Joint Iterative Channel Estimation and Data Detection for MIMO-CDMA Systems over Frequency-Selective Fading Channels.

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Joint Iterative Channel Estimation and Data
Detection for MIMO-CDMA Systems over
Frequency-Selective Fading Channels
Ayman Assra1, Walaa Hamouda1, and Amr Youssef2
1Department of Electrical and Computer Engineering
2Concordia Institute for Information Systems Engineering
Concordia University, Montreal, Quebec, Canada
e-mail: {a assra,hamouda,youssef}@ece.concordia.ca
I. ABSTRACT
In this paper, we present an iterative joint channel estimation
and data detection technique for multiple-input multiple-output
(MIMO) code-division multiple-access (CDMA) systems over
frequency-selective fading channels. Based on the expectation-
maximization (EM) algorithm, the proposed iterative receiver
achieves a performance close to the optimum maximum-
likelihood (ML) receiver. In addition, the performance of the
proposed receiver is optimized through weight coefficients
using the minimum mean-square error (MMSE) criterion.
Compared to the single-user bound, our results show that the
proposed receiver can mitigate the multiple-access interfer-
ence and attain the full system diversity. Furthermore, our
simulation results confirm that the proposed receiver is near-
far resistant and offers fast convergence in severe near-far
scenarios.
II. INTRODUCTION
In multiple-input multiple-output (MIMO) code-division
multiple-access (CDMA)systems, channel estimation plays a
crucial role in determining the system performance [1]. In
order to achieve the promised performance gain of MIMO
systems, the channel coefficients must be known or estimated
perfectly at the receiver [2]. Recently, there has been an
increasing interest in iterative joint channel estimation and
data detection algorithms [3], which take the advantage of
the detected data symbols in the channel estimation process.
In particular, there has been a growing interest in expectation-
maximization (EM) based joint detection and estimation (JDE)
techniques because of its ability to achieve accurate estimation
without wasting the system resources [4]. Therefore, an exten-
sive effort has been focused on employing the EM algorithm
in JDE techniques for MIMO systems [5]. For example, the
authors in [6] proposed various EM-based channel estimation
techniques for MIMO systems. Choi [7] has proposed a
general framework for EM-based JDE schemes considering
different types of MIMO channels (e.g., Rician or Rayleigh
fading). As well, many research efforts have considered the
JDE problem for MIMO systems considering flat [5] and
frequency-selective channels [8]. In this paper, we propose a
JDE technique based on the EM algorithm for MIMO CDMA
systems assuming asynchronous transmission over frequency-
selective fading channels. The proposed JDE receiver structure
is derived, where we show that it can bring an optimum
balance between a single-user matched filter detector and
a parallel interference cancellation (PIC) based detector. To
achieve a fast convergence to the optimum solution, we ini-
tialize the EM estimates based on reliable estimates of space-
time minimum mean-square error (MMSE) separate detection
and estimation (ST-MMSE-SDE) receiver [9]. The rest of the
paper is organized as follows. The system model is described
in the following section. In Section IV, the EM-based ST
receiver is discussed. In Section V, we derive a closed-form
expression for the optimized weight coefficients. In Section VI,
we discuss the initialization of the EM algorithm. Simulation
results and discussions are then presented in Section VII.
Finally, conclusions are drawn in Section VIII.
III. SYSTEM MODEL
Throughout our work, we consider a transmit diversity
scheme with U=2 transmit antennas at the mobile user and V
multiple receive antennas at the base station. We also consider
the original space-time spreading (STS) scheme proposed
in [10] for an asynchronous K-user system over a slow
frequency-selective fading channel with L resolvable paths.
The channel coefficients are, therefore, considered fixed within
a block of M codewords, where each codeword has a period of
Ts= 2Tband Tbdenotes the bit period. The received complex
low-pass equivalent signal at the vthreceive antenna is given
by
rv(t) =
K
?
k=1
L
?
l=1
M−1
?
m=0
hk,v
1l
?
bk1[m]ck1(t − mTs− τk− ˜ τl)
?
