PB 151379

^^Boulder laboratories

^0.20

A PRELIMINARY ANALYSIS OF

AMPLITUDE SCINTILLATIONS

OF RADIO STARS OBSERVED AT

BOULDER , COLORADO

BY ROBERT S. LAWRENCE AND JAMES L. JESPERSEN

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NATIONAL BUREAU OF STANDARDS

technical ^iote

20

July, 1959

A PRELIMINARY ANALYSIS OF AMPLITUDE SCINTILLATIONS

OF RADIO STARS OBSERVED AT BOULDER, COLORADO

by

Robert S. Lawrence

and

James L. Jespersen

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that- are of transient or limited interest. Technical Notes may be

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A PRELIMINARY ANALYSIS OF AMPLITUDE SCINTILLATIONS

OF RADIO STARS OBSERVED AT BOULDER, COLORADO

by

Robert S. Lawrence

and

James L. Jespersen

This work has been sponsored by the USAF, under

D.O. AF Ol*(6V7)-13l<-

TABLE OF CONTENTS

Introduction . . . ......... 1

1. Description of the experiment ............ 1

2. Data Analysis 2

Ionograms 2

Scintillation indices ....... 2

Correlation of IBM cards. . . . . . k

3. Results ............ 5

Distribution of amplitude and flux values ..... 5

Comparison between the two frequencies. ...... 5

Correlation with Ellsworth ionograms. ....... 6

k. Discussion of results ......... 7

Amplitude and flux distributions .7

Comparison between the two frequencies. ...... 9

Correlation with spread F . . 9

Correlation with sporadic E ........... .10

References . . .................. .12

Appendix ; The relation between amplitude and power

indices for a Rice distribution. ........ .I**-

ABSTRACT

Variations in the apparent flux from the radio source Cygnus-A

are recorded at 53 and 108 mc/s using a two-element phase -sweeping

interferometer located at Boulder. An ionospheric sounder operating

at Ellsworth, Nebraska, provides simultaneous vertical-incidence

measurements of the ionosphere along the path. Amplitude scintillations

observed at Boulder over a six-month period are compared with the

ionograms taken at Ellsworth. Positive correlation is found between

amplitude scintillations and spread F, while a small negative

correlation is found with sporadic E. The present result appears to

be significant but a full year's data are being accumulated to disclose

seasonal or diurnal effects. Detailed analysis of the scintillations

indicates that the probability distribution of the amplitude can be

represented by the Rice probability distribution function. The

scintillation rate for 108 mc/s seems to be more rapid than for 53 mc/s.

A PRELIMINARY ANALYSIS OF AMPLITUDE SCINTILLATIONS

OF RADIO STARS OBSERVED AT BOULDER, COLORADO

by

Robert S. Lawrence

and

James L. Jespersen

Introduction

It has been recognized since 1950 that the apparent fluctuations

in intensity of discrete radio sources arise in the ionosphere. The

F region is generally believed to play an important role in the

production of these scintillations, but our knowledge of the effect of

the E region is confused by incomplete and apparently conflicting

evidence. One of the purposes of an experiment now in progress at the

Boulder Laboratories is to improve our understanding of the ionospheric

conditions and levels responsible for the scintillations. A description of the planning and instrumentation for the experiment, along

with a brief summary of the pertinent literature has already appeared?- *

The present report deals with amplitude scintillations observed

at Boulder and with the comparison between them and ionograms taken

at Ellsworth, Nebraska. It must be considered preliminary, in that

it describes data for only six months, February through July, 1958,

and the relationship between sidereal and solar time dictates that

a full year's data are required to distinguish between diurnal and

elevation-angle effects

1. Description of the Experiment

The apparent flux from the radio source Cygnus-A is recorded at

Boulder at frequencies of 53 and 108 mc/s, using a two-element,

phase -sweeping interferometer with an east-west spacing of V?5 meters.

At the time when Cygnus-A is about 15 above the north-east horizon,

the line of sight intersects the E region over Ellsworth, Nebraska.

