# Performance Analysis of CDMA WLL Systems with  Imperfect Power Control and Imperfect

Authors: Sami A. El-Dolil
Journal: International Journal of Vehicular Technology
 (2008)

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## Abstract

Wireless local loop (WLL) provides reliable, flexible, and economical access to the local telephone service using radio technology in the place of traditional wireline. In this paper, an analytical model is derived to evaluate the effect of both imperfect power control and imperfect sectorization on the performance of code division multiple access (CDMA) WLL systems. The results show that the capacity degradation, due to imperfect power control, is about 25.8% and 11.5% for single cell and multiple cell systems, respectively. Increasing the overlapping angle from0° to 5°
causes the capacity gain to decrease from 6 to 5.53, while the corresponding sectorization efficiency drops from 100% to 92.3%.


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## Body

## 1. Introduction

Wireless local loop (WLL) is  a system that connects
subscribers to the  public switched telephone network (PSTN) using radio signals
as a substitute for wireline for all or part of the connection between the
subscribers and the switch. It is believed to be a fast and cost-effective mean
to  provide local phone service in rural
areas and third world countries [1]. Since WLL is a fixed radio communication
system; narrow-beam antennas can be employed at both the base station (BS) and
subscriber's side so that the propagation between BS and subscriber's equipment
is very close to free space propagation. This gives many inherent advantages to
the WLL system over the traditional cellular systems, such as wider coverage
area, reduced interference, higher capacity, no fast fading, and no handoff
[2, 3].CDMA technology has
the potential to provide a significant improvement in the capacity of cellular
radio systems compared with FDMA and TDMA systems [4]. However, this
improvement is dependent upon the effectiveness of the power control system,
especially on the reverse link. In the absence of power control, a BS would
receive a much stronger signal from a subscriber unit that is geographically
close to it than from a subscriber unit that is farther away. This is the
so-called near-far problem [5, 6].This paper presents a
theoretical model to evaluate the reverse-link capacity of CDMA WLL systems in
terms of outage probability, taking into account the power control error.Sectorization in
cellular CDMA systems increases the capacity in proportion to the number of
sectors per cell. In practice, the antenna patterns do not fit the sector area
perfectly, and there are overlapping between sectors, which causes additional
interference on both the reverse and forward link [7]. The imperfect
sectorization effect on the performance of CDMA WLL systems is also considered.


## 2. Effect of Imperfect Power Control in Single Cell CDMA WLL Systems

Consider a CDMA WLL single cell system
consists ofN subscriber units transmitting to a BS receiver on the
reverse-link. A simplified CDMA transmitter is shown in Figure 1.Figure 1
A simplified CDMA transmitter diagram.The signal transmitted from
theith user to its BS is given by [5]
(1)Si(t)=2Aibi(t)ci(t)cos(w1t+θi),where Ai is the transmitted power of the ith user, bi(t) is the
data sequence of the ith user, where each bit has an amplitude of ±1 and
a duration of Tb, ci(t) is the spreading
code sequence of the ith user and each of the M chips per code
has a duration Tc, wi is the reverse
link-carrier frequency, and θi is the   random phase of the ith
user carrier.The received signal at the BS receiverRup(t) consists of the following: interference from other users in the cell which
called intracell interference, the receiver noise n(t), and the received
signal from the desired user.From Figure2, Rup(t) is given by
(2)Rup(t)=∑i=0N−1aiSi(t−τi)+n(t),where ai represents the path loss of
the ith user, τi is the random delay of the ith
user signal at the receiver, and n(t) is the additive white Gaussian
noise (AWGN) of the receiver.Figure 2
Channel and BS receiver block diagram.The signal at the output of the matched filter is
given by(3)Z(Tb)=1Tb∫τ0Tb+τ0Rup(t)2cos(w1t+θ)ci(t−τi)dt,where θ is the
carrier phase angle in the receiver as shown in Figure 2.The intracell interference
at the output of the matched filter is given by(4)Zint(t)=∑i=1N−1Zi(Tb).


