BSTJ 50: 10. December 1971: Some Considerations of Error Bounds in Digital Systems. (Prabhu, V.K.)

8.5 MB·English

Some Considerations of Exror Bounds
in Digital Systems

hy VK, PRADIN

Simple upper ond fer tani on the dstation function of the
un of ao random aos pent frm of hr xargenal ee
ion functions a he sashes. Those bounds ae enw ain upper
tend Env Boul to the evn pba of bout ital sytem ha
presence oj inertial icerfocense ane bie yonsian noite. The
Fe ere expressed in tems ofthe evar probaly aban ith a
‘Bie yu lean, an the bounds te the marginal dsrbutien action of
the resi te tai, Since br dfn Between tke wppe and ioe
Droouda ran be an to ben manntantaly dameceng junction oj tie
‘mente f ude inthe le ue rah Gl Oh wo egal
Teen pba oj th ear oth orlavoly ama evo Alea hen
the samen pevformane owe ey wind tends, te mets
proved now pape a Be word atte ie rar aaa Uy wing
Ife uae tain appearnet

In digital transmission systmna the theslor ehamnetevstos of the
twansmitting ant rociving Eters a fax from ea, andthe eal trans
Inision chanel vsullyexbiite som forza of time dgpesion."* When
fn ideal dal sal fe passe. hgh sue tere o i transmitted
through soca ehanne, the aveeervn ule ovesap; this form of
tistortion ix ually lows indereVebol itererenes. Tutersymbol
Inesferce may also roan nin Ue eve of ronoptinnsm skenping
imlants imperieet demadulairganier phase, improper pulse design,
fhe. Tw ation the sie may be commupted by tema! mise, co
Channel and adjacent chron! iuleforenee, and other forms of noise
that nay be precnt ete elise we i tho syst wel to Eanamit
tiv information

In digital transmis systens 0

cof che min performance ehes-
198 wey ovetnAr Teomeeat FEMAR, ARERR en

Lis sete enor aan oon

seven nce cbt ef ore;
Te expend at rue seni se of ore uP more debtion
Faw

‘Vt antcas hag tse le uve is prubabilily of ere by 2
kale of thes" bat thi igaly mpl probit diab ton
cent seh fe ox4ets enmptd

Sivas eights es sale tsa others
problems are never cet since ono i ears to a only a nite
Ihumber of poles wid mo bounds to the traneaton error have been
feria

‘Another method ian analysis by sean af a weeseeae oF eye
potter’ analysis. Sines the protabilivy ef eecurenee ofa worst sequence
ny Bevery smal this orale usu leads on very peernistio remlta
and suboptimum Systm dosisn

"Recsntly, rome antnors hove derived’ several difleent upper
boule on the probabilty of error wien the avetem ie ubjest ta both
Incerymbel intesforenoe and deter gnssvian noise. Same of these
bonds make wee of the Chemo ineqtinity i tei derivation, aud
hence ne oem rans wetul than the Woretonse bound.” Lioweven,
singe these bond, in eeriin eases, en be shown 4o be leo," ata
ite asf ler al have ben derved, they are not as use
in ayaer design she wvalton af Ue oenet err me ofthe sytem,

‘The thi! reba ovine 'n ing Ue Grits pee tre sepia
tion au enealting the save pb ther hy the direct emer:
io all pile sequen” hy the sores expansion mtd,
"The serene tral, ehh ivolees the eompitation of he
ronients of the intersymbol interferon, in goonies method bt
Ss al inexact ae no traning crane tile die athe res ie
fain have Been derived. Note that fn his etiam af errs
in the nite pale tin de gradually inereneed until the change in
obebility of error is ess than given mame

Tn tis paper Ret presat srl er aa awe Dude the
distribution funetion af the eum of exp eandom variable #y and 25
fn term of their marginal distiburion fnetinss, Tr the spread disc
petson”” of the randoms variable ey is ssaller an tw aro of the

=e Tine ce dhs osha eho ie Be math
mon none pica Hosea sy

random yarshle 2, ne e8 store hae hoe two boc ace Bale
flow to ouch other and tha on ea velo dhe Mishhation fxgtion
sv sua the wanes sv terme a he ian Tasetion af 2
funy boss the asebution Fanesios vy

‘We then ts these bones brain act and leer boxnds on the
cere piobabiiey of binury exleront oigtal ater fs the presense
of interaymin Sferferenee cad alive ysureaa noise. Sines the
ference between the upper aoe ner sae oan Br how to be a
fnovolene cvenaing fmageion of the nursber Na yale the fnte
uly tiny Me home eam be ute ty worspnie le mor probability
tthe svatem with arbiters sa ere

Aly slow Use sytem perfomance is eluate by svcletionteoh-
sigues, fhe vel eset ia our pager es be ui wo estimate
fhe eer cutee by Heng anes pause Goa apypximation.