An ionospheric sounder operates at Ellsworth at five-minute intervals

to provide simultaneous information concerning the ionsophere along

the path. As the source rises above 15 , the intersection point moves

from Ellsworth toward Boulder. Figure (l) is a map of the BoulderEllsworth region showing the plan view of the intersection point for

CO

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(=> «t CO

CP

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CO

C_3

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various ionospheric heights. The numbers along the loci indicate the

local sidereal time (LST) at Boulder.

The 90 and 100 kilometer intersections both pass nearly overhead

at Ellsworth, but at different times. The 300 kilometer intersection

point comes to within less than 100 km of Ellsworth about two and

one-half hours later. Notice that for each ionospheric height there

is a time when any agreement between the scintillations at Boulder

and the vertical measurements at Ellsworth should be most pronounced.

Conversely, the time of best agreement is indicative of a unique height.

2. Data Analysis

Ionograms Ellsworth ionograms are made at five-minute intervals

each day from 1230 until 1330 LST, and are made at half-hour intervals

for the next three hours. Each ionogram is scaled for critical

frequencies, minimum virtual heights, and for the occurrence and type

of special features, especially spread-F and sporadic -E traces.

Absence or presence of spread F (or sporadic E) is indicated by an index which has only two values, 1 and 2. For each ionogram, the

scaled information is punched on an IBM card, and to this card is added

a set of scintillation indices obtained from the scintillation records

for the same five -minute interval.

Sc int illat ion indiee

s

. Figure 2a is a twelve -minute section

of a typical scintillation record. The trace shows the 10 cycle/minute

sine wave generated by the phase-sweeping interferometer as the lobes

pass across Cygnus-A. The height of the envelope of this trace is

proportional to the received flux from the discrete source. The

fluctuation in this envelope is the amplitude scintillation caused by

the ionosphere. Figure 2b shows a record in which the scintillations

are much deeper and more rapid. Many of these records are available

not only as pen recordings, but in digital form on punched paper tape

which can be fed directly to an electronic computer.

The records are classified by eye as to depth and rate of the

amplitude fluctuations. The depth index consists of a scale of five,

described as:

1. no scintillations,

2. shallow scintillations,

3. medium scintillations,

^4-. deep scintillations,

5. scintillations so deep that the

signal disappears.

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Several records bearing each index have been analyzed with the

computer to determine both the power index,

A P

and the amplitude index,

C-^)

s (~)A

where P is the power received from the radio star, A = vP is the

amplitude of the radio signal, and a bar denotes an average over the

record. The results are summarized in the table below.

depth

index

AP/P ( AA/A f

min . m$x . avg

.

min . max . avg

.

1

2

3

k

5

.Ok .13 -09

.06 .Ik .10

.lh .35 -23

•30 .39 -35

.kk .75 -57

.0008 .0071 .0025

.0016 .0083 .00^+5

.0092 .OkkQ .0238

.0377 .0575 .0512

.O838 .3237 .1689

The rate index consists of a scale of four, described in terms

of the apparent "period" of the scintillations, as follows:

1. no scintillations,

2. period greater than 2 minutes,

3. period 1 to 2 minutes,

k. period less than 1 minute.

This classification is more subjective than the depth index because

the scintillations have no unique period. After the rate index had

been assigned by eye, several records from each group were smoothed

with an l8-second running average and then measured to find the

average interval between crossings of P. The apparent "period", t,

is defined as twice that interval. The table below summarizes the

results of this analysis.

rate

index

t

min. max. avg.

1

2

3

k

no scintillations

1.8 k.2 2.7

0.8 1.6 1.1

0.7 i.k 0.9

Correlation of IBM cards . The IBM cards containing the scintillation indices and the readings from the ionograms are used to determine

the correlation between any two of the above-mentioned variables x and

y. The correlation coefficient is defined in the usual way as

r = xy - x y

where the variance

,

0" (7 x y

2

0" =

2

X

-2

X

(1)

If all the data are independent and normally distributed the standard

deviation of the correlation coefficient is

1 - r

nTn (2)

where N is the number of samples. If the data are not independent,

but the serial correlation is significant over a span of n samples,

N in (2) should be replaced by N/n so as to include only independent

samples. A brief inspection of the data indicates that a typical

value of n for the scintillation records is 6.

