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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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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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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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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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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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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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


Decorrelator detectorLMMSE detector

LGDEC-PIC detector (G=2,Ug=10 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



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


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



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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5 10 15 20 25

10-2

10-1

100

Number of LGPIC stages

AverageBER



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


LMMSE detector



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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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

CDMA. IEEE Global Telecommunication Conference, San Antonio, TX, December 2001. (pp. 3148

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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A New Linear Group-Wise Parallel Interference