BSTJ 50: 7. September 1971: On the Intersymbol Interference Problem for the Gaussian Channel. (Wyner, A.D.)

3 MB·English

On the Lutersymbol Interference Problem
for the Gaussian Channel

By a. D. WYNER

veered wi te HeingeCaloe wal for
faceaton sonnet Galeger! pred i

oli schon te
tees i aon ito

ference fae pricier nel

tht fhe Condase-TPagier coe
nj here Se we

Hirt a

1 a8 inpoenut pen pe

lemeutciy sir ne ot deena 99 reponierag he
Ineo, Pally pom what we
eng the sty he me

J thie popes we are ermoemed with the Holinger-Callugr mo‘)
fi the eorsneua-ine Guussinn charne’. allege rover a cod,
Cheoreat “or uh'a channel, and Coca ne? Wane chawe: thal tis
theerem zeman valid wien fhe cert of fterwrabolintcrfereoe fem
Irovious rrurl sage is tam ingo seuosel, We shew hare tha

ndaro-Wouver seit Foils rade euatrlad wren spots.
Further, our pruv" is see eletnertany, afer it core aot depend oo
reproducing serve! Filler epace shuory Vinal s, we pase Naa Re
fost san opertont je pam concerning te eeilily of Ue nde

Tu Une Haleagee-Cullager mesa, che chennel om is

fo aeilar—

w

wos fe ants
sihere 3) ie the shane! inp, A athe impulse osname of a ess
Tincar filter, av a) sw sump from a steiarnry Cretan process
‘vith rworeided spereal denaty (fA cole GW, 7, 8, 9) for tais
Henna be seb of AY Tunes La] wit appr: on tae incerval
[0, Ph whieh sty
2) oR KELL Bre TELEENTEAL JOUNRAL, SMEAR AL

. ®

- [cow

(Qhus 9 i tae alownble everuge sual “oowur"l, vege wish a set
Gf A joint Borel els [f° of fovea dined af, Ph ah Hat
the erm peobabiiies

“Th random feta yi = ¢ = TY iv ehven by UD with 210) — 2
Gallager vas the following eamumptions|

Cxowee

Peuje "nae Foie caem tf) Sue:

{eh Gly ce ha te apy fi sane
th lows averse ig! power 38

ca PT onan Soy? HWDE gh ay

cee Pre finn EOF oh ay

(ee an

where the somber By ie defined (uniquely) by

si .
A et
Fev (Ong bth reagan

AEE aul Ae my

tl 5) oe Foo

How suppose tha: we wisn wip the shansel i succansive second
incarvala If [oe}"y ote seb ol AF waoke Savesione wth apport on
19,7), tben the channel wala Ree 5 € Trl be glen hy

wt = [hit amide 0

where 1 2 &
Mem Dele = nD

repmesnte intersymbol inteaferunoe in he inonval (0 7] sign
foeresponding to previous intervals. We use yt) instad. of (0) to
inten te peveer of satareymbalintenfeenes, With the channel out
‘ut given by (6) we an redefines cod (M1, 7, S, 2) a a et of AF pate
Tet, BIN, nesoly we abwve. Tee, however, we eequire [instead
of (3) ins

a, a
for ll povite Ht. "Pe rendoma funetion yt) i given by (6)

CCordaro end Wagner suevceded ia etublnhing the validity of
Gxikgers Theorers for this modified model by meking the following
‘aldtionelssromptions, Tis Gets that

Pr Aid eB

[ties v4 pa « ®

se hae VG) = [EVUPIE for rome Cy(P) with inverse Fourier trens-
form {#1 =U. €< I The aeons astorapion ia tt (2), the invereo
Fourie teaawform 0° FyIfi/0-4), ie bounded by

ho sae", o

forvome de> 0.

