A New Linear Group-Wise Parallel Interference
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Transcript of A New Linear Group-Wise Parallel Interference
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Wireless Pers Commun (2009) 49:2334
DOI 10.1007/s11277-008-9553-7
A New Linear Group-Wise Parallel Interference
Cancellation Detector
A. Bentrcia A. Zerguine
Published online: 29 July 2008 Springer Science+Business Media, LLC. 2008
Abstract In this paper, a new linear group-wise parallel interference cancellation (LGPIC)
detector is proposed. Four different group-detection schemes are derived, namely, the linear
group matched filter PIC (LGMF-PIC) detector, the linear group decorrelator PIC (LGDEC-
PIC) detector, the linear group minimum mean square error PIC (LGMMSE-PIC) detector
and the linear group parallel interference cancellation weighted PIC (LGPIC-PIC) detector.
The convergence behavior of the proposed detector is analyzed and conditions of conver-
gence are derived. Finally, extensive simulations regarding the convergence behavior andthe effect of the grouping on the convergence behavior of the proposed LGPIC detector are
conducted.
Keywords SIC PIC Group-wise Jacobi Decorrelator Multiuser detection CDMA
1 Introduction
Interference cancellation (IC) structures are the most promising multiuser detector structures
to be implemented in future commercial systems [1]. Linear interference cancellation struc-
tures are commonly used to implement the decorrelator/LMMSE detectors [2]. Two types of
IC structures can be distinguished: successive interference cancellation (SIC) structures [3]
and the parallel interference cancellation (PIC) structures [4]. The linear SIC structure exhib-
its low computational complexity, however, it suffers from relatively long detection delay.
One solution to this problem is group-detection where groups of users instead of individual
users are detected in series, which reduces considerably the detection delay of the linear SIC
detector. Such structure is known as the linear group-wise SIC (LGSIC) structure [5]. The
A. Bentrcia (B) A. ZerguineKing Fahd University of Petroleum and Minerals,
P.O.Box 1387, Dhahran 31261, Saudi Arabia
e-mail: [email protected]
A. Zerguine
e-mail: [email protected]
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24 A. Bentrcia, A. Zerguine
PIC detector on the other hand, is attractive due mainly to its inherent parallelism which, if
properly exploited, reduces considerably the computation time.
On the other hand, as for the LGSIC detector, the convergence behavior of the linear PIC
detector can be greatly increased if group-detection is applied, where groups of users instead
of individual users are detected in parallel. Despite of all these apparent advantages, and upto our knowledge, no linear group-wise PIC structure is proposed in the literature. In this
paper, a linear group-wise PIC (LGPIC) structure is proposed. The LGPIC detector will be
shown to be equivalent to linear matrix filtering which allows the derivation of analytical
expressions for the Bit Error Rate (BER) and Asymptotic Multi-user Efficiency (AME) for
proposed structure. Depending on the group-detection scheme, four different group detection
schemes are obtained. The proposed LGPIC structure is shown to converge to the decorre-
lator detector if it converges and conditions of convergence are derived. Finally, simulation
results are shown to corroborate well with theory.
2 System Model and the LGPIC Detectors Structure
In this paper, we consider a case of an uplink channel scenario where K users transmit
simultaneously over a synchronous additive white Gaussian noise (AWGN) channel using
Binary Phase Shift Keying (BPSK). Each user is characterized by its own pseudo-noise code
of length N chips. The received signal is expressed in vector form as:
r=
SAb+
n (1)
where S = (s1, s2, . . . , sk, . . . , sK) 1/
N, 1/
NN,K
, A = diag(a1, a2, . . . , ak,. . . , aK)RK,K, b= (b1, b2, . . . , bk, . . . , bK)T {1, 1}Kand n= (n1, n2, . . . , nn , . . . ,nN)
T RN.S is a N K matrix of the spreading codes, sk is the N 1 spreading code of the kth
user, A is a K K matrix of the received amplitudes, b is a K-length vector of receivedbinary symbols, and finally n is a N-length vector of independently and identically distributed
additive white Gaussian samples with zero-mean and variance 2.