### 2.1. Perfect Power Control

To reduce the near-far
problem, as well as the interference from other users and hence to increase the
capacity of CDMA WLL system, it is important to apply a power control on the
reverse link so that the received power from each user at the BS is controlled
to be the constant target power,S, [5] where
(5)ai2Ai=Sfori=0,1,…,N−1.
The total noise power is
the sum of the intracell interference power and the AWGN power of the receiver.The AWGN power at the
output of the matched filter is given by [5, 7] as
(6)η=NoRb=(NoTc)TcRb=NoWGp,where Rb=1/Tb is
the bitrate of the message sequence bi(t), W=1/Tc is the chiprate, and N0W is the noise power at the
receiver input.Thus, after despreading,
the noise powerη is the input noise power decreased by the processing
gain Gp=Tb/Tc. The
intracell interference power is given by [5]
(7)Iint=var(Zint(t))=1Gp∑i=1N−1ai2Ai,where var(Zint(t)) is the variance of intracell interference at the output of the matched
filter.By applying voice activity
detection, users transmit only when speech signal is present. We introduce a
voice activity variable (VAF)vi which equals 1 with
probability of μ, and equals 0 with probability of 1−μ. By
multiplying (7) by vi and using (5) so,
(8)IintS=1Gp∑i=1N−1viai2AiS=1Gp∑i=1N−1vi,the intracell interference
to signal power ratio given by (8) is reduced by a factor of Gp after the process of matched filtering. Now, we define the ratio Eb/Io,
which is the energy per bit to interference density ratio, where Eb=STb, I0=I/Rb=ITb,
and I is the total interference power (sum of Iint and η) so
(9)EbI0=SI=1Iint/S+η/S.In (8), the summation of vi over (N−1) users may be expressed as
(10)∑i=1N−1vi=μ(N−1),so that
(11)EbI0=1μ(N−1)/Gp+η/S.
The
bit error rate (BER) for the binary phase shift keying (BPSK) modulation can be
expressed as
(12)Pb=12erfc(EbI0),where erfc(σ) is the
complementary error function [5]. For a required BER, a required Eb/Io,
(Eb/Io)req can be
determined from (12). Given (Eb/Io)req,
the maximum number of active users, other than the ith user, that can be
supported by the system is given by (11) as
(13)m=∑i=1N−1vi=⌊Gp(Eb/I0)req−GpS/η⌋,where ⌊x⌋ represent the largest integer that is smaller
than or equal x. Provided
that the number of active users does not exceed m. However, when the
number of active users
is larger than m, the BER will be greater than the required BER, and
this situation is referred
to as system outage. The outage probability of the single cell system is
defined as
(14)Pout=Pr(BER>BERreq)=Pr(EbI0<(EbI0)req).
The outage probability is defined as the
probability that the number of active users being greater than m, that is,
(15)Pout=Pr(∑i=1N−1vi>m)=∑i=1N−1(N−1i)μi(1−μ)N−1−i.


