JOURNAL OF RESEARCH of the National Bureau of Standards-D. Radio Progapation Vol. 66D, No.6, November-December 1962

Conversion of the Amplitude-Probability Distribution Func­tion for Atmospheric Radio Noise From One Bandwidth to Another

A. D. Spaulding, C. 1. Roubique, and W. Q. Crichlow

Contribution From Central Radio Propagation Laboratory, National Bureau of Standards, Boulder, Colo .

(R ece ived M ay 10, 1962; Revised June 6, 1962)

The a mplit ude-proba bility d istr ibut ion func t ion of atmospheric radio noise can be predicted with reasona ble accuracy for a give n ba ndwid t h us ing on ly t he first two moments of the noise measured at t hat bandwid t h. Thi s pa pcr prese nts a method for predicting t his dist ribu t ion func t ion for any specified ba nd wid t h from t he mome nts of t he noi se measured at a pa rti cula r ba ndwid t h.

1. Introduction

Atmospll eric radio noise is often the limi ting factor in radio communications at frequencies up to about 30 Mc/s. In the design of communications circuits, it is necessary to know the detailed statistical characteristics of interfering noise in order to cal­culate the channel capacity or error rate for a noisy channel. The cumulative amplitude-probability distribution (APD) has been found to be a useful tool in such analyses [Montgomery, 1954; Watt , et a1., ] 958 ; Crichlow and Disney, to be published].

Atmosph eric radio noise is a nonstationary ran­dom process whose characteristics change not only with tim e but also with bandwidth. Th e APD is usually measured as it time sequen ce of several simultaneous levels, ~tJld the necessarily long integm­tion tin'les mn,ke difficult the obLaining of ~1, contin­uous cmve. The need to overcome th is difficulty has led to th e develop men t of a method o( predicting the APD from the three statistical mom ents o( atmospheric radio noise measured by the ARN- 2 noise recorder of the National Bureau of Standards [Crichlow et aI. , 1960a; Crichlow et 11,1., 1960b ; Fulton, 1961]. These moments are measured on a world­wide basis [Crichlow, 1957], and the data 1'rom these measurements are summarized and published quar­terly [Crichlow, Disney, and Jenkins, 1957- 1961].

The three moments are measured for a power band­width of about 200 cis, so the APD derived from them is valid only for this bandwidth. For this reason, a method for converting the APD of atmos­pheric radio noise at a 200 cis bandwidth to a range of other bandwidths was developed.

2 . Parameter Definitions and Basic Assumptions

It has been shown that the distribution Junction for atmospheric radio noise can be determined from its following three statistical moments [Crichlow

648645- 62--6

et aI. , 1960a]:

rms voltage: l'rms

~t\Tenl,ge vol t,age : Vav•

=(f;~ f~~ v2dp(j , t) } /2

f ~~ J~: vdp(j, t)

log of the voltage: log V10g f ~:J ~~ log vdp(j, t )

where v is 1/-/2 times the instantaneous envelope voltage, p is the probability of v being exceeded, .f2-.iI = b is the ba ndwidth, and t2 - t1 is the time intenTal of measurement .

The three moments as measured are expre ed, respectively as:

Fa = the effective noise figure = the external noise power available (rom an

equi valen t , short, lossless, vertical an ten na in decibels above ktb (the thermal noise power in a passive resistan ce at room temperature, t, in a bandwidth, b; where k is Boltzmans' constant).

V d=vav e in decibels below vrms.

Ld=v ,og in decibles below vrms.

Fa is independent of bandwidth for a uniform spectrum (a condition closely approximated in normal communications bandwidths) , but Vrms varies as the square root of bandwidth. V d and La are always positive, since for this type of distribution fWlction, vnns>Vave>V IOg. Because the shape of these distribution curves is dependent only on Va and La*, which have been normalized to Fa, it is possible to construct distribution curves o( a given form factor for various combinations of V d and L a.

Measurements of V el and Ld tend to show that La is a linear function of V d • Figure 1 shows La versus Va for various bandwidths, seasons, and time

'Unless otherwise stated , the moments lIsed in t he an alysis are assumed to be the true moments of the distribution fUll etion.

