Performance Analysis of Multi-cell Systems with Cochannel Interferences and Multiuser Diversity

Yingquan Zou 1,2, Rui Zhao 3 , Luxi Yang 1 Member, IEEE1 School of Information Science and Engineering, Southeast University, Nanjing, 210096, China

2 School of Electric and Information Engineering, Nanjing University of Information Engineering, Nanjing, 210044, China 3 School of Information Science and Engineering, Huaqiao University, Quanzhou, 362021, China

Email: {zouyingquan0309,rzhao.seu}@gmail.com, lxyang@seu.edu.cn

Abstract—The performance of the system with maximal ratio combining and multiuser scheduling is studied in the multiuser uplink in the presence of unequal power cochannel interferences from other cells. The channels of useful and interference signals are assumed to be Nakagami-m and Rayleigh fading. The closed form expressions for outage probability and average symbol error ratio (ASER) of system are derived by using performance analysis method based on probability density function. Simulation results show that, the analytical curves of the derived expressions match well with the numerical simulations, and the increase of the number of receive antennas and users are both of benefit to the improvement of system performance due to multi-antenna and multiuser diversity, and the presence of a predominant interferer deteriorates the outage performance.

Keywords—Cochannel interference (CCI), maximum ratio combining (MRC), multiuser diversity, outage probability, average symbol error ratio (ASER)

I. INTRODUCTION

The signal transmission in wireless communication system is usually subjected to additive channel noise, multipath fading and cochannel interference (CCI), and by using some diversity techniques such as adaptive array processing, we can deal with these impairments. Optimum combining (OC) can maximize the output signal-to-interference-plus-noise ratio (SINR) at the receiver, through designing the combining vectors considering the effect of the multipath fading of expected signals and CCI, but it requires the channel information of all CCIs. In practice, an effective reception way is using maximum ratio combing (MRC), which can acquire high receive signal-to-noise ratio. Recently some authors have analyzed the performance of MRC system in the presence of additive white noise and CCI [1]~[3]. The effect of OC and MRC on the system performance was analyzed in [1]. The performance of MRC system in the presence of CCI with arbitrary power in Nakagami-m channel was analyzed in [2]. And the effect of the channel estimation error of expected signal on outage probability and average symbol error ration is analyzed in [3]. The above studies are all based on SIMO-MRC system. For MIMO-MRC multi-cell system, the outage probability performance of uplink

transmission is analyzed based on more practical different powers of intra-cell interferences and inter-cell interferences in [5], and the conclusion is the multiple antennas at the receiver can improve the system performance and the transmit diversity can not overcome the interference of users within the same cell. Furthermore, the symbol error ratio (SER) of this system was analyzed based on the above study in [6]. In the communication link of multi-point to point, one of multiple users communicates with the base station, and the links between each user and base station are independent with each other. The channel in deep fading for a certain user may be the strong gain channel for other users. The sum throughput of system can be maximized by allocating the limited wireless resources to the users with good channel conditions, known as multiuser diversity, which can improve the system performance dramatically [7]. Till now, we have not seen the study on the performance of MRC system in the presence of CCI with multiuser scheduling. We consider the multi-point to point (MPP) system of multiuser uplink in multi-cell communication networks in the presence of multiple CCI, and each user is equipped with single antenna and the base station is equipped with multiple antennas. The base station selects one user from the intra-cell users through multiuser scheduling, and receives the signal using MRC. Since the transmit powers of inter-cell users are different, this paper studies the interference probability distribution characteristics in the presence of different inter-cell and intra-cell interference power. In addition, the channel of expected signals in most studies is Rayleigh distribution, and we extend it to Nakagami-m distribution, and the interference channel is still the common Rayleigh channel, hence the derived expressions can adapt to more communication scenarios. MPP multiuser communication system is different from single-user system, and the performance analysis of multiuser diversity is more complex and is important to the design of multi-antenna system. Recent studies show that multiuser diversity gain can improve the system outage probability dramatically [7]. We investigate the effect of multiuser diversity on the system outage probability and average symbol error ratio (ASER), considering unequal power CCI in multi-cell environment. We also derive the closed-form expression based on the performance analysis method of probability

This work was supported by the National Basic Research Program ofChina under Grant 2007CB310603, the National Natural ScienceFoundation of China under Grant 60902012, the National Science andTechnology Major Project of China under Grant 2009ZX03003-004, the Ph.D. Programs Foundation of Ministry of Education of China under Grant20090092120013. 978-1-4244-7555-1/10/$26.00 ©2010 IEEE

density function (PDF)[8][9]. Simulation results show that the closed-form expressions of outage probability and ASER can match with the Mont-Carlo simulation results, and the increase of the antenna number of base station and user number will improve the performance of outage probability and ASER. Furthermore, large difference of interference power will also affect the improvement of system performance.