To Beet TT wo ahow thee he Cordara-Wepner resus olde ih
condition (8) malas! hy he ‘ellowing essential y tle oondlition
Fy (8) we car Sad u Gi) such Lusk Uf) = [GUN * ad the Joone
fier coreeponding 0 {1{1 nits aver ner agua Lat #8) be
the inverse Fourier tunnfor of Fe U]/40(}). Then the new condition
Sel for ¢utiecely nae:

le

for omce A, 8 > 0, Note aly (4)

Cr nae esenet

that (2) isnot a very song adsivonal sasumption
a Upen Lrblem: Un wer to ae ieva tae corer probability guarastced
iby Pheopest Jorn nel simi a vanishing ear probability fas Tw)
for A Ci, Caliagoes pat enquires the recsvar to meke arbitrary
preeee metsuromeni« Ge for example Lemma SL in Ref. 2). A
2358 np am svdrene MBENSTUAL JOURNAL, SEDTEMER 7?
racteal sytem, however, inpwees cts Fneatons an thy actu
‘th whieh we ean mae mensieements, Thevtere a neacgnable requ
ron fr te ing reins yf fe th flowing

wish a>», (0)

netinek Crook [td

for somo » > 0, Thus + is w mcacuce of the gecurtey ofthe msosune-
rents at the dour, The channel capacity sill therefore depend on,
fay 046). A uit uf nena ght be in « Cx). Wheihor ot nok
hit Ue ses ws er) sw

Tat i eorsider st the pmublom wie no interferene fen previous
haart apes kon the num 3) bus rapnnrs a Uo lore 1D, PL
Galloger Lemme $5.1, p. {19} shows that knowledge uf the funetion
(0, 0-5 C57, ie vapivaert tu kamal of rt olan ©
fect) coo} Phie verte nas be een i

youn ay

of eacstoally indepondsrs standard Gaussian

ww fallona, Lob

where 2 it a sous
‘rites wed ube weelan is dt

vin ~ [ke einde, am

‘where #6. dened in Section Td (othe chanel input, et be
the subepnee of £3{—=; =) sponned by the erthorome) funetions
160315 elie om pA Role 2 Tad Py a hs psn of
tan the ssp 8. Mla = (a, ys oy ahem ete en
Co 0) i se expansion of Po) in he bain [2.0

‘We il note any pres a the 80 expen Fa tho Fat hich
follows tory Ue eaeliy the flies oarmagunding tw CUD and
BD FOG] thet 8.0) has uppors am the iterval 1, Th

Tat [21% be a sot eles wie pzueners 8: = Sy an
Ta Ty Lela, be thew eneresjading v2) =), al ek =
1, 4 2. Then if the mlniasurdctaice decoder is used

Bu = Prete = Pe Yili yw 1B Iya, |

=P YI ez Me, —w) i

WY lm, ade sll, wT,
Sager 00% EYTERTTRTANE 2358

shone] [* denotes Blea neem ord") dovuter ier prduat
Ta partieuar,

Pos Peng B® Peta, ws the ow IP
Bia —w ID, Ie i)
where $208) — SE Bete 7 dy, the complementary error function,

Now, le us auppose that we nee given s eode (M, Te Sy, 2),
Ie.i)y BOLE for the ninterterone> made We eso sume thal
the 8, corrsipond to the minimumdstance decoder. We now form
now cade (24), 2] ith parameters T= Ts = (LP AYP, aod
8 = 5; = a8\/(t + 5) for use on the channel with intersymbol inter
ference (@). We set

ato fe O,
0 Mee

n
atane=r,

a3}

whore of > 1 and @ > O are eebicary, Neve that ee ve allowed
Eur Ua f with 3 tesa ean) inputs, We wk apeity the
‘leooding ors 5 belo, mentioning Lere only tbat se doealer will
laberve the revived waveform p') enly for 0 < ¢ = T,

"We ean diveretve (ve chartel rsagty a ahove and consider the
hannet output Gshen 120) fe pu be given by the woe

=a, 5)

‘where wand 2 ar
Of provions shanna
Cents i the expansion of Pai in tzz

wo EO

ly ae (10) ant Gh reprsants the fect
‘athe vector whose eaonintces are the cote
(oan

