The Journal of China
Universities of Posts and
Telecommunications
September 2008, 15(3): 1–7
www.buptjournal.cn/xben
Suboptimal resource allocation with MMSE detector for grouped
MC-CDMA systems
AAAA1, BBBBBB1, CCCCC1, DDDDD2
1. School of Telecommunication Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China
2. Lucent Technologies (China), Beijing 100102, China
Abstract
This article puts forward two novel user-grouping algorithms for grouped multi-carrier (MC)-code division multiple access (CDMA)
systems. As is well known, the adaptive assignment for user-grouping plays an important role for link quality of multi-access
transmissions. In the study, the capacity-maximizing problem of user-grouping is formulated. By using the Kuhn-Tucker condition,
the optimal criterion is deduced and found to have a similar form with signal to noise plus interference (SINR). However SINR
includes the signal power that can only be determined after user-grouping. Therefore the optimal criterion will lead to an
impractical application. To deal with it, the user’s equivalent SINR for minimum mean square error (MMSE) detector is proposed
and served as a suboptimal assignment criterion, based on which two kinds of user-grouping algorithms are proposed. In the
algorithms, only partial channel information is needed at the base station, which saves a large part of the bandwidth occupied by
feedback information. Computer simulations have evaluated an excellent performance of the proposed algorithms at both link
quality and data rate. Meanwhile, the proposed algorithms have lower implementation complexity for practical reality.
Keywords MC-CDMA system, resource allocation, user-grouping, MMSE detection
1
Introduction
At present, mobile communication systems intend to meet the users’ demands for high data rate services. Considering the
limited bandwidth resource, it is urgent to devise methods to increase spectral efficiency. Integrating CDMA with MC modulation,
MC-CDMA [1] has attracted a lot of interest in carrier transmission. By combining multiple-input multiple- output (MIMO) [2]
with MC-CDMA, data rate can be highly enhanced [3].
To achieve the maximum bandwidth efficiency, adaptive resource allocation techniques are needed for MC-CDMA systems.
Authors in [4] propose an optimization model for allocating resources to multiple mobile stations (MSs) employing adaptive
modulation, coding, and multicodes. Reference [5] presents the theoretical work of rate and power control based on quality of
service (QoS) constraint models, for direct sequence (DS)-CDMA systems, whereas, Ref. [6] proposes two interference-based
subcarrier grouping strategies.
Grouped MC-CDMA is put forward in Ref. [7] and further investigated in Ref. [8]. Users are adaptively grouped in
MC-CDMA systems to make full use of the radio bandwidth. The purpose of adaptive allocation is to let users occupy the
sub-band that they are mostly fit into. The aim of this article is to derive feasible allocation criterions and algorithms in grouped
MC-CDMA systems to maximize the link capacity.
2
2.1
System model and problem formulation
System model
Received data: 29-06-2007
Corresponding author: AAAAA, E-mail：[email protected]
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The Journal of China Universities of Posts and Telecommunications
2008
Considering a downlink grouped MC-CDMA system is in a hot-spot area, the system bandwidth W is subdivided into N c
subcarriers. Each active user transmits a symbol on a sub-band including N subcarriers, N is the spreading factor (SF) of codes.
Each sub-band can accommodate N active users, which is regarded as full load. Thus the whole band can be subdivided into
N g  N c N groups, and active users are assumed to be adaptively assigned to each band group. U i is used to present the number
Ng
of users in group i . Thus the whole band can accommodate a total of U T =
å
U i users. The block diagram of downlink grouped
i= 1
MC-CDMA systems is illustrated in Fig. 1.
(a)
Transmit end
(b) Receive end
Fig. 1 Block diagram of downlink grouped MC-CDMA system for transmit end and receive end
Let i denote the group index and u denote index user in each band group. The time index is omitted in the following. The
input-output relationship of group i with U i active users for synchronous downlink can be presented as:
(1)
yi  H i S P i xi  εi
where x i is the active users’ symbols in group i and S is the code matrix. P i  diag  p1i , p2i , , pUi  and H i  diag  h1i , h2i , , hNi 
i
denote each user’s transmit power and selective channel fading in group i , respectively. y i   y1i
y2i
yNi 
T
is the
T
 Ni  is the complex Gaussian noise vector in the ith band group. Each element
received signal vector and ε i  1i  2i
of εi has zero mean and  2 2 variance per dimension. Here, aT is the transpose of a . When equal power assignment is
assumed, P i turns into an identity matrix.
2.2
Problem formulation
With respect to the transmission scheme, the total channel capacity can be expressed as:
2

