class B Interference 

C. Degenerate Cases 

7. PRACTICAL CONDITIONS FOR CLASS A AND CLASS B INTERFERENCE 

Glossary of Principal Symbols 

References 

vi 

62 

63 

67 

68 

74 

78 

83 

84 

86 

93 

97 

104 

104 

107 

111 

111 

114 

115 

117 

118 

120 

120 

122 

125 

127 

130 

133 
LIST OF FIGURES 

Figure 

1.1. Schema of the EM interference and desired signal environ- ment vis- ~- vis a typical narrowband receiver. 7 

2.1. Comparison of measured envelope distribution, P1 (e > eo)A, 

with Class A model, cf. (2.2) . Interference from orecrus.hing machinery [data from Adams et al (1974)]. 21 

2. 2. Comparison of measured envelope distribution, P1 (t1'> eo)A, 

with Class A model, cf. (2.2). Interference (probably) 

from nearby powerline, produced by some kind of equipment 

fed by the line [data from Bolton (1972)]. 22 

2.3. Comparison of measured envelope distribution, P1 (€'> eo)B, 

of man-mad~ interference (primarily automotive ignition 

noise) with Class B model, cf. (2.7a). [Data from Spaulding 

and Espeland (1971)}. 23 

2.4. Comparison of measured envelope distribution, P1(e > €'o)s, 

of atmospheric noise with Class B model , cf. (2.7a). {Data 

from Espeland and Spaulding (1970)]. 24 

2.5. Comparison of measured envelope distribution, P1(e> e'o)B 

of man -made interference (fluorescent lights in mine shop 

office) with Class B model, cf. (2.7a) . [Data from Adams 

et a 1. (197 4)] • 25 

2.6 . Comparison of me~sured envelope distribution P1(e'> 6\)s 

of automotive ignition noise from moving traffic with 

full Class B model, cf. (2.7a,b). [Data from Shepherd 

(1974), fig. 14.J 26 

2.7. Comparison of measured envelope distribution P1(€> e'o)s 

of atmospheric noise with full Class B model, cf. (2.7a,b). 

[Data from Esperland and Spaulding (1970), p. 42.] 27 

2.8. Comparison of measured envelope distribution P1(€> e'o)s 

of man-made interference (mining machinery noise) with 

the full Class B model, cf. (2.7a,b). [Data from Bensema 

et al. (1974), fig. 67, p. 115 .] 28 

2.1 II. Typical output envelope of ARI filter output in (narrow- band) receiver, showing Class A and Class B envelope com- ponents (positive halves only); Cl ass C = Class A+ Class B. 53 

2. 2 II. Geometries of source and receiver beam patterns : {a), in 

a plane; (b), in a volume. 64 

2.3 II. Several typical cases (A only) of source and receiver 

domains. 66 

vii 
LIST OF FIGURES (can't) 

Figure 

A 

2.4 II. Schema of the (x,~)-doma1ns of integration for 100 , 

(.2.65), for Class A,B interference . 

3.1 II. The envelope distribution [Prob(e > Co)] calculated 

for Class A interference for AA= 0.1 and various rA 

from eq. (3.7b). 

3.2 II. The envelope distribution [Prop(e> ~'o)J calculated 

for Class A interference for fA = 10~~ and various 

AA from eq • ( 3 • 7 b) • 

A A 

3. 3 II. The envelope distribution [Prob(€ >e\)J calculated 

for Class B interference for a= 1.0 for various ~a 

from eqs. (3.llb, 3.15) . 

3.4 II . The envelope distribution [Prob(8° > €0)] calculated 

for Class 8 interference for Aa = 1.0 for various a 

from eqs. (3.llb, 3. 15) . 

3.5 II . Schema of P1_s, eq. (3.20), obtained by joining the 

two approximating fonns (3.llb, 3.17). 

3.6 II. The (complete) envelope distribution P1(€ > e'o)s, eq. (3.20), calculated for Class B. interference for various 

69 

87 

88 

91 

92 

95 

Aa, given a eqs. (3.11, 3.17, 3.19, 6.9, 6.10). 98 

3.7 II . The (complete) envelope distribution P1(t> e°o)B, eq. (3.20), calculated for Class B interference for various 

a, gi ven Aa (eqs. 3.11, 3.17, 3. 19, 6.9, 6.10) . 99 

4.1 II. The pdf., W1(€)A, of the envelope for Cl?ss A i nterference, 

calculated from eq . (4.2) for various rA, given AA [cf. 

fig. 3.1 (II) for Pi(e > eo)AJ· 105 

4. 2 II. The pdf, w1(~)A, Qf the envelope for Class A interference, 

claculated from eq. (4.2) for various AA, given fA [cf. 

fig . 3.2 (II) for P1(t > e0 )A]. 106 

4.3 II. The (complete) pdf w1(S)s, eq. (4.5), of the envelope 

for Class B interference, calculated from eqs. (4.3, 

4.4) for various Ao., given [cf. (3. 19)]. [See fi g. 3.6 

(II) for the associated Pi(€> ei)8 and parameter values.] 109 

4.4 II. The (complete) pdf, w1(€)8, eq; (4.5), of the envelope Cl ass B interference, calculated from eqs. (4.3, 4. 4) 

for various a, ·given ~ [cf. (3.19)]. [See fig. 3.7 (II) 

for the associated P1(! > So)B and parameter values.] 

viii 
LIST OF TABLES 

Table Page 

1.1. Interference Categories and Classes 8 

2.1. Class A vs. Class B Interference 33 

6.1. Class A Parameters 120 

6.2. Class B Parameters 123 

ix 




decay occur, with the latter predominating. The 

receiver is to varying degrees "shock-excited", 

particularly for inputs of very short duration, so 

t ha t the receiver is said to ''ring". 

Class C Interference: This is the sum of Class A and Cl ass B interference, 

which can occur either because of the presence of 

sources of mixed types (producing Class A, Class 

B emissions vis-a-vis t he receiver), and/or because any received emission is itself strictly 

Class C: there is always a build-up interval and a 

decayi ng transient period in any recei ver front-end 

reac tion to an incoming emission. Effecti ve Class 

C occurs. in this latter instance where the bui ld-up 

and decay times (at comparabl e levels} are themselves comparable . 

For Class A noise the transient decay period is negl igible vis-a~vis the 

emission's duration, while for Cl ass B interference it is highly dominant. 

See, for exampl e, Fig. (2. 1) , Part JI, following. [More precise, quantitative 

conditions specifying Cl ass A, or Class B types, vis-a-vis Cl ass C and each 

other, are derived in Section 7, Part II. ] 

The above three categories for interference, as it impacts on a typical 

(narrow-band) receiver, e.g . , as (the l inear, front-end of) that receiver 

responds to the EM environment, provide a useful way of describing the 

different effects which these different categories have on reception. This 

categorization is useful because receiver response is statistically different for each Class. As will be seen presently, these differences appear 

most generally and explicitly (as far as first-order statistics are concerned 

cerned) i n the experimentally derived, and theoretically determined 

exceedance probabilities (PD's) [also often cal led APD 's (a posteriori probability distri butions , cf. Spaul ding, [1971])] , such as P

1(X>X0 ), or 

P1 (£'>[0 ), wh ich are the respective probabilities that the instantaneous 

ampl itude, or instantaneous envel ope observed at the receiver's IF output 

exceed some threshold \• or €0 , as these l atter are allowed to assume 

5 
values in the interval (-~ ... ), or (o,""). Furthermore, this categorization 

is recommended because the conditions governing the various Classes are 

simple to distinguish, cf. remarks in Section 7 (II). The conditi ons 

"spectrally broader than ", and "spectrally narrower than", cf. Fig. (l.l), 

are to be interpreted as "sufficiently broader or narrower", etc., where in 

any case, care is taken to refer to the definitions of Class A,B, etc., 

in terms of the residual transi ents vs. the "on"-time of the input emission 

which appears at the output of the IF stage of the receiver in question. 

It is instructive to extend our schema of classification further, in 

order to distinguish between man-made and natural interference, and 

between "intelligent" and "nonintelligent" emissions. Accordingly, we 

define: 

( i). "Inte 11 i gent" noise or interference as man-made and intended 

to convey a message or information of some sort; whereas, 

(ii). "Nonintelligent" noise or interference may be attributable 

to natural Phenomena, e.g., atmospheric noise or receiver noise, 

for example, or may be man-made, but conveys no intended communication, such as automobile ignition, or radiation from power 

lines, etc . 

We remark again [cf. Middleton, 1960, Sec. 1.3-5] that by definition, 

"noise" or "interference" is any undesired "signal" at or in the receiver, regardless of origin.] The importance of distinguishing man-made from natural 

noise lies in the fact that the former is potentially controllable, some-

.times to the point of elimination, whereas the latter cannot be eliminated, 

at the source, and is usually not subject to control: one can seek only 

to investigate its effects on the communication process. Moreover, the 

distinction beb1een "intelligent" and "nonintelligent" is always significant with regard to information transfer: the taxonomy of the former 

can have greatly different .iinpl ications and consequences from that of the 

latter. 

We can readily tabu l ate these different varieties of interference, 

6 
(Desired Signals) 

EM 

...., 

11Environm ent11 

r--------, 

I 

I 

• 

I 

I I 

I 

I I 

I 

I 

I I 

I 

I' " I . \. LJ . 

6fN 

I I 

Interference l 

Source(s) l 

1 

I I ---------' 

Receiver 

Narrow-Band 

6fR . 

~Aperture X R 

• 

Class A: 6fN Class B : 6fN Class C: Clas 

Process·n 1 

g 

F x IT -------~

< 6fR (or match

> ~fR 

s A + Class 8 

Figure 1.1. Schema of the EM interference and desired signal environment vis- a- vis a 

band recei ver. 
in a concise way as suggested in Table (l.l) below: 

Table l .l Interference Categories* and Classes 

Type "Intelli gent" Class ''Nonintelligent'' Class 

Man- 1 ) • Compatible A l ) . Automobile ignition B 

Made 

2). Incompatible A,B,G 2). Other EM emissions: A,B,C 

(Communication} power lines, electric tools, etc . 

[3}. Extra-terrestrial A,B,C] 

( Communication) 

Natura 1 1 } • Atmospheric B 

2}. Extra-terrestrial .[A]• solar, galactic, B,C 

cosmic radiation, 

etc. 

* The listing here is not intended to be exhaustive. 

We have included a further refinement through the term "compatible". By 

definition, compatible interference here is one that is appropriately 

matched spectral ly to the receiver band t,.fARI' in the sense of being equivalent to Class A interference vis-a-vis the receiver and occupying a 

spectral region in t,.fARI' and such as to produce ignorable transients 

in the ARI -stages. "Incompatible" may mean that t,.fN > t,.fARI (Class B}, or 

that only a portion of the incident emission is spectrally available to the 

receiver: Class- A again, e.g. t,.fN-effective < ,:,fARI, but now the interference is not wholly in the receiver band t,.fARI" Class C above in the 

Table reminds us that combinations of Class A and B noise can occur, as 

well. 

1.2 Earlier Work: 

For the most part, earlier efforts at modelling man-made and natural 

noise. (principally atmospheric noise} have produced a wide variety of 

8 
analytical results, often wi th t he virtue of mathematical simplicity, but 

severely l imi ted i n usefulness by lack of generality and physical insight, 

and a concommitant dependence on local, empirical data and circumstances. 

Two important excepti ons to the above are t he work of Furutsu and Ishida 

[1960] on oDtaining t he APD's (and associated probability densities [pdf's]) 

of atmospheric noise under rather broad cond itions, and the more recent 

!,tudies of Giordano [1970]. and Giordano and Haber [1972), similarly direct ed to atmospheric no ise. Both sets of intestigations, howeve r, are 

{necessarily) constrained to Cl ass B types of interference (cf. Sec. (1.1) 

above), and neither attempts the canonical formul ation, which is a key 

feature of our current efforts [Middl eton, 1972, 1973, 1974, and . this 

Report; see also comments, Sec . 5. 3 (JI)] followi ng. This canonical formulat i on allows us to apply the new. models formally by Class(A,B, etc.) to 

all types of (EM) interference, unrestricted in general structure by t he 

parti cul ar physica l mechanism invol ved. [These latter, of course, determine 

the generic properties of the model parameters, and must be specificall y 

i ntroduced into model building if the ad hoc and arbitrary empiricism of 

much of the earl ier work is to be avoided.] For a mo re detailed revie~, 

of earli er work vis-a-vis th is newer approach , see Chapter 2 of Spauldi ng 

and Middleton [1975], and references therein. 

1.3 New Results: 

The principal new results of this study may be briefly introduced here, 

in contrast to our remarks above on previous work. Here we obta in canonical, 

analytical , first-order statistical mode ls of both Class A and Class B 

interference, specifical ly for the envelope (E) and phase (~) of the narrowband outpu t of the composite aperture-RF-.I F stages of a typical receiver. 

As noted above , these models are based on a general physical mechanism 

{cf. Section 2, Middleton [1 974], for example), providing, among other 

things, insight into the parameter structure, as wel l as contributing , 

in a broad way to the analytical form of the probability distribution (PD's) 

and probability densi ti es {pdf 's) themsel ves, which are the principa l 

resul ts here. In additi on, the general method of approximating the 

governing {1st-order) characteristic functions (c .f . 's) .is described, wh ich 

9 


2. RESULTS AND CONCLUSIONS 

Let us now summari ze in some detail the principal results of the analysis in Part II of this Report. Accordingly, Section 2 here is organized 

as follows: first, we briefly consider the first-order resu~ts for the 

phase, w, of the narrow- band output of ARI-stages of a typical narrow-band 

receiver immersed in the general EM interference environment under analysis here [cf. Section (1 . 1) above]. Next, we present in Section (2 .2) 

various results for the envelope statistics of Class A interference, with 

a similar presentation in Section (2.3) for the Class B cases. A number of 

comparisons with experiment are given in Section (2.4), both to demonstrate 

the canonical character of our models and to exhibit the excellent agreement between theory and experiment which is obtainable by our present 

analytical approach . Section (2.5) treats the estimation of the physicallybased model parameters; Section (2.6) reviews such other results as moments, 

limiting forms, the existence of Hall models, conditions for the existence 

of Class A, B, and C noise types, etc. We conclude with remarks in Section 

(2.7) on uses, advantages, and limitations of these models and outline a 

number of next steps for their continuing analytical and experimental 

development. 

2.1 Phase Statistics: 

In the general case we may use (2.14), Part II, and the relation 

w1(w) = J~w1 (E,w)dE, to obtain the pdf, and the APO(= 1;w,(w)dw, (0<~<2~)), 

of the i nstantaneous phase~, which will not generally be uniform [on 

(0,2~)]. However, in the truly narrow-band situation of Section (2.2)11, 

we obtain the will-known uniform pdf [(2 . 21), Part II), e.g. w1(w)=l/2~, 

(0,2~)] as in the simpler, gaussian examples. [Higher-order statistics 

of w, on the other hand, are nonuniform and analytically much more complex: 

vide Section 9. 1. 2 of Middleton [1960] in the gaussian cases.] Because 

of this first-order simplicity for the statistics of the phase we accordingly concentrate our attention on the (first-order) statistics of the 

associated envelope, which, as expected, departs radically from a gaussian 

(i.e. rayleigh) behaviour, as our results following, and experiment as well, 

12 
amp1y demonstrate. 

2.2 Enve1ope Statistics: The APO and pdf for C1ass A Interference: 

Our principal analytical results here are: (i), the c.f.; (ii), the 

PD, or exceedance probability P1(E>&0 ); and (iii), the associated pdf, 

w1(8). These are respective1y* 

(c.f.): , a2 = [2n2

A(l+rfl.)f 1 } 

cf. Eq. (3.3), Part II , 

( 2.1) 

with 2;~ = (m/AA+r;...)/(l+rp,), cf. (Eq. (3.5), Part II; and 

(PD): 

Am -£212i A e o mA 

mT 

cf. Eqs. (3. ?a,b), Part II; and 

(pdf): 

(O<f:. <oo), - 0 (2.2) 

(Q _2 £ < oo) ' (2.3) 

cf. Eq. (4.2), Part II. Various curves of P

1 ,w1 are given in Figs. (3 .l), 

(3.2), (4 .1), (4.2), Part II, showing typical behaviour for selected values 

of the (global) parameters (AA,rA). Here c,E.. 0 are normalized envelopes 

(2.4) 

cf. Eq. (3.1 ), Part II, where E0 is some preselected threshold value of the 

envelope E. (Note that our normalization introduces a third parameter n

2

A.) 

The parameters (AA,rA,n2A) which appear directly in our statistical 

results for P

1 ,w1 we ca 11 "global II parameters. The phys ica 1 significance 

of these global parameters (AA,rA,n2A) is briefly stated: 

* See.the glossary of principal symbols, at the end of ·the Report. 

13 
1). · AA = the Impulsive Index (for Class A interference) : this is defined 

as the average number of emission "events II impinging on the 

receiver in question times the mean duration of a typical interfering source emission [cf. Eqs. (2.38), (2.39), Part II and . 

associated discussipn] . The small~r AA' the fewer such events 

and/or their duration, so that the noise properties are then 

dominated by the wavefonn characteristics of a typical event . 

Loosely speaking , we say that such noise is "impulsive", although 

here the mean duration of events is sufficiently long to avoid 

generating noticeable transients in the receiver, i .e . we have 

Class A noise, as defined above, Section (1.1) . As AA is made 

large, one approaches gatlssian (or in the case of the envelope 

here), rayleigh statistics (cf. Sec. (2.4), Part II). 

2). rA - cr~/ n2A = the ratio of the intensity of the independent gaussian 

component a~ of the input interference including received 

11 front-end 11 noise, to the intensity n2

A of the 11 impulsive 11 , 

non-gaussian (or rayleigh) component, cf. (3.la), Part II. A 

portion, cr~, of this normal component [cf. Sec. (2.3.l), Part II] 

arises from the cumulative effect of a large number of external 

sources, none of which is so strong as to be considered part 

of the 11 impulsive 11 interference, which is statistically the 

dominating effect (for small and moderate Indexes, AA) . 

3). n2A = the intensity of the above-mentioned 11 impulsive 11 component, cf. 

Eq. (3.la), Part II. 

[The rayleigh nature of P

1, w1 for large Ihdexes, i.e., when AA + 00 , is seen 

~2 

at once from (2.2), (2.3), as then rA + 00 also, so· that 2crmA + 1, and 

·2 

) -£ 

w1 (£ A + 2E.e (2.5) 

in normalized form; se~ Section (2.4), Part II, for the general case.] 

14 
Characteristic behaviour of the APO Pl-A vs . cS0 , cf. Figs. (3.1), (3.2) 

[and w1_A vs. SJ, is exhibited by the 11 rayleigh 11 form [constant slope n = -2, 

on the linear _by~0/2)log10(-loge[ ]) plots of P1_AJ for the comparatively 

small values of threshold e

0 , i.e., large values of P1(S > e0 )A, followed 

by a very steep rise, after which Pl-A bends over and approaches some 

asymptote with fixed slope n, 0 < n < 2, at large S0 (small P1_A) less than 

that of the rayl~igh behaviour for Pl-A in the 0.1-1.0 region. Thus, we 

have Pl-A+ e-il£'o , £0 + 00 , (O<n<2). 

This limiting, finite, and bounded slope as S0 becomes very large, 

after the characteristic bend-over, reflects the physical condition that 

the interference process has finite total average energy; accordingly, no 

individual source, or finite collection of sources, can emit unbounded 

energy over any finite period. Furthermore, if the number of sources is 

finite, with finite power, e.g. no infinite instantaneous amplitudes (as 

is the case ultimately in practice), then the limiting slope n becomes in-

.finite at some extremely large value of e0 . This effect can show up at 

comparatively small values of threshold £

0 , for example, with a single, 

finite source, of limited peak emissions, whereas with multiple sources the 

phenomenon will occur at larger £

0 . In any case, these (below-)bounded slopes 

(>O) insure, also; that all (firtfte) moments of the envelope exist5 as physically 

required by the condition of finite average emission energy. 

In our models, however, w~ assume that the number of potential 

emitting noise sources is infinite, although the probability of even a large 

number radiating at any given instant is very small, according to the 

fundamental assumption of poissonian "events", e.g., emissions, postulated 

here . In addition, we permit a distribution of emission levels (~ amplitude) 

per ~ource, where infinite magnitudes are possible, similarly with vanishing 

probabilities of occurrence. Thus, w~ may expect a nonzero limiting 

slope for Pl-A as £

0 + 00 : infinite amplitudes can occur, but with 

vanishing probability*. In practice, however, although the Impulsive Index 

"'The poissonian 11 rare-event 11 dominates any "rare-event" from the gaussian 

component. 

15 

(2.7a) 

-A8 00 Am -£2/2;2 

pl (c.>e- )B-II ~ L-2 l -1f. e o mB (2.7b) 0 4G m=O m. . B • A a a A with A = Aa/2 G8, €0 = (£0 N1)/2G8, where 

{ 

0 <a< 2; cf. Eq . (2.82), Part II, et seq.; (2.7c) 

A2 A A 2-a 2amB = (m/A8

+r• 8)/(l+r8), A8=( 4_a)A8, cf. Eq. (3.16a),Part II; (2.7d) 

G~=2-2(1+r8)-l(t~ + r13 ), cf. Eq. (3.12b) 1 Part II . (2 .7e) 

The associated pdf 1 s are (from Eqs. (4.3), (4.4), Part II): 

(£ B 2- [ < oo) , (2.8b) 

with 1F

1, as usual, a confluent hypergeometric function [Middleton, 1960, 

Appendix A.1.2], so that the w1 (£) 8 = w1 (£) 8

_1 for O 2. E- 2. c:8 , while 

w1(£8 ) = w1(£.)B-II when e.~£8. In Rart II, Figures (3.6), (3.7) show 

typical curves of the PD, (2.7), and Figures (4.3), (4.4) for the pdf 

[(2.8.a,b)], for selected parameter values. Again, E., C:0 are normalized 

according to (3.2), Part II, e.g. like (2.4) above, with n2

A replaced by 

n28 , etc. 

