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Delayed two-streams division, a diversity technique to
improve signal transmission in relatively fast flat fading
channels
Mohammad-Ali Khalighi, Laurent Ros
To cite this version:
Mohammad-Ali Khalighi, Laurent Ros. Delayed two-streams division, a diversity technique to improve
signal transmission in relatively fast flat fading channels. Signal Processing, Elsevier, 2005, 85, pp.705-
715. �hal-00078446�
To Improve Signal Transmission In Relatively Fast Flat Fading Channels
MohammadAli Khalighi a
,LaurentRos b;y
a
Institutd'
ElectroniqueetdeTelecommunicationsdeRennes(IETR),France
b
LaboratoiredesImagesetdesSignaux(LIS)deGrenoble,BP46,38402Saint-Martind'Heres,France
Received17November2003;receivedinrevisedform01September2004
Abstract
We consider in this paper a combination of data symbolsthat serves to provide time diversity
and to reduce at fading e ect at receiver, when no other source of diversity is available. This
technique, called delayed two-streams division (D2SD), is particularly interesting in relatively fast
fading channels. By D2SD, the stream of the data symbols is divided into two substreams, the
symbolsofwhicharemixedpairwise together,andtransmitted throughthechannel byasuÆciently
long time delay between each pair. At receiver, a simple detector based on maximum likelihood
criterion isused to \equalize"the symbol combination madeat transmitter. Thepresentedresults
show that with the negligible complexity added to the system, and while implying no loss in the
spectral eÆciency nor any increase in the transmit power, D2SD permits to obtain a considerable
improvementin thereceiverperformance. Selection of the designparametersof D2SD is discussed
fordi erentcasesoffadingstatistics,basedonthebiterrorprobabilitycriterion.
Keywords: Fadingchannels,Rayleighfading,Ricean fading,convolutiveprecoding,modulationdiversity,code-
divisionmultiple-access(CDMA)
1 Introduction
Fading mitigation in wireless channels has beenone of the mostchallenging issues in recent years. To
prevent the degradationof thesignal transmission qualitydue to time-varying multipath propagation,
diversitytechniquesareusuallyemployed [1, 2 ]. When,dueto thelimitation ofcost orsize, mitigation
techniquesbasedon spaceorpolarizationdiversitycannotbe used,theonlysolutionmaybeto take use
PartsofthisworkhavebeenpresentedinInternationalSymposiumonSignalProcessinganditsApplications(ISSPA),
June2003,Paris,France(Reference[25]).
y
correspondingauthor. E-mailaddress: [email protected]
andmodulation diversitytechniques. Convolutiveprecodersinduceanarti cialchanneldelaydispersion
byspreadingthe(equivalent)channelimpulseresponse, andaverage overthefadingprocess [3,4,5,6 ].
Modulation diversity techniques(also called constellation precoding orsignal space diversity) take ad-
vantage ofchanneltimeselectivitybytransformingthesignal constellation[7 , 8,9,10, 11 ].
To e ectivelyreduce fadinge ect when usingchannelcoding,we have touse low-rate codes withinter-
leaving, or to use concatenated codes thatrequire complexdecoding[12 ]. Techniqueslike trellis coded
modulation [13 ] also need complex decoding. On the other hand, convolutive precoders impose long
delaysinsignaltransmission[4 ,5]andrequireacomplexequalizationoftheequivalentchannel[3,4,5 ].
Also,whenchannelisquasi-static,thecomplexityofthereceiverremainsthesame. Modulationdiversity
techniques, inturn,needcomplexdemodulation,whileincreasing thesizeof channelalphabet [14 ].
In this paper, we propose a technique, named delayed two-streams division(D2SD), permittingto
reduceconsiderablythefadinge ect. Itcaninfactbeconsideredasakindofmodulationdiversitywith
a diversityorder of L=2. It consists of a particular combination of data symbols implyingno loss in
spectral eÆciency, and is of special interest in relatively fast fading channels. The idea of D2SD was
taken fromcode divisionmultipleaccess(CDMA) [15 , 16 ] and aprevious work on thesubject[3 ]. The
importantcontributionsof ourworkarenotablyproviding asimplestructurefortheoptimalmaximum
likelihood (ML) detector, as well as providing tight upperand lower bounds on the corresponding bit
errorprobability. Also,westudytheperformanceimprovement fordi erent cases of fadingstatistics.