1 N
pi
log 2 1  u 2 hui , n 

N
i 1 u 1 N n 1


Ng
Ng Ui
Ctotal   C i  
i 1
(2)
It is worth mentioning that the number of active users in each group satisfies U i ≤ N , to combat multi-access interference (MAI).
An equivalent channel fading hui is assumed for user u in group i , which depends on the multiuser detector. For the CDMA
scheme, MAI brings a large impact for the system throughput, which turns the total capacity in Eq. (2) to


2
2

 Ng U i
i
i

Ng
Ng U i
p
h
pi h i 
u u
 
1  u u 
Ctotal   C i   log 2 1 
log
(3)

2
Ui
 
  2  ui 
2 
i 1
i 1 u 1
i 1 u 1
2
i
i
i 
    u  pv hu  




v 1, v  u

 
where ui  ui
Ui

v 1, v  u
2
pvi hui is MAI and  ui is the orthogonality factor for user u in group i .
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WANG Ya-chen, et al. / Suboptimal resource allocation with MMSE…
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In the aforementioned CDMA scheme, each user occupies just a group of subcarriers so that the overall system is able to
accommodate as many active users as possible. Then the optimization of the problem is formulated as:
2
 N g UT

pi h i
i

1  k k
max  R   max

log
 k 2   2   i
ki , pki  i 1 k 1
k



  s.t.

 
N g UT

i 1 k 1
i
k
Ng UNg
≤ UT ,
 p
i 1 k 1
i
k
≤ Ptotal ; ki ≥ 0, pki ≥ 0
（4）
where k is the user index. ki 0,1 is the allocation index for the kth active user, that is,  ki  1 if user k is assigned to
group i , and ki  0 , otherwise. To make the problem tractable, ki can be relaxed to a real number in [0, 1, 9, 10].
3
User-grouping algorithm
3.1
Optimal grouping criterion
For downlink systems, to maximize the overall link capacity, users need to be grouped optimally under the constraint in Eq. (4).
The marginal rate functions with respect to subcarriers are
2

pi h i 
R
1  k k 
(5)

log
2
  2  ki 
ki


and
R

pki
ki
1
hki
2
(6)
2
2
i
pki hki   k
 2  ki
Proposition (grouping criterion) To maximize the data rate for the overall link, active user k should be assigned to group i
by
 hi 2 
k
(7)
 i*, k *  arg max  2 i 

 k 
 i ,k 



Proof Set the Lagrangian function as
 p i h i 2   N g UT
N g UT

 N g UT i

k k
i
  
L ki , pki   ki log 2 1+ 2


U


(8)

  pk  Ptotal 

k
T
i







i 1 k 1
i 1 k 1
i 1 k 1
k






where  and  are non-negative Lagrangian multipliers. Taking account of the Kuhn-Tucker condition, the optimal solution is

i*
k*
, pki** 
L ki , pki 
ki**
L ki , pki 
pki**
N g UT
 
i *1 k *1
i*
k*
N g UT
 p
i *1 k *1

i*
k*
2

pki** hki** 

 0
 log 2 1  2
   ki** 


ki**

1
hui**
(9)
2
2
2
i*
pki** hki**   k *
 0
(10)
 2  ki**
 UT ≤ 0
(11)
 Ptotal ≤ 0
(12)
N g UT

 i*1 k *1

   ki**  U T   0
(13)
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The Journal of China Universities of Posts and Telecommunications

N g UT
2008

   pki**  Ptotal   0
(14)
 i*1 k *1

(15)
≥0
≥0
(16)
From Eqs. (9) – (16), it is found that the k * th user should be assigned to the i * th group when the following expressions have
the same solution:
2


pki hki  



(17)
i
*,
k
*

arg
max
log
1



 2
2
i 




 i ,k 
k 








2


i
i
k hk


 i*, k *  arg max 
2 
i
i
i
,
k


 
 2

pk hk
i

    k   1  2
i 
   k  




The solution in Eqs. (17) and (18) can be calculated by the water-filling power allocation (see Appendix A)

1
p 
U
 T
i*
k*

N g UT 
2
j 
2
i* 
P 
   l      k * 

 total j 1 l 1  h j 2 h j 2   h i* 2 h i* 2 
l
k*
k* 
 l



(18)