There are now six global parameters for our model : (Aa,a,A8,r 1

8,n28 ;NI), 

cf. Section 68 (Part II). The subset (A8,r

8,n28 ) are, just for Class A 

interference above, respectively, 1), the Impulsive Index; 2), the ratio 

of the intensity of the independent gaussian component (a~) to the intensity of the impulsive component ; and 3), the intensity of the impulsive 

component (n28 ), itself. These have the physical significance described 

above in Section 2.2. The additional parameters required here are: 

17 
4). 

5). 

A = 2r 1-a/2 

a r l+a 2 

(s~B) A = 

{ 2n ( 1 + r ' ) } a/ 2 B 

ar. "effective" Impulsive Index 

proportional to the Impulsive 

Index A8 , cf. (2 .38), (2.39), Part 

II, which depends on the generic 

parameter a. Here ( 13~8 ) is the 

a-moment of the basic envelope of 

the output of the composite ARI 

stages, cf. Fig . (1.1) above, and 

E q • ( 2 . 8 7 d ) Pa rt I I. 

2B B 

= 2-μ I 3-μ 

a Y surface; Y vol 

18 

= spatial density-propagation parameter, cf. (2 .82), Part I-I et seq . . 

Hereμ, y are respectively the 

power law exponents associated 

with the range dependence of the 

density distribution of the possibly emitting sources, and their 

propagation. (See Eq . (2.61) et· 

seq; Section 2.5.2, Eq. (2.63, 

Part II). The parameter a provides an "effective" measure of 

the average source density with 

range . Thus, if we standardize, 

for example, the propagation law 

as y = 1 (the usual spherical 

spreading), we haveμ= 2-a, 

(O<a<2), for the source density . . . - μ a-2 ( R d1str,but1on crs"'A =:\ , c11.= = 

distance from a typ ical source to 

the receiver). Knowledge of a 

accordingly gives us a direct measure of effecti ve source density, 

and if y is known or measured, 

separately , then μ=2 -ay gives us 
= 

the actual power l aw for as, cf. 

(2. 63), Part II. [We shall exploit 

this relationship in detail in a 

later study in this series.] 

The scaling factor which insure that 

pl- I'wl - I yields the correct mean 

square envelope 2n28 (l+r

8)(See Sec. 

( 3. 2A). 

the (normalized) "bend-over" point, 

at which the two (approximate} forms 

of PD (and pdf) are joined, according to the procedures discussed in 

Section 3.2, and Eqs. (3 .18)-(3.20), 

cf. Fig. (3.5), Part II. This is 

an empirically determined point, representing the point of inflexion 

(for small P

1_8) at which the experimentally determined PD, or exceedance probability P1(£;E-0 ) 8_expt.' 2 bends, e.g. at which d Pl-exot./d£B 

= a. Examples of this are indicated 

in the next Section, (2.4). 

We note that without an (experimental) £8 we cannot predict the limiting 

form of the PD as e0 + 00 ; we can then only .obtain the subset of global 

parameters (Aa,a; r

8,n

28 ,NI)), cf. Section 6C (Part II) . Examples of 

this are the particular cases of atmospheric and automotive ignition noise 

shown in Figs. (2.3), (2.4), and interference from a fluourescent light, 

Fig . (2.5). 

The si-x parameters (Aa,a,A8 ,r8,n28.,N1) are all physically specified and 

measurable parameters in the analytical model (provided £8 is determined). 

Only £8 itself is an empirical parameter, without explicit quantitative 

relationship to the underlying physical mechanisms involved. This is because the simplest canonical approximation to the exact c.f. [Eq. (2.87), 

Part II] requires a two-part c.f.,approximate for one to the small and 

intermediate values of the envelope, and for the other, to the large values 

19 
of the envelope . This second c.f. , [and PD= P

1(( > £

0 )

8_I 1

J, provides the 

needed "bending" of the APO curves for the rare events, as sketched in 

Fig. (3.5), Part II, for instance, and shown in some of the experimental 

examples of Section (2.4) following . 

Generally, unlike the Class A cases, Class B interference exhibits 

a much more gradual rise (as £0 becomes larger), also with increasing a. 

Similar upward displacement of the rayleigh sections (small ~0 ) of these 

APO curves occurs for an increasing gaussian component (r8), while i ncreasing the Impulsive Index A8(~Aa) also acts to diminish the steepness of 

these curves as ~o is increased. The physical necessity for a suitable 

11 bend-over 11 at the larger values of e0 has already been discussed above in 

Section (2.2) for the Class A noise: a fixed, asymptotic slope (n>O) is 

required, to insure the existence of all moments, which in turn is demanded 

by the condition of finite total average energy. Again, increasing the 

Impulsive· Index and/or increasing the independent gaussian component (a~) 

eventually yields a wholly gaussian process (rayleigh, of course, in the 

envelope), as expected. 

2.4 Comparisons with Experiment: 

In this subsection we include a variety of comparisons of our theoretical models with experiment, for both Class A and. Class B interference, 

cf. Figs. (2.1)-(2.8) following. Four significant features are at once 

evident: 

(1). The agreement between theory and experiment is excellent, i.e., 

the approximating forms are effective, analytical relations for 

predicting the desired fi rst-order statistics; 

(2). The caaonical nature of our models is demonstrated: the fonn of 

the results [here APD's: P1(E>E.0 ),J, is invariant of the specific source mechanism, whether ignition noise, atmospherics, 

flourescent light, etc., man-made or natural, within the distinct 

Class A, or B; 

(3) . Class A and Class B interference are observeably and quantitatively different noise types (vis-a-vis the narrow- band receiver 

used). 

20 

40 

20 

0 

Vl 

E 

... ~ 

I\ 

0) 

-0 

-0 

(j(J -20 

-40 

10 20 40 60 

Denve r 

July I, 1972 

1520-1600 MDT 

200 kHz 

6 kHz BW 

o- Denotes Measured 

Points 

80 90 95 

Percent of T ime Ordinate is Exceeded 

98 99 

Figure 2.2. Comparison of measured envelope distribution, P1(e' > eo)A, with 

Class A model, cf. (2.2) . Interference (probably) from nearby 

~owerline, produced by some kind of equipment fed by the line 

[ data from Bo 1 ton ( 1972)] . 

22 
60 

Central Colo. Springs 

June 5 , 1970 

0857-0903 MST 

250 MHz 

40 4 k'Hz BW 

o - Denotes Measured 

Points 

Nr/2G8 == 4.47 

20 VI 

E 

"'-

(Q,> 

<I,) 

> 

0 

..0 

<l 

CD 

"O 0 

Aa=I.O. 

~a=l.5 

-20 

10 20 40 60 80 90 95 98 99 

Percent of Time Ordinate is Exceeded 

Figure 2.3. Comparison of measured envelope distribution, P1 (€> e'o)s, of 

man-made interference (primarily automotiv·e ign.ition noise) with 

Class B model; cf. (2.7a). [Data from Spaulding and Espeland · (1971).] 

23 
50 

40 

30 

20 

(/) 10 

E 

lo.. 

~ 

Q) 

> o · 0 

.0 

<( 

ID 

-0 

-10 

-20 

-30 

-40 

Figure 2.4. 

0 

RR42A, Tracks 1&3 

10 mi le s northeast 

Laramie, Wyoming 

02 - 17-1970, 0004MST 

5 10 20 40 60 

500 kHz 

4 kHz BW 

o- Denotes Measured 

Po ints 

Nr/2GB = 70.79 

A0 =1.0 

/a=l.2 

80 90 95 

Percent of Time Ordinate is Exceeded 

98 99 

Comparison of measured envelope distribution, P1 (€'> ~o)B, of 

atmospheric noise with Class B model, cf. (2.7a). [Data from 

Espeland and Spaulding (1970).J 

24 
0 

- 6MHz 

(f) -,o Horizontal Component E-W ~ 

a::: 1.2 kHz BW 

... April 24, 1973, 4:15 p.m . (1) 

-(1) Shop Office 

E -20 Grace Mine ... 

Cl> 

a. 

a. o- Denotes Measured 

E 

0 -30 Points 

0 

... 

(.) N1/2GB = 3.98 X 10- 3 

E 

0 -40 -(1) 

> 

-0 A = 10-2 

(1) a ... -50 CD a = 1.85 

-0 

........ 

I ~ 

.c. 

-0\ -60 C 

(1) 

... 

(f) -

-0 

(1) -10 iL 

(.) 

-(1) 

C 

0\ 

0 -80 ~ 

10 20 40 60 80 90 95 98 99 

Percent of Time Ordinate is Exceeded 

Figure 2.5. Comparison of measured envelope distribution, P1 (8' > S'o)B of 

man-made interference (fluorescent lights in mine shop office) 

with Class B model, cf. (2.7a). [Data from Adams et al. (1974)J 

25 

40 

30 

20 

10 V, 

E 

... ~ 

Cl) 

> 0 o· 

..0 

<! 

CD 

"O 

-10 

-20 

-30 

-40 

Figure. 2. 7. 

PR21A, tracts 2 and 4 

A8 =0.005 

07-25-69, 1405 MST f 8' =0.008 

Eggleston Res. No.4 

Boulder, Colorado 

500 kHz 

ea 4kHz Noise Bandwidth 

0- Denotes Meo sured Points 

Aa= 1.0 

a= 1.0 

Nr=·27.38. 

10 20 40 60 80 90 95 98 99 

Percent of Time Ordinate is .Exceeded 

Comparison of measured envelope distribution ·pi(§ > t\)s of atmospheric noise with full Class B model, cf. (2.7a,b). [Data 

from Esperland and Spaulding (1970), p. 42.] 

27 

(4). The governing, physically structured parameters of these PO's and 

pdf's which are likewise also canonical, can be obtained from 

approximate experimental data (usually expressed as an APO). 

The importance of the canonical character of these models cannot be overstressed: with such models we avoid the very limited and nonpredictive 

quality of all ad hoc models; whose structure must be verified and whose 

parameters provide little or no physical insight into the underlying pro-

'· 

cess itself. Second, because these models a're derived from physical principles, their parameters are physically defined, are consequently canonical, 

and are quantifiable in specific instances from empirical data. Their 

structure, however, is independent of any particular meas~rement. 

Figures (2.1) and (2.2) show,APO's, e.g. P

1(€>~)A vs. the normalized 

envelope threshold €

0 , for Class A interference, respectively from orecrushing machining in a mine (data from Adams, Bensema, and Kanda [1974]), and 

and from a powerline (from E.C. Bolton, [1972]). Observe the characteristic 

very steep rise following the rayleigh region (constant slope), followed 

in turn by the expected bending over of the APO for the rarer "events", in 

each case. [Similar examples of Class A interference, but from man-made, 

intelligent sources, have also been observed; current experimental studies 

at ITS, Boulder, are underway to obtain additional such data.] 

Figures 2.3-2. 5 show APD's of Class B interference, respectively for 

(i), primarily urban automative ignition noise [Spaulding and Espeland, 

1971]; (ii), atmospheric noise [Espeland and Spaulding, 1970]; (iii), 

flourescent lights, in a mine shop office [Adams et al, '1974]. Observe 

the more gradual departure from the straight-line rayleigh region, a~d the 

continuing rise, with constantly increasing slope in the Figures (which is 

equivalent to n-+ 0 for exp(-a2c:;), as £

0

-+ 00 ) . [In these particular 

examples the inevitable "bend over" points eB, lie outside the range of 

data taken, e.g. for P

1_B<10- 6, so that we are ·ab.le to obtain all the 

gliobal parameters, except for A8 , cf. Sec. 6C, Part II.] This is 

not the case, however, for the Class B examples of Figs. (2.6)-(2.8), 

e.g., respectively for (i), ignition noise from vehicles moving on a freeway [Shephard, 1974]; (ii); atmospheric noise [Espeland and Spaulding, 1970]; 

29 
and (iii), machinery noise in a coal mine [Bensema, Kanda, and Adams, 1974]. 

Here the required bend-over of the APD's is exhibited, along with the inflexion points, E8. In these cases we can obtain numerical estimates 

all the six global (and hence all the generic) parameters characteristic of each example of interference*, man-:-rnade or natural, by the 

methods briefly cited below in Section 2.5, and in more technical detail 

in Section 6, Part II. 

Figures (2.1)-(2.8) are typical of Class A and Class B interference, 

man-made and natural. They are not intended to be exhaustive. Extensive 

additional APO data (mostly Class B) are available, for example in 

Espeland and Spaulding [1970], and Bensema, Kanda and Adams [1974], for 

example. [We nave not included Class CAPO data, although these appear 

in the references cited, because we limit our analysis and comparisons 

here to the essentially 11 pure" Class A and Class B interference environment, some (analytical) conditions for which are examined in Sec. 7, Part 

II.] Again, a striking feature of the present approach is its ability 

to handle an unlimited variety of noise sources, as long as the dominating 

Class is identified. 

2.5 Remarks on the Estimation of Model Parameters: 

We distinguish two sets of model parameters: (a), the so-called global 

parameters, which appear explicitly in the analytical forms for the APD's, 

etc. and (b) generic parameters, .which are defined directly in terms of the 

underlying canonical, statistical-physical model. To some extent, both 

sets overlap. In any case, once the global parameters have been estimated 

from the data, which usually requires the calculation of the (first-order) 

APO, the generic parameters can be calculated from them. 

The method for obtaining the global parameters is described in detail 

in Section 6, Part II, and will not be repeated here. However, for convenience, we list the two sets of parameters for each interference Class. 

These are [from Tables (6. 1 ), (6.2), Part II]: 

*See footnote added Po 37. 

30 
Class A: Global: 

Class 8: Global : 

(2.9) 

2 (Aa \ /A2) (Aa,a,A8,r8,n28 ;NI); Generic: (A8,a,crG, 808 ;,,808 ,NI) 

(2.10) 

where , in addition to those parameters described in Sections (2.2), (2 .3) A2 A2 above, (80

A),(808) are the mean square envelopes of the basic waveforms 

emitted from the ARI receiver stage, cf. Eqs. (2.64a,b), Part II. Class A 

noise is described by a three- parameter model, while Class 8 interference 

is a six-parameter model globally, and similarly a fjve-parameter 

model generically, since the inflexion point (£8) is empirical and not 

deriveable here from the fundamental physical model itself. Again, we 

stress the fact that (except for&8) al l model parameters are physically 

structured and hence are canonical in form; they are not ad hoc quantities, 

of strictly limited application.* 

' Finally, from an examination of the Class B model parameters vis-a-vis 

those of Class A, w~ note that (AA,A8), (rA,ri/, and (n2A,n28 ) are each 

identical types, of equivalent physical interpretation. The remaining 

Class 8 parameters are (a,(§~8)), which provide additional .information 

about the emitting sources, e.g., source density, etc . ., basic waveshape . 

Accordingly, it is suggested that to assess the interference environment· 

more fully, in addition to Class A measurements, when possible ARI receiver 

bandwidths also be selected, to produce Class 8 interference at the output 

of the ARI -stage, so as to obtain a and (s~8), in addition to (As,cr~,(B~8)), 

which are analogous to the corresponding Class A set, cf. (2.9). [These 

(A8,cr~,(s~8)) are, of course, modified from these Class A counterparts by 

this choice of ARI bandwidth.] In any case, the important new parameter, 

a, is obtained, which gives us an ~stimate of an effective mean source 

density with range, and the actual one (with range), if the governing propagation law (y) is also known, or measured, cf. comments following (2.8b) 

above. Further information about source distributions may be obtained with 

the help of steerable, directional beam patterns, cf. Section (2.5), Part 

II. [We shall reserve these questions, in detail, to a succeeding study.] 

* Nr is not fully dependent .on the other generic parameters and is indepen- dent of £8. Hence it may be regarded as generic, cf. Sec . 68, 6C. 

31 
2. 6 Some Additional Results: 

A brief review of additional results obtained in this Report is now 

presented. We consider: 

(1 ). First-Order Moments,<£e) : These are obtained analytically for both 

Class A and B noise in Section 5 (Part II). They exist for all 

(real,finite) e, although the (approximate) expressions for the 

Class B cases are necessarily more complex then for Class A. 

Al ternative, exact, closed-form relations are also obtained for 

the even integer moments (e=2,4,6, ... ), cf. Section 5.2 (Part 

II). [See also the discussion in Section 5.3 (Part II).] 

(2). Conditions for Class A,B,C Noise: More precise, analytical conditions 

are derived in Section 7 (Part II) mutually to distinguish Class 

A, B, and C interference, than those qualitatively discussed 

in Section (1. 1) above. In general, if the Impulsive Index of 

one component (A or B) greatly exceeds that of the other (B or 

A), then the former (A or B) dominates, and we have in practice 

Class A, or B noise. When this is not the case, the result is·the 

the more general, Class C interference (which we shall treat 

in a subsequent study). 

(3). Approach to Rayleigh Statistics: This occurs when either, or both, 

the Impulsive Index or the independent gaussian component becomes very large, cf. Section 2.4, Part II. (This is a consequence of the Central Limit Theorem in probability [cf. 

Section 7.7- 3, Middleton, 1960]. 

(4). Hall Models: A primary empirical model, constructed earlier by 

Hall [1966], is frequently used for ad hoc representations of 

the interference environment. Our Class B results, upon deletion of the additive gaussian component (both from the impulsive 

and independent sources), can be shown to exhibit a Hall form, 

with Hall parameter (eHall=2) . [See Section 3.2B, Part II; 

also, Spaulding and Middl eton [1975], Chapter 2.] Such models, 

however, have a variety of draw- backs, among them being their 

32 
ad hoc character, with the parameter(s) entirely empirical, and 

the non- existence of the second moment, in many instances, as 

well as the non-existence of all moments <te>, where e>eHal 1-l. 

Their principal advantage is analytic simplicity, ·which, however, does not.ultimately compete with the physical-statistical 

models of the types developed here . These, though analytically 

much more involved, are nonetheless still tractable for the 

purposes of source and system analysis, cf. Spaulding and 

Middleton [1975]. No Hall models are deriveable from Class A 

models, however. 

(5). Class A vs. Class B Interference: Some Summary Remarks: A ·concise 

comparison of some of the salient properties of Class A and 

Class B interference ~s presented in Table (2.1): 

Table 2. 1 Class A vs. Class B Interference 

Class A Class B 

1. New Models and Results; 

3. A11 Moments (se), 02e exist 

4. Insensitive to Source Distribution in Space and Propagation Law, 

Canonical Forms; 

5. Waveform in IF Output: "Gaps" in 

Time [P1 (£=0)>0); 

6. No Gaps in Time if Gaussian 

Background 

f x Gauss P.O. ) as AA+00 ; 

\.[ Rayleigh P. O. and/or a~+oo 

7. No Hall Models Exist; 

33 

"Classical 11 (20 Yrs. Old), But New 

Approach; New Results 

6 Global and Generic Parameters 

(Aa,a,AB2r13•rl._2A•Nr1 

(AB' at ,aG ,(B~B) .(soB) ,Nr) 

t 8: ~mpi~ical paramater of approx- 1mat1on 

All moments {se), 02S exist 

Sensitive to Source Distribution and 

Propagation Law (a); Canonical 

Forms; 

Waveform in IF Output: no "Gaps II in 

Time [e.g. P(e=O)=O]; 

No Gaps in Time (a~O); 

{ 

X Gauss P. D. ) as (Aa ,AB)+oo; 

[ Rayleigh P.O. and/or a~ 

Hall Models for Special Values of~; 

(Gauss Component Absent) 
2.7 General Comments; Next Steps: 

In the preceeding sections we have summarized the principal results of 

our present study of the (first-order) envelope and phase statistics of 

man-made and natural electromagnetic interference, whatever its physical 

origins and characteristics. These analytical models, of Class A and B 

interference, are mathematically tractible and canonical in application: 

the forms of the results, and the number, type, and general structure of 

the associated parameters, are invariant of the particular source. Of 

course, particular parameter values do depend on the specific properties 

of the particular source involved. These are estimated in turn, by general 

procedures outlined here., cf. Section 2.5 above, and Section 6, Part II 

from experimental data, principally the APD [= exceedance probability 

P1(£>c.0 )2. The canonical character of these models and their parameters 

is derived from the general underlying physical structure upon which the 

models are based. This, in turn, is itself a general space-time model of 

propagation, source distribution, and emission [Middleton, 1974, and Section 

2, and principally Secs. 2.1,2,5, Part II here]. 

As expected, the resulting statistics of amplitude and envelope are 

highly nongaussian (or nonrayleigh), as the analysis and examples in Part 

II and the experimental results of Section (2.4) indicate. This necessarily has a critical effect on conventional receiver and sy~tem operation, 

which may be in conventional usage, (approximately) optimized, e.g. 

11matched 11 , to desired signals in gaussian noise (so-called correlation 

receivers and their extensions), but which is radically suboptimum for 

this kind of electromagnetic environment [Spaulding and Middleton, 1975]. 

Here we are concerned with first-order interference statistics themselves, not only for purposes of system design, optimization, and comparison, 

but also for the tasks of measuring and assessing the properties of EM 

interference fields. Excellent agreement between model and observation has 

been found, as the examples of Section 2.4 above demonstrate. In addition, 

explicit ·numerical results are also obtained for the global and generic 

parameters of the interference phenomenon in question, e.g., automotive 

ignition noise, communications, atmospherics, machinery, power 1 ine 

emissions and the like. These parameter values, along with the basic 

34 
physical structure, permit us to deduce general properties of the interference field, such as average source distribution in space (a), emission 

density in time (Impulsive Index, A), mean intensity (n2

L the amount 

of external gaussian noise (a~), etc., and, of course, the associated 

APO, or exceeda.nce probability P

1(c..>t:0 ), as well as various moments ((£ 8>) 

of the interference process. 

First- order statistics of these highly nongaussian EM noise environ- • ments as e~bodied in the APO, P1(E> £0 ), P

1(X>X0 ), for example are, 

however, minimal for the prope·r treatment of the general class of communication systems operating in such environments. In many situations the 

performance bounds established from these first-order statistics are quite 

adequate [the independent sample cases of Spaulding and Middleton, 1975, 

for example], where higher-order time structures are not significant. 

However, when they are, one clearly needs appropriate extensions of the 

present models. In addition, the joint statistics of signals and noise 

are also required, of first- and higher-orders as well. Therefore, as 

part of our continuing effort to ·develop an applicable analytic description 

of-- .the EM nongaussian interference environment, we present the following 

program, in approximate order of undertakfog: 

I. Interference Models (present series): 

1). Report, Part III: First-order statistics of the instantaneous 

amplitude (X) for Class B noise , e.g . , P

1(X>X0 )

8, w1

(X>X0 )

8, 

moments, parameter estimates, etc. (now underway). 

