BSTJ 40: 2. March 1961: Further Results on the Detectability of Known Signals in Gaussian Noise. (Martel, H.C.; Mathews, M.V.)

10.7 MB·English

Further Results on the Detectability of
Kuown Signals in Gaussian Noise

By H.C. MARTEL and MY. MATHEWS
‘Maoneviy revo Seamer 1, 10

‘The detection fw evelyn signal which may ur may wat be
poeont ino fndle sample of yestian noise i conasderod frst ple
UF view, The fv! castes the performance of © asinn Keio! de.
factor operating un fie ac of eisercte mensaroments ofthe sinalua as
the st Gecrmes Rags, The lisa a ihe pal va note oF nate aie
‘mye re presse for sigue va baaiteeted note, unig as meurirs
sundae eguiepaed empl cama a desis af ae eda i
Ure Por bh he dt qe eto oan nthe mai 08
Sent i iorene The aout go of ew bases section a om
Tiowown meoewoner! ionse rmegral nara! which weaciniee he
lively. Solutions hove hon obtained wh the woe Rasa setione poe
seta density, The dtr sire a rzs-corcation lire aalun
tent eignal which swell on und a mechani designated extapoltion
etesion which soles eatvaon of dermten ote atau The em
Init ofthe derivative moassromnte Ho te etait 9 ezamined a6
"he none appmonchea bait? noste end 2 found $n many eae Jo
sr tot Oe

‘rag problezn unde cou desation be she drtetion of & comple ly

iknow: signa vish acy o may noe wea ia finite sete 3
fuasi roke, Ml we mage anion sale to Hig. Lm hr
Sotinulis i made sp of eer gm plus nee oF noise le ml we

inky given 7 seconde of shiv ails, baw accurate” cun wr eile

ttbether or aot the signa i poe. ‘The noes fe Hoop of 3 Having
iy a slags protest so Us the question i really

sel yet esi testing.

This particule press fe been treated eather extensively, and

cortin questions, even snutioceres, he aren. These concern whit
42 rime cue saan cueuieat smn, sane ot

is Dsxcam of sea ner coerton

sonstitstos a pinper desription forthe eimulue, under what einem
‘nees ran he schnuls be chenaeevied by Site numberof samen,
fan under wha eondicona in prefrt detectability obi, i, whe
ie it always posable to detect che preeence or absence of the signal
‘Peterson, Birdsall and Fox” have deserved the stimulus as being Powter
aries handlmitd and by so doing have obteinod quite eiferes: results
from che ther than sho fr the pr urea ane
fun noi, In many este, Relleducution siz ave boen earacte
fncd by atste numbor of ezapes usually chosen so thor ere irdepend:
thi, un asin lke die erseyerating os thaws saris ve
oer developed, This Is le to the eqialenl ua cerrelation delesvon
proces in whic the tes statisti the ing ofthe peodt of the
mls and function dived forthe Sigua Such deectors always
fnaluoe finite delortabity. On th other hata, Stepan’ haa pointed
fut by an aegument involving analytic continuation that many egusle
‘ee paseety separated irom reise provided the noice econsdazed to
hve a bandied spectrum. Uirazly game mechaniem in addition to
oreation detection isinheres: in Stepian's result, and indeed be points
‘ut one eh detector

‘The reals of Peceson, Birdsall und Pox hove been ured extensively
far compare with the peHormtineashioved by Husa dl
seinal, atl questions stn Le aity nf ach comparaons oFiglly
Iholiested thir ‘avertigalioa, Eewever, it moun very dovbtul if the
‘mechanisms whith wil bo developed can hurse anything vo do vith per
‘eptina, Ip addition, we have snern to work with stationary gnuesian
oie eather thau Fourier sts bended noise, the Sommer belig
‘ouch tote stistactory charecesiaaton of eal nose,

‘Two diferent attempts to better understand the questions cited above
hove heen undertaken, ‘The fret examines the performance of a mat!-
rmxim Hlihood detector operating enw fnite ant of diner menae=
rents of the stimuos asthe set heeomes very lane, ‘The rent shen
cases whore the deeetafuity grows without oun, Thus, the ehstee-
lerapiiry oF s10SMS 8 eaunenas sume 435

lovin the simile by ite wt of measurements is ncemplot,
However i stm casos «lar of cinsinishing retums operates £0 that
Ie le of invwase in detectatai'y ows ae the number of samples is
ners