+bk2[m]ck2(t−mTs−τk−˜ τl)+hk,v
2l
?
bk2[m]ck1(t−mTs−τk
?
− ˜ τl) − bk1[m]ck2(t − mTs− τk− ˜ τl)
where bk1[m] and bk2[m] are the odd and even data streams
of the kthuser within the mthcodeword interval. The codes
+ nv(t),
(1)
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ck1(t) and ck2(t) represent the kthuser’s spreading sequences
with processing gain 2N, where N = Tb/Tc is the number
of chips per bit, and Tc is the chip duration. In (1), hk,v
q = 1,2, is the attenuation coefficient corresponding to the
kthuser, lthpath from the qthtransmit antenna to the vth
receive antenna, where hk,v
ql
=
sponding fading channel coefficient and Ek is the kthuser
transmit energy. These attenuation coefficients are modeled
as independent complex Gaussian random variables with zero
mean and variance σ2
noise nv(t) is Gaussian with zero mean and variance No. At
the receiver side, the received signal at each receive antenna is
chip-matched filtered, sampled at a rate 1/Tc, and accumulated
over an observation interval of (2N + τmax+ L − 1) chips
corresponding to the mthsymbol of the received data block
for the K-user system. The (τmax+ L − 1) samples are
due to the maximum multipath delay (i.e., delay of the lth
path) corresponding to the user with the maximum transmit
delay, τmax. Let yv[m] denote the observation vector at the
vthreceive antenna containing all samples related to the STS
symbols transmitted by the K users within the observation
interval. Then, we have
ql,
?
Ek
2αk,v
ql, αk,v
ql
is the corre-
k, where σ2
k=Ek
2σ2
h, and σ2
h=1
L. The
yv[m] = (C[0]B(m) + C[−1]B(m − 1) + C[1]B(m + 1))
× hv+ nv[m],m = 1,...,M − 2,
where C[0], C[−1], and C[1] include the code sequences
corresponding to the current, previous and following STS
symbols of the K-user system within the observation interval
respectively. In (2), B(m), m = 0,...,M −1, represents the
users data matrix within the mthperiod, defined as
(2)
B(m) = diag{B1(m),B2(m),...,BK(m)},
where
Bk(m) = IL⊗˜bk(m),
˜bk(m) =
?
bk1[m]
bk2[m]
bk2[m]
−bk1[m]
?
,
k = 1,...,K,m = 0,...,M −1, and ILis an identity matrix
of L-dimension. Also, hvis a (2LK ×1) channel coefficients
vector defined as
hv= [hv
1
T,hv
2
T,...,hv
K
T]T,
where hv
pose operation. Finally, in (2), nv[m] is a [(2N + L − 1 +
τmax) × 1] vector representing the additive-white-Gaussian
noise (AWGN) samples at the vthreceive antenna, each with
zero mean and variance No. After sampling the received
signal, the matched filter output at the vthreceive antenna
, yv[m], is correlated with the code matrix, C[0], as follows
k= [hk,v
11,hk,v
21,...,hk,v
2L]T, and T denotes the trans-
yv
c[m] = (R[0]B(m) + R[−1]B(m − 1) + R[1]B(m + 1))
× hv+ nv
c[m],
(3)
m = 1,...,M − 2, where R[0] = CH[0]C[0], R[−1] =
CH[0]C[−1], R[1] = CH[0]C[1], and H denotes Hermitian
transpose. In (3), nv
to [11], let
c[m] is modelled as Nc(0,R[0]). Similar
R[0] = F[0]TF[0] + F[1]TF[1],
(4)
and
R[−1] = F[0]TF[1],
(5)
where F[0] is a lower triangular matrix, and F[1] is an upper
right triangular matrix with zero diagonal. If yv
through a filter with impulse response (F[0] + F[1]z)−T[11],
then
c[m] is passed
yv
w[m] =
1
?
