INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 17 |NUMBER: 4 |2019 |DECEMBER
Pe o mance Analysis o MIMO Wa ele Packe
Mul ica ie Mul icode CDMA Sys em wi h
An enna Selec ion
Ma yam Mahmoud AKHO-ZAHIEH, Nasse ABDELLATIF
Depa men o Elec ical Enginee ing, Facul y o Enginee ing & Technology,
Applied Science P i a e Uni e si y, Al A ab s . 21, 11931 Amman, Jo dan
ma y[email p o ec ed], nasse _ab[email p o ec ed]
DOI: 10.15598/aeee. 17i4.3437
Abs ac . Pe o mance o Mul ica ie Mul icode
Code-Di ision Mul iple Access (MC/MCD-CDMA)
sys ems can be imp o ed signi ican ly by using Wa ele
Packe s (WPs) as subca ie s ins ead o a sinusoidal
unc ion. This is because WPs ha e much lowe side-
lobes and negligible sidelobe ene gy leakage compa ed o
sinusoidal ca ie s; his p ope y signi ican ly dec eases
he in e ca ie in e e ence and imp o es he sys em
pe o mance. Fu he imp o emen can be achie ed by
u ilizing mul iple an ennas a ansmi e and ecei e
o cons uc a Mul iple-Inpu Mul iple-Ou pu (MIMO)
sys em. Channel ha dening is he main d awback o
MIMO sys ems. An enna selec ion can be employed
o educe his p oblem. In his pape , we use he an-
enna selec ion in MIMO WP-MC/MCD-CDMA sys-
em. These combina ions o an enna selec ion, MIMO,
and WP in MC/MCD-CDMA imp o e he sys em pe -
o mance signi ican ly and educe he channel ha den-
ing. The pe o mance o he sys em is es ed acco d-
ing o he ou age p obabili y and bi e o a e. Two
MIMO schemes on Nakagami-m ading channel a e
conside ed, which a e: selec i e ansmi /selec i e e-
cei e and selec i e ansmi /maximum a io combin-
ing ecei e. The s udy includes he e ec s an ennas’
numbe , ading pa ame e -m, numbe o use s, and
he h eshold signal o in e e ences plus noise a io.
The pe o mance o he sys em is compa ed o ha o
MC/MCD-CDMA based on a sinusoidal ca ie . The
esul s e eal ha : by inc easing an ennas’ numbe ,
he sys em pe o mance is imp o ed signi ican ly, and
MC/MCD-CDMA sys em based on WPs ca ie s ou -
pe o ms MC/MCD-CDMA sys em based on a sinu-
soidal ca ie .
Keywo ds
Di e si y, MIMO, Nakagami channel, wa ele
packe s.
1. In oduc ion
Mul ica ie /Mul icode-Code Di ision Mul iple-
Access (MC/MCD-CDMA) communica ion sys-
ems [1], [2] and [3], ha e many a ac i e cha ac e is-
ics such as:
•Using MC p o ides high immuni y agains In e -
symbol In e e ence (ISI). This in e e ence esul s
when he channel delay sp ead exceeds he symbol
du a ion [4]. Also, MC can supp ess he e ec o
he na ow-band jamme [5] and [6] by inc easing
he signal bandwid h.
•By MCD scheme, he numbe o ca ie s can be
dec eased; as a esul , he In e ca ie In e e -
ence (ICI) will be educed, and he sp eading gain
will be inc eased. Also, mul i a e se ices can be
achie ed by using MCD [7].
MC/MCD-CDMA sys em pe o mance can be
enhanced signi ican ly by using Wa ele Packe s
(WPs) as subca ie ins ead o sinusoidal ca i-
e s [8], [9], [10], [11], [12], [13], and [14]. This can
be ela ed o he lowe sidelobes and negligible side-
lobe ene gy leakage o WPs compa ed o sinusoidal
ca ie s. This p ope y can u he dec ease he ICI
and mul iple-access in e e ence. Ano he ad an age
o WPs is ha : he e is no need o equency/ ime
gua d be ween di e en use signals because WPs a e
na u ally o hogonal and well localized in ime and e-
quency domain.
c
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To u he educe he e ec o mul ipa h ading and
imp o e sys em pe o mance, an enna di e si y ech-
nique can be used. In di e si y schemes, he ecei ed
signal ia se e al pa hs, ca ying he same in o ma-
ion, a e combined o imp o e sys em pe o mance.
In [8], [10], [11], [13], and [14] h ee schemes o di e -
si y combining echniques we e conside ed, namely Se-
lec ion Di e si y (SC), Equal Gain Combining (EGC)
and Maximal Ra io Combining (MRC). I is obse ed
ha he MRC ou pe o m he o he wo di e si y com-
bining me hods.