713

.D

19

18

17

16

15

14

13

12

I

"0 10 z ~9 --'

I

• li 1170 cIs . • 200 cis BANDWIDTH 1 1 70CI~

- X 10 cis BANDwIDTH • !/ . ® AS LABELED

~ 170 cIs

~ . liT" . '7 cIs

• . ~ .j

X.~ 1170c/5

,-.; X 170 C/5 . ~' 70 C/S :z ., ~950C/S

I .f~ 950 cis iI

/ 6 cIs

;,: 0.64 1C/5

10 12 13

FIG U RE 1. L d versus V d from measured distributions.

14

of day from measured distributions. The relation La= 1.69 Va+ 0.72 seems to hold in gerenal. Because of the correlation between V a and La, one can deter­mine the most probable APD's of atmospheric rftdio noise for the range of V a normally encountered. A family of these is presented as figures 2 and 3.

The APD of atmospheric radio noise can be closely represented by a three-section curve on Rayleigh graph paper, i.e., two intersecting straight lines joined tangentially by a circular arc, figure 4. This particular Rayleigh graph paper has scales chosen so that a Rayleigh distribution function plots as a straight line of slope - h. These scales are labeled noise level in decibles above Vrrn s versus the percentage of time that each level is exceeded.

The lower section of the curve (low levels exceeded with high probabilities) represents the part of the noise composed of many random overlapping pulses, and plots as a straight line Rayleigh distribution [Crichlow et al., 1960aJ. The upper straight line section of slope less than - h (high levels exceeded with low probabilities) represents the part of the noise composed of large, infrequent nonoverlapping pulses. If the exponent in the expression for the Rayleigh distribution function is raised to the power - ~f8, where 8 is the slope of this upper straight line, an expression of the upper portion of the APD will be obtained. For this reason, the upper portion is sometimes called a power Rayleigh.

Four parameters are used to define the distTi­bution as shown in figure 4. A is the decibel dif-

90

80 I I

..::::" 7 11 . ~

6 II :::"

~ 9 c:::, "' 4 7.~

=-- "' J 6

i'- r--..... 1 4

95 98 99

PERCENTAGE OF TIME LEVEL IS EXCEEDED

FIG URE 2. 111 ost probable amplitl,de-probability distribl,tion f unctions for atmospheric radio noise.

ference between the Vrrn s level and the Rayleigh line at 0.5 probability, and determines the amplitude of the Rayleigh section of the distribution. C and X fix the amplitude and slope of the power Rayleigh. C is the decibel difference between the Rayleigh line and the power Rayleigh line at 0.01 probability and X is the ratio of the slope of the power Rayleigh relative to the slope of the Rayleigh, that is, X = - 28. The parameter B describes the circular arc and is defined as the decibel difference between the inter­section of the two straight lines and the line tangent to the circular arc at its center. For atmospheric radio noise, an experimental correlation between B and X has been found, with B = 1.5 (X- I). These parameters have been determined as functions of Va and La.

The following assumptions are used to determine the relationship between the APD 's for a sample of atmospheric radio noise received through different bandwidths. Each of these will be discussed in detail as they arise.

1. The shape of the distribution for the prob­abilities of interest (10- 6 and greater) will be of the above form for any bandwidth considered.

2. The rms value of the distribution will vary as the square root of 'the -power bandwidth, increasing with increasing bandwidth.

714

PERCENTAGE OF TIME LEVEL IS EXCEEDED

w

~ ...J o > (J)

40r-r--,---,-----,---r-----,----------~

x • ABSOLUTE VALUE OF SLOPE OF POWER RAYLEIGH LINE RELATIVE TO RAYLEIGH SLOPE

B·1.5(X-I)

LINEAR By-Iog(-Inp)

20~~-4---+----_+--~~---+~~_+----~

~ OI-:--t---t---\'~----:4---+----_j.----------~ w > o ro « -D CJ

-20 ~+--_t---+_\--_+---+--~.._+----------_t

-40 L--.l.----lL..---'-____ ---1 __ --L ____ --'-__________ ---l

.0001 .01 10 50 10 90 99

FIG URE 4 .