With the depth and rate indices as they have been defined the

distribution of scintillation data is by no means normal, but is

strongly skewed toward the high indices. Figure 3 shows the

histograms of scintillation indices on all the IBM cards used in the

present comparison with the ionograms. (These histograms are not

representative of all scintillations, as the data considered here are strongly biased toward elevation angles of 15 . ) In order to assess

the error introduced by using the simple relationship (2), a more

sophisticated analysis^, involving higher moments of the actual

distribution, was made of a typical correlation coefficient taken from

our data. The difference in °"r amounted to less than .001, and

equation (2) has been used for all subsequent calculations of °"r

.

1400

1200

1000 -

< 800 -

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400 -

200 -

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INDEX DEPTH 108 Mc/s

1 200

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-

400

200 •

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INDEX RATE 53 Mc/s

I

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1200

1000 -

800 -

600

400 " i

////

200

'V/////V///////

! '///31<///§4 INDEX RATE 108 Mc/s

THE DISTRIBUTION OF SCINTILLATION INDICES USED IN THE ANALYSIS OF DATA

FEBRUARY THROUGH JULY, 1958

FIGURE 3

3. Results

Distribution of amplitudes and flux values . Amplitude scintillations have been measured and discussed by various authors in at least

two very different ways. Observations made in Cambridge, England have

been reported in terms of a fluctuation index derived by dividing

the root-mean -square fluctuation in amplitude by the mean amplitude

received from the source, i.e. in terms of the quantity

^TlTTlTya _ ./7a -fr 2

1 A \ A m

<¥)

2

the square root of our (—j— )• The observations from College, Alaska

P

k A P

are reported in terms of —5—. Results from Manchester, England were

reported ' in terms of a fluctuation index which was approximately

3^r times larger than —

. Booker, in his recent discussion of

'A A^ 2

scintillations, uses the quantity (—a- ) and mentions the difficulties

involved in converting from one system of measurement to another.

A P /A AN 2

The relationship between —5— and (—7—) depends upon the

distribution law obeyed by the amplitude, A, of the radio wave. Thirty-one records, selected at random and analyzed with the computer,

gave the values of -r^— and (—r— ) shown by the dots in figure k.

The solid curve, a theoretical one, is described in section K.

Comparison between the two frequencies . Quantitative comparison

between the scintillations observed at 53 and 108 mc/s can best be

made not with the subjective depth and rate indices used for the

statistical analysis, but with detailed analysis of pairs of simultaneous records made at the two frequencies. Several such pairs of A A\2

records have been analyzed with the computer to get both t and (—»—) • The comparison of fading "periods" is displayed in figure 5 a where it can be seen that the 53 mc/s scintillations seem to be consistently

slower than those at 108 mc/s. The amplitude indices, (—r— ) } are related as shown in figure 5h. The two lines show the locations to

2 k be expected for the points if the amplitude index varied as X or X .

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2.0 r2.0 FADING PERIOD AT 53 Mc/s

(SECONDS)

COMPARISON OF FADING PERIODS AT TWO FREQUENCIES

FIGURE 50

0.04 1—

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 /Ab\2 (7A~J AT 53 Mc/s

COMPARISON OF AMPLITUDE INDICES AT TWO FREQUENCIES

FIGURE 5 b

Correlation with Ellsworth ionograms . We have seen from figure t

that the degree to which agreement can be expected between the

scintillation indices and any particular feature of the Ellsworth

ionograms is a function of sidereal time and of the height of the

effective ionospheric region. Accordingly, for correlation studies

the data were first separated into four groups depending upon sidereal

time.

The upper diagram in figure 6 shows the four correlation coefficients for the depth index of 108 mc/s scintillations versus the

occurrence of spread F on the ionograms. The vertical bar through each

point has length 2crr . The abscissa on this diagram is labeled both

with local sidereal time and with the corresponding effective height

for which best agreement would be expected at that time . The

correlation coefficients, though small, seem to be significant and

they increase for high effective heights as would be expected for an F-region phenomenon.