The paper is organized as follows. We present in Section 2 the D2SD combination scheme. Then,
we provide inSection 3 the detector structure and expressionsfor the upperand lower bounds on the
error probability. Performance analysis of the proposed method is performed in Section 4 for some
particularchannelrealizations,aswellasforthecases of Rayleigh and Ricean fadingchannels. Finally,
someconclusionsanddiscussionsconcludethepaper. Weconsidersingle-usercommunicationandBPSK
modulationthroughoutthepaper. Also,weconsidertheconditionsof atfadingwherewedonotdispose
ofthesourceofdelaydiversitythatwehaveinfrequencyselectivechannels[15 ,17,18 ]. Weassumethat
thecommunicationchannelisperfectlyknown at receiver.
2 Delayed two-streams division (D2SD)
Consider the stream of uncorrelated BPSK symbols a 2 f+1; 1g with the symbol duration T
b . It is
rst split into two half-rate streams S
1 and S
2
of source symbols with the duration T
s
=2T
b
. Let us
P+
P-
a [1] a [2] a [3] a [4]
n
(t = nT b )
1 2 3 4 2 3 4 5
T s = 2T b
T b
mapping
a [1]
µ a + [2]
a [3]
µ a + [4]
µ a [1-2N]
-
a [2-2N]
µ a [3-2N]
-
a [4-2N]
n
a [n] b [n]
Figure 1:D2SD:combinationofsourcesymbolsanddivisionintwophaseswithinsertionofthedelay(2N+1)T
b
denote thesymbolscorrespondingto S
1 and S
2 bya
1 and a
2
,respectively. Wehave:
a
1
[m]=a[2m 1] ; a
2
[m]=a[2m] ; m=1;2;::: (1)
Next,thesymbols ofS
1 and S
2
are combined insucha waythatinaphase P +
we transmitthesumof
apair ofsymbolsa
1
[m]+a
2
[m],and ina phaseP thesubtractionofthem, a
1
[m] a
2
[m]. We further
introduceadelayof (NT
s +T
b
) inthetransmissionof P relative to P +
. The combination of symbols
is depicted in Fig.1. Due to a reason related to signal detection at receiver that will be explained in
Section 4, we introduce further a mixture factor (0 1) in our combination. In this way, the
transmitcombined-symbols b +
[m] andb [m] inphasesP +
and P willbe:
8
>
<
>
: b
+
[m] , a
1
[m] + a
2 [m]
b [m] , a
1
[m] a
2 [m]
(2)
When transmitted through the channel, the combined symbols in phases P +
and P will undergo
di erentchannelfades. So,theinformationofa[n],duplicatedintwo phasesP +
andP ,undergoesthe
channelfades [n]and [n+2N +1], respectively. 1
Considerthebasebandtransmit signalasin(3):
b(t)=T
b X
n
b[n] h
e
(t nT
b
) (3)
whereb[n] is thesample ofthetransmit signalat time nT
b :
b[n ]= 8
>
<
>
: b
+
[m] for n=2m
b [m N] for n=2m+1
n=1;2;::: (4)
and h
e
( ) is the half-Nyquist lter at transmitter, designed for a rate 1=T
b
. The basebandequivalent
complexreceived signala ected byfading and complexadditive whiteGaussiannoise(AWGN) n(t)is
r(t)= (t)b(t)+n(t): (5)
1
Noticetheinserteddelay betweenthecorrespondingcombined-symbols.
r +[m]
r -[m]
b + [m]
b - [m]
D2SD
a [n]
a 1 [m]
=
a [2m-1]
a 2 [m]
=
a [2m]
combination
C
S 1
S 2
P +
P -
↑2
2N+1 2N+1
delay ∑
separation
odd/even
t
T b
T s T s T b
T b
↑2
b [n]
1/2-Nyquist
Tb-pulse
h e (τ)
1/2-Nyquist
Tb-pulse
h e H (τ)
α (t) x
n(t) +
b (t)
r (t)
t = nT b
r [n]
↓2
↓2 -2N-1 -2N-1
Channel
Front-end of the receiver
T b
T s
Figure 2:D2SD:symbolcombination,basebandmodeloftransmission,andthereceiverfront-end.
After thereceiverhalf-Nyquist lterand synchronizedsamplingat timeinstantsnT
b
,weobtain:
r[n]= [n] b[n]+n[n] (6)
where n[n] is complex AWGN with the variance 2
. The block diagram of Fig.2 illustratesthe D2SD
transmissionschemeincludingthecombinationofsourcesymbols,thebasebandrepresentationoftrans-
mitted signal,and thereceiver front-end (beforesignal detection).