PERCENTAGE OF TIME ORDINATE IS EXCEEDED

Definition of parameters fOT amplitt.de-pTObabili ty distl'ibution 101' atmospheric radio noise.

I FIGU RE 3. Most pTobable amplitllde-pTobability distTibt.tion Junctions JOT atmospheric radio noise.

3. As the bandwidth decreases and becomes quite small, the APD approaches a Rayleigh distri bution .

Since with the above assump tions the cumulative 1 distribution is de~ermin.ed by its moments (Va a.nd I L a), the mann er 1Il whIch the moments vary wIth

bandwidth needs to be determined. With Vdl and ; L a[ designating the mom ents of the original APD at I bandwidth bl , and V d2 and L'l2 designa ting the mo-

ments of the desired distribution at bandwidth b2 :

, 1. X 2 and C2 must be determined as fun ctions of : Xl, CI , and w, where w= b2/bl and,

I· ,2. V d2 and L d2 I,nus t be determined as functions

of Val' L al' and w from (1).

3. Transformation of the Power Rayleigh Section of the Distribution

The high-amplitude, low-probability section of the APD represents a train of nonoverlapping pulses. If the response of a receiver with bandwidth bl to one of these pulses is a pulse of amplitude a and time duration t, and then t he bandwidth is changed to b2 ,

the response will be ft pulse of amplitude wa and time duration t/w, For this section of the APD Fulton [1961] has shown that every point (p, v) corresponding to a bandwid th bl transforms to the point (p /w, wv) for a bandwidth b2 , as long as the

715

.n 'tJ

100

80

, 60 w o ::> ~

--' a.. ::;; « w > ~ 40 --' w a:

20

o

1\ ~"

1\ ~

1\ ~ 1\\ X,

'f\') \ VI'OI~ K

X,

.0001 .01 10

b, w : -

b ,

"'2 = v, w

V2. = V, + 20 log w

P _ EL , - VI

P, ALWAYS 0.01

LINEA R BY- log (-Inp)

~

50 10 90 99

PERCENTAGE OF TIME ORDINATE IS EXCEEDED (P )

F IG UR E 5. Transfonnation of a power Rayleigh line ,

I'eceiver response maintains the criterion of nonover­lappin g pulses. However , if the transform ation (p, v) --7 (plw, wv) is m ade point by point, the results will no t be another straight line, but a curve. As shown by the dashed lines in figure 5, the curvature is very slight in the probability range for which the power R ayleigh holds, because in the r egion in which the curvature becomes objectionable the criterion of nonoverlapping pulses is no lon ger met.

Since it is des ired to portray the transformed dis­tribution with the same forl11 factor as the original, the curve will be approximated by an appropriate tangent to the curve. The slope and point of tan­gency of this line will be determin ed as a function of the bandwidth ratio w, so that a suitable approxi­mation is obtained; and a reciprocal relationship resul ts, i.e., tr ansformin g from b1 to b2, and then from bz to bl , the original distribution is r egained.

The equation of a cumulative R ayleigh distribu­tion is:

or 11-

loa' V= - - [-loa' (- lnp) 'I+ - loo' v2 to 2 b 2 to ,

where v is Ih/2times the instantaneous envelope volt­age, p is the probability of v being exceeded, and v-::. is the mean square voltage.

Since the slope of the power Rayleigh line is - ~, the equation of the distribution function for the power Rayleigh will be

I x [ 1 ( 1 )] 11 2" oo'V= - - - 00' - np +-2' og Vr , h 2 ' '"'

where v/ is the mean square voltage of t he corre­sponding Rayleigh distribution,

or

p = exp [-(:r2zlJ. Transforming the power Rayleigh by the band­

width rat io w,

VZ X [ lJ wp = exp - ( wzvrz) ,

X l-or log v=2 log (-1 nwp) +2" log w2v/ .