### 2.2. Imperfect Power Control

In a practical system, the power control
is not perfect. So, the received signal power from theith user at its
BS will differ from the target power level S by δi dB.
This power error δi is a random variable with a standard
deviation σe. There are several reasons for δi being nonzero, such as the power measurement error at the BS and the inability
to adjust the subscriber unit transmitted power sufficiently fast to force δi to zero [5, 8]. The signal power at the output of the matched filter for the ith
user can be expressed as
(16)S′=10δ0/10⋅S,and
the intracell interference power is
(17)Iint′=1Gp∑i=1N−1vi10δi/10⋅S.
The
intracell interference to signal power ratio at the output of matched filter
becomes
(18)Iint′S′=1Gp∑i=1N−1vi10(δi−δ0)/10,where δ0 and δi are two mutually independent
random variables of power control errors of the signal and the intracell
interferers, respectively. By setting ε=δi−δ0 (18) becomes
(19)Iint′S′=1Gp∑i=1N−1vi10ε/10=IintS10ε/10,where ε is a random variable with zero mean and a standard deviation σε=2σe.
Following the same procedure as in the perfect power control case, the ratio Eb/Io can be written as
(20)(EbI0)imp=1Iint′/S′+η/S′=1(Iint/S)10ε/10+η/S′.In order to evaluate the system performance, we
introduce the outage probability that is defined as the probability of a
system’s BER being greater than10−3, that is,
(21)Pout=Pr(BER>10−3)=Pr((EbI0)imp=S′Iint′+η<γreq)(22)=Pr(Iint′+ηS′>1γreq)=Pr{10ε/10∑i=0N−1vi>Gp(1γreq−ηS′)},where γreq is the required Eb/Io to ensure that the BER is less than 10−3.
If the number of active users inside the cell is k, then (22) can be
rewritten as
(23)Pout=Pr{(k10ε/10>Gp(1γreq−ηS′))|(∑i=0N−1vi=k)}×Pr(∑i=0N−1vi=k)=P1P2.
The probability P1 in (23) is given by [5]
(24)P1=Q(Gp(1/γreq−η/S′)−kE(10ε/10)kvar(10ε/10)),
where
(25)Q(x)=12π∫x∞e(−y2/2)dy.
The
mean of the term 10ε/10 in (24) can be derived as following:
(26)E(10ε/10)=∫−∞∞exp[εln(10)10]exp(−ε2/4σe2)4πσe2dε=exp(σeln(10)10)2,and
the variance of the 10ε/10 is given by
(27)var(10ε/10)=E(10ε/10)2−{E(10ε/10)}2=exp(σeln(10)5)2−(exp(σeln(10)10)2)2.Now,
consider the probability that there are k active intracell users, P2,
which is given by [5]
(28)P2=Pr(∑i=0N−1vi=k)=∑k=0N−1(N−1k)μk(1−μ)N−1−k.
The performance of the reverse
link in a single cell CDMA WLL system, shown in Figure 3, is evaluated for 20 dB/dec, where WLL has a fixed-to-fixed link so propagation exponent of 2 is
used [2], Eb/Io=5dB [9], W=1.25MHz, Rb=8Kbps, VAF=3/8, a signal-to-AWGN ratio of 20 dB at the
output of the matched filter and a BER outage threshold of 10−3 were
used in the calculations.Figure 3
Outage probability of a single cell CDMA WLL system.For perfect power
control and an outage probability of 2%, the single cell system can support up
to 89 users/cell as shown in Figure3. The outage probability of the imperfect
power-controlled system having different standard deviations of power control
error is also shown in the figure. For an outage probability of 2% and a
standard deviation of power control errors of 2 dB, the system can support 66 users per cell. The capacity degradation, due to imperfect power control, is
about 25%.Table1 shows the number of users per cell for different values of the standard
deviation (STD) of power control errors and the percentage decrease in users
due to imperfect power control, for an outage of 2%.Table 1
Number of users per cell for different values ofσe, and Pout=2%.
STDUsers/cell% decrease0 dB890%1.5 dB6922.4%2 dB6625.8%2.5 dB6230.3%


