BSTJ 50: 8. October 1971: Improved Intersymbol Interference Error Bounds in Digital Systems. (Yeh, Y.S.; Ho, E.Y.)

4.6 MB·English

Improved Intersymbel Interference
Error Bounds in Digital Systems

By X. 8. YEH and FY. 10

ong sl
mansion
sine 0 camention ul
fie Pee han
Testy kev us ah be
treaty swe acne he wana coatvatinn ef he bound tb se
thon she eel on thy Sie Sark IT Ses

Hownnive aie give for 2A-any apt! maeme to demonanate the
aeoarany oad swiputanat ery uf ou Method, The ree aie
fhe

a the wablan of dteraining te nar sae of &

fin ache fo te
datenal imply en 0 gh degre
vs

ing wer or eben ell

ete of evar ete aw ray cera af magus beter

the Chere bar. Far
gt (960 274 pale shop) in sel
Stor evanion of D9 fren ep

rate qe 1 XO

va 2

spe ie eo

Te method eva aia be appli te he enfelaion ef se perlormane
of seta ps a aye atone ave ecicn syste with each

Te many ecees the seanemiston econ a? a clgial epsom is
Lay hinited by vrsysaholsztarerenge carer than by sedi
Ini, datersprobal intorsvanes mage rest foe inpertese dese ef
the fikere. distort “w Hie fanerabslon chat! ideal ezmpliog
ingant, ot nondeal deviodatsiag eamicr Uhese, To ana:yaing such #
tigtal data syvtom. iis pert deterniae he ey ro rate
due te frsereprvl inteteene aad sie an,
2886 rm mt

four wells!" avalnate the ear cuts have bees propo,
+ prove «dine Ine np bound of ths error ete or the error
rite af shawna with trapetad impulse spur

Th Tis paper ve presenta imple mtb uate tue up
runt lows bon of the err rate witout invoking the ice pae-
train eoprasinariar. Ferhicrny ihe she tha for symtoms wth
fs rwenaizes pele distortion la than uvity, Tie pont ar ie
Ipyudaean be made cbiterly aloe es obtaining un cccurte ott
she erm rat of te gatta, 4 eso ews be ili bs Aare
AAMT nd heer phi

iret ryan sel wills dossbod bre ly i Boerion TT. Vuroue
ropseedfecer quer fr eeainy Pes enor probably ea shee ee
Fock ere diseused i> Beeson TH. tn Sesion 1Y, ail, esa a09F
‘per and lover bonds and the computation of the Dou By a serias
txpareio, Applinaioa a Ys oomesegense propecie of the bourds
are desecibed fv Resi ¥. Thur aldiies Ganneion noise cd
indsendenos of invormatnn dig ere aunt

A simple Work siagnara of diitel AML data emsean in eowe in
ig, 1, We cessine tent am impulse 21) having amp nl ou
ivi then the shannel sveer TP sgounds, The seater: trai
Tonle

Rial = REATHITS a

In the chance of ohe-inel ais ean ng a,

Eva ~ a, a

will gonerale gore lp ene,

Eswit— am, i

whove vil ia te Foner tranaform of a}, [ay] 38 seqanes of ine
lopesdent muasi variables, apd 3, — tly 18 1 GM 1)
with egal prohehiliee for all izagors, We ube cxsurs! thot ative
Ganesien noite ie presale the rystem. ‘has ho earrupted rsotve
eoumnae atthe ‘oput te the eecevor Uet-ulor

= Zonta, @
burst syste pa RATE 2587

nee nD is adaitive Gaussian i ol power 2° watte, At the
‘etecen, yt) i aatnpel evore aeoonds co devermine tae amplitule
Uf the teanemitied signe], AU sapling ime the sampled ego] ie

Wed = ante) = Earthy = 1 + wt. o

‘The fines term. is she desir signal while che seaond and the thied
ferme opnesext the intersenl inuerlermie and the Gexasien noise
rapwetvely.

