BSTJ 60: 2. February 1981: Fractionally-Spaced Equalization: An Improved Digital Transversal Equalizer. (Gitlin, R.D.; Weinstein, S.B.)

10.2 MB·English

Fractionally-Spaced Equalization:
‘An Improved Digital Transversal Equalizer

By R. D. GITLN and S. 8. WEINSTEIN
(Manvecrbt recsves Septomber 12, 1980)

Here we deseribe and demunstrate, via analysis and simulation,
the performance improvement of voiee-grade modems which use a
Fractionally Spaced Equalizer (F9&) instead of a conventional sy
chronous equaliser. The reason for this superior performance is that
the rom adaptively reulizes the optimum Knoar receiver; consequently
itean effectively compensate for more severe delay distortion than
the conventional adaptive ryeolizer, which suffer from aliasing
fffects, An additional advantage ofthe £55 is that data tranamission
an begin with an erbitrary ramping phase, since the equalizer
‘nahenizes the correct delay daring adaptation, We show that ant
vor combined with a decision feedback aeetion, which can mitigate
the effet of severe eoplitude distortion, can compensate fora wide
‘range of linear distortion. At 6 Bbit/s the Paw provides a 2103 4B
tain in output signal-to-noise ratio, relative to the rynchronous
‘Gquatizer, over worst-case prinate- tine channels. This translates toa
theoretcat improvement of approximately tea orders of magnitude
fn bit error rate

1 wermopucTiOn

[As ix well own," highspeed (248 Kbit/s) voineband modems
nt employ some 40rt of adaptive equalization to achieve reliable
performance in the presence of Tineae distortion and adtive noise
‘The equalizers are invariably implemented using transvertal filters,
tut the question of how the laps should be spaced has hen, and till
is of grea theoretical ax well practical interest. Conventionally, the
tcjualier laps ara spaced a he reciprocal ofthe signaling rate. While
ithas been known theorially that this synchronous structure does
fot, hy itself, realize the pimar linear filter, i haw up to this time
provided adequate performance, ‘The continuing demand for improved
performance at 9.6 kbit/s has rensved interest in adaptive equalizers
‘whose taps are spaced closer than the reciprocal of twice the highest
‘equency component in the baseband signal? Aa we shall denon
strate, auch Fractionally-Spaced Kquslizers (rats) are able to compen
sate much more effectively for delay distortion than the conventional
synchronous equalizers. Consequently, we will show thatthe perform
lance of PER, with a miffeienL number of taps, is almost independent
of the channel delay diario, and thue of the receiver sampling
‘hase. More generally, the vou is able to adaptively realize, in one
Aevice, the optimum linear rocsiver, which is known to be the cascade
ofa matched filter and a synchronously spaced equalizer

‘The purpose of this paper to report Use reals of an in-depth
comparative anlyLcal sed simulation sady of as and the convene
tonal eynchronous equalize, We aloo evaluate the performance of a
squalizer which results when a decision-fedback section, which is
particularly effsccive in compensating for amplitude distortion, is
Fembined with an eRe. We prezent simulation results that compare
the performance of practical-length syuchronous and fractionally
spaced equalizas over a variaty of voice grade private-line channels.

‘Many years have elapoed between Lay's invention ofthe adaptive
synchronous equalizer Gersho's! and Brady's" carly work on 353,
fand our prosent intercet in factionally-spaced cqualizaton, ‘Thi i
‘deta bath the inetease complexity required to tmplement the PSE,
tnd the relatively aatitactory performance af the conventional syn
‘hronous equalizer. Recent investigators have regarded the far pr
‘marily a5 a moans for mitigating the timing jitter produced by an
‘eavelope-derived timing recovery aystera*" Our viewpoint, however,
ie that this propery i jus an example of the salient feature of the
‘rar—the ability to effectively compensate for an extremely wide range
St delay distortion, and vo deal more effectively with amplitude distor
‘on than the synchronous equalizer,

‘in Section LI wo deseribe why un FS hus the ubilty to compensate
for an arbitrary receiver sampling phast. Performance, as measured
day the equaled meaneaquared errr, ofan infiitely-long passhand
‘a is derived in Section IT, and the coreqponding results fr a finite-
Ingth equalizer are deceribed in Section 1V. Simulation realy for
typical voice-gade chanel, are presented in Section W, nnd these
reeults are used to compare the performance of the syachronous
‘quiver, the i, and the mse with a decision feedback section.