## 2.1. Perfect Power Control

To reduce the near-far
problem, as well as the interference from other users and hence to increase the
capacity of CDMA WLL system, it is important to apply a power control on the
reverse link so that the received power from each user at the BS is controlled
to be the constant target power,S, [5] where
(5)ai2Ai=Sfori=0,1,…,N−1.
The total noise power is
the sum of the intracell interference power and the AWGN power of the receiver.The AWGN power at the
output of the matched filter is given by [5, 7] as
(6)η=NoRb=(NoTc)TcRb=NoWGp,where Rb=1/Tb is
the bitrate of the message sequence bi(t), W=1/Tc is the chiprate, and N0W is the noise power at the
receiver input.Thus, after despreading,
the noise powerη is the input noise power decreased by the processing
gain Gp=Tb/Tc. The
intracell interference power is given by [5]
(7)Iint=var(Zint(t))=1Gp∑i=1N−1ai2Ai,where var(Zint(t)) is the variance of intracell interference at the output of the matched
filter.By applying voice activity
detection, users transmit only when speech signal is present. We introduce a
voice activity variable (VAF)vi which equals 1 with
probability of μ, and equals 0 with probability of 1−μ. By
multiplying (7) by vi and using (5) so,
(8)IintS=1Gp∑i=1N−1viai2AiS=1Gp∑i=1N−1vi,the intracell interference
to signal power ratio given by (8) is reduced by a factor of Gp after the process of matched filtering. Now, we define the ratio Eb/Io,
which is the energy per bit to interference density ratio, where Eb=STb, I0=I/Rb=ITb,
and I is the total interference power (sum of Iint and η) so
(9)EbI0=SI=1Iint/S+η/S.In (8), the summation of vi over (N−1) users may be expressed as
(10)∑i=1N−1vi=μ(N−1),so that
(11)EbI0=1μ(N−1)/Gp+η/S.
The
bit error rate (BER) for the binary phase shift keying (BPSK) modulation can be
expressed as
(12)Pb=12erfc(EbI0),where erfc(σ) is the
complementary error function [5]. For a required BER, a required Eb/Io,
(Eb/Io)req can be
determined from (12). Given (Eb/Io)req,
the maximum number of active users, other than the ith user, that can be
supported by the system is given by (11) as
(13)m=∑i=1N−1vi=⌊Gp(Eb/I0)req−GpS/η⌋,where ⌊x⌋ represent the largest integer that is smaller
than or equal x. Provided
that the number of active users does not exceed m. However, when the
number of active users
is larger than m, the BER will be greater than the required BER, and
this situation is referred
to as system outage. The outage probability of the single cell system is
defined as
(14)Pout=Pr(BER>BERreq)=Pr(EbI0<(EbI0)req).
The outage probability is defined as the
probability that the number of active users being greater than m, that is,
(15)Pout=Pr(∑i=1N−1vi>m)=∑i=1N−1(N−1i)μi(1−μ)N−1−i.


## 2.2. Imperfect Power Control

In a practical system, the power control
is not perfect. So, the received signal power from theith user at its
BS will differ from the target power level S by δi dB.
This power error δi is a random variable with a standard
deviation σe. There are several reasons for δi being nonzero, such as the power measurement error at the BS and the inability
to adjust the subscriber unit transmitted power sufficiently fast to force δi to zero [5, 8]. The signal power at the output of the matched filter for the ith
user can be expressed as
(16)S′=10δ0/10⋅S,and
the intracell interference power is
(17)Iint′=1Gp∑i=1N−1vi10δi/10⋅S.
The
intracell interference to signal power ratio at the output of matched filter
becomes
(18)Iint′S′=1Gp∑i=1N−1vi10(δi−δ0)/10,where δ0 and δi are two mutually independent
random variables of power control errors of the signal and the intracell
interferers, respectively. By setting ε=δi−δ0 (18) becomes
(19)Iint′S′=1Gp∑i=1N−1vi10ε/10=IintS10ε/10,where ε is a random variable with zero mean and a standard deviation σε=2σe.
Following the same procedure as in the perfect power control case, the ratio Eb/Io can be written as
(20)(EbI0)imp=1Iint′/S′+η/S′=1(Iint/S)10ε/10+η/S′.In order to evaluate the system performance, we
introduce the outage probability that is defined as the probability of a
system’s BER being greater than10−3, that is,
(21)Pout=Pr(BER>10−3)=Pr((EbI0)imp=S′Iint′+η<γreq)(22)=Pr(Iint′+ηS′>1γreq)=Pr{10ε/10∑i=0N−1vi>Gp(1γreq−ηS′)},where γreq is the required Eb/Io to ensure that the BER is less than 10−3.
If the number of active users inside the cell is k, then (22) can be
rewritten as
(23)Pout=Pr{(k10ε/10>Gp(1γreq−ηS′))|(∑i=0N−1vi=k)}×Pr(∑i=0N−1vi=k)=P1P2.
The probability P1 in (23) is given by [5]
(24)P1=Q(Gp(1/γreq−η/S′)−kE(10ε/10)kvar(10ε/10)),
where
(25)Q(x)=12π∫x∞e(−y2/2)dy.
The
mean of the term 10ε/10 in (24) can be derived as following:
(26)E(10ε/10)=∫−∞∞exp[εln(10)10]exp(−ε2/4σe2)4πσe2dε=exp(σeln(10)10)2,and
the variance of the 10ε/10 is given by
(27)var(10ε/10)=E(10ε/10)2−{E(10ε/10)}2=exp(σeln(10)5)2−(exp(σeln(10)10)2)2.Now,
consider the probability that there are k active intracell users, P2,
which is given by [5]
(28)P2=Pr(∑i=0N−1vi=k)=∑k=0N−1(N−1k)μk(1−μ)N−1−k.
The performance of the reverse
link in a single cell CDMA WLL system, shown in Figure 3, is evaluated for 20 dB/dec, where WLL has a fixed-to-fixed link so propagation exponent of 2 is
used [2], Eb/Io=5dB [9], W=1.25MHz, Rb=8Kbps, VAF=3/8, a signal-to-AWGN ratio of 20 dB at the
output of the matched filter and a BER outage threshold of 10−3 were
used in the calculations.Figure 3
Outage probability of a single cell CDMA WLL system.For perfect power
control and an outage probability of 2%, the single cell system can support up
to 89 users/cell as shown in Figure3. The outage probability of the imperfect
power-controlled system having different standard deviations of power control
error is also shown in the figure. For an outage probability of 2% and a
standard deviation of power control errors of 2 dB, the system can support 66 users per cell. The capacity degradation, due to imperfect power control, is
about 25%.Table1 shows the number of users per cell for different values of the standard
deviation (STD) of power control errors and the percentage decrease in users
due to imperfect power control, for an outage of 2%.Table 1
Number of users per cell for different values ofσe, and Pout=2%.
STDUsers/cell% decrease0 dB890%1.5 dB6922.4%2 dB6625.8%2.5 dB6230.3%


