Blind Equaliza ion Based on Spa ial and Tempo al Di e si y
in Block Coded Modula ions
F ancesc Rey, Me i xell Lama ca, G ego i Vizquez
Depa men o Signal Theo y and Communica ions,
UPC Campus No d-Mbdul D5, c/ G an Capi i s/n, 08034 Ba celona (Spain)
E-mail:
{
ey,xell,g ego i} @gps. sc.upc.es
ABSTRACT
Linea block codes can be applied in spa ial and/o empo al
di e si y ecei e s in o de o de elop high pe o mance
schemes o blind equaliza ion in mobile communica ions. The
p oposed echnique uses he s uc u e o he encoded
ansmi ed in o ma ion (wi h edundancy) o achie e
equaliza ion schemes based on a de e minis ic c i e ion.
Simula ions show ha he p oposed echnique is mo e e icien
han o he schemes ha ollow simila equalize s uc u es. The
esul is an algo i hm ha p o ides he design o blind channel
equalize s in low EbNo scena ios.
I. INTRODUCTION
The p oposed app oach elies on he a ailabili y o spa ial
and/o empo al di e si y in digi al ecei e s, which enables he
de elopmen o blind channel equaliza ion echniques based on
a de e minis ic design c i e ion. Blind equalize s can supp ess
he IS1 and educe he impac o he inpu noise by he
knowledge
o
he ansmi ed/ ecei ed da a s uc u e o i s
p ope ies. In such sys ems no aining sequence is needed. Fo
his eason he blind echniques a e o p ac ical in e es in
mobile communica ions, whe e he channel is cons an ly
changing and con inuous aining sequences can no be
ansmi ed.
I is known ha i he channel ou pu is o e sampled in he ime
domain and/o in he spa ial domain (mul i-channel di e si y),
channel compensa ion can be pe o med based on he ecei ed
signal second-o de s a is ics.
A
ecen s udy
[l]
used he
Bezou equa ion o in oduce a new blind equaliza ion c i e ion.
Basically, i p oposed an algo i hm ha maximizes he signal-
o-ISI-plus-noise a io
(SINR)
a he equalize ou pu .
Un o una ely, his algo i hm o e ed high eliabili y only in
mode a e o high
SNR’s
scena ios bu showed s abili y
p oblems a lowe
SNR’s.
This pape enhances he esul s o he me hod p esen ed in
[I],
and enables he design o high-pe o mance blind channel
equaliza ion schemes in mobile communica ion scena ios wi h
low EbNo a es. The main idea is o combine he in o ma ion
edundancy in oduced by a sys ema ic linea block coding
echnique wi h he s uc u e inhe en
o
he ‘T ans o m
Modula ion’
([2])
ansmission echniques o imp o e he
s a is ical s abili y o he equaliza ion echnique in p esence o
noise. The esul is a mo e obus scheme ha can be applied o
TDMA, DS-CDMA sys ems in equency selec i e mobile
channels, and OFDM sys ems in equency la ading mobile
channels
([3]).
Mo eo e , he p oposed solu ion
is
a ailable o
bo h spa ial and/o empo al di e si y on -ends, whe e i is
possible o conside a single-inpu mul iple-ou pu o mula ion
(SIMO)
o he ansmission sys em.
In some equaliza ion me hods
([4])
he ansmi e in oduces
edundancy ha he ecei e uses o iden i y o equalize he
channel. Al hough some addi ional edundancy is also added in
ou me hod, he p oposed app oach can be conside ed a blind
echnique because ha edundancy
is
no in oduced o equalize
bu used o co ec de ec ion e o s. Thus he ecep ion is done
in wo s eps. In he i s s ep he ecei e uses he s uc u e o
he encoded ansmi ed in o ma ion (wi h edundancy) o
achie e he equaliza ion. In he second s ep he same
edundancy is also used o co ec e o s. This pape s udies he
i s pa o he whole ecei e , analyzing he pe o mance o
he equalize using he edundancy symbols. I will be he aim
o a la e s udy o choose linea codes de ined o e he complex
ield
([5],[6])
ha , applying he s uc u e p esen ed in his
pape , could be used o co ec decision e o s a he ou pu
o
he equalize .