2). Report, Part IV : First-order statistics of Class C noise, 

envelope (E) and instantaneous amplitude (X), e.g . P

1

(X>X0 )c, 

P1 (C>~)c, etc., with experimental comparisons. 

3). Report (possibly Part V) on measurements, parameter estimates, 

and description of EM interference environments. This will include 

evaluation of selected, earlier data [for example, Furutsu and 

Ishida, l.960~, Espeland and Spaulding, 1970; Shephard, 1974], and 

comparisons with our models. This may also include reconmendations for van usage and area coverage, etc. 

4). Report, on simulation of EM environments, to establish robustness 

and sensitivity of the various models to modifications in their 

35 
structure. 

5). Report, on various special problems and extensions of Part I-IV, 

for example, to develop the analysis for Class B noise when 

a= O,a ~ 2, including other source distributions. Also, we 

need to examine the case of a single source at known positions 

[Shephard, . 1974]. The role of the 11 corrections 11 [Middleton, 1974] 

requires further clarification, etc. 

6). Report on mean and variance of 11 zero-crossings 11 for Class A,B,C 

interference. These statistics are useful adjuncts to the APD's 

to provide some insight into the time-structure of the interference. 

7). Report on the first-order statistics (envelope, phase, and instantaneous amplitude) of Class A and B interference with a general, 

additive signal present at the input to the typical narrowba~d 

receiver. 

8). Report on the development of higher-order (principally 2nd-order) 

pdf's for Class A, B, C noise; possibly including signals as 

well. 

II. Performance and Optimum Systems in General EM Interference Environments: (series with Spaulding and Middleton [1975]). 

1). Report (Part II): Optimum reception with Class B interference; 

this is now underway); 

2). Report (Part III): extension of the above to Class C cases; 

3). Report, on the analysis of other specific systems in Class A 

and B noise; the improvement of performance bounds~ the specification of LOBO receiver structures, and further extensions of 

the evaluation process. 

These lists are not, of course, complete, nor will the program itself 

necessarily be carried out in the order indicated, since some of these 

topics may shift in priority as time goes on. In any case, however, the 

general goal of developing an applicable analytical theory, tested by 

experiment, for these general classes of man-made and natural, highly 

36 
nongaussian, electromagnetic . noise or interference processes, may be considered a major priority task for the future in the science and technology 

of telecorrmunications. 

2.8 Acknowledgements 

The author wishes to thank Or. A. D. Spaulding of the Office of 

Telecommunications for his constructive discussions and criticisms of 

this material and for performing the _humeri cal calculations and developing the figures contained in this report. 

h I _ 2/ 2 *Note t at rs = oG n2B, where crG is the independent gaussi.an component, 

2 2 which is different from the total gauss component ~crG = crG + b2aAB, cf., eq. 

(2.88a). Thus, in Figs. 2.6, 2.8 we must calculate n28 from the data curve 

and then obtain cr~ from rs. From the other parameters in these figures all 

· the remaining generic parameters. are then readily found. On the other hand, 

for Fig . 2.7 n2B occurs at O dB, by normalization. Since P1=0 . 36 determines 

the total gauss component (~cr~ for Class B, cr~ for Class A noise), from the 

data of Fig. 2.7 we get ~a~; -17 dB (=2 . 10- 2 ) and ~a~= r~n2B = 10 10910 

(8.10- 3 ); -21 dB, which gives in turn b2aA8 ; 0.012 (in units of n28 ) . 

Again, all remaining generic parameters are now obtainable, from this and 

the other parameter data on Fig. 2.7 (also in units of n28 here). 

37 
PART II. ANALYSIS 

l . INTRODUCTION TO THE ANALYSIS 

Part II of this Report is devoted to the analytical details of our statistical- physical interference models, where in particular we are concerned 

with the first-order probability density functions (pdf' s) w1 (E) ,w1 (£) and 

exceedance probabilities (PD's) P

1(E > E0 ), P1(t: > ~ 0 ) of the envelope of 

the input noise process following the combined .(linear) aperture - RF - IF 

(or ARI) filtering stages of a typical narrow- band receiver. As we see 

bel ow, three principa l classes of interference process are defined : Class 

A noise, where (in qualitative language) the input noise is spectrally 

narrower than the ARI-filter at the receiver's front-end; Class B interference, where the reverse is true - this input process is spectrally broad 

vis-a-vis the ARI filter; and a general Class C noise, which consists of the 

sum of Class A and Class B components. 

For Part II the material is organized as follows: In Section 2 below 

we develop the various forms of the first-order characteristic function 

(c .f.) for the envelope after reception in the ARI fi lter of the typical 

receiver. Included here are the Class A, B, and C noise types, with their 

associated descriptive parameters, and the modifications introduced by the 

geometrical effects of source distribution and propagation law. Section 3 

is devoted to the determination of the envelope exceedance probabilities, 

for Class A and B interference, and Section 4 gives the associated pdf's. 

In Section 5 we determine the moments (c.13), O ~ s, and the conditions 

for their existence; Section 6 provides procedures for estimating the basic 

parameters of the various noise mode ls from empirical data; and finallly, in 

Section 7, we give quantitative, practical conditions for the applicability 

of our Class A or Class B models. 

38 
i. FORMULATION : THE CHARACTERISTIC FUNCTION 

In this section we obtain the general forms of the first-order characteristic functions (c.f. 1

s) and probability densities (p.d . 's) and distri -

butions (P. O. 1

s) for the "impulsive" interference of the various man-made 

and natural sources described in Part I above . 

Our first step is to derive the desired general forms of the characteristic functions for the envelope of the received wave. The next step 

is to take advantage of the various physical conditions of the model, further to reduce our resul ts to the particular expressions appropriate to 

the Class A and Class B interference, which can then be put in forms suitable 

for evaluation. A number of important parameters of these interference 

processes appear in the analysis and have important physical implications, 

· which we shall develop further in the subsequent sections. 

2.1 The Basic Statistical Model: 

We assume as before [Middleton, 1974] for our basic model that there 

is an infinite number of potential sources in a source domain A, and that 

while the basic waveforms emitted all have the same form, their scale, 

durations, frequencies, etc., may be randomly distributed . Our fundamental 

postulate of this basic interference model is that: (;-), the locations 

of the various possible emitting sources are poisson distributed in space; 

(ii), the emission times of the possible sources are similarly poisson 

distributed in time. Physically, this means that the sources are statistically independent, both in location and emission. Thus, by a slight 

generalization of earlier results [Middleton ; 1967, 1972b, 197~], we can 

write for the first-order characteristic function of the instantaneous 

amplitude, X, of the received interference process 

f.'. isU(t;A,£, . .. ) l \e - ~ - ( 2. l ) 

Here e is an epoch, indicating vis-a-vis the receiver's (i.e. observer's) 

time t when a source may emit. The~= (;l.,6,4>) are coordinates, or a 

vector magnitude , appropriate to the geomtry of the source field, located 

in the region A, and of the -receiver, with d;l. (= dAd4>) for a surface element; ,.,,,., 

(= d;l.d6d4>) for a volume element. The quantity p(1 ,E) is the "process 

39 
density" of this joint space-time poisson interference process, and is nonnegative, and can be regarded as proportional to a probability density 

[cf. (2.28) below]. The< )(i denotes a statistical average, e.g. 

f[a][ Jw1 (~)~, over various random parameters(~) which may be pertinent 

to-our source model, such as doppler, source amplitude and duration, etc. 

The U are the typical waveforms of the emitting sources, after reception by the (assumed linear) aperture - RF - IF stages of our "narrow-band" 

receiver. The received process Xis given by 

X(t) = ~ U(tll)dN(I), Z(=Axe) 

..,., ...., ... (2.2) 

where the {dN} are a poisson point process (inland~), such that 

(X(t)) =;; U(dN) = (fa(~,i)U(t;~,i,~)<Jl df\ , 

- A ~ (2.3a) 

is the process mean (if any), and 

(2.3b) 

is the general second-moment of X(t), under our basic poisson assumption 

above of source location and emission.* Higher moments may be similarly 

obtained. 

Since we are interested here in the envelope of the received process X, 

which is always narrow-band, in as much as the receiver is itself narrowband, we have to consider the new random variables X , X , representing the · C S 

slowly-varying "in-phase" and 11 out-of-phase 11 components of X, viz., 

*Fora general development of process statistics, not necessarily limited to 

the poisson case of independent sources, see, for example, recent work 

[Middleton,1974,1975b] in the development of generalized scattering models. 

40 
where now w0 (=2rrf0 ) is the central (angular) frequency of the final (~IF) 

stage of the receiver, and 

(2.4b) 

with 

Xe = E cos iμ, \ = E sin iμ • 

Here E, iμ are, respectively the envelope and phase of the narrow band received process X. In functiona l form, cf. (2.2), we can write alternatively 

(2.5a) 

(2.5b) 

in terms of an envelope and phase, where 

(2.5c) 

Comparing (2.5) and (2.4) we see at once that 

(Xc·•Xs)= ~(Uc,Us)dN(?) (2.6a) 

"" 

- iiμ f -i~ (ti£) Ee s = e(tlZ)e s dN(Z) z - .,., "" 

(2.6b) 

The characteristic function which we need now is for the random variables Xe, Xs, namely, 

which is the two-dimensional generalization of (2.1) required here. 

corresponding p.d .. is 

jj(( - i~Xc- inX 

W 2 

1(Xc,Xs) = 1 F1(ie,in)X ,X e sd~dn/(2rrJ -~ C S 

41 

(2 . 7) 

The 

(2.8) 
Also, we have, formally, the following expression for the joint firstorder density of envelope E and phase ljJ, in terms of th.e in-phase and outof-phase components Xc,Xs of X. 

E > 0 

(2.9) 

where w1 , . (2.8), is now 

(2.9a) 

with F

1 therein given by (2.7) 

To proceed further, we make use of a number of results from our earlier 

development of the physical model [Sec. 2.2, Middleton, 1974], to write for 

the narrow-band basic waveform U (at the output of the receiver's IF) 

(2.10) 

where 80 (>O) is an envelope, whose detai.led structure we shall consider in 

more detail later and where l is a phase, which has the form 

(2.lOa) 

in which ~S' ~T' ~Rare respectively the typical source phase, and the 

phase angles of the ~ource (T) and receiver (R), complex beam patterns 

[cf. Sec . 2.5 below]. [The quantity £d(=μd-l) is the sum of the relative 

dopplers between sources and the receiver, and is always small, 0(10-S,- 6), 

in our applications, · viz~ £ct= 2v/c = 0(10-6) for v = 0(105 mph) so that the 

envelope B0 is independent of £d. 

From the fact that U b = Uccos w0 t+U sin wt, we see at once from (2.10) n s o · 

that 

42 
I where ~s = ~s+qiT+qiR. We now use polar coordinates 

1; = r cos qi n = r sin 4>; Ja(1; ,n)/a(r,4>)J = r ( 2. 12a) 

and transform from {i; ,n)-space to (r,4>)-space in (2.9a). Thus, we see that 

di;dn F

1(il;,in)x X = r F1(ir,4>)drd4> , O < r < ~; O..::. qi< 2~. (2.12b) 

C; S 

~ 

The c.f. F

1 is F1(ir cos 4> ,ir sin t )x X , (2.7), which with (2.11) now 

C ' S reduces expl icitly to 

(2.13) 

The first-order p.d. for the envelope and phase (E,~) , (2.9), with the help 

of (2.12b), .(2.9a), becomes 

E > 0, 0..::. ~ < 2~ . (2.14) 

This is as far as we can go without further appeal to the physical 

model, in particular, to the statistics governing the locations ('\,~) of the 

sources and the epoch (i) oi interference emissions. We note, however, that 

the p.d. for the envelope alone is readily found, e.g.* the integration 

over~ (in 0,2~), is well- known [cf. (2. 19) fo l lowing)] : 

(2 .15) 

-*-

As usual, functi ons of different arguments are different functions, e.g. 

w1(E)!W1(~)!W1(xc,Xs), et~ . , unless it is otherwise stated . 

43 
In addition, we have respectively for the P.O., and exceedance probability, 

or APO (a posterior probability here, that E exceeds a level E

0 (>0),) defined 

as usual by 

f

E

0 

o1(E0 ) = w1(E)dE 

0 

the following results, where we have used 

viz: 

E > 0 o-

(2. 16) 

( 2. 16a) 

(2.17a) 

(2.17b) 

Our results (2. 13)-(2.17) are generalizations of earlier results 

[Furutsu and Ishida, 1960; Middleton, 1972b; Giordano, 1970], where our 

basic assumptions, so far, postulate only poisson distributions of source 

location and emissions, e.g. essentially independent sources. No restrictions on the specific character of the statistics of the source parameters 

are as yet introduced. It is for this reason that the characteristic A 

function F1 depends on~, as wel l as on r . 

2. 2 First Reduction of the c.f. F1: The Narrow-Band Receiver Condition 

At this point we invoke certain properties of the basic waveform 

B0 cos[~~+μdw0 (A+e)-w0 edt-~J .which appears in the exponent in the integrand 

of (2. 13). We use the facts that (i), B0 , ~s' are both slowly-varying 

functions of A; and (ii), the process density p(A,E) is likewise slowly ~ 

varying, vis-a-vis cos w0 μdA' sin w0 μdA. Employing the familiar expansion 

in Bessel functions, 

44 
In addition, we have respectively for the P.O., and exceedance probability, 

or APO (a posterior probability here, that E exceeds a level E

0 (>0),) defined 

as usual by 

f

E

0 

o1(E0 ) = w1(E)dE 

0 

the following results, where we have used 

viz: 

E > 0 o-

(2. 16) 

( 2. 16a) 

(2.17a) 

(2.17b) 

Our results (2. 13)-(2.17) are generalizations of earlier results 

[Furutsu and Ishida, 1960; Middleton, 1972b; Giordano, 1970], where our 

basic assumptions, so far, postulate only poisson distributions of source 

location and emissions, e.g. essentially independent sources. No restrictions on the specific character of the statistics of the source parameters 

are as yet introduced. It is for this reason that the characteristic A 

function F1 depends on~, as wel l as on r . 

2. 2 First Reduction of the c.f. F1: The Narrow-Band Receiver Condition 

At this point we invoke certain properties of the basic waveform 

B0 cos[~~+μdw0 (A+e)-w0 edt-~J .which appears in the exponent in the integrand 

of (2. 13). We use the facts that (i), B0 , ~s' are both slowly-varying 

functions of A; and (ii), the process density p(A,E) is likewise slowly ~ 

varying, vis-a-vis cos w0 μdA' sin w0 μdA. Employing the familiar expansion 

in Bessel functions, 

44 
00 

exp[ia cos~]= l im~mJm(a) cos m~, 

m=O 

(2.18) 

in (2.13), we see that only for m=O does the integrand (containing the 

exponent') .. contribute, as all the other tenns are highly oscillatory in 

regions where 8

0 , ~s' and pare slowly changing with A, The result is the 

important simplification -0f (2.13) to 

F1(ir,o) = ex~~_,(°p(~.<)[J0 (r80 [t,~1E,~J -1 ]>

0

djdE} = F1(ir), (2.19) 

"" 

which is valid, provided that Unb is truly narrow-band, e.g. 6fARI << f 0 : 

the (composite) bandwidth of the (linear) aperture-RF-IF receiver stages is 

much less than the (IF) central frequency f

0 . [The result (2.19) is now in 

the same form as obtained in earlier work, but still somewhat more .general 

in detail.] 

The important analytic feature of (2.19) is that now, because of the ~ 

narrow-band receiver condition 6fARI << f 0 , the c.f. F

1 is independent of 

i=_ Accordingly, we see that (2.14), (2.15), (2. 17} reduce at once to the 

simpler forms (with the help of (2.18): 

(2.20) 

with 

(2. 21) 

and 

(2.22a) 

(2.22b) 

45 
• 

Respectively for the PD and exceedance probability,. P, for these 

narrow-band interference waves (in the n. b. receiver), the first-order .p .. d. 

of phase~ is seen to be uniform over an (IF) cycle (T0 = l/f

0 ) . The 

results (2.20)-(2.22b) are formally identical to those derived by Furutsu 

and Ishida [1960, Eqs. (2.9)-(2. 11)], and by Giordano [1970], and, Giordano 

" and Haber [1972], for example. Furthermore, F

1(ir) is clearly a Hankel 

transform of w1(E), from (2.21) and the fact that the inverse of (2.21) is 

(2 . 23) 

[This relation is easily established with the help of 

(2.24) 

cf. p. 943, Morse and Feshbach [1953], applied· to (J 0 (rE))E' with (2.21) 

for w1(E).] An equivalent expression, now in terms of the average over E 

and w, is obtained at once from the fact that 

[cf. (2.2.)], and from the relation (2.23), viz: 

(2.24b) 

" from which is seen· the fact that F1 here is also the joint c.f. of envelope 

and phase, as expected. We shall use this result later in the calculation 

of moments, (cf. Sec. 5.2). 

Next, let us look at the process density p(A,~): we write ,,.. 

46 

(2.29) 

where explicitly we have from earlier work [Middleton, 1972b, 1974] for the 

received envelope 80 of a typical emission 

where 

C2R,c2T = (complex) beam patterns of receiver and typical 

interfering source; 

A0T = (real) envelope of the source emission; 

(2.30) 

g(A) = a geometric factor, which describes the propagation law, .,., 

from source to receiver (which are assumed to be in each 

other 1s far field). 

· ( 2. 30a) 

[For this receiver, although the aperture may be comparatively broad-band, 

as may be that of the source, it is the narrowest filter, of the combination 

(aperture x RF x IF) which is controlling. By assumption, one or more of 

these filters is very narrow vis-a-vis f 0 , cf. the corrments following (2.19), 

so that the eff.ecti ve aperture response here is determined essentially by 

the response at (and about) f

0 , cf. (2.30).J 

Next we let 

T

5 = duration of a typical emission, at the IF output 

(which may be ST); (2.31a) 

this can generally be a random variable (one of the a in (2.29)). We now 

"" 

1 et 

t-A-~; y = T sz , (2.31b) 

where Ts is the mean duration of an emission (at the IF output), and z 

is dimensionless, to rewrite (2.29) as 

48 
'(T /T ) 

IT= AAA<,TTs~o s swl(t-,-Tsz)e{Jo[rBO(z.~;~')]-l} Ts·~·~dz (2.32) 

all (t e T). 

,.. 

Here~· are' any other random parameters in B0 , e.g. ampli~ude of the basic 

emission envelope B0 (in the receiver, at output of IF); B0 itself is 

B

0

(zTs,i;i' ). Further reduction is obtained by writing 

where 

(2 .33) 

A

0 

= (peak) amplitude of the received envelope (at output of the IF); 

e

0

Y = a limiting 11 voltage 11 setting (in suitable dimensions), at which 

the receiver will respond to a test signal, above the receiver 

noise,* at output of the IF; 

u

0

(t) = normalized envelope wave form at output of receiver IF. 

Note from (2.33) that the generic waveform u

0 (z) is, of course, required to vanish outside the tim~ interval during which the typical 

emission ("'A0

e

0 

u

0 ) is 11 on 11 , e.g. for T z > Ts, T z < 0. 

(2.34) 

y s ,.. s 

Finally, let us multiply and divide I by T, the observation period, 

and write (2.32) as 

T /T 

IT= (A~E,T) Ts/f

0 

s \Tw1(t-:>--zTs);}[J0 (rB0 (z,~lf ))-1]\, · T dz, \ 1;§, '~' s 

(2.34) 

all (t e T). Also, we _see that 

* The precise definition of e0 y can be determined by standard decisiontheoretical techniques of detection [Middleton, 1.960; Chapters 18, 19], 

with some appropriate c~oice of false alarm rate . 

49 
-

vT: av. no. of emissions per second, in the 

observation period T, 11 density 11 of the process [cf. Sec . 11 .2; Eq. 

(11.74), Middleton, 1960]: (av. no. of emissions per second)x(mean duration of an emission). 

(2. 35a) 

(2.35b) 

Equation (2. 34) is a generalization of earlier results, which permits the 

treatment of nonstationary regimes. 

At this point we restrict our attention to the most corrrnon situation 

of "local stationarity", whereby it is assumed that there are no changes 

in average source numbers and emission properties during the observation 

period T, and that the emission prdbability w1(i) is uniform; e.g. Tw1_~ = 1, 

for all allowed values of z. Thus, (2.34) reduces to the basic form 

[uniform p.d . of i]: 

(2.36) 

which is the one from which we develop our subsequent analysis (beginning 

with (2.38)) in this Report. Furthermore, for the idealized steady-state 

si tuation where T +.., ,we write 

1 im 

T+<» -

\I T = \I 

.., A T+<» e:, T (= A lim (A /r)) } . 

and, accordingly, (2.36) becomes 

[uniform p.d. of e] : 

50 

(2.37) 

A _ y = v i a, Q) a>S 

(2.38) 
This limiting form of (2.36) is the expression which we shall exploit in 

the remainder of the study. 

The quantity A appearing in (2.38) is co 

A {=y ): impulsive index (of the present analysis)* co co (2.39) 

As we have alr~ady noted in our earlier studies [Middleton, ·1972b,1973,1974], 

the Impulsive Index is a measure of the temporal "overlap" or "density", at 

any instant, of the superposed interference waveforms at the receiver's IF 

output. It is one of the key parameters of the interference model, in 

that it critically influences the character of the p.d.'s and P;O.'s of the 

interference, as observed at the output of. the initial (linear) stages of a 

typical narrow-band receiver. With small values of A the statistics of co 

the resultant output wavefonn are dominated by the overlapping of comparatively few, deterministic wavefonns, of different levels and shapes, so that 

the interference has an "impulsive", somewhat structured appearance. For 

increasingly large values of A the resultant approaches a normal, or co 

gaussian process, as one would expect from the Central Limit Theorem 

[Middleton, 1960, Sec. 7.7], as we shall see in more detail later [cf. 

Sec. 2.4]. 

2.3 Interference Classes A, B, and C: The R61~ of Input and Receiver 

Bandwidths: 

We are now ready to examine the basic form, (2.38), of Ico(r) [=log F1(ir)]. 