I. BASEBAND DESCRIPTION OF FRACTIONALLY.SPACED EQUALIZERS.

‘We begin with a brief discussion of the ability of an sito compen-
‘ante for any receiver timing phase. To do this we nood the transfer
function of» hnscband factonallysmaced equalizer. Consider the

276 THE BELL SYSTEM TECHNICAL JOURNAL, FEBRUARY 1001
ood signal
P10) = Boaflt mT) +10), o

where (an) isthe discrete taukilavel data aoquenee, 1/'is the symbol
Tate, 70 is the system pulse resporar, and w(0) i additive mois. As
Shown in Fg. 1, we denote excess bandwith ofthe palze fle) by a
‘The inpat toa cunventional ynchronau digital equaliver are eampls
of the fillred reoeived signal atthe istanca¢= n+, Le,

Hn +9) = rst) = ZagfinP— mT 2) + vin +8).

1 noiseless output ofthis nonrocursive digital filter, with tap weights
(e), brthe sample sequence

uaF 42) = Zonhind — mT + *, @

‘where the equalizd pale samples, h(n), have a (Nyquistequivalent)
Fourier transform

aw =zoe ry e(uee Beal (ork

= Cut Pot “i
Here, Fe(a i Ube liad spectrum of Fa), Co(u) isthe perindicy
‘runsfer function of the equalizer, end ideally, the equaller output is
‘he data symbol ic, win +3) —

"Recal that the Nyquist equivalent or folded (aliased) spectrum is
the relevant transform when dealing with sampled-data ayrtams. In
particular, since Cr(o) = Cro + A2s/T), the synchronously-spaced
equalizer can only act to modify P(t, 2s opposed to deetly modify
Jag Fle)!" In other words the syrelronous equalizer cannot exorcise
independent control over both sides of the rollotf region abut

PSE: DPROVED DIGITAL TRANSVERSAL EQUALIZER 277
OM

gt real ‘eeenegein ep
2/71, because ofa severe phase characteraic and x por choie of
nulls ereated inthe rll portion® ofthe folded apectrum Ft),
then all the conventional oquiler ean do to compensate for thi nll
inco synthase a rather large xin inthe affeced ren eis leads to
a cevere performance degradation because of the nose enhancement
st then frequencies.

‘Consider, on the other hand, a reesiver which uses a fractionady
paced equalizer with taps spaced T”< T/(1 + a) aeoonds apart. This
esurlizer has the (periodic) transfer function:

Cte) Zee o

Note thatif/7" = (1-+ e)n/1; then tho frst repetition interval of the
transfer function Cr(a) includes the rollof portion of the epectzum,
as show in Fig. 2 We assume that, fr digital implementation pur
poses, 7" is gnerally an appropriate rational fection of T: For an vot
receiver the equalizer input i sampled at the rate 7", but the equalizer
‘output is il sampled atthe rate 7, since data decisions aze made at
symbol intervals. ‘The equalized. speetrum, just prior to the ouput
sampler ia periodic (with period 2n/°) anda given by

te) =rt ga(oor3oi(ooa%)]. 0

and fo systems where #/7’= (1 + a)n/Ponly the k= 0 term survives,

Brie) ColelFiaie, [ol ©

‘The salient nepect of (7) is that Cr(a) acts on Flee before aliasing,
swith reapect to the output sampling rte, is performed. Thus Cr (0)
can compensate for any timing phase—or phase ditortion—by ayn

Tei the eaguoey range ale 2 ala

278 THE BELL SYSTEM TECHINICAL JOURNAL, FEBRUARY 1981
thesining a transfer characeriacc of tke form e°". Cleary, such
‘compensation highly desirblesinee it minimizes noise enhancement.
nd neoids the extreme weritvity te ciming phase ascialed with the
Conventional equalize,* Afler sapling the equal? output at che
fate 1/P, the output spectrum is periedic with period 2/7 and is
fewen by

was =pie(oe 8)

ger(en'E)o(oron|(eere)y]
ott festa sho a of in sa
“ptt én sae ot
Seen

een aati ch men coon
santana Wik op ssa ae
ener tint ete plier tne et
ee ae, esate im eee nat
oll wt