In the following we assume that the K users are partitioned into G groups, where the gth
group consists ofUg users such that: K = U1+U2+ +Ug+ +UG and thus the matrixS can be partitioned as S = (S1, S2, . . . , Sg, . . . , SG ) where Sg = (sg,1, sg,2, . . . , sg,ug , . . . ,sg,Ug )
1/
N, 1/
N
N,Ug. We define: R = STS as the cross-correlation matrix of
the spreading codes, Ri,j = STi Sj asthe(i th, j th) submatrix ofR, and Ag as the gth diagonalsubmatrix of the matrix A. We assume that R and Rg,g (for g = 1, 2, . . . , G) are nonsingular(the spreading codes are assumed to be linearly independent).
The LGPIC detector consists of interference cancellation units arranged in a multistage
structure as shown in Fig. 1. The internal structure of each interference cancellation unit isillustrated in Fig. 2. The vector of decision variables of the (p 1)th stage, gth group yp1,gis first despreaded added to the vectors of decision variables of the other groups to form the
interference due to all users at the (p 1)th stage, that is, Ip1 =G
j=1 Sj yp1,j . Theinterference Ip is subtracted from the received signal r to obtain a purified received signal
(r Ip) where all users exhibit less mutual interference. The vector of decision variablesof the pth stage, gth group yp,g is obtained by despreading the purified signal, multiplying
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A New LGPIC Detector 25
PICU
1
PICU
2
PICU
p
y1,1 y2,1
r
PICU
P
yp,1
yp,2y2,2y1,2
y1,g y2,g yp,g
y1,G y2,G yp,G
yP,1
yP,2
yP,g
yP,G
Fig. 1 Multi-stage structure of the LGPIC detector
the result by a transformation matrix and finally adding the result to the vector of decision
variables of the previous stage, that is:
yp,g = Fg STg
r Ip1+ yp1,g (2)
This process is repeated in a multistage structure as shown in Fig. 1.
Note that for the case of a CDMA multipath fading channel, the structure is the same, only
the effective spreading code Sg is substituted by Sg where Sg = (sg,1, sg,2, . . . , sg,ug , . . . ,sg,Ug ) and sg,ug = sg,ughug . Here hug is the vector of the complex fading coefficients ofthe ugth users channel and is the convolution operator. Moreover, all transpose operations(T) should be replaced by the conjugate operation (H).
In the ensuing, we evaluate the computational complexity of the proposed detector. The
computational complexity is usually evaluated in terms of floating point operations (flops).1
Thus the total computational complexity of a certain detector is usually evaluated in terms
of the number of flops required per decision per user. However, if these operations can be
handled in parallel then the computation time can be greatly reduced. Hence, parallelizable
algorithms are highly desirable due to their ability of reducing the computation time, which
is of paramount importance in real-time implementation.Here, we consider the case of an asynchronous multiapth-fading channel and therefore the
received signal is not processed in a symbol-by-symbol approach due to the asynchronism of
users; instead, a processing window of length W symbols is used. Eventually, it can be shown
1 A floating point operation denotes usually, an addition, subtraction, multiplication or division.
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26 A. Bentrcia, A. Zerguine
yp,1
r
+
T
gS
T
GS
2
TS
1
TS
yp,2
yp,g
yp,Gyp-1,G
yp-1,g
yp-1,2
yp-1,1
1S
2S
gS
GS
F1
Fg
F2
FG
Ip-1
Fig. 2 The pth stage interference cancellation unit of the LGPIC detector
that the computational complexity of the proposed LGPIC structure using the decorrelatordetector as the group-detection scheme is given by:
P W
2G
g Ug +
W N+ max1kK
k+ max
1kK
lkG
g=1
2Ug 1+
2W N+ 2 max1kK
k + 2 max
1kK
lk 1
Gg=1 Ug
+ P (W G 1)
W N+ max1kK
k+ max
1kK
lk
+ 2P W N+ max1kK
k+ max1kK lk+ WGg=1 11U3g + 32 U2g + Ug
+2W
W N+ max1kK
k+ max
1kK
lkG
g=1
Ug2
(3)
where lk is the delay of the lth path (L resolvable paths are considered) of the kth user and
k is the relative delay of the kth user.