A Mul iple-Inpu Mul iple-Ou pu (MIMO) sys em,
which uses mul iple an ennas o ansmi ing and
ecei ing, o e s ad an ages o e Single-Inpu Single-
Ou pu (SISO) sys em because di e si y can be used
a bo h ansmi e and ecei e [15], [16] and [17]. The
MIMO sys ems a e classi ied in o closed-loop sys ems
o open-loop sys ems acco ding o he Channel S a e
In o ma ion (CSI). In he closed-loop sys em, CSI is
a ailable a he ansmi e while in open-loop sys ems
i is no . In he closed-loop MIMO mul i use sys em,
an enna Selec ion (SC) a he ansmi e can be em-
ployed o educe channel ha dening and enhance he
sys em pe o mance.
The analysis o an enna selec ion a ansmi e
wi h maximal- a io combining a he ecei e (SC-
TX/MRC-RX) is discussed in [18], [19], [20], [21], [22],
[23] and [24]. In [18], a e age ou age capaci y o mul-
iuse MIMO-SC-TX/MRC-RX sys em is de i ed. The
au ho demons a es he e ec s o MIMO con igu a-
ion, numbe o use s, and mul iuse di e si y on he
ou age capaci y pe o mance. The amewo k analy-
sis o Mul iuse Di e si y (MUD) gain in selec ion an-
enna MIMO sys em o e la - ading Ralyleigh chan-
nel is de eloped in [19]. Only poin - o-poin commu-
nica ion links a e conside ed in his pape . As an ex-
ension o [19], he au ho s in [20] ocus on he pe o -
mance analysis SC-TX/MRC-RX sys em in poin - o-
mul ipoin communica ions.
O e la - ading Ralyleigh channel, he au ho
in [21] demons a es he analysis o a e age Bi E -
o Ra e (BER) pe o mance o mul iuse MIMO-SC-
TX/MRC-RX. Two scena ios we e conside ed which
a e:
•He e ogeneous: independen non-iden ical dis-
ibu ed Signal o Noise Ra io (SNR).
•Homogeneous: independen iden ical dis ibu ed
SNR.
The analysis o he impac o eedback delay be ween
he ansmission selec ion pa ame e s and he ans-
mission ime on mul i use sys em employing o hog-
onal space- ime block coding wi h a e-adap i e mod-
ula ion and use selec ion scheme is p esen ed in [22].
The au ho s analyze he BER, a e age spec al den-
si y, and bi e o ou age p obabili y in he e ogeneous
ime- a ying MIMO Ralyleigh ading channel.
In [23], he au ho s de i ed he BER and he ou age
P obabili y (Pou ) o bina y phase shi keying o he
SC-TX/MRC-RX sys em in Rayleigh ading channels.
By analysis and simula ion he au ho s p o e ha he
SC-TX/MRC-RX sys em signi ican ly ou pe o ms he
space- ime block coding, which has he same di e si y
o de and he same numbe o ecei e an ennas. The
au ho s in [24] pe o m he analysis o Symbol E o
P obabili y (SEP) o se e al cons ella ions o pe ec
CSI eedbacks. In hei analysis, hey showed ha he
an enna di e si y imp o es he sys em pe o mance bu
he eedback delay deg ades signi ican ly in he SEP
o SC-TX/MRC-RX scheme. In [25], exp essions o
Pou and BER on a la - ading Rayleigh channel o
some MIMO sys ems wi h MUD we e p esen ed. The
au ho s use he uni o m powe alloca ion p ocedu e a
he ansmi e and assume ha he CSI o each use
is known o he schedule a he base s a ion.
The Nakagami-mdis ibu ion is known o span ia
i s pa ame e ma wide ange o mul ipa h ading.