(19)
where  x   max  x,0 . By substituting Eq. (19) into Eqs. (17) and (18),

   hi 2
k
 i*, k *  arg max log  2 i
i
,
k


k
 
 

and
 i 
 i*, k *  arg max  k 
i ,k    





(20)
(21)
where

1
UT

N g UT 
2
j
P 
  +l
2
 total 
 hj
j 1 l 1
 l



 

For each user in the link,  has the same value. When ki achieves the maximum value, it states that  hki
(22)
2

2
 ki 
should have the maximum value among all the optional items, and user k * is assigned to the group i * . Unlike uplink systems,
Eqs. (20) and (21) have the same solution and they can be rewritten as a uniform expression
 hi 2 
k
(23)
 i*, k *  arg max  2 i 

 k 
 i ,k 



that demonstrates the conclusion in the Proposition.
The conclusions in the Proposition indicate that the assignment is not only up to the channel fading, but also that the multiuser
interference MC-CDMA allocation works according to the equivalent channel fading over the band groups, and not just the fading
on a subcarrier. The equivalent fading relies mainly on the detection algorithm employed in the system. The grouping criterion
derived in the Proposition is somewhat similar to the form of SINR, expect that the signal power is included. Actually, such a
criterion is difficult to work because users’ power cannot be determined before user-grouping. In the following, a feasible MMSE
detection is considered and the math expression of the equivalent SINR is derived.
3.2
Equivalent SNR with MMSE detector
An equal power allocation is assumed here, to investigate the math form of the equivalent SINR in each group. Meanwhile, as the
Issue 3
WANG Ya-chen, et al. / Suboptimal resource allocation with MMSE…
5
effect of MAI differs because of the different number of users in each group, it is assumed that the system has a full load for
simplicity, which means that the number of active users in each group equals N . Albeit this, the proposed criterion can also serve for
fewer users. The feasible MMSE detection algorithm [11] for multiuser detection is considered by
xˆ u  wuH y i
(24)
where
wuH  suH
 H 
i H

H i   2IN
1
H 
i H
(25)
The estimated symbol for user u in group i is written as:
 H  H   I   H  y  s  H  H   I 


s  H  H   I   H   H  s x  ε 


1
i H
xˆ ui  suH
i
i H
2
i
1
i H
H
u
i
i
1
2
N
 H   H Sx
i H
i
i
 ε i   suH
 H 
i H
H i   2IN

1
H 
i H
H i su xui 
Ui
i H
2
i H
H
u
N
i
N
i
v v
v 1, v  u
i
( 26 )
Terms in Eq. (26) at the last equal sign are useful signals and interference plus noise respectively. According to Eq. (26), the
equivalent SINR of user u for group i is expressed by
ui 
R  suH H i H i su xui 
(27)
Ui


R  suH H i H i  sv xvi   R  suH H i ε i 
v 1, v  u



wher H i   H i  H i   2 I N
H

1
H 
i H
E  AAH  . To obtain an accurate expression of equivalent SINR, mathematical
and R  A 
analysis is given for each term in Eq. (27). The useful signal at the numerator of Eq. (27) is

R  suH H i H i su xui   E suH H i H i su xui  suH H i H i su xui 
H
  s H H s E  x  x   s H H s 
H
u
i
i
i
u
u
i *
u
H
u
i
i
u
H
 suH H i H i su
2
(28)
The first term at the denominator of Eq. (28) denotes MAI, which can be rewritten as:
Ui


R  suH H i H i  sv xui   R  suH H i H i Sx i   R  suH H i H i su xui 
v

1,
v

u


Similarly with the deduction in Eq. (28),

R  suH H i H i Sx i   E suH H i H i Sx i  suH H i H i Sx i 
H
(29)
  s H H SE  x  x   s H H S 
H
u
i
i
i
i H
H
u
i
i
H
suH H i H i SS H  H i H i  su
H
 suH H i H i S  suH H i H i S  
H
(30)
Recalling that SS  I nT for full load,
H
R  suH H i H i Sx i   suH H i H i  H i 
H
H 
i H
su
(31)
The noise term can also be rewritten as:

R  suH H i εi   E suH H i εi  suH H i εi 
H
  s H E  ε  ε   s H 
H
u
i
i
i H
H
u
i H
  2 suH H i  H i  su
H
（32）
As H i is a diagonal matrix, H i is also a diagonal matrix and H i H i can be rewritten by
2
2
 hi 2

h2i
hNi
1


H H  diag
,
,
,
i 2
2 
 hi 2   2 hi 2   2
h


2
N
 1

i
i
Let Gni = hni
2
i 2
n
(h
)
+ s 2 . By substituting Eq. (33) into Eqs. (28), (31) and (32),
(33)
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The Journal of China Universities of Posts and Telecommunications
2
i 2
hn
1 N
1 N

