Tài liệu GSM, cdmaOne and 3G systems P5 pdf

G c asymptotic coding gain of convolutional codeg controls the rate of increase in the step size by the adptive power control algorthm gr j=γ;σ variance of 10ζ = 10with the constraint fu

Chapter 5

Analysis of IS-95

5.1 List of Mathematical Symbols

a i j path loss and shadow fading between the zeroth BS and the

ith MS in the jth microcell

a0 (t) path loss and shadow fading multiplicative factor

b i(t) data sequence for the ith user

b0 value of b(t)over the zeroth symbol period(= 1)

C PN(t) down-link pilot codes

C W(i+ 1 )

(t) (i+1)th Walsh code

c i j spreading code of of ith MS in the jth cell

c i(t) code for the ith user

cc(n;k;K) convolutional code

D j distance between the zeroth BS and the adjacent jth cell

site

(E b=I0)im SIR in the presence of power control errors

E[(
)] expectation of(
)

erfc[(
)] complementary error function of(
)

F a factor used in estimating E b=I0

f(r) PDF of an MS being in a ring of area 2πrdr

f(r j=r;σ) expectation of 10ζ = 10 with the constraint function

φ(ζ;r=r j)

285

Print ISBN 0-471-49185-3 Electronic ISBN 0-470-84167-2

G c asymptotic coding gain of convolutional code

g controls the rate of increase in the step size by the adptive

power control algorthm

g(r j=γ;σ) variance of 10ζ = 10with the constraint functionφ(ζ;r=r j)

h j(t τj) jth component of the impulse response

I total interference power at the output of the matched filter

I ext intercellular interference power at the output of the

matched filter

I0

ext I ext in the presence of power control errors

I0

int I int in the presence of power control errors

I j interference power from all MSs in the jth cell to the zeroth

k CDMA capacity as channels per cell per MHz

L number of match filters, or number of resolvable paths by

the RAKE receiver

M0

spreading factor in the presence of convolutional coding

n I;m(t); n Q;m(t) inphase and quadrature components of the multipath

inter-ference

n ext(t) equivalent baseband intercellular interference

n I;ext(t); n Q;ext(t) inphase and quadrature componenets of n ext(t)

n int(t) equivalent baseband intracellular interference

n I;total(t); nQ;total(t) inphase and quadrature components of the total

interfer-ence noise

n(t) receiver noise

P i transmitted power of the ith MS, or from the BS for the ith

MS

P i j transmitted power from the ith MS in the jth cell (up-link),

or the transmitted power allocated for the ith channel at the

jth BS (down-link)

p o outage probability, i.e probability of the BER>10 3

P T transmitted power of an MS in a power control system

P tar target received signal power at a BS

R I(t); R Q(t) inphase and quadrature components of R(t), the received

baseband signal at the BS

R0 distance from a BS where ‘near-in’ MSs are present

R up(t) received signal at the BS from an MS

pres-S I(t); S q(t) inphase and quadrature components of the wanted signal

S p received pilot power compoment for the zeroth MS on the

down-link

s dn(t) signal transmitted from a BS

s i j(t) transmitted signal from the ith MS to the jth BS

s dn j (t) signal transmitted from the jth neighbouring BS

s0 (t) spread BPSK signal for zeroth MS

W chip rate and bandwidth of the CDMA signal, also the

width of a street in a street microcell

X distance from a street microcellualr BS to the end of the

microcell

X b a break distance in a street microcell where the propagation

path loss exponent changes

x i j(t) transmitted baseband signal from the ith MS to the jth BS

Z ext(T b) intercellular interference component of Z(T b)