This includes one-sided Gaussian model (m= 0.5), he
Rayleigh model (m= 1) and Ricean model (m= 3)
[26]. In [27], he au ho in es iga es he pe o -
mance analysis o SC-TX/MRC-RX MIMO sys em on
Nakagami-m ading channel. The exac closed o m ex-
p ession o he SNR and BER a e de i ed by ob aining
s a is ics as such as Cumula i e Dis ibu ion Func ion
(CDF), he P obabili y Densi y Func ion (PDF) and
he Momen Gene a ing Func ion (MGF). The le e
in [28] ocuses on he s udy o he di e si y ad an age o
using MIMO an ennas by de e mining he PDF o sin-
gle and MIMO channel by which he s a is ical cha ac-
e is ics o MIMO sys ems can be desc ibed. By Mon e
Ca lo simula ions, he au ho inds o mulas ha de-
ine he channel gain and he di e si y o de o single
and mul iuse MIMO sys ems. Also, by using a Chi-
squa e goodness-o - i es , he au ho p o es ha he
dis ibu ion o he channel gain can be app oxima ed
by a Nakagami-mdis ibu ion. Uni ied simple o mulas
o BER and Pou o some mul iuse MIMO sys ems,
such as SC-TX/SC-RX, SC-TX/MRC-RX, and space-
ime block coding sys ems, on Nakagami-mchannels,
a e p esen ed in [29]. The pe o mance limi s o mas-
si e MIMO sys ems unde p ac ical an enna selec ion
algo i hms a e in es iga ed in [30]. In his sys em,
he ansmi e selec s a subse o he a ailable an-
ennas wi h he s onges channel gains. They show
ha wi h only 30 % o an ennas being ac i e, mo e
han 90 % o he e godic a e ob ained by ull an-
enna selec ion, can be achie ed. The analy ical ex-
p ession o he numbe o selec ed an ennas ha can
maximize ene gy e iciency is also de i ed. In [31], he
au ho s in es iga e he MIMO echniques in a ehicle-
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o- ehicle communica ion sys em. The pe o mance o
SC-TX/MRC-RX and SC-TX/SC-RX schemes was an-
alyzed o e n*Rayleigh ading channels; "n*Rayleigh
can be de ined as a p oduc o nindependen Rayleigh
andom a iables connec ed ia na ow pipes". The
au ho s de i e a closed- o m exp ession o he ou age
p obabili y and he amoun o ading. Thei nume -
ical esul s show ha SC-TX/MRC-RX ou pe o ms
SC-TX/SC-RX, bu i s pe o mance is limi ed as nin-
c eases. Using Cumula i e Densi y Func ion (cd ) an
exac closed- o m exp ession o he Symbol E o Ra e
(SER) o SC-TX/MRC-RX wi h elay and use selec-
ion o MIMO sys em o e non-iden ical Nakagami
ading channels is de i ed in [32]. They show ha
he SER is imp o ed by mul i- elay di e si y, mul i-
use di e si y and he sum o non-iden ical ading pa-
ame e s o di e en ansmi an ennas, elays, and
des ina ions. In [33], Signal Space Di e si y (SSD) is
employed in o a MIMO sys em wi h MRC and ans-
mi an enna selec ion. The sys em pe o mance wi h
Phase-Shi Keying (PSK) modula ion is in es iga ed
unde a slow la Rayleigh ading channel wi h he co -
ela ed ecei e an ennas. The exac closed- o m ex-
p ession o pai wise e o p obabili y is de i ed using
he ealis ic exponen ial co ela ion model. I is shown
ha he e o pe o mance o he sys em can be im-
p o ed wi h almos no ex a complexi y o cos .
Many esea ch pape s combine bo h echnologies o
MIMO and wa ele . In [34] o imp o e he qual-
i y o he ansmi ed image in he MIMO sys em,
adap i e comp ession echniques based on wa ele
ans o m in a MIMO communica ion sys em is p o-
posed. Wa ele comp ession is used in [35] o
educe CSI eedback da a size in MIMO sys em.
In [36], [37], [38], [39] and [40], he au ho s s udied
he pe o mance o MIMO-O hogonal F equency Di-
ision Mul iplexing (OFDM) based on wa ele ins ead
o Fas Fou ie T ans o m (FFT) by using Space-Time
Block Code (STBC) unde di e en channel condi ions.
All hese s udies demons a ed ha using wa ele in
MIMO-OFDM sys em ins ead o FFT will imp o e he
BER pe o mance. The au ho s in [41] compa e he
BER pe o mance o STBC- amele OFDM sys em us-
ing M-PSK and M-QAM modula ion echniques wi h
a ious numbe o cons ella ion poin s. The amele s
elimina e a po ion o he cons ain s o wa ele s such
as shi -sensi i i y, poo di ec ionali y, and lack o
phase in o ma ion. In [42], he au ho p oposes a Dis-
c e e Wa ele (DWT)-Join An enna Selec ion (JAS)
sys em o mi iga e he pe o mance loss due o an-
enna selec ions in MIMO sys ems. Ex ensi e simu-
la ions demons a ed ha he DWT-JAS imp o es he
capaci y in he p esence o co ela ion a he ansmi -
e and he ecei e .
In ou p e ious wo k [8], [9] and [10], we p opsed
MC/MCD-CDMA sys em ha uses a WP as a sub-
ca ie o educe ICI and mul iple-access in e e ence.
The sys em pe o mance is es ed using he signal o
in e e ence plus noise a io in [8] and [9]. To u -
he imp o e he sys em pe o mance di e si y is used
in [8] and [10], he sys em pe o mance is esed using
BER and Pou pe o mances wi h and wi hou di e -
si y. The e ec s o wa ele amily ype, wa ele il e
leng h, di e si y ype, di e si y o de , mul ipa h in en-
si y p o ile, and Nakagami pa ame s a e in es iga ed.
The au ho s in [11], [13] and [14], use in e e ence sup-
p ession il e a he ecei e o mi iga e he e ec o
na ow-band jamme in e e ence and hus imp o e he
sys em pe o mance. In [12] and [14] o accommoda e
in o ma ion sou ces wi h di e en da a a es, we p o-
pose he use o MCD scheme o mul i a e se ices.
