A study on detection techniques for downlink in multi-carrier cdma system

 OFDM symbol ([16]). The
complex channel fading coefficients are supposed to be independent for each sub-carrier and
constant during the transmission of each OFDM symbol. Under the above assumptions, the
IFFT and guard interval insertion block, the Rayleigh fading channel along with the FFT and
guard interval removal block are considered as an equivalent frequency channel, as illustrated
in Figure 1.
At the receiver the guard interval is removed and the inverse OFDM operation is performed.
Then, the received signals can be expressed as
r = [r1, r2, ..., rNc ]
T =HS +N =HCd +N (3)
where H is a Nc × Nc diagonal matrix containing the complex channel attenuation of each
sub-carrier
H =


H1 0 · · · 0
0 H2 · · · 0
...
...
...
...
0 0 · · · HNc

 (4)
and N = [N1, N2, ..., NNc ]
T is Nc × 1 vector of Additive White Gaussian Noise (AWGN)
components with Nn representing the noise at the sub-carrier n
th which has variance σ2n =
E[|Nn|
2], n = 1, 2, ..., Nc.
3. EQUALIZATION TECHNIQUES
Detectors for MC-CDMA can be classified into the two basic categories: Single User De-
tection and Multi-User Detection.
3.1. Single user detection (SUD)
In the first category, the receiver has knowledge of spreading code employed by the user of
interest only, which means that it has no knowledge of the spreading codes employed by other
users. Interference from other users is assimilated to additive channel noise and no attempt is
made to compensate for it. In SISO MC-CDMA mobile radio system, SUD is realized by one
tap equalization to compensate for the distortion due to fading on each sub-carrier, followed
by using specific despreading.
After equalization the receiver signal can be written as
Y = [y1, y2, ..., yNc ]
T =Gr =GHCd +GN (5)
The estimated symbol of the kth user is equal to:
Q(d˜k) = Q(c
∗
k,nGnrn) (6)
A STUDY ON DETECTION TECHNIQUES FOR DOWNLINK 19
where Q(.) denotes quantization operation. The Nc×Nc matrixG contains complex equaliza-
tion coefficients obtained from channel estimation, which can be known through transmitted
pilot symbols inserted between the OFDM signals.
In the sequel, we will describe different basic equalization techniques of this category in
details.
3.1.1. Maximum Ratio Combining (MRC)
This technique corrects the phase shift by multiplying the receiver signal with the conjugate
complex channel coefficient
Gn = H
∗
n (7)
where (.)∗ denotes complex conjugation, and Hn(n = 1, 2, ..., Nc) are the diagonal components
of H. The drawback of MRC in the downlink of SISO MC-CDMA system is that it destroys
the orthogonality between spreading codes and thus, additionally enhances the multiple access
interference ([12]).
3.1.2. Equal Gain Combining (EGC)
EGC, also called phase equalization, compensates only for the phase rotation caused by
the channel by choosing the equalization coefficient as
Gn =
H∗n
|Hn|
(8)
EGC is the simplest single user detection techniques, since it only needs information about
the phase of the channel.
3.1.3. Orthogonal Restoring Combining (ORC)
ORC inverses the channel transfer function and can eliminate multiple access interference
by restoring the orthogonality between the users with an equalization coefficient chosen as
Gn =
1
Hn
(9)
In the literature, ORC is also called Zeros-Forcing (ZF). The drawback of ZF equalization
is that for small amplitudes of Hn the equalizer enhances noise Nn in such a way that the
signal to noise ratios γc (the average SNR per carrier at the input of the data detector γc =
E [|sn|
2]
σ2n
=
Ecarrier
σ2n
) may reduce to zero on some sub-carriers.
3.1.4. Minimum mean square error combining (MMSEC)
Equalization according to the MMSE criterion minimizes the mean square value of the
error εn between the signal Sn transmitted on sub-carrier n