The role of the duration\ of a typical emission (as perceived at the output 

of the ARI (= aperture x RF x IF) stages of the narrow-band receiver) is 

critical in detennining the fonn of Ico(r). 

Let us consider first the important special case when the .emission 

duration\ is fixed. From Eqs. (2.63a,b), (2.70), (2.72a) of Middleton 

[1960] we may write for the envelope B

0 , cf. (2.10), (2.30), (2.31a), (2.33), 

* Ar(= AAA£,T=vTT, cf. (2.27c),(2.35)) was designated "impulsive index", A, 

in the author's earlier treatments [Middleton, 1972b,1973,l974]. 

51 

(2.40) 

Here h

0 , y0 are real, and hARI(t) = 2h0 (t)ARI cos [w0 t-y0 {t)J is the 

weighting function of the composite ARI fi lte·r. The (narrow-band) system 

function Y0 is obtained from the fourier transform Yo-ARI= J {h

0 e-iro} 

and w0 [,,;,wc-w0 ] measures the amount of "detuning" of the ~nput signal 

(at we' shifted to the IF region) from the (trial) central frequency (f0 ) 

of the ARI stage. 

With Ts fixed, we have in general the situation shown in Fig. (2 .l)II 

for the envelope of the narrow-band output of the ARI filter, produced 

by a typical interference emission of finite duration, Tin" The output 

envelope (~u0 (z)) produced by a typical input interference envelope [shown as 

a rectangular pulse in Fig.(2.l)II], always consists of two parts: a part 

which we shall call Class A with normalized enveJ ope u

0

A(z), which is 

produced by the input emission (~Eo-in), which is "on" during the interval 

0 ~ z ~Tin(= TsA = TsA); and a part we shall term Class B, with normalized 

envelope u

08 (z-l) (/u0A(z)), which represents the transient decay of the 

output of the ARI fi lter, following the termination of the input emission 

[~E0 (z)in]. The sum of Class A and Class B envelopes is cailed Class C, 

e.g. u

0

C(z) = u0A(z) + u08 (z-1 ), [cf. Fig . (2.l)II, where, of course, u

08=o, 

z < l, u

0

A = O; z < 0, z > l in our definition . Thus, all receiver outputs 

are typically Class C, with variable amounts of Class A and Class B, 

depending on the duration of the typical input interference waveform 

vis- a- vis the response time of the ARI filter at the front-end of our 

receiver. Equivalent,~, the relative extents of the Class A and B components 

depend generally on the ratio of the bandwidth of the input (6f;n) to the 

52 
u, 

w 

0 

------- --- Class C 

I 

: LJ (z) I . OC I 

- - -----------T- -----

I 

I 

I 

---------- Class A---------

z= 1 14- .... - T -

I SB 

1.. T -

\""' SB 

Figure 2. 1 (II) . Typi cal output envelope of ARI filter output i n (narrowband) receive

and Class B envelope components (positive halves onl y) ; Class C = Cl
bandwidth tifARI of the linear 11 front-end 11 of the receiver. With tifin » tifARI' 

for example, the usual case of very wide band interference (automobile ignition, fluorescent lights, atmospheric noise), TA is very small vis-a-vis - s Ts8

: the build- up time (=Tin) is very brief for Class A, while the duration 

of Class B depends on the decay time(~ tif-}RI·) of the ARI stage, which is 

much longer than Tin' . On the other hand, with narrowband inputs of long 

duration [tifARI » tifin], the transient at _the termination of the typical 

input is ?f negligible effect vis-a-vis the Class A component. For comparable bandwidths (tifARI ~ tif.

1

n) both Class A and Class B make comparable 

contributions, e.g. neither can be ignored vis-a-vis the other, so that we 

have then generally the Class C waveform in the receiver. [In all cases 

J~uoA ,B(t)kdt, k > 0, are finite.] A 

From (2.38) we see accordingly that I~(r) can now be written as the 

sum of the Class A and Class B components, viz: (Tin= TsA fixed for the 

moment) : 

on changing variable z' + z in u08 (z'-l) + u08 (z), e.g. 

(2.41) 

(2.41a) 

(2.41b) 

from (2.40). In terms of .the characteristic function (2.19) we see at once 

that 

A A A 

Fl ( i r) = Fl ( i r) A• Fl { i r) B - Fl'( i r ) C (2.42) 

with the important result that Class C interference consists of the 

54 
independent sum of Class A and Class B components, as defined above. Note, 

also, that the limiting voltages e~~) f e~~), generally, as the receiver 

responds to "narrow-band" inputs (A) differently from "broad-band" (B). 

Specifically, when we can ignore the Class B component [is8 (<00 ) << TsA' 

e.g. sufficiently narrow-band input vis-a-vis the receiver~], we have here, 

from (2.42), (2.41), in _(2.19), 

(2 .43a) 

(2.43b) 

(2.43b) 

where th·e averages < ) · z,)..,6 1 

"" "" 

are explicitly 

< ), AS' = fa\, t, ,\N,W t-' 

w1(e')p(;..) . _, ..., [ ] d).d6 I 

A~ - - (2.43c) 

Similarly, when the Class A component is ignorable [TsA = Tsin << T

58 

e.g. very broad-band inputs vis-a-vis the receiver's ARI stages*] we get 

(2. 44) 

A B»A A' oo, oo, 

with the averages given by (2.43c), (without the average over z). Note 

that when r-+ 00 , Fl-A+ exp(-A00 ,A), whi.le Fl-B,C-+ 0. This means, as we 

shall see in detail in Section 3.1 later, that for Class A interference 

* The precise conditions for effectively Class A or Class B interference 

alone are developed in Sec. 7 later. 

55 
there will bea non-zero probability of 11 gaps-in-time 11 , i .e. finite (nonzero) 

intervals in the receiver's output when there is no waveform present, while 

for Class Band C interference there is always a nonvanishing wavefonti 

level and hence no· 11 gaps-in-time 11 • [Of course, physically there is always 

some inherent system noise, which makes it st(ictly impossible to have a 

true 11 gaps-in-time 11 situation.] 

We remark, again, that Class A [and consequently tlass CJ interference 

models are new. The earlier "classical" analyses [Rice (1945); flHddleton 

(1953); Furutsu and Ishida (1960); Giordano (1970; Giordano and Haber 

(1972)], for example, all dealt with Class B interference, and for the most 

part in much less general terms and by different modes of approximation.* 

2.3.l Some Extensions: 

Usually, there is an accompanying gaussian background noise, which may 

arise in a number of ways: 

(i) . as system noise in the receiver; 

(ii). as external interference, which is the resultant of 

many independent sources, none of which is exceptionally 

dominant with respect to the others (so that the Central 

Limit Theorem applies); 

(iii). as a mixture of· receiver noise and (independent) gaussian 

external interference . 

From (2.12b) and the gaussian counterpart of (2.7), viz . 

-(~2+n2),r212 

Fl (i~,in)x X :·gauss= e G 

C' S 

we readily find that 

(2.45) 

(2.46) 

(2.47) 

* Technically~ Giordano [1970] and Haber [1972] express their results in a 

Class A format, whenever sample size (T) is finite, cf. remarks in 

Section (5.3) following. 

56 
cf. (2. 13), where a~, a~ are respectively the receiver and external noise 

variances. 

Applying (2.47) to (2.15), (2.17) shows directly that* 

2 2 -EUa2 

f

oo -r 0

6

/2 E ye. G 

w,(E)G = E rJO(rE)e · dr = ~ , E ~ O 

0 a6 

(2.48a) 

so that 

(2.48b) 

As expected, the· p.d. and P.O. here are rayleigh. 

Our results of Sec. 2.3 above are readily extended to include the 

more general situation of interference inputs -0f random duration, e.g. 

Tin= T5A (r TsA) generally. O~ly the Class A portion of f

00 (r), (2.41), 

is modified. Letting z0 = T;n/TsA' we have at once, for the desired 

extension of (2.43), 

Fl (ir)A = e-Aoo,A exp{(.Co Jo(rlloA) dz\ .) . . 0 -~z0 ,~,~ 

(2.49) 

Combining (2.49) and (2.44) with (2.47) gives us the desired characteristic functions with which we shall be concerned here, and subsequently, 

in this report: 

Class A Interference and Gauss: 

2 2 

A -aGr /2-A A ~ < rz A > } F1(ir)A+G = e 00

' exp A00 ,A J~ 0J0 (rB0A)dZ 1 0 z0 ,~,~ 

Class B Interference and Gauss: 

2 2 

F1(ir)B+G = .-aGr /2 exp~oo,8 ~\[Jo(rlloB)-ll)~.~·ctz) . 

* Use Eq. (A. l-49) [Middleton (1960)], for example. 

57 

(2.50) 

( 2. 51) 
[We shall reserve the analysis for Class C interference and gauss noise, e .g. 

based here on 

F1(ir\+G = .exp [-a~r2;2+A"",.A:(J.

20

[J0(r80A)-l]dz\ , O . ~z0 ,l,~ 

(2.52) 

to .a subsequent Report.] 

2.4 Large Impulsive Indexes: 

When the impulsive index, A"", is large, we expect asymptotically· gaussian 

statistics for the instantaneous amp.litude X [Secs.3,p.26; 5,p. 39, Middleton, 

1974], and rayleigh statistics· here·, cf. (2.4e), for the fnstantaneous ,.. 

envelope E. This latter is easily shown by developing I""(r), (2.41) or 

(2. 52), as a power ieries in r about r = 0, in the usual way.* Thus, the 

c.f. (2.52) ·for our general Class. C case, with gaussian background noise 

in addition, becomes 

where 

b (B) = f ""! 132n) dz 2n 0. \ oB A 6 • ...., '.,., 

(2. 53a) 

(2.53b) 

(2.53c) 

* Provided we consider for the moment finite observation intervals T(<""), 

i.e. finite upper limits on the z-integrals in (2.51), (2. 52), so that 

these integrals are uniformly. convergent, proper integrals, permitting 

a series expansion of their integrands. Then, we ultimately have 

F1 (ir)C+G: +-1: F1(irlT)C+G' where (i.::) is invoked for each term of the 

resulting expansions. See the comments in B, Sec . (5.2) below. 

58 
[We shall reserve the analysis for Class C interference and gauss noise, e .g. 

based here on 

F1(ir\+G = .exp [-a~r2;2+A"",.A:(J.

20

[J0(r80A)-l]dz\ , O . ~z0 ,l,~ 

(2.52) 

to .a subsequent Report.] 

2.4 Large Impulsive Indexes: 

When the impulsive index, A"", is large, we expect asymptotically· gaussian 

statistics for the instantaneous amp.litude X [Secs.3,p.26; 5,p. 39, Middleton, 

1974], and rayleigh statistics· here·, cf. (2.4e), for the fnstantaneous ,.. 

envelope E. This latter is easily shown by developing I""(r), (2.41) or 

(2. 52), as a power ieries in r about r = 0, in the usual way.* Thus, the 

c.f. (2.52) ·for our general Class. C case, with gaussian background noise 

in addition, becomes 

where 

b (B) = f ""! 132n) dz 2n 0. \ oB A 6 • ...., '.,., 

(2. 53a) 

(2.53b) 

(2.53c) 

* Provided we consider for the moment finite observation intervals T(<""), 

i.e. finite upper limits on the z-integrals in (2.51), (2. 52), so that 

these integrals are uniformly. convergent, proper integrals, permitting 

a series expansion of their integrands. Then, we ultimately have 

F1 (ir)C+G: +-1: F1(irlT)C+G' where (i.::) is invoked for each term of the 

resulting expansions. See the comments in B, Sec . (5.2) below. 

58 

where 

2 2 4 2 2 3 4 x6 

1F1(-2;1 ;x) = 1-2x +x /4; 1F

1(-3;1 ;x ) = l-3x + 2 x - ~; 

2 2 4 2 6 x

8 

1F1(-4;1 ;x) = 1-4x + 6x - 3 x + 24 , etc., 

with x

2 

= E

2

/2a6 here. 

with 

Similarly, we get for P1(E > E0 )C+G, (2.22b), here 

1

F

1(1 ;2;x2) = (ex

2

-l)/x2 

1

F

1(-l ;2;x2) = 1 - x

2

/2 

1F1(-2;2;x2) = 1 - x2 + x4

/6 

[Eq. (A.l.19b), Middleton (1960)] 

1F1(-3;2;x2) = l - 3x2

;2 + x4/2 - x

6

/24, etc . 

The first two terms of (2.57) reduce to 

2 2 -E/2cr0 Pl(E>Eo)C+G ~ e 

60 

(2.56a) 

( 2. 57) 

(2.57a) 

(2.57b) 
as expected, for this cumulative rayleigh P.O., cf. (2.48b) . 

Finally, the above results apply also for the purely Cl ass A or Class 

B interference, whenever voo becomes very large (i.e. the impulsive index 

is large).. The variance cr~, (2.53a), is then s~itably modified, as is 

(2.54a) for the correction terms. Since B2n is O(A00 ), while cr~ is also 

O(A00), it is clear that the correction coefficients B4(cri, [B6/cr~, (Bi)cr~8

] 

are O(A-l, A- 2), etc., showing the rate of fall-off of the correction 00 00 

terms with increasing index A. co 

" 2.5 Second Reduction of the c.f. F

1: The Role of Source Distribution and 

Propagation Law: 

Our major problem, as stated earlier in Part I, is to obtain 

analytically tractable results, as well as a pertinent physical foundation 

for our models of man-made (and natural) electromagnetic interference. 

Technically, our central problem now is to evaluate the probability densities and cumulative probabilities (2.21), (2.22), when the interference 

is Class A or Class B, accompanied by gaussian noise, with the respective 

characteristic functions (2.50), (2. 51). [The detailed study of Class C 

interference, with the more general c.f. (2.52), is reserved to Part IV 

of this series of Reports.] 

The desired evaluation may now be achieved by recalling (as in 

Section 3 of Part I [Middleton, 1974]) that the general character of the 

p.d. (and hence of the P. O. ) of a random variable at large values is 

controlled principally by the behavior of the associated characteristic 

function at and near zero values of its argument. Thus, the behaviour of ~ 

F

1(ir) at, and in the vicinity of, r=O, is determined by the largest 

r-dependent contribution which establishes the large-amplitude structure 

of w1 (E), P

1

(E), etc., i .e. as E + 00 • In fact, for these general classes 

of non-gaussian noise this corresponds to the expected slower fall -off of 

~

1(E), as E + c:o, than t~e rayleigh p.d. (2.48a), for e~ample here. [See 

also the discussion in Section (2.7)A following.] 

Our preliminary procedure for obtaining the required development of the /\ c.f. F

1 in the neighborhood of r=O consists of: (i), expressing J 0 -l as an 

integral; (ii), using an ~xplicit class of propagation law and source 

distribution; (iii), reversing the order of integration in (i), (ii) 

61 
as expected, for this cumulative rayleigh P.O., cf. (2.48b) . 

Finally, the above results apply also for the purely Cl ass A or Class 

B interference, whenever voo becomes very large (i.e. the impulsive index 

is large).. The variance cr~, (2.53a), is then s~itably modified, as is 

(2.54a) for the correction terms. Since B2n is O(A00 ), while cr~ is also 

O(A00), it is clear that the correction coefficients B4(cri, [B6/cr~, (Bi)cr~8

] 

are O(A-l, A- 2), etc., showing the rate of fall-off of the correction 00 00 

terms with increasing index A. co 

" 2.5 Second Reduction of the c.f. F

1: The Role of Source Distribution and 

Propagation Law: 

Our major problem, as stated earlier in Part I, is to obtain 

analytically tractable results, as well as a pertinent physical foundation 

for our models of man-made (and natural) electromagnetic interference. 

Technically, our central problem now is to evaluate the probability densities and cumulative probabilities (2.21), (2.22), when the interference 

is Class A or Class B, accompanied by gaussian noise, with the respective 

characteristic functions (2.50), (2. 51). [The detailed study of Class C 

interference, with the more general c.f. (2.52), is reserved to Part IV 

of this series of Reports.] 

The desired evaluation may now be achieved by recalling (as in 

Section 3 of Part I [Middleton, 1974]) that the general character of the 

p.d. (and hence of the P. O. ) of a random variable at large values is 

controlled principally by the behavior of the associated characteristic 

function at and near zero values of its argument. Thus, the behaviour of ~ 

F

1(ir) at, and in the vicinity of, r=O, is determined by the largest 

r-dependent contribution which establishes the large-amplitude structure 

of w1 (E), P

1

(E), etc., i .e. as E + 00 • In fact, for these general classes 

of non-gaussian noise this corresponds to the expected slower fall -off of 

~

1(E), as E + c:o, than t~e rayleigh p.d. (2.48a), for e~ample here. [See 

also the discussion in Section (2.7)A following.] 

Our preliminary procedure for obtaining the required development of the /\ c.f. F

1 in the neighborhood of r=O consists of: (i), expressing J 0 -l as an 

integral; (ii), using an ~xplicit class of propagation law and source 

distribution; (iii), reversing the order of integration in (i), (ii) 

61 
and observing the bounds imposed by the fact that u

0

A is of finite duration, 

while u

08 (z) f 0, O<z<00 , cf. Fig. (2.1) above.* As we shall see below, it 

is this latter condition (on u

0A vs. u08 ) which critically affects the 

explicit form of the needed development of F

1• 

We begin with the identity -J~(x) = J1(x), from which it follows by 

integration that 

(2.58) 

A 

Then, the exponents of the c.f. 's .FlA,B are (without the contributions, for 

the moment, of the background normal noise) but with the help of (2 .58), 

(2.59) 

2.5. l Propagation Law: 

We now introduce the somewhat restrictive condition that the source 

distribution and propagation law are expressible _in the factored form: 

a(A)[b(~) or b(e,~)J. The beam patterns are always independent of 

distance (CA), e.g. 

e.g. 

JORT(l;f~)I = 1aRT[(iT'iR)f~/c]I '(lR=-iT'cf. Fig. (2.1), 

Middleton [1974]), 

where specifically 

(2.60a) 

(2.60b) 

* Our procedure here is a generalization of that used by Giordano [1970], 

who, however, considered what in the limit (T+oo) is ultimately only Class 

B interference, and only special choices of source distributions and 

propagation law. 

62 
~ A ~ A 

iR = ixcos <PRsin aR+fysin <PRcos eR-izcos aR ( 2. 60c) 

in which <Pr is an azimuthal angle and aT a polar angle, as sketched in 

Fig.(2.2)II . Thus, for the propagation l aw, g(!) in (2.41b) we ~ri te 

(2 . 61) 

where g

5 V are angular factors, usually taken to be unity in the conman 

' propagation models. In general, y > 0, and, in fact, y ~ 1/2: y = 1/2 

corresponds to the "wave guide" modes often encountered in long-distance 

propagation in the atmosphere, while y = 1 applies for the usual spherical 

spreading of less distant sources. For practical applications, sources 

and receiver in a corrrnon plane, Fig . (2a)II, is typical of most mobile land 

transport cormiunication environments, while the 11 volume 11 situation of Fig. 

(2b)II is characteristic of ground/air, or ground/satel l ite, or air/air 

environments. Also, for practical purposes, atmospheric noise may .often 

be regarded as essentially coplanar with the surface (and y .; 1/2), unless 

the principal discrete sources are comparatively near to the receiver, i .e. 

aR(=ar) is·large, e.g. [>0(5-10°)]. 

2.5.2 Source Distributions: 

For the moment we continue to assume that the source distributions· 

are factorable into the form a = a(A)b(a,<P),' cf. remarks at the beginning 

of Sec. (2.5.1) above. Then, the density w1(A) required in the averages 

( )A in (2.59) is now from (2.26), (2.28) 

a5(A)C2Aos(<P) 

(2.62) 

.. for the surface and volume regimes, where ·the normalizing factors As,V 

are given by 

63 
z 

CA 

Figure 2. 2 (II) . Geometries of source and receiver beam patterns: (a), in a plane; (
Av= f oy(A)os(a,<p)c\2

sin a dAdad<1> 

AV-eff 

(2.62b) 

The Aeff are the effective domains of the possible interfering sources, 

namely, those capable of registering at our receiver [OR, cf. Fig. (2.2)11], 

i.e. observeable in the receiver packground noise. The receiver accordingly 

has a limiting range CAmax' which depends on e01 , cf. (2.34), e.g. 

Amax= Amax(e01 ). Several cases are distinguished, as shown in Fig.(2.3)11, 

[as far as dependence on A is concerned]: From Fig.(2.3)II it is clear 

that the source domain to be used for A(SorV) is A(SorV)-eff: the domain 

of sources perceivable by the receiver. This is detennined by either 

Aeff = Amax or AA [Cases I, II)], whichever is the lesser, or by a pair 

of A1 S, e.g. Ao~ A~ Amax' Case III, for such distributions. [Case IV, 

not shown in Fig.(2.3)II, is a combination of regions (irA!ax and A) which 

partially overlap. Here we must consider the overlapping and pertinent 

nonoverlapping regions separately, which will involve the angles <I>, or 

(a,<1>) explicitly.] In our present applications, however, we shall assume 

Case I, e.g. Aeff = .Amax < AA' which is by far the more prevelant situation in practice: the potential source domain always exceeds that of the 

receiver's acceptance region. 

Finally, we shall, where necessary, postulate the following range 

dependence of source density: 

er (A) = 1/Aμ s,v 0 < μ (2.63) 

for both the volume and surface situations, in accordance with our remarks 

at the beginning of Sec. 2.5.1. 

65 
A= A effective : 

c'A.A < c'A.max. 

Aef f ective : 

c'A.A>c'A.max. 

