EXPLOITING TRANSMISSION SPATIAL DIVERSITY IN FREQUENCY SELECTIVE
SYSTEMS WITH FEEDBACK CHANNEL
An onio
Pascual
Ise e,
Ana
I.
Pi ez-Nei a, Miguel
A.
Lagunas
He ndndez
Depa men o Signal Theo y and Communica ions
Poly echnic Uni e si y o Ca alonia (UPCj
Telecommunica ions Technological Cen e o Ca alonia
ClTC
-
Edi ici NEXUS
I
C/ Jo di Gi ona
1-3
(Campus No d UPC
-
mhdul DS),
08034
Ba celona (SPAIN)
e-mai/:{ onip,anuskaj
@gps. sc.upc.es.
[email p o ec ed]
,
ABSTRACT
In
his pape we add ess he design o a mul iple ansmi an enna
sys em in which he Channel S a e In o ma ion
(CSI)
a he ans-
mi e is
no
pe ec . Two di e en app oaches a e analyzed
one
based
on
he Minimiza ion o he Mean Squa e E o (MMSEj and
he o he based
on
he applica ion o he Maximum Likelihood Se-
quence Es ima ion (MLSE).
In
bo h cases a Bayesian c i e ion is
used
in
o de o ake in o accoun he e o be ween he CSI and
he eal channel. Finally, some simula ion esul s and conclusions
a e p o ided, showing which is he gain o hese app oaches when
he e o be ween he CSI and he eal channel is ei he Gaussian
o uni o m, whe e his las case co esponds
o
a quan iza ion o
he channel ime esponse in o de o ansmi he CSI h ough a
eedback channel om he ecei e
o
he ansmi e .
1.
INTRODUCTION
Spa ial di e si y is
an
e icien me hod
so
as o comba he impai -
men s p esen in he wi eless channel.
In
cellula communica ions
o
Wi eless LAN's he ecei e an enna di e si y is no a ac i e
o he downlink channel because he mobile s a ion should
be
equipped wi h mul iple ecei e an ennas. Fo his eason, he use
o ansmi an enna di e si y o he downlink is mo e desi able.
Exis ing ansmission schemes o exploi ing he po en ial o -
e ed by ansmi an enna a ays a e gene ally conce ned wi h in-
c easing he di e si y o de . The e a e se e al examples o such
echniques, such
as
he
delay di e si y
s a egy,
a
special case
o
a
mo e gene al solu ion p esen ed in
[I].
O he possible app oaches
ha inc ease he di e si y o de consis in he applica ion o
spnce-
ime
coding,
echnique p esen ed in wo ks such as
[Z]
and
131.
Space- ime codes
do
no
exploi channel knowledge a he
ansmi c . In o ma ion abou he channel,
i
a ailable, should
be used
o
imp o e he pe o mance by means
o
op imal e minal
il e ing.
I
can be shown ha unde a ze o o cing c i e ion, he
maximiza ion o he Signal
o
Noise Ra io (SNR) esul s in a de-
coupled o spa ially scalable solu ion whe e each ansmi b anch
can be designed independen ly. Fig.
1
shows a gene alized a chi-
ec u e ha allows
a
no maliza ion
o
he il e s dynamic, while
he beam o ming weigh s
{ uq):=,
a e in cha ge o adjus ing he
ansmi powe . Modula ion can be seen
as
a quan iza ion p ocess
and i s e ec s can be s udied as quan iza ion noise. This quan i-
za ion a he ou pu o each il e a oids ins abili ies, and
so,
IIR
designs could also be used. By depa ing om his a chi ec u e,
0-7803-7663-3/03/$17.00 02003
IEEE
IV
-
85
FILTERING MODULATION BEAMFORMING
2,
Q
[.I
WI
.z
Q
b
QPI
b
WQ
5(4
Fig.
1.
T ansmission di e si y a chi ec u e.
a clea ade-o be ween op imali y and c ucial aspec s like o-
bus ness, p ac icali y, so -deg ada ion o he
QoS,
eliabili y, e c.
can be easily aken in o accoun . Conce ning p ac ical conside -
a ions, pa ial and quan ized Channel S a e In o ma ion (CSI) a
he ansmi e can be in oduced in a na u al way by adap ing, o
ins ance, each complex weigh
wp
o he s onges channel pa h
a each b anch o , i
a
con olled uni a y dynamic is desi ed, jus
by compensa ing he phase o he s onges pa h.
