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NBS MONOGRAPH 23

Amplitude-Probability Distributions

for Atmospheric Radio Noise

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U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

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UNITED STATES DEPARTMENT OF COMMERCE • Frederick H. Mueller, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

Amplitude-Probability Distributions

for Atmospheric Radio Noise

W. Q. Crichlow, A. D. Spaulding, C. J. Roubique, and R. T. Disney

National Bureau of Standards Monograph 23

Issued November 4, 1960

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington 25, D.C. Price 20 cents
Contents

Page

1. Introduction 1

2. References 2

3. Appendix 15

1. Introduction 15

2. Measurements of the amplitude-probability distribution 15

3. Definition of parameters and statistical moments 16

4. Determination of the parameters as functions of the measured

moments 17

5. Errors in noise measurements and their influence on the calculated

distribution v 18

6. Bandwidth considerations 20

7. References '_ 22
Amplitude-Probability Distributions

for Atmospheric Radio Noise

W. Q. Crichlow, Q. D. Spaulding, C. J. Roubique, and R. T. Disney

Families of amplitude-probability distribution curves are presented in a form such that

by using three statistical parameters of atmospheric radio noise, of the type published by

the National Bureau of Standards, the corresponding amplitude-probability distribution

may be readily chosen. Typical values of these parameters are given.

1. Introduction

A knowledge of the detailed characteristics of

atmospheric radio noise is essential to the design of

radio communication systems operating at frequencies up to about 30 Mc/s. These characteristics can

conveniently be expressed in terms of an amplitudeprobability distribution which has been found to

be an extremely useful tool in the analysis of the

expected interference to a communication system [l]. 1 However, since the measurement of the complete

amplitude-probability distribution on a continuous

basis for many frequencies and locations is prohibitive in both manpower and equipment needs, a method of obtaining this distribution from three

easily measured statistical moments has been developed at the National Bureau of Standards [2]. The three moments are the average power, the

average envelope voltage, and the average logarithm

of the envelope voltage, and are expressed, respectively, as: Fa=the effective antenna noise figure,

=the external noise power available from an

equivalent, short, lossless, vertical antenna

in decibels above ktb (the thermal noise

power in a passive resistance at room

temperature, t, in a bandwidth, b),

Va=ihe .voltage deviation in decibels below Fa , L d=the log deviation in decibels below Fa

. Fa can be expressed in terms of the root mean square

field strength by means of the nomogram in figure 1

.

It should be noted that although Fa is independent

of bandwidth, the root mean square field strength

is proportional to the square root of the bandwidth

used.

The shape of the distribution curve is dependent

only on Vd and Ld , and since they have been nor- malized to Fa , it is possible to construct [2] distribution curves for various combinations of these parameters, independently of the value of Fa

. Such curves

are given in figures 2 through 21 for the range of

values of Vd and Ld for which the distribution is valid. These curves give the percentage of time

1 Figures in brackets indicate the literature references on page 2.

the ordinate is exceeded. In order to minimize the

number of graphs required, several curves have been

drawn on each graph at arbitrary levels. The circle

on each curve corresponds to the value of Fa (the

root mean square voltage for the distribution) and

by shifting the ordinate scale so that zero decibel

corresponds to the circle on the proper curve, the

ordinate can be determined in decibels above Fa

. Values of Fa , Vd , and Ld are recorded continuously

at ten of the stations in the worldwide network

established by NBS [3]. These recordings are made at eight fixed frequencies between 13 kc/s and

20 Mc/s, using a bandwidth of about 200 c/s, and

the data are published [4, 5] in tabular form. Values

of Fa only are recorded at six additional stations in

the network and are included in the publications.

In addition, predictions of worldwide values of Fa have been published by C.C.I. R. [6]. Typical values of the three parameters are shown

in figures 22 and 23. Figure 22 gives Fa versus frequency for summer nighttime and winter daytime, which are the periods of highest and lowest

atmospheric noise levels, respectively. Curves are shown for the estimated values of Fa for each type

of noise (atmospheric, galactic, and manmade) by

taking into account propagation conditions as well

as the recorded values.