The nex sec ion illus a es he scheme and es ablishes he
p oblem. Sec ion
3
desc ibes he linea block coding
cha ac e is ics and shows how he s uc u e o
he
ansmi ed
encoded da a can be used in he design o he equalize . Sec ion
4
applies hese esul s
o
imp o e
he
[l]
cos
unc ion.
Finally
sec ion
5
p esen s some simula ion esul s, whe e i is possible
o see he imp o emen o he p oposed solu ion.
*
This
wo k
has
been
pa ially suppo ed by
he
Spanish
Resea ch
Council (TIC-95-1022-CO5-01)
and
by
ACTS
11
TSUNAMI-I1
0-7803-4872-9/98/$10.00
0
1998
IEEE
1265
11.
PROBLEM STATEMENT
The p oposed app oach elies on he a ailabili y o spa ial
and/o empo al di e si y in digi al ecei e s, which enables he
use o single-inpu mul iple-ou pu o mula ion
(SIMO)
o he
ansmission sys em. I consis s o an algo i hm o linea
equaliza ion o he ecei ed da a ha is based on a de e minis ic
design c i e ion.
Figu e la shows a disc e e- ime model o a spa ial di e si y
ecei e . The same in o ma ion signal
T[k]
is ansmi ed
h ough
B
di e si y b anches.
I
is dis o ed by di e en channel
esponses
Ci[k/
and inally deg aded by AWGN e ms
wl[k/.
In a simila way, Figu elb shows he model o a empo al
di e si y ecei e . The in o ma ion signal
T[k]
is ansmi ed
h ough a channel, which dis o s he signal wi h a channel
esponse
C[k]
and in oduces an AWGN,
W[k].
The ecei ed
signal is o e sampIed, a
B
sampIes pe symbol, and in oduced
in
B
di e en b anches as in he p e ious s uc u e.
1
.
/
47
CB kl
: yBM,Fg
Figu e
la.
Block
diag am
o
he spa ial di e si y sys em.
The equaliza ion p ocess o spa ial and empo al di e si y
schemes is simila , and bo h can be combined in o he same
ecei e sys em. Thei associa ed equa ions can be w i en in he
z- ans o m domain as:
(1)
Simila equa ions can be de i ed in
OFDM
signals h ough
equency- la ading channels in he ime domain (see [I], [7]).
Y'(z)
=
T(z)C'(z)+
W'(z)
i
=
1,
...,
B
As shown in [l], he equaliza ion p ocess can be designed
ollowing a blind c i e ion. The mul iple empo al o spa ial
di e si y b anches a e combined by means
o
FIR
il e s
E'[k]
o gene a e an ou pu
R[k]:
B
R(z)
=
ZY'(z)E'(z)
=
i l
B
B
=
T(z)CC'(z)E'(z)+C~'(z)E'(z)
,=I
,=I
The pe ec equaliza ion c i e ion equi es
R(z)
=T(z),
and
he e o e (neglec ing he noise e ec ):
(3)
The Bezou equa ion [8] gua an ees ha he p e ious equa ion
has solu ion i and only i he
B
channel esponses ha e no
common ze os, a esul well known in he li e a u e ([9]), and
he e o e (igno ing he noise e m):
T(z)
=
g.c.d.@'
(z,}
(4)
A ma ix o mula ion o he me hod can be ound in
[
101 and is
b ie ly summa ized he e in o de
o
app oach he p oblem. As
shown in
[lo],
equa ion
(2)
can be w i en in ma ix no a ion as:
-
R=Z
(5)
whe e
g
is he equalize ou pu ec o ,
y
is a gene alized
Syl es e ma ix wi h he ecei ed da a and is he equalize
weigh ec o . The pe ec equaliza ion case in
(3)
can be
w i en as:
-
whe e he ecei ed da a ma ix
y
has been spli in wo pa s,
2:
is he ansmi ed da a ec o and
a
is an a bi a y cons an . The
op imum equalize coe icien s (i he noise e m is no
conside ed) can be es ima ed wi h:
and once he equalize has been es ima ed, he ecei ed da a can
be il e ed o yield an es ima ion o he ansmi ed da a:
An in e es ing algo i hm, o muIa ed in
[
11,
ies o maximize
he
signal- o-ISI-plus-noise- a io
(SINR)
a he equalize ou pu ,
ha is:
(9)
and he equalize coe icien s a e designed o maximize he
p e ious quo ien .
