Multicode DS-CDMA with joint transmit/receive frequency-domain equalization

Abstract

Bit error rate (BER) performance of multicode direct-sequence code division multiple access (DS-CDMA) in a frequency-selective channel severely degrades due to strong inter-chip interference (ICI). Frequency-domain equalization (FDE) based on the minimum mean square error (MMSE) criterion can take advantage of the channel frequency-selectivity and significantly improve the BER performance. However, the performance improvement is limited by the residual ICI which is present after MMSE-FDE. In this paper, we apply a joint transmit/receive MMSE-FDE (called double MMSE-FDE in this paper) for multicode DS-CDMA signal transmissions to further improve the BER performance. We derive the conditional BER for the given channel realization and evaluate the achievable average BER performance by Monte-Carlo numerical computation method. The BER performance is confirmed by the computer simulation.

Full text

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Multicode DS-CDMA With Joint Transmit/Receive
Frequency-domain Equalization
Kazuki TAKEDA† Hiromichi TOMEBA† and Fumiyuki ADACHI‡
Dept. of Electrical and Communication Engineering, Graduate School of Engineering, Tohoku University
6-6-05, Aza-Aoba, Aramaki, Aoba-ku, Sendai, 980-8579 JAPAN
†{kazuki, tomeba}@mobile.ecei.tohoku.ac.jp, ‡adachi@ecei.tohoku.ac.jp
Abstract—Bit error rate (BER) performance of multicode direct-
sequence code division multiple access (DS-CDMA) in a
frequency-selective channel severely degrades due to strong
inter-chip interference (ICI). Frequency-domain equalization
(FDE) based on the minimum mean square error (MMSE)
criterion can take advantage of the channel frequency-selectivity
and significantly improve the BER performance. However, the
performance improvement is limited by the residual ICI which is
present after MMSE-FDE. In this paper, we apply a joint
transmit/receive MMSE-FDE (called double MMSE-FDE in this
paper) for multicode DS-CDMA signal transmissions to further
improve the BER performance. We derive the conditional BER
for the given channel realization and evaluate the achievable
average BER performance
computation method. The BER performance is confirmed by the
computer simulation.
Keywords-component; FDE, DS-CDMA
by Monte-Carlo numerical
I.
INTRODUCTION
Broadband wireless channel comprises many propagation
paths having different time delays [1]. This results in a severe
frequency-selective fading channel. In the third generation
mobile communication systems, direct-sequence code division
multiple access (DS-CDMA) using coherent rake combining is
used [2]. Coherent rake combining can achieve the path
diversity and improve the bit error rate (BER) performance in a
moderately frequency-selective channel. However, as the chip
rate increases, the channel frequency-selectivity gets stronger,
producing severer inter-chip interference (ICI), and the bit error
rate (BER) performance significantly degrades.
The use of simple one-tap frequency-domain equalization
(FDE) based on the minimum mean square error (MMSE)
criterion can significantly improve the BER performance of
nonspread single-carrier (SC) signal transmission and DS-
CDMA signal transmissions [3, 4]. Reference [4] reviews one-
tap MMSE-FDE for DS-CDMA signal transmissions.
Replacement of rake combining by MMSE-FDE reduces the
ICI and hence significantly improves the BER performance
compared to that with coherent rake combining. However, the
performance improvement is limited by the residual ICI after
MMSE-FDE [5]. Reducing the residual ICI can further
improve the BER performance.
One approach to reduce the residual ICI is to introduce an
iterative ICI cancellation technique [6-8]. The residual ICI
replica is generated using the log-likelihood ratio (LLR) and is
subtracted from the received signal.
In this paper, we take another approach to reduce the
residual ICI. One-tap FDE is jointly used at both the transmitter
and receiver. In [9], a joint transmit/receive FDE was proposed
for SC transmission, where the transmit and receive FDE
weights are optimized using an adaptive algorithm. Since the
transmit and receive FDE weights interact each other, it is quite
difficult if not impossible to theoretically derive an optimal set
of transmit and receive FDE weights in the closed form. In this
paper, we present a suboptimal set of transmit and receive FDE
weights based on the minimization of the total mean square
error (MSE) of the received DS-CDMA chip block after
receive MMSE filtering. We derive the conditional BER of
DS-CDMA using joint transmit/receive MMSE-FDE (called
double MMSE-FDE in this paper) for the given channel
realization. The achievable average BER performance is
evaluated by Monte-Carlo numerical computation method
using the derived conditional BER and compared with that of
conventional receive MMSE-FDE (called single MMSE-FDE
in this paper). The achievable BER is confirmed by the
computer simulation.
The rest of this paper is organized as follows. Section II
presents the system model. The MMSE weight matrix for
double MMSE-FDE is presented in Sect. III. Section IV
derives the conditional BER expression. The BER performance
is discussed in Sect. V. Section VI concludes this paper.
II.
SYSTEM MODEL
Figure 1 illustrates the transmission system model of DS-
CDMA using double MMSE-FDE. Throughout this paper,
chip-spaced discrete-time signal representation is used. It is
assumed that the perfect channel state information (CSI) is
available at both the transmitter and receiver.
Spreading
code
cu(t)
cscr(t)
Nc-point FFT
Data symbol
block
Transmit FDE
weight Wt(k)
Nc-point IFFT
CP-insertion
s
CP-removal
r
Nc-point FFT
Receive FDE
weight Wr(k)
Nc-point IFFT
Decision
variables
(a) Transmitter
(b) Receiver
S/P
Scrambling
code
x
Despreading
{cscr(t)cu(t)}*
Copy
Σ
P/S
x ˆ

