42

CHAPTER 3

SYSTEM MODEL AND DESCRIPTION

3.1 INTRODUCTION

The increasing demand for transmitting information over a wireless

channel has led to the emergence of Multiple Input Multiple Output (MIMO)

technology. This technology has materialised its promise of providing high

information rates without additional spectral requirements, which has been well

explained in the pioneering works of Foschini and Gans [13] and Telatar [14].

There is a considerably large amount of literature on Rayleigh fading which

considers only Non-Line-Of-Sight (NLOS) components. However, in reality,

there are Line-Of-Sight (LOS) components between the transmitter and

receiver which are best described by the Rician fading distribution. In [15], the

author investigates the capacity limits of MIMO communication systems

following Rician distribution. In [16], the authors arrived at an exact expression

for average mutual information rate of MIMO Rician fading channels when the

fading coefficients are independent but not necessarily identically distributed.

Research work in [17] has established that the presence of strong LOS

components correlates with channel sparsity, thereby reducing the number of

Degrees of Freedom (DoF). The presence of NLOS components reduces the

correlation between the signals thereby increasing the rank of the channel

43

matrix. Capacity of spatially correlated MIMO channels has been obtained in

[18]. Both, single-sided and double-sided correlation has been considered in

[18]. In [19], the author analyses ergodic capacity for MIMO channels with

rank-1 mean matrices. Upper and lower bounds on the ergodic capacity have

been presented in [19]. Upper bound on ergodic capacity for a system

undergoing Rician fading for arbitrary Signal-to-Noise Ratio (SNR) and rank

of matrix is derived in [20].

Researchers have analysed ergodic capacity for MIMO channels with

rank-1 mean matrices, and upper and lower bounds on the ergodic capacity.

Moreover, upper bound on ergodic capacity for a system undergoing Rician

fading for arbitrary Signal-to-Noise Ratio (SNR) and rank of matrix was also

derived. In this Chapter, closed-form expressions for asymptotic tightness and

upper bound of the system incorporating Von Mises Fisher (VMF) distribution

are obtained and plotted. Also, system model have been developed when the

channel is subjected to interference.

3.2 CAPACITY ANALYSIS OF MIMO SYSTEMS

Consider an Nt x Nr MIMO system with Nt transmit elements and Nr

receive elements. Input-Output relation is given as y = Hx + n, here y is the

received signal vector, x is the transmitted signal vector, H is the channel

response matrix and n is the Additive White Gaussian Noise (AWGN) vector.

A MIMO system with Nt transmit and Nr receive elements has capacity given

as

+Ε=

T

t

NN

ICr

HHρ

detlog 2 , (3.1)

where H is the Nr x Nt channel matrix, and ( )⋅Ε denotes the Expectation. It

consists of LOS and NLOS components. Under Rician fading conditions, the

channel transfer function matrix, H, consists of a spatially deterministic

44

specular component, HL, and a randomly distributed component, HW. The

channel matrix is defined as

,

1

1

12

12

1 ttttWWWWrrrrLLLL RRRRHHHHRRRRHHHH+

+

+

= KKKH

(3.2)

where Rr is the receiver correlation matrix, Rt is the transmit correlation matrix

and K is the Rician factor.

In wireless communication systems, the signal propagating through the

channel under consideration is affected by channel properties of the link called

Channel State Information (CSI). It includes signal fading, scattering, power

decay with distance etc. The knowledge of the CSI at the terminal points

enables subsequent transmissions to adapt to the current channel conditions,

which is crucial to achieve reliable communications with high date rates in

multiple antenna systems. Consider a MIMO communication system with Nt

transmit and Nr receive antenna elements. The input-output relationship of the

MIMO channel given in Section 3.2 is rewritten as

nHxy += ρ , (3.3)

where y is Nr x 1 output vector, x is Nt x 1 vector of transmitted signals, H is

the Nr x Nt matrix denoting channel gains for each transmit–receive antenna

pair, and n is the Nr x 1 vector of independent complex Gaussian noise terms.

The signal vector satisfies the power constraint ( ) 1≤ΕH

xx . Here, ρ can be

interpreted as the SNR at each receive antenna element if the channel does not

introduce additional power. It is assumed that the channel matrix, H, is fixed,

i.e., the channel is deterministic. Using Singular Value Decomposition (SVD),

the H matrix can be written as

HHHHVVVVUUUUHHHH ΣΣΣΣ= , (3.4)

45

where U and V are Nt x Nt and Nr x Nr unitary matrices, i.e., rNI=UU

H and

tNI=VVH

, and Σ is an Nr x Nt non-negative diagonal matrix whose diagonal

elements are the singular values of H.