Δm=0
F[Δm]B(m − Δm)hv+ nv
w[m],
(6)
where nv
and covariance matrix NoI2LK, and I2LKis an identity matrix
of dimension 2LK. Note that both yv
the same information about the transmitted data. Due to the
whitening noise property of (6), our subsequent analysis will
be based on yv
w[m] is a complex Gaussian vector with zero mean
c[m] and yv
w[m] have
w[m].
IV. SPACE-TIME RECEIVER STRUCTURE
In this section we describe the details of our proposed EM-
Based ST Receiver. The receiver is derived by first decom-
posing the observed data into its signal components using
the approach presented in [12] for the estimation problem
of superimposed signals. Then, using the decomposed signal
components which are coupled with the corresponding un-
known channel coefficients of different users, we define the
complete data set. Subsequently, we alternate between the E-
step and the M-step of EM algorithm until the convergence
is achieved. Now starting with the approach proposed in [12],
we decompose the whitening filter output, yv
of K statistically independent components, i.e.,
w[m], into a sum
yv
w[m] =
K
?
k=1
gv
k(m),
(7)
where gv
Fk[Δm] is 2LK × 2L matrix including the 2L columns
corresponding to the kthuser in the matrix F[Δm], nv
is a complex Gaussian vector with zero mean and covariance
matrix βv
the constraint?K
EM algorithm parameters:
1) Observed data, yw, which includes the outputs of the V
whitening matched filters within a frame of M codes is
given by yw = [y1
ww
[yv
2) Parameters to be estimated, b, which include the
transmitted data bits from the K users within the
frame period b = [bT
[bk[0]T,bk[1]T,..., bk[M −1]T]T,k = 1,...,K, and
bk[m] = [bk1[m],bk2[m]]T,m = 0,...,M − 1.
k(m) =?1
Δm=0Fk[Δm]Bk(m−Δm)hv
k+nv
wk[m],
wk[m]
kNoI2LK and βv
k=1βv
kis a non-negative value satisfying
k= 1. Our goal is to obtain the users’
data estimates using the EM algorithm. First, we define the
T,y2
w[M − 1]T]T,v = 1,...,V.
T,..., yV
w
T]T, where yv
w=
w[0]T,yv
w[1]T,...,yv
1,bT
2,...,bT
K]T, where bk =
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3) Complete data, G, which are defined based on the com-
plete data definition in [9], where the unknown channel
coefficient vectors are included as a part of the com-
plete data as follows: G = [G1T,G2T,...,GVT]T,
where Gv= [(gv
1,...,V, and gv
1,...,K.
1,hv
1),(gv
k(0),gv
2,hv
k(1),...,gv
2),...,(gv
K, hv
k(M − 1)],k =
K)],v =
k= [gv
Since the components of G given b are statistically inde-
pendent, the complete log-likelihood function is given by
Φ(G|b) =
V
?
v=1
K
?
k=1
Φ(gv
k,hv
k|bk),
(8)
where
Φ(gv
k,hv
k|bk) = Φ(gv
k|hv
k,bk) + Φ(hv
k|bk).
(9)
The second summand in (9) can be ignored as it is independent
of b. Therefore, (9) is reduced to
Φ(gv
k,hv
k|bk) ∝ −
M−1
?
m=0
?
?H?
gv
k(m) −
1
?
Δm=0
Fk[Δm]
× Bk(m − Δm)hv
k
gv
k(m) −
1
?
Δm=0
Fk[Δm]
× Bk(m − Δm)hv
k
?
.
(10)
By ignoring the terms in (10) which are independent of b, the
conditional likelihood in (10) can be simplified to
Φ(gv
k,hv
k|bk) ∝
M−1
?
m=0
2Re{hv
k
HBk(m)Fk[0]Tgv
k(m)
+ hv
k
HBk(m)Fk[1]Tgv
×Bk(m−1)hv
k(m + 1) − hv
HBk(m+1)Rkk[−1]Bk(m)hv
× Bk(m)Rkk[0]Bk(m)hv
k
HBk(m)Rkk[−1]
k−hv
kk}−hv
k,
k
H
(11)
where Rkk[−1] = Fk[0]TFk[1], and Rkk[0] = Fk[0]TFk[0]+
Fk[1]TFk[1]. At the ithiteration, the E-step of the EM
algorithm is implemented by taking the expectation of the
complete log-likelihood function defined in (8) with respect
to the observed data vector, yw, and the current EM data
estimates, bi, i.e.,
Q(b|bi) =
K
?
k=1
Qk(bk|bi),
(12)
where
Qk(bk|bi) =
V
?
v=1
E?Φ(gv
k,hv
k|bk)|yw,bi?