Fig. 1 Transmission system model.
A. Transmitter
At the transmitter, an information bit sequence is
transformed into a data-modulated symbol sequence, which is
serial-to-parallel (S/P) converted into U parallel streams {du(i);
i=…, −1, 0, 1, …}, u=0~U−1. Then, each stream is spread
using an orthogonal spreading code with spreading factor SF
{cu(t); t=0~SF−1}, u=0~U−1. U chip sequences are summed up

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to form the multicode chip sequence, which is further
multiplied by a scramble code {cscr(t); t=…, −1, 0, 1, …} to
obtain the multicode DS-CDMA chip block. The resultant chip
block can be expressed using the vector form as x=[x(0), …,
x(t), …, x(Nc−1)]T, where x(t) is given as
(
∑
=
0u
The chip block x is transformed by using an Nc-point FFT
into the frequency-domain signal X=[X(0), …, X(k), …,
X(Nc−1)]T as
X =
,
⎣⎦
)
−
=
1
)() mod(/)(
U
scruu
tc SFtc SFtdtx . (1)
Fx
(2)
where F is an Nc×Nc FFT matrix given as
⎡
−
e1
1
?
⎥
⎦
⎥
⎥
⎥
⎥
⎤
⎢
⎣
⎢
⎢
⎢
⎢
=
−×−
) 1
π−
×
) 1
−
π−
−×
π−
×
) 1
π
c
N
c
N
c
N
j
c
N
c
N
j
c
N
c
N
j
c
N
j
ee
e
Nc
))1(((
2
) 1((
2
))1(1 (
2
1 (
2
1
111
?
???
?
?
F. (3)
One-tap transmit FDE is carried out as
XWS
t
C⋅=
, (4)
where Wt=diag{Wt(0), …, Wt(k), …, Wt(Nc−1)} is an Nc×Nc
diagonal transmit FDE weight matrix. C is the transmit power
normalization factor, which is introduced to keep the average
transmit power intact, and is given as
][/
H
ttctrNCWW
=
. (5)
An Nc-point IFFT is applied to S to obtain the transmit
signal s=[s(0), …, s(t), …, s(Nc−1)]T=FHS. After the insertion
of Ng-sample CP into the guard interval (GI), the signal block
is transmitted.
B. Receiver
The propagation channel is assumed to be an L-path
frequency-selective block fading channel. The complex-valued
path gain and time delay of the lth path are denoted by hl and τl,
l=0~L−1, respectively. The CP-length is assumed to be equal to
or longer than the maximum channel time delay τL−1. The
received signal block r=[r(0), …,r(t), …, r(Nc−1)]T after the
CP-removal can be expressed as
n hsr
+=
ccTE /2 , (6)
where Ec and Tc are the average transmit chip energy and chip
duration, respectively, h is an Nc×Nc circulant channel matrix
given by
⎡
−
10
h
?
0
h
⎥
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎤
⎢
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
=
−
−
−
01
1
?
1
10
1
1
?
hh
hh
hh
hhh
L
L
L
L
??
??
?
??
?
0
, (7)