In his pape , which is an ex ension o [8], [9], [10],
[11], [12], [13], and [14], we employ mul iple an ennae
a he ansmi e and ecei e o he WP-MC/MCD-
CDMA sys em. Ins ead o STBC which is used
in [36], [37], [38], [39], [40], [41] and [42], in ou sys-
em we use he selec ion di e si y a he ansmi e o
educe he channel ha dening which is he main d aw-
back o MIMO sys ems. The pe o mance analysis o
MIMO WP-MC/MCD-CDMA sys em in slow ading
Nakagami-mchannel is p esen ed. Two schemes o
MIMO sys em a e in oduced, which a e: SC-TX/SC-
RX and SC-TX/MRC-RX. The pe o mances o hem
a e in es iga ed acco ding o bi e o a e and ou -
age p obabili y. The pe o mance analysis includes he
e ec s o an ennas’ numbe , numbe o use s, ading
pa ame e , and he h eshold signal o in e e ences
plus noise a io. The pe o mance o he MIMO-WP-
MC/MCD-CDMA sys em is compa ed wi h he pe o -
mance o Sinusoidal (SIN) based MIMO sys em, which
is deno ed by MIMO-SIN-MC/MCD-CDMA.
The o ganiza ion o he pape is as ollows. In
Sec. 2. , we p opose he anscei e sys em o
MIMO-WP-MC/MCD-CDMA. Signal o in e e ence
plus noise a io is illus a ed in Sec. 3. In Sec. 4.
and Sec. 5. , espec i ely, he BER and Pou o SC-
TX/SC-RX and SC-TX/MRC-RX sys ems a e gi en.
Speci ic nume ical esul s o he sys em BER and Pou
pe o mances a e p esen ed in Sec. 6. A las , in
Sec. 7. , he conclusions a e gi en.
2. Sys em Model and
Desc ip ion
In his s udy, he channel o he sys em unde consid-
e a ion consis s o N ansmission an ennas and N
ecep ion an ennas and cha ac e ized by a N ×N ma-
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ix gi en by:
Hk=






h11
kh12
k. . . h1N
k
h21
kh22
k. . . h2N
k
.
.
..
.
.. . . .
.
.
hN 1
khN 2
k. . . hN N
k







=
=hk
1hk
2. . . hk
N .
(1)
The channel is assumed o be slowly ading Nak-
agami channel [26], and i s elemen s a e independen
and iden ically dis ibu ed (i.i.d.) complex Gaussian
andom a iable wi h µ= 0 and σ2= 1 . The impulse
esponses om each ξ h ansmi an enna o each χ h
ecei e an enna can be w i en as:
hχξ
k( ) =
L−1
X
l=0
hχξ
k(l)δ( −lTc),(2)
whe e Lis he numbe o p opaga ion pa hs and hχξ
k(l)
is he pa ame e o he l h pa h which depends on he
gain, phase and ime delay o he pa h be ween he
ξ h ansmi an enna and χ h ecei e an enna. Each
use can ansmi using only one an enna, he op imal
an enna, and i s signal is ecei ed by N an ennas.
The anscei e o he sys em unde conside a ion
is shown in Fig. 1. Each o he ansmi e and ecei e
consis s o wo pa s, which a e he mul icoding pa
and he sp eading/WPs pa . The da a o k h use :
dk( ) = dI
k( )−jdQ
k( ) =
∞
X
i=−∞
di
kΠT
JV −iT
JV !,(3)
is a andom complex sequence, whe e Tis he bi du-
a ion, Jis he o al numbe o mul icode sub-s eams,
Vis he o al numbe o he WPs supe -s eams and
Πx ep esen s a ec angula pulse wi h du a ion x. A
he mul icode pa o he ansmi e , he k h use da a
is Se ial- o-Pa allel (S/P) con e ed o gene a e he J
sub-s eams, which hen coded by an o hogonal signal:
aj( ) =
Nc−1
X
i=0
ai
jΠTc( −iTc),(4)
o educe he in e e ence be ween he sub-s eams
hemsel es. A las , he codded sub-s eams a e added
be o e being ansmi ed o he sp eading/WPs pa .
A he sp eading/WPs pa o he ansmi e , he ou -
pu signal o he coding pa is S/P con e ed in o V
supe -s eams. These supe -s eams a e sp ead by he
Pseudo-Noise (PN) sequence:
ck( ) =
Nn−1
X
i=0
ci
kΠTn( −iTn),(5)
modula ed by he wa ele packe wp ( ), added, and
inally modula ed by a sinusoidal ca ie , exp(jω0 ).
No e ha :
• he o hogonal signal, aj( ), has leng h =Ncand
chip du a ion =Tc=T
HNc
,
• he PN sequence, ck( ), has a chip du a ion =Tn
and leng h =Nn=T
Tn
,
•wp ( ) = N
TnX
i
p N·
Tn−iTn!is he h WP.