The sco) study 0s detetion on a eonsinuous measurement
inca incre apenatary, whe Ue solution of an optimising integral
taal. The tee statistic 0 oblained has se parse, ane elar to
Corrlalion detacton, the ether bused en mescuremente of Ue doa
Uses the stisales, The onsrbtinn of this letter term it ural he
nally ni ihe to, but, here The nviw apeetam appmoeskes «baad
five Jonsn se may grow itt Round. In addition, ie ey
prt i te srzad s very srt

Toth anism Hkellhood Wetevtine wil a Enive aumber of aanples
and the istegeeleqation forthe cantionous statiste have been pre
Vinny preeented. he nes routine arse frm the more camels
Shittinae whieh have Icey shine, Tap aaoet significant resi Wn
oubtedl the soluinw of the isepra egustion in eapd fran that
ite chanictcintes wn pareulay ie asymptotic properties far any
pve nese ean be sem. 'Th derivative detector, whieh ‘il be terme
mmputaton dteton, ee apparent tot th Tain

I his section we il deve the pin lth deter for de
tecting kramer ste mute of sasples
Gf the sto an app hiv detetae te 39 pee problems invals-
ing bardli'tn sae acl snp vetla fom eome Hizear operation
fn the ins al the stele rel wot he inéepmndent. ‘Ue devivu
tian of the detent essen dies ony lighsly fron: previeusy pubs
Tsted work, sn ilu bul elogey into the spec problems,
tehich are the principal new rls: ve probleme the bebevor of
the elector ie tudied 9 the number of samples heen lng, fist
‘when the somples consist simply of omplitde mesure of the
‘imulss und sens whet the simples urea se of derivatives at one
‘init am san.

21 Abin iil dio

he atl
$25 aE ELL evaTEW oneUNleat JOURNAL, MAME sot
‘neither gausan moive NCC) or thal nui ploy kr eignal 8¢0)
sed is observed forthe terval) 2 ¢ 2.7, The m names,

Ye

fon whiea"the detetion se ace ace chtsland By linear operations
yay di 2 cho tiie

meni f=,

esas oftheir finery,
TWEE = SiN] = LINO] + USO]

ad 2, wll be gemtsion eundois. varices whieh sag Is cumplesely
‘huchotrid ty their wack 6 of carelation ceeliiate,
by = Hep
‘and by thet speaus which for sanity wil be ania to Ie wt,
bee = 0,

‘The demsity foneton of the P, evaaples when the scuba fs noise
Aloe rn he 3 iten

Pela ah) = BM | eww bY oF 8s
‘whem go's are the cima arguments of the density forstinn
comesponding to ¥,. Ys and [81 is the determinant of 3, with el
fumes going the Snage 1 to toes especially insted other.
Tee donsiy Tureion uf, fer sigue plus noise ie simpy
Feeley ooutel =e = Spe BD
beste the signal is aitive, ‘Lust Walihood ratio Zs ==)
\ ya Fac
eC
whieh when evahate for ew deity fe tioms bees
Lin. ads alo EB, S86 owl a, Sod.

A ruuximum likelihood deicetar sys thet signal ie presone if tee
statist Jo Vy oY goer Ubi ne thers rad wal ma
‘nine the contol prabaliiiyof deleeing signal shea Hs present
fora given vonditional probably uk iesting sigusl for aie alone
However bis smmnavonie fation ofthe etic,
en Dae 8, a

and coset a qu: gu tee > a, wheon ay a eqiva
Tent threshold. v macy be chartered by tw density funesons, one
ifthe stimulus i uve ale the ol for signal pls woe. For noise
lar ey {the sateen 8 desire nie ale, SN" algal pie
ties yess with gore ean and variance

Bees? = Fay '88s.

ior sgual ple oer gy we joann with rhe same vevianee but
wth mtn

Bees?