and n=[n(0), …, n(t), …, n(Nc−1)]T is the noise vector with
each element n(t) being a zero-mean additive white Gaussian
noise (AWGN) having variance 2N0/Tc (N0 is the one-sided
noise power spectrum density).
An Nc-point FFT is carried out on r to obtain the frequency-
domain received signal R=[R(0), …, R(k), …, R(Nc−1)]T as
NXHWFrR
+⋅==
tcc
CTE /2, (8)
where N=Fn and H=FhFH. Due to the circulant property of h,
the Nc×Nc channel gain matrix H=FhFH is diagonal. The (k,
k)th element of H is given by
∑
=
0
l
One-tap receive FDE is carried out as
Wr=diag{Wr(0), …, Wr(k), …, Wr(Nc−1)} is an Nc×Nc diagonal
receive FDE weight matrix. Then, an Nc-point IFFT is carried
out on Xˆ to obtain the equalized DS-CDMA chip block
x ˆ =[ x ˆ (0), …, x ˆ (t), …, x ˆ (Nc−1)]T as
ˆ
ˆ
−
τπ−=
1
)/2exp()(
L
cll
NkjhkH . (9)
RWX
ˆ
r
=
, where
NWFXHWWFXFx
r
H
tr
H
cc
H
CTE
+⋅==
/2 . (10)
Finally, despreading is applied to x ˆ to obtain the decision
variable )(
ˆ
idu
for du(i) as
∑
=
iSFt
−+
=
1 ) 1(
*
u
*
scr
) mod()() ( ˆx)/ 1 ()(
ˆ
SFi
u
SFtctct SFid . (11)
III.
TRANSMIT AND RECEIVE FDE WEIGHTS
A. Total Mean Square Error (MSE)
The relative error vector e=[e(0), …, e(t), …, e(Nc−1)]T
between the transmit chip block x and the received chip block
x ˆ is defined, similar to Ref. [12], as
CEtrTNE
CTE
H
ccc
cc
⋅⋅
⋅−
=
)]([) /(2
/2
ˆ
x
xx
x
e
. (12)
The total MSE is given, using Eqs. (10) and (12), as
],[][))/)(/((
]))([(
)]([),(
1
0
H
rr
H
tts
H
trtrc
U
H
rt
trtrNE SF
trN
E tre
WWWW
I HWWI HWW
eeWW
−
+
−−⋅=
=
(13)
where Es is the average transmit symbol energy.
B. Transmit and Receive MMSE-FDE Weight Matrices
A concatenation of the transmit FDE Wt and the
propagation channel H can be viewed as an equivalent channel
HWt. The MMSE solution to the receive FDE weight matrix
can be derived from Eq. (13) as
.]][
))/)(/)((/ 1 (
+
[
1
1
0
HH
t
H
tt
sct
HH
t
tr
r
NESFUN
HWIWW
HWHWW
⋅⋅×
=
−
−
(14)
Substituting Eq. (14) into Eq. (13) and representing
(1/Nc)((U/SF)(Es/N0))−1 by Ω gives
],)][ [(
][)(
1
−
⋅⋅Ω+×
⋅Ω=
IWWHHWW
WWW
H
tt
HH
tt
H
ttt
tr tr
tre
(15)
which shows that the total MSE is a function of WtWt
e(Wt) is a convex function of |Wt(k)|2, k=0~Nc−1, the global
H. Since

Page 3

optimum solution that minimizes Eq. (15) under the transmit
power constraint can be found.
Let us introduce a power
tr[WtWt
)(. min
t
Ntr t s
=
WW
constraint condition
H]=Nc. The optimality condition can be expressed as
eW