The wa ele unc ion p (•)has a suppo leng h
=Nand is de ined ecu si ely by a pai o quad a-
u e mi o lowpass il e h0(k)and highpass il e
h1(k)[8],
•ai
jand ci
k· ∈ {±1}a e he i h bi s wi h p obabili-
ies P(1) = P(−1) = 0.5.
Assuming iden ical powe , P, o all use s and he
ξ h an enna is he op imal an enna o k h use , he
ansmi ed signal o he k h use using he ξ h an-
enna is gi en by:
sξ
k( ) = √2P
V
X
=1
J
X
j=1
Re[dξ
kj ( )aj( )·
·ck( )wp ( ) exp(jω0 )],
(6)
whe e dξ
kj ( )is he da a symbol on he ξ h ansmi ed
an enna o k h use , j h sub-s eam, h supe -s eam
and wi h a du a ion =T. The abo e signal, sξ
k( ), is
de ec ed by he ecei e a e passing h ough a noisy
channel. The ecei ed signals on he χ h ecei e an-
enna o he k h use is gi en by:
χ
k=
N
X
ξ=1
hχξ
k( )∗sξ
k( ) + nχ( ),(7)
whe e nχ( )is a ze o-mean Addi i e Whi e Gaussian
Noise (AWGN). The o al ecei ed signal o k h use
can be ep esen ed in ec o o m as ollows:
k=hξ
k∗sξ
k+nk,(8)
whe e k= [ 1
k, 2
k, . . . , N
k]Tis he ecei ed signal ec-
o and nk= [n1
k, n2
k, . . . , nN
k]Tis he noise ec o .
In his pape , wo MIMO sys em will be conside ed:
•SC-TX/SC-RX Scheme
In his scheme, a any bi du a ion, each use can
choose he link wi h he highes Signal o In e e ence
plus Noise Ra io (SINR γk) om he N N possible
c
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1
)exp(
o
�
N
�
)( dk
)(
1 a
)()( 1 wp ck
PS /
J�1
PS /
V�1
T ansmi e
)( aJ
�
pa gmul icodin
)()( wp c Vk
pa WPssp eading/
�
�
)exp(
o
�
1
N
N
�
�
combine
Di e si y
)( ck
)(
1 wp
k
b
ˆ
1�V
1
x
)( aJ
�
)()( wp c Vk
��
Td
0)(
��
Td
0)(
V
x
��
Td
0)(
��
Td
0)(
)(
1 a
k
d
ˆ
1�J
1
z
J
z
Recei e
pa co ela o / WPsdesp eding
pa co ela o mul icode
selec edisan ennaone
eedbackCSI
use o h
k
SP /
SP /
�
Fig. 1: T anscei e o MIMO WP-MC/MCD-CDMA sys em.
an enna combina ions. Thus, a any bi du a ion, he
ule o signal de ec ion om he χ h op imal ans-
mi an enna o he k h use is: k∗= max
N |kχξ
k|. Ac-
co dingly, he e ec i e SINR γka he ecei e an enna
combine ou pu o k h use is:
γSC/SC = max 






γ11
kγ12
k. . . γ1N
k
γ21
kγ22
k. . . γ2N
k
.
.
..
.
.. . . .
.
.
γN 1
kγN 2
k. . . γN N
k







,
(9)
whe e γχξ
k=γ|hχξ
k|2and γis he a e age SINR o
each use .
•SC-TX/MRC-RX Scheme
In his scheme, he e ec i e SINR γka he ecei e
an enna combine ou pu o he k h use a any bi
du a ion wi h espec o he ξ h ansmi an enna is
gi en by:
γSC/MRC =
N
X
χ=1
γχξ
k=γ
N
X
χ=1 |hχξ
k|2.(10)
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The ou pu s o he combine a e demodula ed using
a locally gene a ed ca ie .
Then, a he disp eading WPs co ela o pa o he
ecei e , each k h use ecei ed signal om he desi ed
N N possible an enna combina ions, χξ, is dep essed
and demodula ed by ck( ), and wp ( ), espec i ely.
The signal a e being co ela ed o e a pe iod Tis
Pa allel- o-Se ial (P/S) con e ed o eco e he supe -
s eam. I he i s use is ou e e ence use also he
i s pa h, and he i s WP is ou pa h and WP e -
e ences, espec i ely, hen he ou pu o he i s co -
ela o , i s P/S con e e in his pa o he ecei e ,
x1, is gi en by [8] as ollows:
x1=ZT
0
χξ
k( )c1( )wp1( )[cos(ω0 )−jsin(ω0 )]d =
=x1
DS( ) + x1
MP I ( ) + x1
CDI( )+
+x1
WPI( ) + x1
MUI( ) + ˆn1( ),
(11)
whe e:
•x1
DS is he desi ed use signal,
•x1
MP I ,x1
CDI,x1
WPI, and x1
MUI a e he mul i-
pa h in e e ence, he mul icode in e e ence, he
wa ele packe s in e e ence and he mul iuse in-
e e ence, espec i ely, due o use s o he han
he desi ed use a he desi ed N N possible an-
enna combina ions,
•ˆnis he co ela ed AWGN.