The corresponding graphs for the rate index at 108 mc/s, and for

the depth and rate indices at 53 mc/s, are not shown, but they all

exhibit similar behavior when correlated with spread F. When the

data are divided into two three-month sets, each independently shows

the same effect, but naturally the standard deviations are slightly

larger. The significance of these correlations is discussed in

section k.

The lower diagram in figure 6 shows the result of correlating

the depth index at 108 mc/s with sporadic E. Here the correlation is

negative and, as would be expected for an E-region phenomenon, its

absolute value is a maximum at low effective heights. The 53 mc/s

depth index gives the same result. The rate indices for both 108 mc/s

and 53 mc/s scintillations appear to be uncorrelated with sporadic E.

In order to check more closely the negative correlation with

sporadic E, the data were separated according tp the type of sporadic E.

Of the standard IGY types in accordance with which the ionograms were

scaled, one (Type "i") is described as "a flat Es trace below the

normal E layer minimum virtual height." This type, which we refer to

as "low-level" sporadic E was separated from all other types (referred

to as "high-level"), and the curves shown in figure 7 result. The

negative correlation appears in both types, and the effective heights

vary in the proper direction.

This apparent statistical connection between scintillations and

sporadic E stimulated a search for specific cases in which the

scintillation records change character at the time when sporadic E

appears or disappears at Ellsworth. Spectacular results cannot be

expected from such a search for several reasons, and the statistical

significance is difficult to assess. The sporadic -E appearance must

0.5

80 105 H—h170 EFFECTIVE HEIGHT (Km)

260 360

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1300

80 105

1400

170

1500

260

1600

360

1 1 i 1 —i———

SPORADIC E

-

i >

< •

" 5

I

1—

h

L___H —

1

1300 1400 1500 1600

LOCAL SIDEREAL TIME

CORRELATION OF 108 Mc/s SCINTILLATION DEPTH WITH SPREAD F AND SPORADIC E

FIGURE 6

80 105 170

EFFECTIVE HEIGHT

260 360

(Km)

0.5

LJO

till. UJo

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-0.5

0.5

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80 105 H 1—

1400

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1500

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1600

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HIGH-LEVEL

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1300 1400 1500 1600

LOCAL SIDEREAL TIME

CORRELATION OF 108 Mc/s SCINTILLATION DEPTH

WITH HIGH-LEVEL AND LOW-LEVEL SPORADIC E. FIGURE 7

occur within about an hour of the proper sidereal time since it is

known° that vertical-incidence measurements of sporadic E only 100 km

apart often show only slight correlation. The elevation angle of the

radio star is changing at this time at a rate of about 10 per hour ? and this causes a continuously changing appearance of the scintillation

record even in the absence of ionospheric changes.

From the six months of available data there are two cases where

the scintillation record seems to change in synchronism with the

sporadic E on the ionograms, and two or three others where the effect

is more questionable or absent. Figure 8 shows the scintillation

record from 1302 to 1^01 LST on April 17, 1958. Sporadic E was present on ionograms taken at 1317* 1322, 1327, and 1332. It was absent at 1312 and earlier, and also at 1337 an<i later. The character

of the scintillation trace appears to be different during a period

roughly coincident with the apparition of sporadic E. The principal

effect seems to be a reduction of the amplitude of the scintillations,

although there is some evidence of a decrease in average flux.

k. Discussion of Results

Amplitude and flux distributions. Figure k shows that the

A AV , M , AP observed values of amplitude index, (—7—) > and power index, -^=— , lie close to a smooth curve. Since the relationship between these

two indices depends upon the distribution function of the amplitude

of the radio wave, this result gives us some information about the

distribution functions themselves. Any particular probability A A\2

density function would produce a single value of (—7— J and a A P \ " /

corresponding single value of ——. The fact that the points on

figure k lie near a single smooth curve implies that the family of

probability density functions from which our scintillation samples

have been taken can be described by a single parameter, the position

along the line depending upon the value of that parameter.