To
sum up, o-
bus ness
[4]
implies insensi i i y
o
de ia ions om he heo e ical
assump ions, being he impe ec CSI one o he possible sou ces
o de ia ion.
So
a we ha e commen ed
on
ansmi e design based ei he
on coding o linea p ocessing depending
on
he channel knowl-
edge, and how impe ec ions o bad knowledge can be aken in o
accoun by mo ling
o
obus a chi ec u es. Ano he al e na i e
exis ing in he li e a u e o inco po a ing bad channel knowledge
in
ansmi space- ime p ocessing is by means
o
a Bayesian poin
o iew, ha is, modeling he side channel in o ma ion using a
pu ely s a is ical app oach. P e ious and ela ed wo k includes he
pe o mance analysis o
[SI
o la ading channels
o
he design
p oposed in
[6]
o OFDM sys ems.
.In
[7]
he bene i s o ans-
mi beam o ming and o hogonal space- ime block coding o la
ading channels a e combined.
[8]
conside s he e o in he CSI
om a MAXMIN poin o iew di e en om he Bayesian one.
In
his pape we analyze he gene al case o
a
equency selec-
i e channel and p opose wo space- ime p ocessing solu ions ha
ollow he Bayesian app oach in o de
o
inco po a e he e o in
he CSI. Simula ions compa e he p oposed echniques wi h space-
ime p ocessing designs ha assume pe ec CSI and schemes ha
do no need CSI such as delay di e si y. Al hough he Bayesian
app oach will ei he esul in spa ially non-scalable solu ions, o
non- obus when he s a is ical assump ions a e no ue, i se es
as a use ul benchma k
o
analyze and compa e he commen ed o-
bus a chi ec u es.
ICASSP
2003
Fig.
2.
Gene al scheme o ansmi di e si y wi h eedback chan-
nel and impe ec
CSI
a he ansmi e .
2.
PRE-FILTERING WITH FEEDBACK CHANNEL
In
his sec ion we ocus ou a en ion on he case o a eal sys em in
which a digi al eedback channel is implemen ed om he ecei e
o he ansmi e . The single-an enna ecei e
is
esponsible o
es ima ing he channel, quan ize his es ima e. code i in o a digi-
al o ma and send i o he ansmi e ia he eedback channel.
By means
o
his, he ansmi e has an es ima e o he channel,
possibly impe ec
CSI.
In his wo k we exploi his
CSI
in o de
o design linea il e s a he ansmi e om a Bayesian poin o
iew and wi hou o cing he design o he spa ially scalable. The
gene al scheme is p esen ed in Fig.
2.
Ou goal is o design he
il e s
{z,},"=,
o he
Q
ansmi an ennas aking in o accoun he
impe ec ions in he
CSI
a he ansmi e .
2.1.
Sys em and Signal Models
Le
us
conside a equency selec i e channel wi h
Q
ansmi
an ennas, whe e each o he channels in he Mul i-Inpu -Single-
Ou pu (MISO) link has
L
aps.
h,
=
[
h!p)hp).
.
.
h.!)
ep esen s he ime impulse esponse o he q h ansmi an enna.
I
is
possible o collec
all
hese ime impulse esponses in
a
sin-
gle ec o
h
by means o his no a ion:
h
=
[
hTh ..
.
hz
]',
whe e he numbe o componen s
o
h
is
K
=
QL.
The channel
is
modeled as a complex andom Gaussian ec o , whe e i s co a i-
ance ma ix
Rh
collec s he spa ial co ela ion and he powe delay
p o ile
o
he channel. In case ha he e is a di ec line o sigh ,
hen hese andom ec o would ha e a ce ain mean
m
di e en
om ze o. The e o e, he P obabili y Densi y Func ion (PDF)
o
he channel ollows he s a is ical law:
h
-
G
(m,
Rh).
Ou
goal is o design he ansmi il e s while conside ing e y
simple de ec o s a he ecei e .
As
i is seen
in
Fig.
2,
wo possible
ecei e s a e conside ed:
a
symbol-by-symbol de ec o and a Max-
imum Likelihood Sequence De ec o (MLSE) based
on
he appli-
ca ion o he Vi e bi algo i hm. Le
zp
=
[
zj )z?)
. . .
zk)
1'
be he ime M- aps impulse esponse o he q h ansmi il e .
Once again, i is possible o ep esen in a single ec o
z
all hese
il e s:
z
=
[
zTzT.
.
