BSTJ 10: 3. July 1931: The Statistical Energy-Frequency Spectrum of Random Disturbances. (Carson, John R.)

3.1 MB·English

‘The Statistical Energy-Frequency Spectrum of
Random Disturbances,
Fy JONTE R. CARSON

Den eae eee at ey ts ath sami
Conte ac ovis donor heap agsttend sae!

N a yiner ented "Selective Circuits ant Siac Taterference”™
(ST Ty Apel, 1025) the weter dscns the "ene
frequency spectrom” hereinafter precively ened) of feregular
random dieturbances extending over slong interval of time, In
ew of our lace of even slalites! infant engasting sate oF
Aisrogphere uae the sperifeation of tie eneruserequency
omctrm, cenoted by Ral, wye reseeesvily qvalitative, ard ft wae

snecely postulated tha
(a) is a coatiauots finite fuzction of w which converges to ze:0
at infty and is everywhere postive, It poieeace no sip saan
Sr eminima and its variation with respect «> sl ~ 2ef), wheee it
fait, ia elatively Sow.

Ti & paper entitled "Tne ‘Theory of the Setnoteteke!" 1. C, Kay
deals wish'a sims problem, rene y, the energy or “nose” aborted
ina vacutm zube from a stra of eecerane ith randem time die.
Iihution. The method of attack ie widely diffrent from that ofthe
present paper. Iie move recent paper on "The Analysis of Teele
Motions with Applications to the Lnergy-Lvequency Spsctruct of
Suatic and of ‘Teegrapa Signals” (Phi. Magy Jan., 1999), CW
Kenrick. by making certain bypatheses regarding the wave-fori of
the elemencary disturbeaces whose aggregate is suppsced 20 represent
tent tnterferene, and by suing fealty analy, arse mt
suet fora for the "atalnical” or" expects” value of (4)
fora mame of efereat cos,

1

In the present nape the statics” or Nexperted”” eneray-Geguency
spectrums A) of ramlom dlisiirinces is invectgnced by a metho
‘hich is believed le surnehat mone general and direct than tial nf
Kenrick: ‘The results are anglivalte to the Schreck, tlegeapl,
STATISTICAL ENEREY-FREQUENCY SPECIAL 378

siensis and similar Qatuctunces. Tl writes, however, corchedes that
Uhr applicction to Maaie” oF Malawephere distwrbances it of
“oven onable vale owing potay toot lack of the necrvary sarstical
informition epacding sch disturbances and als tthe fact that they
feznnor be expected 19 have the "usa-ayrtenilic™ charazceristies
rresssary t the apoicaion of peebability theory

“Tae etegytrequecsy spertnam nf divsrbance, 26 the concept
fee enpleyed, wil now De ened, ee disturbance 440) exis in
the epoca O = ¢= T'and lot

Ha) = Cha 1 Sa

eee

“Then, a8 chown ia my paper eeerred to above,

1 [recipe = [

“The energy frequency spectrum ie defined by the equation

eo that

Its on this last equation thatthe physfea application af che conceyt
of the evergy featency spectrum rests; namely shat ct detrei
the menscuare value ef (7, a> the ejuch Ts exule irtenitely gre
Iie pricipal applestion in cleccotechaica depen urn Ue farther
face that, 1) represents an electromotive foree applied to 2 net
‘work of fmpedance Z(inl, the mean square cusient J aloceal iy th
neswork is given by ®

{We nove auppise that she function or eistuanee
cf a number Auf eemeacary dieturlsies; the

2. mend nor ml! armas gies she mean oversized, See my
aed SIE De ha ror
the pith elementary disturbance Seng sinner zero wll

we now write

Sali) = Ga + fhe -f gale, o
ik i easy to show hy che methods employe! iw my previous paper
that

[meant = Sat fatien|”

+25" F entalents + S48) 008 al

=n @

PARTS asec

This i moce compactly expresible a5

Latte Latin)

y

42 tonoati fa— Sah an (60)

Now, obwiomly. if dhe amplitues es, «++ 04 and the wave form of
the elementary functions gy. --+, ds are specihe, Glu) ix wiiquely
dfs wd determined by the. preseding forma, This, however, ix
fot the race in che prislem wader consideration, where at beac the
factions are speifial oly tatieically by probability considerations,
Unler such clevsmtauses, when the probton ia cosrectly sot and
Siulicent etotisecal information is forashed for its solution, we
inteoduce the oa of the srt! energy frequency speetcien is)
Adatined ns follows:

‘Te htsial energy frequency spestrone (ol paul to he woe
erage nf a) for il sie salues of Gah, the aighng begs
fuconlace withthe fralubiity of the ocuerence af each parscwar
post ite

For exarape, the statistical value af @ fsnction lsu 27s 30
were the voriiles ay -=-y 4 ate dened only by prabaiciny eon
Tidcracons is, in accordanot wth ce foregoing defston,

dsgpalenlilan sa 8d

shore Patni hs he peobaiity thay ty Ties Letwecn ag ad
wa tae
STATIsTIvas EvENeL-ARSpUENEY SPE

