BSTJ 60: 9. November 1981: Adaptive Equalization and Phase Tracking for Simultaneous Analog/Digital Data Transmission. (Lim, T.L.; Mueller, M.S.)

8.4 MB·English

Adaptive Equalization and Phase Tracking for
Simultaneous Analog/Digital Data
‘Transmission

By T LLM and M.S. MUELLER
(Manucernt recived May 14,188)

“The general problem of equalization for date tranamission where
one of the two data sources produces continuous amplitude date
imple i sluied. There are various ways ta configure a modem for
such a transmission scheme, and we deseribe Row standard quad:
Feature amplitude adulation structure can be modified to qperate in
(his made, Tis solution eam be spectalized to include various linear
tmotlolotion scheme, euch as single videband and vestigial sideband,
‘Theoretical analysis shoun that adaptive equatization and adaptive
phase tracking can be achieved with wiilar quality owin Ove fiiar
thigital only modem. We provide extensive computer simulation re.
‘sults which eonrm the uality ef our theory.

1 wtRoueTion

cently, considerable interest araso in Finding ways to Lens ani
receive digital and anslog dnta simoltancomly aver che Davie voice
band telephone channel. We investigate a ayer wating quedeacine
amplitude modulation (qa) whore dixie and analog dal modulate
the two quadrature channels. The same scheme wax independently
proposed in Hef 1, The wffwof various channel impsinments, a well
fs an imbalance of signe] powers Ietween the digital and’ analog
‘Sqnata,on the ertr probabilicy of the ayatem has been studied in
Rar,

"a thie paper, we onalyze an adaptive equalizer for this type of
Dbybrid modulation using a erese-voupled transversal fer as deseribed.
im detail by Faleonor in Teefs 9 und 4. The diffoenec here is that-one
of he tve quadrature channels transmits analog, ic, continuous am
plitude, dats, I the womminieaion vtgnncl be tiene invariant, i ix
possible to toxin Uhe onwaliver sith gil daa in ot squadentre
hannels and freeze the equalizer tape aller convergence. Analog data
‘om ten be sent: However, posible change i channel characteristicn
‘warrant the use ofan adaptive equalizr wo update the taps continually
Sinoe references forthe analog daca are nut wvalable, especially when
the recehver i in a dackion-dirocted mod, the wpdale algorithms
reported in Ref. 3,4 az0 not upplieble inthis ease. We propose a
todifeation whieh only minimizes he mesn-aguare error (me) in cho
sfigital datapath, Thus ony de error sigaal from this path i required
(er adaption. This nnolysis ix stla co that in Ils. 3 and 6

‘Reaults dram compute sinulations are given co verify our analytical
resulta, We abecrvod thal because the analog data doos net sd in the
‘equalization, t otnilly acux as an incerferer. As auch it would sem
ftdeantageows ls resuc Use analog signal powcr hus, unbalancing the
fystem On the other hand, this would degrade the analog 3/3. There
fore, there fa ead off in allocating diferent power level che two
shannels depending on which signal core important.

[nthe acheme doterthe hero, we astume that data symbols aro sent
every Tsooonds, Ths the nalog signal as co be limited in bandwidth
to 1/27 in order to avoid alinsing, Alteratively, the to quadrature
‘channels can be unc to transmit primarily analog data with occasional
figital data for purposes of equalize updating. Then we could have
fone high-rave analog chanel ain alow data rate digital channel.

MATHEMATICAL MODEL

‘The general Qaw tennamission acheme af Fig. 1 considared, Two
data sequences, {an} andl (2c), are applied to the in-phase and quad
Tnre inputs of the eress-coupled cranaaiscion vere with impulse
reepomes £3) and (0. Their output signals are modulated by sine
and cosine waves of carrer frequency ac 0 form the passband signal

ws = FO

Fig 1—Model of curated tana

‘2040 THE BELL SYSTEM TECHIOAL JOURNAL, NOVEMBER 1981
sn-ne[pr.cu-atrotia.
os
ou

a4lt) + Ja @

Tm the above equations, (Dy) is the complox sequence of data symbola
and G0 i the complex impulse reyponse ofthe tranamiasion filter,
"These parameters can be specialized to represent any Enear modula.
tion scheme,’ og, amplitude/phase modulation, single sideband (es),
vestigial sideband (ven), and Qa