TH tie symbole sy gy ele, wy a sane Kink anther srt of
‘upper ard lower bowls wat be dosved forthe probably of er ofa
fata rahjoee to interesmbol itetereacs and eas gveinn none

"Dhe asalress of the bounds ia Lustreed by tou ovale

ev ue gerurs thas w tlons sate ¢ tbe mom oF on et
voriables3y te 25

fe o
fn that re are indorsed in the sistem fame of &

Faigt = Wale Sel — Prise be Sal &

In thie seetion > hal alo stair that ey aad re rately
rncependent ach vara

is prnbublity nf sace of ange nurstor of yal seats sjene
fo varia ine Woe ean often ke sepreceal a arighted Sum: af
Biot Ts fe the sam ol an fete marebor of tndon Watbles, aee
Se apmesh s pte amo the lira Note we some aes eam,
fvelte fo, ln et ean seers be eraputed vxney. Iv sich a
feet i Gite eae ohtein Z3per and lows? hou to
Fito) tore ni Fla) ad some iu guramsine eaoraced wih
(ovahom eatisbleq the sum of tae reneiniog tema tmz TE Hho
iors nbworn the ure bowtie ie seedy morutanedoere'ng
neti of A oe ean Ure lew Fon wit apberaly small #18
“Wichous lows of gonora"ts we sia sostce thet tee meat of 2 Ie
SISO cae i ersten muustead Youu, er 18r)

zero, From (2) we can write ee Big.

Beh — Wale naan ‘i

Tet uso sles a interes
veatable gg Pron (4

shore

et
a ase mo 8 a1

nec ste 2) hat
oe hs [arty =F a0, 9
Ost 5 Pale an) [aPaG) = Pale ~ anit — Batt

va

(1 = Feyfand}, a)

fez Pate a faa) ~ Pla ~ A01F las = Fa

snd

Beg Fast ab | dali) ~ Peso + abi ean ~ Flat

as,

Coming (6 it. (10)-(1, 90 ee
Baio Ao fP, a) — F(A)
5 Fo) SFA an
2 Fal-ad | Polat aby ag

fo ant — F(ab)

In gonone ite now ony ts eompate Ff, Howevae we may be eble
te burl Ff) 90 thas

02 FOAN = Mele ear ST,

set 45)
NS P= Phe a ets 1 a0)

ond

12 Pasta) — Fila) = Prat < 24 Sau)

Ap Clad eo an)

AE these bonds ea be fl, (04) -(17) ee

(oy) Ped

SF@t ads Bal ad. 8)

te to yield

Fifa — aut Lala ~

“These are the Base bounds tht we hall ae i the sof ti pape

1 th ru of the distribution of ry is very roeh eooeen trated
around y 0, one feshnique of computing bf) from (18) elie o
the assomprion thet we eam rel too mumhors aw ned sue that
312 mus ott evirem cenaTeaL. soma, wxCRMES SL

ail

fg Sept tin ),
Ti, spins inten Hn! ea | a ug tin
pit? cenit me oy stn Ea

av lel, ats << Poste au
Fuge aa) =
“The diterenes Dian, af between se ype ad Tower ourds ea

Dea,

2) SAW. Cah

-Prle aw ery get al
File suilLag(—al + clam) + taal ad. 0)

1 anand A ea bow ean ch hey nee y monotoneederrasing
finetions of Ny Su-—0, af», an 4 — =, and ifthe bounde on the
Aislin sy ane sue dha, for eaedonty lange 3, L(A)
and U,q(4) sa be made smualer than any ga number We et
fstinnds Fo) from (U8) with arbitrary small ero”