The solid line in the figure shows the theoretical relationship

which would occur if the observed signal were composed of a steady

(undiffracted) component and a random component consisting of a number (three, or more) of independent, randomly-phased, diffracted

waves. The probability density function from which this curve has been

derived was first described by Rice9 and has been discussed in terms

of its application to ionospheric problems by RatcliffelO, This

probability density function, which we shall call the Rice distribution,

is really a family of distribution functions parametric in the quantity

CYGNUS-A APRIL 17, 1958

53 Mc/s

i i i

i

§ i i i

i

i i i i i

i i i i

i i i 1305 1310 1315 1320 LST

SPORADIC E ABSENT

I I I I I I I I I I I I I I I I

I I 1325 1330 1335 1340 LST

SPORADIC E PRESENT

i i i

i

i i i i ii - 1 i i

i

i i i i i

i 1345 1350 1355 1360 LST

SPORADIC E ABSENT

A SCINTILLATION RECORD TAKEN DURING THE PASSAGE

OF A SPORADIC-E CLOUD

FIGURE 8

a amplitude of the steady component

Norton, et al

r.m.s. instantaneous amplitude of the random component

\12 r.m.s. instantaneous amplitude of the steady component

r.m.s. instantaneous amplitude of the random component

, have defined the quantity

k

2 r.m.s. instantaneous amplitude of the random componentr.m.s. instantaneous amplitude of the steady component

-i 2

which has the advantage of being a ratio of mean squares, k may be

thought of as the r.m.s. noise-to-signal ratio of the radio wave. In

the present application "signal" means the steady component unaffected

by the irregularities of the ionosphere, and "noise" means the

combination of the various diffracted waves.

The Rice distribution degenerates into the familiar Rayleigh

distribution in the limit as k approaches infinity, and approaches a displaced Gauss distribution as k approaches zero. Several values of

the parameter k are indicated on the curve . The details of the

derivation of this curve are given in the appendix.

The apparent agreement between the scintillation records and the

Rice distribution means that the arbitrary amplitude and power indices

may be replaced by the more physically meaningful parameter k. Each

depth index naturally covers a considerable range of values of k.

Extreme and average values are approximately as shown below.

depth

index

k

min. max. avg.

1

2

3

k

5

.Ok

.05

.13

.28

.12

• 13

•35

• 35

> 3

.05

.07

.20

• 31

1.1+1

9 The family of Rice distributions as defined by Rice applies to

the amplitude of the radio wave and is thus quite convenient for the

study of communications signals. For radio-star observations it is

usual to employ a square -law detector and thus obtain records which

are linear in power. The Rice distributions, G(p), plotted in terms

of power, are shown in figure 9- These curves accurately match

the distributions of chart deflections which are found experimentally.

On these curves the long tail which exists for large values of k is

particularly noticeable. The corresponding large deflections of the

record are indeed prominent, and may easily give the erroneous

impression that they are caused by some special mechanism unrelated to

random diffraction.

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Comparison between the two frequencies . It Is generally reported

that the scintillation rate is independent of frequency. A casual

inspection of the records upholds this belief, but the more detailed

analysis made with the computer seems to show that the 108 mc/s scintillations are consistently more rapid than those at 53 mc/s. The

reasons for this are not clear; in fact, it would be reasonable to

expect just the opposite behavior since the phase index of the

diffracting region is presumably greater at the lower frequency. This

effect warrants further study > and will be investigated by inspection

of the power spectra of the scintillation records.

7 Booker has pointed out that the mean square amplitude fluctuation,

^r— J

, may be expected to vary as X or as X , depending upon the

size of the irregularities compared to a Fresnel zone. Figure 5b shows

that, for the few simultaneous records which have been studied, the

dependence upon wavelength lies between these two limiting cases with

no apparent preference for either. This indicates that significant

irregularities must be present with a 0.5 autocorrelation distance of

about one kilometer, in good agreement with the scale of irregularities

obtained by other methods.

Correlation with spread F . The apparent correlation between

amplitude scintillations and spread F is not surprising; similar

results have been reported?

>

12~17 by most other workers in the field.