## 3. Effect of Imperfect Power Control in Multiple Cell CDMA WLL Systems

In addition to the intracell interference, there is
now interference from neighboring cells, which called intercell interference.
The received signal at the BS includes: the desired signal, intracell
interference, the AWGN at the receiver input, and intercell interference. Figure4 shows the reverse-link  communication system, where the arrangement for the
transmitter and BS receiver is the  same as those shown in Figures 1 and 2, respectively.Figure 4
Block diagram of the multicell CDMA reverse-link system.The received signal at the BS is given by [5]
(29)Rup(t)=∑i=0N−1aiSi(t−τi)+∑j=1J−1∑i=0N−1aijSij(t−τij)+n(t),where
the intercell interference from the J−1 surrounding cells is
(30)∑j=1J−1∑i=0N−1aijSij(t−τij),where aij represents
the effects of path loss, τij is the random time delay of
the ith user in the jth cell, and sij(t)
is the signal transmitted by the ith user in the jth cell.For a particular user, say the zeroth one, the signal
at the output of the matched filter is given by(31)Z(Tb)=a02A0b0cosφ0+Zint(Tb)+Zext(Tb)+Zn(Tb),where φ0 is the carrier phase difference. The first term is the desired signal, the
second term is the intracell interference component, the third term is the
intercell interference component, while the last term is the AWGN component.
The intercell interference at the output of the matched filter is given by
(32)Zext(t)=1Tb∫0Tb∑j=1J−1∑i=0N−1aij2Aijbij(t−τij)cij(t−τij).


