BSTJ 20: 2. April 1941: Steady State Delay as Related to Aperiodic Signals. (Hartley, R.V.L.)

4.9 MB·English

Steady State Delay as Related to Aperiodic Signals

the consents of phase al ervlone delay, 2s apie to ary Hoe
{Gripen oem seen: of he cen, The aga spt.
Ur Gaesal ses the in of ety and spec es Noles Use
tie arcs ofthe sston ive ths the of ex

The amplitudes conan: over Lae sagan, eis bow tht bes
sheers he Uo af eatin eave a Ue dia the
sec dl Meal wh the ensope dds When i tbe
{ie msn abi at, the Gla eno om eign sve
{tum the phase sit rhe ser set teeta el, to
pines fey

ite apd vss rapidly wih sone, the compan of
spc datubence whi coestnds to natow suet the
‘run pera o lng tha the suing over lapping lglg

ipoes faut eit nerpreation cl

THE exis aplcalons of stay stte theory to Wsaswishr pa
leas the emphasis ws paced on the veriation of ampitude wih fe-
tyvony. The ae of hig Headed ie wade it neces oo take acon
pls stort! aswel. With de development of tepastogrply and
eleva, the phase caisctevade wae found to provice wef ade for
dicing theoverapping of sdiecent picture laments. For these purpse:
Fs oun convenient to expeess the pase character in ier
thse oe envelope delay, Thave ay be cll “steady late dey!” since
{hey are defined sod measured ia teas of sinusoidal dsturbanoss of ad
joutle equancs. However, the sguals for wales they aye iatended ue
fuaaish a Sndex ace apeinde innate Ht eeemed worth, theron
toevarine more dene te teem exiting setveon “aperinli eye
Sede i em of seal gnc sal ey late da,
Ee fea evi the developceat of the covcept fees state delay.
ity i the sady ofthe propagsion of snasoidal saves dscinetion nas
mate hetween pe atl frou velacy. Thee oa patil diane
Df Irsnamon the rat of this dstce ta each nf Use two veloc
tay be inept delay ase with the tranisen1D the

REEL Sra
commurieation art, Use deve hove Bota cad pas a envelne lay,
respectively: Iv he nechim eeu dispersion tay vary with ‘reqem

Tel ta four attention on the condtins hougaout he rca m8
parol instn: cuvirg th tenemision ofa euscidel ceachanae We
hey ete tie Ina kaye hase pean fora the put ta the

foulgeL This gy Le ure ans sgl cys Uren oe tide this
hiss eile ty the Frecumey, epeceed “a he seme anuvlar uote, We get
the te hich wil be tect for che phase at che soto pga
tas eueput, of she phase cay. Mio ‘Lay liye ore that the
evivative of ais laze aif. wth repect tv feeuancy iz equa he
cried i eng aloe fa tems of the group veoeity. The
FFrplettraimert of Fic brad nthe eomidenaion ef tio sina
saves of eqn arplnide ae elghly err: sequels

Wl ta elas Ivan inverted oe most medi, ule
aia inthe sate! thie altace lies exe arenes depen,
Her, ip she neighborhood of eet fequensie, the phave sis varies
Bp with requey, a cfter agpens <2 Ye disoutivums, ete ly
che appatent dxcntiny ecg of wry rail deve of eve with
frequeney, mia lena o a negative velue of envio deve, 1a the sme
segio: the uasanssion varies vapid wilh Urasy, aml selesto reo
Hine accu at Lie lari seanear. Tere ean he ep nod ter
cf reonance in te deneata which wake xy the Ene stzccze of the

"The nest slap was to diac the ie of dea fom that of velay

fn taadum, ad aecociate fe wil seals tne Irv on lame
Touwcen cay tke pains ota sa stem, "This we pew app ie
sen to tl ste of canabcies ustvods is hich wnloem propoestion

‘cunot be rewiy visuslise. Mere too types wf rhurasteitie ear
be distoguishse. One, whichis aticatel will ws yt he eae
Segoe ajseme exhibits areas peas verti. gL th pase
shits vl terion wth Frequency This is the ae of © aon
hing nares sire. The ner, which fcemicee with “canst
solamente Ue pheveteny secinee wth ncn dpe,
the cage a ior a ella wae pies Ver ress iv se 2
eit Lise sean ae rete ilance, whic me the ene
‘the tapes of igh absaplicr alse tive rsreetow a the bound
Ulamediin. Trapping be san det in tetres the, ec expect

Uh eae bv ag ils a Halse ee keep eva fom
Ae ert Scie 0 eaomart system

Te camputig or massaceg Le plese aif a x aie, a» dig
qency, che sents ieteriness ao fou ay Uc of vies