(13)
From (11), the expectation of the individual log-likelihood
function is reduced to
Qk(bk|bi) =
V
?
v=1
M−1
?
m=0
2Re
?
E
?
hv
k
HBk(m)Fk[0]Tgv
k(m)
+ hv
× Bk(m − 1)hv
×hv
k
HBk(m)Fk[1]Tgv
k(m + 1) − hv
k− hv
?
k
HBk(m)Rkk[−1]
k
HBk(m + 1)Rkk[−1]Bk(m)
HBk(m)Rkk[0]Bk(m)hv
k|yw,bi
??
−E
hv
kk|yw,bi
?
(14)
.
To find the joint conditional expectation in (14), we evalu-
ate E?gv
find E?f(hv
sity function, Pc
[12]
k(ms)|yw,bi,h?, ms ∈ {m,m + 1}, where h =
[h1,h2,...,hV]. Then the subsequent expression is used to
k)|yw,bi?, where f(hv
?gv
1
?
+ βv
k
yv(ms) −
j=1
Δm=0
k) denotes the resultant
function in hv
k. By noting that the conditional probability den-
k(ms)|yw,bi,h?
is Gaussian with mean
E?gv
k(ms)|yw,bi,h?=
?
Δm=0
?
Fk[Δm]Bk(ms− Δm)ihv
k
K
?
1
Fj[Δm]Bj(ms− Δm)ihv
j
?
(15)
,
ms∈ {m,m + 1},
The conditional expectation of the likelihood function in (14),
after some algebraic manipulations, can be expressed as
Qk(bk|bi) =
V
?
21(l,l?) + Tkv
K
?
× Bj(m + 1)i+ Rkj[0]Bj(m)i
v=1
M
?
m=1
Re
?
22(l,l?) − Skv(l,l?) + βv
?
L
?
l=1
L
?
l?=1
Tkv
11(l,l?)
+ Tkv
12(l,l?) + Tkv
?
k(hv
k)iH
×Bk(m)
yv
c,k[m]−
j=1,j?=k
Rkj[−1]Bj(m−1)i+Rkj[−1]T
??hv
j
?i??
,
(16)
where
Tkv
qw(l,l?) =
?
(1 − βv
k)
?
ρkki
qw,ll?,m(−1) + ρkki
qw,ll?,m(1)
+ ρkki
qw,ll?,m(0)
?
− ρkk
?
qw,ll?,m(−1) − ρkk
?i∗ ?
qw,ll?,m(1)
?i
?
×
hk,v
ql
hk,v
wl?
,q,w ∈ {1,2},
Skv(l,l?) =
L
?
hk,v
1l
l=1
L
?
?i∗ ?
l?=1,l?>l
ρkk
11,ll?,m(0)
?
hk,v
1l
?i∗ ?
?
hk,v
2l
hk,v
1l?
?i
+ρkk
12,ll?,m(0)
?
hk,v
2l?
?i
22,ll?,m(0)
+ρkk
21,ll?,m(0)
hk,v
2l
?i∗ ?
?i∗ ?
hk,v
2l?
hk,v
1l?
?i
?i
+ ρkk
?
,
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where ρkki
the cross-correlation coefficients and the kthuser’s current and
next data estimates. Also, yv
ing the outputs corresponding to the kthuser at the despreader
output during the mthsymbol interval, defined as yv
Fk[0]Tyv
expectation of the attenuation coefficients given yw and bi
is given by (hk,v
?
of the assigned vector, and (hk,v∗
ql
(Ωi
{1,2}, l,l?∈ {1,...,L}, k,j ∈ {1,...,K} and Ωi
E
conditional distribution of the channel vector, hv, given yw
and biis Gaussian with mean
?M−1
m=0
+R[1]B(m + 1)i?+ NoΣ−1
qw,ll?,m(mp), mp∈ {−1,0,1}, are defined in terms of
c,k[m] is a (2L×1) vector includ-
c,k[m] =
w[m]+Fk[1]Tyv
w[m+1] [13]. In (16), the conditional
ql)i= E?hv|yw,bi?