### 3.1. Perfect Power Control

Similar to the
approach in deriving the intracell interference power, the intercell
interference power at the output of the matched filter,Iext, can be shown to be
(33)Iext=E((Zext(Tb))2)=1Gp∑j=1J−1∑i=0N−1aij2Aij.
The intercell interference to signal ratio is given by
(34)IextS=1Gp∑j=1J−1∑i=0N−1vijaij2AijS=1Gp∑j=1J−1IjS,where Ij/S is the interference to signal power ratio from the jth
cell.Let us consider one of
the interfering cells, saycellj, where its BS is at a distance d from BS0 as shown in Figure 5. The interference term Ij in (34) is the interference power from all the users in cellj to
the BS0, [5, 10].Figure 5
CDMA interference calculation.If the interfering user incellj is located at a distance r from its BS and ro from the
BS0, the interfering user, when active, produces an interference to
the BS0 given by [11]
(35)I(ro,r)S=(10(ζ0/10)r0α)(rα10(ζm/10))=(rr0)α10ζ/10≤1,where the first term is,
due to the attenuation, caused by distance and blockage to the given BS, while
the second term is the effect of power control to compensate for the
corresponding attenuation to the BS of the out-of-cell interferer, ζ, ζ0, and ζm all are random variables with zero mean and standard
deviation σ, (ζ=ζ0−ζm), and r0 is given by
(36)r0=d2+r2+2drcosθ.Replace
the summation in (34) by an integration over the area of the cellj,
so Ij/S will be
(37)IjS=∫02π∫0RvijI(r0,r)S⋅φ(ξ,rr0)ρda,where ρ is the user  density, assuming N users are uniformly
distributed in a circular cell of radius R, ρ=N/πR2, da=rdrdθ is the unit area in Figure 5, Φ(ζ0−ζm,r/r0) is the constraint function for the interfering users in the cellj,
which can be defined as [5, 11]
(38)φ(ξ,rr0)={1,if(rr0)α10ξ/10≤1,0,otherwise.It is necessary to calculate
the mean and variance of the intercell interference power in order to calculate
the outage probability. From (35) into (37) and by taking the mean, we obtain
[11]
(39)E(IjS)=ρμ∫02π∫0R(rr0)αE(10ξ/10⋅φ(ξ,rr0))rdrdθ,where E(vij)=μ. The variance of Ij/S can be given by [11]
(40)var(IjS)=∫02π∫0RE[vij(rr0)α10ξ/10⋅φ(ξ,rr0)]2−{E[vij(rr0)α10ξ/10⋅φ(ξ,rr0)]}2ρrdrdθ.
The total
interference-power-to-signal-power ratio for all the surrounding cells, Iext/S, in (34) has a mean and variance of the following:
(41)E[IextS]=1Gp∑j=1J−1E[IjS],var[IextS]=(1Gp)2∑j=1J−1var[IjS].The
ratio Eb/Io at the output of the matched filter
can be written as
(42)EbI0=1Iint/S+Iext/S+η/S.The system performance in terms of the outage
probability that has a BER greater than 10−3 is
(43)Pout=∑k=0N−1(N−1k)μk(1−μ)N−1−k×Q(1/γreq−η/S−k/Gp−E(Iext/S)var(Iext/S)).


### 3.2. Imperfect Power Control

The received signal
powerS′ from a user at its BS will differ from the target power level S by δ0 dB. This error power is a random variable with
standard deviation σe. Using  (16) and (35), the interfering power to
received signal power ratio will be
(44)I′(ro,r)S′=(rr0)α10ζ/1010δij−δ0=(rr0)α10ζ/1010ε/10,where δij is the power error for
the ith user in cellj. The total intercell
interference-to-signal ratio is
(45)Iext′S′=IextS10ε/10.
The ratio Eb/Io can be written as
(46)(EbI0)imp=1Iint′/S′+Iext′/S′+η/S′+1(Iint/S)10ε/10+(Iext/S)10ε/10+η/S′.
Following
the same procedure as employed in the perfect power control case, Pout is found as
(47)Pout=∑k=0N−1(N−1k)μk(1−μ)N−1−k×Q(1/γreq−η/S′−k/Gp−E(Iint′/S′)−E(Iext′/S′)var(Iint′/S′)+var(Iext′/S′)),where Iint′/S′ and Iext′/S′ are two independent
Gaussian distributed random variables, whose mean and variance may be expressed
as
(48)E[Iint′S′+Iext′S′]=E[IintS10ε/10]+E[IextS10ε/10],var[Iint′S′+Iext′S′]=var[IintS10ε/10]+var[IextS10ε/10].The
performance of the reverse-link CDMA WLL system is shown in Figure6. For an
outage probability of 2%, the perfect power-controlled system can support 52
users/cell for a VAF of 3/8. The number of users per cell for different values
of the standard deviation of power control errors and the percentage decrease
in users, due to imperfect power control, are displayed in Table 2 for an
outage of 2%.Table 2
Number of users per cell for different values ofσe, and Pout=2%.
STDUsers/cell% decrease0 dB520%1.5 dB479.6%2 dB4611.5%2.5 dB4415.4%Figure 6
Outage probability of the multiple cell CDMA WLL system.


