Chapter
5
Analysis of IS-95
5.1 List of Mathematical Symbols
a
ij
path loss and shadow fading between the zeroth BS and the
ith MS in the jth microcell
a
0
(
t
)
path loss and shadow fading multiplicative factor
b
i
(
t
)
data sequence for the ith user
b
0
value of b
(
t
)
over the zeroth symbol period
(= 
1
)
C

(
n

k

K
)
convolutional code
D
j
distance between the zeroth BS and the adjacent jth cell
site
da area housing an MS
d
f
minimal free distance
(
E
b
=
I
0
)
im
SIR in the presence of power control errors
E
()]
expectation of
()
erfc

φ
(
ζ

r
=
r
j
)
285
GSM, cdmaOne and 3G Systems. Raymond Steele, Chin-Chun Lee and Peter Gould
Copyright © 2001 John Wiley & Sons Ltd
Print ISBN 0-471-49185-3 Electronic ISBN 0-470-84167-2
286
CHAPTER 5. ANALYSIS OF IS-95
G
c
asymptotic coding gain of convolutional code
G
p
processing gain
=
T
b
=
T
c
g controls the rate of increase in the step size by the adptive
power control algorthm
g

I total interference power at the output of the matched filter
I
ext
intercellular interference power at the output of the
matched filter
I
0
ext
I
ext
in the presence of power control errors
I
int
intracellular interference power
I
0
int
I
int
in the presence of power control errors
I
j
interference power from all MSs in the jth cell to the zeroth
BS
I
MAI
multiple access interference
I
0
interference PSD

n
I

m
(
t
)
; n
Q

m
(
t
)
inphase and quadrature components of the multipath inter-
ference
n convolutional code length
n
ext
(
t
)
equivalent baseband intercellular interference
n
I

ext
(
t
)

t
)
inphase and quadrature components of the total interfer-
ence noise
N
s
number of sectors per cell
5.1. LIST OF MATHEMATICAL SYMBOLS
287
n
(
t
)
receiver noise
P
i
transmitted power of the ith MS, or from the BS for the ith
MS
P
ij
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
m
power allocated to each MS by a BS
p
o
outage probability, i.e. probability of the BER
>

b
bit rate of the message sequence
R
c
chip rate
R
dn
(
t
)
received signal at an MS
R
I
(
t
)
; R
Q
(
t
)
inphase and quadrature components of R
(
t
)
, the received
baseband signal at the BS
R
0
distance from a BS where ‘near-in’ MSs are present

t
)
signal transmitted from a BS
s
ij
(
t
)
transmitted signal from the ith MS to the jth BS
s
j
dn
(
t
)
signal transmitted from the jth neighbouring BS
s
0
(
t
)
spread BPSK signal for zeroth MS
T
b
bit duration
T
c
chip duration
288
CHAPTER 5. ANALYSIS OF IS-95

)
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
Z
w
(
T
b
)
wanted component of Z
(
T
b
)
β

)
delta function at time u
η AWGN power at the output of the matched filter
γ
b
E
b
=
I
0
, or energy per bit per interference PSD
γ
c
E
c
=
I
0
, or energy per symbol per interference PSD
γ
req
required
(
E
b
=
I
o
)
for BER

θ
0

ˆ
θ
φ
(
ζ

r
r
j
)
constraint function
5.2. INTRODUCTION
289
ρ density of MSs in a cell
σ
ε
standard deviation of ε
σ
e
standard deviation of δ
i
τ
i
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
τ
ij

0
overlapping angle of adjacent sectors
4
i
adaptive step size used in the power control algorithm
var
()]
variance of
()
εδ
j

δ
0
random variable having a normal distribution
ξ error in estimating P
R
in the power control
ζλ
ij

λ
0
5.2 Introduction
In CDMA many mobiles use the same RF bandwidth at the same time, and a CDMA re-
ceiver is able to separate the wanted signal from the other mobile signals if it knows the
spreading code used in the generation of the wanted CDMA signal. This demodulation
process 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.
In this chapter the capacity of a CDMA system in tessellated hexagonal cells and city

1
t. Spread-
ing occurs when the BPSK signal is multiplied by the code c
i
(
t
)
. This is equivalent to
multiplying the data signal, b
i
(
t
)
,byc
i
(
t
)
and this spread data signal modulates the carrier
cos ω
1
t. Figure 5.2 shows the arrangement.
Let us consider a particular user, say the zeroth one. The spread BPSK signal s
0
(
t
)
is
applied 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 a