II. SYSTEM MODEL

Consider the uplink communication scenario with Ksingle-antenna users and one base station with rNreceive antennas. The base station selects one user as the expected user from K users in the presence of L intra-cell and inter-cell cochannel interferences. Assume the k -th user is selected and transmits signal to the base station, then the receive signal vector kr of base station can be expressed as

0 01

L

k k i i ii

P s P s=

= +�r h h (1)

where 0s is the desired signal at the base station, and isis the interfering signal, and both of them are independent zero mean with unit power, i.e., { }2

0 1s =� , { }2 1is =� . 0P

and iP are the powers of desired signal and interfering signal. kh is the selected 1rN × channel vector from user to base station,

1, ,r

T

k Nh h� �= � �h � , and ih the 1rN ×

channel vector from the i -th interfering source to user, which is Rayleigh fading. Since the system is interference limited, and the strength of interfering signal is usually much larger than that of noise, we can neglect the effect of additive white Gaussian noise on the performance. The base station design the receive vector by employing MRC to maximize receive SNR, then the signal processed by receive vector is

20 0

1

LH

k k i k i ii

z P s P s=

= +�h h h (2)

The receive signal-to-interference ratio at the base station is

4 20

2 2

1 1

k kk L L

Hi k i i i

i i

P

P wγ

α= =

= =� �

h h

h h

(3)

where 0i iP Pα = and Hi k i kw = h h h . Since the

interference channels are Rayleigh fading, iw follows zero mean unit variance independent identical complex Gaussian distribution and is independent to kh [1].

22 iw follows 2χ distribution [1], i.e.,

( )2 22 2iw χ� (4)

So the SIR of user can expressed as the ratio of two independent variable x and y

kxy

γ = (5)

where 2 2

12 2 rN

k iix h

== = �h is the power of expected

signal, which is equivalent to the SNR of SIMO-MRC system without interference. 2

12 L

i iiy wα

== � is the power

of interfering signal. When the channel of expected signal kh is Nakagami-m fading channel, x is Gamma distribution variable [4], i.e., ( ),2rx G mN m� , and the PDF is give by

( ) ( )1 2

2

r r

mxmN mN

xr

m x ep xmN

−−� �= Γ� � (6)

where ( )Γ ⋅ is Gamma function, and m is Nakagami fading factor. In the following, we discuss the distribution of y . In an uplink cellular environment, there will often be multiple interferers from users within the same cell (intra-cell interference), as well as from other cell users (inter-cell interference). Intra-cell interferers are power controlled by the same base station as the desired user, therefore, intra-cell interferers have equal powers. In the case of inter-cell interferers, however, the power control adjusts their average power at another base station. Hence, inter-cell interferers have unequal powers[5][6]. For analyzing the performance of SIMO-MRC system with multiple interferers, therefore, it is reasonable to consider the case that the interferers can be divided, according to their power, into groups with each having the same power. Hence, for generality, we assume that L interferers can be divided into p groups, each of them having the same power iP . Each group contains it , 1,2, ,i p= �

1

pii

t L=

=� .From (4) and the multiply characteristic of independent Gamma variable, we get ( )2

12p

i iiy tα χ

=�� with the characteristic function as

( ) { } ( )1

1 2 ip

tzyy i

iz e zφ α −

=

= = −∏� (7)

Then (7) can be further expressed as

( )( )11 1 2

i

i

tpij

y ji j

zz α

βφ

= =

=−

�� (8)

The coefficient ijβ can be determined by the following as

( ) ( )1

2

1 1lim! 2

ii

ii

tt j

ij yt jzi i

d z zt j dzα

β φα

−

−→

�� �� �� �= −� �� � − � �� �� �� �� �

(9)

By using polynomial theorem, (9) can be simplified as

( )( )

( )( ), 1

1 212 1

k

i k kk

i

qpk k k

ij t t qi j k ki k i

t qq α

α

αβ

α +Θ =

≠

+ −� �= − � � −

�∏ (10)

where ( ),i jΘ is the set of p-tuples with nonnegative integers such that

( ) ( ){ }1 2 0 1, , , , : , 0, p

p k i k iki j q q q q N q q t j

=Θ = ∈ = = −�� [10], 0N

denotes nonnegative integral set. Performing inverse Fourier transform on (8), the PDF of y is given by

( ) ( )( )

21

1 1

11 !