IW. PERFORMANCE AND STRUCTURE OF THE OPTIMUM FSE

"The receiver msimiring the mean-aquared eror is kann to consist
ofa mched filter fellow! by @aynchronous sarplee* Our discussion
fn Section Il demonstraced the equivalence of an appropritely si
pled fractionally spaced equslier with an analog receiver, We begin
by writing the tanemitted sil a(0), im an i-phase and quadrsture,
fF quidrature amplitude modulaced (qa), data ransmiscion system
fs the real pare of the analytical siznal

He) = st + Jat) = Ye

nie nTe o

where da denotes the complext diserete-multilovel data sequence, a
Titus pith the (generally rl) haseband transmiter pulse shaping
It's the syinbol rate, wb he vadien earier frequency, and (0) is
the Hilhert tanaform of #2) In ose presentation we wil make exter
sive use of comples notation to devote either passband or in phame
Sand quadrasure signal, a wes svatem palee responses. A ascussion

say stern rel hn oe pea
‘pede ep ping eh et he a oe lnc at emnte

SE: IMPROVED DIGIIAL TRANSVERSAL EQUALIZER 279
‘ofthis approach is presented in the nppendix. As shown in Fig 8, (1)
is transmitted through the passband farcund e,} channel x(0), with
Impulke remonce

(0) = #10) comet — a(t) sat

= Re(tase) + s()e) = Refine), (10)
where che compler batchand-oquivalent channel is defined by
Sale) = a) + Jel ay

‘Thus th received analytic signal hus the representation
Hl) = le) + JHC) =F dafale— TOM"? 4 stEVe, CD)

‘where (¢) and 7(¢) are the in-phase and quadrature components of
the received signal (and are a Tilbere transform pain), fs(¢) i the
bbatoband-equivalen received pee which ngiven by the convolution
af Zu(0) with pl), 8a the channel phase shite and (0) is the complex
noice signal

"At thin point we may eonsider wither « petshand equalizer, which
‘operates deol on #10), or @ baseband receiver which processes
710) expl—jle-t + 6)]—aseuming that carrier phase coherence" bas
been established. From a mathematical viewpoint both systems tre
squivalent, and hare we find it eonveniant ta filer the demodulated
igo,

Ge) = Fee mY afl ~ nT), (a3)

wi

1g. 2s tse gem, Yar wea pe

280. THE BELL SYSTEM TECHNICAL JOURNAL, FEBRUARY 1901
day the rcelving filter
0 = 8d + Je. as
In writing (14) we have used the notation g(t) and got) rather than
ett} and gf) to emphasize thal the ceiving filter does at correspond
fo en analytic pula, Note, however, that tho equalized signal st che

filter output ix piven by the analytic signal

Ce) = GUE) B BLE = (ated + HHO) B Carl + sled
= (9101 gil = die) Seal0)
+ H1GlE) © gue) + gle) Osten)

= ue) +300, 15)
lies u(t) and a) are » Hilbert tranaform pair. Aa shown in Fig. 3 cho
inphaso and quadrature output signals u(e) and ue) are synchro-

nously sampled at t = nT" + and quantized to provide the data
‘ecivions dean by

‘82 The mean-squared ear

‘ur attention now curns to finding the linear fier, 2(0), which
minimizes the output mear-aquared error. The ourpat oF equalized,
rnean-equared error (Mee), which ie the performance messure com:
Tronly used for in-phase and quadrature data tranamiasion eystoms, ic
faven as

4 = BUI) = Beds &)

= (ul) — an)* + (HHT) — BA)?

