BSTJ 50: 3. March 1971: Spectral Density of a Nonlinear Function of a Gaussian Process. (Neal, Scotty)

8.2 MB·English

Spectral Density of a Nonlinear
Function of a Ganssian Process

By SCOTTY NEAL

Saunist resid Septenier I, 3)

We deroe av expovsioa whi ona be ant tn @ reat soight

orwaed mawias to erat ets te awarseloion junction oF the peel

dent wf ay “onacnte™ retin way atin gener yore

"Tne copecaon io uied 0 ealy tho antral Aen of sional ace
hick phan node bya boned gown pa,

A bund-Jimted scassian proress Aasing € woerangeite power apes
teu i assume! tor some purposes vo ce ca adoquale apseciasion
for eeveralelcees of trusting spas reeour even pcan trot
Jon 1PM systers! Loweyer:n ap uectcl imp-c=cntotion of © EM
stem Ye rapdulating sige neicytheoagh cnculry whieh eaten
veh Lie sigeal ciate ove w Laee vel, ie sipping the, Exiting
level is wsatly acjusted snr sigh ‘uomina.y at cous tines the mms
st che mediating signal) ard tao ignored in ary autanqent alysis
ft te aslemn Comespagally, tre abet of tie study be em datere

tiny the yn ine Pook 0” arte sueing the odcting siane! im
a PW astern,

From a mathemacical yisecin, ww um oblain an undoesanding
of The steending querion 2 invtstierting he falling yuan
Tied Iie rowrlel cersty of ermal wie while phate tec
laced pg Sunotion ge (oe alalavvary anussoir oratn Xp OF

frwia, his version of she pve cas se be viewed as Geli tbe
eetel cezaty et & Ieomconie' nownesr Sanolion uly ganec
oceas; a provlem origiva.'y aun ay & Gl Tews D), Midltont
bra. R, Brcnetes Ip feet, we Jo ase itete azpooneh ‘eoeesentne
Tbe uanlivetily oa serne of 9 siczefocmt to dares an expzeaien
\whioh i oeabelly te sorting peint ui our omtyai, However, wing
fur relsiw aig eanw af the somplesity asebsacee with the rrante
‘eich feo evunplen Te the tind ection, we ubleia specie remus Zor
‘he hant-liitine cus,

Tat WCE) deuoie romsantesrsptnds sinawidal wave which i
hace wrdulated bya roul=vuloed allow gs) of the etetiomry
stan prooeee XII,

Wit = A om ial + ok) + 8 wo

ishutt A ia tke muve split, J. ~ a2 ia the enrice frequency,
thal @ is asunder variable wh probability dacetyfuretion

[btw neces
wil = \ * @
bona
Tue spel ey oe (0), cnn: eee
i a wr copia

Wig ~ Ke Le! ”

FO = A capita + HFS, ®

8 = cap bh o

binge 2, is salivary and # ie uniformly dseibuted, Doth TC a
Vip arp widereiee -tnhivnary.* Manoores, tbe spectl density 51)
of HY in given by

BAD ASAE 1) 1 SC = 19 Cc

30 ie Ue spesteal dani of V0.
ine Rats} — (VC HAP), the Winer Khinchine theorem

‘wher

lensing

8) = fn exp ieee o

Tae (ou Tato dee ener the we acy par
ce AL cE abe Mapa ean os
So, assume that Xoge and X; are gauesian with joint probaly
density fonetion

ates td
raced saemaeel Haat").

cacacn, ©

Fain 1 lore

whore Mefe} = Aya, P= Rit) ant

at

Ro) = | eeuieind ~ stearate,

In Appendix A, we show thet $7 Gisi is ef axperential over and
oF bounded variation on bounded inervales tam

a
Ried = Boye a

tee
| Tew aton[ ne | oof ay
ec tha he Mert prea! ee satin the

Bowe

= We) exp (2)

t noy tee whey 2 fotos Hay equations (1) nd
1028 ams amas avermneTRoMBIcud suLsal, MABOHE 72