Case I, Case n Case IIT· .· 

Figure 2.3 (II). Several typical cases (;\ only) of source and receiver-· do
2.5.3: Role of In.put Signal Duration, T5 

We begin by developing in fuller detail the structure of the: basic 

" received envelope B0A,B' cf. {2.41b). Using (2. 60), (2.61) in (2.41b) 

allows us to write 

" (A B) B0A,B = e0/ A0 u0A,8(z)laRT(<1>;or e,<1>lf~)lgs,v(<1>;or e,4>)(4ircfY/xY 

· ( 2. 64a) 

= G0A,8(z,A0 ,e0 y,I i2RTI; 4>,or e,4>)/xY, (2.64b) 

with 

GoA,B = e~~,B)AouoA,BIC2RTlgs,v(4irc)-y · (2. 64c) 

containing Ure (possibly) random parameters A , e , QRT' for both surface 0 Oy 

and volume r~gimes and Class A and Class B interference. Next, we use 

(2.62)~ (2.63) to write (2.59) explicitly as 

\ 

z0 ,CX) ~2/xμ-l 

I,Jr)A B = -ACX)IA B f dzf As\crs vd(e,4>)[ dX · ' ' o ~(6,4>) ' ' [X] c3/Xμ-2 

x=rG /xY · 

• ~ 0 

· J1 (x)dx)e , _ (2.65) 

... 

with i = z

0 ,A0 , etc. , where the upper term applies for surface sources and 

the lower for those distributed in the volume. 

Next, we implement the key step, (iii), in order to interchan·ge the 

order of integrations over x and x in (2.65). This permits us to develop 

" ICX) explicitly as a function in r, to which we can then apply the approach 

indicated at the beginning of Section 2.5, to obtain the controlling 

term(s) at and near r = 0 for the characteristic function. Since from 

(2~59) 

X = rG 1,Y., .·.' ~ (r,G )11Yx-l/y d G /'y o " " o ; an :. xo = r o "max (2.66) 

is the value of x corresponding to xmax' which establishes the domain of 

67 
sources perceivable by the receiver [in the present Case I, cf.Fig . (2.3)II]. 

Now we use the fact that u

0A is nonzero only for (0.::. z.::. z

0 ), while 

u08 f 0, (0.::. z < 00 ). Since G0A,B ~ u

0A,B' cf. (2.64c), we see at once 

that for 

Class A: 

{ 

O < x < x

0 , since u

0A = 0, z > z0 ; 

Amax~ A~ 0, cf. Sec. (2.5.2). 

Class B: { 0.::. x < oo, since u

08 ~ 0, z < oo (u08 + 0, z + oo); I 

Amax~;\~ 0, cf. Sec. (2.5.2). 

(2.67a) 

(2.67b) 

Fig. (2.4)II shows the allowed domains of x and A for these two classes of 

interference. [For Class C interference, we use (2.42), with the c.f. 's of 

Class A and Class B determine9 separately, with the help of (2.67) and 

the resu1ts of Section 2.6, 2.7 below. The details are reserved for a subsequent Report.] In Sections (2.6), (2.7) following we obtain the desired 

c.f. 's (= exp{! )) for these two basic classes of interference, with the 00 

help of (2.67) in (2.65) and the observations presented at the beginning of 

Sec. 2.5. 

2.6 The C.F. for Class A Interference: 

With Case I conditions (cf. Fig. (2 .3)11) on the source distribution 

vis-a-vis the receiver range (Amax), and (2.67a) applicable here, we see 

that x + x0 for the upper limit on the integrand (for x) in (2.65) . 

Accordingly, since As,v is now precisely equivalent to the indicated integration over (A,6,~) therein, we see that (2.65) becomes at once 

ioo(r)A = -Aoo,A/~zodz~xoJl(x)d~ ; xo = rGo/A~ax =·rBoA, \o o "/z0 ,i 

( ~- 68) 

Next we use (2.58) to reexpress (2.68) as 

68 
sources perceivable by the receiver [in the present Case I, cf.Fig . (2.3)II]. 

Now we use the fact that u

0A is nonzero only for (0 .::._ z .::._ z

0 ), while 

u08 f 0, (0 .::._ z < 00 ). Since G0A,B ~ u0A,B' cf. (2.64c), we see at once 

that for 

Class A: 

{ 

O < x < x

0 , since u

0

A = 0, z > z

0 ; 

Amax~ A~ 0, cf. Sec. (2.5.2). 

Class B: { 0 .::._ x < oo, since u08 f 0, z < oo (~OB+ 0, Z + oo); 

Amax~>..~ 0, cf. Sec. (2.5.2). 

(2.67a) 

(2.67b) 

-Fig. (2.4)II shows the allowed domains of x and A for these two classes of 

interference. [For Class C interference, we use (2.42), with the c.f. 's of 

Class A and Class B detennine9 separately, with the help of (2.67) and 

the resu1ts of Section 2.6, 2.7 below. The details are reserved for a subsequent Report.] In Sections (2.6), (2.7) following we obtain the desired 

c.f.'s (= exp(I )) for these two basic classes of interference, with the 00 

help of (2.67) in (2.65) and the observations presented at the beginning of 

Sec. 2. 5. 

2.6 The C.F. for Class A Interference: 

With Case I conditions (cf. _ Fig. (2.3)II) on the source distribution 

vis-a-vis the receiver range (Amax), and (2.67a) applicable here, we see 

that x + x

0 for the upper limit on the integrand (for x) in (2 .65). 

Accordingly, since As,v is now precisely equivalent to the indicated integration over (A,6,~) therein, we see that (2.65) becomes at once 

Ioo(r)A = -Aoo,A/~Ozodz~OxoJl(x)d~ ; Xo = rGo/A~ax =·rBoA, 

\ ; z

0 ,§, 

( 2. 68) 

Next we use (2.58) to reexpress (2.68) as 

68 
I 

X 

0 

Class 8: 1.._ __ ..... 

Closs A:~ 

Figure 2.4 (II) . Schema of the (x ,A) -domains of intergration for I 00 , (2 .65) , for Class 
(2.69) 

sin~e z0 = (r5

)/Ts = 1), and then employ· the steepest descent method of Report I [Middleton, 1974, Sec. 3, Eq. (3.10) et. seq; as mentioned earlier 

[cf. Section 2.5)] it is the largest contribution to the exponent .in the 

neighborhood of r=O, which controls the behavior for E >> OD to obtain the 

desired development in r iri the neighborhood of r = 0. For this we use 

the identity 

in (2.69). From (3.11), [Middleton, 1974] we have 

) 

r2(A2)/4 (X) < 2 .e, 2.t \J 0 (rA) e = 1 + l A~/ / 1F1(-t;l ;A2

/\A2

))) (2.72) t=2 2 t! \ 

(2.56a) . The result is that 

(2.73) 

in which 

" Equation (2.73) in (2.69) now gives us the desired expansion of r(X),A 

about r=O, which governs the p.d . (and P.O.) for large values of the 

70 
envelope E. We have explicitly 

(2. 74a )' 

{2 .74b) 

this last if we neglect the correction terms [O(r4 in I..,)J .. 

When we consider correction tenns [in a later Report] it is convenient 

to define, as we did in Report I [Middleton , 1974] a set of coefficients, 

now extended to include the average over z

0 : 

c21 = 1!(-1 )

1

~~z

0

,F,(-i ;l;B~A/( B~A))dz\ = (- 1)1

1! Fi, . ~o ~z0 

(2.75) 

A 

cf. (2.73a), which contain the 21, 21- 2, ... , moments of the envelope 8

0

A 

at the output of the ARI stages of the rece iver. Specifically, we have 

here 

( 2. 75a) 

(2.75b) 

Ca= r(s~A ) - 16(B~A)(s~A ) + 72(B~A x s~A)2 - 72(s!A)

4

J;(s~A )

4

; etc., 

(2.75c) 

(2.75d) 

Thus, (2.74) becomes here, equivalently, 

A - r2(s~A );4~ .., (s~A/ (-1 l A 2j I..,(r)A = -A.., A+ A.., Ae 1 + l 2 ~ c2 r . ' ' 1=2 2 1(1!)~ 1 (2.76) 

71 
Combining (2.74), or (2.76), with the practical situation involving 

an additive background gaussian component, cf. Sec. (2.3.l), we get finally 

for the desired form of the c.f. for Class A interference 

so that 

(2.77) 

where [l+O(r4)J in (2.77) is given explicitly by the expression { } in 

(2.76). The final step in the reduction of these c.f.'s to the desired 

technically manageable form, particularly for the smaller values 0(~100) 

of the impulsive index A00 ,A, (2.39), which are typical of this class of 

interference, is now the following direct expansion of the t~rms conta·ining 

A00 ,A in (2.76) or (2.77): 

.. .) (2.78) 

[Equation (2.78) is formally the same as our earlier result (3.16) 

[Middleton, 1974], which is n?t surprising, since the c.f.'s have the same 

72 
form, cf. (2.50) . The parameters differ somewhat in detail, of course.] 

However, in approximating the exact (2.74a) c:f. by the approximate 

forms (2.74b), (2.77), (2.78), we obtain ultimately (cf. Section 3 

following) approximate P.O. 's, and pdf's (Section 4, ff), which may .not be 

properly'··n.or_malized, in the sense of yielding a mean square value, (E2\, . 

of the envelope, different from the exact relation (E2)A = 2n2

A(l+rA), cf. 

Eq. (5.12b), which is derived from the exact c.f. (5.10), or (5.10a). 

Accordingly, we choose the c.f., here and subsequently the leading 

term of (2.78), and from it detennine the associated P.O. and pdf. Then, 

from the pdf we determine 

E • (2.78a) 

If w1(E.)A is properly normalized, (£2\ should be unity. If not, and 

<£2)A is bounded, such that {f-,2)A = N~ (<~,11), then the properly normalized 

pdf (:, PD, also) is given by 

(2.78b) 

so that 

2 r~ -2 2 r~ 2 (£ )(A) = Jo wl ([) ANA E.; dS = ) 0 wl (E) A-norme de-= l , (2.78c) 

as required. From (5.3) following we have (cf. Sec. 5.1): 

(2.78d) 

exactly, showing that w1 (E)A here is indeed suitably normalized (and 

:. N! = 1 ), as is the PD, P1 (E. > E:'0 )A, consequently. Accordingly, the Class 

A cases discussed subsequently need no scale or other adjustments (unlike 

the Class B model, where the approximating forms do require scale and 

level adjustments, cf. remarks following Eq. (2 . 94), anu the discussion in 

73 
Section 3.2-A. 

An interesting feature of our results here, for Class A interference, 

is that the c.f., and hence the p.d .. , and P.O., are not explicitly dependent on the interference source distribution [for the usual Case I, II, 

and not so common Case III source-receiver conditions, cf. Fig. (2.3)11]. 

Furthennore, these statistics are insensitive to the propagation law 

(~ \ - y) involved, which merely affects scale, thrdugh the average 

<s~A)a~ (G0

/AY). The source distribution does appear, but in averaged 

form and only in the impulsive i ndex A00,A, [cf. (2.35), (2.38), (2.39) , 

in conjunction with (2.27)] . Physically, this is understandable, since 

it is only the average number of emissions per second times the mean 

duration of these finite emissions (in the receiver, cf. Fig. (2 .l)II) to 

which the receiver can respond. It has no way to distinguish where and in 

what concentration, or by what propagation law, the sources may be acting 

[for a given position of the receiver beam c".2R' or for any position if 

QR is omnidirectional]. The only thing that counts here in determining 

the (first- order) statistics of the received input is total input level 

and process "density",A A" As we shall see in Section (2.7), and in 

m, 

later Sections, this insensitivity to source distribution and propagation 

law is definitely not characteristic of Class B interference, and, · consequently, Class C noise, to the extent that its Class B component is 

significant. 

2.7 The C. F. for Class B Interference: 

Here we use (2.67b} for the exponent .(2.65) . The result is a tenn 

like (2.68), plus an additional term for x

0 < x < 00 , with\ ma,.- v>A>O, viz: -

(2.79} 

74 
Section 3.2-A. 

An interesting feature of our results here, for Class A interference, 

is that the c.f., and hence the p.d .. , and P.O., are not explicitly dependent on the interference source distribution [for the usual Case I, II, 

and not so common Case III source-receiver conditions, cf. Fig. (2.3)11]. 

Furthennore, these statistics are insensitive to the propagation law 

(~ \ - y) involved, which merely affects scale, thrdugh the average 

<s~A)a~ (G0

/AY). The source distribution does appear, but in averaged 

form and only in the impulsive i ndex A00,A, [cf. (2.35), (2.38), (2.39) , 

in conjunction with (2.27)] . Physically, this is understandable, since 

it is only the average number of emissions per second times the mean 

duration of these finite emissions (in the receiver, cf. Fig. (2 .l)II) to 

which the receiver can respond. It has no way to distinguish where and in 

what concentration, or by what propagation law, the sources may be acting 

[for a given position of the receiver beam c".2R' or for any position if 

QR is omnidirectional]. The only thing that counts here in determining 

the (first- order) statistics of the received input is total input level 

and process "density",A A" As we shall see in Section (2.7), and in 

m, 

later Sections, this insensitivity to source distribution and propagation 

law is definitely not characteristic of Class B interference, and, · consequently, Class C noise, to the extent that its Class B component is 

significant. 

2.7 The C. F. for Class B Interference: 

Here we use (2.67b} for the exponent .(2.65) . The result is a tenn 

like (2.68), plus an additional term for x

0 < x < 00 , with\ ma,.- v>A>O, viz: -

(2.79} 

74 
where we have used (2.66) and reversed the order of integration, according 

to the rigime of (iii) , Sec. 2. 5, and Fig.2 .411) above: Note, in particul ar, 

the order of the limits on the A-integration, which correspond to the variation in x, from x=x

0 

to X-+<», cf. Fig.(2.4)II again. [The average over z

0 

is unity, as none of the arguments contain z

0 ; also, i ' = A0 , eoy' etc . , as 

before.] The integrals I¢ ' l e,¢ over~ (or 6,¢) , with As:v , become 

expli ci tly from (2. 62a,b)* 

_ _ 2-μ) μ- 2 1¢ - 6s/As - (-2 Amax ' 

C 

I =A/A = (3-μ3) Aμ-3 e ,¢ uv V max 

C 

where specifically here 

0 ~ μ < 23} 

, 0 ~ μ < 

(2.80) 

(0 < μ < 2); c

3 3-μ Av= 6y 3-μ Amax (0 < μ < 3), 

(2.80a) 

so that 6

5, 6y are respectively the integrals J a

5

d¢, J ayded~ in (2.79). 

With the above we readily find for (2.79) the (exact) relation 

with 

A 

XO= rGO/A~ax = rBo,B (2.81a) 

as before, and the new parameter 

* For this Report we shall limit the allowed values ofμ as shown in 

(2.80}; extension to other values (μ>2,3) will be considered in a sub- sequent study, as is t~e analysis for a>2. 

75 
Cl -- £:1!_1 

Y surface 

or 3-μ1 

y VO 1. 

(.2.82) 

This parameter a. we call the spatial density-propagation parameter, since 

it depends on the interacting spatial effects of source density and source 

propagation 1 aw·. 

The lower limit on a. is established by the present conditi,an, i.e., l 

upon the upper limit onμ (=2,or 3), y > 0. Analytically, for the integral 

over x in (2.81) to be convergent, we require that a.> -1/2. There is, 

however, no (finite) upper limit on a., so that we can write 

-l/2<a.<co ] (2 .82a) 

For the purposes of the present Report we shall, however, further restrict a. 

to the range (O<a.<2), which covers many of .the practical cases encountered 

in applications, at least down to quite small values of the exceedance probabilities P1(c < £0 ). In a later Report we shall develop the analysis in 

detail for (a.~ 2). 

The first integral in (2.81) is readily evaluated by expanding the 

Bessel function, followed by termwise integration . We get 

(2.83) 

For the second integration, over x in the second integral of (2.81), we 

use a Barnes integral representative of J1(x) [cf. Middleton, 1960, Eq. 

(13.10)]: 

C > 0 , (2.84) 

76 
so that 

I = a 

= ~a rmi-c r(-s)ds~2:i [x2s+l-adx\ • 

\( 0 J-mi-C r(s+2)2 S l x

0 '/z,~' 

(2.85a) 

This becomes, on choosing c = -5/4 (as a result of the condition a> -1/2) 

and carrying out the x-i ntegration 

2s+2-a m 

X 

2s+2-a = 

XO 

2s+2-a 

-x 0 

2s+2-a { 

Re(2s+2-a) 

a-2 

s < 2. 

The resulting integral I is now ~xplicitly a 

< 0, or 

m1 -C -r -S X S n1 

~ . ( ) 2s+2-ad 12 . ) 

10 = x~~~i-c r(s+2);2s+l(2s+2-a) z,~·' 

(2. 85b) 

(2.85c) 

which has a simple pole at s = (a-2)/2 (>- 5/4), and at s=O,l ,2,3, . .. 

(0 <a< 2). The residue at s = (a-2)/2 is -r(l-a/2)/2°-1r(l+a/2), while 

those of r-function are (-l)i/t! at s=t (=0,1 ,2, .. ). The result is 

1 = -r(l-a/2)(x~) ~ (-1) (x~1

+

2

) 

a 2°-1r(l+a/2) - i~O t!(t+l) 22i+l(2t+2-a) · (2. 85d) 

A 

The exponent Im of the c.f. for Class B interference thus becomes, on 

combining I

1, (2.83), and I

0 , (2.85d), in (2.81): 

A {r(l-a/2)<X~) m (-l)i<x~i+2) [ 4t+4-a l\ 

Im,B(r) = - Am,B 2

a-lr(l+a/2) + i~O i! (t+l)! 22t+l (2t+2)(2t+2~a)~' 

(0 <a< 2) , (2.86) 

with <)-= () 8 , =(!~()dz\,, etc. 

z , ..... With the additive, .accompanying gaussian background, cf. Sec. (2 .3.1), 

we have at last the desired c.f.: 

77 
(2.87) 

which, like (2.81), is also exact, so far. Here we have explicitly 

_ r(l-a/2) /, a ) -ay _ r(l-a/2) li( 8

;

2

,B)~) (>O) (2.87a) bla = 2a-lr(l+a/2) \Go,B Amax - 2a/2-lr(l+a/2) \1 rL 

b _ (4-a) /G2 )A -2 = (4-a) \

13~ ,B) (>O) 2a 2-a ~o,B max 2-a 2 ' (2.87b) 

/1321+2) 

b _ (41+4-a) \ o,B ( O) (B G ; .,Y ) (21+2)a = 1!(1+1)!(21+2~a)(21+2) 2

21+1 > ; o,B = o,B Amax ' 

(2.87c) 

since -0 <a< 2, and from (2.65) we write 

<G~,8) = <•!~la) (A~) (1 °iirl t ( g~,v)( 4,c )-ay { u0 (z)IJdz (,0), 

(2.87d) 

and formally \G~:;2) is given by (2.87d)·on replacing a by 21+2, etc. 

A. The Approximating C. F.'s for (0 <a< 2): 

Unlike Class A interference [Sec. 2.6], where the c .f. is solely a 

function of r

2 [cf. (2.76)-(2.78)], and where a single ''steepest-descent" 

approximation [cf.(2. 72)-(2.76)] provides a good fit for both large, small, 

and intermediate ·values of r2(and hence for E ,E), Class B noise requires 0 

a pair of approximating c.f. 's, one of which will at least insure suitably 

bounded behaviour of the exceedance probability P

1 (E > E

0 ) as E0 + ~, 

including the existence of all finite moments of the envelope (Es)(O<s<~), 

and the other of which will provide a satisfactory account of P

1 for small 

78 
and intermediate values of E (>E ). It is the presence of the term 

0 

O(ra) in the (exponent of the) c.f. (2.87) for Class B interference, in 

addition to the typical development in powers of r2 (analogous to that for 

the Class A noise), which forces this double approximation for our canonical c.f.'s, and P.D. 's. P

1(E>E0 ), pdf's, w1(E), here. 

with 

At this point we define the gaussian variance 

~o~ - a~+ b2aA~,B = b2aA~,B(l+o~/b2aAoo,B) 

- = n(G\1+r(G)) - 28 B ' 

(G) - . n2B = b2aA00 ,B . rCG) = a2;~(G)= indep. gauss intensity ' B - G 28 . Rimpulsive 11 gauss Intensity 

(4-a) = 2_a n

28 , cf. Eq. (3 .2a)ff. 

(2.88a) 

(2.88b) 

(2.88c) 

where ~1i)is the "impulsive" contribution to the gaussian component arising 

from the Class B noise alone, and where o~(=o~+o~) are the (independent) 

inherently gaussian contributions from potential externa l (gaussian) sources 

and from the receiver noise (essentially al l arising in the initial linear 

input stages), cf.(2.47). ·(Note that n~~.r~G)are also functions of a here.) 

For the c.f. which is appropriate to the intermediate range of envelope 

values, including the very small (E,E0

+0), the controlling term in the 

exponent of the (exact) c.f. (2.87) is the smallest power of r with negative 

coefficient, e.g., ~b1aA 8ra here, so that this approximate form remains 

1 im ~~ 1 im ,.. . a proper c.f., e.g. r+O F

1 = 1, r-+oo F1

+o. The form of the associated pdf. 

and P. O. for small and moderate values of E, E0 is governed principally by 

the behaviour of the c .f. is r becomes large. Thus, as a first approximation which ignores any gaussian contributions , we have from (2.87) 

- b

1 

A 8ra 

• e a °', 0 <a< 2. (2.89) 

79 
However, practically there is always at least an observeable gaussian system noise component, and as noted above, cf. (2.87), (2.88a), an additional 

gaussian term (""b 2 A 8) contributed by the "impulsive" Class B noise, so CL a,, 

that the more realistic intermediate c.f. here is now 

, (0 < CL < 2), (2.90) 

where the subscript (-I) indicates the c.f. for the range (0..::.. E, . E0 ..::.. E8) 

of envelope values. (The precise definition of E8 will be given presently, 

cf. Sec. 3.2). 

For values of E, E

0 

> E8 we require a c.f. approximating the exact 

relation (2.87) where the largest (r-dependent) contribution to the 

exponent about r = 0 and in the finite (nonzero) neighborhood of r=O is 

the controlling term. For this we seek again a "steepest descent" form 

for the exponent of (2.87), exclusive of the term in rCL, which as we shall 

see below is always here smaller than the fonner (for O..::.. r..::.. €) and thus 

does not control the character of the c.f. at small r (and hence for large 

E, E

0 ). Accordingly, as in the Class A cases, cf. Section 2.6 above, we 

wish to represent the class B tenns (exclusive of. rCL) in (2.87) by a series 

of the fonn 

r

2 "" t+l 2t+2 -ar

2 r ; 2kJ A..,,B-b2CLA"",B "2 + I,(-1) A..,,Bb(2t+2)CLr = Ae Ll+k;lBkr ' 

t- .. ( . 91 ) 

where the "steepest-descent" nature of the approximation is exhibited 

not only by the exponential factor but by requiring the vanishing of the 

B1-term in the right hand series, where the nearest "correction" term 

(k=2) is O(r4) and quite ignorable vis-a-vis unity. This condition and a 

tenn by tenn comparison of (2.91) detennine all the parameters A, a, Bk 

(k ~ 2), which are r.eadily found to be 

80 
A = A QO ,B a = b2

/2 ; (B1 = 0) 

- 2 B2 - b4a-b2/8 

···-- b b 

B = b + 4a 2a 

3 6a 2 

b3 

+ --1.£. 48 t , e c. 