In most cases the sounder was not situated in the optimum position

for best correlation, and in several of the reports it was only the

agreement in diurnal behavior of scintillations and spread F which

indicated the possible connection. Briggs has recently made a detailed study of the connection between scintillations and three

reasonably well-situated ionospheric sounders. He concludes that

spread F occurs in patches having dimensions of about ^50 km northsouth and larger east-west. As Briggs points out, the variation of

the correlation coefficient indicates that the scintillations are produced by irregularities at a height of about 300 km, although the

figure is admittedly rough.

Our data seem entirely consistent with the results of Briggs.

The effective height cannot be read from figure 6 with any accuracy,

but it is consistent with 300 or ^-00 km. The fact that the

correlation is already noticeable at 1300 LST, but is not so high

as at l600, indicates that the patches of spread-F ionization are of the order of ^00 km in extent in a NE-SW direction.

Correlation with sporadic E . A connection of the type seen here

between scintillations and sporadic E has not been reported previously.

Most observers have found little or no correlation with sporadic E.

Bolton , Slee, and Stanley ^ found a positive, day-to-day correlation

with low-angle scintillations in Australia. This appeared even though the iono sonde was several hundred kilometers from the point of

penetration of the E region by the line of sight to the radio star.

Wild and Robertsl5 found that this effect occurred only for daytime

(or summer) scintillations. (Using a single radio star, it was

impossible to distinguish between diurnal and seasonal variations.)

Because of the patchy nature of sporadic E and the hundreds of

kilometers between the ionosondes and their optimum position, Wild

and Roberts were forced to consider only those days which showed

significantly large or small values of fEs in agreement at two wellseparated locations. Hartz^ noted that the diurnal variations of

scintillations and sporadic E in Canada are opposite but he found

neither positive nor negative correlation on a daily or hourly basis.

In this country, Dueno " found some correlation between sporadic E

and low-angle scintillations in the winter.

For E-region effects, the present experiment is potenti411y more

sensitive than any of those just mentioned because the Ellsworth

iono sonde is located in precisely the optimum position, but the

limited sample of data (six months) now available cannot disclose

diurnal or seasonal effects, and is short enough to demand a careful

evaluation of the significance of whatever correlations do appear.

Some of the correlation coefficients shown in figures 6 and 7 differ

from zero by two or three times the computed standard deviation.

This fact alone would not be sufficient to inspire much confidence in

them, but it seems most significant that the peaks (or dips) in the

correlation appear at the proper times, and therefore at the proper

effective heights, for the various layers. This concerted behavior

of the correlation coefficients lends much more confidence to the

entire picture than any significance test on the individual coefficients

could do.

At first glance it seems paradoxical that scintillations, an

effect of irregularities in the ionosphere, should be negatively

correlated with sporadic E. Actually, it is quite in accord with the

behavior of sporadic E that at least some types may be indicative

of unusually calm or quiet conditions in the ionosphere. For example

>

it is common knowledge that the fading rate of a signal propagated

by ionospheric scatter drops markedly when sporadic E appears. It

may be that the normal E region is in a state of turbulence and

contributes a small part to the total scintillation of radio stars > and that sporadic E appears at times when the region is unusually

quiet, allowing the formation of thin layers with sharp ionization

gradients, and the scintillations are correspondingly reduced.

Reflection from the sharp gradients might also explain a decrease in

average flux.

10

Acknowle dgment

This experiment could not have been performed without the

cooperation of several people. Among them are E. R. Schiffmacher

and H. A. Erickson who have been responsible for the maintenance and

operation of the scintillation receivers } R. F. Carle who has

established and operated the ionosonde at Ellsworth, and R. C. Lamb

and Mrs. C. I. Enfield who have assisted with the analysis of the data.

11

EEFEKENCES

1. Lawrence, R. S.; "An investigation of the perturbations imposed

upon radio waves penetrating the ionosphere/' Proc. I.R.E.,

vol. k6, pp. 315-320; (1958).