Z int(T b) intracellular interference at the output of the matched filter

Z n(T b) receiver noise at the output of the matched filter

Z(T b) output of the matched filter at time t=T b

system parameter in the power control algorithm

δi normally distributed received error power random variable

at a BS for MSi

δi j power control error for the ith MS in the jth cell

δ(t u) delta function at time u

γb E b=I0, or energy per bit per interference PSD

γc E c=I0, or energy per symbol per interference PSD

γreq required(E b=I o)for BER<10 3

λi j normally distributed random variable with standard

devia-tionσ and zero mean

νi voice activity variable of the ith user

νi j voice activity variable of the ith user in the jth cell

φi j carrier phase between the interference signal from the ith

MS in the jth cell and the zeroth MS in the zeroth cell

φ0 carrier phase difference ˆθ0 θˆ

φ(ζ;

r

r j

ρ density of MSs in a cell

τi random delay of the ith user signal at the BS on the up-link,

or the random offset at the BS on the down-link

τi j relative propagation delays of the ith MS in the jth cell

with respect to the zeroth MS in the zeroth cell

τp time offset of the pilot signal at the BS

θ received carrier phase angle at the BS, or the transmitted

phase angle of the carrier at the BS

θi random phase angle of the trnsmitted ith mobile carrier

ˆ

θi change in the phase angle of the ith MS=ω1 (τ0 τi) +θi

θi j carrier phase of ith MS in the jth cell

4i adaptive step size used in the power control algorithm

var[(
)] variance of(
)

ε δj δ0random variable having a normal distribution

ξ error in estimating P Rin the power control

5.2 Introduction

In CDMA many mobiles use the same RF bandwidth at the same time, and a CDMA ceiver is able to separate the wanted signal from the other mobile signals if it knows thespreading code used in the generation of the wanted CDMA signal This demodulationprocess occurs in the presence of interference generated by other mobile users This inter-ference is a major limitation on the capacity of a CDMA system

re-In this chapter the capacity of a CDMA system in tessellated hexagonal cells and citystreet microcells is investigated The system performance in terms of outage probabilityfor a bit error rate (BER) larger than a minimal required level is analysed The number

of users that can be supported by a cell for a given outage probability is evaluated Thecorresponding capacity in terms of channels per cell per MHz is calculated according to thisnumber of users per cell Our discussion concentrates on the capacity evaluation rather than

on other issues, such as code synchronisation We begin by examining a single cell CDMAsystem before moving on to a multiple cell CDMA system Since the arrangement of theup-link, or forward link, is different from the down-link, or reverse link, the performances

of both the up-link and down-link are considered The effect of sectorisation and channelcoding on CDMA systems is also discussed

5.3 CDMA in a Single Macrocell

Consider a single cell CDMA communication system using binary phase shift keying (BPSK)spread spectrum modulation As shown in Figure 5.1, the BS uses the angular carrier fre-quencyω2on the down-link to communicate with all its mobiles, while mobiles transmit totheir base stations (BSs) via the angular carrier frequencyω1

5.3.1 The up-link system

The CDMA single cell system consists of N mobile users transmitting to a BS receiver on

the up-link We consider a simplified mobile transmitter consisting of a BPSK modulator,

formed by multiplying the data sequence for the ith user b i(t), by a carrier cosω1t

Spread-ing occurs when the BPSK signal is multiplied by the code c i(t) This is equivalent to

multiplying the data signal, bi(t), by ci(t)and this spread data signal modulates the carriercosω1t Figure 5.2 shows the arrangement.

Let us consider a particular user, say the zeroth one The spread BPSK signal s0(t)isapplied to the radio channel shown in Figure 5.3 We have separated this channel into a part

that allows for path loss and slow fading and is represented by the multiplicative factor a0

The fast fading is represented by a number of impulse responses h j(t τj);j=0;1; : ;L.

The input of the receiver consists of: interference from the other users in the cell and is

known as intracellular interference; the receiver noise n(t); and the received signal for the

zeroth user The sum of these signals, Rup(t), is demodulated by multiplying by a recoveredcarrier having the same frequency but different phase, relative to the transmitted carrier.The resulting signal is applied to a RAKE receiver that may be considered to be composed

of L matched filters, one for each significant path in the impulse response of the channel.