‘Throughout this paper ic wil be assumed that at least one data
sequence, for exumple (oy) is digtal. In particular, we sill report
sulle fora sytem with GU) real ad ehare (ay) and (8n) are digital
fd analog data sequences, eapertively. These scenes are nested
telhave zor mean and the flloving correlation properties:

E (data) = Pade (4)
a)
(eo

Afler passing through a noisy. dispersive, and phase-jitered channel,
‘the signal atthe input tothe receiver enn be expressed az

aorta gosta) =

‘where Ut) ithe combined complex impulae response of the trans
‘iting fer and che baseband component of the pase-band channel
‘dO with respect to the carier frequency x. The noise proces) it
independent of the dita soquonces while the phase stift #(¢ caused
by the channel ia mune to vary tach more eowly than the channel
Impulse eqponte, f(t, a is typiclly about 10 degrens peak-to-peak.
{eis mutually independent of the additive noise process, ax wel a of
the dara gymbola.

"Tho genoral qau receive ashown in Fig. 2 The received signal ¢(0)
4s bandpeoe filtered by the phase splitter pair with impulse remponse
FU) = 10 + iF where /) denotes the Hilbert oanaform of 7

DATA TRANSMISSION 2041
ifr

“The pair of ont af the phage splitter wt line RT is written asthe
somplex signal sale
Meath
= empLilak? + ONY Das + Nas 6

ore [Ife = Hie + 1) are the samples of the overall complex
Hbesbund equivalent impulse response, and (Ny] are the complex
samples ofthe Eltered-noise process. Tho later process isuncorelated
‘ith che signal and hus an aucooorrlation cy, Thus, we have

(NNR = Rovl ln mT] (6)
(NN) = (oy)
R(aNu) = 0 (ee)

EfBnNy) =0 (6a)

for al icogens mand m.

"Tie complex signal sequence (%s}iapassed through a ervescoupled
pussbund equatizer with 24+ 1 camplex tap, the outpnc of which at
time nie given by

a
where we write the complex tapas
Com ea+ jy oxamd real
and define the vectors
LETC ao Cad ®
Kh = hoe oo Xe ®

2042 THE BELL SYSTEM TECHMICAL JOURNAL, NOVEMBER 1951
‘We use the * to denote conjugution for wslare ant coniogate trane-
position for vectors and matrices. The symbol ¢ denotes ranapoaition.

"The eignel Q. is demedulsted lo hasehand by multiplication wich
exp(= fe — jaan), where dy is dhe estimated phase olf (or jer)
fat tine 07. renlting signal Y, ean thon he wrieten a

Yn = Quexpl— J, — jounT) (atin)
owt ihe om)

‘The demodulated outputey, and jy aro ottimates of the tranemited
data sample, In th following eacton, ma optimum equalizer ap vector
in derived which minimiges ihe nse of an appropriate cose function

1M. OPTIMIZATION OF EQUALIZER COEFFICIENTS
24 Anatole of the minimum mae cterion forthe i-phase branch

In Refs. J and 4 an oplinwm equalizer chat rinimis an mae
citer was derived, The mi wa defined 9
E{\¥o— Daf ~ FUREY, ~ Da) + la?¥4 Bal]

Five ad] + EUG bk (a)
which isthe rn ofthe mse in hath branches ofthe equalizer output.
Teen foond thatthe optimum equalizer cosfiienta ean be caeulated
adaptively provided the copter ontnet ore Vr— Da eral
tha recsiver, While this i the case for a canamission ayetom sith
algal data in both branelos where rofernces ean be esimated easily,
tia not forthe wystem considered here. In this application only one
reference sequonce is assumed to be avaliable. Consequently, only the
‘error signal in hisbranch can he made available for updating purposes.