In high EbNo scena ios he p e ious cos unc ion pe o ms
co ec ly, bu in noisy en i onmen s many s abili y p oblems
wi h he
yo
ma ix appea . Hence, he denomina o o (9) is
c i ical and new design c i e ia shouid
be
ound.
- -
Figu e
ib.
Block
diag am
o
he empo al di e si y sys em.
id
1266
111.
LINEAR BLOCK CODING STRUCTURES
Equa ion (10) de ails he cons uc ion
o
linea block codes
using ma ix no a ion. I desc ibes how o encode
k
symbol da a
in o ma ion using he code gene a o ma ix
G,
and ob ain he
n
coded ou pu sequence:
-
-
Conside ing a sys ema ic code, he encode ma ix becomes:
7
1
Lk
1
G,
=
................
-
iG
=(n-!+k
1
and he code wo d is di ided in o wo pa s. The i s k-symbol
pa i ion is always iden ical o he in o ma ion sequence o be
ansmi ed, while in he second po ion each o he (n-k)
symbols is a linea combina ion o he in o ma ion da a
acco ding o
G,
:
-
-
T
i
I
The second s ep in he coding p ocess applies he pa i y check
ma ix
c,
o e he mobile channel ou pu da a. This ma ix is
de ined as:
- -
pa i y o edundancy symbols allow he ecei e o de ec he
e ec s o he IS1 and noise,
so
new cons ain s can be
in oduced o ind he equalize weigh ec o
B.
As
in equa ion
(8),
i he coe icien s equalize co ec ly he
ecei ed da a, he il e gi es a good es ima ion
o
he
ansmi ed sequence:
T'=Y,E
(16)
- -
-
hence, applying he check pa i y ma ix o a pe ec
equaliza ion exp ession can be ound:
The p e ious equa ion can also be seen as a :measu e o he
quali y o he equalize . The close o ze o i is, he be e he
equalize pe o ms, whe eas any non-null esidue in (17) will
deno e a poo channel equaliza ion.
IV. THE COST FUNCTION IMPROVEMENT
As i was said be o e, equa ion (7) can exhibi poo
pe o mance in e y low
SNR
scena ios. In his pape , a new
me hod o design equaliza ion coe icien s and
o
o e come his
p oblem, is p oposed. The cos unc ion equa ion ies o
maximize he
SINR
a he
ou pu ,
and we ha e p e iously
discussed ha i
is close o ze o i means ha he equalize is educing he
IS1
e ec and il e ing he addi i e noise co ec ly.
(13)
1
=("-k)d
.
=(n-k)x(n-k)
L
J
and
G,
and
c,
ma ices a e de ined
so
ha :
C,
G,
=
0
-
Thus he pa i y check ma ix can de ec changes be ween he
ansmi ed code and he ecei ed in o ma ion. I bo h
sequences a e iden ical, he null ec o will be ob ained. On he
con a y, he e ec o he noise and he IS1 channel in oduces
di e ences in he ecei ed sequence, and he p oduc o ha
sequence wi h he pa i y check ma ix di e s om he null
ec o .
Acco ding o his conside a ion he new cos unc ion is
o mula ed as:
Wi h his imp o emen he unc ion o be op imized is mo e
obus
in
p esence o AWGN, i only needs he signal subma ix
y,
,
a oiding he use o he esidual
yo
ma ix, which uses o be
ill condi ioned. No ice ha he changes ha e been done in he
denomina o , which was he c i ical pa o he quo ien :
-
-
-
-
Equa ion (19) co esponds o a ypical Rayleigh quo ien
o m
(see [ll]), and he e o e he equalize
(E
ec o ) ha
Because o he cons uc ion o
G,
and
c,
ma ices, any ull
-
-
- -
ank
ma ix can be used
as
pa i y check symbol ma ix
G,
.
Fu he mo e o ensu e ha he encoding p ocess main ains
cons an he symbol ene gy o all he ansmi ed symbols we
a e in e es ed on hose ans o m such ha :
/&I2
=lg212
=.-=18n12
=1
(15)
maximizes i co esponds o he gene alized eigen ec o
associa ed wi h he maximum gene alized eigen alue:
(20)
-
_.