The channel properties affect the characteristics of the independent

channels differently. In most cases, the receiver is aware of the instantaneous

state of the channels from the ongoing transmissions. However, the transmitter

does not keep track of the CSI. Each independent parallel channel is allocated

the same power irrespective of the fading coefficient. This causes unnecessary

wastage of power at channels with low signal energy and high noise content.

To overcome this inefficiency, the transmitter is fed back the appropriate

channel information so that the best channel (channel with less fading) can be

given proportionately more power than the poor channels. This is given by the

waterfilling algorithm [1].

3.2.1 Waterfilling Algorithm

MIMO channel can be decomposed to its characteristic independent

parallel channels for a fixed channel matrix, H, known at the transmitter and

receiver. The transmit power is optimally allocated between these channels.

The MIMO capacity with CSI at the Transmitter (CSIT) and CSI at the

Receiver (CSIR) is given in Equation (10.9) of [1]

( )∑=

≤

+

∑=

H

i ii

R

i

iiBC1

2

2:

1logmax ρσρρρ

, (3.5)

where HR is the number of nonzero singular values, 2

iσ of H. Since HR is the

number of independent paths, it also gives the number of degrees of freedom.

Equation (3.5) signifies that MIMO channel capacity equals the sum of

capacities on each of the independent parallel channels at the received SNR on

46

the ith

parallel channel, iρ , satisfying the condition that ,ρρ ≤∑i

i where

2σ

ρP

= is the average SNR per received antenna under unity channel gain.

3.2.2 Asymptotic Tightness and Upper-Bound of capacity

Shannon’s capacity of a fading channel with receiver CSI for an

average power constraint is given as

( )∫∞

+=

0

2 )(1log γγγ dpBC , (3.6)

where γ is the instantaneous SNR and B is the bandwidth in Hz. Using

Jensen’s inequality,

( )( ) ( ) ( ) γγγγ dpBB ∫ +=+Ε 1log1log 22[ ]( ) ( )γγ +=Ε+≤ 1log1log 22 BB , (3.7)

where γ is the average received SNR. The PDF of the received SNR,γ , varies

as a function of the Rician factor (K) as

( )( ) ( )

+

+−−

+=

γ

γ

γ

γ

γ

γ1

21

exp1 KK

IK

KK

p o , (3.8)

where ( )⋅oI is the zeroth order modified Bessel function of the first kind. The

channel capacity unconditioned on the received SNR is expressed as

( ) ( )∑∞

=

−

−

∂+

−

=

1

1 1)!1(

1

2ln m

m

mmmK

IKKm

e

B

Cξ , (3.9)

where γ

ξ1

= and γ

K+=∂

1.

The integral of the VMF PDF is obtained as

( )

+π

ϑκ

+

ϑκ

+

ϑκ

+πκ= A2

cos

15

4

3

4

!2

sin

3

cos22CP o

22

o

22

o

22

p1 , (3.10)

where ...8

3

2

cos

105

16

15

16

!4

sinA o

22

o

22

+π

ϑκ+

ϑκ= , ( )κpC is the

normalization vector, 0ϑ is the mean co-latitude angle of the cluster of

47

scatterers and κ represents the accumulation of the distribution. Equation

(3.10), the integral of VMF PDF, and is obtained by the simple method of

substitution and grouping of the double integral given as

( ) ( )[ ]( )∫ ∫π π

ϕϑϑϑϑ+ϕ−ϕϑϑκκ=

2

0 0

000p1 ddsincoscoscossinsinexpCP .

(3.11)

The right hand side of Equation (3.11) is the double integral of the VMF PDF

which un-conditions it on the basis of the two angles, viz., Azimuth and

Elevation angles. The upper bound of channel capacity under Rician fading

conditions using VMF directional data is given as

( ) 1

0

1!

1

11ln PlK

le

KB

C

l

lKUB

+

+

+= ∑∞

=

−γ

. (3.12)

Equation (3.12) is obtained by using Equation (3.8) in ( )[ ]γΕ+1ln where

( ) ( ) γγγγ dp∫∞

=Ε

0

is the Expectation of the SNR. The Expectation is used since γ

represents instantaneous SNR.

The asymptotic tightness of capacity of the system under consideration is

expressed as,

( )( ) ( ) ( )( )

( ) ( ),ln1

!1

11

11

1P

n

PPEPKK

me

B

C M

n

nMn

m

M

mmK

∂∂

+∂+∂−−∂−+

−

= ∑∑=

−

∞

=

−−

∞

(3.13)

where E = 0.5772156659 is the Euler-Mascheroni constant, M is the diversity

order, and ( ) ∑−

=

−

=

1

0 !