R  suH H i H i su xui    2
   Gni 
i
2
N n 1 h  
 N n 1 
n
1 N
R  s H H Sx   
N n 1
H
u
i
i
i
R  suH H i ε i  

N
2 N

n 1
h

hni
4
2
hni   2
i 2
n

2
2
hni

2

2008
2

(34)
2
1 N
Gni 


N n 1


2
2 N
N

(35)
G 
i 2
n
hni
n 1
(36)
2
Accordingly, the SINR of user u in group i can be rewritten as:
2
 1 N i
 N  Gn 
 n 1 
i 2
 2 N  Gn 
 
i
u
2
1 N
  Gni   N
N n 1

n 1
hni
2
N
1 N

   Gni 
N
 n 1 
2

G
i
n
n 1
(37)
N
N   Gni
n 1
Eq. (37) shows that instead of obtaining full channel state information (CSI), the base station (BS) gets the feedback of users’
equivalent SINR value. Thus feedback information from the subscriber devices only uses 1 N of radio resource compared to full
channel information feedback.
3.3
Two novel user-grouping algorithms
Two user-grouping algorithms are proposed for grouped MC-CDMA systems.
1) Uniform user-grouping algorithm
Uniform user-grouping (UUG) algorithm computes the equivalent SINR values by Eq. (37) and equivalent-average SINR by
averaging a user’s equivalent SINR over all the band groups, the ratio of which accounts for how urgently a user needs to be
assigned into a band group. By sorting priorities, users are grouped based on a uniform order. Steps of UUG algorithm are
elaborated as follows:
Step 1 Let Ai   denote the vector including the indices of active users assigned to group i , and # Ai  0 denote the
number of users assigned to group i .
i  S  1, 2,
Step 3
Step 4
1 Ng
æNg
ö
è i= 1
ø
In each band group, calculate r ui and sub-band equivalent-average SINR for user k , r k = ççÕ 1+r ki ÷
÷
÷
çç
÷
Step 2
, N g  and k  T  1, 2,
. Here
, UT  .
Compute users’ priority order by d ki = (1+r ki ) r k in each band group.
Choose the pair
i*, k *  arg max dki  and
assign user k * to group i * if # Ai ≤ N  1. Let Ai*  Ai*
k* ,
iS ,kT 
# Ai*  # Ai*  1 .
Step 5 Update the number of unassigned active users by subtracting the index of the assigned ones in the cardinality of
T.
Step 6 Repeat Steps 4 and 5 until all active users are allocated into a band group.
2) Iterative user-grouping (IUG) algorithm
IUG algorithm gives an updated priority of active users by detracting the effect of the users that are assigned in the last iteration.
Different from the UUG algorithm, it considers how urgently a group needs a user. Only one user is assigned in one iteration. The
following describes the steps of the IUG algorithm.
Step 1 Let Ai   denote the vector including the indices of active users assigned to group i , and # Ai  0 denote the
number of users assigned to group i .
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WANG Ya-chen, et al. / Suboptimal resource allocation with MMSE…
7
UT
çèk = 1
i  S  1, 2,
Step 3
1 UT
æ
ö
÷
In each band group, calculate r ui and the user equivalent-average SINR on the sub-band i, r i = ççÕ 1+r ki ÷
÷
ç
Step 2
÷
ø
. Here
, N g  and k  T  1, 2, , U T .
Compute user’s priority order by d ki = (1+r ki ) r i in each band group.
Step 4 Choose the pair
i*, k *  arg max dki  and
assign user u * to group i * if # Ai ≤ N  1 . Let Ai*  Ai*
k* ,
iS ,kT 
# Ai*  # Ai*  1 .
Step 5
Set r ki * = 0 , i  S  1,2,
, N g  . Update the number of unassigned active users by subtracting the index of assigned
ones in the cardinality of T .
Step 6 Repeat Steps 2–5 until all active users are allocated into a band group.
4
Performance analysis
To show the advantage of the novel user-grouping algorithms, computer simulations are performed in multi-path propagation. In
this simulation, the carrier frequency is 2 GHz, the channel bandwidth is 1.25 MHz, and the multipath channel model is a six-path
typical urban channel [12]. The radio bandwidth is divided into 128 OFDM subcarriers. Thus the total OFDM symbol duration is
107.7 µs, and the guard interval duration is 21.04 µs. The vehicular speed is 44 km/h and the maximum Doppler frequency is
approximately 83.1 Hz. Quadrature phase-shift keying (QPSK) constellation and the afore- mentioned MMSE detection is used at the
transmitting end and receiving end respectively. The downlink is assumed to be ideally orthogonal and synchronized between users
within the cell, that is, no intra-cell interference is considered. Partial channel state information is assumed to be well known at BS.
Rate-craving greedy (RCG) algorithm is a famous subcarrier assignment in OFDMA systems because of its high achievable data
rate [13]. It can also be employed in grouped MC-CDMA systems. However, it is computationally intensive as it involves finding the
optimal pair of users who would bring a minimum capacity decrease. The MMSE assignment criterion proposed in [14] is
considered combined with RCG algorithm as the base line for comparison.
Figures 2 and 3 show the capacity performance of the novel algorithms with two other kinds of grouping algorithms, that is,
random allocation and RCG algorithm. The spreading factor (SF) is 16, whereas, the numbers of active users are 64 and 128
respectively. Computer simulations show that UUG algorithm has the same performance with RCG, and IUG algorithm is prior to
RCG and UUG. By investigating the detailed results of RCG and UUG, it was found that they have almost the same assignment.
Fig. 2 Data rate comparison among UUG, IUG and existing ones ( U T = 64)
8
The Journal of China Universities of Posts and Telecommunications
Fig. 3
2008
Data rate comparison among UUG, IUG and existing ones ( U T =128)
Figures 4 and 5 exhibit BER performance of the proposed algorithms with RCG and random algorithm. SF equals 16,
whereas, the numbers of active users equals 64 and 128 respectively. Clearly, UUG and RCG have the best BER
performance, and the performance of the IUG is a bit worse.
Fig. 4 BER performance comparison among UUG, IUG and existing ones ( U T = 64)
Fig. 5 BER performance comparison among UUG, IUG and existing ones ( U T = 128)
Figure 6 gives the comparison of implementation complexity between proposed algorithms and existing ones. Y-axis denotes
the operation flops. It is found that the proposed algorithms have an attractive reduced-complexity in implementation.
Issue 3
WANG Ya-chen, et al. / Suboptimal resource allocation with MMSE…
Fig. 6
9
Complexity comparison among UUG, IUG and RCG
Therefore, the IUG algorithm can achieve a larger data rate with some expense in link quality and practical complexity,
whereas, UUG has a desirable trade-off among them.
5
Conclusions
This study mainly considers the capacity optimization criterion of subcarrier allocation for the downlink grouped MC-CDMA.
MAI was put under consideration with MMSE detection, to propose a novel criterion and two user-grouping algorithms.
Computer simulations show that the novel algorithms are effective in improving spectral efficiency and link quality. Meanwhile,
they are also feasible for real implementation.
Acknowledgements
This work is supported by the DoCoMo Beijing Labs Co. Ltd., and Program for New Century Excellent Talents in Beijing University of Posts
and Telecommunications (04-0112).
Appendix A
Water-filling power allocation
Considering the derivative expression in Eq. (10),
ki
1
pki hki
2