.
z:
1'.
In he design o he ansmi
il-
e s, he ollowing ansmi powe cons ain mus be ul illed
I'
112//2
=
ZHZ
=
P,
(1)
-
A he ansmi e side, only a channel es ima e
h
o pa ial
CSI
is a ailable.
e
ep esen s he e o be ween he channel es i-
ma e
h
and he eal channel ealiza ion
h: h
=
h
+
e.
In he
conside ed sys em, his channel e o is due o he own es ima ion
p ocess a he ecei e and/o he quan iza ion
o
he channel es-
ima e
so
ha i can be ansmi ed h ough he eedback channel
om he ecei e
o
he ansmi e . In gene al, we model his e o
s a is ically by means
o
i s PDF, which is assumed o be known:
e(e).
In case ha no quan iza ion is ca ied ou , hen he s a is ics
o
he e o would usually co espond o a Gaussian PDF, whe eas
in
case ha only quan iza ion is conside ed, he PDF would be
uni-
o m. By making use
o
his no a ion, i is possible o o mula e he
PDF
0:
G
condi io_ned o he eal channel ealiza ion
h
as
ollows:
Glh(hlh) = c(h-h).
-
-
3.
SYSTEM DESIGNS
In his sec ion we p esen he wo conside ed design s a egies.
The i s one co esponds o a symbol-by-symbol de ec o based
on
he
Minimum Mean
Squa e
E o (MMSE) c i e ion, whe eas
he o he one makes use o a
MLSE
de ec o by means o he ap-
plica ion o he W e bi algo i hm.
3.1.
Symbol-by-Symbol De ec o
When applying a symbol-by-symbol de ec o a he ecei e ,
an
ad-
equa e design c i e ion is MMSE, as i akes in o accoun he noise
powe and also he signal dis o ion o In e Symbol In e e ence
(1%).
In
case ha he
CSI
was pe ec , he equi alen channel im-
pulse esponse a he ecei e would be almos equalized.
As
his
is
no he case in a eal scena io,
we
add a il e
hn
a he ecei e
esponsible o equalizing he esidual
ISI.
When designing his
il e , he MMSE c i e ion is conside ed and i is assumed ha he
eal channel impulse esponse
h
is
known a he ecei e , as i is
also assumed in
171.
E(h, h)
is he Mean Squa e E o
(MSE)
o a conc e e chan-
nel
h
and
o
a conc e e collec ion o il e s
z(G)
and gain ac o
a he ecei e
aR(h):
-
whe e i is assumed ha he symbols s(n).a e no malized
so
ha
E{ls(n)l'}
=
l,H
=
[
HlHz...Hq
]
isama ixcan ain-
ing all he Toepli z con olu ion ma ices
{Hq}:=,
co esponding
o
he
Q
channels
{h,}$,,
1
is
an
all-ze os ec o excep a
I
in
a posi ion ep esen ing he desi ed empo al esponse o he equal-
ized channel
akHz,
and
a$
ep esen s he powe
o
he Addi i e
Whi e Gaussian Noise (AWGN) a he ecei e . The con olu ion
ma ix
H,
is
de ined as he
(M+L-
1)
x
M
dimensional Toepli z
ma ix, whe e he i s ow
is
an all-ze os ec o excep he i s el-
emen which
is
equal o
hp),
and he i s column
is
an all-ze os
ec o excep he i s
L
elemen s which a e equal o he ec o
h,.
Ou goal
is
o
design he il e s
so
as o minimize he MSE
a e aged o e he
eal
channel s a is ics and he e o s a is ics.
This can be exp essed
as
ollows:
IV
-
86
As h: il e s
z
and gain ac o
an
depend only on he channel
es ima e
h,
he minimiza ion
o
E
is equi alen o he minimiza ion
o
C(g)
subjec o he ansmi powe Cons ain
(I).
The op imum
solu ion co esponds o he ollowing equali ies:
whe e
a
is
a cons an such ha he ansmi powe cons ain
(I)
is
ul illed, X(c)
=
E,,s
{HHHIG}
and
M(G)
=
Eh,c
{HI';}.
In
gene al i
is
di icul lo ob ain closed exp essions
o
X
and
M.
In he Appendix
we
show how o ob ain a closed exp ession o
he case
in
which he e o
e
is assumed o be Gaussian.
3.2.
MLSE De ec o
I is
also
possible o use o he kind o de ec o s wi h
a
highe com-
pu a ional load bu wi h a be e pe o mance, such as he MLSE
based on he Vi e bi algo i hm
(see
Fig.