We note that in general the number of matched filters and the number of channels will not

be the same, but it is desirable if there are at least as many matched filters as there aresignificant paths in the channel The RAKE receiver is a maximum ratio diversity system

if it can obtain accurate estimates of the complex impulse responses h j(t τj) The RAKEreceiver is described in Section 2.3.2.6

A CDMA system has other attributes to combat the effects of fast fading on the signal

s0(t) These include symbol interleaving, forward error correction (FEC) coding, spacediversity reception, power control, and so forth Using this battery of techniques we caneffectively compensate for the effects of fast fading The channel model is now reduced to

the multiplicative factor a0which accounts for path loss and slow fading The BS receivermay now be configured for our analysis as one having despreading followed by a matchedfilter, i.e single stage RAKE, which is an integrator and dump circuit for each mobile Oursimplified model of the radio channel and the BS receiver is depicted in Figure 5.4

Each user has a unique spreading code that is known to the BS The spreading codes are

carrier generator

i-th user b i (t)

si(t)

cos(ω1t+θi)

2P i

Figure 5.2: A mobile’s CDMA transmitter diagram.

of length M chips, or an M-chip segment from the long psuedo noise (PN) sequence [1, 2].

As the mobiles are in different locations within the cell, the transmission delay for each

mobile is different The signal transmitted from the ith user to its BS is

s i(t) =

p

2P i b i(t)c i(t)cos(ω1t + θi); (5.1)

where P i is the transmitted power of the ith user, bi(t)is the data sequence of the ith user

where each bit has an amplitude of1 and a duration of T b , c i(t) is the spreading code

sequence of ith user and each of the M chips per code has a duration T c, andθiis the random

phase of the ith mobile carrier and is uniformly distributed in [0;2π) All the mobilestransmit their signals to the BS receiver over the same radio channel, and the received signal

matched filter

combiner

matched filter matched filter

RAKE Receiver cophaser

h1(t-τ1) +

Figure 5.3: The up-link representation.

where a i represents the path loss and slow fading of the ith user, τi is the random delay

of the ith user signal at the receiver and is uniformly distributed in[0;T b), and n(t)is theadditive white Gaussian noise (AWGN) of the receiver noise The signal at the output of thezeroth matched filter is given by

of the radio path associated with the zeroth user, the integration is done from t = τ0 to

t = τ0 + T b Letting t = t +τ0in Equation (5.3), we have

Owing to the stationary property of the AWGN, n(t + τ0 ) in the above equation can be

substituted by n(t), and Equation (5.5) can be rewritten as

c i(t τi0)becomes c0(t)and from Equation (5.6) c i(t τi0)for i=0, multiplied by c0(t)

yields unity, and therefore the wanted component of Z(T b)is

2P i b i(t τi0)c i(t τi0)cosφi (5.12)

is the equivalent baseband intracellular interference The receiver noise term is, from tion (5.6),

other N 1 users We are cognisant that c i(t τi0)are the independent spreading codesfor different users and that the relative time offset of the data transmitted from each mobile

is a random variable, i.e.τi0is an independent random variable that is uniformly distributedover[0;T b) We further assume that b i(t)represents random independent binary data, and

as a consequence the intracellular interference is a stationary random process From the

Central Limit Theorem, the summation of N 1 independent random process means that

n int can be approximated as a Gaussian random variable [3, 4]

5.3.1.1 Perfect power control

Since all users are sharing the same radio frequency, a strong signal from mobiles close

to the BS will mask weak signals from distant users To reduce this so-called near–farproblem, as well as to reduce the interference from other users, it is important to exercise

a power control on the up- link of CDMA transmissions so that the received signal power

levels from all users remain close to a target power, P tar Identically, the received power from each user at the BS is controlled to be the constant target power, P tar, namely

a2i P i = P tar; for i = 0;1; : ;N 1: (5.14)

With the aid of Equations (5.14), (5.7), (5.9) and (5.13) we may express Equation (5.3) as

Z(T b) =

p

2P tar b0cosφ0 + Z int(T b) + Z n(T b); (5.15)

where the first term is the desired signal, the second term is the interference from the N 1users in the cell, and the last term is the AWGN component The bit error probability atthe output of the bit regeneration circuit depends upon the bit-energy-to-total-interference

power spectral density (PSD) ratio or signal-to-total-interference power ratio (SIR)

Accord-ing to Equation (5.15), the average power of the wanted signal component is

T b

0

n(t)c0 (t)cos(ω1t + θˆ)dt

 2

=

4

T b2E

 Z

whereδ(t u)is a delta function at t=u So,

where Rb=1=T b is the bit rate of the message sequence bi(t), W =1=T c is the chip rate

and we assume it is also the bandwidth of the CDMA signal, and W N0 is the noise power

at the receiver input Thus, after despreading, the noise powerη is the input noise power

decreased by the processing gain G p=T b=T c The intracellular interference power is