Je ieturenrael, ‘THe soos rot aot the envelepe €eay, ith deel
Ee ete svSmeAr BocMPLCAL WENA

nly on the derivate, ands ths typeof dle ova be woneaized dire
(e laclade ste toons charasteragse of ubitory welts. To give ar
fac memng tn pve ely sme convertion wd ave he aczpunt
for decerining lols if uy, alle of 2 tobe added to tho ems
pose lor the Faquaney in question. Ayurely ae such convention box
boon agra yon obich i of genen! wicaon. Rar daaxpod pesos
sich rani Trequenie dowa to ery Lie eustomary > wire ie
phase svi phe ao at ae requerey, wl [ow higher fequences, toa)
ultipls al 2r oo dhat tae pase dis res soinwoaty wt sequen.
Th ten, # is nope phase shit, brave ~ wand e, vo ray repre
tent the onatnaoualy aspng poate sil oy B 2a Whee me i re
tree of iacontauities fF whic lave sce chiminated i pang fon
ero tho tequeney in stan, Tho hewn lay roy he be dined as

_ Btn @

Dy

Any simile sowvmaten fer e2zzat systems sl To less simple, al
‘Snes oy wl snr Ii, paeee dey hes Te Leary om apedidic
‘Sons it seme ure to stsemps to forall such @ ceavention hee
Sy nme! wth steady ete delay, let none caine Lia cay of em
apetilic evel 1F the sigal is tananlted sibout dtorton the eam
Cept of delay’ af se signe whole snp 1, becmape of ee,
{Pe seal vecived signals ace elena L we may’ sti agree upon come
Temata Sasi ot eich ae determining he Use of occurence, the
“stn in consicerable eels ay vy qealy ita te dings
‘lsracatiaie beeen, For exenpir it depends on the Eebavier of cor
wonwncs high feeqaenoy the dak aay he ute een frm oat
Je depeads ov thier ho. Te the Exes eee she sl wel be ie
fMfected i, before tarsminsn, uke gach were se rg ghee
flee and in the second, Hit ent trough low mas te Teac ease
fre mene a sly estate with w ditrianee which cmprisce ony
thoee Fourier raat of the sins) which vont paticuler limited
Inoge of fequeany. We any cong thi dea “aethr aad make go of «
‘rey nore un pas flee, Dy mirying ste ifeegucuey of tai fond
tye cbtan ray wich is a funecion of frees. Tis alae, et any
Fogaency, Ci uiy, ae defred by cur cnaveuton, of © dicubeace
Sshicrsexpubds to the pat of the spctraza ofthe sigeal wie i fr
the irvmalia ueighborkcod of the ‘eyes ix quetio, Our prebler
then bo ad sccog'eahle fares ofa lturbarce ofthis kad ih bal,
Dun Uiey ane asd oe exter of ray, the result cea be rete dinetly
To the se ue envelpe dele os iene in len sof peindic distarbanes
Compared wilt the pair of rqu sinusoid used in the deximition of
STRADY STATE DBLAY AND APERIODIC StGwAtS Be

caveloe delay, this dstusbence dirs in tain any finite cange of fre
fqueney, cere tre en Isinty of slusos, the amplltdes of which nae
tot all he che same, Tor simlisty, we same te accual ker to be
replaced iy sn ied one in which there is ro dscortion within the base
Aiud no Lavamibsion oats it.I he sgn] wae waoie be represented 2p
8 Fourier ing ve may chin the dese? lsurbance, for an angular
requeney, ay by Sategrating from my ~ 5 tows} 2. "Tae dieurbsnce may
be seprescnteé by

f= reiyst tat [amin bslt Pla, 0)
wer ia ent hee he se
{aU ae hoa ey aon dae pe
cipal tin oe ye

a ta nor trai nn en xe he
rng at feet fm Yo et

Since we ar interested only in emall valuce af « xe may replace « by

shure a af te the values of ad

‘We deine an instant, Pb

sand tine, by

Sbstitating these in 2) ane peforing he inegraion, we get

sinh

20) = ro part of 28 exp lane — Oy — 08)
ewe inte te angle,

ae we tan =

sod

task (ot

sre tan
suid tke the reil pt, we act

{ermB snd) (inh Bt com.
. naa

2 exe

cesar — Os — a) 19 si

Lec us conser fist the exten wn ete the epectmam of UE sgl

— a

Vig. BsnctaySatcnace worming i on sme of he sete

‘ifr ia spline 9 he nelgiboehood! of oy 20 that of fezer. ‘Thon

J) = 2840 en (09 EB con fur =O ald.)

