BSTJ 50: 3. March 1971: On the Design and Analysis of a Class of PCM Systems. (Heffes, H.; Horing, S.; Jagerman, D.L.)

5.6 MB·English

On the Design und Analysis of a
Class of PCM System:

Dy I, HBFFES, §, HORING, 0. E. JACRRMAN
(aan ied Ste 20,180)

This paper cmsiers the problem of kransmaiting Debilimitad signals
wing binary signating over a niefvor channel, An enaltea famenor:
{e proened forte design and ennigts of alae of PCM ayes whee
pak ern ao primey tntars. Bur psf late of pu signals which
Fctuies determine anpi-onstasned lawainted funtion? as
Telltale wideenaestatonar, wconuender, random pence,
Tous vu clainad whieh unt lenge beeen the warn rae
‘ati ed room ctin

‘This peace come the poder: of transmitting bandied ei
ue wang binary sig ing over a goisectue chavwel. A funtios]
block diagnars of the tye ow POML syst user eonsieratinn it
a is Fis.

Tae toe mor differen letween the prodiom consid heze

nt previous sor ures nessune 92 stem: performnner at signal
fleests considered. The swsasnse of system performance saucy eon
filered. ie. rol lo the aitegral-mextexared sor! This

{xne of performnos sass aesell to a lrequeney-domsin
alysis. While this evterion i widely sed, it does’ provide direst
ifornetionrogaring ti ivf the ever as a funeton of time. To
Tivetigate tht tive belvesinr we vee a fnseomein appaeh and
tap the main error over Le we sneseure of syetem performance

resus of the desize to got il: igus thet engineers would
orally eal bandied few, eiaamete polynomiahy, snl a
‘well ex functions ia C4(—=, 0) with five bandwidth] we (at a

ste oye CM
ied Tae ete ws na
918 gre syanentmcuemsenl souaMaL, ste gar

woe eH st a etn

Fig HEPOM at

somewhat broader class of signale chan orally eonsidered in the
Teeeator, (Che input else ia usually conidered to be = wide-conee
stationery random proves.)

Ts tis papar, 1 fachnigue ie precented which cimultancousy slocts
the carpling rte, quantizer end reconatrution filter in euch 2 way

o minioiae-w bourl om he pee eeor between the reconstructed
tnd teunsmited-senals. Since our interest center on studying the
trade-offs between the variour ayer parameter, we desire prviee
rosthomatieal results whith eae vad over the entze range of posible
syeiom parameces. By using ax upper bourd or ayetem performanee
tie sim was achieved Tar sevren! claw of input sigs. Using 2
cera wach evaluates 2 piven enooding decoding wactie in tonne
bf performten for Che worst xo ia Ue palolney, ape Be
tsoding-dovodirg algoritm, which wil be ealed Caitorm PCM or
CECH, a aipgceted ane. evaluated, The acelts ary peeeced in the
foe of & set 0! nonmaliged curves wi Plot an wpe Hed st
the pereentage ero> catoeicted with the aeeposed UPC spre ee
‘function of a rormlized parazczer,p, which ropronent the eetio of
the bic vere vo the bexcwdth of she input clic. Te epeimizee values
of the various poramaiars wie deine the erstm eam aleo be deter-
‘ined trom che plot (these inslnde the aezapling ents, the ramber of
quantizing loves snd the delay sesocaved with the decoder)

‘The elas, of Funokions emis af wtineFanotions of exparentil
‘onda one ead type e whieh are bousided om the nel aes" Tt inlides
fonctions in (2) having Sita mdian si, wn
‘wigonowietie polyuomial of degree

The penforaes of 9 giver sjete: ie detined by

lem Segre ba fre

1 dots eta par A eno

ss Sincontt tant
rox sree a9

= gap sup Ih ue — a9 |) ©

ete) Ue wa of then rignal a timo nd) nepresents
{he vaio of the output signal at time ¢ eaxresponding to the input
feneton ws 'The int fanetinn aia a element of B, and the values of
tho put ead ou tout igual a any time fe 0, bolong to « zormed
Tinoar sae, <2 By oper enoion of Bund the nom or) w aot of
‘Sffenent ingot classe ead porfrmines mcuniees one Tie tented
Spwitsly, deine the sot BAM) by

BAM) = las e Be 0) €8 || 210 || SAE

{i arriving at the proposed algorithm, the folowing represeratin
fs ged for secant of Bonde BAAD

aig = visnean e

where

ay = ®

5) = ele o

"Thie representation eval for any 3 (0 1), henge 8 muy be cueen
avoropsntely Zor exe appeation. ‘The parameter & wilh elle the
Tenet) guandbanal snae the time belwern samples ia x(k ale
ici lee then the time berwsun stones x whith: eorerpond to
Ue Nain eae