## 3.1. Perfect Power Control

Similar to the
approach in deriving the intracell interference power, the intercell
interference power at the output of the matched filter,Iext, can be shown to be
(33)Iext=E((Zext(Tb))2)=1Gp∑j=1J−1∑i=0N−1aij2Aij.
The intercell interference to signal ratio is given by
(34)IextS=1Gp∑j=1J−1∑i=0N−1vijaij2AijS=1Gp∑j=1J−1IjS,where Ij/S is the interference to signal power ratio from the jth
cell.Let us consider one of
the interfering cells, saycellj, where its BS is at a distance d from BS0 as shown in Figure 5. The interference term Ij in (34) is the interference power from all the users in cellj to
the BS0, [5, 10].Figure 5
CDMA interference calculation.If the interfering user incellj is located at a distance r from its BS and ro from the
BS0, the interfering user, when active, produces an interference to
the BS0 given by [11]
(35)I(ro,r)S=(10(ζ0/10)r0α)(rα10(ζm/10))=(rr0)α10ζ/10≤1,where the first term is,
due to the attenuation, caused by distance and blockage to the given BS, while
the second term is the effect of power control to compensate for the
corresponding attenuation to the BS of the out-of-cell interferer, ζ, ζ0, and ζm all are random variables with zero mean and standard
deviation σ, (ζ=ζ0−ζm), and r0 is given by
(36)r0=d2+r2+2drcosθ.Replace
the summation in (34) by an integration over the area of the cellj,
so Ij/S will be
(37)IjS=∫02π∫0RvijI(r0,r)S⋅φ(ξ,rr0)ρda,where ρ is the user  density, assuming N users are uniformly
distributed in a circular cell of radius R, ρ=N/πR2, da=rdrdθ is the unit area in Figure 5, Φ(ζ0−ζm,r/r0) is the constraint function for the interfering users in the cellj,
which can be defined as [5, 11]
(38)φ(ξ,rr0)={1,if(rr0)α10ξ/10≤1,0,otherwise.It is necessary to calculate
the mean and variance of the intercell interference power in order to calculate
the outage probability. From (35) into (37) and by taking the mean, we obtain
[11]
(39)E(IjS)=ρμ∫02π∫0R(rr0)αE(10ξ/10⋅φ(ξ,rr0))rdrdθ,where E(vij)=μ. The variance of Ij/S can be given by [11]
(40)var(IjS)=∫02π∫0RE[vij(rr0)α10ξ/10⋅φ(ξ,rr0)]2−{E[vij(rr0)α10ξ/10⋅φ(ξ,rr0)]}2ρrdrdθ.
The total
interference-power-to-signal-power ratio for all the surrounding cells, Iext/S, in (34) has a mean and variance of the following:
(41)E[IextS]=1Gp∑j=1J−1E[IjS],var[IextS]=(1Gp)2∑j=1J−1var[IjS].The
ratio Eb/Io at the output of the matched filter
can be written as
(42)EbI0=1Iint/S+Iext/S+η/S.The system performance in terms of the outage
probability that has a BER greater than 10−3 is
(43)Pout=∑k=0N−1(N−1k)μk(1−μ)N−1−k×Q(1/γreq−η/S−k/Gp−E(Iext/S)var(Iext/S)).