,][ . .
c
H
tt
(16)
whose solution satisfies the Karush-Kuhn-Tucker (KKT)
condition [13-15]. To derive the optimal Wt, first, diagonal
elements of H are permuted in descending order of |H(k)|,
k=0~Nc−1, and the permuted diagonal matrix is defined as
G=diag{G(0), …, G(q), …, G(Nc−1)} (i.e., |G(0)|≥ |G(1)|≥ …≥
|G(Nc−1)| and G(q)=H(q’), q=0~Nc−1, q’=0~Nc−1). The same
ordering permutation as H→G is done to the diagonal elements
of WtWt
P=diag{P(0), …, P(q), …, P(Nc−1)}. Then, the optimality
condition given by Eq. (16) can be rewritten as
H and the permuted diagonal matrix is defined as
. 1
−
~0 for)(0 and][. .t s
)()()(/)(
])][[(][)(.min
1
0
1
0
2
1
0
1
=≤=
⎟
⎠
⎟
⎞
⎜
⎝
⎜
⎛
′
⋅Ω+
⎟
⎠
⎟
⎞
⎜
⎝
⎜
⎛
′
⋅Ω=
⋅⋅Ω+×⋅Ω=
∑
=
q
∑
= ′
q
∑
= ′
q
−−−
−
cc
NNN
H
NqqPNtr
Below, P0=diag{P0(0), …, P0(q), …, P0(Nc−1)} represents
the global optimum solution that satisfies the condition given
by Eq. (17). Without loss of generality, we assume that P0 has
(Nc−m) zero diagonal elements (0<m≤Nc) and has m zero
diagonal elements. Since e(P0) is a monotonic decreasing
function of |G(q)|2, q=0~Nc−1, it can be said that P0(q)≥0 for
q=0~m−1 and P0(q)=0 for q=m~Nc−1. Lagrangian function J
can be expressed as [15]
∑
−
=
Ω+
qGqP
qPqGqPqP
trtrtre
ccc
P
IPGPGPP
(17)
, }0)({
0)()(
)()()(
)(
1
0
11
0
1
0
1
0
2
1
0
∑
∑∑
∑
∑
−
=
−
=
−
=
−
= ′
q
−
= ′
q
−−⋅ψ+
⎭⎬⎫
⎩⎨⎧
−μ+
⎭⎬⎫
⎩⎨⎧
−⋅κ+
′
′
Ω
=
c
c
c
c
c
N
q
q
N
mq
c
m
q
N
q
N
N
qP
qPNqP
qP
qP
J
(18)
where κ, μ, and {ψq; q=0~Nc−1} are the Lagrange multipliers.
P0 must satisfy the KKT condition [13, 14]. We obtain
0) )((
)()(
0
=
qPqP
1
0
=−
∑
=
q
1
0
=
∑
=
mq
1~0, 0)(
0
−=≥
c
NqqP,
1~0 , 0
−=≥ψ
cq
Nq
,
and
1~0 , 0)(
0
−==ψ
cq
NqqP
=∂∂
qPJ for q=0~Nc−1, (19)
0)(
0
−
−
c
m
NqP, (20)
0)(
c
N
qP , (21)




(22)
(23)
. (24)
Using Eqs. (18)~(24), we obtain
2
1
(
0
)(
)
)(
qG
N
qG
qP
cm
Ω
−
Ωθ
=
−
,
q=0~m−1, (25)
where
1
1
0
1
0
2
1
)(
)(
1
−
−
=
−
=
−
⎟
⎠
⎟
⎞
⎜
⎝
⎜
⎛
Ω
q
⎟
⎠
⎟
⎞
⎜
⎝
⎜
⎛
Ω
q
+=θ
∑
q
∑
q
mm
cm
G
G
N
(26)
and P0(q)=0 for q=m~Nc−1. After finding the value of m that
satisfies Eqs. (20) and (21), we can find the global optimum
solution P0 that satisfies the optimality condition given by Eq.
(17).
After applying the converse permutation to P0, Wt
obtained as
~
max[
1
−
⋅Ωθ=
HWW
mtt
~=diag{|H(0)|, …, |H(k)|, …, |H(Nc−1)|}.
The above joint transmit/receive MMSE-FDE is called
double MMSE-FDE (note that the use of receive MMSE-FDE
is called single MMSE-FDE). As seen from Eq. (27), Wt is a
diagonal matrix and therefore, the low complexity property of
simple one-tap FDE is kept even with double MMSE-FDE.
HWt is
] 0,)(
11
−−
⋅Ω−
HΗ
H
c
H
N
, (27)
where H
IV. CONDITIONAL BER
When using the above double MMSE-FDE, Eq. (10) can be
rewritten, from Eq. (2), as
ˆ
/2
ˆ
NWFFxHFx
r
HH
ccTE
+=
, (28)
where
matrix
(
ˆkH
elements of
components, respectively. The tth element
as
1
E
tx
cc
⎝
∑
=