Let usnow usethepolyphase representationof thereceived signal. In accordance withourprevious
notations,weusesuperscripts: +
and: to distinguishbetween twophases. So,samplesofthereceived
signalscorrespondingto thecombined-symbolsb +
[m] and b [m] willbe 2
8
>
<
>
: r
+
[m] , r[n=2m] =
+
[m] b +
[m]+n +
[m]
r [m] , r[n=2m+2N+1] = [m] b [m]+n [m]
(7)
where +
[m] = [2m] and [m] = [2m+2N +1]. Also, n +
and n are the noise samples in two
phases. The block diagram of Fig.3 shows the polyphase representation of D2SD transmission. To
simplifyfurtherournotations,hereafter,wewillnotspecifythe(symbol)timeindex[m]. Assumingthe
conditionsof relativelyfastfading,witha reasonablevalueof N (regardingtherequireddelayinsignal
transmission), +
and will be independent random variables. De ning the vectors r =
"
r +
r
#
,
2
Noticethatalthoughweintroducedatimedelayof(2N+1)T
b
betweenthecorrespondingcombinationsofeachpairof
sourcesymbols,thedetectionproblematreceiverbecomesinfactthatofaninstantaneousmixtureofthemwhileimposing
adelayinsignaldetection(thenegativedelayshownonFig.2and3).
N
x
1
x
µ
x
-1
x
µ
∑
∑
stream S 1
stream S 2
phase P +
phase P -
x
α +[m]
+
n +[m] r +[m]
x
α -[m]
+
n -[m]
r -[m]
a 1 [m]
a 2 [m]
b +[m]
b -[m]
delay
channel
D2SD
-N
Figure3:PolyphaserepresentationofD2SDtransmission
b=
"
b +
b
#
, anda=
"
a
1
a
2
#
, andusing(7) and(2), wecan write:
r=G 2
6
4 a
1
a
2 3
7
5 +
2
6
4 n
+
n 3
7
5
with: G= 2
6
4
+
+
3
7
5
(8)
We furtherde nethe matrixC asfollows.
C = 2
6
4
1
1
3
7
5
; G= 2
6
4
+
0
0
3
7
5
C (9)
Thereisatight analogybetweentheD2SDandCDMA signaling[16 ]: wecanconsiderourtransmission
scheme like thecombination of the symbols of two users ina multiple access channel. In thisway, the
symbols in streams S
1 and S
2
belong to users #1 and #2, respectively. The phases P +
and P can
hence be regarded aschips#1 and #2. However, incontrast to CDMA, herewe introduced a delayof
(2N+1)T
b
inthetransmissionofP +
andP ,andalsointroducedthemixturefactor. Bythisanalogy,
C can infact be regarded asa matrix of non-binary orthogonal codes 3
,whereas Gcan be seen as the
matrixofnon-orthogonalcodesduetothefadinge ect,althoughwehave atfadingconditions. Hence,
at receiver, ourproblem issimilarto that ofmulti-userdetection; withmultiple-access interference but
withoutinter-symbolinterference.
3 Detector structure
Foranon-fading channel, ora veryslowlyvaryingchannelwherethechannelcoherence time
c
NT
s
for a pre-de ned N, we have +
= and so, G remains orthogonal. The detection of symbols is
easy inthis case, and is done inan optimum manner by the matched lter (MF) G H
=
C T
, where
3
Concerningchips#1and#2,wehaveunequalpowerforeachusercode,butequalpowerfor theensembleofusers.
superscripts : , : and : denote transpose, transpose-conjugate, and complex conjugate, respectively.
Inother words, D2SDis transparent to non-fading channels.
Let us consider the general case of fading channel, i.e., NT
s
c
. We assume that the channel
is estimated perfectly at receiver, and hence, +
and are known exactly. We choose the optimum
ML detector to detect a
1 and a
2
from thereceived signalr and proposea very simpleimplementation
for it. As we will see later in Subsection 3.1, the ML detection is done at the output of the MF to
\channel+code",G H
,which givesthesampledsignalat rate 1=T
s
on therecovered streamsS
1 and S
2 :
y= 2
6
4 y
1
y
2 3
7
5
=G H
2
6
4 r
+
r 3
7
5
= 2
6
4 a
1
a
2 3
7
5 +
2
6
4 n
0
1
n 0
2 3
7
5
(10)
where,
G H
= 2
6
4
+
+
3
7
5
; , G
H
G= 2
6
4 j
+
j 2
+ 2
j j 2
j +
j 2
j j 2
j +
j 2
j j 2
2
j +
j 2
+j j 2
3
7
5
: (11)
n 0
[:]isthe noisesampleat the MFoutputand j:jdenotesmodulus.