T b

0

n int(t)c0(t)dt

 2

E[n int(t)n int(u)] E[c0(t)c0(u)]dudt: (5.22)

Since nint(t)is Gaussian distributed having a power of E[n2int(t)]and a double-sided

band-width of W , its double-sided PSD is

where the intracellular interference power is

because the expectation of cos2φiis 0:5

By applying voice activity detection (VAD) and thereby discontinuous transmitting (DTX),the mobiles transmit only when speech signal is present We introduce a voice activity vari-

able vi which is equal to 1 with probability of µ, and to 0 with probability of 1 µ, where µ

is defined as the voice activity factor (VAF) By multiplying Equation (5.25) by v i, and withthe aid of Equations (5.14) and (5.16),

I int S

Equation (5.26) is also reduced by a factor of Gpafter the process of matched filtering

The energy per bit E b measured at the output of the matched filter is a random variablebecause of the variations in the path loss, slow fading and fast fading of the mobile chan-

nel The interference PSD I0measured at the output of the matched filter is also a randomvariable because it depends on the interference being generated by mobiles roaming within

the cell We therefore need to take the expectation of the ratio of E b to I0, namely E b=I0, in

determining the probability of symbol error Now E b = ST b and I0 = I=R b = IT b, where

I is the total interference power at the output of the matched filter Consequently,

From Equation (5.30), the bit error rate (BER) for the BPSK can be expressed as

where erfc(σ) is the complementary error function [5] For a required BER, a required

E b=I0, namely (E b=I0)reqcan be determined from Equation (5.31) Given (E b=I0)req, themaximum number of active users, other than the zeroth user, that can be supported by thesystem is

wherebxcrepresents the largest integer that is smaller than x Provided the number of active users does not exceed m, the required BER is secured 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

p o=Pr(BER>BER req) =Pr E b

Since users in a cell are not active all the time, the number of active users is less than the

number of potential users Consequently, a cell can support more than m users, but the

system will experience outage at those instances when the number of active users exceeds

m The outage probability is then the probability of the number of active users being greater

In practice, the received signal power P R from the ith mobile at its BS will differ from the target power level P tar byδi dB This error powerδi is a random variable that is normallydistributed with a standard deviationσeand is discussed in detail in Section 5.6 and in Ref-erences [6]– [8] There are several reasons forδibeing non-zero, such as the inaccuracies in

measuring the received power, S, at a BS, and the inability to adjust the mobile transmitted

power sufficiently fast to forceδi to zero The relationship between P R and P tar for the ith

mobile may be expressed as

= Pr 1

G p10

ε 10

η

S0

 )

whereγreqis the required E b=I0to ensure that the BER is less than 10 3 If the number of

active users inside the cell is k, i.e.∑N 1

i= 0 v i = k, then Equation (5.42) can be rewritten as

η

S0



The outage probability p o is the product of two probabilities, p1 and p2 We will first

consider the probability that there are k active intracellular users,

where p here is the probability of a head being tossed In our case we replace p by the VAF,

µ, and observe that k can range from 0 to N 1 Hence,

G p

 1

7 7 5

h ε p

2 σe

ln ( 10 ) p

# 2

h ε p

2 σe

p

2 σeln ( 10 ) 5

i 2

From Equations (5.53) and (5.54), the variance of the 1010ε becomes

# 2

=

;

The performance of the up-link in a single cell CDMA system having a processing gain

of 128 was evaluated over a channel having an inverse fourth power path loss law andslow fading whose standard deviation was 8 dB A signal-to-AWGN ratio of 20 dB at theoutput of the matched filter was assumed and a BER outage threshold of 10 3was used inthe calculations Figure 5.5 shows the outage probability from Equation (5.35) for perfectpower control and VAFs of 3/8 and 1/2 For an outage probability of 2%, the single cellCDMA system can support 48 users and 38 users for a VAF of 3/8 and 1/2, respectively.The outage probability of the imperfect power controlled system having different standard

deviations of power control error in E b=I0is show in Figures 5.6 and 5.7 for a VAF of 3/8

and 1/2, respectively We observe that a standard deviation of the measured E b=I0was found

to be 1.7 dB in a particular set of measurements [7] For an outage probability of 2% and a

standard deviation of power control errors in E b=I0 of 2 dB, the single cell CDMA systemcan support 37 users and 28 users per cell for a VAF of 3/8 and 1/2, respectively Thecapacity degradation due to imperfect power control is about 46% This highlights the needfor an accurate power control technique for the up-link in this type of CDMA system