th and the assigned output yn of
the equalizer.
εn = sn −Gnrn (10)
The mean square error is
Gn = min
Gn
E[|εn|
2] (11)
The receiver signal at nth sub-carrier is:
20 NGUYEN NGOC TIEN, NGUYEN VIET KINH, SEONG RAG KIM
rn =
K∑
k=1
Hnck,ndk +Nn = Hnsn +Nn (12)
where sn =
∑K
k=1 ck,ndk and sˆn = yn = Gnrn are the transmitted signal on n
th sub-carrier
and receiver signal after equalizer, respectively. According to the Wiener - Hopf equation, the
equalization coefficient matrix Gn is equal to
Gn = R
−1
n pn (13)
where R−1n is the autocorrelation of the received signal rn and pn is the cross-correlation signal
between the desired signal on nth sub-carrier sn and the received signal rn.
Rn = E [rnr
∗
n] = E
[( K∑
k=1
Hnck,ndk +Nn
)( K∑
k=1
H∗nck,nd
∗
k +N
∗
n
)]
=
K∑
k=1
Echip|Hn|
2 + σ2n
= KEchip|Hn|
2 + σ2n = Ecarrier|Hn|
2 + σ2n = Ecarrier
(
|Hn|
2 +
1
γc
)
(14)
or
Rn =
K
L
Eb|Hn|
2 + σ2n (15)
pn = E [r
∗
nsn] = E
[( K∑
k=1
Hnck,ndk +Nn
)
∗
( K∑
k=1
ck,ndk
)]
= KEchipH
∗
n = EcarrierH
∗
n
pn = EcarrierH
∗
n (16)
where |ck,n|
2 =
1
L
; E [|dkd
∗
k|
2] = Eb; Ecarrier, Echip and Eb are the energy per sub-carrier;
the energy per chip and the energy per symbol before spreading, respectively, the relation
between them is given by
Echip =
1
L
Eb, and Ecarrier = KEchip =
K
L
Eb (17)
The equalization coefficient based on MMSE criterion applied independently per carrier
are given as
Gn = R
−1
n pn =
EcarrierH
∗
n
Ecarrier
(
|Hn|2 +
1
γc
) = H∗n(
|Hn|2 +
1
γc
) (18)
where γc is the SNR per sub-carrier and relates to the SNR per symbol γb =
Kγc
L
. Then the
optimal coefficients of the equalizer are equal to
Gn =
H∗n
|Hn|2 +
L
K
1
γb
(19)
When the system has only one user, the equalization coefficient matrix Gn is equal to
Gn =
H∗n
|Hn|2 +
L
γb
=
H∗n
|Hn|2 +
1
γchip
(20)
A STUDY ON DETECTION TECHNIQUES FOR DOWNLINK 21
and when the system is full load (K = L), the formula (19) becomes the formula (18). And
then the estimated data symbol of the user kth is
d˜k = c
∗
k,nGnrn =
Nc∑
n=1
c∗k,nH
∗
n
|Hn|2 +
L
K
1
γb
rn (21)
The MMSEC equalization corrects the phase shift and the attenuation of the channel
fading, taking into account the number of active users K and the present signal to noise ratio.
For all these basic techniques, the matrix G is diagonal and the receiver sequence is equal-
ized by using a bank of Nc adaptive one tap equalizers. This means that the complexity of the
equalizer is low. Among all the SUD techniques, the MMSEC equalization per sub-carrier can
offers the best results. However, MMSEC equalization per carrier method is still not optimal
because it does not take into account the despreading process and thus does not minimize
the mean square error at the input of the threshold detector. Thus, to this end, we analyze
an improved method base on the linear MMSE per user (MMSE MUD) technique to detect
multi-user interference.
3.2. Multi-user detection (MUD)
In this section we introduce MinimumMean Square Error MUD equalization method which
belongs to the second category of MC-CDMA detector. The basic idea of MMSE MUD is to
minimize the mean square error between transmitted data symbol dk and the estimated data
symbol dˆk.
dˆk =W
H
k r (22)
where W k = [w
1
k, w
2
k, ..., w
Nc
k ]
T is the optimal weighting vector. We have
min
W k
E [|dk −W
H
k r|
2] (23)
Applying to the Wiener-Hopf equation, the optimal weighting vector is equal to
W k = Rrr
−1prd (24)
whereRrr is the autocorrelation matrix of the received vector r and prd is the cross-correlation
vector between the desired symbol, dk and the receiver vector r. Rrr is given by
Rrr = E [rr
H ] = E [(HCd +N )(HCd +N )H ] = E [HCddHCHHH ] +E [NNH ]
=HCE [ddH ]CHHH + σ2nINc×Nc (25)
prd = E [d
∗
kr] = E [d
∗
k(HCd +N )] =HCE[d
∗
kd] = E [|dk|
2]HCk
prd = E [|dk|
2]