Figure 23 gives Vd and Ld versus frequency for

summer nighttime and winter daytime. The curves were not drawn through all of the measured points,

since some signal contamination has been encountered and the smoothed curves are considered more

representative of true conditions. The portions of

the Vd and Ld curves that result from each of the

three noise sources will be the same as shown for

Fa in figure 22.

Although the distribution curves given in figures

2 through 21 are considered valid for a wide range

of bandwidths, the values of Vd and Ld used in determining the distribution must correspond to the

bandwidth to be used. The values given in figure

23 are for a 200 c/s bandwidth and must be adjusted

for any other bandwidth. Methods of converting

these statistical moments from one bandwidth to

another are under development at NBS and will be

published in the near future.
2. References

[1] A. D. Watt, R. M. Coon, E. L. Maxwell, and R. W. Plush,

Performance of some radio systems in the presence of thermal and atmospheric noise, Proc. IRE 46, 12, 1914

(1958).

[2] W. Q. Crichlow, C. J. Roubique, A. D. Spaulding, and

W. M. Beery, Determination of the amplitude-probability distribution of atmospheric radio noise from statistical moments, J. Research NBS 64D, 49 (1960). (See

appendix.)

[3] W. Q. Crichlow, C. A. Samson, R. T. Disney, and M. A. Jenkins, Radio noise data for the International Geophysical Year July 1, 1957-December 31, 1958, NBS Tech. I Note 18 (PB 151377) (July 27, 1959).

[4] W. Q. Crichlow, C. A. Samson, R. T. Disney, and M. A. Jenkins, Radio Noise Data for the Season March, April, i May 1959, NBS Tech. Note 18-2 (PB 151377-2)

(March 14, 1960).

[5] W. Q. Crichlow, Noise investigation at VLF by the

National Bureau of Standards, Proc. IRE 45, 6, 778

(1957).

[6] Report on Revision of Atmospheric Radio Noise Data,

C.C.I.R. Rept. No. 65, VHIth Plenary Assembly, War- saw (Intern. Radio Consultative Comm., Secretariat,

Geneva, Switzerland) (1956).

2
(Megacycles)

80 -ir

-11-70

60 Hh

Hr 50

40 -r—30

20--

9 Hh

8

7 - L e

--2

I -r

0.8

0.09

0.07

0.05

0.03 -f0.01 -1-

•0.9 Hh 0.7

0 6 -§-

0.5

0.4 -jjL 0.3

Q2--

0.1

0.08

0.06

0.04

0.02

Fa (db above ktb)

-r-20

0--

-- 20

40--

-- 60

80--

120 —

160 —

100

--I40

En =Fa + 20 log, 0

fMc/s-65.5

En

(db above I /xv/m)

---50

-40--

---30

-20--

0--

20-

40--

60--

--30

--50

-r70

Figure 1. Nomogram for transforming effective antenna noise figure to noise field strength as a function of frequency.

/•' a =effective antenna noise figure = external noise power available from an equivalent short, lossless, vertical antenna in decibels above ktb. En = equivalent vertically polarized ground-wave root mean square noise field strength in decibels above 1 /iV/m for a 1-kc/s bandwidth.

JmcIi = frequency in megacycles.
0001 09 01 J 5 I 5 10 20 30 40 50 60 70 80 85 90 95 98 99 PERCENTAGE OF TIME ORDINATE IS EXCEEDED

Figure 2. Amplitude-probability distributions.

Lineor by —£log

|0(-lnp)

PERCENTAGE OF TIME ORDINATE IS EXCEEDED

Figure 3. Amplitude-probability distributions.

4
PERCENTAGE OF TIME ORDINATE IS EXCEEDED PERCENTAGE OF TIME ORDINATE IS EXCEEDED

Figure 4. Amplitude-probability distributions. Figure 5. Amplitude-probability distributions.