“Tent: he paoceed algosehm 3s seems for upprosinaling any
emet 0” BfME bone nite ace of oroprately cere Fame
tions, Tse fauelions ate determined by fy tating the itn
wie, 2). This erineation proces proiiene nn apyomleastion to

eam ae a
This fo penal ene afm seamen cles af

{"Fhe meule resus the aber ie opleabie he nso gna havin
seston (evn al WT cl Lae Us Bi
900 ee mt ssren TeenvteN okay Sea Tor

‘Thos we have replueed the xeuirewent of trunsuilting am inte
fsuber uf sump values by Ube problem of teaneaitting «Site
tiie. Tw aebieve our wlitmte objeetive, chess sample Faluen are
‘en tized and the quent samples ere used to -eepnsrack

esx

oft = "SE" sane, c)

‘where i) cupresents Hae quai elt of wh
‘The eacoder will sherefore ceusist of approprite quantination ofthe
ioyut samples wile the dover simply mest thy roonieuetion

17 te Load vies ing (se yuatized esmplie |i, the eg
(2) gies io equation (An sra’og itarpretation of the deco g
Drow is dria i Seeley WU boll bo not at ie one

onstria g(t) gesording ta enue, (8), «delay of P ~ 12 evan
reauirod, The ticeofe between thie delay gnd the aecuraey of 2
onetration will become war a the revlle uae oreunlal

Having cuaslrsined the pram Tore of Ue pmmpenel sylem, we
now sont to detarnine tke vations pararoters which define i ow
costing sete and quantizer) 1 seeh nse me te mvsiine as Upper
bound en tke velue e+ Deas: of the binary nota of whe sigvaline
‘hich is =eing consiered, che crimber o: quantzation level is eax
fesained 19 bo 2 whine» fe an ingen Ths» repre the nurher
Of bits per eample. The bit rue is then xiven br

Ho The o
tis convenient define & normalised parameter gy which given br
zB ®

oT

shore wis the cuian basi of the put olan an fi the band
‘vk in Ha. Ta tera other parameter, Fram equation (2), we ave

y-1-8. o

lees

9)
ese serra oa
In tems of these rameters @ simple box on Ue reeonsteuction
error reals

"Lhe onnlyss of the reoonsirsotion oreor, given in che Appendix,
hows tha 6 dsned by equi (1, sabes

WK
+09 oy
where
a
K uw
+1
a) = up ep | wR) as

‘nd. 6 i given by equation (@). Taus the setual valu of e which is
‘nieve By the proposed algorithm may be lee Tau Pe value «given
by eauition (1). Infact, since the remnstruction formule (8 i inter
polstors, the ern at sarple points can never exceed the rast quantion-
le oror, ase). Tha fst term on i ight-hand eid of equation (11)
Iney be view asthe eror dies teuesting ui) as even by equation
{2 while to seoond orm represents the qutatmaiea eur. ‘Te alot
Of the dey on the eer em be easily sen "mm equations (11) naa (12

As n design proceduze, on might fast choote the quarsizer whieh
rinimiaes eo) and then rletermine the value of » [subject wo (10)
‘whieh minimis the aor bound wiven by [pe equation (11)

the nest sections, ever ios apeei ens ave eozaired
and exes ergo esrvee are jvvenced fer cece exec

Th this ease, © =R the seal line), anda]

fl. The
BG) ~ fw eB, WHER [ad | SVEN

"The ayantioe ebicheinimians gly] for this eee ie own in Fig 2
‘The optimal value ef aa) 38 gives 2

4

6) as

fe infematineemton” bows LAT = aC A sheet ei
eT eine ae ee
922mm. averen omnes soma, ane 18T

1g Union cuanine
Tsing equation (14), equation (14) besomes

as
For inter une, we deine

a9

or the present cae Ky ~ 1nd equation {15} thus becomes «8 a &

‘Pao bouud expresetad by equstion (16) ia vad tor any allowable
set of systom parameters For fixed valucs cf», P, and Fy che value of »
‘rich minimises @ enn easly be found. If ‘ee denn the soevelaed

reontng orror ws 100(¢/207 RK), then Ki. 8 plot this as w tunetion
‘of sing the onimized values of . Those values of» @enoted +) ae
indicated on the curves. Curves ae plotted for KuL/K = 5, 10, 20,3,