ii

j ytpij j

yi j

p y y ej

αβ −−

= =

−=

−�� (11)

Combining (6) and (11), the PDF of SIR kγ in (5) can be derived as follows

( ) ( )0

,k k xy kp yp y y dyγ γ γ

∞= �

( )( )( )

1121

01 1

12 1 !

k ir irir

j mmN tmN p yij mN jk

i jr

m y e dymN j

γ ααβγ +− −∞ + −

= =

−� �= Γ −� ��� � (12)

( )

( ) ( ) ( )( ) ( )

1

1 1

1 22 1 1 !

rr ir

r

j mN jmN tmN pij i rk

mN ji jr k i

mN jmmN m j

β αγγ α

+−

+= =

− Γ +� �= Γ� � + −�� (13)

where (12) can be derived from the independence of random variable x and y , and (13) can be derived from (3.381.4) in [11]. The cumulative distribution function (CDF) of kγ can be derived as follows

( ) Prk k k

xPyγ γ γ� �= ≤ � �

( ) ( )0 k

x yxp x dx p y dy

γ

∞ ∞= � �

( )( )( )

1 221

01 1

12 1 !

r iri

k

mxj ymN tmN p

ij j

xi jr

m x e dx y e dymN j

α

γ

β−− −∞ ∞ −

= =

−� �= Γ −� ���� �

( )( ) ( ) ( ) 1 2

01 1

22 ,

1 ! 2

r ir

mN mxtpjij mN

ii jr k i

m xx e j dxmN j

βα

γ α∞ −−

= =

� �= − Γ Γ − � �

�� � (14)

( )( ) ( ) ( ) ( ) ( )

21 1

1

2 22

1 ! 12 2

1, ; 1;1

r i

r

mN jtpjij k i r

i mN ji jr

rk i

k ir r

k i

m mN jmN j mmN

mF mN j mNm

β γ αα

γ α

γ αγ α

−

+= =

Γ += −

Γ − � �+

� �� �

⋅ + + +� �

��

(15)

( ) ( )( )( ) ( )( )

1

1 1 0

21 1

rri

r

mNmNtp jj r i k

ij i mN qi j q r i k

m mN qq mN m

α γβ α

α γ

−

+= = =

Γ += −

à + à +��� (16)

where (14) and (15) is derived from (3.381.3) and (6.455.1) in [11] respectively, and ( )2 1F ⋅ is the hypergeometric function [11], which can be expressed as limited series expression, thus we can get (16). When

1m = , Nakagami-m channel can be degraded as Rayleigh channel. When all Rayleigh cochannel

interferences have the same power, i.e., iα α= and 1p = , the CDF of kγ can be simplified as

( ) ( )( )

1

0

11

r

k r

NLr k

k N qq k

N qP

qγαγ

γαγ

−

+=

+ −� �= +� �� (17)

III. OUTAGE PROBABILITY PERFORMANCE ANALYSIS

The outage probability is an important performance measurement in wireless communication systems over fading channels. For good codes and long block lengths, outage probability gives an approximation of the frame error rate. We consider the outage probability that corresponds to the probability that the SINR falls below the specific SINR threshold, i.e.,

( ) ( ) ( )0

Pr th

out th thP p dγ

γγ γ γ γ γ≤ = �� .

1,2, ,maxs kk K

γ γ=

=�

(18)

From probability theory, the CDF and PDF of system effective SIR sγ with K random variables { } 1,2, ,k k Kγ = �

can be given by, respectively

( ) ( )s k

KP Pγ γγ γ� �= � �

( ) ( )( )( ) ( )( )

1

1 1 02

1 1

rri

r

KmNmNtp jj r i

ij i mN qi j q r i

m mN qq mN m

α γβ α

α γ

−

+= = =

� �Γ += −� �

à + à +� �� ����

(19)

( )( )( ) ( ) ( ) 1

s

s k k

Kd Pp Kp P

dγ

γ γ γ

γγ γ γ

γ−

� �= = � �

( )

( ) ( ) ( )( ) ( )

1

1 1

1 22 1 1 !

rr ir

r

j mN jmN tmN pij i r

mN ji jr i

mN jmKmN m j

β αγγα

+−

+= =

− Γ +� �= Γ� � + −��

( ) ( )( )( ) ( )( )

11

1 1 0

21 1

rri

r

KmNmNtp jj r i

ij i mN qi j q r i

m mN qq mN m

α γβ α

α γ

−−

+= = =

� �Γ +⋅ −� �

à + à +� �� ���� (20)

From (19), the outage probability closed-form expression of MPP multiuser selection MRC system in Nakagami/Rayleigh channel is