5
6
Lmeor by - y

log |0(-lnp)

--jlj MINIM 1

1

1 — TTTTTT (Circle -z- '« 1 a —

-i —— J — 1

tttttt — —— TTTTTT

TTTTTT

Tt'Sr jl '— '! U — TTTTTT \ \ V \ — TTTTTT vR -if — TTUTT — TTTTTT * A — TTTTTT — I

"

V — TTTTTT" "in i iu — 1

" v i

1 \ —T^fH-H-

-

, A \ V

- -)- —-—r— \ i TTI -

E"fr\ -

— 1mtr T7 ' \

:\ r "W

t-- in V

ilH \ k \ i\

- s— - «^ aNJ

> \ ! V L IX V XT v v\ \

\

V

H .— . .

i

L ^ 1 \ -f-t+rt \ \ nrt>.

-J vj- -Vh— V \

I "1 ~ M ,---&>»

\i\ — *H4lJ

"

1 ' ^SUTT" - TH* * V

— tffri"

TTIT* JT

i S/L S i r i

0001 09 .01 J 5 I 5 10 20 30 40 50 60 70 80 85 90 95 98 99 PERCENTAGE OF TIME ORDINATE IS EXCEEDED

Figure 8. Amplitude-probability distributions.

Lmeor by - ^

log

jQ

( -In p)

PERCENTAGE OF TIME ORDINATE IS EXCEEDEO

Figure 9. Amplitude-probability distributions.

560873 O - 60 - 2 7
PERCENTAGE OF TIME ORDINATE IS EXCEEDED PERCENTAGE OF TIME ORDINATE IS EXCEEDED

Figure 10. Amplitude-probability distributions. Figure 11. Amplitude-probability distributions.

8
Lineor by - j log^t-lnp) Lineor by log

|Q(-lnp)

0001.001.01 .15 1 5 10 20 30 40 50 60 70 80 85 90 95 98 S9 1001 001 01 .15 1 5 K) 20 JO 40 50 60 70 80 85 90 95 98 PERCENTAGE OF TIME ORDINATE IS EXCEEDED PERCENTAGE OF TIME ORDINATE IS EXCEEDED

Figure 12. Amplitude-probability distributions. Figure 13. Amplitude-probability distributions.

9
1001 OH .01 J i I 5 10 20 30 40 50 60 70 80 85 90 95 98 PERCENTAGE OF TIME ORDINATE IS EXCEEDED

Figure 14. Amplitude-probability distributions.

0001 OCH 01 .15 1 5 10 20 30 M 50 60 70 80 85 90 95 98 99 PERCENTAGE OF TIME ORDINATE IS EXCEEDED

Figure 15. Amplitude-probability distributions.

10
Lineor by log

lo(-lnp) Untor by -j iog

lo(-inp]

JXMIJBMJSI J 5 I 5 K) 20 30 40 50 BO 70 80 85 90 95 98 99 O00I0OI.0I J 5 I 5 10 20 30 40 50 60 TO 80 8 5 9 0 95 9 8 99 PERCENTAGE OF TIME ORDINATE IS EXCEEDED PERCENTAGE OF TIME ORDINATE IS EXCEEDED

Figure 16. Amplitude-probability distributions. Figure 17. Amplitude-probability distributions.

11
Linear by lag

l0(-lnp) Linear by log

|0(-lnn

0001 .001 01 .15 1 5 rO 20 30 40 50 60 70 80 85 90 95 98 99 0001 001 01 15 1 5 K) 20 30 40 50 60 TO 60 8 5 9 0 9 5 9 8 99 PERCENTAGE OF TIME ORDINATE IS EXCEEDED PERCENTAGE OF TIME ORDINATE IS EXCEEDED

Figure 18. Amplitude-probability distributions. Figure 19. Amplitude-probability distributions.

L2
13
-3 JUpper Decile Median t Lower Decile

01 I 10

f Mc/s

(a) Summer Nighttime (20 - 24 hr)

Upper Decile Median Lower Decile m l i i ii m i n i

-4

a-:

'Mc/S

(b) Winter Daytime ( 08- 12 hr)

Figure 23. Radio noise statistical parameters.