2L(k−1)+q+2(l−1)=
(hv)i?
hh)2L(k−1)+q+2(l−1),2L(j−1)+q?+2(l?−1), where q,q?
?
2L(k−1)+q+2(l−1), where [·]sdenotes the sthelement
hj,v
q?l?)i= (hk,v
ql)i∗(hj,v
q?l?)i+
∈
hh=
(hv− (hv)i)(hv− (hv)i)H|yw,bi?
. Similar to [13], the
(hv)i=
?
B(m)i?R[0]B(m)i+ R[−1]B(m − 1)i
?−1
hh
×
M−1
?
m=0
B(m)iyv
c(m),
(17)
and covariance
Ωi
hh= No
?M−1
m=0
?
B(m)i?R[0]B(m)i+ R[−1]B(m − 1)i
+R[1]B(m + 1)i?+ NoΣ−1
hh
?−1
,
(18)
where Σhh = diag{Σh1,Σh2,...,ΣhK}, Σhk = σ2
From (12), the M-step of the EM algorithm is performed
by maximizing the individual likelihood functions Qk(bk|bi),
k = 1,...,K, as
kI2L.
bi+1
k
= arg max
bk
Qk(bk|bi).
(19)
V. OPTIMIZED WEIGHTS (βm
In (16), we notice that Qk(bk|bi) is a sum of V sta-
tistically independent terms given b and its EM estimate
bi, which are related to the V receive antennas. Since the
spatial channels corresponding to the links between transmit
and receive antennas are considered independent, βv
separately optimized. In this case, we choose the weight
coefficients to minimize the linear mean-square error between
the true signal vector, gv
E?gv
βv
βv
k
where
Eg= Fk[0]T?gv
k)
kcan be
k(ms), and its estimate (gv
k(ms))i=
k(ms)|yw,bi?, ms∈ {m,m + 1}, after being projected
k= argmin
on Fk[0] and Fk[1] respectively as
E?? Eg?2?,
(20)
k(m) − gv
k(m)i?+ Fk[1]T(gv
k(m + 1)
−gv
k(m + 1)i?,
(21)
and ? · ? denotes the vector norm. In order to simplify our
analysis, we assume that M → ∞, i.e. the random channel
coefficients are assumed to be known at the receiver. It follows
that
E?gv
+ βv
k(ms)|yw,bi?=
?
1
?
1
?
Δm=0
Fk[Δm]Bk(ms− Δm)ihv
k
k
yv
w(ms) −
K
?
j=1
Δm=0
Fj[Δm]Bj(ms− Δm)ihv
j
?
(22)
,ms∈ {m,m + 1}
Substituting gv
(21), we have
k(ms) and (gv
k(ms))i, ms ∈ {m,m + 1}, in
Eg= Rkk[0]?Bk(m) − Bk(m)i?hv
×?Bk(m + 1) − Bk(m + 1)i?hv
× nv
k+ Rkk[−1]
k+ Rkk[−1]T
k+Fk[0]Tnv
? K
j=1
×?Bk(m − 1) − Bk(m − 1)i?hv
wk[m]+Fk[1]T
wk[m + 1] − βv
kyv
c,k[m] + βv
k
?
Rkj[0]Bj(m)ihv
j
+ Rkj[−1]Bj(m − 1)ihv
j+ Rkj[−1]TBj(m + 1)ihv
j
?
(23)
where
Fk[0]TFj[0] + Fk[1]TFj[1].