( ) ( ) ( ) ( )( )( ) ( )( )

1

1 1 02

1 1

rri

s r

KmNmNtp jj r i th

out th th ij i mN qi j q r i th

m mN qP P

q mN mγα γ

γ γ β αα γ

−

+= = =

� �Γ += = −� �

à + à +� �� ����

(21)

The following gives the system outage probability closed-form expressions under some special circumstances: 1) when iα α= and 1p = , i.e., all interferences have the same power, (21) can be simplified as

( ) ( )( )( ) ( )( )

1

0 1 1

rr

r

KmNmNLr i th

out th mN qq r i th

m mN qP

q mN mα γ

γα γ

−

+=

� �Γ += � �

à + à +� �� �� (22)

2) when 1m = , iα α= and 1p = , Nakagami channel is degraded into Rayleigh channel, thus the CDF and PDF of the effective SIR of MPP multiuser selection MRC system under Rayleigh/Rayleigh channel is simplified as

( ) ( )( )

1

0

11

r

s r

KNLr

N qq

N qP

qγαγ

γαγ

−

+=

� �+ −� �= � � +� �� �� ��

(23)

( )( )

( )( )

11

1

1 0

1 1

1 1

r

r r

s r r

KNL Lr rN N

N j N qj q

N j N qjp Kj qγ

αγγ α γ

αγ αγ

−−

−+ +

= =

� � � �+ − + −� � � �= ⋅ � � + +� � � �� �� � � �

� �

(24)

IV. AVERAGE SYMBOL ERROR RATIO ANALYSIS

In this section, we derive the ASER by averaging the instantaneous SER over instantaneous SIR, i.e.,

( ){ } ( ) ( )0e e eP P P p dγγ γ γ γ∞

= = �� [12]. In the presence of a

large number of interfering signals, which is typical in a wireless environment, we make the Gaussian assumption with respect to the interfering signal and express the instantaneous SER conditional on the SIR and evaluate the ASER over the PDF of the SIR. The instantaneous SER using Gaussian Q -function is valid for the Gaussian assumption. The conventional form of the central limit theorem does not apply here since the interference term in (2) consists of independent, but not identically distributed term. Cramer’s central limit theorem, however, can be applied here. According to the theorem, the sum

i NX X+ +� of a large number of independent variables is approximately normally distributed if the following conditions are satisfied: 1) every component iX has a zero mean; 2) every component has a finite variance { }22

i iXσ = � ; 3)

( )2 21 0i Nσ σ σ+ + →� and 2 2

1 Nσ σ+ + → ∞� as

N → ∞ [13]. It can be easily shown that the interference component in (2) satisfies these conditions from the properties of the interfering symbol assumed above, hence, it can be assumed Gaussian distribution. Previous works for the calculation of the ASER are also based on the Gaussian assumption for the interference by applying the central limit theorem[1]. For commonly used modulation format, the ASER over a SIR γ can be given by

( ){ }1 22eP Qγ β β γ= � (25)

where ( )Q ⋅ is Gaussian Q function with definition

( ) ( ) ( )21 2 exp 2Q x dxα

α π∞

= −� .1β and

2β are modulation

constant, modulation formats include BPSK1 1β = ,

2 1β = , BFSK with orthogonal signaling

1 1β = ,2 0.5β = , and M -PAM ( )1 2 1M Mβ = − ,

( )22 3 1Mβ = − [12] For other modulation format, this

expression is asymptotic expression, e.g., M-PSK( 1 2β = ,( )2

2 sin Mβ π= )

We can express (22) as the integral form of CDF of system effective SIR [14]

( )2

1 2

02 s

x

eeP P x dx

x

β

γβ β

π

−∞= � (26)

When 1m = , 1L = , iα α= , and 1p = , by substituting (23) into (26), the ASER of MPP multiuser selection MRC system with only one interfering source under Rayleigh/Rayleigh can be expressed as

( )2

12

1 2

02 1

rr

r

KNKN x

e KNe xP dx

x

ββ β απ α

−−∞=

+�

211 22

21 222

r

r

KNr KNKN e D

βαβ β

απ−

−

� �� �= à + � � � �

(27)

where (26) can be derived from (3.383.6) in [11], and ( )pD z is the parabolic cvlinder functions with definition:

( )1 2

1 24 21 1,4 2 4

22

p

p pzD z W z

+ −

+ −

� �= � �

(see (9.240) in [11]), and

( ),W zλ μ is Whittaker function.