Tn the Gllowing dicussion, we dafite an aprimuam tap vector which
toininites che mae in that Branch whars reference signal i able
br ean be estimated easily. The reeulting ap vactor will be compared
wvich tho Teeult for the case where bath referencts are available
Acouming we have e reference for (ty). we define the mee in that
branch aa the cost function to be minimised

= Bil. asl coy
wih

B= Ref CK ex fa— ji a3
Wis convenient to express yin vector-macex notation as follows:

~eTaIAx., aw
where we partitioned the complex vectors of che passband equalizer

DATA TRANSMISSION 2083
‘cotticients and X.exp(—/0, — jenn7? of the ideally demodulated
received signal fo set ren veciors C and X with twee the ovginal
Sinennn

= RCH |IMCI] a5)
FRAN expt ~ jenn]

= [TX ennimse— fae (16)

B= be an

neq, {14}, T(AM) isa transformation matrix expressing the affect of
the demodulating phase vreor AM. and i defined as

°
Te- a)
o
[Note chat this matrix i orthogonal that i,
Pea) x Ta) = Ta) x Tal =F (8)
Furthermore,
Tea) = 3%) (a0)
ond
Tha + Bh = Tay x PUM = Tie) x Tra) (aR

In order to ael the mee 69. (12) aa an explicit function of the even
sect ©, we icroduce the autovozrelstion mitt A of the demod
Thc racived nigral

AEX) aa)

and the crss-curelacion vector V betmeen the demodulated recsived
signal ana che reference

V = O60 @)
‘With ge (4,09), and (20), we van express 09. (12) a fellows:
A= CTUMAT(-ARIC~ 207Q0IV + Bla, — ca)

Secting the partial derivatives with repect (uC ls vero yields the
‘ectar equation for the opemum tap veetor Cae

ATIM-A9)0.u = V. )

‘2044 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 108
‘The definition of A in eq. 18) ensures Ural i x positive definite,
Consequently, the equation hn

Gay PRIARY. 3)
Inserting eq (23) ino eg (21) yields forthe minimum mse
Gp = Bla) Cy TARIV = Ela) VAY. (20)

In Apperali A, we ahow that, for stationary data sequences (ax) and
(G.) uncorrelated with the nose, the qulocortelation matrix A and the
«ross cortelation vector V are independent of che tie instant n. Tis
important eo note that the mininim mae i independent of the
‘constant demodulation phase error 34, This isa consequence of ey
(2) and indicates that even by minimizing the mae in one of the (eo
‘swale tpae, che optimum coelicent vectors can tako ene of any
‘hase errr, in the eame manner 43 ina <rose-coupled equsliver which
frinimis the coal mew ut che opt

"These facts have been reported in Ret and 6 for vsu- and ss5-
mudulated pute amplitude modulation signals. Our analyeis shows,
hhowever, that thia boda in gener] for siationary sequences {a,) and
(2). Thus, the independence ofthe evinireum mee an the demodula-
tion phave iva property of the cross-coupled equalizer which is not
adversely affected by thy parlicula selection ofthe cost function nor
by the nature of the sequence {@)

‘When the power of the te data sequenoca ix balanced, Pe = Po
jeune ace fram ea, (0) that the resling equation for che optima
tap vector coined exacdy with the cquation resulting from minimis-
‘ng the total mse im boc branches. Tw thi case, bath methods wil give
the same optimal coefficient vector and the same total mse.

“ivall our analysia wo have assumed uncorcelated daca ax described
by eas, (a) through (4c) [fiesta of eg tb}, we have

E (babel = Rata — my,

Aten the expressions for A.C, 2) and As(k ©) im Appendix A would be
‘more complicated but they wowld remain stationary matrices Then,
‘asumaing correlated dala, eq. (24) it in general stil valid, except that
‘Vand Aare more complicated than the expremions derived in Appen.
din A

“Although we have lel our analysis ona symbol spaced equalize,
‘we can also handle fretional sparing and derive similar rosuls. Asan
‘euample, we can view the 7/2 equalizer aa two parallel aymbol-spaced
‘quallzers where che fist ofthe two data samples during each baud is
‘proceed hy one of tho equalizers while the second is processed by the
bother equalizer, Let ws denote the two equalizer tap vectors as an