(2. 92) 

Clearly, A Be-ar

2 

dominates -A Bbl ra A Be-ar

2

r4, etc., at and in the QO' c,o, a QO' neighborhood of r = 0, and this is the determining element for this approximation to the exact c.f. (2.87). 

Accordingly, we have finally for the c.f. appropriate at least to the 

large values of E, E

0 , i.e. for the "rare events 11, 

• 2 . 

A • (-A B Ii -b2 r /2 2 2· ~} 4 F1(ir){B+G)-II = e QO' exp~QO,Be a -a6r /~ [l+O(ra,r )], 

(O<a<2). 

(2.93) 

Comparison with (2.77) shows at once that this approximate c.f. for 

Class B interference has the .same (approximate) form as that for Class A 

hoise, ant thus will yield the same type of pdf's and P.O. 's, etc., cf. Sec. 

(3.1 ). As we shall see later, in Section 4, this has the important consequence of insuring th~t all (finite) moments of the envelope, (E~), 

exist, as required by the physics of the situation in all cases. 

A 

We note in passing that a more elaborate approximation 

be obtained by a combination of (2.89) and (2.93), viz: 

to F

1 here may 

A -b1 AQO 8ra-AQO 8-a~r

2 /2 ( -b2 r

2 /2) 

Fl(ir)(B+G)-III = e a ' ' exp AQO,Be a , 

(2.94) 

which may be used for intermediate ranges of r for improved fits to the 

corresponding intermediate ranges for the envelope. However, since the 

resulting pdf's, and P. O. 's, are considerably more complex and since the 

simpler forms of c.f. above , e.g. (2. 90), (2.93), appear ultimately to 

provide excellent agreem~nt with observation, we shall not pursue the consequences of using (2.94) further in the present Report . 

81 
We remark. that both approximating c. f. 1 s (2. 90), (2.93) for the 

true Class B c.f. are such as to give pdf's which are not properly 

normalized; each pdf,. w1(c:.) 8

_1 ,w(£) 8_1p (4.3),(4.4) does not yield 

(E..-2

)

8 = 1. The former gives an infinite value, while the latter, although 

Class A-type, cf. (2.78), yi.elds (£2

) 8_11 = 4G~ (.Fl), where G8 is given 

by (3.12b). Thus, w1(t:)B-II,norm = (4G~)-~w1(£) 8_11 , whil~ the normalization of w1 (E) 8

_1 requires, instead, a change of scale for the argum·ent 

€...(and:. £

0 

in the associated PD). How this is done is described in Section 

3.2-A. 

Finally, it is important to observe that unlike the Class A interf~rence 

discussed in Section 2.6 above, the (first-order) statistics of Class~ 

noise are obviously sensitive to the combined effects of source distribution (μ) and propagation law (y), through the density-propagation parameter a, cf. (2.82). Physically, this ·may generally be explained by the 

fact that now the receiver itself largely determine.s the waveform of its 

response to the (relatively) short input excitations, unlike Class A noise, 

where apart from amplification (for fixed aperture bearing) the receiver 

negligibly influences the structure of the received wave trains. The. 

composite sum of the "tails" of the transients in the receiver, generated 

by the Class B input, depend on the (relative) times of arrival of individual wave trains (~source distribution) and on the level of the · 

various wavetrains (~source distribution and propagation law). The 

(relatively) longer time-pedestal provided by the transient decay of in- , 

dividual impulses provides a wider range of possible total amplitudes of 

overlapping transients and hence a.more gradual transition to given 

thresholds (E0 ) of the exceedance prob~bilities P

1(E > E

0 ), than that 

occuring with Class A interference, as can be seen subsequently in Figs.(3.5, 

3.6)II vs. Figs.(3.l)II,(3.2)II. [These effects accordingly influence 

the instantaneous waveform in the receiver's ARI stage, and hence the 

statistics of that waveform.] In any case, the sensitivity to a is 

thus a receiver bandwidth phenomenon, which is illustrated by the 

experimental and theoretical results shown in Part I of this Report. 

82 




"' E 

.... ~ 

A 

CD 

"O 

0 

~ 

20 

0 

-20 

-40 

Closs A lnterf erence 

At,0.1 

-60 .........._.....__.._ _..___._....,___.____.__..,__,__,_.____._ _ ___J,. ___ L__ _ ____J. _ __.:::i,,, 

io·6 w·410·3 .01 OJ 0.2 0.4 0.6 0.8 0.9 .95 .98 .99 

Pi(e > €0) 

Figure 3.1 (II). The envelope distribution [Prob(e' > t 0 )] calcul?ted for 

Class A interference for AA= 0. 1 and various rA from 

eq. (3. 7 b) . 

87 
40 

20 

0 

If) 

E 

.... (,0 

(I) " 

"O 

0 

~ -20 

-40 

AA =0.5 

AA=10-1 

/ 

0.1 0.2 0.4 0.6 

Closs A Interference 

f'=I0-4 

A 

0.8 0.9 .95 .98 .99 

P1 (e > eo) 

Figure 3.2 (II). The envelope distribution [Prob(§ > €0 )] calculated for 

Class A interference for rA = 10-~ and various AA from 

eq . ( 3. 7 b) • 

88 
These, in turn are applied to (3 .4) to give us Pl-I ,II respectively . 

Starting with (3.10a), we expand the exponential in Aa and use (2.55) 

to get 

A ~ ~ ~ oo (-1 )hAn 

( 3 .1 Oa) 

( 3. 1 Ob) 

= P,(£>£0) ~ 1-[2 I a r(l+ an) F (l+ an 2 {2) - o n=O n ! 2 1 1 2; ; - o ' with (3.llb) 

" 

AB<~ ,:~,B(l+r'))") 

28 8. 

and ( 3. 12a) 

G2 = l(l+r' f 1 (4-a +r ,) B 4 B 2-a B ' ( 3. 12b) 

cf. (2.88a,b,c), (3 .3a), where NI is a scaling factor which scales Pl - (B-l)' 

wl-(B- I) !o insure that (£2)

8 = 1, cf. (2.94) ff, where a~~cr~ = 2G~. The 

quantity A is the Effective Class 8 Impulsive Index, which is proportional a . 

to the Impulsive Index A8 , for this Class B interference. In addition, it 

depends spatia l ly on the spatially sensitive parameter, a, and on the 

relative gauss component rs, (3 . 2a) . 

With the help of Kumrner's transformation [Middleton, 1960, Section 

A.l.2, p. 1073, Eq . A.1 - 17] we can write (3.llb) alternatively as 

_:t.2 oo (-l)nAn 

i,'l (t >~)B- I ~ e of,_t,2 l n! a r(l 1~n)1F1 (.l-a~;2;C~)) ~ ,o n=l 

89 

(3.13) 
where we have used Eq. (A. l-19b) [Middleton , 1960]. For large~ we obtain 

formally, with the help of the asymptotic re l ation [Middleton, 1960, 

(A . l-16b), p. 1073], 

F (a.·D ·-x) I\, ~r_(s_) x- a. [i+ a.(a.-s+l) a.(a.+l )(a.- 13+1 )(a.- 13+2) J 1 l ,μ, - ~ . l !x + 2 +. · · ' 

· 21 x (3.14) 

the following expression for P1: 

" /.::- . °" A~(-l)n+l 

pl(£\& )8- I ~ L --,-- 0 n=l n. 

r(l~) A 

2 r - na. [1 + ( l+a.n/2) ( a.n) + J 

V A2 ••• ' r ( 1-a.2n) o 2£0 

c2 >> l e-o . (3.15) 

This shows that i::., P1_1 + O(~;a.) + 0. However, as explained in A, Section 2.7, for £0 greater than some (large) value f:s, which is determined 

from Eq. (3 .19e) below, we must use the second form of c.f., (3 .. lOb). 

Figs.3.3II,3.4II here are based on (3 . llb), (3.15), and are valid representations, provide.d % is not too large, e.g.~ 2 £8. 

For the "rare events", or large £.0, we apply (3.10b) to (3 .4), as 

discussed earlier (cf. A, Sec. 2.7), to obtain 

.... -AB <» A~ [°" -a~Ba 2;x. 

2 /2 

Pl ( S >6c,)v-:-II ~ l-£oe m~O. mt Jo Jl (C:o;>..)e d;>.. , (3 . 16) 

with 

( 3. 16a) 

from (3.2a) ,(2.88c), cf. (3.5), (3 . 6): thus omB has the same form as amA' 

(3.6) . Accordingly, we may use the result (3.7b), rewriting it here for 

this large-magnitude approximation for Class B noise, as* 

- A m 2 A2 

e B oo A8 -S/2omB --I I e 

4G2 m=O m. 

B 

(3.17) 

* This PD is now properly normalized [remarks after (2. 94) and], cf. (i), 

(3. 17a,b), and discussion following . 

90 
80 

60 

40 

20 

0 

-20 

I0-6 10-4!0-3 .01 .05 0.1 0.2 

Closs B Interference 

a= 1.0 

0.4 0.6 0.7 0.8 .85 0.9 .95 .98 

~l (e > e o) 

.99 

Figure 3.3 (II) . The envelope distribution [Prob(§ >§ 0 )] calc~lated for 

Class B i-nterference for a = 1.0 for various Aa from 

eqs. (3.llb, 3.15). 

91 
O'.) 

"O 

~o 

80 

60 

40 

20 

0 

a =0.2 

__/ 

0.1 0.2 0.4 0.6 

Class B Interference 

A A0 =1.0 

0.8 0.9 .95 .98 .99 

~ "' Figure 3.4 (II). The envelope distribution,JProb(e' > e0 )] calculated for 

Class B interference for Aa = 1.0 for various a from 

eqs. (3.llb, 3.15) . 

92 
[Figures 3.1II, 3. 2II for Class A interference i l'lustrate the character of 

(3.17), which, of course, is only applicable here , Class B, for the larger 

values ·of t:

0 (~8).] 

A. The GQmp?site Approximation : . 

The problem with the approximating results for P

1

_

8 in the Case B 

model, cf. (3 . llb) and (3 . 17),is that these forms , stemming as they do from 

approximate c .f.'s [cf. (3 . lOa,b)], are not properly scaled , or "normalized", 

in the sense that each ~pproxim;ting form, P1_I,Pl-II' does not yjeld the 

correct mean square value of(€ ) 8 = l or (E2)8 = 2n28 (l+r8), cf. (5.14) 

with (3.2), and the remarks following Eq . (2.94) above . The approximation 

Pl-I' and its associated pdf, wl-I' (4.3), in fact, do not possess a finite 

mean square on (0, 00 ), cf. Sec. (5.3)ff., while Pl - II' the "Type A" form and 

.its pdf., wl-II' (4.4), yields (t:2)8_II fl . 

Accord i ngly, since the precise mean square is finite and is known to 

be (t3)8 = 1, by calculation from the exact c.f. [cf. .(5.lOa), and Section 

5. 2-8], we must suitably scale (or "normal i ze") w1

_1, wl-II (4. 5) so that 

(s2

) 8, cf. (5.6c), exists and is equal to unity. This is done as follows : 

(i). Let us consi.der first·wl-II' (4.4), and calculate (£2)

11 on (O<S<00 ) 

according to (5 . 1). The result is easily seen to be 

A ( /AA I ) 4-a I 

2 - B "' m B+rB m 2- a +rB 2 

(S >s-rr = e m~o(l+r13)m! AB = l+rB = 4GB(;l)' (3.17a) 

where G~ is given by (3.12b), so that here we require the normalization 

2 ( 2)-1 factor NII = 4G8 , e.g. 

(3.17b) 

(Henceforth in the text we write w1(c:) 8

_

1

I,Pl- II in normalized form, 

which are then used for anlytical and numerical calculations in the 

remaining sections of Part II here.) 

(ii).The case of w1_1,(4.3), requires a different approach, since (&2) 8

_

1 on 

(O<E<1i>) becomes infinite (0<a<2), cf. Section 5.3: [(e2\_11 on 

93 
(0<[<~<00 ), of course, is :finite, cf. (5.6c)]. Her: we need to scal1e 

e0 according to (3.llb) above: S

0 +[0

N

1 (and:. [ 0 = (e0

f~r)/2G8). 

The rationale for this is the observation that Pl-I (and w1_r) must 

have the same values in the rayleigh region (E..~<<l ), where P

1

_r ~ 0.9, 

or 0. 99, etc., as does the precise distribution, P

1_8, based on the 

(intractable but) exact c.f . (2.87), hence (3.llb). The scaling factor, 

Nr, is to be determined by fitting the two approximate forms Pl-I' 

Pl-II together by the procedure outlined below, which is based on the 

canonical properties of the Class B model generally. Note, finally, 

that the 11Cl ass A" form (I I) is coupled to the Cl ass B form (I) 

through the Class B parameter a, and vice versa through the "Class A11 

parameter r

8, appearing in G8, common to both approximations I,II. 

To combine the suitably scaled Pl-I and normalized Pl-II to form the 

composite approximation for Class B interference which is valid for all 

E0 ~ 0 we now use the following desired pro~erties of Pl-composite' which 

is sketched in Fig. 3.5II: 

(i) . Pl-I= Pl-II in the rayleigh region, e .g. 0 .::_S0 small. Equality 

(ii). 

of the two approximatfons in the rayleigh region is . 

required, sjnce both must represent the same (small) 

amplitude behaviour, characteristic of all these 

PD' s. 

dPl-II = --,--- in the rayleigh region. dco 

(iii). Pl-I Pl-II at the "bendover" or junction point £ 8 of the two 

approximations, · cf. Fig.(3 . 5)II. This point, c8, 

is empirically determined from the data, e .g. from 

the experimental APO or exceedance probability curve 

P

1 (€. >G ) , as described below, cf. (vi). o exp. 

94 

{ dPl-I -- dPl-II) (iv). \ de.. de- £ the (finite) slopes of the approximating Pl-I,II 

are equal at cS8: this insures a common tangent, 

i . e. , a smooth fit; moreover, we have 

( V) • 

. B 

d2P 

= 1

2

11 ). (r.O): this follows as a consequence of (iv) , 

d~ c

8 

and the continuity of the derivative at 

£8 , insuring that the associated pdf's 

are continuous at the joining pointC8. 

However, note thatt:'8 is not usually a 

point of inflexion of Pl-I,II' 

(vi). £ 8: this is the point of inflexion (d2

P1/d£~ = 0) of the actual 

P and is determined as such (by inspection, usually), of 1 ' . . 

the empirical exceedance probability Pl-exp.' cf. Fig. 

(3.5)11. (3.18) 

Accordingly, from (3.llb), (3.15), (3.17) we have explicitly for (i)-(v) 

above: 

( i ) . 

A n 

( -1 ) n (-a-) A 

2

a'G~ 2 - B m 

B Sae = AB "2 -1 

--n..,..!--=-- f (l+ ~n) . 2 l mi"" (2crmB) 

4G8 m=O · ( 3. 19a) 

[(ii). [Same as (3.19a), without the 2£:: factors common to both members 0 

of the equation; however, (ii) is here implied by the form of 

(i) and do~s not provfde new information.] (3.19b) 

A r ( 1 + / 2 ) o N -a -A m 2 A 2 

a (e,B I_\ 2 2 B = A -~/2cr 

-2~-G-~r-(-1--a-/-2) ~ 1 +qi i d[£BN!r~£BNI) ]= ~: mlo m'I- e mB; 

A ar(l+a./2) (£ N )-a-1 E. e-As = Am -£2;2"2 

a BI [l O (r£ N]-ar ]3 B B B 0mB 

2o.Gar(l-a/2) 2GB +(iv)!.! B I LE.ON! )]~ 4G2· mrOrn! e ; B B -

(iii) . 

( i V) • 

( 3. 19d) 

(v),(vi): £8 cannot be determined analytically from either approximating form Pl-I,II' It must be established as a point of 

inflexion from the empirical PD, as noted above. (3.19e) 

96 
[In using (iii), (iv), w.e may need at least the next set of "correction" 

terms in the asymptotic developments of Pl-I' dP1_I/dE8.J We note that 

(given £ 8) these three relations [(i),(iii),(iv)] are sufficient to 

determine in principle, any three of the six parameters NI,a,Aa,A8,n28 ,r8 

(cf. (3.l6a)), when the other three are specified . Later, in Section 6, 

we shall show how (3.18), (3.19) may be extended to permit us to obtain, 

from the experimental exceedance probability P1(f>c0 )xpt' the~six 

parameters NI,a,Aa,A8,n28 ,r8 (or, more fundamentally, [a,A8,(B~,8),n28 ,r8J, 

cf. (3.12)). 

For the illustrative calculations of Figs. 3.6II, 3.7II, it is convenient to preset £8;N 1,a,Aa, and determine A8,r8,n28 from (i),(iii),iiv). 

Other possibilities are: Fix (c8;N 1,a,r8), determine (Aa,n28 ,A8); fix 

(£8;Aa,a,r8), determine (N 1,A8,n28 ); fix (e8;A8,r8,n28), determine 

(NI,a,Aa); fix (c8;N1,A8,r8), determine (Aa,a,n28 ) etc. In any case, 

we have now 

(3.20) 

with Pl-I' Pl-II given respectively by (3.llb) and (3. 17). The curves of 

Figs. 3.6II, 3.7II are equivalent to Figs. 3.1 .1, 3.1.2 of Furutsu and 

Ishida [1960], with (v/o:)F+I + A8, (a/a)F+I + (r8r 1, and (R/a)F+I + en 

and exhibit the same kind of "elbow" in the transition region from the 

rayleigh behaviour (for£;<<l), with a bend-over to a constant slope 

(P1'\,e-at:B, .n>O), as for Class A noise, when Co+ co, cf. Figs.3.lII ,3 .2II, 

3.5II. 

B. Remarks on Hall-Type Models: 

Finally, we observe that a Hall model [Hall, 1966] may be obtained 

formally from the Pl-I form for the rayleigh and intermediate region 

(O.::,~,::,c8), provided we neglect the gaussian contributions (e.g . ~a~+ O), 

so that the c.f. (2.89) now applies. From (2.89) in (3.4) we accordingly 

97 
-

CD 

-0 

.._.. 0 (,0 

60 

40 

20 

0 

a=I, Nr/2G9=I 

, _____ _ A a = I, a= I , NI = 1.7 31 

As =5x 10-4, fa' = 10-3 , il2s = 3162 

____ A a= 10- 1• a= I , NI= 1.729 

As = 10-4, fs '= 5x 10-3, il2e = 199.5 

~Aa=lo-l, a=I, N1= 1.723 

As =I0- 3,f a'=l.6x 10-2, il2s = 70.8 

-----Aa = 10-2, ~=I, NI= 1.53 

As= 10-4, f 8 '=0.5, il2s= 2.01 

0.1 0.2 0.4 0.6 .0.8 0.9 0.95 0.98 0.99 

Pi(e > e'o )

8 

Figure 3. 6 (II). The (complete) envelope distribution P1 (e > €0 )s , eq. 

(3.20 ), calculated for Class B interference for various 

Aa , given a eqs. (3 . 11, 3. 17 , 3.19, 6.9, 6. 10) 

98 
CD 

"O 

-

<J 

70 

60 

50 

40 

30 

20 

10 

0 

-10 

Aa= I, a=0.6, Nr= 1.56 

As= 10-2 , rs'= 1.6 XI 0-5 , fl. 28 = f.99x 105 

Aa = I, a=0.2, NI= 1.45 

Aa=0.35, fs=2xl0-5, il2s=8.9xl04 

A a=I, a= 1.2, NI.= 1.86 

As =5x 10-~ rs'= l.6xlo-2, il2s = 158.5 

Aa= I, a=l.8, N1 =!.63 

-- . As= ,o-4, rs·= 5.o, n28=0.47 

-20 ........ ....._.....__.._~ .......... --..__. ..........__.__,___.~_.__.____.~~._~~-'-~-

10-6 10-4 103 0.01 0.1 0.2 0.4 0.6 0.8 0.9 0.95 0.98 0.99 

P1(e> eo)B 

Figure 3.7 (II). The (compl~te) envelope distribution P1 (€ > e'o)B, eq. 

(3.20), calculated for Class B interference for various 

a , given Aa (eqs. 3.11, 3.17, 3.19, 6.9, 6.10). 

99 
obtain* 

(3.21) 

This integral may be evalu~ted in several ways, 

or large €0 . We start with the case convenient for 

employ the following transformations: 

convenient for small 

the small values, and 

B = bl ABa0 (=A [-a.); a. a. a. : . >.. = 

d>. = dz 

so that (3.21) becomes now 

£* 00 1-a. 

Pl (C:>l'o)B-I ~ 1- a.of Jl (f;~z 1/a)z a. e-zdz 

0 

C'* - C I Bl I a. 

' (...0 = co a. ' 

Next, let us use the Barnes integral representation of J

1: 

r ( )£*2s+l 

J (S*zl/a.) = j -s o 2 (2s+l)/a. ds. , 1 o r r(s+2)22s+1 2~1 

(3.22) 

(3.23) 

(3 . 24) 

cf. Eq. (13.106) and Fig. 13. 22 of [Middleton, 1960], where r is the contour 

(-00 i+c, ioo+c), with c(<O) chosen so that the integral over z in (3.23) is 

convergent at z = 0, e.g. 

(3.25) 

* Equation (3.21) was obtained earlier by Giordano [1970, Eq. 3.66 therein ; 

Giordano and Haber, 1972, Eq. 24] but was not analytically evaluated. 

Moreover, P1 1, here (as wel l as the earlier forms [Giordano , 1970], etc.) 

is not scalea, e. g. ,(£2)8- Irl, as discussed at the beginning of A. above. 

See comments at end of B nere . 