Measured during summer 1959 and winter 1958-1959, Ounbarrel Hill, Boulder,

Colo.
3. Appendix

Determination of the Amplitude-Probability Distribution

of Atmospheric Radio Noise From Statistical Moments *

W. Q. Crichlow, C. J. Roubique, A. D. Spaulding, and W. M. Beery

(July 23, 1959)

During the International Geophysical Year, the National Bureau of Standards established a network of atmospheric noise recording stations throughout the world. The ARN-2

noise recorder at these stations measures three statistical moments of the noise: average

power, average voltage, and average logarithm of the voltage. An empirically-derived

graphical method of obtaining an amplitude-probability distribution from these three

moments, and its development, is presented. Possible errors, and their magnitudes, are

discussed.

1. Introduction

The interference to reception of radio signals

caused by atmospheric noise depends not only on

its average level bub also on its detailed characteristics. A complete description of the noise received

at a particular location would require an exact

determination of the variation of its instantaneous

amplitude as a function of time. Because of the

complexity of the noise structure and the fact that no two samples of noise have identical amplitudetime functions, it is necessary to resort to simpler

types of description of a statistical nature.

Various methods of measurement have been investigated by experimenters throughout the world

for many years, and the relationship between the

measured values and the interference caused to

various types of service has been studied.

In order to determine the most significant characteristics to be measured, an international subcommittee was formed within Commission IV of URSI 1 to study the problem. A recommendation [1],* based

on the subcommittee's report [2], was endorsed by

Commission IV at the XIIth General Assembly of

the URSI in 1957. In the Annex to this recommendation, it is stated that "it would be highly desirable to

obtain detailed statistical information on the amplitude-probability distribution of the instantaneous

envelope voltage, the various time functions, and the

direction of. arrival of the noise at many locations

throughout the world. However, since continuous

detailed measurements of this type at all frequencies

and a large number of stations become prohibitive

because of the complexity of the equipment involved

and the large number of personnel necessary to carry

out the observations, it is recommended that these

complete studies of the detailed noise characteristics

be confined to a few selected locations with continuous

measurements of one or more simple parameters at a

large number of stations."

•Reprinted from J. Research NBS, Vol. 64D, No. 1, Jan.-Feb. 1960.

» International Scientific Radio Union.

* Figures in brackets indicate the literature references at the end of this paper.

During the International Geophysical Year, the

National Bureau of Standards established a network

of 16 noise recording stations distributed throughout

the world [3]. Measurements are made with the

ARN-2 recorder which was designed at NBS to

record the average noise power in a bandwidth of

about 200 cps on eight fixed frequencies between

13 kc and 20 Mc. In order to obtain additional

information on the character of the noise, two other

statistical moments, the average envelope voltage,

and the average logarithm of the envelope voltage

are recorded at ten of the stations. These data are

available from IGY World Data Center A, NBS,

Boulder, Colo., at a nominal fee. The results of

data analysis will also be published in an NBS

Technical Note for public sale.

The amplitude-probability distribution is very

useful in assessing the interference potential of noise

and has been measured at several locations [4, 5, 6, 7].

Since data on the moments measured by the ARN-2

can be recorded continuously over the full frequency

range with much less equipment and fewer personnel

than would be required for a comparable amount of

data on the complete amplitude-probability distribution, an investigation was made at the NBS of

methods' of deriving the complete distribution from

these three moments.

It is the opinion of the authors that mathematical

models for the distribution [1,5,7,8] have proved to

be extremely cumbersome and of doubtful value in

deriving the distribution from measured moments;

therefore, an empirically derived graphical method

has been developed.

2. Measurements of the AmplitudeProbability Distribution

Simultaneous measurements of the amplitudeprobability distribution and the three moments

measured by the ARN-2 were needed over a wide

frequency range with a variety of types of noise.

There were two specific purposes in making these

measurements:

15
a. The distribution measurements, alone, were

needed to establish typical shapes that could be ex- pressed by numerical parameters, easily related to

the measured statistical moments.

b. The simultaneous measurements of the distributions and the moments were needed to check the

accuracy of the ARN-2 noise recorder in measuring

the moments.

Accordingly, a distribution meter was constructed

to be used in conjunction with the ARN-2 recorder.