.I;
.I;g
=
A.
Y
c,
i
c,
;E
--
"2
__=
-
--
The equaliza ion pe o mance can be also imp o ed i a delay is
allowed in
R[k],
and he bes equalize is selec ed as ha one
whe e
lgj
I
is he no m
o
he ow ec o s
o
G,
.
- -
which yields he g ea es
2".
Finally i is also wo h ema king ha he compu a ional load o
he solu ion does no inc ease i adequa e ma ices a e selec ed
o
G,
.
In ou s udy Sub-Hadma d ma ices ha e been chosen.
The in oduc ion o he edundancy P o ides a mo e ich
s uc u e in he ansmi ed in o ma ion, which can be used in
he design o he equalize . P e iously, we ha e seen how he
=
1267
V.
SIMULATIONS
Figu es
2
o
6
compa e he pe o mance o he cos unc ion in
[
11
wi h he new cos unc ion p oposed in his pape .
The simula ions display he pe cen age o eaIiza ions
(IO00
we e a e aged) o which he equalize ou pu EbNo was highe
han he alue indica ed in he x-axis. In all cases he
ansmi ed in o ma ion consis ed
o
118
QPSK
da a symbols
and 10- edundancy symbols gene a ed wi h a sub-Hadama d
ma ix
G,
.
Each b anch equalize had ou coe icien s and ou
b anches
(BA)
we e simula ed.
- -
Figu es
2
and
3
show he pe o mance o he algo i hm using a
TDMA
bu s ansmission o e a equency selec i e channel
wi h spa ial di e si y (Figu ela). The
ou
s a iona y channel
esponses we e:
c’(z)
=
(1+j)+(-O.1-0.2j)z-’+O.4~-~
+z-~
+0.5z4
C~(Z)=O.~+~~-~+~~Z-~+O.~Z-~+Z-~
C3(2)
=0.1j+z-’-0.4jz-2+0.2z-3
-0.5~-~
C4(z)
=
(1+0.8j)-2jz-’
-0.4jz-*
+O.2zS3
+(1-0.5j)~-~
No ice ha he ou channels ha e no common ze os. In o de o
conside a di icul mobile scena io, he ou channels ha e
been selec ed wi h high a enua ion in ce ain equencies (in
special channels
1
and
3),
wi h some close ze os and wi h a
non-minimum phase beha io .
linea
block codes.
In Figu e2 he channel EbNo is 12dB, while Figu e
3
illus a es
he simula ion wi h EbNo=lSdB. In bo h cases he pe o mance
o he imp o ed cos unc ion
(I)
is highe han he pe o mance
o
he algo i hm wi hou linea block codes
(II).
No ice ha he
new me hod gua an ees ha wi h
a
EbNo highe han 12dB,
mo e han he
95%
o
imes he ou pu EbNo is o e 7dB, which
ep esen s a BER210”.
0
2
4
6
8
10
12
14
16
18
20
EbNo
ldBl
Figu e
3.
Algo i hm pe ommce
in
spa id
di e si y
wi h
EbNo=ISdB
Compa ison
o
he
algo i hm
wi h
( )
and
wi hou
(II)
he use
o
linea block codes.
F equency
espanse
chnnnek
....
....
....
....
....
. <.
-*5
:
::;
..
2w
400
600
800
1000
1200
-30
k
Figu e
4.
Channel
1
and
3
equency esponse.
Figu es
5
and
6
show he pe o mance o he algo i hm in a
TDMA bu s ansmission wi h empo al-di e si y (Figu e
1
b).
Two di e en non-minimum phase channels we e simula ed:
C’[z];
C3[z].
Thei equency esponses a e illus a ed in
Figu e.
4.
Figu e
5
illus a es he simula ion wi h EbNo=12dB,
and Figu e
6
wi h EbNo=15dB. The dashed line cu es a e
ela ed o channel
3,
and he solid cu es o channel
1.