V. SIMULATION RESULTS AND ANALYSIS

In this section, we verify the derived closed-form expression through numerical simulations, and investigate the effect of user number K , antenna number

rN , and interference power iP and Nakagami parameter m on the performance of system outage probability and ASER. First of all, we simulate the system outage probability performance for different receive antenna number rN of base station. Assume the number of cochannel interferences is 6, and interference sum power is

1

LI ii

P P=

=� with same iP ( 1, ,i L= � ), i.e.,

1 6α α α= = =� . Nakagami parameter 1m = , user number 2K = , and the receive signal-to-interference ratio threshold of base station 12th dBγ = . The signal-to-interference ratio is defined as 0 ISIR P P= .

0 5 10 15 20 25

10-4

10-3

10-2

10-1

100

SIR (dB)

Out

age

prob

abili

ty

analytical,Nr=2analytical,Nr=4analytical,Nr=6simulation,Nr=2simulation,Nr=4simulation,Nr=6

Fig. 1. Comparison of system outage probability for different receive antenna number rN

In Fig. 1, the analytical curves match well with the Mont-Carlo numerical simulation curves under three kinds of antenna number, which verifies the correctness of the derived outage probability expression (given in (21)) of MPP multiuser selection MRC system. The system outage probability performance improves with the increase of antenna number, which is caused by the increase of antenna diversity gain.

Next, we consider another condition of interference power: iP ( 1, ,i L= � ) is not all same,

1 5α α α= = =� , 6 5α α= , i.e., the sixth interference power is much larger. Other simulation parameters are the same with the above simulation. We compare the system performance for same interference power and different interference power.

0 5 10 15 20 25

10-4

10-3

10-2

10-1

100

SIR (dB)

Out

age

prob

abili

ty

equal interference powerunequal interference power

Nr=2

Nr=4

Nr=6

Fig. 2. Comparison of system outage probability for different interference power iP

Fig. 2 shows that, the system outage probability performance deteriorates for different interference power compared with same interference power for three conditions of antenna number. To investigate the effect of user number on system outage probability, we simulate the outage probability changes with system SIR threshold as user number is 2, 6, 10 respectively. 2m = and 3rN = .

0 5 10 15 20 25

10-4

10-3

10-2

10-1

100

SIRth (dB)

Out

age

prob

abili

ty

analytical,K=2analytical,K=6analytical,K=10simulation,K=2simulation,K=6simulation,K=10

Fig. 3. Comparison of system outage probability for different user number K

In Fig. 3, system outage probability improves with the increase of user number due to multiuser diversity gain. In addition, the analytical curves have a good match with the numerical curves, which verifies the correctness of (21).

Fig. 4 simulates the effect of different receive antenna number on system ASER performance. Simulation parameters: same power CCI, the number of CCI is 6,

2m = , 2K = , modulation format is 4PAM( 1 3 2β = ,

2 1 5β = ). In Fig. 4, ASER performance improves with the increase of receive antenna number rN , which indicates that multiple antennas can bring to diversity gain. Fig. 5 simulates the effect of different user number on system ASER performance, when 2m = , 2rN = ,and BPSK modulation is applied( 1 1β = , 2 1β = ). In Fig. 5, ASER performance improves with the increase of user number. From Fig. 4 and Fig. 5, for three conditions of antenna number, analytical curves match well with numerical simulations, thus verifying the correctness of (27).

0 5 10 15 20 2510

-6

10-4

10-2

100

SIR (dB)

Ave

rage

Sym

bol E

rror

Rat

io

analytical,Nr=2analytical,Nr=4analytical,Nr=6simulation,Nr=2simulation,Nr=4simulation,Nr=6

Fig.4. Comparison of system ASER performance for different receive antenna number

rN

0 5 10 15 20 2510

-6

10-4

10-2

100

SIR (dB)

Ave

rage

Sym

bol E

rror

Rat

io

analytical,K=2analytical,K=6analytical,K=10simulation,K=2simulation,K=6simulation,K=10

Fig. 5. Comparison of system ASER performance for different user number K

VI. CONCLUSIONS

This paper considered the uplink communication of multiple base stations and multiple users in the presence of unequal power CCI, and the channels of expected

signal and interfering signal follow Nakagami-m and Rayleigh distribution respectively. We analyzed the performance of MRC system with multiuser scheduling, and derived the system outage probability and ASER closed-form expressions by using PDF based performance analysis method. Simulation results verified the correctness of derived closed-form expressions, and the increase of receive antenna and user number will improve the system outage probability and ASER performance, and show extinct multi-antenna and multi-user gain. Simulation results also indicated that the performance of the system with same interference power is superior to that with different interference power.

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