The operation of the distribution meter is based

on counting techniques. The total number of IF

cycles in the ARN-2 which exceed a specified amplitude in a given time interval are counted. By proper

choice of the time interval, the percentage of time a

level is exceeded can be read directly from the

counter. Three counters are used to provide simultaneous measurement at three levels, separated in

amplitude by 6 db. By use of a calibrated, manually-operated attenuator, the total possible range

of 100 db can be covered in levels separated by

multiples of 2 db. The time duration of each sample can be preset to provide an automatically timed

sample having an interval ranging from 1 sec to 1 hr,

in 1 sec increments.

The actual intervals were chosen of sufficient duration to provide a statistically significant sample of a

slowly varying phenomenon such as thunderstorm

activity. Longer intervals were required for the

high-level, low-probability noise pulses than for the

low-level, high-probability noise. The time neces- sary to obtain a complete curve ranged from about

15 min to 30 min. The longer period was necessary

for measuring noise with a large dynamic range.

A check for accuracy of operation was made by

obtaining a distribution of thermal noise. The

Rayleigh distribution thus obtained confirmed the

accuracy of measurement. Figure 1 is a typical

amplitude-probability distribution of atmospheric

radio noise at 13.3 kc. This was plotted on Rayleigh

graph paper, whose coordinates were chosen so that

a Rayleigh distribution will plot as a straight line

with a slope of —){? The coordinates are shown as noise level in decibels above the root mean square

voltage versus the percentage of the time that each

level is exceeded. This measurement was made

with the distribution meter at the Boulder Laboratories. The dynamic range between the 0.0001 percerit and 99 percent intercepts is approximately 84 db.

It was found that these amplitude-probability

distributions for atmospheric radio noise can be adequately represented by a three-section curve which

has been drawn through the data points in figure 1. This curve has a characteristic shape that can be

described by means of four numerical parameters to

be defined later. The accuracy of fit using this particular kind of curve composed of three sections is typical of more than 100 distributions experimentally

measured at Boulder. In the particular case of

thermal noise, this curve becomes a straight line.

« These coordinates are log of voltage versus— J4 logio (—In of probability).

50

40

30

-30

-40

V

is:

i Frequen

S7 Oc

Boulde

1 cy - 13 tober

r, Colo

3 kc

5, 1958

ado

\

\

\

\1

0.0001 Oil 0.1 I 5 10 20 30 40 50 60 ?0 80 90 95 98 99

PERCENTAGE OF TIME ORDINATE IS EXCEEDED

Figure 1. Measured amplitude-probability distribution of atmospheric radio noise.

Data obtained by other experimenters [2] in Alaska,

Panama, England, and Florida when plotted on

Rayleigh paper also fit curves of this form.

The lower portion of the curve, representing low

voltages and high probabilities, is composed of many

random overlapping events, each containing only a

small portion of the total energy. Therefore, this

portion of the curve must approach a Rayleigh

distribution.

The section representing very high voltages ex- ceeded with low probabilities is, in general, composed

of nonoverlapping large pulses occurring infrequently.

From experimental measurements of atmospheric

noise distributions, this section has been found to be

well represented by a straight line on Rayleigh graph

paper. On this graph paper, the remaining section

of the distribution has been found to correspond

quite closely to an arc of a circle tangent to the

above two straight fines. Based on experimental

evidence, it is assumed that no pulses of amplitude

greater than those for which the probability of being

exceeded is 10~6 are present in the distribution.

3. Definition of Parameters and Statistical

Moments

Since on the coordinate paper used for the plotting

of amplitude distributions for atmospheric radio noise a Rayleigh distribution will always have a slope of — and since the center of the circular arc menL6
tioned above will lie on the bisector of the angle

formed by the Rayleigh and high-voltage, lowprobability lines, four parameters are needed to define the distribution. A point through which the

Rayleigh line passes, a point and a slope for the

high-voltage, low-probability line, and a parameter

of some kind to determine the radius of the circular arc are needed.

These four parameters are defined as follows:

a=point at which the Rayleigh line intersects the

0.5 probability coordinate.