5.3.2 The down-link system

The CDMA down-link, namely the forward link, has a coherent BPSK communication tem where the coherent demodulation is facilitated by a pilot signal As shown in the system

sys-arrangement of Figure 5.8, the BS transmitter adds the CDMA signals from the N 1 trafficchannels with a CDMA pilot, then transmits this combined signal to all the mobile users inits cell A mobile can recover the portion of the signal intended for itself by coherentlydemodulating and despreading the signal with its own code The signal transmitted from

Figure 5.5: Outage probability of a single cell CDMA system in the presence of a perfectly power

controlled up-link, with VAFs of 3/8 and 1/2

Figure 5.6: Outage probability of the single cell CDMA system in the presence of imperfect power

controlled up-link, a VAF of 3/8, and different values of the standard deviation of power

control errors in E b=I0

Figure 5.7: Outage probability of the single cell CDMA system in the presence of imperfect power

controlled up-link, a VAF of 1/2, and different values of the standard deviation of power

where P i and P p are the transmitted power allocated for the ith mobile and the pilot signal,

respectively,τi is the random time offset of the ith user, ω2 is the down-link carrier

fre-quency, cp(t)is the pilot code sequence,τpis the time offset of the pilot signal andθ is an

arbitrary phase angle Let us assume that the pilot signal is transmitted on the Nth channel,

then Equation (5.57) can be simplified to

During the down-link transmission there is no relative time delay between each user’sCDMA signal For convenience we will set the signal delay on the down-link to zero.While the signal is transmitted by the zeroth BS to its service area, the signal received byone of its users, say, the zeroth mobile, has the form

R dn(t) = a0s dn(t) +n(t)

= a0

p

2P0b0(t τ0 )c0(t τ0 )cos(ω2t+θ) +

2P i b i(t τi)c i(t τi)cos(ω2t + θ) +n(t); (5.59)

where the first term is the signal for zeroth mobile, the second term is the intracellular ference, and the last term is the AWGN component Assuming that the receiver is correctlychip synchronised to the zeroth user, we can setτ0 to zero without loss of generality Af-ter demodulating and despreading, the signal at the output of the matched filter is, afterfollowing a similar procedure to that in Section 5.3.1,

The performance of the down-link can be obtained by following the same procedure asused in the up-link From Equation (5.60), the received signal power component for thezeroth mobile receiver is

and the power in the received pilot is

If discontinuous transmission is applied to all the traffic channels, then the interference

is the summation of 2a20P i v i for i ranging from 0 to N 1 The pilot channel is usuallytransmitted at a higher power level than a traffic channel and also at a constant power level.The intracellular interference is therefore

If each traffic channel and the pilot signal have the same power, i.e P i = P p for all i, we

ob-tain the average bit-energy-to-interference PSD ratio, or the average signal-to-interferencepower ratio, as

G p



E b I0

The performance of the down-link in a single cell CDMA system in terms of the BER iscalculated using Equation (5.35) For an inverse fourth power loss law, a slow fading whosestandard deviation is 8 dB, a signal-to-AWGN ratio of 20 dB, and a processing gain of 128,the outage probability as a function of the number of users per cell for two different values

of VAF is displayed in Figure 5.9 For an outage probability of less than 2%, the single cellsystem can support 47 and 37 users for VAFs of 3/8 and 1/2, respectively

5.4CDMA Macrocellular Networks

In the previous section we addressed the performance of the single cell CDMA system Wenow consider the performance of the multiple cellular arrangement shown in Figure 5.10 Inaddition to the intracellular interference, there is now interference from neighbouring cells

This interference is referred to as intercellular interference The effects of intercellular

Figure 5.9: Outage probability of a single cell down-link system.

interference must be determined for both the up-link and the down-link communicationsystems