Ry tron hae forthe spe of wing the ae of the wet
sot aa Far cw atau Ci eon a ae

ft aa get syria mene, JocINAL, wane nL

snd os, Sage LZ Ks 2 ‘ora given yale of the eaia L/
fmocinie nry value of Lalslsing L/L 27K. Using
(12), the eoeresponing valve of 7 eon then be eumputed. The delay
‘amiated with rsoastruction in wecordknee with equation (8) ia then
iaven By Zy = T+ Lh. Av diueuwed in Seeton V, when an anslog
fnplemennation e used forthe reconstruction, value of K— shan
Tyre atl the delay esapeaied nth the futer wea aroomaplies ths
reconstruction i etvon by Ts = (Lt Wit

‘Ta lustre tke wee of those curves consider che oroblem deSind by
‘the foloving parameter yelus

ene x 1
B83 x Why
‘Using thee vais, we comoute p12. From the eves, for L/K=20,

wwehare SDG:M = 67 petoess, ot — 8, Urag if ~ fe have # 0.08%,
Since 8 = 2h, we have le = 070 X 10-* seconde ae 18 = 1.26. 10°
saplae/seou. For L ~ 40, the delay esociated ith Une eto (ang

fouation (@) to aesompish the raeonetraction) is sbout 32 ge or the
cormcponding aalog implementation (Se Section V) these remlts
forepond tn n vain of Fo 20 (and AC = 1) with x eomosponding
Ahly oF 188

Teds clear that she reemnlned sugar the eure in ig. fait
‘hein use in aide variety af seaye, Foe example one ul case
Ansar questions such a

For given i
‘ce al gua
Tevet?

(6) Tor a given bit ets, rhan Za the Inger alas of signals chat can
be aandind wath « mscimute eror not exceeding a pmvanrined
Invel?

In the nxt seetion, anwlugous ceslts ae developed fer some

important clases of randoms spit signa

of Signals, whet inthe smallest Wit rate tha
em a entrant exounling peated

Ia this eoeton we conde the eaze whens @ ia the apace of sero
ren seoond-order tenders werinblee* with norm given by

we
res evens os

Then

BAAD = ju sw eB, wld) eB, ee © MI as

‘hus AyCM) conta of weed unr utlors prowess (9th ey) © AD)
svith semple Sotions hie all i B,~ Por Th impostant speci
ove of akdestam etationsrr avian proses, toy — 2, Kor a
fase {ve quantizer willbe charweterzed bey Ube propery Unt i elai-
fnizes she mvan-equared err al Ihe suxipie timed. The problem of
Tlsgning sack quanticr for» given probability distibstion of the
Jngat saiplitade Bes beow emeideend by B. Smith An approximate

tes bur ne pins aie ore ea by
aseca-2h[frma} oo)

whem pi) ete pes cauiby fonetion fe). Th enresponding
quarter # deer in Ret. 6

‘For the mpm esaee hers pn) is afer, yoann, a expor
ential 2 on be etiam the fe

~ he en

‘whasy Ke ie a constant cresting aa Ee pualiatee tition fans
finn, Teale I gives Ue value 0° Ka “er fae of fa input clases of
innsiest The osnd ov the saeon perfomance wat $8 given by
rian {LL} ean sas be wrile> me

_an aM. F
verees U4 Ie ea
wich is onsen form o the corres: Imi for the deter=

Chine save fore eqnatier (1B), Tae aga of fe vhrirnaalseie
furves of Fis, & tov these eeser is thus ienaea, ly te preci
Aveoribed de-ariastie es

ts ineerssting to wots stl for toe ease of a uniform smplibule
diseibation aswel ts the wostlnde eoslnined spat elas, = 2,
td the optimal quasars i sani qvantiver. Tu earh of hee

Tr the next seein, the isersreation wal implemestatiss of the
own sigan wil be drone Twa he shows thatthe wale
Tiniaretation eke the sere: of « PCM system where the sapling
026mm asuuaveree meus suum, acon oT

‘Panu 1_ Len Sree Desenaerroy

bo sat) xo = a *

ate, quancser and Jon-pas: filter wbich is used vo accomplish the
Arooding ave ceronty charen

Tn Wis sention wp dian an arog recerstmiction in the UPCM
system. For ‘lw analg yeeneseuctign, eve apeeem tees the corm
‘hows in Fig. 4. The iseanfeation of tae loa-page Bar eslts “am
he flowing analysis

Thowe ft aise the seporae, BF, ot eaura,etatimnary fer
[itn impee response 21 | 2 the Saput

wie Som aa oa
wn

a9 = "SE saan am coy

We now conser h(t to bea delayed, coasted (for negative tru

ee ee [=| =

Mig Analg TPOM ete,