“xe— matte ana Saw, an

where 1
"The decoding regions BY will correspond to the minimum-distenee
eee, ie a" Be i the eoreeponling brow in Buen orm
tora, then to lle, (joo) Th for yen
Pristenth - PUL

Now let a S; > 0, and RO S 2 < Cy.) be given,
the (VF, Ty By hs} ende [Os 15117, doused above, be a act of ones
which tiny Theure: 1; Lat is

aM

eo aul exp | U

S)- 8h beltal
2080 sue weLL AArRA mIerSTEAL SS, SRPTENED Ant

Wo will show chat for 1 eufciontly Large, the derived cede bas
cvarameter A 25 2, Thus we pill have ‘ound e sub of code (M, Ts,
8, A) fas the model wath imrsyauelinloarenee with

5 = wi tm)

Sa giiy. weenie tan}

and

sam LMR Be 8
oP a+
Sinn A, ) continous avd w may fe cise airy elo 50 1
tnd barat che fosexo new ave coals Tien 1 for
th intenynba intreence come which oer rnin rut Th
trains estaba hat tne eo prbebli ere eve code.
‘We wl do fhe by shoving tie or, slice To) Tor each

Pe Davee ard al patie

Te GMOeRH Pell eB a
nog (18) wil ello sets fo: th tlloving lente (i
Trovit ash col hy rn)
ina fo Tey (15) soa, i

th

09

Lari 8: Bor th code If
nin | ae
Lenn te day 2, PS OMT).

From Tammans 2 and 8, condition (1) in Lemme 1 il be entised
foe, sulin Wag. Ts wl (15) bested for Paufiersly
Proof of Femme fs Since B, ond. 2% are the minimum-distance decoder,
he Tell mere of (18)

Pr iyteat) = PUI et ome |

Ist an, ill 0)

“The right mombur of (18), Pr fy # Bu} ie pvon by (13). Consider the

Allvet— au, |] = lv ~ ot, |)

wegiw—al en
svar anv 2861
‘But by the Spotless of Lema 1 (5
Qe wile lables:

“thus the event Sn (20)

flaw, 09 2% ue — a, |

emma 1 now flows fram (29), (18) andthe above
Proof of emia 2: Tor tho codes ( (3. B
Dr lye Be) S emp (— ER, 8) — OT + TDI
a that irom (1)
SG ||u,—¥.[) S exp 1 BE, 8) — AT, + oT

Bios, ee Hy
ila, — a IP = Stet, 8} — en, + oft,

‘hich ples Tema 2

Proof of Lenora ds Lev eb» te union alias with P05) wit

tal Ley son hat ie) = Pathe Pat) | wee |

Inline the 34m,

to rsn=| J Roiea

Dot singe all functions is and therefore & have auppost on (0, 2)

ate a

sisilfeoay
vas 8 [f'xoa]
here (i die in 1

Consider fine the = —1 term i the shuve summation, For
osren,
202 rit nm aySEEN sucuniow SCRA, sueranUEE Tt

Changing the saraie of integration to» =~ 1, wo hevo
= e5in [es 60

mci (9 fvaiety e

sd therefore

PATS, gent
Tran

[eons e

Next consider -2 = > <0. For 0
hove sie wr obeain

5 T,, puclleling the

aged |

splsing condition (07, we be

aorsese [~ 1

ant
A Lis wha)
ean
weal
- ast
ay asa + ay am be

important fut hone in that enn
the tear

wl

jn 1), Taeefor,

est
w+ ama t
Sabatitating (28) and (24) neo (22) we here

fens

en
smash unstated a8

Hens rea, Sy hfe eas ot

Sinoe the suumation eonverges,
Lema 3

ve have |]? O1T*), whieh ic

1 Gage, rman Dry and Haile Commi, Ne Yor
2 Conhma dT) ent Wagner, TZ, "uemaubol Interference 9p u Cuntinuyae-
"Tine Gigaiat Cousecl® i Tene Tne "bare TPIS Sur

4 1h tomer, C. Lin Son Thre, Now Yok: Me