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A New LGPIC Detector 27
3 Algebraic Approach to the LGPIC Detector
In this section, we show, using an algebraic approach, that the LGPIC detector is equivalent
to matrix filtering of the received chip-matched signal. This enables the determination of
analytical expressions for the BER and AME of the proposed detector.The vector of decision variables of the pth stage, gth group yp,g in Eq. 2 can be written
in matrix form as:
yp = yp1 + FST
r Syp1
(4)
where F = diag F1, F2, . . . , Fg , . . . , FG. Hence (4) is equivalent to:yp = FSTr FSTSyp1 + yp1
=FSTr
+ I FSTS yp1
= FSTr +
I FSTS
FSTr +
I FSTS
yp2
= FSTr +
I FSTS
FSTr +
I FSTS2
yp2 (5)
= FSTr +
I FSTS
FSTr +
I FSTS2
FSTr +
I FSTS
yp3
= FSTr +
I FSTS
FSTr +
I FSTS2
FSTr +
I FSTS3
yp3
Proceeding in the same way and taking in consideration that y0 = 0, we obtain:
yp =p
i=1
I FSTS
i1FSTr
= GTp r (6)
where Gp can be partitioned as : Gp =
gp,1 gp,2 . . . gp,k . . . gp,K
. Therefore the LGPIC
detector can be described as matrix filtering of the received chip-matched signal vector. Thus,
if the spreading codes and grouping of all users are available, the decision variables of all
users could be obtained without explicitly performing parallel interference cancellation.Using the same approach as in the case of the matched filter detector [6], the BER of the
kth user at the pth stage can be evaluated as:
Pp,k () =1
2K1
all b
bk = 1
Q
gTp,kSAb
gTp,kgp,k
(7)
where Q(.) is the Q
function.
Similarly, the AME [6] for the kth user at the pth stage is given by:
p,k =1
gTp,kgp,kmax2
0, gTp,ksk
Kj=1j=k
aj
ak
gTp,ksj (8)
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28 A. Bentrcia, A. Zerguine
4 Group Detection Schemes
Similar to the case of the linear GSIC detector proposed in [5] and depending on the
transformation matrix Fg, different group detection schemes can be obtained, namely: the
LGMF-PIC, LGDEC-PIC, LGPIC-PIC and finally the LGMMSE-PIC detectors.
4.1 The LGMF-PIC Detector
This is the simplest scheme, and it is obtained by choosing [5]:
Fg = diag
STg Sg
1(9)
which is in fact the conventional linear weighted PIC detector.
4.2 The LGDEC-PIC Detector
For this detector, the linear transformation is given by [5]:
Fg =
STg Sg
1(10)
Note that if the group size is equal to one, we obtain the conventional linear PIC detector.
4.3 The LGMMSE-PIC Detector
For this detector, the linear transformation is given by [5]:
Fg =
STg Sg + 2A2g1
(11)
4.4 The LGPIC-PIC Detector
The linear transformation for this detector is given by [5]:
Fg =NP I C
i=0
I diag
STg Sg
1STg Sg
i(12)
where NP I C is the number of PIC stages used in the group-detection.
5 Convergence Behavior and Conditions of Convergence
Before discussing the convergence behavior of the proposed scheme, let us establish the
connection between the LGPIC detector and the Jacobi/block-Jacobi iterative method [7].
The matrix R can be decomposed into three parts, that is: R = DLLT, where D is blockdiagonal matrix, that is D = diag
R1,1, R2,2, . . . , Rg,g, . . . , RG,G
, and L and LT are the
remaining lower-left and upper-right block triangular parts of R, respectively. On the otherhand, the block-Jacobi iterative method is given by [7]:
yp = yp1 + D1
y Ryp1
(13)
By comparing (4) and (13), it easy to notice that if Fg =
STg Sg
1then the LGPIC
detector is in fact a realization of the block-Jacobi iterative method. On the other hand, if
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A New LGPIC Detector 29
Fg = diag
STg Sg
1then the LGPIC detector is in fact a realization of the Jacobi relaxation
iterative method.