From Eqs. (11) and (29), we obtain the decision variable
for the ith transmit symbol
du
⎛
=
∑
=
where μICI and μn denote the residual ICI and the noise after the
despreading, respectively.
It can be seen from Eq. (30) that
∑
sequence is used to make the multicode DS-CDMA chip
sequence white noise-like and therefore, μICI can be
approximated as a zero-mean complex-valued Gaussian
variable. The sum of μICI and μn can be treated as a new zero-
trHWW
whose
=Wr(k)H(k)Wt(k). The diagonal and off-diagonal
FHF
ˆ
produce the desired and residual ICI
H =
ˆ
is an Nc×Nc equivalent channel gain
kth diagonal element is given by
)
H
) ( ˆ tx
of x ˆ is given
).(2exp)()(
ˆ
1
2
)()(
ˆ
1
2
)( ˆ
1
0
1
, 0
=
0
tn
N
t
kjxkH
NT
E
txkH
NT
cc
c
N
k
N
∑
t
ccc
c
N
∑
=
k
c
+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
τ−
πτ+
⎟
⎠
−
⎟
⎞
⎜
⎜
⎛
=
−
≠

ττ
−

(29)
ˆ
du
)(
i

)(i
as
,)()(
ˆ
12
)(
ˆ
1
0
nICIu
N
k
cc
c
u
idkH
NT
E
id
c
μ+μ+
⎟
⎠
⎟
⎞
⎜
⎝
⎜
−
(30)
)(
ˆ
idu
1
H
is a random variable
with mean
)()(
ˆ
)/ 1 (/2
0
idkNTE
u
N
k
ccc
c
−
=
. A scramble

Page 4

mean complex-valued Gaussian noise μ. The variance of μ is
given by
]| [|]|[|2
μ+μ=σ
μ
nICI
EE
.)(
ˆ
1
)(
ˆ
1

)(
1
2
1
2
1
0
1
0
2
0
1
0
2
0
222
⎥
⎦
⎥
⎥
⎤
⎟⎟
⎠
⎟
⎞
⎜⎜
⎝
⎜
⎛
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
⎢
⎣
⎢
⎡
=
∑
=
k
∑
=
k
∑
=
k
−−
−
cc
c
N
c
N
c
s
N
r
cc
kH
N
kH
NN
E
SF
U
kW
NT
N
SF
(31)
The conditional BER using QPSK data modulation for the
given channel realization H can be given as
(
/ 1 ([ 5 . 0,/
0
H
erfcNEp
sb
×=
[ ]
∫
x
error function and γ(Es/N0, H) denotes the conditional signal-to
interference plus noise ratio (SINR), given as
)()],/) 4
0
H
NE
s
γ×
, (32)
where
∞
−π=
dttxerfc) exp()/2 (
2
is the complementary
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
−×
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
γ
∑
=
k
∑
=
k
∑
=
k
∑
=
k
−−
−
−
2
1
0
1
0
2
0
1
0
2
2
1
0
0
0
)(
ˆ
1
)(
ˆ
1
)(
1
)(
ˆ
1
2
,
cc
c
c
N
c
N
c
s
N
r
c
N
c
s
s
kH
N
kH
N
N
E
SF
U
kW
N
kH
NN
E
N
E
H
. (33)
The achievable average BER can be numerically evaluated by
averaging Eq. (32) over possible realizations of H.
V.
PERFORMANCE EVALUATION
An L=16-path frequency-selective block Rayleigh fading
having exponentially decaying power delay profile with decay
factor α (dB) is assumed. A block transmission using Nc=256
and Ng=32 is considered. The spreading factor is set to SF=256.
QPSK is used. Ideal channel estimation is assumed.
A. One shot observation of the equivalent channel
Figure 2 shows one shot observation of the equivalent
channel seen after receive MMSE-FDE (i.e., |Wr(k)H(k)Wt(k)|)
for single MMSE-FDE and double MMSE-FDE when the
transmit Eb/N0 (=0.5(SFEc/N0)(1+Ng/Nc)−1)=8dB and α=0dB.
The double MMSE-FDE reduces amplitude variations in the
equivalent channel Wr(k)H(k)Wt(k) compared to those in the
original channel H(k) when U=256, as seen in Fig. 2 (a). The
amplitude of Wt(k) given by Eq. (27) drops at some frequencies.
However, the frequency-selectivity of the equivalent channel
can be made weaker by using the double MMSE-FDE.
As the code multiplexing order U decreases, the total
energy of the transmit chip block tr[xxH]/Nc reduces for the
fixed transmit Eb/N0. Therefore, most of the total energy tends
to be allocated to the frequencies having a good condition by
using the double MMSE-FDE and therefore, the received
signal-to-noise power ratio (SNR) improves.
2.0E-02
2.0E-01
2.0E+00
0 32 64 96
Frequency index, k
(a) U=256
128 160192224256
Equvialent channel gain amplitude
Double
Single
QPSK,
Eb/N0=8dB,
SF=256,
U=256