As
in he
p e ious case I ep esen s he esul s using linea block codes
and
I1
he
esul s wi hou hem. No ice ha he pe o mance
wi h spa ial di e si y is be e han wi h empo al di e si y. Two
easons can jus i y ha . Fi s , he spa ial di e si y uses B
an ennas, and
so
he maximum ou pu EbNo could be:
mm{(EbNo),,,,}=
(ELINO),,
lOlog(B)
meanwhile in empo al di e si y:
“DbNO).,,,
I=
(EbNO),”
Second, when he Nyquis pulse is o e sampled (in empo al
di e si y), addi ional IS1 is in oduced in o he ecei e , causing
he pe o mance o he sys em o be educed.
1268
VI.
CONCLUSIONS
In
his pape linea block codes in he complex ield ha e been
in oduced o blind equaliza ion o mobile channels in spa ial
and empo al di e si y ecei e s.
The p oposed c i e ion, combining a blind equaliza ion
echnique
([
11) wi h he edundancy in oduced by a sys ema ic
linea code, can be applied o e TDMA s uc u es wi h
equency selec i e mobile channels, DS-CDMA sys ems and
OFDM modula ion wi h equency la ading channels.
Spa ial and empo al di e si y ecei e s o e TDMA s uc u es,
in equency selec i e mobile channels, ha e been conside ed.
The esul s show he pe o mance imp o emen
o
his new
equaliza ion me hod o e he p e ious scheme p esen ed in
[
11.
Be e esul s ha hose p esen ed in he cu en pape can be
achie ed in oducing codes de ined o e he complex ields
([5],[6]) ha , ollowing he s uc u e p esen ed in sec ion 3,
could be used o co ec mos o he e o s a he ou pu o he
equalize .
REFERENCES
M.
Lama ca, G. V izquez, ‘Di e si y echniques o blind
channel equaliza ion in mobile communica ions’,
P oceedings.
o
PIMRC’97,
pag. 1079-1083, Helsinki
(Finland), Sep . 1997.
M. Lama ca, G. Vhzquez, ‘T ans o m modula ions in
mobile communica ions’, P oc.
o
PIMRC’97, pag 462-
466, Helsinki (Finland), Sep . 1997.
R
S eele,
Mobile Radio Communica ions,
Pen ech P ess.
A. Scaglione, G.B. Giannakis,
S.
Ba ba ossa ‘Sel -
eco e ing mul i a e equalize s using edundan il e bank
p ecode s”,
P oceedings
o
ICASSP ’98 Sea le
May 1998.
T.G. Ma shall, ‘Coding o eal-numbe sequences o e o
co ec ion: a digi al signal p ocessing p oblem’
IEEE
Jou nal on Selec ed A eas in Communica ions,
V01.2,
Ma ch 1984.
R.E. Blahu , ‘Algeb aic ields, signal p ocessing, and e o
con ol’,
P oceedings
o
he IEEE,
Vo1.73, no 5, May
85.
M.
Lama ca,
G.
Vhquez, ‘Channel es ima ion o
ans o m modula ions in mobile communica ions’,
P oceedings
o
EUSIPCO’96,
T ies e (I aly), Sep .
1996.
Figu e
5.
Algo i hm pe o mance in empo al di e si y wi h EbNo=lZdB,
h ough di e en channels. Compa ison o
he
algo i hm wi h
(I)
and wi hou
(U)
he
use o
linea
block codes.
.--,
Figu e
6.
Algo i hm pe o mance
in
emponl di e si y
wi h
EbNo=lSdB,
h ough di e en channels. Compa ison
o
he algo i hm wi h
(I)
and
wi hou
(II)
he use
o
linea block codes.
[8] T. Kaila h,
Linea Sys ems,
P en ice-Hall, 1980.
[9] Y.Ii1, Z.Ding, ‘Blind channel iden i ica ion based on
second o de cyclos aciona y s a is ics’,
P oceedings.
o
ICASSP
’93
Minneapolis (USA)
[lo]
M. Lama ca, G. Vgzquez, ‘Mul ichannel ecei e s o
OFDM and TDM in mobile communica ions’,
P oc.
o
ICASSP
’97,
Munich (Ge many), Ap il. 1997.
[
111
G.H.
Golub,
C.F.
Van Loan,
Ma ix Compu a ions,
Johns
Hopkins Uni . P ess,
1989.
1269