J=point on that tangent to the circular arc which

is perpendicular to the bisector of the angle formed

by the Rayleigh and high-voltage, low-probability

lines and on the probability coordinate through the

vertex of this angle.

c =point where the high-voltage, low-probability

line intersects the 0.01 probability coordinate.

s =slope of the high-voltage, low-probability line.

The three statistical moments experimentally

determined are defined as follows:

rms voltage: e rms=-y'

J*

e*&V

average voltage: eave =

J* edp

antilog of the mean log of voltage:

e log antilogj* log e dp.

''

where e is 1/^2 times the instantaneous envelope

voltage, and p is the probability of e being exceeded.

Since the ARN-2 noise recorder measures the db

difference betweeen the e rms and eave and between the

« ra8 and eiog, and in order to minimize the necessary

graphical calculations, the above parameters are

altered to relate them to the Rayleigh section of

the distribution as follows (see fig. 2)

:

^4=20 log-^-=db difference between Rayleigh at

^rms

0.5 probability and the eTma value.

B—db difference between the point of intersection

of the Rayleigh and high-voltage, low-probability lines and the tangent to the circular arc. (The tangent is drawn perpendicular to the

bisector of the angle formed by the Rayleigh

and high-voltage, low-probability lines.)

C—db difference between c and value of Rayleigh

at the 0.01 probability coordinate.

X=absolute value of slope of high-voltage, lowprobability line relative to Rayleigh.

X=-2s

The moments actually measured are labeled as

follows

:

Vd =20 log-

;20 log

'log

0.0001 Oil 0.1 I 5 10 20 30 40 50 60 70 80 90 95 98 9 9

PERCENTAGE OF TIME ORDINATE IS EXCEEDED

Figure 2. Definition of parameters for amplitude-probability

distribution atmospheric radio noise.

The erms is also measured.

The following relation will always hold true for the

distribution

:

^log <C^ave"\^rms

therefore, Vd and Ld are always positive.

4. Determination of the Parameters as

Functions of the Measured Moments

Since four parameters are needed and only three

statistical moments are measured, a dependency

between two of the parameters must be determined

in order to be able to obtain the remaining three

independent parameters as functions of the measured

moments. From experimentally measured distributions at eight frequencies and various bandwidths,

the following linear relation between X and B is

found to hold in general (fig. 3)

.

£=1.5(X-1)

Distributions were employed in which C was

allowed to vary from 0 to 40 db and X from 1 to 12.

These distributions were numerically integrated in

order to determine their three moments. From these

integrations, Vd and Ld were obtained as functions

of X and C. A was also obtained as a function of

X and G (fig. 4). These functional relations were

solved simultaneously by graphical means to obtain

X and G as functions of Vd and Ld (figs. 5 and 6).

17
10 -

-Hb 5(> -' h-—

/

/

16 - //

11 -

12 - :a

-4r 9

•

10 -

»

• r

8 - 4-

•

6 -

4 -

2 -

n

o X Figure 3. Experimental correlation of B and X.

Figure 7 shows the distribution obtained by the

above procedure for the following measurements

:

T

7

d =4.5 db below

Ld =9.2 db below e^.

On the X versus Vd and Ld curves, the above measurements are seen to give an Z of 3.7, and on the C

versus Vd and Ld curves the above measurements

give a 0 of 15.8 db. Using the A versus X and C

curves, this pair of values for X and C gives an A of

12.3 db. Point a is located on the 50 percent coordinate 12.3 db below the Ems and the Rayleigh line

with slope —)i is drawn. Point c is located 16 db

above the point where the Rayleigh line intersects

the 1 percent coordinate, and through point c the

high-voltage, low-probability line with slope s=

—X/2= — 1.8 is drawn. The angle formed by the

Rayleigh line and the high-voltage, low-probability

line is now bisected and B is calculated from the

relationship B=\.b{X-\). B is found to be 3.6 db.

Point b is located 3.6 db above the intersection of the

Rayleigh and high-voltage, low-probability lines and

through this point the tangent line is drawn perpendicular to the angle bisector. Distance 2 to 3 is made equal to distance 1 to 2, and at point 3 on the

Rayleigh line a perpendicular to the Rayleigh line

is constructed. This perpendicular intersects the

angle bisector at the center of the circular-arc portion

of the distribution. The complete distribution is now determined with the construction of this arc.