3.1 Maximum likelihood detection of data symbols
Let ^a
1 and a^
2
be the hard decisions made by the ML detector on the transmitted symbols a
1 and a
2 .
It isstraight forwardto showthat forjointlyML detection of (a
1
;a
2
),wehave to look for( ^a
1
;^a
2 ) that
minimizethefollowingexpression.
4
! =
r +
+
^ a
1 +
+
^ a
2
2
+
r
^a
1
a^
2
2
(12)
Aftersome manipulations,whileneglectingconstantterms thatdonotdependona^
1
noron^a
2
,theML
detection reducesto themaximization ofthe function:
=^a
1
<fy
1 g + ^a
2
<fy
2 g ^a
1
^ a
2
j +
j 2
j j 2
(13)
<f:gdenotesthereal part operator. Letus de ne
0
=
j +
j 2
j j 2
: (14)
It can be shown that the detector outputs, or in other words, the jointly optimal hard decisions on
(a
1
;a
2
),are obtainedfrom (15), inwhich sgn(:)is thesign functionand j:jtheabsolutevalue [20 ].
8
>
<
>
:
^ a
1
=sgn
<fy
1 g+
1
2
<fy
2 g
0
1
2
<fy
2 g+
0
^ a
2
=sgn
<fy
2 g+
1
2
<fy
1 g
0
1
2
<fy
1 g+
0
(15)
Noticethat j( ^a
1
;a^
2
)j can be consideredasa measure ofreliabilityofthe decisionin(15).
4
HeretheMLdetectionisequivalenttoGLRT(GeneralizedLikelihoodRatioTest)detection[19 ].
y 1 [m]
r +[m]
r -[m]
x
α +* [m]
x
α -* [m]
C T y
2 [m]
Re
ρ ’
Re
+
+
+
-
+ +
+
| . |
| . |
+ x
1/2
]
1 [
ˆ m
a
Matched Filter ML detection
decision
+
-
Ts
Ts
Figure 4:MLdetectorgivingharddecisionsonthetransmitted(source)symbolsofstreamS1 (a1).
3.2 Error probability
A closed form expression forthe exact error probabilityP
e
can notbe obtained. We provide, instead,
expressions for an upper and a lower bound on P
e
. For a given pair of source symbols (a
1
;a
2 ), the
errorprobabilityon each one willbe di erent,dependingon the channel gains( +
; ) corresponding
to thecombinedsymbols(b +
;b ). Letusdenotetheerrorprobabilitieson a
1 and a
2 byP
e;a1 andP
e;a2 ,
respectively,resultingfrom thejoint MLdetection of thispair ofsymbols. Inspiringbytheapproachof
Verduinthecase ofmultiuserdetection [16], weprovideinthe following,boundson theseprobabilities
asa functionof( +
; ) [20 ]. Let us rstde ne thefollowingprobabilities:
P 0
1;a
1
= 1
2 erfc
r
j +
j 2
+ 2
j j 2
2
!
(16)
P 0
2;a
1
= 1
2 erfc
r
j +
j 2
(+1) 2
+j j 2
( 1) 2
2
!
(17)
P 0
3;a
1
= 1
2 erfc
r
j +
j 2
( 1) 2
+j j 2
(+1) 2
2
!
(18)
whereerfc(x)= 2
p
R
1
x e
t 2
dt, and 2
=2N
0
=T
b .
The upperand lowerboundson P
e;a
1
,denoted byP
upper;a
1 and P
l ower;a
1 ,are:
P
upper;a1
=P 0
1;a
1 +
P 0
2;a
1
2 +
P 0
3;a
1
2
; P
l ower;a1
=max n
P 0
1;a
1
; P
0
2;a
1
2
; P
0
3;a
1
2 o
(19)
By interchanging +
and in (16), (17), (18) we obtain P 0
1;a2 , P
0
2;a2
, and P 0
3;a2
, the corresponding
probabilitiesfor a
2
. Meanwhile, we notice that P 0
2;a
2
= P 0
3;a
1
and P 0
3;a
2
= P 0
2;a
1
. The upper and lower
boundson P
e;a
2
arethen:
P
upper;a
2
=P 0
1;a2 +
P 0
2;a
2
2 +
P 0
3;a
2
2
; P
l ower;a2
=max n
P 0
1;a2
; P
0
2;a
2
2
; P
0
3;a
2
2 o
(20)
Theerror probabilityP
e
,averaged on a pairof (a
1
;a
2 ) is:
P
e
= P
e;a
1 +P
e;a
2
2
(21)
upper l ower e
P
upper
= P
upper;a1 +P
upper;a2
2
; P
l ower
= P
l ower;a1 +P
l ower;a2
2
(22)
We willseethat these boundsaretight enoughand quiteusefulinstudyingthereceiverperformance.