“To analy the firegsing socupt an cbicion of the statistical
value ef a function 19 ae prabiem a ad tis necessary to suppose
tha the tye! ine ee Fanci of waned eerale pevermeters
Dade and chat these parameters ae statistically spiel iy
probably considerations. Thus we suppose thal fu(hyDRy ix the
Probability tha: by ies Hetvecr hy abd Ny hah fa) ll he be
2 furetion of wand %y, des ++, Xp the amplitues a, ey being
egarced as pacameters, when Celia Lay pratab iy fawesons, We
then have, fa accordance wilh Use Tureen

xpos [ena

wf

Xo [panera hn

u
Toyo the amzing che sinplest pose cave let ws supose
hac the eleiestary plas sr all sens —mg eae
She thet teh saben in tine w pusely soon Wh ewe

Eton allows eter ou thn
rus = Eyanfbae Sao ORE re ag)
Ru be Pane se. (8

160) #0, hi hoe a digslty a6 = Oshewever

Lin 1 [we = [Rede

woes ”

ao Powel for]
Fowl ula ein nt ren with Fry ear
re Schroneet)

To eo ect se a eb, we ll spe
igus erating he nlite the aman
ad ronda" fer deo he pry that the
Chr ny peace random, sec Nad
WE by anda Conon tue dees te pbsy
‘hes tes taser oo Se "Porshe he

"Ke tow yt tnt eo Stn eo mbes
the snes fom

wo = Ee emg 60

all) = 40), OE he
arerere
feed bh te

and we denote the meaa frequency of occurrence, 27/7, b
‘Sulsitstion inthe prenesing Corman ail straightfor

i per
Roa = 2 oaisids [Ikea FeO
a eet prt

[Cow] [Cada nenoeeas [I saapoa,

xtigh £,co[fomeal oo

Heme wie

[C posen = aia ap

swe have by etraehtfonvard procedure

Bene foment

Ria) E [" entate [ie orcas

20 (* ayanaa | «| Sts}
-2Lf ateide || ES) at GD
where :
dots = oka) 0,4
Go, noida oy

fa, poe.
STATISTICAL ENERGY FREQUENCY SPECTRUM m9

1, oe the eee ad we spoon that he eben Fondo!
syataealy acca i spar uly ely he de
‘au the lle sutmaton ses 9 sales snd

[eva

raat [eden [one as

‘This follows feom che fact that the amplitude a is equally ely
be postive or negative. Consequently the integration with respect,
Avo mit he ac enfe fun on Em ame sare by hypottesis
2-9) ~ fo} i flow tat

Proves

“Topol the noeceding formulas to actual calculations, ie neces
tevlenow the foeton fm, 2) and in ardion the probability funetions
involved, ‘These later may be supposed knuwa trom statistical data
or caleule on theories! asumpious. For exasspl, if we asa
that the Lines of iacideace of dhe cementaiy disturlances are dis
tribuced entively a random, tke application of well-krown probability
thesry ves HO) =o

A thinl ease is of interest. ene, insted of poetlating chat he
lecrivasion of one sie evinces wits the ata of the next Ge,
eee ~ fe heh we snppowe thae the cimes of icidence are entice
related, and that the amples are en Fly to be pitve
fe negative, Bor this cose the form's for Rio) ie formally identical,
wi she

Mm

‘The Zones amas wl now be applied eo deriving what eepre-
sons more of less accurately the static energy frequeney spectrin
of telegraph signals, To thisea! we sil suppose tha the elessentary
‘fsturbanee may have any ene of “hues posite values fall equally
Urubable), charaeteraed by durations Ay sy Oy and amplitudes
i diy ‘The comesponding spectra af the clemeatary disturbanoe
are then deternized by the eqwatioss,

sites — [scree
0 BeLL SYSTEM TSCHRICAL s01RNAT
he applicetion of the preci aatsi co this cae gives

PF ae ai! Hot LEHI

Eeela’
=

plus the mul fer of

Be ouhitinie® + axfiianer® + estinne™)

xehinie tan Ww tmA— we) OD
stig BEE ie bee bee

1u sto he anderson thatthe eta ofthe eon rer ie lone

Insets

aH beh 1 AN,

togeea et

‘Theae is thereon sn Haficisy at =O. a we shuld exzect, te
measure, however, is Anise

The preceding is merely a exanple which adiuits of extension tn
more vimplvaied types of fal, a Wil, be cavis te the reader.
‘Por example, the prububilitie of the lementary xgvals wel ot Be
‘he sane acd thelr sumer end not be restricted! to

W

Ta all the cows dissed above ie wil hr wierved that. che di
uibusce is uuusbeystematin” in the sense that the ckenentary
cures ave all of the me waveform dering only in duration
nl smpitude. ndee’ seme ac eauraptions a8 theee are ecental
fo ahe anplicaion of the matzematcal theory. Tn the e23e af atmos
here disturbancee we have #0 reazon to suppene say ue que
Spatenatie exaricter mint, Puthermere, even if for che sake of
Tryusment, we supposathat the elemetary disturbances, which make
tp state, have & comimon wave fon al the point at which they
mech eBgURNCY spRCTHU 8h

originate, sey would vay ley fw this respect after arriving st a
Common vereives, The writer & therclore of the opinion that the
‘Suotaton fom hie previous per appearing a the start of this atl,
Tepresen ll that ean safely be wid segarcing the spectrum of sade
fre that owe present knowlege ie insuicient to justify the xpplicaion
EC jeohability analysis to the problem. All that we can say is tz
She pect of la} whic conesihures co "etic Saedevence" ie simply

sort eine from {8} and ia agreement wits the co3e"sson of
Ingraiginal paper leh Tex hesesuppose thse te times of incidence
sre lstoated at room. This forala, however, supplies no used
information ia the absence of data regurding the wave forms and
niplitides of the india «sta baazes.