It should be noted from the curves (figs. 5 and 6) that for a given Va only a certain range of values of

Ld will result in a distribution of the above form.

These ranges are enclosed approximately within the

dashed lines. Also, the combination of the above

given Vd and the minimum allowable Ld results in a nonunique solution, i.e., there is, essentially (within

the range of accuracy of measurements) , an infinite

number of distributions, all with the same Vd and Ld of the above special combination. Fortunately

these special combinations almost never occur. The above facts allow us to determine to some

extent the validity of data received from recording

stations, and whether a measured sample is true

atmospheric noise or contaminated noise.

5. Errors in Noise Measurements and Their

Influence on the Calculated Distribution

The accuracy with which a distribution can be

determined from measured moments depends upon

(1) the validity of the form factor which has been

taken to represent the distribution, and (2) the

accuracy of the measured statistical moments.

The validity of the form factor is very difficult to

check because of the fact that 15 to 30 min are required to measure the distribution, and the statistics of the noise do not always remain constant

for this period of time. As a check for changing

noise characteristics, each series of distribution

measurements was immediately repeated and any

changes noted. Further, the moments recorded by

the ARN-2 provided information on the stability

of the noise statistics. Figure 1 represents an

example of the fit in the case where the noise moments

were measured precisely and the noise characteristics

appeared to be constant during the interval required

for the measurement of the entire distribution.

Methods of recording noise samples on magnetic

tape are under development and these samples will

enable a more critical study of the form factor as

well as other detailed characteristics.

Errors in the ARN-2 could be detected by comparing the recorded three parameters with the same

three parameters as numerically integrated from the curve recorded by the distribution meter. The most

likely source of error in the ARN-2 was the possible

saturation of the square law detector with a resultant

lowered value for Ems. Since the average voltage

and average logarithm of the voltage are measured

in decibels relative to E^, their values would be in error by the same amount.

Approximately sixty sets of distribution measure- ments were .analyzed for the error made by the ARN2 recorder in measuring the moments. These

ARN-2 errors were divided into class-intervals and

plotted as error-probability curves on arithmetic

probability graph paper in figure 8. The fairly

straight lines obtained indicated largely random errors, with some bias due to square-law detector saturation. For the Erma values, the standard deviation

18
inE

w

uio

_JUJ

ffi

73

Figuee 4. A versus X and C.

was about 1.1 db, with an average error of —0.4 db.

Ninety percent of the values were within ±2 db

of tbe mean. For the E^ values, the standard

deviation was about 0.8 db, with an average error

of +0.2 db. Ninety percent of the values were

within ±1.5 db of the mean. For the Eloe values,

the standard deviation was about 1.4 db, with an

average error of +0.6 db. Ninety percent of the

values were within ±2.5 db of the mean.

An effort was made to correlate the error with

dynamic range between the 0.0001 percent and 99

percent intercepts of the amplitude-probability distributions. The same data were divided into classintervals and the medians of the class-intervals which

contained sufficient samples for statistical signifi- cance were plotted against dynamic range in decibels

in figure 9. The best straight-line fit indicates good

correlation of error with dynamic range, a negative

error for ETmB and a positive error for Eave and

Eloe . The error in E^ at 85-db dynamic range

varied from —2.7 to +2.1 with a median value of

about 1 db.

19
Figures 10 and 11 show the influence of errors in

the moments measured by the ARN-2 on distributions determined from the moments. In these

figures, the measured moments are indicated as £"rm8 etc., and the moments obtained from the

simultaneously measured distribution are indicated

by the same symbols without the primes. The

examples chosen contain errors typically measured

by the ARN-2. In figure 10, the errors in the

measured moments are small and the distribution

derived from the slightly erroneous ARN-2 moments

fits the measured points with negligible error. In

figure 11, the dynamic range is wider with corre- spondingly larger errors in the ARN-2 measured

moments. Thus, the departure of the distribution

derived from the ARN-2 measured moments is greater. These departures, though significant are not considered excessive in view of the fact that

they are of the same order of magnitude as the

changes in the noise itself during an hour.