= B80) - dP, as)
‘where denotet the ensemble average with respect to tho data
‘ymbols and the edlive noize, ¢y is the complex error sample, andy
land 6, ate the in-phase and quadrature erors, raspectvely, Far com
‘enonce, e have sbworhed the receiver's sampling phase, into the
pre-equalizer pulse tesponge by incorporating the tanefer function
Pinto the tranaform of ft). To terms of the equalized pulse lt),
defined by

= hi ate, an
the fer ouspat is
He) =FA,Ale— a7) +510, sy

‘whee the fitered noise, 6), ia defined by
He) = HO BUD. a

FSC: IMPROVED DIGITAL TRANSVERSAL EQUALIZER 281
‘With these definition in mind we can write the MSE as
8 = Blan) ~ Aaa — a)

= Elan PEAT) — dy" WP) aRaInT) + yds).

Using the independence ofthe dota symbole and the independence of

the noi sample, Hi), the terma in (20) are readily canted, The
frat corm isthe quadratic form

21) = BL da LE [doe 8 + BLOT EAT]

rut {dma

+ otble— g tale) ed, en
[107 9/*1/2E1 dy, andthe Hermitian kernel, At,

ACG, = S faln = O70 - 0. ay

Nole that (21) can be written compactly ae the quadratic form (i,
i), shore coneaponds to (0), isthe Hermitian integral operator
‘with kernel Aig, 7) + o'8(¢— 1), and (ay) denotes the inner product
J <(09(8) dt, Ty straightorened evaluations, the aecond and third
erm in (20) are seen co be

seaareary = 214) | Tanat0 a

edawear = B14 f flO a es
Combining 20 10 2) the BE, normale by BIA? bos te
compart peat
8B) — EB, BO) — Uf +L (4)

whichis quadratic form, where a! is recognized a being a Hermitian
‘Operator its kemel is conjugate symmetric.

ivan by (24), ia minimized by taking the gradient with
‘optimum fite is given asthe solution of the integral

respect to 1
uation,

[Aun sete ottin ara ie-n, ee
‘or the equivalem: operator equation
abe (25)
Tt cums out chat the solution
Bo = de 5)
can be explicitly determined. This is accomplished by writing the ef-
hand side of (28) a

J :rur-onnr-»+e8.-n)aoe

se fH net i mi

cquating (27) to the right-hand side of (25) gives
Léfatnt 0) + ole) = felt, om

‘which indicates that the optimum filter has the representation
doth = Séxfaln — 0, es)

where the sare to be determined. The solutions as represented by
(G2). is recoynined ae che canende? of 4 aynchronoualy-sampled Elter
matched to f@), and a synchronously-spaced tapped delay line with
‘eights {é,)- To solve for the {é) we aubstitte (29) into (25), giving

J pita nian ay etinr 0 dr

Sez Ohler 0 = ten, 0
and if we define he cha-comelation fancton,

Beas [ior ntinr= 0 a om
then we can evr (0)

EDA adolf bo" Beaded 0) falta

“Taking Fourier transforms on both vides of (32), with respect to the

SE: PROVED DIGITAL TRANSVERSAL EQUALIZER 263
continooue varinhle vee
LE AnbaPaiele™ + ot Lene7P 0) = FH), (00)
where Fle) ia the twansiomm of fa(t). Dividing through by

Fh(o), aver the region where the channel does not vanigh, we ca
rewite (38) a8

Flu} Crle) + oFCrta) = 1 a)

‘where the following Fourier transforms, with respect to the discrete
‘Gin variables are Mentified by

Calw) = Fee,

2a) |*
nfo). oo
1 tanto Fu) copmpns e enehwnulyaed
map hsv TE a Cl Tie mae
Senn nett tee ta ae Po
scinrene

1 1
MO Fo FF" TY Rovere
nd hus the optimum ear receiver ho tens
" Frio
tal) = Ph arse TTE
“The Fl ransom af nee hn of the aid bcd
cian pale, che
ae? ce)

MO)~ py yTFolo + 127

Since He) in veal, the roel part (0) and imaginary part A(t) of fet
inverse Fourier transform are even and edd functions of tm, respec
tively. Moreover, ato” —+O ite ao clear thal Hoya) = (1/7) Sa Hs
+ 25/T) = 1, be, not murprisingly, che equalized charmel is Nyquist
rom (24) te (25) i-fllows thatthe minimized iit

fan 1 ~ (hyad Be) #1 — in

=1= [Ateneo

284, THE BELL SYSTEM TECHNICAL JOURNAL, FEBRUARY 1961

te 39)
Be