5.4.1 The up-link system

The received signal at a BS includes the desired signal, intracellular interference, the AWGN

at the receiver input, and intercellular interference Figure 5.11 shows the up-link cation system where the arrangement for the mobile transmitter and BS receiver are exactlythe same as those shown in Figures 5.2 and 5.4, respectively The signal received at thezeroth BS is given by

where the intercellular interference from the J 1 surrounding cells is

Figure 5.10: Hexagonal multicell arrangement.

and where ai j represents the effects of path loss and slow fading, τi j is the random time

delay of the ith mobile in the jth cell, and s i j(t)is the signal transmitted by the ith mobile

in the jth cell Assuming that the receiver is correctly chip synchronised to the zeroth user,

we can setτ0 to zero without loss of generality After the received signal goes through theprocess of demodulation and despreading, the matched filter output is calculated followingthe methodology given in Section 5.2.1, as

Z(T b) = a0

p

2P0b0cosφ0 + Z int(T b) + Z ext(T b) + Z n(T b) ; (5.72)whereφ0is the carrier phase difference The first term is the desired signal, the second term

is the intracellular interference component, the third term is the intercellular interference

component, while the last term is the AWGN component In Equation (5.72), Zint(T b)and

Z n(T b)are given by Equations (5.9) and (5.13) , respectively, while

where n ext(t)is the equivalent baseband intercellular interference defined as

in the jth cell and the zeroth mobile in zeroth cell Similar to the intracellular interference,

n ext(t)is also a random variable with a Gaussian distribution

Power control, discussed in Section 5.2.1, is also applied in multicellular systems For

per-fect power control, we can find the signal power S, the AWGN powerη, and the intracellular

interference-to-signal ratio I int=S from Equations (5.16), (5.20), and (5.26), respectively,

Similar to the approach in deriving the intracellular interference power, the intercellular

interference power at the output of the matched filter, I ext, can be shown to be (see tions (5.24) and (5.25)),

Equa-I ext = E

n [Z ext(T b)]

2 o

By applying voice activity detection (VAD) and thereby discontinuous transmitting (DTX),the mobiles transmit only when speech is present We introduce a voice activity variable

v i j to the intercellular interference power Then the intercellular interference-to-signal ratiocan be derived from Equation (5.76) as

up-cells, the jth cell say, where the cell site is a distance D jfrom the zeroth cell site as shown

in Figure 5.12 The interference term I j in Equation (5.78) is the interference power from

all the mobiles in the jth cell to the zeroth BS We will calculate this interference power I j and then sum the interference for all the J 1 significant interfering cells

From Equation (5.78), I j=S is found as the summation of N terms corresponding to N

mobiles in the jth cell We will replace this summation by an integration over the area of the jth cell, assuming that the mobiles are uniformly distributed The active mobiles in the

jth cell produce an interference power of I(r j;r)at the zeroth cell site Under perfect power

control, the mobiles in the jth cell have their power controlled by their own BS to be P tar

In order to track the relative path loss and slow fading variations, the transmitted power P i j from the ith mobile in the jth cell is made inversely proportional to a2i j, whence

P i j =

P tar

a2

i j

= P tar rα10 10λi j = Srα10 10λi j ; (5.79)

whereα is the path loss exponent, and λi j is a normally distributed random variable withstandard deviationσ and zero mean, while r is the distance from the interfering mobile to its

own BS The arrangement is shown in Figure 5.13 Consequently the interference-to-signal

power ratio at the zeroth cell site due to the mobiles in area da who are communicating to

: 2nd tier cells

Figure 5.12: Intercellular interference geometry in hexagonal cells.

the jth cell site is [9]

where r j, the distance between the interfering mobile and zeroth BS, is

r j = q

D2j + r2 2D j r cos(ϕ); (5.81)andζ is the difference between λi jandλ0, the independent random variables with zero meanand standard deviationσ Hence ζ is also a random variable with zero mean and variance of

σ2

ζ = 2σ2, while D j is the distance between the zeroth BS and the jth co-channel BS Note that a2

0 j in Equation (5.80) is the path loss and slow fading between the interfering mobile

in the jth cell and the zeroth BS.