4 Performance analysis of D2SD
To studytheperformanceof D2SD,we rst considersomeparticular channelrealizationsinSubsection
4.1, before treating thecases of Rayleigh and Ricean fading inSubsections4.3and 4.4, respectively.
4.1 Particular channel realizations
4.1.1 Performance evaluation
We provide performance curves in terms of the error probabilityP
e
versus E
b
=N
0
for a given channel
realization, that is, for a particular pair of channel gains +
and . From (8) the \instantaneous"
received energyperT
b
includesapartrelatingto a
1
andtheother part relatingto a
2
. We separate the
energiescorrespondingto a
1 and a
2
,thatwecallE
b1 and E
b2
,respectively:
E
b1
= T
b
2
j +
j 2
+ 2
j j 2
; E
b2
= T
b
2
2
j +
j 2
+j j 2
: (23)
Thelocal average received energy(overa
1 and a
2
symbols)perT
b
is then:
E
b
= 1
2 E
b1 +E
b2
= T
b
4
j +
j 2
+j j 2
1+ 2
(24)
and thelocal average errorprobabilityP
e
is given by(21).
4.1.2 Impact of the mixture factor
The mixture factor has an important impact on the receiver performance and should be chosen
appropriately. It can in fact be regarded as a \degree of freedom" that we do not dispose in binary
CDMA coding,forexample. Here,a trade o shouldbe consideredinthe choice of : Thecloser is
to 1,thebetterwecan pro t intermsof fadingreduction,buttheinterference of theothersymbolwill
bemore importanttoo (interference ofa
2
inthedetection ofa
1
,forexample).
Let usde ne thefactor f =
+
. We want to see thee ect of f on theerrorprobabilityfora given
. Notice that f close to 1 signi es a small change in the channel gain from the time reference T
b to
(2N +1)T
b
, whereas a large f signi es a deep channel fade in the latter reference time. Fig.5 shows
0 5 10 15 20 25
10 −6
10 −5
10 −4
10 −3
10 −2
10 −1
10 0
f = 100
f = 10
f = 5
f = 2
f = 1
µ=0,1
µ=0
µ=1
P e
E b /N
0 (dB)
(a)
0 5 10 15 20 25
10 −6
10 −5
10 −4
10 −3
10 −2
10 −1
10 0
P e
E b /N 0 (dB)
f = 2
f = 1
f = 5
f = 100
(b)
Figure 5:ImpactofonD2SDperformance,particularchannelrealizationwithf=
+
;(a)=0;1;(b)=0:5
curvesof (upperand lower boundson) P
e
versusE
b
=N
0
for=0;0:5;1 and di erent valuesoff. 5
=0: No mixture is performed on the symbols of S
1 and S
2
actually, and hence, we see purely the
e ect of the channel fading on P
e .
6
So, the worst performance is obtained for the same f (> 1) as
compared to thetwo other cases. Forf !1, inhigh SNR we can decidecorrectly on a
1 (ora
2 ) only,
and so,P
e
!1=4.
=1:Wehavethemaximumfadingreductionbuttheinterferenceismaximumtoo. So,theperformance
isbetterthanfor=0,butworsethanfor=0:5forlargef. Forf !1,inhighSNR wecan decide
correctly on a
1 anda
2
onlywhenthey areequal, and hence, P
e
!1=4.
=0:5:Forlarge enoughf theperformanceisbetterthanthatfor=0 and=1,andthemaximum
degradationwithrespecttotheno-fadingcaseisabout3to4dB.Forf !1whereb islost,thedecision
ona
1 and a
2
shouldbe performedusingonlyb +
,whereinthepresenceof noisethebestperformanceis
obtainedfor
opt
=0:5.
It could be seen that for f > 1 the performances of ZF or MMSE detectors degrade considerably as
comparedto ML detector, sincethese solutions consist ininverting(exactly orpartially)the matrix .
5
Noticethatfor f =1(nofading), theperformanceof thesystemis independentof ,andthe optimumdecisionson
a1 anda2 aremadeaftertheMFG H
simply. InthiscaseweobtainPe= 1
2 erfc
p
E
b
=N0
.