From (6), it easy to show that as the number of stages tends to infinity the vector of decision
variables tend to that of the decorrelator detector, that is,:
limp yp = limp
pi=1
I FSTS
i1FSTr
=
FSTS1
FSTr (14)
=
STS1
F1FSTr
Hence if the matrix F is nonsingular then, (14) is equivalent to:
limp yp =
STS
1 STr (15)which is the decorrelator detector, therefore, if the proposed LGPIC detector converges, it
converges to the decorrelator detector.
To determine the conditions of convergence, we use the following theorem [7]:
Let B be a square matrix such that |max (B)| < 1, then the iteration bp+1 = Bbp + fconverges for any fandb0 where ma x is the largest eigenvalue of the matrix B.
By rewriting Eq. 4 as:
yp = FST
r + I FSTS yp1 (16)and using the above theorem, it is easy to show that the iteration matrix B of the proposed
detector is given by:
B =
I FSTS
(17)
Hence, the proposed scheme converges if and only if:
0 < max
FSTS
< 2 (18)
If the above inequality is not always satisfied then the LPIC and LGPIC schemes will diverge.However, as in the case of the weighted LPIC scheme [8,9], a weighted LGPIC scheme can
be used here as well to ensure convergence.
6 Simulation Results
In this section, the convergence behavior of the proposed LGPIC multiuser detector is sim-
ulated and the results obtained are detailed. Two different scenarios are considered: a syn-
chronous CDMA AWGN channel and an asynchronous CDMA multipath Rayleigh fadingchannel. The simulation parameters are depicted in Table 1.
Figure 3 depicts the average BER (average of all users) versus the number of linear
group-wise PIC stages. Four different detection schemes are considered. For the LGPIC-PIC
detector, a 2-stage PIC detector is used. It is easy to notice that the LGDEC-PIC converges
faster than the other group-detection schemes. As can be seen from this figure, the LGDEC-
PIC needs only 7 stages whereas the LGPIC-PIC detector needs 8 stages, the LGMMSE-PIC
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30 A. Bentrcia, A. Zerguine
Table 1 Simulation parametersSynchronous CDMA AWGN
channel
Asynchronous CDMA multi-
path Rayleigh fading channel
K = 20, N = 31 (Gold codes),S N R = 4 dB, perfect powercontrol.
K = 10, N = 31 (Goldcodes), S N R = 4dB, powercontrol. Vehicular A outdoorChannel power delay profile
for WCDMA [10] is used per-
fect
2 4 6 8 10 12 14 16 18 20
10-1.6
10-1.59
10-1.58
10-1.57
10-1.56
Number of LGPIC stages
AverageBER
Matched filter detector
Decorrelator detector
LMMSE detector
LGDEC-PIC detectorLGMF-PIC detector
LGMMSE-PIC detector
LGPIC-PIC detector
Fig. 3 Convergence behavior of the LGPIC detector
detector needs 10 stages and the LGMF-PIC detector needs 11 stages; however, the linear
LGMF-PIC and the LGMMSE-PIC detectors achieve the lowest average BER level among
all detection schemes. Moreover, it is important to notice that lower average BER levels areachieved prior to convergence, this is more noticeable for highly loaded systems and it has
also been reported in [3].
The effect of grouping is analyzed and depicted in Figs. 47. It can be seen that while
convergence speed of the LGDEC-PIC, the LGMMSE-PIC and the LGPIC-PIC detector
increases with decreasing number of groups, the convergence speed of the LGMF-PIC detec-
tor is independent of grouping and is constant for any grouping. This is because the LGMF-
PIC detector is equivalent to the conventional LPIC detector and hence the grouping in this
case is G = K. However, the average BER difference between different groupings is small
and of theoretical importance only.In Fig. 8, the convergence behavior of different LGPIC detection schemes is evaluated in
an asynchronous CDMA multipath fading channel. Ten users are divided into two equally
sized groups. In addition, a two-stage PIC detector is used for the group-detection in the
LGPIC-PIC detector. The simulation parameters are depicted in Table 1.