2. Cramer, H. ; "Mathematical methods of statistics," Princeton U.

Press ; p. 359; (195*0-

3. Hewish, A.; "The diffraction of galactic radio waves as a method

of investigating the irregular structure of the ionosphere,"

Proc. Roy. Soc. A, London, vol. 2l4, pp. ^9^-51^; (1952).

k. Little, C. G. , et al.j "Radio properties of the auroral ionosphere*

Quarterly progress reports Nos. 1-5 > 1, 8," Geophys. Inst.,

U. of Alaska; (1958).

5. Little, C. G. , and Maxwell, A. ; "Fluctuations in the intensity of

radio waves from galactic sources," Phil. Mag., ser. J, vol. k2f pp. 267-278; (1951).

6. Little, C. G.; "The origin of the fluctuations in galactic noise,"

Ph. D. thesis, Manchester U. , England; (1952).

7. Booker, H. G. ; "The use of radio stars to study irregular

refraction of radio waves in the ionosphere," Proc. I.R.E.,

vol. 46, pp. 298-311

!-; (1958).

8. Thomas, J. A.; "Sporadic E at Brisbane," Austr. J. Phys., vol. 9>

pp. 228-246; (1956).

9. Rice, S. 0.; "Mathematical analysis of random noise," Bell System

Tech. J., vols. 23 & 24, pp. 282-332 and 46-156; (1944 and 1945),

10. Ratcliffe, J. A.; "Some aspects of diffraction theory and their

application to the ionosphere," Rep. Prog. Phys., vol. 19>

pp. 188-267; (1956).

11. Norton, K. A., et al.j "The probability distribution of the

amplitude of a constant vector plus a Rayleigh-distributed

vector" Proc. I.R.E., vol. 43 pp. 1354-1361; (1955).

12. Ryle, M. and Hewish, A.; "The effects of the ionosphere on radio

waves from discrete sources in the galaxy," Mon. Not. Roy.

Astron. Soc, vol. 110 pp. 381-394; (1950).

12

13. Bolton, J. G. ; Slee, 0. B.; and Stanley, G. J.; "Galactic radiation

at radio frequencies VI; low-altitude scintillations of the

discrete sources," Austr. J. Phys., vol. 6, pp. k-^-h^l; (1953)

»

1^. Hartz, T. R.; "Radio star scintillations and the ionosphere,"

Can. J. Phys., vol. 33, pp. hj6-k82; (1955).

15. Wild, J. P. and Roberts, J. A.; "Regions of the ionosphere

responsible for radio star scintillations," Nature, vol. 178,

PP. 377-378; (1956).

16. Dagg, M. ; "Diurnal variations of radio-star scintillations,

spread F, and geomagnetic activity," J. Attn. Terr. Phys.,

vol. 10, pp. 2C4-214; (1957)-

17. Koster, J. R.; "Radio star scintillations at an equatorial

station," J. Atm. Terr. Phys., vol. 12, pp. 100-109; (1958).

18. Briggs, B. H.; "A study of the ionospheric irregularities which

cause spread-F echoes and scintillations of radio stars,"

J. Atm. Terr. Phys., vol. 12, pp. 3^5; (1958).

19. Dueno, B.; "Study and interpretation of low angle fluctuations

from the radio star Cassiopeia as observed at Ithaca, N.Y.,"

Tech. Rept. No. 27, Cornell U. School of Elec. Eng.,

Ithaca, N.t.; (1955)

20. Hoel, P. G. ; "Introduction to mathematical statistics," p. 213;

John Wiley & Sons. (1956).

13

APPENDIX

The relation between amplitude and power Indices

for a Rice distribution .

Rice^ has shown that the probability distribution function (pdf

)

of the resultant instantaneous amplitude (A) of the sum of a unit

reference vector which represents a sinusoidal voltage of constant

amplitude and phase, and a Rayleigh distributed vector which

represents a random component with r.m.s. voltage k ( = v2/a, in

Rice^ notation) is given by

where I e is the modified Bessel function of the first kind. It is

possible to compute from (l) the quantities

(¥/ (2) = 1

N

N

i=l

ICYJ

)

a and

A P

P

1

N

N

1 Pi - P I (3)

2 - where A = P; and A and P are the means of A and P respectively.

Before proceeding to this, it will be helpful to make some preliminary computations.