H1 0 · · · 0
0 H2 · · · 0
...
...
...
...
0 0 · · · HN




ck,1
ck,2
...
ck,Nc

 = E [|dk|2]


ck,1H1
ck,2H2
...
ck,NcHNc

 (26)
Since the user signals have the same power and are independent, we can haveE [|dk|
2] = Eb
and E [ddH ] = EbU , where U = {uik} is the diagonal matrix with the term ukk = 1, if the
user k is active, and ukk = 0 if the user k is inactive.
22 NGUYEN NGOC TIEN, NGUYEN VIET KINH, SEONG RAG KIM
WHk =H
HCHk (HCUC
HHH +
σ2n
Eb
INc×Nc)
−1 (27)
The optimal weighting vector can be expressed as
WHk = C
H
k G (28)
Hence, the equalization coefficient matrix of the MMSE MUD per user is equal to:
G =HH(HCUCHHH +
σ2n
Eb
INc×Nc)
−1 (29)
When the system is full load (K = L), the quantity CUCH is equal to the identity matrix and
the equalization coefficient matrixG is a diagonal matrix with the nth sub-carrier equalization
coefficient is calculated by equation (18). On the other hand, in the non full load case (K < L),
the equalization coefficient matrix G is no longer diagonal.
In the special case, with K = 1, the U is given by 11T , where 1 is a L-by-1 vector with all
the entries equal to one, and the autocorrelation matrix Rrr becomes
Rrr = EbHCk11
TCHk H
H + σ2nINe×Ne (30)
The cross-correlation vector is
pk = EbHCk1 (31)
Defining xk = HCk1 and using matrix inversion lemma, the optimal weighting vector is
equal to
W k = R
−1
k pk =
Eb
σ2n
(
I−
γb
1 + γb‖xk‖2
xkx
H
k
)
xk =
Eb
σ2n
(
xk−
γb‖xk‖
2
1 + γb‖xk‖2
xk
)
=
( γb
1 + γb‖xk‖2
)
xk
Then the optimal weighting vector can be expressed as
WHk,n =
γbc
∗
k,nH
∗
n
1 + γb
L∑
l=1
|Hn|
2|ck,n|
2
=
c∗k,nH
∗
n
1
L
L∑
l=1
|Hn|
2|+
1
γb
(32)
Comparing equations (32) and (20), we observe that there is difference between the equal-
ization coefficient of the MMSEC technique and that of the MMSE MUD one. As confirmed
by simulation results due to that difference, the MMSE MUD offers better BER performance
than the MMSEC when the system has only one active user.
From equation (27), if we want to implement the MMSE MUD algorithm we must know
H and U . Furthermore, the inversion of HCUCHHH +
σ2n
Eb
INc×Nc matrix may be a time
consuming operation, particularly for large length L of codes.
4. SIMULATION RESULTS
4.1. Performance comparison of the detection techniques
The following results are obtained using Monte Carlo simulations Matlab code. In Figure
2, 3 and 4 the graph of BER versus SNR in dB of the MC-CDMA system of MRC, EGC, ORC
and MMSEC equalizations with different number of active users are shown. The simulations
A STUDY ON DETECTION TECHNIQUES FOR DOWNLINK 23
were performed without channel coding and interleaving. Each of the independent sub-carrier
is QPSKmodulated at the transmitter and then multiplied by an uncorrelated Rayleigh fading.
We assume that the estimation of the frequency channel response for each sub-carrier is correct
and the channel matrix H is perfectly known to the receiver, and therefore, it is possible to
calculate the optimum weights by a direct matrix inversion operation.
Figure 2 shows the performance of a full-load system where the number of active users is
equal to the length of the WH codes, K = L = 64, (maximum user capacity). Without using
an equalizer, the performance of the system is very bad. Even in the case MRC equalizer is
presented, the performance is still not good enough for practical use. The loss of orthogonality
of the WH codes is heightened in the receiver when applying MRC. The ORC, zero-forcing
equalization restores the orthogonality between the user signals and avoids MAI. However, it
introduces noise application which is especially high at low SNRs. EGC avoids noise appli-
cation but does not counteract the MAI caused by the loss of the orthogonality between the
user signals, resulting in a high error floor. The single user detection based on minimum mean
square error per carrier equalization offers good results. The matched filter (MF) bound is also
given for reference. The MMSEC technique outperforms the other techniques because it avoids
an excessive noise application for low signal to noise ratios, while keeping the orthogonality
among users for large SNRs.
0 2 4 6 8 10 12 14 16 18 20
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
SNR (dB)
B
E
R
K=L=64 and Nc=64 (Full load system)
No Equalizer
MRC Combining
ORC Combining
EGC Combining
MMSEC/carrier
MMSE MUD/user
MF
MMSEC and
MMSE MUD
is coincident
Figure 2. Different detection performance for K = L = Nc = 64
(full load system) in the MC-CDMA
4.2. The relationship between MMSE MUD per user and MMSEC per sub-carrier