Putting the above transformed curve in terms of our coordinates with

or

and

x=-log (- lnp)

p = exp [-10- "],

y = log v,

1.

1.

1

I

0

9

8

7 ><1 '-1.6 ,f 15

14

1.3

12

I

I

f-I /

v-

----15 1 10 15 10 30 50 70 100 BANDWIDTH FACTOR w

FIGURE 6. Ratio of X 2 to Xl versus bandwidth factoT w.

it is seen that

dy _ X ( .. -::-_IIlL) . dx - 2 lnp+ lnw

If 8 1 (81 = - X) /2) is the slope of the original power Rayleigh and 8z(8z= dylclx) is the slope of the result-ing power R ayleigh , I

X z Inp X 1= lnpw'

The point for the best tangent approximation will change with each transformation w. The point of tangency must be chosen such that t he above men­tioned reciprocal relationship is maintained, and such that the transi"ormed curve is well approxi­mated by its tangent in the interval of interest. This tangent point for each transformation was de­termined to be p~O.OOOl for w>l and bl = 200 cis. The above requirements will be met if the tangent point is chosen at p = O.OOOl for w> l, and the ap­propriate value for w< l to maintain t he reciprocal relation .

In order to transform the power Rayleigh, t hen , t he point (O.Olw%, v) is translated to the point (0.01 %, wv) and through this point the desired power Rayleigh with a slope given by figure 6 is drawn (w> 1). Figure 6 shows the ratio of X 2 and XI as a function of w.

4. Transformation of the Remaining Section of the Distribution

Having obtained X 2 as a function of XI and w, I

G2 must be determined as a function of GI , Xl , and w. This is obtained by choosing various Xl and GI

for a particular w, transforming the power Rayleigh portion according to the above, and then determining G2 (that is , locate the Rayleigh portion of the desired distribution) such that the Vrms values of the original distribution and the desired distribution are related by the ~w. An example of the proced ure is given in figures 7 and 8. In figure 7, t he distribution to be

716

tra.ns[ormed is chosen (XI = 6, CI = 10 db), and from t he known relationship between A, C, and X (fig . 9), A I is seen to be 9.9 db and t he relative Vrms level is arb itrarily set at the 20 db level.

Transformin g by the band width factor 20 , the point (PI, VI) is transformed to the point (P2 , V2 ),

where P z is 0.01 percent , P I is 20 (0.01 %) or 0.2 per­cent , VI is 36.5 db , and V z is VI + 20 log 20 or 62.5 db. X 2 is found to be 8.88 from figure 6 and the power Rayleigh section is transformed. Th e Vrm s

level of the desired distribution transforms as the -V20 or 13 db, so Vrm s2 will be 33 db. Using figure 9 and choosing various C2 , the correct C2 is found to be 9.6 db (fig. 8). Figure 10 shows C2 as a fun ction of CI and XI for a bandwidth factor of 0.5.

From previous worl.;: [Crichlow et aI. , 1960a] and t he above, the following fun ctional relat ions are now known :

These may be manipulated graphically to obtain Vdz as a function of V dl and w. The results are shown in figures 11 an d 12.

In order to detennine the shape and amplitude of the APD for atmospheric rad io noise at bandwidth

80

.D -0

~

-;'

I I

I VR MS

1\ --' ~ vw vR M S I

V RMSZ = VRMS1 + 10 log w

LI NEAR By-log (- Inp)

LL 0

60 {Pz,Vz l

w 0 ::J r-

i\ :J Q.

Z <!

w > ;:: 40 <! -' W ex

.D -0

~ 40

\' ~ B.88

~ .........

l-+-VRMS,

~ r ....

I ....

1\ A, I' ..... ,

I'-- ~

:> LL 0

W 0 ::J r-:J Q.

20 Z <!

w > ;:: <! -' w a:

(PI ,VI

\ r....i'- r-- .........

v-- .......... .........

............... ......... Cz=O

I' ~ bo-.............

1\ ...... r---........... C, ~9.6

'-... ~ :~

1'- ..... v .......... ......... C2 "20 ......... ...............