?K
vector including the noise samples corresponding to the kth
user at the vthdespreader output during the mthsymbol inter-
val, and
in (23), we have
Rkj[−1]=
Fk[0]TFj[1],and
Rkj[0]
yv
−
c,k[m] is (2L×1)
=
=
+
Substituting
Rkj[−1]Bj(m
c,k[m], where nv
c,k[m]
1)hv
j=1Rkj[0]Bj(m)hv
Rkj[−1]TBj(m+1)hv
j
+
j
j+nv
?βv
knv
c,k[m] = Fk[0]Tnv
wk[m] + Fk[1]Tnv
wk[m + 1]
Eg= Rkk[0]?Bk(m) − Bk(m)i?hv
−Bk(m + 1)i?hv
×?Bj(m) − Bj(m)i?hv
k+ Rkk[−1]
k+ Rkk[−1]T(Bk(m + 1)
? K
j=1
j+ Rkj[−1](Bj(m − 1)
j+ Rkj[−1]T?Bj(m + 1) − Bj(m + 1)i?
× hv
×?Bk(m − 1) − Bk(m − 1)i?hv
k+
?
βv
knv
c,k[m] − βv
k
?
Rkj[0]
−Bj(m − 1)i?hv
j+ nv
c,k[m]
?
(24)
In
samples,
are mutually independent, as well as E
σ2
k
forsimilar
otherwise.Inaddition,
No(Rkk,11[0] + Rkk,22[0] + ... + Rkk,2L2L[0]),
Rkk,ζζ[0], ζ ∈ {1,2,...,2L}, represents the ζthdiagonal
oursystem
nv
model,
and
we
the
assume
channel
that
coefficients,
?
c,k[m]
thenoise
hv
?
zero
c,k[m],
k,
=
hk,v∗
ql
hj,v
q?l?
and
?
channelcoefficients
?
E
nv
c,k[m]Hnv
=
where
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Page 5

element of Rkk[0]. Using (24), (20) can be expressed as
?
+Rkk,22[0] + ... + Rkk,2L2L[0]) − 8σ2
L
?
,2l − 1) + rkk,−1(2l,2l) + rkk,1(2l − 1,2l − 1) + rkk,1(2l
K
?
+rjj,0(2l,2l) + rjj,−1(2l − 1,2l − 1) + rjj,−1(2l,2l)
+rjj,1(2l − 1,2l − 1) + rjj,1(2l,2l))
βv
k= argmin
βv
k
No
?
βv
k− 2βv
k
3
2+ βv
k
2?
kβv
(Rkk,11[0]
?Pv,i
k
ek1+ Pv,i
ek2
?
×
l=1
(rkk,0(2l − 1,2l − 1) + rkk,0(2l,2l) + rkk,−1(2l − 1
,2l)) + 4βv
k
2
j=1
σ2
j
?
Pv,i
ej1+ Pv,i
ej2
?
L
?
l=1
(rjj,0(2l − 1,2l − 1)
?
.
(25)
where
{1,...,2L},mp
ζth
diagonal
Pv,i
ejq
{1,...,k},q ∈ {1,2},mp∈ {−1,0,1}, where the probability
of error, Pv,i
that αk = Rkk,11[0] + Rkk,22[0] + ... + Rkk,2L2L[0], and
θj
1) + rjj,−1(2l,2l) + rjj,1(2l − 1,2l − 1) + rjj,1(2l,2l).
By differentiating (25) with respect to βv
x =
rjj,mp(ζ,ζ),j
∈
{−1,0,1},
of
Rjj[mp]TRjj[mp],
{1,...,K},ζ
represents
∈
∈
the
andelement
?1 − E?bjq[m + mp]bjq[m + mp]i??,j
=
1
2
∈
ejq= P?bjq(m + mp) ?= bjq(m + mp)i?. Assume
l= rjj,0(2l − 1,2l − 1) + rjj,0(2l,2l) + rjj,−1(2l − 1,2l −
kand substituting
?βv
⎛
k, we have
⎝2Noαk+ 8
× αk)x +
K
?
j=1
σ2
j
?
Pv,i
ej1+ Pv,i
ej2
?
L
?