## 3.2. Imperfect Power Control

The received signal
powerS′ from a user at its BS will differ from the target power level S by δ0 dB. This error power is a random variable with
standard deviation σe. Using  (16) and (35), the interfering power to
received signal power ratio will be
(44)I′(ro,r)S′=(rr0)α10ζ/1010δij−δ0=(rr0)α10ζ/1010ε/10,where δij is the power error for
the ith user in cellj. The total intercell
interference-to-signal ratio is
(45)Iext′S′=IextS10ε/10.
The ratio Eb/Io can be written as
(46)(EbI0)imp=1Iint′/S′+Iext′/S′+η/S′+1(Iint/S)10ε/10+(Iext/S)10ε/10+η/S′.
Following
the same procedure as employed in the perfect power control case, Pout is found as
(47)Pout=∑k=0N−1(N−1k)μk(1−μ)N−1−k×Q(1/γreq−η/S′−k/Gp−E(Iint′/S′)−E(Iext′/S′)var(Iint′/S′)+var(Iext′/S′)),where Iint′/S′ and Iext′/S′ are two independent
Gaussian distributed random variables, whose mean and variance may be expressed
as
(48)E[Iint′S′+Iext′S′]=E[IintS10ε/10]+E[IextS10ε/10],var[Iint′S′+Iext′S′]=var[IintS10ε/10]+var[IextS10ε/10].The
performance of the reverse-link CDMA WLL system is shown in Figure6. For an
outage probability of 2%, the perfect power-controlled system can support 52
users/cell for a VAF of 3/8. The number of users per cell for different values
of the standard deviation of power control errors and the percentage decrease
in users, due to imperfect power control, are displayed in Table 2 for an
outage of 2%.Table 2
Number of users per cell for different values ofσe, and Pout=2%.
STDUsers/cell% decrease0 dB520%1.5 dB479.6%2 dB4611.5%2.5 dB4415.4%Figure 6
Outage probability of the multiple cell CDMA WLL system.


## 4. Effect of Imperfect Sectorization in CDMA WLL Systems

In the case of perfect directional antennas, there is
a sharp separation between the sectors. Due to overlap and sidelobe of
practical antenna, the BS still receives some interference from users in other
sectors [11, 12].Figure7 shows a sectorized
cell arrangement  having Ns overlapping sectors, each with an
angle of (2π/Ns)+θ0, where θ0 is
the overlapping angle. If there is no overlapping, that is, θ0=0, then 1/Ns of the total interference is received,
and the capacity gain, due to sectorization, is Ns times that
of an unsectorized cell.Imperfect sectorization: (a) three sector; (b) six sector per cell.
(a)(b)In a sectorized cell,
only(2π/Ns)+θ0/2π of the total
interference from the surrounding is received. For this condition, the capacity
gain, due to sectorization, is 2π/(2π/Ns)+θ0 times
that of the unsectorized cell.We define the
efficiency of sectorizationes as the ratio of the capacity
gain with the sector antennae pattern having an overlapping angle θ0 to the nonoverlapping antennae pattern of 2π/Ns, so
(49)es=2π/(2π/Ns+θ0)2π/(2π/Ns)=2π/Ns2π/Ns+θ0.
So, the imperfect sectorization capacity gain Gimp will be
(50)Gimp=Nses.
In WLL system, due to
fixed-to-fixed link, six sectors per cell arrangement can be used, the capacity
gain due to sectorization and its corresponding sectorization efficiency for
different values of overlapping angle θ0 in degrees is shown
in Figure 8. From the figure, the interference on the reverse link increases as
the overlapping angle θ0 is increased, which causes the
capacity gain and sectorization efficiency to decrease proportionally.
Increasing θ0 from 0° to 5° causes the capacity
gain to decrease from 6 to 5.53, while the corresponding sectorization
efficiency drops from 100% to 92.3%.Figure 8
Reverse-link sectorization gain and efficiency as a function of the overlapping angle.


## 5. Conclusion

CDMA
technology   has the potential to provide a significant improvement in the
capacity of WLL systems compared with FDMA and TDMA systems. However, this
improvement is dependent upon the effectiveness of the power control system,
especially on the reverse link. In this paper, a theoretical model to evaluate
the reverse-link capacity of CDMA WLL systems in terms of outage probability,
taking into account the power control error, is obtained.The results
show that the capacity degradation, due to imperfect power control, is about
25.8%, and 11.5% for single cell and multiple cell systems, respectively. The
effect of imperfect sectorization on the performance of CDMA WLL systems is
also considered. The interference on the reverse link increases as the
overlapping angle is increased, which causes the capacity gain and
sectorization efficiency to decrease proportionally as shown in Figure8.



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*Source: 413821-2008-09-30.xml*