Lass,

‘The deveity fatten fr 2 ane pietured on Fig, 2 The eletiveness of
this devrtor as indicated by the eignal detection probabil given
alos alriy rae cae tseterined by singe amber d hich the
ratio he emer rea the sgzal prs note distribution to the
Pune of eitaer dsttbuion, ‘The langin fs, Ue me vompletely
Sepaneted are the distributions on Tp. 2nd the higher Wl be Ue do
tection probubility. This mraber ie tea

ae Dats ry

A altonmate fons forthe stavstc wees ta ive ia (2)

Dar, a

SS
228 rut wnt eMC MROANUCAL JOURNAL, MRE

“whote the Zs ae achatons to the equations
Bote 8 iw ww

and d may be expres

a=Da8 o
“Thi frm 6 wally pee fr compataions sve i noe the
soliton om ear eine rater Ban the iene of ab 0 3
Tease Tate thi rn er lly eserbin the integral ich
‘ilappear when conic sane art rove

“Yo summarise, = etiieg shies operates on at of wcomeited
‘omplee and which i quan: tot masizum Ielboed wate bos
been develops. Sgr innate Ig greater han ome ‘heaold.
i formed asa linear sum of the ample, basa gosséan dstrb ion,
Snd it has the ene vasacee fr ts re slone lig! lo Ene
‘aera, "The performace tthe deacon may be eaartecacd Ky ingle
umber d = [Beper)*/Beoy' she lng lh te ble he por

9 Dalton nf Sint Ranritet Nei with Tan Sain

“Whe argonunt peste ay Sinn? inves hat cworecealy, be
snag of te aye nalute of ls oe, neni an always he dee
(evled in sportu: inl aie, Honors, sis beat saya using
bout how fest the detectability increase with the eomplesity ef the
ddesector. In this eeton en cxazple is euained in which the stimulus
Js time tampled with m sampler nqully sed over the ienval 1)
fT and seectdilily be orginal sx fone of To aii 8

Ue gerem’ havior of this Calin, it ie of meal inert note
‘whether any peelinilinsacear ale. 21 (rhe Nynist ate), IV
Tg the nse acd, sinew Uh ee sasee mati fn
pend samples which ry frre. "Te ocelot nian
the nie

Bin) = wevieiwie + vy ~ B2atee,

ar
de 0 the mate af eo

‘vote the anico Has unit moan equnee arp
Felton vuTen 0 be wile
Dunuersioure ni SOHAL TS GARESAN Soin BD

with
ny = 207.

Volortusatey, no aoulytie way for eter inverting this matrix or sy

ing (51 i known, henee Hap setotabity wae computes? noericaly

“This eompcation sg ex nk on an II FOU acne for ial

sth ode centered Hie ise bend

4 ncing the arid and gh belie an arbitrary’ pase chosen for
‘ompitetionel sonvenience, ‘The tnd results of @ eatin of (31
tl) ane piven on Tig, shows oi given ata function of
the nace es iy ge af Ube ie durin is arm of
the umber of independent sarspes The curves exhibit knee, no:
nm ny hl fve a Bt large than Deverill continues coi
roe bt nal f isneaae becomes impoeeepie, The curves are
tll cari out ro matric of sine 128 X12, whieh ie the Limit of the
Fapavity of he eomaputer progr. Double precision arithmetic and 1
"fen erbor angie were id a iar The accueney of ho rss
‘The inenease ila bon Py oe aly eae
‘hath wn in bal ey inornsing Ptah 277 wed unig
tt the Nsquis site. Hocratiealle wr etn say That, by adding ext
pints nse he iene tute ene o prediet WC) two independent
ple tine lyon each end of the inter, very hard to predict
further, In as unpubl poo! Slop haa shown tha: the quadratic
forsn fort given By (3) done oom inate for handled noise usw
Wears iit, However, she proven example indicts 3€ mereases
aban exccdingiy sow te, Clay sstistie whieh prot more
fapily isda ancl ch i evant in the ie etn