2).
This de ec o
is
he
op imum one in case ha he channel is known wi h no e o a
he ecei e , which co esponds wi h ou assump ions and
as
p e-
sen ed in
[7].
The pe o mance
o
his de ec o is di ec ly ela ed
o he
SNR,
which is de ined
as
ollows:
(6)
-1
SNR(h,h)
=
-zH(c)HHHz(';)
whe ei isassumed ha E{ls(n)l'}
=
1andzandHa ede ined
as
in he p e ious subsec ion. Ou
goal
is o maximize he
SNR
a e aged o e he eal channel s a is ics and he e o s a is ics.
This can be exp essed
as
ollows:
ai
SNR
=
g(g)SNR(K)&
(7)
s
SNR(6)
=
Ehls
{SNR(h,G)IK}
=
/SNR(h,K) ,,i;(hlc)dh
The maximiza ion
o
he
SNR
is
equi alen o he maximiza-
ion o
SNR(c)
subjec o he powe cons ain
(1).
The solu ion
o his op imiza ion p oblem is ound
as an
eigen ec o p oblem:
z(G)
=
a"
(8)
x,,,u
=
xu,
lull
=
1
(9)
-
whe e he ma ix Xis de ined as in he p e ious subsec ion:
X(h)
=
{HHHlc}.
A closed exp ession o his ma ix is p esen ed
in he Appendix when he e o
is assumed o be Gaussian.
I can be shown ha his design c i e ion is equi alen o he
minimiza ion o he e o powe
P,
=
E
{Ie(n)I'}.
The equi -
alen ime impulse esponse
o
be used
when
applying he Vi e bi
algo i hm
is:
hD
=
aRHz.
The gain ac o
a~
is
a bi a y and
does no a ec he pe o mance o he sys em. Usually,
a.q
is cal-
cula ed
so
ha he mean powe
a
he inpu o he
MLSE
block
is no malized o he uni y. I mus be said ha in his sec ion
we
assume ha he Vi e bi decode admi s any leng h o he equi a-
len esponse
ho,
and he e o e, he complexi y
and
compu a ional
load can be e y high. Fu he wo k will analyze o he kinds o de-
ec o s based on MLSE bu wi h
a
lowe compu a ional complexi y
by means
o
a
sho ening o
ho.
. ;...;.
A
.,
,
...... ..
'
. .
..{
.
<
-.
.~
Fig.
3.
Simula ions esul s o he MMSE echnique.
Fig.
4.
Simula ions esul s o he echniques based on
a
MLSE
ecei e .
4.
SIMULATION RESULTS AND CONCLUSIONS
In
his sec ion we p esen some simula ions ha help o unde s and
he bene i s
o
using he designs p esen ed in his wo k. We ha e
simula ed no malized channels
(E{~~hq~~z}
=
1)
wi h
a
delay
sp ead o
3
symbol pe iods, an angula sp ead o
3(P
and
BPSK
symbols. The leng h
L
o
he channel o he MMSE echnique is
5,
whe eas o he case o MLSE
is
3.
In Fig.
3
we p esen some esul s o he
MMSE
echnique,
in which he ecei e is based on
a
symbol-by-symbol de ec o .
The ansmi il e s
zp
ha e
7
aps, whe eas
bq
has
IO
aps.
In
hese simula ions,
we
ha e always made use o he ma hema ical
exp essions p esen ed in he Appendix, i. e., we ha e assumed
ha he e o is Gaussian.
Two
di e en si ua ions ha e been an-
alyzed. The i s one co esponds o an e o which is ac ually
Gaussian, whe eas in he second case he e o is due o he quan-
iza ion
o
he channel impulse esponse.
so,
in his las case, he
1V
-
87
s a is ical model o he e o does no co espond wi h he eali y.
We ha e simula ed di e en powe s
o
he Gaussian e o and di -
e en numbe o bi s o
ca y
ou he quan iza ion ( he numbe
o
bi s in he igu e ep esen s he numbe o bi s wi h which each ap
o he channel is quan ized).
As
i can be seen, he solu ion ha
does no ake in o accoun he e o in he CSI and assumes ha
he channel es ima e is pe ec , is no able o dec ease he BER
al-
hough he SNR
is
inc eased, whe eas in he case o he solu ion
based on he Bayesian poin
o
iew (Eq.