2.0E-02
2.0E-01
2.0E+00
0 326496
Frequency index, k
(b) U=16
128 160 192224 256
Equvialent channel gain amplitude
SF=256, U=16, QPSK,
Eb/N0=8dB
Double
Single

2.0E-02
2.0E-01
2.0E+00
0 3264 96
Frequency index, k
(c) U=1
128160 192 224 256
Equvialent channel gain amplitude
Double
Single
QPSK,
Eb/N0=8dB,
SF=256,
U=1

Fig. 2 Equivalent channel.
B. BER Performances
The achievable average BER performance is evaluated by
Monte-Carlo numerical computation method. The set of path
gains {hl; l=0~L−1} is generated and the conditional BER for
the given average transmit Es/N0 is computed using Eq. (34).
The average BER is obtained by averaging the conditional
BER over all possible channel realizations. Figure 3 compares
the BER performances of DS-CDMA using single and double
MMSE-FDE as a function of Eb/N0. The computer simulation
results are also plotted to confirm the validity of our analysis
based on the Gaussian approximation of ICI. A fairly good
agreement between the numerically computed and computer
simulated results is seen. The double MMSE-FDE provides
better BER performance than single MMSE-FDE irrespective
of the degree of the channel frequency-selectivity.
When U=256, the residual ICI after the receive MMSE-
FDE is a predominant cause of BER degradation. In this case,
since double MMSE-FDE further reduces the channel gain
variation compared to single MMSE-FDE (see Fig. 2 (a)), the
residual ICI after the receive MMSE-FDE is further suppressed.
Therefore, better BER performance can be achieved by double
MMSE-FDE than by using single MMSE-FDE. As U
decreases, the noise after the receive MMSE-FDE becomes a
predominant cause of BER degradation. In this case, double

Page 5

MMSE-FDE allocates most of transmit power to the
frequencies having a good condition to improve the received
signal-to-noise power ratio (SNR). Therefore, double MMSE-
FDE provides better BER performance than single MMSE-
FDE.
1.0E-01
α=0dB, QPSK, SF=256
1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-02
48 12 16
Average BER
Average transmit Eb/N0(dB)
(a) α=0dB
Numerical
Double
Single
Simulated
U=1
4
16
64
256

1.0E-06
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
8 121620
Average BER
Average transmit Eb/N0(dB)
(b) α=6dB
α=6dB, QPSK, SF=256
Numerical
Double
Single
Simulated
U=1
4
16
64
256

Fig. 3 BER performance comparison.
C. Impact of code multiplexing order, U
Figure 4 plots the required Eb/N0 of double MMSE-FDE for
achieving BER=10−3 as a function of code multiplexing order
U when SF=256. For comparison, the required Eb/N0 of single
MMSE-FDE is also plotted. For U=256, double MMSE-FDE
reduces the required Eb/N0 compared to the single MMSE-FDE
by about 1.0dB (0.5dB) when α=0dB (6dB). Also, it is obvious
that double MMSE-FDE can significantly reduce the required
Eb/N0 for a low value of U. An Eb/N0 reduction of 4.2dB
(2.7dB) is achieved when α=0dB (6dB) when U=1.
18
QPSK, SF=256
4
6
8
10
12
14
16
116256
Required Eb/N0for achieving BER=10-3
Code multiplexing order, U
α=0dB
α=6dB
Double
Single

Fig. 4 Impact of code multiplexing order, U.
VI. CONCLUSION
In this paper, we evaluated the BER performance of DS-
CDMA using double MMSE-FDE. A suboptimal set of
transmit and receive one-tap MMSE-FDE weight matrices was
presented based on the minimization of the total MSE. For a
large total transmit chip energy, the transmit MMSE-FDE can
reduce the variations of the equivalent channel H(k)Wt(k) and
can suppress the ICI. On the other hand, for a small total
transmit chip energy, most of the energy is allocated to the
frequencies having a good channel condition and can improve
the received SNR. The average BER performance was
numerically evaluated using the derived conditional BER
expression to show that the double MMSE-FDE can
significantly improve the BER performance.
In this paper, it was assumed that the CSI is available at
both transmitter and receiver. There have been many studies on
the CSI estimation at the transmitter [10, 11]. How the
imperfect CSI affects the achievable BER performance by the
proposed joint transmit/receive MMSE-FDE is an important
future study.
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