6
Theerrorprobabilityfor=0isgivenbyPe=(Pe;a
1 +Pe;a
2 )=2=
1
4 erfc
q
E
b1
N0
+ 1
4 erfc
q
E
b2
N0
.
Theperformanceimprovement obtainedbyD2SDdependsonthefadingstatistics. We expecttoobtain
amoreimportantgaininthecaseofRayleighfading,ascomparedtotheRiceanfading,sincetheformer
case undergoesmoresevere channelvariations.
4.2.1 Performance evaluation for a random channel
RememberthatregardingthechoiceofN,i:e:thedelayimposedinsignaltransmission,D2SDissuitable
forrelativelyfastfadingchannels. So,itisquitereasonabletoconsidertheconditionsofergodicchannel,
and to considertheaverage errorprobability
P
e
=EfP
e
gforthe performanceevaluation (Ef:g denotes
expected value). We should hence use the average received SNR
E
b
=N
0
in the performance analysis,
with
E
b
=EfE
b
g the average received energy per T
b
at the receiver input. According to our previous
de nitions,
E
b
=(1+ 2
) T
b
2 j j
2
,where j j 2
=Efj [n ]j 2
g.
4.3 Rayleigh fading
4.3.1 Error probability computation
The upper and lower bounds on
P
e
can be computed by means of Monte Carlo simulations using the
expressions given inSubsection 3.2. We can, however, obtainan approximateanalytical expressionon
theupperbound,asexplainedinthe following.
We have erfc(x)e x
2
. By replacingerfc(:)function in(16), (17), (18)bye x
2
we obtaina somewhat
looser but stillusefulupperbound, aswe willshow. We can now easilyaverage the upperboundover
+
and ,from thePDF of=j j,given by(25)forRayleigh distribution:
P()=2 exp( 2
): (25)
After some manipulations,we obtainthefollowinganalyticalexpressionforthenew upperbound.
P 0
up
= 1
4
1+ 1
2
1
1+
2
2
1
+ 1
4
1+
(1+) 2
2
1
1+
(1 ) 2
2
1
(26)
We can furthersimplifythisboundinhigh SNR wherewe obtain:
P 0
up
1
4
E
b
N
0
2
2
+(1 2
) 2
1+ 2
2
; E
b
N
0
1 (27)
4.3.2 Choice of mixture factor
From (27) we obtainthe optimal choice of =0:53 minimizing P 0
up
, independentlyof SNR. However,
remember that (27) is valid for high SNR. For low SNR, P 0
up
from (26) as well as P
upper
and P
l ower
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10 −6
10 −5
10 −4
10 −3
10 −2
10 −1
10 0
E b /N
0 = 3 dB
10 dB
20 dB
25 dB
µ
P e
P upper
P’ up
P lower
Figure6:E ectofontheperformanceofD2SD,Rayleigh atfading
from (22)do dependonSNR. We have showninFig.6 boundson
P
e
versus forfourvaluesof
E
b
=N
0 .
Remember the trade o in thechoice of between the inducedinterference and thefading reduction.
ForE
b
=N
0
10 dB, theoptimalchoice isbetween0.52 and 0.57. InlowerSNR theexact
opt
depends
on SNR,varyingbetween 0.4and 0.6;however,
P
e
hasa poorsensitivityto inthisinterval.
We willtake
opt
=0:55. Notice thatthis
opt
isa littlelargerthan0.5, which wastheoptimumchoice
fora channelwithno-or-severefading (see Subsection4.1.2 forf !1).
Letuscompareourresultwiththosein[10 ]and[8]forthemodulationdiversitywiththediversityorder
L = 2 under Rayleigh fading. Here, our criterion to nd
opt
was to obtainthe minimum
P
e
. In [10 ]
the optimization criterion for thedesign of the constellation transformation has been to maximize the
minimumproductdistancebetweenanytwopointsofthetransformedconstellation. Withournotations,
by interchanging the columnsof C, we would obtain
opt
=2=(1+ p
5) = 0:618 (see [10 ], Paragraph
VI.A).In[8]theoptimizationhasbeenintermsofthechannelcut-o rate. Thetransformationconsists
of arotation matrixwiththeoptimum rotationangle of 29:63 Æ
(see [8 ],SectionV). Thiscorresponds
to amixturefactor of
opt
=sin( )=cos( )=0:57. We seethatinbothworkstheproposed
opt
isvery
closeto thatwe obtainedforD2SD.