The ou pu s o he o he co ela o s in he WP pa
a e gi en by:
x =x
MP I ( ) + x
CDI( ) + x
WPI( )+
+x
MUI ( ) + ˆn ( ), = 2, . . . , V. (12)
The ou pu signal o he i s P/S con e e in WPs
pa o e e ence use is gi en by:
ˆ
b=x1
DS +
V
X
=1
[x
MP I +x
CDI+
+x
WPI +x
MUI + ˆn ].
(13)
A he mul icode co ela o pa , he second pa o
he ecei e , he P/S con e e ou pu signals a e de-
sp ead by he use code ajand co ela ed o e a pe iod
T o eco e he subs eams o use ’ signal. The ou -
pu signal o he e e ence use o he i s co ela o ,
z1, is gi en by:
z1=ZT
0
ˆ
b·a1( )d =
=z1
DS +z1
MP I +z1
CDI +z1
WPI +z1
MUI + ˜n1.
(14)
The ou pu s o he co ela o s o he all Jsub-
s eams a e P/S con e ed o eco e he da a signal,
ˆ
dk( ).
3. Signal- o-In e e ence Plus
Noise Ra io
Two me hods will be used o es he sys em pe o -
mance: he a e age bi e o a e and he ou age p ob-
abili y. The wo pe o mances depend on he ins an a-
neous signal o in e e ence plus noise a io (SINR γ),
which depends on he desi ed use signal powe , he
a iances o he in e e ences and noise a iance. The
desi ed powe signal, he in e e ences, and he noise
e ms consis o wo pa s, he inphase pa and he
quad a u e pa [8]. In his pape and wi hou loss o
gene ali y, he inphase pa was conside ed.
The desi ed inphase signal powe (S), is he powe o
he signal o he i s use , o he i s wa ele packe
which p opaga es ia he i s pa h o he desi ed χξ
an enna combina ion. This powe is gi en by [8]:
S= [z1
DS]2=P(NnT)2
2|hχξ
11 |2.(15)
The in e e ence a iance consis s o ou a iances,
which a e:
•σ2
MP I = a [z1
MP I ]: mul ipa h in e e ence a i-
ance due o pa hs o he han he desi ed pa h,
•σ2
MCDI = a [z1
MCDI ]: mul icode in e e ence
a iance,
•σ2
WPI = a [z1
WPI]: wa ele packe s in e e ence
a iance,
•σ2
MUI = a [z1
MUI]: mul iuse in e e ence a i-
ance.
To calcula e he abo e a iances o BPSK modu-
la ion, i is assumed ha all he in e e ence and he
noise e ms a e Gaussian ze o-mean independen an-
dom a iables. In oking he esul s in [8], he o al
in e e ence a iance, σ2
T I , can be shown o be equal
o:
σ2
T I =σ2
MP I +σ2
MCDI +σ2
WPI +σ2
MUI =
=P(NnT)2
2MI, (16)
and MI is gi en by:
MI =Ω
12Tn(NnV)2·
·"QK
J
V
X
=1
V
X
0=1
ψ 0 −1
J2
V
X
0=1
ψ 01#,
(17)
whe e:
•ψ 0 =ZTn
0n( 0 (ρ))2+ (ˆ 0 (ρ))2odp,
0< ρ < Tn, wi h 0 (ρ) = Zρ
0
w ( )w0
( )d
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and 0 (ρ) = Zρ
0
w ( )w0
( )d being he pa ial
c oss-co ela ion unc ions be ween WPs [8],
•Ω = a [hχξ
k1]and Qis he a iance o Mul ipa h
In ensi y P o ile (MIP).
In his pape , he uni o m MIP will be used. In his
MIP, all mul ipa h componen s ampli ude le els a e
he same =hχξ
k1. Thus:
ΩQ= a "L
X
l=1
hχξ
kl #=
L
X
l=1
a hχξ
kl =
= a hχξ
k1
L
X
l=1
1=ΩL.
(18)
The noise a iance is gi en by [8]:
σ2
n=P(NnT)2
2




VΩ
NnEs
N0




=
=P(NnT)2
2NI,
(19)
whe e N0
2is he double-sided powe spec al densi y o
he AWGN, Es= 2PΩTis he mean ecei ed symbol
ene gy. Using Eq. (15), Eq. (17), and Eq. (19), he
ins an aneous SINR, γ, can be w i en as:
γ=S
σ2
T I +σ2
n
=|hχξ
1l|2[MI +NI]−1=
=|hχξ
1l|2γ.