  2  ki 
hki
2
(A.1)
 2  ki
After some mathematical transmutation,
pki 
ki  2
i

 k2
2
 hki
hki
(A.
2)
By substituting Eq. (A.2) into Eq. (12),
N g UT
l j
 
j 1 l 1

2
hl j
2

l j
hl j
2
≤ Ptotal
The expression of Lagrange multiplier  is


N g UT 
1
1 
2
l j  

Ptotal   

 h j 2 h j 2  
 U T 
j 1 l 1
l
 l


Therefore, p ki '' has a water-filling form all over the bandwidth.
(A.3)
(A.4)
10
2
T h e J o u r n a l o f C h i n a U n i v e r s i t i e s o f P o s t s a n d Te l e c o m m u n i c a t i o n s
0
0
8



U
i
2
j
2
i
p  k  Ptotal   
 l 2  
 k2
2
2
Ng
T
i
k
 hj
U T 
j 1 l 1
hl j   hki
hki
 l


If user k * is assigned to group i * in the transmission,


N g UT 
1 
2
j
2
 i*
pki** 
Ptotal   
 l 2  
 k* 2
2
2
 hj
UT 
j 1 l 1
hl j   hki**
hki**
 l


(A.5)
Eq. (A.5) can be expressed as:
(A.6)
where k*  1, 2, , UT .
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