In Figure 2, the system is full-loaded (K = L), the K user signals are supposed to be
transmitted with the same power E[|d1|
2] = · · · = E[|dK |
2] = Eb, then the algorithm of the
MMSE MUD per user contains the quantityCUCH which will be equal to the identity matrix.
Thus, equation (29) is similar to equation (18) and the equalization coefficient matrix G is a
diagonal matrix with the nth sub-carrier equalization coefficient being calculated by equation
(18). In that case, the performance of the two MMSE approaches are the same and the curve
of the MMSE MUD coincident with the curve of MMSEC detection technique (see Figure 2).
On the other hand, in the Figure 3 and Figure 4, for the non-full loaded systems (K < L),
24 NGUYEN NGOC TIEN, NGUYEN VIET KINH, SEONG RAG KIM
the equalization coefficient matrix G is not a diagonal matrix. Therefore, the equalization co-
efficient matrixG of the MMSE MUD obtained from equation (29) outperforms the algorithm
MMSEC per sub-carrier based on equation (18). This preeminent advantage can be observed
in Figure 3 and Figure 4 with the case K = 56 and K = 32 (corresponding to a system load
equal to 88% and 50% respectively).
0 2 4 6 8 10 12 14 16 18 20
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
SNR (dB)
B
E
R
K=56 and L=Nc=64 (Non full load system)
No Equalizer
MRC Combining
ORC Combining
EGC Combining
MMSEC/carrier
MMSE MUD/user
MF
Figure 3. Different detection performance for K = 56 < L = Nc = 64 (Non full load system)
Those results are logical because MMSE MUD algorithm minimizes the decision error by
taking into account the despreading process instead of minimizing the error independently on
each sub-carrier, thus both the interference and the noise enhancement are minimized.
Furthermore, it allows adjusting the coefficients using decisions on the information symbols
assuming that those decisions are correct.
0 2 4 6 8 10 12 14 16 18 20
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
SNR (dB)
B
E
R
K=32 and L=Nc=64 (system load is equal to 50%)
No Equalizer
MRC Combining
ORC Combining
EGC Combining
MMSEC/carrier
MMSE MUD/user
MF
Figure 4. Different detection performance for K = 32 < L = Nc = 64
(system load equal to 50%)
A STUDY ON DETECTION TECHNIQUES FOR DOWNLINK 25
In order to compare the performances of MMSEC and MMSE MUD in more details, we
range the number of active users from 1 to 64 then estimate the required SNR to achieved a
BER = 10−3. The relationship between the number of active users and the required SNR is
shown on Figure 5. Again, the Zeros-Forcing technique, which is better than MRC and EGC,
is outperformed by MMSEC and MMSE MUD. The difference between MMSEC and MMSE
MUD can be easily observed in this figure. In the case the number of active users is full-loaded,
both equalization techniques give the identical performance and the user/SNR curves meet
each other at a point. On the other hand, when the system is non-full loaded, the MMSE
MUD based on the MMSE per user criterion achieves a gain of more than 2 dB with K = 32
which corresponds to system equal to 50%. Particularly, when the system has one active user,
the MMSE MUD still performs better than the MMSEC, as illustrated in Figure 5.
Figure 5. Comparison of the number K of active users between MMSEC
and MMSE MUD versus SNR with BER = 10−3, L = Nc = 64
5. CONCLUSIONS
The bit error rate performances of single user detection and multi-user detection techniques
for the downlink of a MC-CDMA system are presented in this paper. Also, the relationship be-
tween MMSEC and MMSE MUD detection are compared and evaluated. It was seen that the
MMSE MUD outperforms all other detection techniques, especially for high bit rate scenarios,
whereas the MRC, EGC, ZF detections result in very poor performances. The MMSE MUD
per user approach offers for non-full load systems a significant gain compared to the MMSEC
per carrier technique. In particular for L = 32 (50% of the system), the MMSE MUD per user
criterion achieves a gain of more than 2 dB in comparison with MMSEC. For a MC-CDMA
system, the probability that the maximum number K of active users are working at the same
time is usually small. Hence, the MMSE MUD is a good choice for non-full load system.
However, the MMSE MUD per user is computationally excessive. It was also observed that
the MMSEC could provide a better trade-off between performance and complexity, especially
under high load conditions.
26 NGUYEN NGOC TIEN, NGUYEN VIET KINH, SEONG RAG KIM
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