100 
Thus we find that (3.23) becomes 

" , £~ f r l-sl (2+2s) (~)Zs+ 

1 

ds 

Pl (c..>Eo)B-I ~ l- ~ ~ r -a- 2 2'1Ti 

r 

ZE2 <><> (-l tr ( 2+2n )£2n 

l o I a o ~ - ~ n=O n!(n+l)!A(Z+Zn)/a 

Cl 

(3.26a) 

(3.26b) 

this last on evaluating (3.26a) at the simple poles s = n = 0,1 ,2, ... and 

using B = A 2-a, cf. (3.22). Equation (3.26b) exhibits the characteristic 

Cl Cl 

rayleigh form (~£~), when £~«1, as expected. 

Next, for large values of e0 (or small values of Aa), we return to 

(3.21) and use the Barnes integral representation for exp(-B Aa), viz: a 

to reexpress (3.21) as 

-2 < Re(as) 

Re(s) > -1. 

To evaluate the z-integral we use [Watson, 1944,p. 391]: 

f <><>J (t)tμ-v-ldt = r(μ/2)/r(v -μ/2+1 )2v-μ+l, Re(μ)<Re(v+3/2) Q V 

(3. 27} 

(3.28) 

(3 .29) 

with v=l ,μ = s+2 here, and :. O<Re(as+2)<2<Re 5/2, as required. Equation 

( 3. 28) becomes 

I r(-s)r(l+ ~s) *-as d 

pl (e>£o)B' -I ~ 1- 2as.eo _2s. 

r r(l - ~s) 1r, 

(3.30a) 

~ I n=l 

r ( l + ~) (-1 ) n+ 1 An 2 a (3.30b) 

101 
which shows how this approximation behaves as [ 0 + 00 , viz. O(c~a) , e.g.: 

r(l+a/2)A A2 

p (£>e )' '\, a - rl.J+aJ _ _£_+ 0(£-3a) '0 <a< 2. 1 o B-I - r(l-a/2)£a

0 

~ £

0

20. o 

( 3. 31 ) 

Now in the special case o. = 1, we may sum the series (3.26b) or (3.30b). 

Choosing (3 .26b) we get 

oo (-1 )nr ( 2+2n )C:2n 

P,(E>£o)B1 I"' l-2c2 l 2+2° (3.32a) - - 0 n=O n!(n+l)1A1 n 

and since (n+l)! = (2)n' r(2+2n) = 2

2

n(l)n(3/2)n [Middleton, 1960, (A.l-46b), 

p. 1078], we find that 

2 

Ea 2 2 P., (t.>E.o)B-I ~ 1-2 2 l1 (l ,3/2;2;-2e/A,) . (3.32b) 

Al 

From [Middleton, 1960, Eq. (A.l-40c)], the (gaussian) hypergeometric function in (3.32b) is explicitly 

h <1 ,3/2;2;-x2) = <21-x2)(1+x2)-112~ -P}. 

so that (3.32b) reduces explicitly to 

(3.33) 

which is a special case of the Hall model (eHall = 2), for envelopes 

[Spaulding and Middleton, 1975, Eq. (2.33)]. [Note that Pi~ £~1, £0 + 00 , 

which checks with (3.31): in fact, both (3.31), (3.33) give Pi~ A1/2c.0 , 

as expected. Observe also that P1(c~£0=0)8_1 = 1, as required: P

1 is a 

proper P.O., although it is an inappropriate approximate form when ~crG 

is at all comparable to (b1

o.Ala)l/a, cf. (3.10a); it is also not applicable 

for very large e0 , as explained earlier in A of Sec. 2.7 above. In any 

102 
case, Hall-type pdf 1

s and P.O. 's are not possible for Class A interference.] 

Finally, we note that although the above PD's and pdf's, Eq . (3.26b), 

(3.31), exhibit the .correct behaviour as t~ ~ 0, , they are not scaled 

(in €0 ) properly, to provide the finite mean square needed, e .g. <E 2

> = l, 

(cf. comments at the beginning of A above). Accordingly, as for P1_

1 

above generall y, cf. (3 . llb) et seq. , we must repl ace c0 by [ 0 N1, where the 

scaling factor N1 is determined, along wi th the four other parameters 

(A0 ,a ,r8,n28 ) of the distribution, by the procedure outlined in Section 6C 

following. [For the Hall model, a= ·1 here, and there are then only four 

parameter values (N1,A0 ,r8,n28 ) to be established.] 

103 
4. PROBABILITY DENSITIES: w1(e)A,B 

It is now a simple matter to determine the probability densities (pdf's) 

(pdf's) associated with the exceedance probabilities (PD's) derived in 

Section 3 preceeding. Because the PD's are continuous, at least through 

the second derivative (0.::. E:, <~),and because 

we may apply this to (2.7), (3. 11) and (3.17), etc., to obtain directly the 

desired pdf's. We have first: 

4. 1 Class A Interference: 

From (3.7b) and (4.1) we find that 

o < C. (4.2) 

Thus, as expected from our earlier r ~ . .:lt(3.7b),w1(~)A (in its principal 

contribution*) is the weighted sum of rayleigh pdf's, whose variances~~ 

cf. (3.5). increase with order (r~:_l. Figs.(4.l)II and (4.2)II show w1(&)A 

for various combinations of the controlling parameters AA, rA. With AA 

small the pdf's are seen to be highly nongaussian (e.g. nonrayleigh in C:), 

unless rA is very large, in which case the gaussian (e.g. rayleigh) component (here) dominates. As the Impulsive Index AA gets larger, the pdf 

approaches the purely rayleigh form, cf. (2.57b) . Also, for rA > 0, the 

pdf near£.= 0 has finite width, shouldering off into a broad, rather 

low level (in w1) form as C+ ~, which represents the strongly nonrayleigh structure of this class of noise. The larger rA, the wider and 

less "peaked" is the "sptke" at£.% 0, and the more "shoulder" there is 

to the rest of the pdf. 

2 When rA = 0, e.g. crG = 0, i .e., when there is no independent, 

* See the conments following Eq . (3.7b). 

104 
...... 

0 

u, 

........ <t 

w 

-

100 

10 

10~ 

Figure 4.1 (II). 

-50 -40 -30 -20 -10 0 

e(dB) 

The pdf, W1(t)A, of the envelope for Class A interference, 

various rfl., given AA (cf. fig. 3.1 (II) for P1(e > eo)A]. 

10 

calculated 
r·=10-4 

A 

A =0.5 

A =10-I 

A 

A =10-2 

A 

-5 10 -s~o~J..._--s~o~J..._--4~o~J..._ __ 3~0~~--2Lo~.L..---1Lo~l__-ol____J_~,Lo~L.L~2uo

e'(dB) 

Figure 4.2 {II). The pdf, w1(e)A, of the envelope for Class A interference, calculated 

various AA, given rA [cf. fig. 3.2 (II) for Pi{e>e'o)A]• 

When S > £,8 we obtain at once from (3.17) in {4 .1) 

-A Am -ff /2~2

8 e B °" B £e m 

W1 (£)8-II ~ -2- l m! ~2 • (£8 ~ £, < oo) • 

4G8 m=O cr mB 

(4 .4) 

analogous to (4.2) in the Class A cases. Observe from (ii),(iv) of (3 .18), 

and [(ii), (iv~ of (3.19)], that w1(£)8, viz,: 

( 4. 5) 

is continuous at£ = £

8, with continuous first derivative , so that w1(£) 8, 

as well as P(€)8, has no break or "jump" at the bend-over point £-8, 

where the two approximations are joined. Furthermore, unlike the Class 

A interference, when r

13 = 0 there are no "gaps in time", cf. (4.2a) vs. 

(4. 3): there is always a non-zero probability (density) for£= 0. 

Figs. {4.3)II,(4.3)II show typical curves of w1(e) 8, analogous to Figs. 

(3.6)II, {3.7)II for the P.O. 

108 


5. MOMENTS 

The general first-order moments (£.a), (0 ~a<=) , are now easily 

obtained from the results of the preceeding Section. S~nce 

( 5. 1 ) 

(a real and nonnegative), we may apply (4.2) for Cla~s A interference, and 

(4.5), with (4.3), (4.4), for Class B noise, respectively. 

5.1 Existence and Direct Calculation (Approximate Forms): 

For Class A interference we get directly 

-A ~ A2 S/2 a m (f!) ~ e A I (2amA) r(-2 +l)AA/m! 

A m=O 

(5.2) 

cf. (3.5). The sum in (5.2 ) is clearly finite, since by Stirling's 

theorem (m·! ~ mme-ml21rm) for sufficiently large m (» AAr~_) the summand 

is dominated by A~/mm+l- a. Accordingly, all (finite) moments* exist for 

Class A interference, and are given by (5.2), approximately. [We recall 

that we consider only the principal development of P1A,wlA' cf. (3.7) , 

(4.2) above.] Typical moments here are, from (5.2) : · 

(~°)A= 1 , as required; 

* Of the envelope, and of the instantaneous amplitude [cf. Middleton, 1974, 

Section 4.5], by the same argument. 

111 
(5.3) 

= 1, (as required). 

For the Class B interference we have, from (4.3), (4.4) in (5.1): 

~ 2 (X) (-l)n l n! n=o 

( 5.4a) 

(5.4b) 

[The integration of the hypergeometric function directly from its series 

form yields 

N 1£.B (X) (-l ltB+2k+2(1 +an/2) (N2k+ 1 

l [e+l F (l+na/2;1 ;-°t2)dE = I . 8 k 1 ) (5.5a) 2GB O l l k=O (kl )2(e+2k-2)(2G8

)2k 2GB 

which is probably the most direct and convenient form for numerical integration here.] In addition, the second integral in (5.4b) may be expressed 

as an incomplete gamma function, Ic, which is tabulated [K. Pearson, 1951], 

e.g. 

(5.5b) 

112 
where Ic is defined by 

. _ 1 y-1 -y l

x 

IC ( x , y ) = rfvT O y e dy . (5.5c) 

Again, direct integration of the integral itself in (5.5b) is probably the 

most convenient numerical procedure.] Clearly, by the same argument used 

above for the Class A noise [cf. (5.2) et seq.], all (finite) moments exist 

for Class B interference, as well. Some typical moments here are, from 

(5.4): 

(£0

) 8 = 1, as required; (5.6a) 

(5.6b) 

(5.6c) 

[For numerical calculation we may also ~valuate the integrals (left member 

of (5.5b)) directly, replacing 8 by 8+1, $+2, respectively for the mean and 

mean square, (5.6b), (5.6c).] 

As required from physical considerations, e.g. finite energy in the 

Class B (and A) interference; the second moment, in particular, is itself 

finite. This is not the case if we use the approximate forms w1([)8_1, 

or w1(£)8_

1 (=-dPl(B-I)/d8, cf. (3.21) et seq.), for all values of the 

envelope. Then, we have, in effect 

€s (cf) ::..fa e~,(Ds_roc:+ 2r l+a/2 A i 00

£J3-a-ld0 r -a/2 a ' £s 

(5.7) 

113 
this last tenn from (3.31). The second term of (5.7) is finite only if 

0.::. S < a (<2). Accordingly, no S moments exist for the Class (B-I) 

distributions, unless sis less than the spatial density - propagation 

parameter a. Thus, (S,2

>B-I +~,in contradiction ~o the physical situations 

we are attempting to model. In some cases (e .g. atmospheric noise) 

wB-I' PB-I are quite satisfactory for even very large values oft:, cf. Figs. 

(~.3) II, :(3.4) It, but there is always some finite €

8 beyond which a 

suitable form of wB-II' or PB-II must be used, in keeping with the ·bounded 

nature of all the moments. 

5.2 Clas~ A and B Moments: Exact Forms (Even Moments O~ly): 

As we have seen above in Sec . (5.1) all (first-order) Class A and B 

moments of the envelope (and hence of the instantaneous amplitude) exist 

and are given, an approximate form, by Eqs . (5.2), (5 .4). These results 

are approximate, albeit good ones. 

However, an alternative development is possible, which can provide 

us with exact expressions, for the even-order moments.* For this we use 

(2.24b) and take its [(-i)k dk/drkj derfvative at r=O, to get 

l(i)k d\ F1(ir0 = (Ekco/i/J>E .,: = <Ek)(co/1"), k = 0,1,2, ... , L dr ~r=O >o/ 

(5.8) 

this last, since w1(E,i/J) = w1(E)W1(wL cf. (2.20), (2 . 21), a result of 

the narrow-band nature of the output from the (aperture x RF x IF) stages 

of the receiver. Since w1(i/l) is uniform in (0,2n), all k-odd moments of the 

phase vanish, e.g. (cosk1"') = O; k = 1,3,5, ... , and only the k-even moments 

remain. With the help of 

< 

2k) 2k 2k 2 cos i/J 1/J = 2kck/2 = (2k)!/2 (k!) (5.9) 

[Middleton, 1960, Eq. (5.26)] we accordingly obtain from (5.8) the general 

relation 

* No such results are available for the odd-order cases; one must use the 

approximate forms (5 .2), (.5.4) . 

114 
(E2k) = (k_! )'"2~r- (-1 / _d - F (ir) 2 2k ~ 2k A ~ 

\2kJ! dr2k 1 r=O (5.10) 

for the even moments, when they exist (as they do here, but see the comments below in Section 5.3). In normalized form [with the aid of (3.3)] 

we can write (5.10) equivalently as 

( 5. 1 Oa) 

Our general result (5.10), (5.10a) is equivalent to the procedures used by 

Furutsu and Ishida [1960, S~ction 6] and Giordano [1970, Appendix II, p. 175 

et seq.], which is derived in our study by a different process. 

A. Exact Class A Even Moments : 

For Class A, even-order moments, including an independent gaussian 

component, the exact fonn is now obtained by usi ng (2 .50) in (5 .10) or 

(5 .lOa), rather than from the approximate relations (2 .77), (2.78). The A 

simplest procedure here is to expand the c.f. F

1(ia\)A as a power series 

in \

2

, which is pennitted, since the integral (in the exponent) is a 

definite integral, continuous in \

2. The functional form of F1(ia\~A 

is seen to be precisely that of the c.f. F

1(i~)A=P+G derived in the 

earlier study for the statistics of the instantaneous amplitudes (C~ass A 

noise), [Middleton, 1974, Section 4]. Accordingly, we use that expansion 

of F

1 to write at once here (exactly) 
where we have 

( A2k\ k 

n2k-A = AA BoA//2, (cf. (2.75d)) 

with 

() 

Applying (5.11) to (5.lOa) and observing that the expression in the square 

brackets [Jin (5.10a) is precisely the coefficient of (-l)k(aA) 2k/(2k)! 

in (5. lla) [or of (-llA2

k/(2k)! in (5.llb)], we obtain 

[(E..0)A = 1] (as expected) ( 5 .. 12a) 

(5.12b) 

(5.12c) 

n 9n 

= 6A + 4A + 6 n3 (l+r' )3 n2 (l+r')2 · 2A A 2A A 

etc. ( 5. 12d) 

(This is given in .!:!!!.normalized form, e.g. with £/a= E, by (5.12) on 

replacing[,. by E and deleting a therein.) These results (5.12) are to be 

compared with the approximate (and sometimes exact) forms (5.3). For 

example, observe that when rA-+- 0, ~, (02) = 1, as required, cf. (5.12b), 

with similar equivalences for (e4

), etc., from (5. 2) vis-a-vis (5.12c), 

(5.12d), etc. For intermediate values of AA, rA, we may expect modest 

departures from the exact values above. Finally, using (5.llc) in (5. 12) 

we can write alternatively 

116 
ls4) = 2 + (84 >/A (82 )2(l+r' )2· ~ A oA A oA A ' 

(5.13) 

which shows how the (normalized moments behave as th~ Impulsive Index AA-+a>' 

or as the independent gaussian component becomes dominant (rA-+«>). 

For the odd moments of £(or E) our procedure above, of course, is not 

applicable, and we must go directly to the calculation on the pdf, w1.(£.)A, 

cf. ( 5. 1 ) , ( 5. 2) , far k= 1 , 3, 5, . . . . 

B. Exact Class 8 Even Moments: 

The r.elations (5 .1 0) , (5 . 10a) apply here also, but the explicit 

differentiation of F

1(iaA)8, based on (2 .51) cannot make direct use of a 

power series expansion of the integrand in the exponential, because the 

integral is now an improper integral (0, 00 ) which is not uniformly convergent (in A) over the entire domain of integration [Courant·, 1936,II Sec. 4, 

Chapter 4, p. 307 et seq.], so that term-wise expansion (in A) of the 

integrand, as for Class A interference above, is not permitted. However, 

let us temporarily consider the case where (z0 )max< 00 (e.g., output signals 

of finite duration). Then, the term-wise expansion is permitted, as the 

integral is now both p·roper and uniformly convergent; (in fact, the 

resulting c.f. belongs formally to Class A). We proceed as in Class A 

a~ove and next apply lim(z0 )max-+a> to the c.f., e.g. F

1(iaA)8= 

g:)max-+ooF1(iaA l(z0 )max<00 )11 8,., and hence to each term of ~5.11), (5.12), 

etc., now specialized to the Class B parameters, A8 , Ar8, AB08 , etc. 

Thus, (5.11) applies again here, with rA+r8, AA+A8, .B0A+B08 , etc. We 

obtain the analogue of (5 .12), for example : 

117 
(E.o)B = 1;} 

as 

I. 2) .. . \8 B = 1 , 

expected; 

(s4)B = n4s/n~B(l+rB)2 + 2; 

(s.6

) 8 = n68;n~8 (l+r8)

3

+9n48;n~8 (l+r8)

2

+6; etc. 

where specifically, 

A <s2k> . - B oB [e• - A Q2k-B = k wB - ·o' 2 

A 

)d~e• /wB , 

(5.14) 

( 5. 14a) 

cf. (2.87d) and B08 in (2.87c) . Again, for the odd-moments (k=l ,3,5 , ... ) 

we must use the approximate fonns (5.4), (5.6), etc. 

5.3 Remarks: 

For Class B (and~ Class C) interference (0 <a< 2), when (2.89) 

is used as an approximation for the c.f., it is clear that if we use (2.89) 

in (5.10a), then (£2

) + 0(Aa- 2)A=O~ 00 : the second moment does not exist. 

Of course, this divergence is simply the consequence of the inadequate 

approximation, a behaviour which is alleviated by the alternative approach 

using the results of Section 4, Eqs. (4.3)-(4.5) in (5.1), cf. (5.4)- (5·.6). 

Of perhaps greater interest is to note that, in terms of our general 

classification [Sec. (2.3) and Sec.(2.5-3)], Giordano and· Haber [1970,1972], 

in effect, postulate a fi nite period of observation (O,T) for each member 

of the ensemble, e.g. Eq. (2.36) above is in force. This is equivalent to 

Class A operation, since itamounts to an abrupt truncation of the basic 

signal wavefonn u0 (z) as emitted from the ARI stages of the typical narrowband receiver, cf. Fig.(2. l)II. This , in turn, means that the receiver 

bandwidth is large enough vis-a-vis that of the input to pass it with 

negligible transients, a defining characteristic of Class A noise . Then 

all moments exist [cf. Sec . (5 . 1)] and the proper approximation for the 

118 
PD is (3.7b) . This Class A, or truncated case, goes over into a Class B 

model as the observation period (0,T) becomes large ~is-a-vis receiver bandwidth, e.g., as T6fARI>>l. The type of approximate c .f. employed is then 

that given by (2.89), or more suitably, (2.90), which includes an independent gaussian component. These,as we have seen [cf. Sec . (3.2)] yield 

satisfactory approximations for small and intermediate ranges of envelope 

€, or thresholds &

0 , but fail at some point ("large"€, &0 ) to give the 

more rapid convergence needed to insure the physically required finite 

moments of all (positive) orders, cf. Sec. (5.1). Thus, the results of 

Giordano and Haber [1970,1972] (for suitably large T6fARI), while practically useful as long as the statistics of very large values of the envelope are not demanded, are analytically incomplete as Class B models of 

the full range of possible values of the random envelope e and exceedance 

probability P1(C > S.0 )

8. 

On the other hand, the important analysis of Furutsu and Ishida [1960], 

which represents a subclass of our Class B model in that a specific emitted wavefonn [u0 (z)8

] is chosen, i.e. an exponential ~e-az, and several, 

particular spatial distributions of sources with a given propagation law 

(~1/A) are assumed, along with an exponential distribution of input signal 

amplitudes, does yield analytic forms of the PD and pdf which permit the 

existence of all orders of envelope moments, and which confonn closely to 

the statistics of the atmospheric data studied therein . The approach 

of Furutsu and Ishida [1960] is similar to ours in that approximate c.f. 's 

are obtained, suitable for the small and large values of [(and ~0 ), while 

the intermediate ranges (of£) are evaluated by numerical techniques. 

The canonical methods of our approach, however, are not invoked . Of course, 

neither Furutsu and Ishida, nor Giordano, and others, consider or distinguish Class A interference, which is a new category, as far as its 

statistical-physical description is concerned, considered originally by the 

author [Middleton, 1973, 1974]. 

119 
6. DETERMINATION OF THE BASIC FIRST-ORDER PARAMETERS 

In this section we outline procedures for determining the basic parameters of Class A and Class B interference models developed in the preceeding 

sections. A variety of overlapping procedures is available. We shall 

select what appears at this stage of the study to be ~he most direct and/or 

convenient, (later efforts may suggest modifications, for .particular 

situati ans). 

We begin with: 

A. Class A Interference: 

The first-order PD (and pdf) are governed by three parameters. It is 

convenient to distinguish two levels of parametric description: the first 

level, which we shall call 11Basic-I 11 , consists of global parameters, which 

appear directly in the expression for the P.O., cf. Eq. (3.7), and the 

second, or "Basic-II" level, contains the associated generic parameters, 

which are defined directly in tenns of the underlying statistical-physical 

model. The two groups, as we shall see, overlap to some extent. Table 

6.1 below gives the global and generic parameters of Class A interference~ 

Table (6.1): Class A Parameters 

Basic I: Global: (AA,rA,n2A)-+ {Practical Global: 

(6.1) 

The generic and global parameters are related by 

(6.2) 

Furthermore, the intensity of the independent gaussian component is 

2 2 2 aG = crR+aE , [Eqs. (2.47)], (6.3) 

120 

( 
where o~ is the intensity of receiver noise (at the output of the initial 

ARI-stages of the receiver) and o~ is the intensity of the independent 

external gaussian component, if any, likewise observed at the output of 

the ARI-stages of the receiver. By blocking the input to the receiver, 

(i.e., insuring that o~ = 0) one obtains o~, at the ARI output. Consequently; as o~ is found by actual reception in the (here) Class A noise 

environment, one then at once determines o~ from (6.3). 