XI" 6 W ~ I ..........

~ I o .0001 .01 10 50 70 90 99

PERCENTAGE OF TIME ORDIN ATE IS EXCEEDED

F IGURE 7. The determination of C z as a flmction of C I, XI, and w.

717

b2 from Vrm si and Val measured at bandwidth bl , then, V d2 is obtained from V al and w from figures 11 or 12, and the shape of the APD for this V dz is found from figures 2 and 3. The amplitude of theAPD at band­width b2 is determin ed [rom the rrlat ion Vrm so =

Vrmsl + 10 log w. -

37

36 "" '" N (J)

::. 35 a: "'" > i',. lL 0 ..J 34 w > W ..J 33 w > f: « 32 ..J w

'" ................

i"---I~I'-....

....... r-- --I--0::

31 I

I ! 30

o 16 18 12 14

F I G URE 8. R elalive V"n s2 level versus C z for /igme 5 (:'( 2= 8.88) .

11

E 16 :> :;: '3 10 uJ m .t:l u

« 14

18

31

36

40 o

"" ! ~

_Z -

~F~ I

'-,.1 " I I

~ '\

~-b j ~ ~.~ ",-"'-

IV '" '" ''\j.

I , Ie ~ol = d"- '" rn -[!I -

I f":-' rn = 1]] =

0Ql = I I @J =

I Oil = 1\ [j§] -

CIill = ~ @= '\ '\

+ ~ = - '\ ~

I 00'\ ~ ~

10 12 14

x

FIGU RE 9. A as a function of C and X.

5 . Accuracy Considerations

As the bandwidth is increased indefinitely, the above analysis becomes inaccurate since the tangent approximation is no longer sufficient. However, since Va increases with bandwidth, and for large V a the integrals defining Va are determined almost com­pletely by the power Rayleigh portion of the cumula­tive distribution; the behavior of V a for large band-

18 18

16

16 14

12

14

10

12

8.

10 ..0 U. 6

N U

4

o

°0~~~---L~--~----~----ILO----~12--~14

FIGURE 10. C 2 as a f1tnction of C l and Xl for w= 0.5.

20

18

16

14

12

l width ratios may be determined by investigating the l beh~vior of the power R~yle~gh. I

Smce the power RaylClgh lS composed of nonover-i lapping pulses, th e character of the repsonse of a l bandwidth -limited circuit to this train of pulses can l be determined from the response to an individual I pulse. Therefore, the average vA,lu e is independentl of bandwidth, and t he rms value varies as the square l root of the bandwidth ratio. Va (t,he deviation in db between the average and rms values) will vary as the square root of the bandwidth ratio , or V a will increase 10 db per decade of bandwidth ratio w. The above may also be shown by actual integration of th e !' power Rayleigh and the transforrned power Rayleigh . .

The above also gives a good eheck on the accuracy of the p.revious analysis. The results must approach /' this 10 db per decade law as w increases and also as Val increases. Figure 12 shows that this is indeed I the case.

For a bandwidth of 200 cis, it has been found i experimentally that the distribution saturates at a \ probability of 10- 6, i .e., there will be almost no pulses I with amplitude greater than that for which the i probability of being exceeded is 10- 6• This is not ) true for other bandwidths. In general , the distribu- ' tion will saturate at slightly lower probabilities for larger bandwidths and slightly higher probabilities for smaller bandwidths. By assuming the same form of the distribution for all bandwidths (saturation at p=10-6) , an errol' is introduced. This error is : indicated by the shaded portions of figure 13. 1

An indication of the maximum error in Va can be obtained by evaluating the average and rms values , for a 1?ower R3:yleigh in the probability ranges [O'. ~] i (that lS, assummg no saturatIOn) and [10- 6,1]. Thls \ will determine the maximum possible error in the average value and the rms value. Since these two l errors will be in the same direction , the difference between them (in db) will be the error in Va. )

VV

/' / vI."

V /' / vV :~

./

V V VV 4;- ~ :/ l/

//' l/ ):/ l/ V VV I ....