4.3.3 Fading reduction; comparison with SISO and SIMO
Taking
opt
= 0:55, we want to see how much we gain in terms of fading reduction. Forthis purpose,
we compareD2SD witha simplesingle-antenna system(withoutanysourceof diversity)and a double-
receiveantenna systemusingMRC(MaximalRatio Combining) detection. We willcallthese two cases
SISO (single-inputsingle-output) and SIMO (single-inputmultiple-outputs), respectively. Considering
0 5 10 15 20 25
10 −6
10 −5
10 −4
10 −3
10 −2
10 −1
10 0
P’ up,approx
P’ up
P upper
P lower
SIMO,
2 antennas
SISO
SISO & D2SD
µ=0.55
P e
E b /N
0 (dB)
Figure 7:PerformancecomparisonofD2SD,SISO,andSIMO,Rayleigh atfading
Rayleigh atfadingconditions,Fig.7contrasts theperformancesofD2SD,SISO,andSIMO.ForD2SD,
the upperand lower bounds on
P
e
are shown versus
E
b
=N
0
, whereas for the two other cases, we have
showncurves ofasymptotic
P
e
inhigh SNR,given below[2 ].
7
P
e
1
4
E
b
=N
0
M
R
2MR 1
MR
!
(28)
M
R
=1 forSISO and M
R
=2 for SIMOand
P
Q
= P!
Q!(P Q)!
. Notice that no coding isused for either
system. From Fig.7 we see that oursystem outperforms the simpleSISO inhigh SNR. It hasa degra-
dationof about 3dB, ascompared to theSIMOsystem. This is,infact, thevery reasonable pricepaid
forthemixture of symbols. Theinterestingpoint is thatthisdegradationdoesnotdependon SNR. In
other words,theslopesof
P
e
curves, whichsigni estheorder of diversityof thesystem, arealmost the
same forD2SD andSIMO (see(27) and (28)).
8
To see theutilitytheanalyticalexpressions, we have also shown inFig.7 curvesof P 0
up
and its approx-
imation in high SNR from (27). It is seen that P 0
up
is quite useful and close to P
upper
; however, its
approximation (forhigh SNR)islooseforE
b
=N
0
<10dB.
7
WeassumetheconditionsofuncorrelatedfadingontheantennaelementsforthecaseofSIMOsystem.
8
Noticethatsincehereweconsiderthe performanceintermsof
Eb=N0 atreceiver,the correspondingcurvefor SIMO
systemcanalsobeconsidered astobelongto adouble-transmit-antennassystem,performingMRCdetectionatreceiver
by usingAlamouticoding,for example[21,22 ]. Noticealso thatD2SDwith=1 maybe regardedas atransposition of
theAlamouticode[21 ]tothecase ofasingle-antennasystem.
For Ricean fading channels, the received signal can be consideredto be composed of two components;
onefromline-of-sight(LOS)andtheotheronefrommultipathre ections. Theformercomponentcanbe
assumedto bealmost deterministicand constant, whereasthelatter isa randomlyvaryingcomponent.
We considerthechannelgain
Rice
as[23 ]:
Rice
= p
RF+ p
1 RF
Ray
(29)
where
Ray
is aunit-variancecircularlysymmetriccomplexGaussianrandomvariablerepresenting ran-
domchannelvariations. RFistheratioofthepowerreceivedfromLOStothetotal received power. The
usuallyemployedRicean K-factor [24 ]isinfactequaltoRF=(1 RF). ThePDFof =j j,forthiscase
isgiven by:
P()= 2
1 RF exp
2
+RF
1 RF
I
0
2
p
RF
1 RF
; 0 (30)
whereI
0
(:)isthemodi edBesselfunctionof rstkindandzeroorder. AswedidforthecaseofRayleigh
fading, we replace erfc(:) in the expressionof P
upper by e
x 2
and average it over the PDF of . After
some manipulations,we obtainthefollowinganalyticalexpressionforthenew upperboundP 0
up
,which
islooserthan P
upper .
P 0
up
=
4
=4
2
+(1 RF)
2
+ 2
(1 RF)
exp
RF
2
+(1 RF)
2
RF
2
+ 2
(1 RF)
+
4
=4
2
+(1+) 2
(1 RF)
2
+(1 ) 2
(1 RF)
exp
(1+) 2
RF
2
+(1+) 2
(1 RF)
(1 ) 2
RF
2
+(1 ) 2
(1 RF)
(31)
In highSNR thisboundis simpli edto thefollowing.
P 0
up
1
4
E
b
N
0
2
2
+(1 2
) 2
1+ 2
2
exp
2RF
1 RF
; E
b
N
0
1 (32)
Notethat forRF=0 (Rayleighfading), we ndthesame expressionof(27).