"The a of ling owe!
trike

door tem — rte o

‘wert om the decision levele given BY ene
‘wenarstted signal lev

sion {6}, foe # purtoulae
te eundsone) env panty x

(Pista) & —2teUred!, ay — Cw —
JPisieg 2m Uvteo!, gy Bm
Jeunes & fees Meteo

Peon

lay Hitt,

40m — 2,
whore Ii the non ofthe eves 4 al B
‘titnting equation (6) into (7,9 aban

Pi Dart ~ a) 4 mle) Er}, we = ew — 1

PZ arly alt) Sw, a Bm 1
Peja — “i
PLZ mn = my 2 OIL

IZ ete, — er) 4 wt

rid1, ac = Hm =,
2588 rm me wn cutuniee. JoCRRAE, eomaREE TL

Since Soon rly — Hand nf) ane ally They tobe positive or
negative, ryuation (8) reduces to

[Peete Man EHO}, w= zed
Pela) = oy
3 E Hd, a6 4 m= 0

[er dgenta— m+

‘The errr rae of th spam is
Pom & Pete

= Kem — Hy mPL EZ arty = 9 mld BHI. AO)

We notie thet in ouation (10) the variables m, a; , nnd n(l) hve
slreaciy bot eine, The sequence 7, ) i aesumned to be known™
inthe following nen:

ri — IM) iefniteand kona Woe Sy, an

where Sy ew eel af N= U distin: ingore (including |=
yrw- m=

Define
x= Dawe - m0. as
From equation (12) we conclude lr che infaite sum A ocrvergee

absolutely to a random variyhle sa nqttion (10) ear be sleerantaly
writen ae

Pua’

Jay arn). (8)

"The existing moatheds of evaluating eyustion (10) ean be divided
ino he following eategorice.

sa Worst Case Setimate

A wort caso soquenes! or "ove plier” analysis i frequently uscd
to analyze a dats spslrn. The evar probability is estimated by setting

“gabe as gift erst tern or evn
Ties tbl Tp ee tain of Parser thorn teu
lars etarenn ee 26 2589

Drew wily — 1) to ts aorst eae wan i oqnion (10). En rng
cen, fis estimate Se exendingly posite sie the onwartenes of
such «worst eave sequenoe in extremely ras

2.2, Chern Roun

Recently, Zalteberg* sad Lusso ypnlied the Chsbyelier iar
ality to sqstion (10) to obtain the upper bound on error prob
iis: We have shown in Hef. 6 that these upper hound ar in tay
fee ll fa asst by ender of magni

tua, Pinte Prunoated Pulse Train A ppencination®

When (6) dssrewses rapidly eelatiew oe sami peti we
rey approrineete the chasnel bys Bnitely traneated pe train
"The error rate cen be eau? fy enumerating all the posible com
binations of intersymbol iterereuee, However, sino each elealation
of the condiions. simer probabikty taker a great deal of computer
‘in, che momber of mst be Held co sevens toueend.” ‘hie limitae
‘ios leads to 4 poor approsimctian of the true chant, ard the eeeor
probability eo ebtorned ot ror very sof, Iesnt, It has reported
thst by computer sinulation of the ces Tanetion of Ny the come
uration time ea bs relueed

24 Sorta Bepasaion Method

Heesuly, Ho and Yet" and, indapeadons, Sambo anc Ce-bilee
éerayried that scuation (16) esa Be caleuaceé ia tare cf an ab
folstaly convergest series involving momoats of the intersymbol
ftevfawase:* Frathemices, the moments ean be obtaiaed roadtly

tesough recumvner rolalars, al tho en'outtion imo. grestiy
reaeedl A bsttse aponosivtinn of Abe eel lnanel ean be bedned
bby snereesing the ember u" fern in te pee tein. apoteximation,
HRowovun, the error i the Py os inte falmalie hy te tres ton

fof the erstem impute reson iy tll were,

Tn this sooo we sll vive ne upper and lower burns on te
feove rate al die the range uf snytescy of sr msethaa Ni
tucson of the intersymbol inesference is reqdited, Fuscheentore,
this mothod wil give an accarate astimare of the stor nate with

eghigie ann eompstaton ine
2500 TR mALL SYSREN HHH NCAR ZORRRAL, oCrOMA 181

ss Upper and Deore Bound of
“Lot ihe intersymbol interference be puritoned into two disjoint
ees where

Xe= Earle 0, 5)

and
Xe - Dany— Im. as

Taunton (1) ean be rewritten ne

p,

tow -m ff ert

[exe bly = nts +N + Taya

06
Prepoiion $s Be over bal hy
Pro Wom tied [eer

[leek wre 1 Parti aron, A

rrsvided sho trunested syst hse “pen exe pattern,

Hid — Zire 120. os
Proof: Toe complimentary ere function iy guneee ups Tor

following lationship
cfe®, 220 (18)

tuegative valigs of fs angannnt and atic

Joriele ta) + bein te a)

Since Xx is symmetrically i ound toro und Ty sane
fequation (1S), ne wba, by applying equation (09) that

[og Pew tera 04 sur aaa

[ooo

Substituting equation (20) into eatin (6). we obtain the Tower
hound af equation (17),

by — rh + ME /Re dy. 0)
Dorray snentat uauon mast 500

Proposition 2: P, is upper bounded by

tem nmi fy?