V~ V/ V V ~ /

V" I/V V~ 1/ V

v p 1/1--1--V ~~ ~ V ~ ~ / V V / ...... k' I--

~ o 0.001 0.002

... ~

0.005

---=::::~VV ~ I--- ~ I:::: _~ I-- I--- I--- ::,.... r-_ ~ i ~~~ ;:..-~ E

0.01 0.02 0.05 0.1 0.2

w = b2 / bl

FIGURE 11. Effect of bandwidth on Vd •

718

~ ~ ~ V ~ ~ ~ l-- l---- j...

F ~I~ ~ [;.....- I- ~

." :3

1.5 f:O'IS

0.5 1.0

L./ V v- 1// V V v Vv V/ / v ./ ./ V

V V V/ V ~ V V v- V/ / v V , ,,,0/ V V- ./ ./

-.\b' V IV V/ V V IV vV- V/ 17 ./V ,,<> V / / ,,, V V- / V/V/'V t}-V V- V/ ~ V V LY V/ l/ VV

V/ V V V PI'

,~ ~/ V/ V LY V./ V V /' /

V/ V V V ~

1V/ / 17 V V- '<> / V LY / ./ / "" V/' V./ V; V V lY'./ /

,/ V / VV v

28

26

24

22

20

18

16

// ~ V V Vv !-,Vy .,y /' V V / /'

V/ v./ V V v-"" V/ .,/ vV- ",V ./

~ V V ~V ./ IV // V V V- V

~ V V V ~.Y V~ Y ..J~ V V V V ./

12

10

~ 8 ~ V f- --~ l----l--- \.5

~ t::== ~ e- ,...- - I-----1.049

o I I I I 10 20 50 100 200 500 1000

W' b2 / b l

FIGURE 12. EjIeet of bandwidth on V d.

110 LIN EAR BY -lo g (-Inp)

~@~

90 1\

80

1\ .D

w:s 2: ~ 70

\ f--:o <!f-- 1\ w ) 1 --' -we[ ~ 0::;: 60

<! \ \ W~~ 50 r\

\ \ 1\ "" 40

\ w 'l .~ 1\ '" 200 CPS ~

30 "-r-,w (I

~ ~

"" i'--10 i"--

~I'--i'--

0 ~ 0.0001 0.1 I 10 10 40 60 80 90 95 98 99

PERCENTAGE OF TIME LEVE L IS EXCEEDED

FIGURE 13. Saturation of amplitude-pTobabili ty distributions for various bandwidths.

'Ve must evaluate

where

p = exp [ -Cr22)~J and PI assumes the two values 0 and 10- 6• Under the change of variable

For PI = 0, then

For PI = 10- 6, Vav e cannot be easily evaluated in closed form in terms of X. If X is chosen corresponding to the power Rayleigh with the largest slope likely to be encountered, Vav e can be evaluated. Choosing X = 12 or a slope of - 6,

r = [_~ 2 - z ~ 720 z"J 13 . 82 ,Vavc Vr e L..J , n=O n. 0

Vavc=708.51~V/ .

719

For the probability interval ro,1], and X = 12,

Vave=~vT2 r(7) = 720 ~V/' Therefore, the maximum relative error in 'Dave is 0.13 db.

In similar fashion, for the probability range [0, 1],

v;ms=~ r (X + 1)

vrrn s= 2.19 X I04 ~V2, . For the probability range [10 - 6,1], and X= 12

v2 = -v2e-Z L::: -=.:. Z n [ - 12 1 ~1 J 13.82

rros T n=O n! 0

Vrms = 1.726 X 104 .J~. The maximum relative error in Drms will , therefore, be about 2.07 db and the maximum possible error in Va somewhat less than 2 db. The maximum error in La will be about the same, since the error in V IOg will be even less than that in Vave·

In practice, the error introduced will be much smaller than the 2 db, as it was assumed that the distribution never saturates in calculating the 2 db. Even so, an error of 2 db in Va for large Va will not noticeably change the shape of the cumulative distribution, but only the relative amplitude of' the distribution.