4.4.1 Choice of mixture factor
We have shown in Fig.8 curves of upper and lower bounds on
P
e
versus for ve values of RF and
E
b
=N
0
=10dB.RF=100%representstheLOSchannel. Itisseenthattoobtainthebestfadingreduction,
should be increased for increased RF. If the LOS contribution is not too high (RF < 90%)
opt is
between0.55and 0.65. Fornegligiblyfadingchannels(RF>90%)
opt
iscloseto 1. Asamatteroffact,
with less signal fading,the MF matrix Gwill be \more orthogonal." So, the e ect of the interference
will be less important, and a larger mixture factor can be chosen. This is in accordance with the
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10 −6
10 −5
10 −4
10 −3
10 −2
10 −1
1
P’ up
P upper
P lower
µ
P e
RF=30%
80%
90%
95%
100%
Figure8: Choiceof,Ricean atfading,
Eb=N0=10dB
resultsofSubsection4.1: wecan seefromFig.5 thatforf=2,=1resultsinabetterperformancethan
=0 or =0:5. We have also shown in Fig.8 curves of P 0
up
. We see that thisbound is quite useful as
long as RF is not too large. The important di erence between P 0
up and P
upper
for RF=100% is due to
thedi erenceof erfc(x) ande x
2
.
5 Discussion and conclusions
For a simple single-antenna communication system D2SD with 0:5 provides an interesting gain
in the system performance, whatever the channel statistics. This improvement is achieved at the cost
of a negligible complexity and without any additional cost in the transmit power and with no loss in
spectral eÆciency. The choice of the imposed delay in signal transmission depends on the rapidity
of channel variations. For this delay to be reasonable, D2SD is appropriate for relatively fast fading
channels. Notice that this is also the case for the techniques of interleaved coding and modulation
diversity. D2SD may not be consideredas an alternative to channelcoding. However, as compared to
thecaseofinterleavedcoding,byD2SDweneedmuchlessredundancytobeaddedtodatabitstoobtain
thesameperformance. Ontheotherhand,D2SDisofconsiderablereducedcomplexity,ascomparedto
concatenated codes.
In D2SD we focused on the diversity order of L = 2 since by this choice we add a very negligible
complexity to the system. Specially, this choice implies a delay of only
c
in the signal transmission
(emission/reception). For larger L values, a better performance can be obtained, but the resulting
systemwouldnotbesuitableforareal-timeorduplexsignaltransmission. Asanexample,foranindoor
0
and a speedof about 5km/h,
c
which is theimposeddelayforL=2,is intheorder of 40 ms. In the
followingwe contrastour work withtheconvolutive precodingand modulation diversityapproaches.
Comparison with convolutive precoders
D2SD is quite advantageous to the precoding techniques of [4 , 5] regarding the complexity and the
induced delay in signal transmission/detection. If for example, we compare the results of Fig.7 with
those of [5 ], to obtain
P
e
10 4
at
E
b
=N
0
20dB, the required precoder length precoder is about
20
c
. D2SD,however, needsa delayof
c
only. Anotheradvantage of D2SDis thatdespite convolutive
precoders that require the noise variance in the fading equalizer section, it does not require the noise
powerfor the detection of symbols. So, D2SD does not su er from a mismatch inthe noise power, as
it may be the case forthe convolutive precoders. Also, D2SD needs the channel gain inonly two time
instants for the detection of each pair of symbols, in contrast to much more time instantsrequired by
convolutiveprecodersforfadingequalization. So,itshouldreasonablybemuchlesssensitiveto channel
estimationerrors.
Comparison with previous works on modulationdiversity
Although D2SD can be regarded as a special case of modulation diversitytechniques, in thiswork we
providedseveralinterestingcontributions. WeprovidedtheML detectorwith asimplestructure,which
permitstodetectthesymbolsseparately. Incontrast,theuniversallatticedecoderin[11],whichismore
general and more complex to implement, is based on the (joint) detection of vectors of symbols or in
other words, thedetection of \signal points". The work of[8 ], on the other hand, considersthe bound
onthecut-o rateoftheML detector, buttreats onlylinear(suboptimal)detectors such astheMMSE
detector. We also provided upperandlowerboundson thesymbolerrorprobability, and showed that
theyaretightenough, andhence, very usefulfortheperformanceanalysisofthedetector. We provided
precisionson the choice of the mixture factor in di erent fading conditionsincluding Ricean fading.
Inmostof thereferences, onlytheclassicalcase ofRayleigh fadingis considered.
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