Consenuenty,

a

Bal [os tinene( Bape ae

Sinoe the eceticents (0) are independent of g(r], an jnfiite
cevies reprecenatien for Ue stral Coney Sf) ean be obit
rom equations (7) atl (11). To obtain the sec, Let

A) = [L809 eit de as

sul dune 5.4) to bs the mold comvoation of S,() with tl ie
sif) Buff fur & 2. Te ollons that

8

Suh = Gain + -ecfes. 0s)

Ui
seven by

Une easier power

sivas (18) and (U5), ene enn eee

6. = [oan LioorOD ” 16)

21 Lsemples

"To ub beteerurderitarég of che ponding 2eult, we prose
torn exscapae, Lae fom oxruyoe r) — = (end srven os
thicken our revltesiace thie cae ie well kaown*l
‘sample seeures eres ing

yf > for x20,
wm -]
Lo tw 20,
imsiting ster preven in

and server ny a preview for 186 ha
Seotion LLL

aaa Phase Modulatioa

or this exueole 2) ~ # fora. Coneoquenty, frm venation (135

wash, fo 8

wail, [ont eatin wo(-ayae"s on

coe (V2 ETab oxp
emermat, maNETT 1109

or
PY aT
(Gor Rel, 6, Seation 788-8),

“sian Tai Oe
lesion that

C= eG a9)

ean) 0°) /2n 1) hich

emir fe nz0. ey

Now, sistitute equation (20) inte ceuation (11) ¢o obtxin the well-
Ienown teal

Ri) — ex TU ah ey
4 Phos Mosulalion bya Matem-Lamile Gaueian Proese
For this owe,

gonfe ters ts

Using equation (28) in equation 183, uve oben

ee
Gof Haden ese) eo

nd
G 1(3- [tanto piv x), 05)

forn & 0. Sinoe

[ep out @9 = ty

for net,

vee ave
e. = or yy, en
CuO for neh, 2)
1080 sauna eremmarmzctsueab svOnNA, MAREE PL

wed

Gin yt

ag sil Ye ny al

3

Using heft ha 0) = La H.) = (NWA = 8D)
form bro ea

a =2em oF
st

Cons = Bein oy ER te EL OD

Substitating equation (2), (28) and (80) into equation (11) and
reetling the series Fxpanrinn foe Aria z, ne obttine

240 = test n"2ar(2), ey

foc-=<r<e.
Notice that tae carrier pce, Co= (om 2), vniehes i = /2-Hhe,
1,1, 0+ white all she peer so ino the once whenever — ke,

1, LTT athe eontinioue part of te spsetrum isms «
fealed version of che epscrun: for an ansiitudeunodulatad wave hen
the moviniator ia an extremo-Liite} qwureian sors, This problom was

hulled by 4.1L VanVleck.*

2 Function g represants an ial

[A problem of intorstarnoe when
barat;

b for >,
oi) —} = for [2] 55, @
<b for 2 <I

for aone b 0

‘an Carrion Power
[Noting equstions (14) sea (18), one ena are thas che cartier power
is given by
omer wserry oa.

G

Dofling « = By (le aati oipping vel, we have

[°° ee tmyespt 340+ en te) coo
a '
wie 8 owt on

isthe extorfuneson# and enfe Ga) — 1 = ee
A mate sai exyrreinn far Cy om he obtained in terse of the
error fune‘ion sith fomple argument. Apparentty

[onsen ont

cr
en {9

o> [v0 (-E) nef

can be caloulaial auneriealle™ Mowever, we eam obtain @ briter
fasuytical urderstartieg of tho earner power by eapirseing the
Integral in equation 186) a

£8} ee ate()) om

Date and i) = HE)
ool-Byue Yel (eel)

= ool S)a{f anen a) aw
Since exp {=z is an ere faetion, th integral eqution 40) it