28 Daeton ofa Constant x Rawinned Nu Ting Dsiatives

“The slot forthe optiman intl oporater deestor eased ont
wn ihe neat secon prluece n eoile involving devas of the
nts, ‘Thi ent atggess tryin devivative for andimated no,
Darvculely ence sll corvatvos of bundimited sciazalus exist. Cou
atuctly the detectability mnie by samples, whic app th i
Teen its ~ T deivativs evaluated atone pin i ie i: ti
Ths, a wil be sen, ve the plesant eharwsteritien of ing
tnctyt rally saher tha only ttre dererminabte und of actesing
Ailey ith fe eater then extioling the kis eurvre of the ime
file Atti property ie that the duswien ui stirs 9 m0
4) emma svonime mecusieaL Jouianety MAA AM

longer a fata in der tailiry sw, theory 6 esha me ber
of derivatives van be mess (ran ax sb mde os sed
Delewaility an nga be rowed from (3) wat (8), wher

By = BNO

we
Dumerainisry a siowAre TS @ACRETAN NOR SAL

‘he corrosion

1 sperinan ofthe nies

sab Ge fe de

ta) — FP wewenines 4 eve Hae

1 bundinited oie with » fat spesirum from — 1 tn +4 vad /eecond
rns scuplioe seneo hn (7) sme

' itr teiceven

alee
fetes

Asai far 13) an (6 wth Han ecefcees an be eft, ee
ie av resi to form wit a aoltion at

‘The ananrean pool he wl on 2 large enough

Inet

Inne t0 Ca

staf pager or sg ang spe derialiver neh a ania

‘ren eespsaly compar fo cona fo Whie
PS gg 2a,

sro) = OSH

sia) =k
‘ree evalucon, camel oat in Appeal A, yi for

where

+) cornea

far rg an aa by subtitting

‘the arymptetin behavior
St Ong 0 EM

fo fastine (8), the lacing eto

oat,
os approxima ie vin 2 pes se Sar
42 tue nua, wvenaas roeuMeAL sOUHeNAL, MAME EL

équations (8) and (exhibit the behaioe of wstitistie in whieh
fuerawos Lineriy with fhe rumer of samples, eae simple bo

derivative, Asner behavior vill be show for rao! ise here
tone torm inthe delortatity depots nearly on the number of diva

tives which exist wad for park nt the stalatie, The baeliited ie
{iene Sn the rind neni Ua allie ovation aerial
tit and he desea. ean fe wade at Iss hwoneieally, a gon
tees by ring me aege enough Obsionsly, i ay patie ease
the mune of derivatives which an beentinated iinet fw alton,
the charaeteriztion of the mdom process a2 gautsian undoubtedly
Fale for hig ennugh dsivssiors,

igationa (8) asd (0 ace devived only for « sigaal which 3 «con
stant However «similar dependence on me would pralubly eeeur lar
‘Stesoidi inl

"The prominence of derivatives ar an effective tate for hath band
ited and acina! se gives & posh ineation why decetabilty
Ine i nully spc tne rmplea ineerageon loly. Thea, eg
uniformly distribued, give poor eetimates of devivativer. A more alee
{ve dite night wll bended sample spared unforany
rose ie terval a le rest the somplon elusive ehenly a8 pes
fle abou exo pista ct each end ofthe interval, Nish urtangoreat 2
oggested by eats forthe rational nolo eu.

‘The preding verti dincweal che delection af & hao signal in
Indie nae wing finite sain of apes of the stiles
asi. i een we oer the deoeion of han ial
itis cue using an tesa eoulnana eens othe nas
fs over em interval 7 in lenge. The noise ia now salen co have a x8
tional power spectral density the sts power spectrum een be repre-
fentod at the mato af fn palsronnla in 2. Buh noise run he thought
es retin fron the passe rf ideal ‘chile gin ie Thea
Finite iar Kane eorent flee although i ned nok stanly Te
Then predueed in thi nay. For the purpcens of the sal i oo
‘enim br Think of he stuntion a shown i igs 4 While ganas
oie se huge a Fler whe (cede Fanti Fs), Ea
ransorm of it apie sponse} ond ta this eyo may wo be ded
the aw wignal S(0) P seca of the eorbinatin fara he lias
(0). The peblem ie Lo deci oun an esoiminalion of Tye abil
ihethee cent the signal wae pteent