W- A - A A4

- 1 (k) 2 2

A A

A P

To evaluate -^- we recall from statistics that the expected

value E[g(x)] of the quantity g(x) is given by

E[g(x)] = / f(x) g(x) dx where f(x) is the pdf of x.

Ik

Thus,

E (J421

)

1_

P

A P

P — / |P - P | f(p)dP

P Jo P ^

(P - P) f(p)dP - / (P - P) f(p)dP

/ (P - P) f(p)dP - 2 / (P - P) f(p)dP

4- [

2 P F (P) - 2 Pa (5)

where

F (P) / f(p)dP and P6 = / Pf(p)dP.

da d

We are interested in obtaining the pdf of the power "p" from (l)

The following theorem!9 will allow us to make this transformation.

"If x = h(y) is an increasing or decreasing function of y and if

f (x) is the pdf of x, then g(y) is the pdf of y where

Thus for the transformation

L<y)l h'(y) |."

A

2

- P,

h(P)

i = P

2

;

h'(p)

i _ i p"2.

g(P), - *&) . 1 • j 2)P2 or in terms of A,

g(P) = g(A2

) = 4^- <«>

We may now proceed to the computation of A A

and A P

'-/-«- = ?P"[-^KS) dA

15

* L\ k

y ^2ky*?^ (T)

Pg(p)dP

2 Too 3 / A

2 + 1

75 J

exp

V" "T2~ k

= 1 + k (8)

Then from (k) , (7) and (8)

A A

k (1 + k

2

) exp(^)

k

2

« -? + ^2 h (—2

k v2k

-t 2

-l (9)

As k -> oo¥) - --i A / n (10)

A P

For -=- we must evaluate

,Vp

3 A exp

A

2

+ l 2A dA (11)

(ll) cannot be integrated in closed form; however, Rice has shown that

4k when k < —

f(A) % -4- exp -tt-A

) ^ W* v k

(12)

16

the Gaussian pdf } for which

A

2

.

•h Tims when k < — , from (8)

1 + (13)

Then from (k)

A A 2 fiL + D . !

k + 2

- 1 r^P

.2 /A-lf .. _ . . (A - A)

Then P — /

2

(kt + A)

2

exp (-t

2

) dt

where t,

k

y

2 +

2

l. l^2

and t. ,2 2

r p o r For k < -|, / (kt + A)

2

exp (-t

2

) dt fc / °(kt + A)

2

exp (-t

2

) dt.

J-t,

Thus P =

s r

/

2

(kt + A)

2

exp (-t

2

) dt

+ / (kt + A)

2

exp (-t

2

) dt

17

1. [|f (^ .t

2 exp(-t|)) * (l + |f J

(2 - e*p(-t|))

vit /Ik +

2

and finally,

(l + ^) + (l + k2

) jf

2 exp(-t

2

)dt (11.)

F (P) = -4- f

?

exp y^-J )dA = -1- f

2exp(-t2

)dt

_xi

* -i- f

2 exp(-t2

)dt + I

(15)

A P "v2

From (5), (8), (l^) and (15) we may compute -^— for k < —- .

In the general development f-j-j was found in the limit as

k -»oo. This is the case where the unit vector becomes negligible

compared to the Rayleigh distributed vector. Then (l) becomes

I

A P

simply the Rayleigh pdf . To complete the evaluation of lim -=jk-^00

ve find P

s = \ J

P exp (- \ J

dP = k

2

(l - §

f (A) = 2

^

exp (.- ±r ),

18

and F„(p) = \ I exP (" ~2 )

dP = 1 - -

e (17)

Thus from (5), (l6) and (17), lim A P

k—> 00

A P /A A\

The table below gives values of -=- and (—r— ) for the

Rayleigh limit (k->«>) and for the Gaussian region (k < —^ ) .

k A P

p m

4~2

10 •153 i010

5

.295 .038

^2

• 359 .059

00 | = .735 .27^

From these points, the curve shown in figure k has been drawn, using

the fact that there should be a smooth transition from the Rayleigh

f

2

2

: r v2 n/2 point to the Gaussian region. The points for k = v/2, —-, and —

have been placed on this curve at the proper value of (—7— ) A

computed from (10)

19






AUG I960