Since the total intercellular interference-to-signal ratio is the sum of all the interference

from the J 1 surrounding cells, then due to the Central Limit Theorem, the interferingpower tends to be Gaussian distributed with a non-zero mean In other words, the intercel-lular interfering power varies around a mean power It is necessary to calculate the mean andvariance of the intercellular interference power in order to calculate the outage probability

We commence by replacing Equation (5.78) by

I j

Z

2 π 0

Figure 5.13: Intercellular interference in a multicell environment.

whereρ is the density of mobiles over the jth cell; da = rdrdϕ is the unit area in ure 5.13;φ(ζ;r=r j)is the constraint function for the interfering users in the jth cell; and R

Fig-is the cell radius Because each mobile communicates with the cell site having the smallestpath loss and slow fading attenuation, the constraint function can be defined as



= ρµ

Z

2 π 0

where E[v i j] =µ, f(r j=r;σ)is the expectation of the product of 10ζ = 10 and the constraintfunctionφ(ζ;r=r j) The term f(r j=r;σ) can be derived noting thatζ is log-normally dis-tributed, namely

10 αlog



r j r

Note that the upper integral limit 10αlog(r j=r)is a consequence ofφ(ζ;r=r j)being set tounity; see Equation (5.83) Hence,



∞

exp

 1 2

h ζ p

2 σ

ln ( 10 ) p

2 σ 10

i 2

 (



= var

 Z

2 π 0

From Equation (5.80),

var



I j S



= Z

2 π 0

2 π 0

2 π 0

ρ dr drϕ

= Z

2 π 0

where f(r j=r;σ)is given by Equation (5.85) and g(r j=r;σ)is

10 αlog



r j r



r j r



∞

exp

 1 2

h ζ p

2 σ

p

2 σln ( 10 ) 5

i 2

 (

By knowing the mean and variance of Ij=S we can now calculate the mean and variance of

the total interference-power-to-signal-power ratio for all the surrounding cells The

expres-sion I ext=S in Equation (5.77) has a mean and variance of

E



I ext S

Note thatη=S is a constant, and I int=S is a function of the voice activity variable which is

binomially distributed, while I ext=S is a Gaussian random variable with a mean and variance

of

E



I ext S

and

has been determined before; see Equation (5.46) We now consider the first probability term

in Equation (5.96) On applying Equation (5.50),



E [

Iext

S ] q

The received signal power S0

from a mobile at its BS will differ from the target power

level P tar by δ0 dB This error powerδ0 is a random variable that is normally distributedwith standard deviationσe The received signal S0

at the BS for the zeroth mobile, and theintracellular interference-to-signal ratio are given by Equations (5.37) and (5.40), respec-

tively Using Equations (5.37) and (5.80), the mobiles in area da in the jth cell produce an

interfering power-to-received-signal-power ratio at the zeroth BS of

I0 (r j;r)

I0

ext

S0 + η

whereε is the error in Eb=I0due to imperfect power control Following the same procedure

as employed for the perfect power control case, the outage probability is found as

I0

ext

S0 +

η

S0

>

1γreq

 +E



I ext

S 10

ε 10

 +var



I ext

S 10

ε 10



Before we derive the outage probability, we have to calculate the mean and variance of theintracellular interference-to-signal ratio, and the intercellular interference-to-signal ratio.The mean and variance of the intracellular interference-to-signal power ratio are

2σe10

! 2 3

2σe10

! 4 3

5 9

 2

 

E



I ext S

and g(r j=r;σ)is given in Equation (5.88) From Equations (5.107) and (5.108) the

expec-tation and variance of I0

ext=S0

can be calculated

The outage probability of Equation (5.103) can be written following the same procedure

as used for the perfect power control case:

h

I0

ext

S0 i 1

C C A

For a processing gain of 128 and a signal-to-AWGN ratio of 20 dB at the output of thematched filter, the performance of the up-link CDMA system is shown in Figure 5.14 for aVAF of 1/2, and in Figure 5.15 for a VAF of 3/8 For an outage probability of 2%, the perfectpower controlled CDMA system can support 23 users and 30 users for a VAF of 1/2 and 3/8,respectively The number of users per cell for different values of the standard deviation of

power control errors in E b=I0, and the percentage decrease in users due to imperfect powercontrol, are displayed in Table 5.1 for an outage of 2%

From Table 5.1, the imperfect power control system having a standard deviation of 2 dBcan only support 22 users and 28 users for VAFs of 1/2 and 3/8, respectively The reduction

in capacity caused by power control error is about 6.3% and 4.3% for VAFs of 3/8 and 1/2,respectively, for a standard deviation of 2 dB For a 2.5 dB standard deviation of powercontrol error, the percentage of capacity loss increases to 13%

Figure 5.14: Performance of the multicellular up-link system with a VAF of 1/2.