This Figure shows that while the LGMF-PIC, which is equivalent to the LPIC detec-
tor, and the LGPIC-PIC detectors are divergent, the LGDEC-PIC and the LGMMSE-PIC
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A New LGPIC Detector 31
2 4 6 8 10 12 14 16 18 20
10-1.6
10-1.59
10-1.58
10-1.57
10-1.56
Number of LGDEC-PIC stages
AverageBER
Matched filter detector
Decorrelator detectorLMMSE detector
LGDEC-PIC detector (G=2,Ug=10 for all g)
LGDEC-PIC detector (G=10,Ug=2 for all g)
Fig. 4 Convergence behavior of the LGDEC-PIC detector for G = 2 and G = 10
2 4 6 8 10 12 14 16 18 20
10-1.6
10-1.59
10-1.58
10-1.57
10-1.56
Number of LGMMSE-PIC stages
AverageBER
Matched filter detector
Decorrelator detector
LMMSE detector
LGMMSE-PIC detector (G=2,Ug=10 for all g)
LGMMSE-PIC detector (G=10,Ug=2 for all g)
Fig. 5 Convergence behavior of the LGMMSE-PIC detector for G = 2 and G = 10
detectors are convergent and they need few stages to converge to the decorrelator detectors
performance. This indicates that while the LPIC detector is divergent, its counterpart LGPIC,
particularly the LGDEC-PIC and the LGMMSE-PIC detectors is convergent. This shows thatgroup-detection can be used to stabilize a divergent LPIC detector. Hence, it is in fact an alter-
native to the conventional way of stabilizing a divergent LPIC detector where a weighting
factor can be used for this propose.
Finally, Fig. 9 shows the convergence behavior of both the LGDEC-PIC and LGDEC-
SIC detector for two different groupings, namely for G = 2 and G = 10. It is obvious fromthis figure that the LGDEC-SIC detector converges relatively faster than the LGDEC-PIC
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2 4 6 8 10 12 14 16 18 20
10-1.6
10-1.59
10-1.58
10-1.57
10-1.56
Number of LGPIC-PIC stages
AverageBER
Matched filter detector
Decorrelator detectorLMMSE detector
LGPIC-PIC detector (G=2,Ug=10 for all g)
LGPIC-PIC detector (G=10,Ug=2 for all g)
Fig. 6 Convergence behavior of the LGPIC-PIC detector for G = 2 and G = 10
2 4 6 8 10 12 14 16 18 20
10-1.6
10-1.59
10-1.58
10-1.57
10-1.56
Number of LGMF-PIC stages
AverageBER
Matched filter detector
Decorrelator detector
LMMSE detector
LGMF-PIC detector (G=2,Ug=10 for all g)
LGMF-PIC detector (G=2,Ug=10 for all g)
Fig. 7 Convergence behavior of the LGMF-PIC detector for G = 2 and G = 10
detector, i.e., for G = 2, the LGDEC-SIC detector needs only 5 stages while LGDEC-PICdetector needs around 7 stages to converge to the decorrelators performance. This is because
the LGDEC-SIC detector uses the most updated estimates of the decision variables at the
expense of more detection delay.