(20)
4. A e age Bi E o Ra e
The BER, Pe, is ob ained by a e aging he ins an a-
neous Pe(γ), o e he channel ading unc ions. Tha
is:
Pe=Z∞
0
γmax (γ)Pe(γ)dγ, (21)
whe e:
•Pe(γ) = Q√2γ, whe e Q(•)is he Gaussian Q
unc ion,
• γ(γ)is he p obabili y densi y unc ion o γ. This
unc ion depends on channel gain dis ibu ion,
which is assumed o be Nakagami in his pape ,
and also on he MIMO scheme ha is used.
Based on [29], he P
SC−T X
SC−RX
eand P
SC−T X
MRC−RX
ea e gi en
by Eq. (22) and Eq. (23), espec i ely.
P
SC−T X
SC−RX
e=B1m, m
γ, KN N ,(22)
P
SC−T X
MRC−RX
e=B1mN ,m
γ, KN ,(23)
whe e m≥1
2is he Nakagami numbe and B1(a, b, c)
is gi en by [26]:
B1(a, b, c) = 1
2√π·
·
∞
X
n=0
anbac+n
L
(ac +n+ 0.5)
(1 + bc)ac+n+0.5; eal a≥1
2,
(24)
wi h a0=h1
L
(a+1) ic, and:
an=1
n
n
X
j=1
L
(a+ 1)[j(c+ 1) −n]
L
(a+1+j)an−j;n≥1.
(25)
No e ha
L
(•)deno es he gamma unc ion [43].
5. Ou age P obabili y
The Pou is he p obabili y ha he channel capaci y,
C, all below speci ic h eshold capaci y, C h [18]:
Pou =P(C < C h).(26)
The channel capaci y is gi en by [18]:
C= log2(1 + γmax)bi s ·s−1·Hz−1.(27)
F om Eq. (26) and Eq. (27) we ge :
Pou =P(C < C h) = P(γmax <[2C h −1] = γ h).
(28)
Acco ding o Eq. (26), we can de ine Pou as he
p obabili y ha γmax alls below a ce ain h eshold
SINR, γ h. Thus:
Pou =P (γ < γ h) = Zγ h
0
γmax (γ)dγ. (29)
In oke he esul in [29], he P
SC−T X
SC−RX
ou and P
SC−T X
MRC−RX
ou
a e gi en as ollows:
P
SC−T X
SC−RX
ou ="˜
Gm, mγ h
γ
L
(m)#
KN
N
,(30)
P
SC−T X
MRC−RX
ou ="˜
GmN ,mγ h
γ
L
(mN )#KN
,(31)
whe e ˜
G(•,•)is he incomple e gamma unc ion [43].
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6. Resul s and Discussions
Using he abo e analy ical esul s, he BER and Pou
pe o mances o he sys em we e e alua ed. The e-
sul s a e based on he nume ical e alua ion o Eq. (22),
Eq. (23), Eq. (30), and Eq. (31) using MATLAB pack-
ages.
The pe o mance o he sys em is es ed o di e -
en combina ions o N and N . Also, he e ec s o
he numbe o use s, Nakagami pa ame e , and h esh-
old alues a e es ed. The pe o mance o he sys em is
compa ed o ha o MC/MCD-CDMA based on a sinu-
soidal ca ie . Unless o he wise s a ed, he nume ical
esul s we e p oduced using:
•Daubechies wa ele packe s wi h o de 3 (db3),
•Numbe o subs eams J= 4,
•Numbe o supe s eam V= 3,
•Ω = 10 dB, L= 3 and Nakagami-m pa ame e
= 2,
•P ocessing gain leng h Nn= 32 wi h chip du a ion
Tn= 10−6s,
•The h eshold o SINR γ h (dB),
•The numbe o use s K= 2.
No e: in he ollowing Figs.; SCξ/SCχand
SCξ/MRCχmean ha : he numbe o an ennas a he
ansmi e is ”ξ” and he numbe o an ennas a he
ecei e is ”χ”.
6.1. E ec o Numbe o An ennas
on BER and Ou age P obabili y
Figu e 2 and Fig. 3 illus a e, espec i ely, he BER
and Pou pe o mances e sus Es/N0 o SC-TX/SC-
RX and SC-TX/MRC-RX using di e en numbe s o
N and N combina ions. F om hose wo igu es, i
is clea ha o he same numbe o an ennas, he SC-
TX/MRC-RX scheme ou pe o ms he SC-TX/SC-RX
scheme. These esul s a e expec ed since he MRC is
he op imum combina ion di e si y. Also, o he wo
schemes, he pe o mance imp o ed as he o al num-
be o an enna inc eases. Fo he SC/MRC sys ems
wi h a ixed o al numbe o an ennas a ansmi -
e and ecei e , he di e si y gain dec eases as he
numbe o ecei e an ennas inc eases. Fo example,
SC2/MRC3 scheme ou pe o ms SC3/MRC2. In all di-
e si y echniques, his is a ypical diminishing e ec .
This con i ms he obse a ions in [18], [27] and [28].