‘Porany given A’ Hen thexgh i ad al ean be chosen by optimising
{he bound in (18), his optiantion lala to very oorapen equations
rence we think that un algoivin should be developed to ehaose
‘and! for say giv zy nnd ap. The develnpoens af this algorithm
tv. be ilestratelhy mena in Beton TV

VWetsumettat Brie ~@ cay S20 =O
enon pote os SHerEA, neSnEa ssa

2 Tae Bont Fesaaton sith Canes Fata)

We shall uow drive simpler lower {nee} baud to Pile) i la}
fs a conver (eonoure} funtion and zy ie an even random vatlable,

1 eo
From cen 7 he meat of i 2686, om et ite

‘Wit, sation the ection

£00. en

oldies

ery is wx eve radon variable, we shall vet ay = aP ie (8)

Tenor wenn test evr esos vas atl hat
Be come uve the rare — An, eu) where fan, Sul i Te
rook of Binge sy soma tesa rainy vale
0) — APasla ~ Yuadoe = Foal + aes ey
Bla) = (UR ga wad 1 Fila wally cay

= Pao ey

ince Fa) x ware aver the mange (2 —

ME se ~ 6.0) + B+ el
From (23) wd

1) se Jae
Pala) © Foe) en
Since this bot des det eontain av and af it simpler to eels

‘an ut glen in (18s lb tighter thou the lower Bou 1S}
Ie Us care oe le Lave

Fula) © Bile} © P

fo) a 4 begat, on

1f Fa) is concave over the domain (@ — i, + av) nd if ae
‘a ou even random vasable, we enn sie show tha

Fi anil 5,

Pele 3 Pad

2.2 Konto af Anctes (per Bown te Fla)

(en we Hn thas ¢enetainss ganssian random variable and ean
be writen ae

seb beget

fered, 08)

voy ean ae sto stately dependent cardonn varies
SIA mie ae avec TEURIICA SOUREAG, BEER wt

We have alteedy aamumod thet tae mean of ry ie zero, Without lost
of generality ee shall now masse thatthe mess of ie to, and ite
‘Fom (3) ore eum show thw

Fate) = co

soba

ete) = oy

Hoses have’

Peas — bee (=8) +

Secwmscevniovaat, 60

‘whens fte} # Ube Ath onder Hstaive polynomial sad gy ie the Beh
memento

ey

Tt nay (Re 4) of ge ite and 2 Hoots he avin
bolts vg that eat be abt by ayy we wa show tha
2z0, 9

eae Sm

= fear 6

‘is alle hth absolut enemant of wy
"E the frst moments are word in estimating
ssuncation errr Tis given by

(a) from (31, he

Teo te emtcetiads Bm winless 69)
Since it em be ahows” th

| TEA) | < bo /a) exp (6/2), b a 1.0985,
dat

oH,

cons can ho fom (B5)-(96)
‘nn HENGE OaGAA, ATH as

8 a ‘

ee ee

Devil ves ©

i 1) al) on may bee is 7 nay eit

situ an easy sn dnte th gee es
Teun es ever ance, warn ow th"

py KEL 5)

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tae |< te exe tt —
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(aie em | yet | a 1.

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[Mo |S Le eeaiek + 96 — 1h ay
se oo se th

1s Sh exw (ahs BE

3 ae {80 - coeliac +2".

iatnk eben oe,

Bale = Here (et te + nde econ
FP Merie nt rec teh Denee AN
Since one ean show free Agpenis A} ut

Eee on ever redo vaciable, 2 awe

Borel tN bere is A) Berle), 220, a
Fla) & Hort
B,

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er n
S108 rap ema grees risusiewr ran, renee en

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oe wa
os . = erie inh. abe, tse,
= Before trains Goat agit
o
cop-out
[oot we
nie a.) (few -wtiesale) em

(fene 8as

ae) on

~ Cee ar

athe monet gesting Fsaaion of the random varie
Tye un Cnt raebeae ny sad eg oH hab

8) S exp ene HORAN WS
1 amano: oor Sh that

U1 eset any (nM ob

Do. aren oo

‘he derivation the unsehousl ‘a {BH i Raed rane gow
Ref 3.
Wis new Ue

if S Pile) & Riley me wd

man BG ekorn,. @