Peet xe 1 xsd ayaran, em
Ber elie" cy
cE- Gam Dem + Det Ie)

sudo i dine in onntion (1
Proofs Avplying Uv Following impli,

esp J=XE2"| 8 1 PD
ve eatin Qi), we tain

Ps [em

es [ole ait ira)

[Snowing from equation (258),
Ka= ov!

‘the average over Xp ean be performed, we thus have

-4

eal 2 | Malte} ated

= [ise

td 4 Kylewity — Ee,

‘whe t9}}. moun exportation of wl. Th Ins be ahown® thot ke

foto inennlity Woe
os four ay S exp (08/2) — esp ba" @n = 1G = HM, CAD)

Subatituring equation (2b) into 244) we obtain

[ogo tale a4 Kana wm

Sew lly (6) | Xofebias'l, 0)
2602 nm mer, sen TECULSCAL JOERNAE, OOMOER FL

sone of given by equnon (210). Suaticing equation (24e) into
(23) we obtain the upper bout ueuualion (2a).

Tis inforsting to note thnt the upper hound difrs from the lower
bout sly thnoagh a modiieation nf Uw nase power by the eraneated
terms, Por a aystem wit peak dinrinn® In than unity, by taking
the set By lnge enough wd anoraachess8z0, of approaches onthe
hipper besa eoxverges fo he wer bound. Therofore, the exe crue
Hbability ean be loeyted within » small range. The compatation
time involved for lange enough ie rather minital whan « digitl
tent sel ae wl etre in Becton V.

42 Boauation of Pend P,

We hove elnanly shown in Ref, 6 that equations (17) and (21) com
be expanded ints isp alsolately convergent series involving moments
cof the tevatid intersymbol snterferones

Wy sven expanaon of equation (17) is

oa

Py = [2m ~ 1yfm) ore [ore

Peay exp 7

itary

+ mat Ey
Merri Mey (a)
wes

iy «in tw Horie pokyaonil
Moy fe the 24th comet of te mondo sine

“Th vores expansion ef wouativn (21) i sue to equation (173,

Po = (nin — HP2ml ee [reid

tan = no Sartre 0m

se caaitbepts} eo

‘Te moments (Mya) ean be abuxined throng the chamtavisti
ores, avormae ennon BATE 2595

function of Xy 9
function, The rerence orm Ie A ie

[Dem (coy

howe the axpliit evaluation of the distribution

2 y)itex of DEPP — 2H |e |

where By ore the Bernt mm

48 Truncation Rerar Bowl of Rove Pepnsion

The enor inoureed By truresting the series of equation (25) at
1} tera in ioe by

Be = Hem — aye) See 20) dF ew FEIT
aN Ha es
tet
DEI ml ew
1 et a sy
Haan 6 Hes PMA a

or (J — 1 >> 4, the Howie yolsaonils are apoer bounded by
[iyi | 22 NB NVR Tex el. an

Sustain equate and C31) favo eyaion EAN} we wan Ue
fall

TR | 2 (Gm = Neon i@e) Ease LL

abner Sayer — rte

= eoidert exp LP

1S.on

= Pep — 94
2504 yin re, Sve oremeAT. roRMMAR, oemOnR Ter

FRR

HL ven ae!

On Be

‘whore pina integer which ie chosen to sain'y O'2pe") <1. Sina
‘rcnexcisn error bounds can be abtened fur F

‘The error probability of « 2M-ury digital AML sytem with wn ideal

bonds iting pubs igh” operating ovr ido rind selena
by ecuitons 25) and (26) to determine the eaxvongeats ofthe method.
‘Tae recived binory puss ie assumed to be

bin ee PVCU es)
"The syne SNR in dned ny
SNR ~ ay"

oye @
1Tke eaneergesee ofthe series expansion method je illstated in Fig. 2
‘The avetom is binary sith the sampling instant deviated by OOP
fron its noonial sariling instant. The SNIR is 10-aR. The set
Indes (2 elements, Ley fl, BT nerve tak