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 are
significant 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

τ
j
)
. The RAKE
receiver is described in Section 2.3.2.6.
A CDMA system has other attributes to combat the effects of fast fading on the signal
s
0
(
t
)
. These include symbol interleaving, forward error correction (FEC) coding, space
diversity reception, power control, and so forth. Using this battery of techniques we can
effectively compensate for the effects of fast fading. The channel model is now reduced to
the multiplicative factor a
0
which accounts for path loss and slow fading. The BS receiver
may now be configured for our analysis as one having despreading followed by a matched
filter, i.e. single stage RAKE, which is an integrator and dump circuit for each mobile. Our
simplified model of the radio channel and the BS receiver is depicted in Figure 5.4.

i
(t)
s
i
(t)
cos(
ω
1
t
+
θ
i
)
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
(

i
(
t
)
is the spreading code
sequence of ith user and each of the M chips per code has a duration T
c
,andθ
i
is the random
phase of the ith mobile carrier and is uniformly distributed in

0

2π
)
. All the mobiles
transmit their signals to the BS receiver over the same radio channel, and the received signal
at the BS receiver is
R
up
(
t
) =
N

1
∑
i
=

(
t

τ
i
)
c
i
(
t

τ
i
)
cos

ω
1
(
t

τ
i
)+
θ
i
]
+
n
(

0
)
h
0
(t-τ
0
)
fast fading
n(t)
a
0
s
0
(t)
R
up
(t)
h
L
(t-τ
L
)
interference from other users
2cos(ω
1
t+θ)
path loss &
slow fading
carrier
recovery

is the
additive white Gaussian noise (AWGN) of the receiver noise. The signal at the output of the
zeroth matched filter is given by
Z
(
T
b
) =
1
T
b
Z
T
b
+
τ
0
τ
0
R
up
(
t
)
c
0
(
t

τ

0
a
i
p
2P
i
b
i
(
t

τ
i
)
c
i
(
t

τ
i
)
cos

ω
1
(
t

τ

T
b
+
τ
0
T
0
n
(
t
)
c
0
(
t

τ
0
)

2cos
(
ω
1
t
+
θ
)
dt
)

(t)
c
1
(t)
decision
decision
decision
b
N-1
(t)
i
b
i
(t)
b
1
(t)
2cos(ω
1
t+θ)
Base station receiver
decision
b
0
(t)
s
0
(t)
s
1

τ
0
to
t
=
τ
0
+
T
b
. Letting t
=
t
+
τ
0
in Equation (5.3), we have
Z
(
T
b
) =
(
1
T
b
Z
T
b
0


τ
i
)

cos
(
ω
1
t
+
ˆ
θ
i
)

c
0
(
t
)
2cos
(
ω
1
t
+
ˆ
θ
)

ˆ
θ
)
dt
)

(5.4)
where
ˆ
θ
=
ω
1
τ
0
+
θ and
ˆ
θ
i
=
ω
1
(
τ
0

τ
i
)+

N

1
∑
i
=
0
a
i
p
2P
i
b
i
(
t

τ
io
)
c
i
(
t

τ
i0
)
cos
(

Z
T
b
0
n
(
t
+
τ
0
)
c
0
(
t
)

2cos
(
ω
1
t
+
ˆ
θ
)
dt
)
:
(5.5)

i
=
0
a
i
p
2P
i
b
i
(
t

τ
i0
)
c
i
(
t

τ
i0
)
cos
(
ω
1
t
+

(
t
)
c
0
(
t
)
2cos
(
ω
1
t
+
ˆ
θ
)
dt
:
(5.6)
Assuming the receiver is chip synchronised to the zeroth mobile, then for the zeroth mobile,
c
i
(
t