Since the amplitude of the distribution (vrms) is measured experimentally , we are only interested in the effect of the error on the shape of the distribu­tion; therefore, the assumption of a uniform poin t of saturation for all bandwidths is a valid one, since the shape of the distribution for the probabilities of interest will not be changed .

In using the above, it should be remembered that the transformation depends on La being a linear function of V a. Figure 1 shows this is generally correct, however, there is some variation of La for a given Vd, and this variation increases as Va in­creases. That is, in any particular case, it is pos­sible for the true Lal and the assumed true La[ (from fig. 1) to be significantly different, especially for large V dl • Also, V al itself is subject to error in measurement, and this error will be propagated by the transformation. The possible error in Vdz for a given V d l subject to error can be quickly deter­mined from the transformation curves. For ex­ample, suppose Vd] was measured to be 5 db subject to a possible errol' of' ±% db (4.5 < V II < 5.5) and it is desired to transform by W = 100. Val of 5 transforms to VdZ of 22.7 , 11a ] of 4.5 goes to V d2

of 21.7 , and Val of 5.5 to V d2 of 23.7. In this case, then, the ± % db error in measurement has trans­formed to a possible ± 1 db error in Vd Z (21.7 < Vd Z

< 23.7) . On the other hand, if we transform the above Vdl by w = O.OI, the ±?f db error is seen to transform to a possible ± 0.07 db error in Vd2 • For

this reason, then, each transformation should be in­vestigated as to the effect of measurement error in Va] on Va2 and the effect of this possible error in Vd2 on the shape of the resulting APD.

8 . Conclusions

Since measurements of the APD for atmospheric radio noise have shown the assumed form factor to hold over a wide range of bandwidths [Watt and Maxwell, 1957 ; Watt et al., 1958 ; Crichlow et al., 1960a], the above transformation will usually o·ive good results. However, in using· the above, it shguld be remembered that an error in measurement of Vdl will generally propagate as a larger error in Vd z as the bandwidth is increased and a smaller error in VdZ as the bandwidth is decreased. Also the linear relation between Va and Ld is only a a·ood a'ppro~imation and subject to possible error, ~spe­clally for large V«(. If a Val and Ld[ are encountered significan tly different horn the pairs corresponding' to the APD's of figures 2 and 3 the APD for this pair of Vd] and Ld] may be obtained from NBS Monograph 23 [Crichlow et al., 1960b]. This APD may then be transformed by the method outlined above.

. E;xperimental veri.fication of this method of pre­dlCtlllg the changes III Vd with bandwidth has been obtain~d, but with some doubts as to accuracy since the nOlse changed character in the period covering successive ba.ndwidth measurements.

An improved method of measurement is currently under development using simultaneous magnetic tape recordlllgs of the noise through different bandwidths. Results of the new measurements will be published upon completion.

7 . References Crichlow, W . Q., Noise investigat ion at VLF by the National

Bureau of Standards, Proc. IRE 45, 778 (1957) . Crichlow, VIr. Q., ll. T . Disney, and M. A. J enkins Quarterly

radio noise data, NBS T ech. Notes 18 thro'ugh 18- 11 (1957- 1961) .

Cri chlow, W . Q., C. J .. Roubique, A. D. Spaulding, and w.. M. Beery , Detenmnation of the a mplitude-probability dlstnbutlOn of atmospheric radio noise from statistical moments, J . Research NBS 64D (Radio Prop.) No.1 , 49 (J 960a).

Crichlow, 'IV. Q., A. D. Spaulding, C. J . Roubique, and R. T. Disney, Amplitude-probability distributions for at­mospheric radio noise, NBS Mono. 23 (1960b).

Crichlow, W. Q., and R. T. Disney, Predictions of radio Hoise a nd systems applications, (to be published).

Fulton, F. F., Jr. , E ffect of receive r bandwidth on the am­plitude di st ribu tion of VLF atmos pheric noise, J. R esearch NBS 65D (Radio Prop. ) No. 3, 299- 304 (1961).

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(Paper 66D6- 231 )

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