‘ere the anita neues a const faevor which proportional to
the bardith, 26, snl he agniande, 3 expe, at Ube Feequency,
shuardaFirstin of linea plat of hick ies in Fig 1. Thi funtion
Corsies of a roi’ navel Seaquezeyjc the clio vars
with tine, re ervey Iwing symm eooat the instars, 2.
AL whic iL maim. Ta, the Hine of maxinuan cove, hem 8
Unique inetast whe ie stale o> deine the treat whi the dae
‘urease occu. T ieterminel ley Phe slope uf the pase eequeney
fare forthe sperm,

‘The ins, may be intexpete, in accordance with the prince
cf setcnany phase the one hich Lie easel components of

foe at near inte some pee, and hve he as deste ter

FREEUENEY (o

fezence. ‘This soliton ill Inkl when the ostanareons phase angle
chong least ply wis Cegaenty, Ua iy when

Rone
fom which

Theanale, 8, ~ eal, in (6), ives the pase of she save ne the insta,
1, whet Ra civehpe sa asiinen. The berceation of this ele wl
bo iced by che genetic soagirution af Fig 2 hich ie sn to thet
ms bese S¥Sreur YACHONICAL FOURNAL

feploye for phase and group veel". The abscicar are value of
Sin the ordinates ate values of phe in rlians. portion of the futon,
fin the neighborhood of ig bow. Theditaace, OB, iss. The lope
tthe tangent, (it, ts Gi carve at A ie. The dstarce, CB, i wu
CCoasequenty, OC, or the intercept of tie tangeat on the Base ws,
fan li xv alan in he Be, the sbeotte vale of this interey
{2 greater i wwe ray teuloan {6} tom oem in which che angle
lass tom 9 by the substitation

= breath Bae, a
vee sc ategr nd
lel<e
Th Pig. 2, mis and isthe distance DC. (6) then heron

f= mt oy) 2H ctor = 9

aud pf the eednnry pls lag of saws, live to an oxg f time
five by the instant of masinun envelope

‘We iy ein asthe inlant st wich Te discsharce aecuss oe Ty
as whieh the envelope i a naninum, bat 7, a whish the istancancoas
‘ave of the function sie eas ablate volue. Since 8's small
Comiare wit, thie wil occur very tel atthe smallest ste value
Sie ler ubich ia — e} isl, This wll occur or

eS mn “Saved,
sual for
wn -rey cha Teves

rom (2, (and (7,
tt he

t

nue ea tego sue) shat

P< annatth <5,

“The signfcane of thie case oan fra ig, 3. Here ination to che @
furve of Hg. 2, tere aoe plot Seis oF eaves whose online der

‘tam, ‘Hounalyonic" Ces, res 1016p. ST
from it by multiple of = Ino ‘ar as anyone purely sinusoidal exmponent
of te dstucbance concerned, wales af lave eternal y Ue twee
hie ier ya even valle af ld he fisting.
Aiteing by an ad musiple would repreext a reversal of siz, Let at

tw ers tha cuve Sr whic the tanger ab inttrects the an le
eat the origin, and elit #-} Br. Since, for the case dra,

vel <5,
st were greater we should have
ba tnt

135 chen obvious shat he Lief sani absante value, Ty is glven
by theeope ofthe line OF, Telfer rors Phy he derence in slope 0:
the lines Ol and

‘We nove ta ced mn Ske epeccram ofthe éstusbane te tire of
cecurrnce n term of twa deitions of the ater The nest step isto
fumnpare ‘bee time for Ce int and oatpae and determine the come
Sponaing dehy, Tec us onsier fst the rave where ie attenuation
indepencent af cient f er in The own signal also. We
nay then cafe vas acorsion t Eee sae, e. Tet deepen is alae
Ae pat by ft the phase wt of the sytem by A. "Then at tbe
Guipnt il 92 exeal t9 Ff BT we take the tire of accureace as

Stein by the asimmum etvelpe, these times eb ad eutput
Tank
Tham

“The aly ie then
We Ty Pym By

chic sy leon the envelop day af tho eyetem,
‘fee ie she me ov arene bused on tke mesma alse value,

we ave, at the input,

Denes

At the opt,

where
STRADY STATE DELAY AND APRRIODIC SIGNALS mL

‘The delay,

Wile theres. sypercial siaity ecco: his and Lae phase delay),
ft fe of fit eral sizineane; ix C2, ia uetennined br Le eggrnae
Increase in-phase sift with Srequarcy, shed in deter naire bye
te of socrease aéw:. Mi exciple ux staien fo shch the two delays
fre very eset, Is fucnshed! by & wave gue shen Ue fecuancy only
Jit exer the ctol. "Phe pase deni hea smet seco wile she ate
Uf edacge ps hit with requste is very lg.

‘Thus the delay haves on maim ateninr value depends on eth the
ave lela anc the phise sit ot te ster, but ot on the phase
‘day. There romaine to eximite this cepencence fn more detall. The
‘alu of & depends on the spect of te igual as wel a the characte
ofthe eystem. Tt of inovst to See i 1 ca Lemley a panty
ceived trom the system characteristic ala: The west ovine Ping
lay fea celsy whieh ie derives fom she pase sit of dhe syste i the

terse deve sn Cas of the

signal epootram. Tai wile

pabohe,
shee

eee aes
Toe dieser ewes this al the aperini iy baton ate
salto

Dem Do = Ele,
Beet t= 49.

Sion 8 thr en ingen stn

“Femtac§

De= DB. =.