τ
i0
)
becomes c

) =
1
T
b
Z
T
b
0
a
0
p
2P
0
b
0
(
t
)

cos φ
0
+
cos
(
2ω
1
t
+
ˆ
θ


1, and φ
0
=
ˆ
θ
0

ˆ
θ. Note that the term Z
w
(
T
b
)
contains the original data sequence b
0
scaled by a
0
p
2P
0
cos φ
0
.
The intracellular interference at the output of the matched filter is
Z
int
(
T

i
(
t

τ
i0
)
c
0
(
t
)


cos
(
φ
i
)+
cos

2ω
1
t
+
ˆ
θ
+
ˆ
θ

2P
i
b
i
(
t

τ
i0
)
c
i
(
t

τ
i0
)
c
0
(
t
)
cosφ
i
dt

(5.9)
where
φ

n
int
(
t
)
c
0
(
t
)
(5.11)
where
n
int
(
t
)=
N

1
∑
i
=
1
a
i
p
2P
i
b

T
b
0
n
(
t
)
c
0
(
t
)
cos
(
ω
1
t
+
ˆ
θ
)
dt
:
(5.13)
Let us consider the intracellular interference shown in Equation (5.12) that comes from the
other N

1 users. We are cognisant that c
i
(

296
CHAPTER 5. ANALYSIS OF IS-95
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–far
problem, 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
a
2
i
P
i
=
P
tar

for i
=
0

1
 :::
N


where the first term is the desired signal, the second term is the interference from the N

1
users in the cell, and the last term is the AWGN component. The bit error probability at
the 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
S
=
P
tar
b
2
0
=
P
tar

(5.16)
while the total noise power is the sum of the interference power coming from other users
and the AWGN power of the receiver. The AWGN power at the output of the matched filter
is given as
η
=
var

Z
n
(
T



(5.17)
where n
(
t
)
is the AWGN component. Consequently,
η
=
4
T
2
b
E

Z
T
b
0
n
(
t
)
c
0
(
t
)
cos

n
(
u
)
c
0
(
t
)
c
0
(
u
)
cos
(
ω
1
u
+
ˆ
θ
)
cos
(
ω
1
t
+
ˆ

0
(
t
)
c
0
(
u
)
cos
(
ω
1
u
+
ˆ
θ
)

cos
(
ω
1
t
+
ˆ
θ
)

dt du

297
where δ
(
t

u
)
is a delta function at t
=
u.So,
η
=
4
T
2
b
Z
T
b
0
Z
T
b
0
1
2
N
0
δ
(

t
+
ˆ
θ
)

dudt
=
2N
0
T
2
b
Z
T
b
0
c
2
0
(
u
)
cos
2
(
ω
1
u
+

0
=
2N
0
T
2
b
1
2
T
b
=
N
0
R
b
=

N
0
T
c

T
c
R
b
=
N
0

=
T
b
=
T
c
. The intracellular interference power is
I
int
=
var

1
T
b
Z
T
b
0
n
int
(
t
)
c
0
(
t
)
dt

int
(
t
)
c
0
(
t
)] =
0. Because the n
int
(
t
)
in Equation (5.21) is a Gaussian random vari-
able, its variance can be found as
I
int
=
1
T
2
b
Z
T
b
0
Z
T
b

(
t
)
is Gaussian distributed having a power of E

n
2
int
(
t
)]
and a double-sided band-
width of W , its double-sided PSD is
E

n
int
(
t
)
n
int
(
u
)] =
E

n
2
int

n
2
int
(
t
)

W
δ
(
t

u
)
E

c
0
(
t
)
c
0
(
u
)]
dudt
=
1
T

(5.24)
298
CHAPTER 5. ANALYSIS OF IS-95
where the intracellular interference power is
E

n
int
(
t
)]
2
=
E
"
N

1
∑
i
=
1
2a
2
i
P
i
b
2
i

i
P
i
(5.25)
because the expectation of cos
2
φ
i
is 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 v
i
which is equal to 1 with probability of µ, and to 0 with probability of 1

µ,whereµ
is defined as the voice activity factor (VAF). By multiplying Equation (5.25) by v
i
, and with
the aid of Equations (5.14) and (5.16),
I
int
S
=
1
G
p
N

tar
of Equation (5.16) for this case of perfect power control.
Thus the intracellular interference-to-signal power ratio given by the summation term in
Equation (5.26) is also reduced by a factor of G
p
after the process of matched filtering.
The energy per bit E
b
measured at the output of the matched filter is a random variable
because of the variations in the path loss, slow fading and fast fading of the mobile chan-
nel. The interference PSD I
0
measured at the output of the matched filter is also a random
variable 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 I
0
, namely E
b
=
I
0
,in
determining the probability of symbol error. Now E
b
=
ST
b
and I

1
I
int
S
+
η
S
:
(5.28)
In Equation (5.26), the summation of v
i
over
(
N

1
)
users may be expressed, upon taking
its expectation, as
E
"
N

1
∑
i
=
1
v
i

From Equation (5.30), the bit error rate (BER) for the BPSK can be expressed as
p
b
=
1
2
erfc

r
E
b
I
0


(5.31)
where erfc
(
σ
)
is the complementary error function [5]. For a required BER, a required
E
b
=
I
0
, namely
(
E
b

6
6
6
4
G
p

E
b
I
0

req

G
p
S
η
7
7
7
7
5

(5.32)
where
b
x
c
represents the largest integer that is smaller than x. Provided the number of active

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
than m,i.e.
p
o
=
Pr

N

1
∑
i
=
1
ν
i
>
m
!