?
l=1
θj
l
⎞
⎠x2+ (−3No
?
?
Noαk− 8σ2
k
?Pv,i
ek1+ Pv,i
ek2
L
l=1
θk
l
?
= 0.
(26)
Solving(26) with respect to x results in two possible solutions
for βv
ST receiver with V =1 receive antenna will converge to the
single-user (SU) bound with perfect channel state information
(CSI) assuming transmission over frequency-selective fading
channels. Then the probability of error of the odd and even
data bits [10] are defined as
k. Assuming that the performance of the EM-based
Pe1=1
2Q
?
Re{G(1,1) + G(1,2)}
?
?
2
G(1,1)
?
+1
2Q Re{G(1,1) − G(1,2)}
?
2
G(1,1)
?
and
Pe2=1
2Q
?
Re{G(2,2) + G(2,1)}
?
?
2
G(2,2)
?
+1
2Q Re{G(2,2) − G(2,1)}
?
2
G(2,2)
?
,
where Q(x) = 1/√2π?∞
G
xe−x2/2dx,
= [(s1ch1− s2ch2)(s2ch1+ s1ch2)]H
× [(s1ch1− s2ch2)(s2ch1+ s1ch2)],
and s1 and s2 include the multipath versions of the two
code assigned for every user, and chq, q = 1,2, include the
multipath channel coefficients from the qthtransmit antenna
to the receive antenna. Substituting with the EM channel
estimates defined in (17) in the single-user bound, Pe1and
Pe2, we obtain an approximation for Pv,i
ek1, and Pv,i
ek2.
VI. EM INITIALIZATION
Since the EM algorithm is sensitive to the initialization
of the parameters to be estimated [14], as well as due to
the high computational complexity of the joint estimation
and detection in MIMO systems, our proposed EM-based ST
receiver is initialized by reliable estimates, for example, by
using the ST MMSE-SDE technique [9]. This guarantees that
the performance of our proposed receiver converges to that
of the optimum receiver with a fast rate. Furthermore, this
will also ease the maximization of the individual likelihood
functions Qk(bk|bi) in (19). Instead of using the Viterbi
algorithm [11], which increases the complexity of the proposed
receiver, we consider that, at the first iteration, the ST MMSE-
SDE estimates provide reliable estimation for the previous
and following codewords, bkq[m − 1] = bkq[m − 1]0, and
bkq[m+1] = bkq[m+1]0, q ∈ {1,2}, while for the subsequent
iterations, we consider that bkq[m − 1] = bkq[m − 1]i, and
bkq[m+1] = bkq[m+1]i−1. Consequently, the maximization
of (19) is performed over 4 possibilities for the current kthuser
data bits, bk1[m] and bk2[m], (i.e, {(1,1),(1,-1),(-1,1),(-1,-1)})
considering binary phase-shift keying (BPSK) transmission.
VII. SIMULATION RESULTS
In this section, we examine the bit-error rate (BER) perfor-
mance of the proposed EM-JDE receiver in MIMO CDMA
systems. In all cases, we consider a system with two transmit
and V = 1,2 receive antennas. We also consider an uplink
asynchronous transmission of a data block of 40 codewords
(M=40) over a frequency-selective fading channel. Without
loss of generality, we consider a 5-user system, where all users
are assigned Gold codes of length 31 chips. We assume the
first user as the desired one. A training codeword of eight
training bits is used for the initialization of the EM receiver.
As a reference, we include the BER performance of the
single-user STS system assuming perfect channel estimation.
Fig. 1 presents the BER performance of the proposed ST EM-
JDE receiver using 2 × 1 antenna configuration. The results
show the proposed JDE receiver achieves a performance close
the single-user bound. We also notice that the full system
diversity is attained at high signal-to-noise ratio (SNR) values.
Similar argument applies to Fig. 2 where we consider a
MIMO CDMA system with two receive antennas. In Fig. 3 we
examine the near-far effect property of the proposed receiver
for V = 1 receive antenna. We fix the received SNR of the
first user γ1at 15 dB, while the interfering users have equal
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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2010 proceedings.