DATA TRANSMISSION 2045
Cy and the input vectors as Ky and Xt: Thon we can define
© = [Re(Ct) iC |Re(C2) [mC]
7 (REX exp(— 78. — jo TY] [Xs expl— js — fonTY]
[RefXsun expt-j0, ~ jon)
Himidtheya exp(-i85 — jan)

noe

‘where 7) is givon in eg (18). With these definitions, al the previous
etl foF Coy nd ey (28) and (24 follow,
182 Mean squire eer in the quadrature branch

‘We now analyzed the mse in the gecond branch of the equalizr.
‘The ourpar ofthis branch a

ofA expt — jenn] (5

Since

(26) using the transformation properties in eq. (18)

~or(ss
“Therefore, the ime in Use socond branch canbe expeatd in vector
rats notacon as fl

= Elie = b0'D (28)
)an(-ane3)e

207 (u-3) onss+500.

en

er (um

Using eq (18) and an approach similar to 6g (68) through e.(T0) in
Appendix A, we show that

7(

2046 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1981

(sh) = ViPL/P, eo)
‘On subsctuting ea. (90) into eq, (29) we obtain

-or(au-3)ar(-s

Py
Ree aD

Fert ned yc Ps Ps Aen A hows that
r(-3)an()-a
sot ota

PaCTWMATEARIC-20TIHIV4+P. (3)
"Thinis exnetly the same expression as for ein eg. (21), Consequenty,
é oo

‘Thus, we conclude thatthe mac's in both branches are equal. In this
special ease, minimizing the mse in one branch aleo minimizes tbe mse
inthe other branch

20718040

‘3.3 Anatai of an lnfintety long equalizer

‘While the formal solution ey, (24) forthe minimure nse sleady
show the independence of a eanstaephaso orto, ic does not rovenl

anything about the inflanes of channel parameters (arpliude, or
phase distortion) or of tho sampling instant, To obtain further insight
{nto this dependence, we analyze nn infinite length equalizer.

We shore in Appendix B hal the resulting minimum mse for an
Infinite tap equalizer can be writen ws

1, c Pai) + 26

— aa, (35)

co

to 08 a) he Fun tr spe np
y= 2000 (oa) (snd
Fale) Ft (: a 7)oP Paks, (a7)

DATA TRANSMISSION 2047,
and |) is the baseband component ofthe noise power spectrum
3B ResikTexpl-jla + aNkT (8)

ra

‘The forrula given ia og. (85) can be evaluated for all the diferent
‘modulation schemes which ean be modeled by a linear transmiscon
system. The only frequeney-dependent term appearing in eq. (85) ix
Z(o), the a/n of the sampled rosived signal. According to eq. (37) this
‘vill, in general, depend on the sampling instant, ¢, apd the phase
characteristic of tho overall channel transfer function, (a). If the
‘ampling theorem is stisied, ie, if H(e) = 0 for all w outede the
interval [oy 01 + 20/7), where oi arbitrary, combining eqs (37) and
£08) shoe thas the minimum rae e only dependent om the amplitude
characeristc of the channel tranafer function and ofthe noise power
‘density spectrum. A qaattranmiasion aystam with no excess bend:
‘width ia en example; @ system tranamitting only ono data sequence
tind sing vab-modulation with les than 100 percent excess bandwidth
{is another, more realistic example. In case of balanced pawer in the

tronamiod data oquenesy LPs = Pv or for Z(o) symmetric around
soeke, Ze +o) =Z(-w + othe moe given by,

. TP, ("7 de

Son Bf «Weert (=

{In eq (100) of Appendix B we show chat the partial derivative of ey
‘wich respect to Py ia nonnegative. Therefore, an increase in che analog
‘signal power P, cause an increase in ey. So the analog signal act as
‘a inerferer to the digital signa.

IV. ANALYSI6 OF THE GRADIENT ALGORITHM FOR JOINT
EQUALIZATION AND PHASE TRACKING

Tn nn adaptive receiver, the equalizer is assumed to know the
reference data for sartup and lo operale in & decision directed mode
‘when random data is being sent. In either cas, the tap weight vector
{5 being updated continuovaly. Similay, the phase offet, or jiver,
‘would he omntinuouely tacked in order to remain in synchronism, Ax
FFaleoner di, in Reta Sand 4, we asuame gradient algorithms are used
in these updatings aa flloms (C i now time-varying),

00)

ou)

‘2048 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1981