Because we do not a priori know the normalizations (3.1) by which the 

threshold E0 and the envelope E are scaled, it is necessary to convert our 

analytic expressions (3.7) for the PD, for example, into forms more 

directly conformable to experimental evaluation. For this purpose we 

write* 

(6.4a) 

with 

(6 .4b) 

the new conversion factor, between c:,' and[, where;~ is an a priori determined reference quantity, used to scale the (absolute) values E , E, etc. 

To obtain the generic parameters (AA, o~, s;A) we first must

0

determine the global parameters (AA,rA,n2A). Practically, this means we must 

initially find the "practical" global quantities (AA,rA,KA), cf. Table 

(6.1) above, and then use (6.4b) to eliminate the conversion factor KA. 

Three relations involving the practical global parameters are needed. 

Perhaps the simplest are the first and second moments of£', and the PD 

of£' in the rayleigh region (c~<<l), where the slope (dP1A/dc~) is constant, cf. Figs. (3.l,3.2)II. Accordingly, from the exact expression 

(5.12b) and (6.4a) we write 

(6.5) 

which gives us KA and hence _n2A(l+r11.) in terms of the known;~ and ((s•}2\, 

*-o:rcourse, one can always measure (E2)A=2n~l+r}i.) and. then normalize, so 

that KA =l. 

121 
this last by measurement in practice. From~the approximate expression for 

(£/A' viz . (5.3) with (6.4a), we obtain 

' "' -1 [-AA ;; °' (m/AA+r Ji) 1 /2 A; J (s) A = (c.)A/KA = KA e T L l/2 mi"" ' 

m=O (l+rA) · (6.6) 

and from (3.7b) specialized to the rayleigh region, e.g. (3 .8), we have 

[ 

-A °' ( 1 + r ' ) Am J 

p (£ ' >[') ~ 1- e A L A A (['K )2 

1 o A m=O (rA+m/AA)m! o A 

(6. 7) 

In practice, of course, (£') A' ((£' )

2)A' and P1 (l.'>l~) are estimated 

from the experimentally derived data, i.e. (E-')A' ((£' )

2

)A' and P1(C'>£~)A 

are respectively replaced by thei:r estimates from the necessarily finite 

empirical data, so that (6 . 5)- (6.7) are three relations for joint estimation 

of the 11 practical 11 global parameters (AA,rA,KA), a proceedure requiring 

a modest amount of computational assistance, particularly when the expressions in brackets [ J have been programmed. With the help of (6,4b) for 

KA involving n2A' rA we next get directly the (estimates of the) gaobal 

quantities (AA,rA,n2A). Then it is a simple matter to use (6. 2) to 

obtain finally the (estimates of the) desired generic parameters 2 (A2) • (AA,oG, B

0

A ). The desired estimates to be used ,n (6. 5)-(6.7) are 

l n n (c) +-Is~; A n i=l , ((8)2) +l 'i' (£'. )

2 ; P(C £') P (c.' l') A n i;l 1 > o A+ 1 > o A-expt 1 1· 

(6.8) 

B. Class B Interference : 

Here we have a six- parameter model for the Class B cases. Table (6.2) 

su1T111arizes the global and generic parameters involved: 

122 
Table 6. 2: Class B Parameters 

The global and generic parameters are related by 

2 - r• aG - n2B B' 

and 

[Eq. ( 3. 2a)] } 

[Eq . (2 . 47)] 

, [Eq. (3.2a)J 

[ E q • ( 3 . 1 2a ) ] 

The common global and generi c paramete rs are clearly (A8,a). 

(6.9) 

(6.10) 

The fact that there are six generic parameters for our statistical -

physi cal model of Cl ass B interference stems di rectly from our pair of ap- A A 

proximations F1_I,Fl - II' E'qs. (2 . 90), (2. 93) , to the exact cf. (2.87): 

(i) , the Impu lsive Index A8 [(2.38), (2.39) in (2.51); (ii), the _spatial 

density-propagation parameter a , (2.82) ; (iii), the independent gaussian 

component a~, [(2 .47) , (2.88c)]; (iv). the a-moment of the generic , fil-

(

~a. tered envelope waveform 8

08), _cf. (2.87a), (2 .87d); (v), the mean-square 

of this generic wave_form, (s~8) ; and finally, (vi), the scaling factor 

123 
N1(Aa,a,A8,n28 ,r8

/E8), cf. remarks in Sec. 3.2-A. This factor N1 is 

functionally involved with but not solely determined by the other global 

(and generic parameters, through the APO form, and is independent of s8 

[cf. remarks below in Sec. 6C]; hence it is regarded as a generic parameter 

here also. The quantity N1 ranges from O(lOdb) to 0(50,60db) in practice. 

For example, comparison of Fig. (2.4) with Fig. 3.3(II) (A ; 1, a~ 1), a 

for the same P

1

=0. 9 gives N1 ; -6-(-44); 38db. The point of inflexion, 

or 11 bendover 11 point[8 , cf. Fig. (3.5)II, ~t which the PO's (and 

pdf's) corresponding to the two approximating c.f. 's, F1_1, Fl-II' are 

joined, to give us the desired composite PO (and pdf), is purely empirical 

[(~i), (3.18)]. The conversion factor Ks is here 

(6.11) 

cf. (6.4b), where, again,;~ is a known (measured) gaussian noise reference 

level. Also as before, we may obtain the components of cr~ as indicated 

above, cf. (6.3) et seq. [See, also, footnote, Eq. (6.4a).J 

Now to obtain the desired global parameters of our model from observed 

data we need six convenient nonidentical relations involving these parameters 

in various, sometimes simple ways. First, we use the exact expression for 

the mean square envelope (5.14), with the renormalization (6.4), to write 

<( , )2) _ (c2'> K-2 _ l/ 2 _ (l , )/"2 _ 2

1--2 £ B - L 'B B - KB - n2B +rB 0G - AcrG 0 G ' (6.12) 

cf. (6.5). Since the expressions for the Class B moments are analytically 

quite involved, and because the PD contains all moment infonnation here, 

we use [(i), (iii), (iv)] of (3.18), (3.19) where P

1_

8 is empirically 

determined from the data. Accordingly, we have the additional five relati ans: 

rayleigh region: 

[2 relations] 

(£ I K N )2 00 

p (£' >[') ~ 0 8 I L 

l - o B 4G2 n-Q B -

(-1) nA ___ a r(l+ an ) n! 2 

-AB oo Am ~ fc'K )2 e \' _Q_ ( 2;2 )- 1 

'C.o B 462 m~O m! mB ; 

B 

124 

( 6. 13a) 
large thresholds: 

[2 relations] 

at bend-over point: 

A ar(l+a/2) (2G )a+l 

a B 

r (l-a/2) ~ )a+ l 

(c.BN I 

[l+O( ) ] 

As noted from (3.18), (v) ,(vi), c:8 is the joining point (or point of continuity (through the second derivative of Pl-I,II' at least) for the approximations to P

1

_8• €8 is the point of inflexion of P

1

_

8 , obtained from 

(Pl-B)expt. Here 

(6.14) 

In practice, s 8

• is available from inspection of·the experimental* PO, P

1 8 , - ,exp 

so that in addition to (6 .12) only the five relations (6.l3a·,1b,c) are then 

required for. the remaining si~ global parameters. Once these have been obtained, we may use (6.10) to determine the six ultimately generic parameters 

of the Class B interference under study. Of course, in practice, our 

data are finite and P1

_8 is an empirical function;({£' )2)

8 is an estimate, 

cf. (6.8), based on sample values, and c8 is likewise an estimate by inspection, so that all parameters actually obtained are necessarily themselves estimates. We do not include €8 in our list above of global 

parameters, and exclude it from the basic, or generic parameters, cf. (6.9), 

since it is in ·effect, an empirical quantity resulting from the procedure 

of joining P1

_

1 ,P1_

11 in approximation to the true P

1

_

8, at l = c8. 

C. Degenerate Cases: 

When E8 (or £.i3) is not known -- i.e., is not evident from the empfri-

* However, see the important situation discussed in C following. 

125 
cal PD, Pl-B -- we can only work with the Pl-I form of the PD, namely the 

approximation suitable for small and intermediate values of ~o (and S~) 

< f'B(£8)~ This is the case, for example, of much of the atmospheric noise 

data, cf. Fig. 2.4, where no bend-over point is at all evident~ The model 

now reduces from six to. a five-parameter approximation, in (ACL,CL,r13 ,n8,N1) 

[or (Aa,a,r8,K8 ,NI)] for the global parameters and in (CL,cr~,N1 ) for the 

generic parameters, cf. Table (6.2). BecauseEB is not known, we are unable 

<

A2) (ACL) to obtain AB' and hence we-can det:rmineAonly n2B = AB B0B /2, ACL ~ 80B AB, 

and not their individual factors (B~8),(s;8 ). For these five global parameters we need accordingly five equations. The conversion factor K8 is 

again given by (6.12), and for the four other parameters we use 

Eg. (3.llb): rayleigh region: 

2 (-l)nAn 

P,(f'>f')s I~ (£.~KB~1r) r I CL r(l+ a2n) = 0.99, say; - 0 - 2G n=O n. 

B 

( 6. l 5a) 

and 

Pl-I in the "bend-up" region, where P

1_1 departs from the "straight line" 

rayleigh fon11, so that (3.llb) fully applies, and two points P1_I=P3,P 4 

with 

Eg. (3.15): large{~ (<£13 1..:_ 

A 

A r(l+a/2)(£B'KBNirCL 

p (€.I>£' ) ~ CL ( 1 +O ] = p p l - o B- I - r ( l -a/2) ( 2GB )-a (iii) 3' 4 ( 6. 1 5b) 

where the PD's are empirically determined. Without the turnover point €8 

we cannot join the large-threshold approximation Pl-BI! to Pl-BI for 

(E 0 <£B), and are thus unable to deten11ine the generic parameters, except for 

a,cr~,NI. This indicates the importance of obtaining the 11 rare-event 11 data 

(E.>c8) , so tha.t the fundamenta 1 ( i . e. , generic) parameters of the interference model may be estimated, as the fundamental descriptors of this 

noise environment, as specif.ied, of course, by our statistical-physical 

model in this case [cf. Section (2.1 ) et seq.] . 

126 
7. PRACTICAL CONDITIONS FOR CLASS A AND CLASS B INTERFERENCE 

In our preceeding analyses we have postulated limiting forms of interference which are strictly Class A or Class 8, neglecting the very small 

contributions of the associated other components (e.g. Class B where Class 

A is said to occur, etc.). Here we shall establish quantitative conditions 

which permit us to neglect these other-component effects and to assert that 

our analytical forms may be applied to the corresponding physical situation, e.g., essentially only Class A, or Class B noise is present. 

To do this we $tart with the relation (2.52) for the general Class C 

interference and use the results [Eqs. (2.77), (2.78)] for Class A, 

[ E q s . ( 2 . 9 0) , ( 2 . 9 3 ) J for C 1 ass B , to wr i te 

~ A ~ 

F1(ir)C+G = F1(ir)A+GF1(ir)8 (7.la) 

(7.lb) 

where we include er~, (-2 .47), in the Class A form ('\,rft.) here . 

When Class A noise heavily predominates, we use the transformation 

r=aAA' cf. (3.3) et seq . , and expand the Class B components of the c.f., 

to get the pair of approximations 

and I 2 2 4 4 -[(m/AA+rA)n2A+b2aA8]aAA /2{ 2 aAA } 

e 1 +b2aAB -8- + ... 

(7.2) 

127 
For this to reduce to the principal form (3.5), we have ·at once the two 

pairs of conditions 

Class A: 

b A ) <s2 

)A A 2a B = (4-a. AoB B (1+2i;A <s2 \)-1 , n2

A 2-a. I 8

2 \A G A oAI « r A' 

\ oAI A 

( 7. 3a) 

with 

(7.3b) 

and 

(

4-a. )2 2-a. « 1 , for [ > ls (7.3c°) 

- From (2.37), (2.38), we also note that AA,B=y~Ts(A,B)· Equation (7.3a) is 

usually the weaker condition and (7.3b,c) the stronger, with the Impulsive 

Indexes AA,B not too large. A useful, rough rule for considering t~e in:erference to be Class A only is, in effect, the condition A8<<AA' or Ts8<<TsA' 

the latter representing the fact that the amount of time, on the average, 

that the Class A component is present, is much larger than that for the 

Class B term, a not at all surprising condition from an intuitive viewpoint. 

The conditions (<<1 ,rA) in (7.3) are, of course, matters of judgment, usually 

0(10-2,10-3) is sufficient, unless we are concerned with the v~ry rare 

events, i.e., extremely large values of E. In general, we shall adopt the 

stricter conditions (7.3b,c). (We shall pursue the detailed anatomy of 

these and the Class B conditions further in a later study.) 

When Class B noise is heavily dominant, on the other hand, we rewrite 

fl.lb) to include the independent gaussian component embodied in rA with 

F

1(ia8

\)8, (2.90) and (2.93), so that we have now for the Class A contribution 

128 
Accordingly, we get 

F1 (ia8;>..)B+G ; [Eqs. (2.90) and (2.93)] , (7.5) 

provided the single condition 

Class B: 

(7.6) 

is obeyed. A~ain,_ the simple, intuitively obvious condition is that 

AA « A8 (or \A <<T~B): the Cl ass B interference is 11 on11 for a much 1 onger 

period than the Class A component. The amount of Class A noise is 

negligible vis-a-vis the Class B contribution. As expected, this condition is, not unexpectedly, just the reverse of that (roughly) required 

for Class A dominance. [Note, too, that the more precise condition (7.6) 

is a kind of inverse of conditions (7.3b,c) above for the Class A cases.] 

129 
Glossary of Principal Symbols 

Equation numbering in Part II does not contain a 11 II 11 suffix; when 

equations from Part II are referred to in Part I, they are so 

designated, e.g . Eq. (3.5), Part II, etc. 

Figure numbering in Part II is similarly indicated by a 11 II 11 fol -

lowing the number. 

A. A0 = peak amplitude of typical input signal 

AA,A8,A ,A A'A 8 = Impulsive Indexes, (Class A,8 interference) co or:>, co, 

A = effective Impulsive Index 

a 

aA,a8 ,a = normalizing factors 

APO = a posteriori probability; here 1-0istribution = P1 

a 

bla'b2a'b2t+2 la 

B 

C. c.f. 

E. E ,E0 

e Oy A 

E ,[o '~ ,£.~ ,E.o 

= combined aperture-IF-IF receiver input stages 

= source, receiver beam patterns 

= spatial density-propagation parameter 

= generic or typical envelope of waveform from ARI 

receiver stage 

= weighted moments of the generic envelope 808 

= exponent of moment 

= characteristic function 

= probability distribution 

= delta (singular) function 

= instantaneous envelope 

= limiting receiver voltage 

= normalized (instantaneous) envelopes; £ 0=envelope threshold 

= 11 bend-over 11 point (Class B), empirical pt. of 

inflexion in P

1_

8 

130 
n 

A 

F. F

1 ,F1 

H. 

l Fl 

t.fwt.f ARI 

f 

r(x) 

y 

L. I\ 

A 

A 

""' 

N. n. b. 

= an exponent 

= impulse epoch 

= normalized doppler 

= characteristic functions 

= confluent hypergeometric function 

= noise, receiver bandwidths 

= frequency 

= a basic waveform 

= geometrical factor of received wavefonn 

= rati.o of (intensity of) gaussian component to that 

of the 11 impulsive 11 , or nongaussian component 

= ga1T111a function 

= exponent of propagation law, with range 

= exponent of characteristic function 

= incomplete r-function 

= unit vector 

= Bessel function, 1st- kind, (0,1 order) . 

= jacobian 

= conversion factor, for arbitrary normalization 

= domain of integration 

= argument of the c.f. 

= (A,6,~), coordinates 

= exponent of source density law with range 

= normalized doppler 

= narrow-band 

131 
0. 

P. 

Q. 

R. 

s. 

T. 

u. 

V. 

w. 

X. 

Y. 

z. 

n2A ' n2B 

w,wo 

pl 

pdf 

'I' ' 4> 

4>r ,<PR 

r 

p 

= mean intensity of the nongaussian component 

= angular frequencies (w0=carrier angular fr . } 

= APO or exceedance probability 

= probability density function 

= phase of narrow band wave 

= aperture phase 

= c.f. variable 

= poisson "density" 

A A 2 2 2 cr,crG ,cr,crmA,B'6crG,crA,crR,crE = variances 

crs,v = source density 

- T s'Ts;A,B 

t, tl , t2, 

e,e • 

U,Unb 

uo,uoA,B 

wl ,w, 

X 

XO 

zo 

= emission duration 

= times 

= sets of waveform parameters 

= basic waveforms out of ARI receiver stage 

= normalized envelope waveform at output of ARI 

ARI stages 

= probability density function 

= instantaneous amplitude 

= a c.f. variable 

= a normalized tim·e 

132 
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Vehicular Techn. Vol. VT- 23, No. 3, August, 1974, pp. 72-83 . 

23. Spaulding, A.O. (1971), Amp l itude and time statistics of urban manmade noise", in Proc. Int 1 l . Commun. Conf., Montreal, Canada, June 

1971, paper No. 37, pp . (37-8) - (37-13); also, IEEE NTC 1 72 Conf. 

Record, Vol. 72 CH0- 601 - 5-NTC, Dec. 4-6, 1972. 

134 
24. Spaulding, A.D., and L.R. Espeland (1971), "Man-made noise characteristics on and in the vicinity of military and other government installations", Office of Telecommunications, Technical Memorandum 

OT-TM- 48 (available only from authors). 

25. Spaulding, A.O., and D. Middleton (1975), "Optimum reception in an 

impulsive interference environment", Office of Telecomnunications, 

Technical Report OT-75-67, June, 1975 (U.S. Govt. Printing Office, 

Washington, O.C., 20402) . 

26. Watson, G.N., (1944), Theory of Bessel Functions, MacMillan (New York). 

135 

FORM OT-29 U.S . DEPARTMENT OF COMMERCE 

13 · 73 ) OFFI CE OF TELECOMMUNI CA TIONS 

BIBLIOGRAPHIC DATA SHEET 

1. PUBLICATION OR REPORT NO 2. Gov 't Accession No. 3, Recipient's Accession No. 

OTR 76-86 

4. TITLE AN D SUBTITLE 5. Publication Date 

Statistical-Physical Models of Man -Mada and Natural Anril 1976 

Radio Noise Part II: First Order Probability Models 6. Performing Organization Code 

of the Envelope and Phase 

7, AUTHOR(S ) 9. Project/Taslc/ Worlc Unit No. 

Dr. David Middleton 

8. PER FORMING ORGAN!ZA TION NAME AND ADDRESS 

Uo So Department of Commerce 

Office of Telecommunications, ITS I 0, Contract / Grant o. 

Bou ldei Laboratories 

Bou lder, Colorado 80302 OT-0124 

11. Sponsoring Organ izat ion Name and Address 12. Type of Report and P,:riod Covered 

U.S. Department of Comme rce 

Office of Telecommunications 

Wa sh,ington, D. C. 20005 13. 

14. S PPLEMENT ARY NOTES 

Prepared under contract by D. Middleton, 127 E 91stst., . New York, N.Y. 10028 

15. ABSTRACT (A 20Q·word or less taccual summary of most s ignificant in forma tion . I( document includes a significant 

bibli ogra phy ex literature survey, mention it here. ) 

Most man -made and natural electromagnetic interferences are highly nongaussian random processes, whose degrading effects on system performance can be 

severe, particularly on most conventional systems, which are designed for optimal or near optimal performance against normal noise. In addition, the nature, 

origins, measurement and prediction of the general EM interference environment 

are a major concern of an adeq uate spectral management prog ram. Accardi ngly, 

thi s second study in a continuing series [cf., Mi ddleton, 1974] is devoted to 

the development of analytically tractable, experimentally verifiable, statistical· 

physical models of such electromagnetic interference. 

Here, cl assi fi ca t ion into three major types of noise is .. made: Class A 

( narrowband .. vis the recei ver) , Class B ( broadband vis-a-vis the re cei ver), -a-vis 

and Class C (=Class A+ Class B). First-order statistical models a re constructed 

for the Class A and Class B cases. In pa rti cul ar, the APO (a pas teri ori probability distribution) or exceedance probability, PD, viz. P 1 (e > eo) A ,B, and the 

16 . Key worlls (Alphabelice l order, separated by semicolons) 

Man-made radi o noise, Radio noise models, Statistical communication theory 

17. AVAILABILITY STATEMENT 18, Securi tY Class (This report) 20. u_m ber of pages 

~ UNLIMITED. Un classified 142 

19. Security Clnss (This page) 21. Price: 

0 FOR OFF ICIAL DISTRIBUTION. 

Un classified 

LI SCOMM-DC ZG716·P73 
associated probability densities, pdf's, W1(e)A B, of the envelope are obtained; [the phase is shown to be uniformly distributed (0,2rr)J. These results are canonical, i.e., their analytic forms are invariant of the particular noise source and its quantifying parameter values, levels, etc. 

Class A interference is described by a 3- parameter model, Class B noise 

by a 6-parameter model. All parameters are deducible from measurement, 

and like the APO's and pdf's, are also canonical in form: their structure is based on the general physics underlying the propagation and reception processes involved, and they, too, are invariant with respect to 

form and occurrence of particular interference sources. 

Excellent agreement between theory and experiment is demonstrated, 

for many types of EM noise, man-made and natural, as shown by a broad 

spectrum of examples. Results for the moments of these distribution are 

included, and more precise analytical conditions for distinguishing between Class A,B, and C interference are also given. Methods for estimating the canonical model parameters from experimental data (essentially 

embodied in the APO) are outlined in some detail, and a program of possible next steps in developing the theory of these highly nongaussian 

random processes for application to general problems of spectrum management is presented.