7 Conclusion
In this work, a new linear group-wise PIC multiuser detector is developed. The proposed
scheme exhibits inherent parallelism compared to the group-wise SIC detector, which
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A New LGPIC Detector 33
5 10 15 20 25
10-2
10-1
100
Number of LGPIC stages
AverageBER
Matched filter detector
Decorrelator detector
LMMSE detector
LGDEC-PIC detector
LGPIC-PIC detectorLGMMSE-PIC detector
LGMF-PIC detector
Fig. 8 Convergence of different LGPIC detection schemes in an asynchronous CDMA multipath Rayleigh
fading channel
2 4 6 8 10 12 14 16 18 20
10-1.6
10-1.59
10-1.58
10-1.57
10-1.56
10-1.55
Number of LGSIC/LGPIC stages
AverageBER
Matched filter etector
Decorrelator detector
LMMSE detector
LGDEC-PIC detector (G=2,Ug=10 for all g)
LGDEC-PIC detector (G=10,Ug=2 for all g)
LGDEC-SIC detector (G=2,Ug=10 for all g)
LGDEC-SIC detector (G=10,Ug=2 for all g)
Fig. 9 Convergence behavior of both the LGDEC-PIC and the LGDEC-SIC detectors for G = 2 and G = 10
reduces the computation time. Depending on the group-detection scheme, four different
group-detection PIC structures were derived. First, we used a matrix algebraic approach todescribe the proposed structure. This approach enabled us to derive analytical expressions
for both the BER and AME for the proposed detector. Second, we proved that the proposed
structure converges to the decorrelator detector if it converges. Moreover, we showed that
the proposed structure, like the conventional linear PIC detector, is not always convergent
and that a weighting factor can be used to stabilize the scheme. Finally, extensive simulations
were performed to uncover different the convergence behavior of the proposed structure.
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References
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IEEE Transaction on Communication, 49(10), 18241834.3. Rasmussen, L. K., Lim, T. J., & Johansson, A. (2000). A matrix-algebraic approach to successive inter-
ference cancellation in CDMA. IEEE Transaction on Communication, 48(1), 145151.
4. Guo, D., Rasmussen, L. K., Sun, S., & Lim, T. J. (2000). A matrix-algebraic approach to linear parallel
interference cancellation in CDMA. IEEE Transaction on Communication, 48(1), 152161.
5. Johansson, A. L., & Rasmussen, L. K. (1998). Linear group-wise successive interference cancellation
in CDMA. IEEE 5th Proceedings of the International Symposium on Spread Spectrum Techniques and
Applications, 1(24), 121126.
6. Verdu, S. (1998). Multi-user Detection. Cambridge University Press.
7. Saad, Y. (2003). Iterative Methods for Sparse Linear Systems, (2nd ed.). publisher: SIAM
8. Rasmussen, L. K., & Oppermann, I. J. (2001). Convergence behaviour of linear parallel cancellation in
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3152).9. Rasmussen, L. K., & Oppermann, I. J. (2003). Ping-pong effects in linearparallel interference cancellation
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Author Biographies
A. Bentrcia was born on May 9, 1976 in Arris, Algeria. He obtained his
Ingnieur dtat certificate from Batna University, Algeria. He obtained
his M.Sc. degree in Electrical Engineering from King Fahd University of
Petroleum and Minerals (KFUPM) in June 2003. Currently, he is a lec-turer at KFUPM and an active member of the TRL laboratory at the same
university. He published more than 18 journal and conference papers.
His main field of research is wireless communication and particularly
multi-user detection in CDMA systems.
A. Zerguine received the B.Sc. degree from Case Western Reserve
University, Cleveland, Ohio, in 1981, the M.Sc. degree from King Fahd
University of Petroleum & Minerals (KFUPM), Dhahran, Saudi Arabia,
in 1990, and the Ph.d. degree from Loughborough University, Lough-
borough, UK, in 1996 all in Electrical Engineering. From 1981 to 1987,
he was working with different Algerian state owned companies. During
the period from 1987 to 1990, he was a research and teaching assistant,
Electrical Engineering Department, KFUPM. Dr. Zerguine is presently
an Associate Professor, Electrical Engineering Department, KFUPM,
working in the areas of Signal Processing and Communications. Dr.
Zerguine is a Senior Member of IEEE. Dr. Zerguine research interests
include Signal Processing for Communications, Adaptive Filtering, Neu-ral Networks, Multiuser Detection, and Interference Cancellation.
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