(5.34)
and because ν
i
is a random variable having a binomial distribution, the outage probability
is
p
o

i
(
1

µ
)
N

1

i
:
(5.35)
5.3.1.2 Imperfect power control
In practice, the received signal power P
R
from the ith mobile at its BS will differ from the
target power level P
tar
by δ
i
dB. This error power δ
i
is a random variable that is normally
distributed with a standard deviation σ
e
and is discussed in detail in Section 5.6 and in Ref-
erences [6]– [8]. There are several reasons for δ
i
being non-zero, such as the inaccuracies in

:
(5.36)
According to Equation (5.36), the signal power at the output of the matched filter for the
wanted zeroth mobile is
S
0
=
P
tar
10
δ
0
10

(5.37)
and the intracellular interference is
I
int
=
1
G
p
N

1
∑
i
=
1
v

δ
i

δ
0
10

(5.39)
where δ
0
and δ
i
are two mutually independent random variables of power control errors for
the signal and the intracellular interferers, respectively. By setting ε
=
δ
i

δ
0
, we have from
Equations (5.39),
I
0
int
S
0
=
1
G

the signal-to-interference power ratio can be written as

E
b
I
0

im
=
1
I
0
int
S
0
+
η
S
0
=
1
I
int
S
10
ε
10
+
η
S

=
Pr

E
b
I
0

im
=
S
0
I
0
int
+
η
<
γ
req

5.3. CDMA IN A SINGLE MACROCELL
301
=
Pr

I
0
int
+


η
S
0
!
=
Pr
(
10
ε
10
N

1
∑
i
=
0
v
i
>
G
p

1
γ
req

η
S

=
Pr
(

k10
ε
10
>
G
p

1
γ
req

η
S
0







N

1
∑
i

o
is the product of two probabilities, p
1
and p
2
. We will first
consider the probability that there are k active intracellular users,
p
2
=
Pr

N

1
∑
i
=
0
v
i
=
k
!
:
(5.44)
The variable v
i
is either 1 or 0 depending on whether the ith mobile user is active or not.
The probability that a user is active is µ, and is called the voice activity factor (VAF). It will


1. Hence,
p
2
=
N

1
∑
k
=
0

N

1
k

µ
k
(
1

µ
)
N

1

k

p
1
=
Q
2
6
6
4
G
p

1
γ
req

η
S
0


kE
h
10
ε
10
i
r
k var
h
10

2
:
Pr

X
>
x
]=
Q

x

µ
σ

:
(5.50)
Because ε is a random variable with normal distribution, the mean of the term 10
ε
10
in
Equation (5.48) can be derived by following the same procedure as used in Section 3.3, i.e.
E
h
10
ε
10
i
=
Z

ln
(
10
)
10

2
Z
∞

∞
exp


1
2
h
ε
p
2σ
e

ln
(
10
)
p
2σ
e
10

e
ln
(
10
)
10

2

(5.51)
where σ
e
is the standard deviation of the power control error. The variance of the 10
ε
10
is
var
h
10
ε
10
i
=
E
h
10
ε
10

E

10
i
=
E
h
10
ε
5
i

"
exp

σ
e
ln
(
10
)
10

2
#
2

(5.53)
where
E
h
10

=
exp

σ
e
ln
(
10
)
5

2
8
>
>
<
>
>
:
Z
∞

∞
exp


1
2
h
ε

σ
e
ln
(
10
)
5

2
f
1

Q

∞
]g =
exp

σ
e
ln
(
10
)
5

2
:
(5.54)
5.3. CDMA IN A SINGLE MACROCELL

(
10
)
10

2
#
2
:
(5.55)
The outage probability may now be expressed as
p
o
=
N

1
∑
k
=
0

N

1
k

µ
k
(

10

q
kvar

10
ε
=
10

3
5
9
=

:
(5.56)
5.3.1.3 Performance of the up-link
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 and
slow fading whose standard deviation was 8 dB. A signal-to-AWGN ratio of 20 dB at the
output of the matched filter was assumed and a BER outage threshold of 10