Figure 5.15: Performance of the multicellular up-link system with a VAF of 3/8.

Table 5.1: Number of users per cell for different values of the standard deviation of power control

errors in E b=I0 The outage probability is 2%

5.4.2 The down-link system

The single cell down-link has been discussed in Section 5.2.2 We now consider the ticell down-link system that has the same system arrangement as the single cell system.However, unlike the single cell system, there is intercellular interference from neighbouringBSs This interference depends on the mobile’s location [10] The nearer the mobile is toits BS, the better the performance, and consequently the worst case is when the mobiles arelocated at the cell boundaries In the following analysis we consider two particular locations

mul-at the hexagonal cell boundary to examine the performance of the down-link CDMA tem with power control and without power control The two locations are shown in Figure5.16, where location A is at the corner of a hexagon, and location B is at the middle of ahexagonal periphery

The CDMA down-link system is a coherent BPSK communication system where the ent carrier is provided by sending a pilot signal Figure 5.8 shows the system arrangement.The BS sums up all the signals for all its users, together with a CDMA pilot signal, andtransmits the combined signal to the users in its cell A mobile recovers the portion of thesignal intended for it by coherently demodulating and despreading the received CDMA sig-

coher-nal with its own spreading code The transmitted sigcoher-nal sdn(t)from its own BS (zeroth cell)

is given by Equation (5.57), while the signal transmitted from the jth neighbouring BS is

2Pi j b i j(t τi j)c i j(t τi j) cos(ω2t + θj); (5.111)

where P i j is the transmitted power allocated for the ith channel andω2is the down-link radiofrequency carrier When the BS transmits the signal to all its mobiles within its coveragearea, the signal received by one of its users, say, the zeroth mobile in the zeroth cell, is

B

BS

Figure 5.16: Multicellular down-link interference geometry.

the first term is the signal for zeroth mobile, the second term is the intracellular interference

from the N 1 mobiles, while the third term and the last term are AWGN noise and

in-tercellular interference, respectively Note that a0and a j represent the path loss and slowfading for the paths between the zeroth BS and the zeroth mobile in the zeroth cell, and for

the jth BS to the zeroth mobile in the zeroth cell, respectively.

Assuming that the receiver is correctly chip synchronised to the zeroth user, we can set

τ0to zero without loss of generality After demodulating and despreading, and on applyingthe same procedure as used in Section 5.3.1, the signal at the output of the matched filter is

whereφj is the phase difference between the neighbouring BS carrier and the zeroth BScarrier.

The performance of the down-link can be analysed by following the procedure used in theup-link analysis The signal power and the AWGN power are given by

10ζ j10

where P i j is the transmitted power from the jth BS for the ith mobile in the jth cell, v i j is

the mobile’s voice activity variable, N is the number of users in the cell, R is the cell radius,

r j is the distance between the neighbouring jth BS and the zeroth mobile in the zeroth cell, and r0 is the distance between the zeroth mobile and the zeroth BS Note that there is a

factor of 1/2 in Iext=S, due to the carrier incoherence between the zeroth and jth BSs.

Because the system should be designed to give a required performance for any mobilewithin its cell area, the performance of a mobile located at the cell boundary is particularlycritical in the system analysis Therefore we consider two locations, A and B, at the bound-ary, as shown in Figure 5.16, to examine the performance in the down-link By combiningEquations (5.115), (5.116), (5.117), and (5.120), the output of the matched filter has abit-energy- to-interference PSD of

S

in capacity caused by power control error is about 6.3% and 4.3% for VAFs of 3/8 and 1/2,respectively, for a standard... arrangement.

and where j represents the effects of path loss and slow fading, τi j is the random time

delay of the ith mobile in the jth cell, and s i j(t)is