(4)),
he BER dec eases
as he SNR inc eases. I can be also concluded ha , al hough o
he case o quan iza ion he e o model does no co espond wi h
he eal s a is ics, he Bayesian design is able o inc ease he pe -
oimance in on o he non-s a is ical based solu ion, ha is, he
solu ion ha assumes ha he CSI is pe ec .
In
Fig.
4
he equi alen esul s o he MLSE echnique a e
p esen ed o
4
an ennas and a Gaussian channel es ima ion e o .
The ansmi il e s ha e only
I
ap. We ha e also made compa -
isons be ween his solu ion, which needs CSI. and he delay di-
e si y echnique o
I
and
4
an ennas, also de ec ed by means o
MLSE (Vi e bi). Delay di e si y is a linea p ecoding echnique
ha does no need any CSI a he ansmi e .
As
i can be seen,
in his case he gains ob ained by means o he Bayesian app oach
a e less impo an han he ones ob ained wi h he MMSE ech-
nique. The eason o i is ha he Vi e bi decode is no sensi-
i e o non-equalized channels and ha he gains o mean SNR by
means
o
he Bayesian app oach a e no ex emely impo an and
do no ha e a di ec impac on he BER. I can be also concluded
ha , al hough he noise powe
in
he CSI is e y high and he qual-
i y o he channel es ima e is e y bad, he solu ion based on he
MLSE echnique pe mi s inc easing impol an ly he pe o mance
o he delay di e si y scheme.
5.
APPENDIX
In his Appendix we
de9
he exp essions co esponding
o
he
ma ices
M(G)
and
X(h)
when he e o e is assumed o be Gaus-
sian wi h he ollowing PDF
E
-
G
(0,
E). Unde his assump-
ion i can be easily p o ed ha
Glh
-
G
(h,
E) and
hjc
-
G
( ,
C),
whe e
and
C
a e
de ined
as
ollows:
C
=
(Rhl+E-’)-’
(10)
=
C
(-
R,’m+E-’h
-1
(11)
Deduc ion
o M:
le us w i e he ec o
=
Eh,~
{hlc}
de-
ined in
(11)
as ollows:
=
[
y ?.
- 6
]’,
whe e
,
=
[
j‘l g)..
. g’
1’
and
j‘)
=
Ehl~
{hj’)lG}.
The ma ix
M(G)
is de ined as
M
=
[
MiMz..
.
MQ
1,
whe e
M,
=
Eh,i;
{H&}
is he condi ioned mean o he Toepli z con olu-
ion ma ix
H,
associa ed o he channel co esponding o he q h
ansmi an enna. The ma ix
M,
is he
(M
+
L
-
1)
x
M
Toepli z
con olu ion ma ix, whe e he i s ow is an all-ze os ec o ex-
cep he i s elemen which
is
equal o
@’,
and he i s column is
an all-ze os ec o excep he i s
L
elemen s which a e equal o
he ec o
,.
Deduc ion
oJ
X:
le us w i e he ma ix de ined
in
(IO)
C
=
Eh,,{(h- )(h- )HIK}as:
By means o hese equali ies and ela ionships, he inal ex-
p ession o he ma ix
X(K)
can be ob ained.
6.
REFERENCES
[I]
G.
W.
Womell and M. T. T o , “E icien Signal P ocess-
ing Techniques o Exploi ing T ansmi An enna Di e si y on
Fading Channels,”
IEEE T ans. on Signal P oc.,
ol.
45,
no.
I,
pp. 191-205, Jan.
1997.
[2]
V.
Ta okh, N. Seshad i, and A.
R.
Calde bank. “Space-Time
Codes
o
High Da a Ra e Wi eless Communica ion: Pe o -
mance C i e ion and Code Cons uc ion,”
IEEE
T ans.
on
In-
o m.
Theo y,
ol.
44,
no.
2,
pp.
744-765,
Ma ch
1998.
[3]
G.
Fanesan and
P.
S oica, “Space-Time Block Codes: a
Max-
imum SNR App oach,”
IEEE
T uns.
on
In o m. Theo y,
ol.
47,
no.
2,
pp. 1650-1656, May
2001.
[41
P-J.
Hube ,
Robus
S a is ics,
Wiley Se ies,
1981.
[SI
Jinho Choi, “Pe o mance Analysis o T ansmi An enna Di-
e si y Wi hnVi hou Channel In o ma ion,”
IEEE T ans.
on
Veh.
Techn.,
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