3
was used in
the calculations. Figure 5.5 shows the outage probability from Equation (5.35) for perfect
power control and VAFs of 3/8 and 1/2. For an outage probability of 2%, the single cell
CDMA 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

304
CHAPTER 5. ANALYSIS OF IS-95
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
=
I
0
.
5.3. CDMA IN A SINGLE MACROCELL
305
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
control errors in E
b
=
I
0
.
the BS is given by
s
dn
(
t
) =
N


θ
)
+
p
2P
p
c
p
(
t

τ
p
)
cos
(
ω
2
t
+
θ
) 
(5.57)
where P
i
and P
p
are the transmitted power allocated for the ith mobile and the pilot signal,
respectively, τ
i


τ
i
)
c
i
(
t

τ
i
)
cos
(
ω
2
t
+
θ
) 
(5.58)
where
P
p
=
P
N

c
p

one of its users, say, the zeroth mobile, has the form
R
dn
(
t
) =
a
0
s
dn
(
t
)+
n
(
t
)
=
a
0
p
2P
0
b
0
(
t

τ
0

(
t

τ
i
)
c
i
(
t

τ
i
)
cos
(
ω
2
t
+
θ
)+
n
(
t
)
(5.59)
where the first term is the signal for zeroth mobile, the second term is the intracellular inter-
ference, and the last term is the AWGN component. Assuming that the receiver is correctly
chip synchronised to the zeroth user, we can set τ

0
(
t
)
dt
+
1
T
b
Z
T
b
0
n
(
t
)
c
0
(
t
)
2cos
(
ω
2
t
+
θ
)

t

τ
i
)
(5.61)
is the equivalent baseband intracellular interference. The first term in Equation (5.60) is the
desired signal, the second term is the intracellular interference, while the last term is the
AWGN component.
The performance of the down-link can be obtained by following the same procedure as
used in the up-link. From Equation (5.60), the received signal power component for the
zeroth mobile receiver is
S
=
2a
2
0
P
0

(5.62)
and the power in the received pilot is
S
p
=
2a
2
0
P
p

t
+
θ)
cos(
ω
2
t
+
θ)
c
0
(
t
)
b
0
(t
)
cos(
ω
2
t
+
θ)
c
1
(
t
)
2cos(

N
-1
(
t
)
Base station transmitter
2
P
p
a
0
channel
2P
1
c
i
(
t
)
2
P
i
b
i
(
t
)
R
dn
(

308
CHAPTER 5. ANALYSIS OF IS-95
I
int
=
2a
2
0
∑
N

1
i
=
1
v
i
P
i
G
p
+
2a
2
0
P
p
G
p
(5.64)

The AWGN power, η, is exactly the same as in Equation (5.20). Hence,
E
b
I
0
=
1
I
int
S
+
η
S
:
(5.66)
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-interference
power ratio, as
E
b
I
0
=
1
1

S
:
(5.67)
Adopting a similar approach to the one used for the up-lnk, the outage probability of the
single cell down-link system is
p
o
=
Pr

N
∑
i
=
1
ν
i
>
m
!
=
N
∑
i
=
1

N
i
!


E
b
I
0

req

G
p
S
η
7
7
7
7
5
:
(5.69)
The performance of the down-link in a single cell CDMA system in terms of the BER is
calculated using Equation (5.35). For an inverse fourth power loss law, a slow fading whose
standard 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 cell
system 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. We
now consider the performance of the multiple cellular arrangement shown in Figure 5.10. In
addition to the intracellular interference, there is now interference from neighbouring cells.
This interference is referred to as intercellular interference. The effects of intercellular

τ
i
)+
J

1
∑
j
=
1
N

1
∑
i
=
0
a
ij
s
ij
(
t

τ
ij
)+
n
(
t

cos

ω
1
(
t

τ
i
)+
θ
i
]+
n
(
t
)
+
J

1
∑
j
=
1
N

1
∑
i


τ
ij
)+
θ
ij
]
(5.70)
where the intercellular interference from the J

1 surrounding cells is
J

1
∑
j
=
1
N

1
∑
i
=
0
a
ij
s
ij
(