10 12 14 16 18 20
10−35
10−30
10−25
10−20
10−15
10−10
10−5
Es/No(dB)
Bi E o Ra e
SC2/SC2
SC2/SC3
SC3/SC3
SC2/MRC2
SC3/MRC2
SC2/MRC3
SC3/MRC3
Fig. 2: BER pe o mance e sus Es/N0 o SC-TX/SC-RX and
SC-TX/MRC-RX wi h di e en numbe o N and N
combina ions.
4 5 6 7 8 9 10
10−20
10−15
10−10
10−5
100
Es/No(dB)
Ou age P obabili y
SC2/SC2
SC2/SC3
SC3/SC3
SC2/MRC2
SC3/MRC2
SC2/MRC3
SC3/MRC3
Fig. 3: Pou pe o mance e sus Es/N0 o SC-TX/SC-RX and
SC-TX/MRC-RX wi h di e en numbe s o N and N
combina ions
6.2. E ec o Numbe o Use s on
BER
Fo SC-TX/MRC-RX scheme wi h Es/N0= 16 dB,
m= 1 and using ou di e en N and N combina-
ions, Fig. 4 illus a es he e ec o he numbe o use s
(K) on BER pe o mance. I is clea om he igu e
ha : he highe he numbe o use s, he be e he
sys em pe o mance. This is he inhe en bene i o
he mul iuse sys em, which indica es ha we can use
mul iuse di e si y in a mul i-use sys em. These e-
sul s a e iden ical o hose ound in [19] and [29]. Also,
as in Fig. 2 and Fig. 3, as he numbe o an ennas in-
c eased, he sys em pe o mance imp o ed.
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1 2 3 4 5 6 7 8 9 10
10−45
10−40
10−35
10−30
10−25
10−20
10−15
10−10
10−5
No. o Use
Bi E o Ra e
SC2/MRC2
SC3/MRC2
SC2/MRC3
SC3/MRC3
Fig. 4: E ec o he numbe o use s on BER pe o mance.
6.3. E ec o Nakagami-m
Pa ame e on BER
Figu e 5 shows he e ec o Nakagami-mpa ame e on
BER pe o mance. The SC-TX/SC-RX scheme is used
wi h Es/N0= 16 dB and di e en (N N ) combina-
ions. F om he igu e we can no ice ha : inc easing
mpa ame e and he o al an ennas’ numbe imp o es
he BER. This is expec ed since he highe alue o
mmeans less ading and be e pe o mance. Fo he
an ennas’ numbe , his con i ms he esul s in Fig. 2,
Fig. 3, and Fig. 4.
1 1.5 2 2.5 3 3.5 4 4.5 5
10−20
10−18
10−16
10−14
10−12
10−10
10−8
m
Bi E o Ra e
SC2/SC2
SC2/SC3
SC3/SC3
Fig. 5: E ec o Nakagami-mpa ame e on he BER pe o -
mance.
6.4. E ec o he Th eshold Value on
Pou
Figu e 6 illus a es Pou e sus γ h o he SC/SC
and SC/MRC schemes using Es/N0= 6 dB. As ex-
pec ed, SC/MRC scheme has be e pe o mance han
he SC/SC scheme. Also, as he o al numbe o an-
ennas inc eases, he pe o mance ge s be e . F om
he igu e we can no ice ha : inc easing γ h deg aded
he sys em pe o mance which is expec ed since by in-
c easing γ h he e is a chance ha he channel capaci y
may go below he equi ed h eshold le el.
2 4 6 8 10 12
10−20
10−15
10−10
10−5
100
Th eshold γ(dB)
Ou age P obabili y
SC2/SC3
SC3/SC3
SC2/MRC3
SC3/MRC3
Fig. 6: Pou e sus γ h o SC-TX/SC-RX and SC-TX/MRC-
RX schemes.
6.5. Pe o mance Compa ison
Figu e 7 and Fig. 8 show BER and Pou pe o mances,
espec i ely, o wo MIMO sys ems, which a e MIMO-
WP-MC/MCD-CDMA, ou sys em, and MIMO-SIN-
MC/MCD CDMA sys em. F om hese wo igu es, i
can be no ed ha o Es/N0= 8 dB he MC/MCD
CDMA sys em ou pe o ms ou sys em sligh ly. Bu
as Es/N0inc eases, ou sys em ou pe o ms he o he
sys em and he di e ence be ween he wo sys ems pe -
o mance inc eases as Es/N0inc eases.
6 8 10 12 14 16
10−20
10−15
10−10
10−5
100
Es/No(dB)
Bi E o Ra e
SC2/MRC2 MIMO−SIN−MC/MCD−CDMA
SC2/MRC3 MIMO−SIN−MC/MCD−CDMA
SC2/MRC2 MIMO−WP−MC/MCD−CDMA
SC2/MRC3 MIMO−WP−MC/MCD−CDMA
Fig. 7: BER pe o mance o wo MIMO sys ems.
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