Thus, it can be concluded that this is a satisfactory method of deriving the complete amplitudeprobability distribution from statistical moments,

since the departures from the true distribution will

be small when the moments are measured accurately.

6. Bandwidth Considerations

The foregoing method of obtaining an amplitudeprobability distribution from the three measured

moments results in a distribution which is only valid

for the bandwidth in which the moments were

measured. Fulton [8] developed a method of bandwidth conversion which was accurate for the highamplitude, low-probability portion of the distribution, but failed to convert the lower portion of the curve as accurately since it was an empirical method

based upon insufficient measurements. A study is

now in progress for a more accurate method of

bandwidth conversion.

20
21
55 60 85 70 75 80 85 90 95 100 DYNAMIC RANGE , db

Figure 9. Error of measured moments versus dynamic range.

50

30

20

O

m

<

-10

-20

-30

-40

1 I

1

1 Frequency - 10 Mc

1435 September MD, 1958 III Boulder, Colorado

i

D Meosur Obtome momen

ed dist d from

s, Vi

ribution ARN-2

ond L'(j

E

-rms - ns T T

vd 1 we-*- L(j \

Eloal

iJ

E "log-L

0.000! Oil 0.1 I 5 10 20 30 40 50 60 70 80 90 95 98 99 PERCENTAGE OF TIME ORDINATE IS EXCEEDED

Figure 10. Comparison of measured distribution with that obtained from moments measured by ARN-2.

Boulder, Colo. (Paper 64D1-37)

22

J0

20

10

l/l e

-10

-30

-40

1

'

1

1 Frequency — 113 kc 1550 September 12,1951

1 Boulder, Colorcdo

1 l i

\

) Meosur

Obtoire momen

I ed distribution d from ARN-2

s Vj and \l a

i

1

1

1 L d

t

d r

L

o

o

0.0001 0.01 0.1 I 5 K) 20 30 40 50 60 70 80 90 95 96 99 PERCENTAGE OF TIME ORDINATE IS EXCEEDED

Figure 11. Comparison of measured distribution with that obtained from moments measured by ARN-2.

7. References

[1] International Scientific Radio Union, Recommendation I and its annex, Proceedings of the Xllth General

Assembly, Commission IV, Pt. 4, XI, 99 (1957). (See

also: H. E. Dinger, Report on URSI Commission IV

Radio noise of terrestrial origin, Proc. IRE, 46, 1366

(1958).)

[2] W. Q. Crichlow, Report of subcommission on the question:

what are the most readily measured characteristics of

terrestrial radio noise from which the interference to

different types of communications systems can be

determined, URSI Rpt. No. 254, Proceedings of the

Xllth General Assembly, Commission IV, Pt. 4, XI, 9

(1957).

[3] W. Q. Crichlow, Noise investigation at VLF by the

National Bureau of Standards, Proc. IRE, 45, 778

(1957).

[4] A. D. Watt and E. L. Maxwell, Measured statistical characteristics of VLF atmospheric radio noise, Proc. IRE, 45, 55 (1957).

[5] A. W. Sullivan, The characteristics of atmospheric noise,

(Atmospheric Noise Research Laboratory, Engineering

and Industrial Experiment Station, University of

Florida, Gainesville, Florida).

[6] H. Yuhara, T. Ishida, and M. Higashimura, Measurement

of the amplitude-probability distribution of atmospheric

noise, J. Radio Research Labs., 3, 101 (1956).

[7] F. Horner, An investigation of atmospheric radio noise at very low frequencies, Proc. Inst. Elec. Engs., Pt. B,

103, 743 (1956).

[8] F. F. Fulton, Jr., The effect of receiver bandwidth on

amplitude distribution of VLF atmospheric noise,

Prepublication Record, Symposium on the Propagation

of VLF Radio Waves, National Bureau of Standards,

Boulder, Colorado, III, Paper No. 37 (1957) (to be

published in J. Research NBS, Sec. D).

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