Incepondeut of Ue pall ea foun (D/D} tae! 2 + TYME). Te
$ret Ce a emia eositing of three sri Rnmsnprenta he
ist from 7/92) to 0 (tho imuginare as, Ue reel from 0 to
fax fo Une soa sds) al te Inst fon WH ta V2 HEY
Trteeatig ale tee Hoos, eget t obs ths

welt foe

once}

oon (—) at (5)
vow (8) 3. suoten ew (LE s'}}ar en

Going eptions (98) and 81) yids

1 ons eee)

sin Bada (LE — aF}}as) 2)

When 1&1 Goaeind ease, the Jae ler ini she brake
velit a Voab

eu (ovo (—U) oe () + ent ate (35) hon ec

It is intceting to oleeye Lin nf (of vi che probability that
inde tar eippig eves [i] [ol ar fa? 9D 3 tho probebiity
that AL, $e outide Ue alizing Teveh. Heaee, ke car ronson Cal
xp (18/2) I ara do wlsnge asonated wits PC) shen Xie
‘ie the cliping levee wile enw (i the average voleage wen X,
Sroutsde the Tits One ean s60 (ran equation (8) tant th erier

wer Cg = | iV

Pecaiog Use livasits fester, one wight consiude that te
fntogtel in sqvation (12) presente a high-order eerretion (lo th
sowernan e9sare roa

preci anntosimaice} associ with tnursitioms af Xy across the
Ulin thesis, When Xie exal, ese tron se ait ra
the eorretion tf the integral) is smal, Iameves a9 Ponte
Inanstions ane slower (ot leet whe X, i beadlsacell andl
integee in equssicn 421 man irae rence for earata velues
or Moneow

siden el

HVE Pan Lo? = 2] a ie Dan's tavtion
(Ea, tas hig en)

whee
Gonsequetly H talons Kiet when C3

Tl 3
sone a[- as
Thay when ese a
com fom ote) a or a
Te eiy ‘ie he ess 9 idl ft Hes
Reto eo ates fei tel Peach il ephly eall
threats power Cr elle alew sgung tra ee ecto rok
Gane
my tte eh 7
wean = oD) ne far ts i owt one (5)
se copuced gp (1/21). nn fw foe ea ao
Se of asst ar kaj Pp toic he dame ge

thi funetiae iva inp ve al on sate Batsean sed Do the
vedinate asi ad a Logaitisie ale others)

"The geswval features of tac theee Sunetions ore ws iaientnd above
Sines (Pi) — T whoarver e — U, eaeh curve arte at agp 11°72)
fit ~ 0 Pop 2 Liner sour sv bg 1) tert derromed mono
canal yt gr, For € = 2, this relive erie romans pie but
trellates slighty beans emveying te une, Howexer, 2" 2he high-
innlox ear, the rive mes swt vw bumps sind (ek
102d tees sroome zeouces ovesaL, stncit te

Tig, Reeve ium of PAD ae a Rate of lain oping leak we
neki hee 8 ee edt de

usted abu). "Mis oxi uilaton about zero apparently occurs
for P23. Nien Gol the valor of a rized for tke retire ean
{and thezefore she earn power) vo tabling ineronser us Tinea.
loner, fer highvindex madultion, setting a = 4 sex tol noosa
oumatter thatthe eaeterpowes is unaceted be the clipping

svoun Pare of the Seton
‘Nunaercal vcknigtee are equine! te sain the eamelnas
the eprctrum, However, one ean ste Trwn mentions 128) and. (31)

pent of

that he occionte {Cy} in tae esries are lodepetid® of ye) an
resi only bo computed one for any pac.ieular coufguration of the
fn Hiniter (and eloive of 13, Cowsteuen ly, there worth ta

derive an algcnth Zor ion: coupuraion oF (£4)
‘Noting ecuutios (11), (7 wd 1, oe aan #60 that

oe

where