K

j

j

Kj

ePµ

µµ

is the Poisson distribution of order K. Equation

(3.13) is obtained by computing the unconditional capacity, which is a triple

integral over the area of interest, given as

48

( ) ( )( )

×γ

γ

γ+Ι

γ

γ++−

γ

+γ+κ= ∫

∞

dK1K

2)K1(K

expK1

1logCB

C0

0

2p

[ ]

∫ ∫π π

ϑϑ+ϕ−ϕϑϑκ

ϕϑϑ

2

0

2

0

coscos)cos(sinsinddsine 000

. (3.14)

In (3.14), integrals I1 and I2 are evaluated separately. Here, I1 is

evaluated after expanding the modified Bessel function of order 0. The

emerging steps are further reduced using Poisson distribution and Exponential

integral, ( )x1Ε of first order. Also, I2 is evaluated by the simple method of

substitution and grouping. Capacity can also be considerably increased by

using the appropriate antenna for transmission and reception.

3.2.3 Numerical Results

In Figure 3.1(a), capacity versus SNR (dB) of a MIMO system

incorporating VMF for varying levels of spatial diversity is depicted. The

power level, Po, is kept at a constant value of 5 W. The wave number is

18.85m-1

and the Rician factor is 7. It is observed that as the diversity order is

increased, i.e., as (Nt, Nr) is increased, and takes values given as (3, 4), (3, 5)

and (4, 7), the capacity increases. As the number of antenna elements increases,

more spatial streams are available. This enables data to be sent through more

independent channels. All the paths do not suffer the same amount of

shadowing and multipath. Some signals are less affected by noise than the

others. This enables better reception of the strong signals than the others, which

in turn increases the capacity.

In Figure 3.1(b), Capacity vs SNR (dB) is plotted for varying power

levels. The number of transmit antennas is 3 and the number of receive

antennas is 4. The radius of the UCA is taken as 0.5 m. The graph shows that

with increase in SNR (dB), the capacity also increases. This has been plotted

49

for three different power levels, Po = 5 W, Po = 10 W, and Po = 15 W. It can be

observed that for a fixed SNR (dB), with an increase in the power level, the

capacity increases. The difference in capacity at low SNR (dB) values is

compared to that at high SNR (dB) values. This is because at low SNR (dB)

values, noise power is comparable to the signal power. As SNR (dB) increases,

signal power dominates and this reflects as an increase in capacity.

Figure 3.2(a) and Figure 3.2(b) show the PDF and CDF of capacity at

different antenna configurations, respectively. It can be observed that as the

configuration increases, the relative frequency of occurrence of high data rate

also increases. However, it can be observed that when the number of antenna

elements at both the transmitter and receiver are identical, the magnitude of the

(a) (b)

Figure 3.1. Capacity vs SNR in dB (with VMF) for (a) varying antenna

configurations, (b) varying transmit power levels.

0 5 104

5

6

7

8

9

10

11

12

13

SNR (dB)

Spectr

um

Eff

icie

ncy (

bps/H

z)

P0=5

k=18.85

K=7

Nt=4,N

r=7

Nt=3,N

r=5

Nt=2,N

r=3

0 5 101

2

3

4

5

6

7

8

9

10

11

SNR (dB)

Spectr

um

Eff

icie

ncy (

bps/H

z)

Nt=3 N

t=4

Radius=0.5m

P0=15

P0=10

P0=5

50

Figure 3.2. (a) PDF of capacity for varying antenna configurations (b) CDF of

capacity for varying antenna configurations.

PDF is lower than the case when the number of transmit antenna

elements is lower than the number of receive antenna elements. This is because

all signals radiated from the antenna elements have to be received and decoded

at the receiver. The probability to successfully receive and decode more

number of signals by the same number of receive elements is lower than that

for lower number of signals. Hence, the relative frequency of occurrence of

higher rate is lower.

Figure 3.3(a) and Figure 3.3(b) show the variation in capacity for

channels undergoing Rician fading. The cluster of scatterers causing multipath

fading is directionally distributed using VMF distribution. Capacity equation is

plotted for different configurations, viz., 2 x 2, 2 x 3, 3 x 3 and 4 x 4. It can be

0 5 10 15 20 250

0.05

0.1

0.15

0.2

Rate (bps/Hz)

PD

F o

f C

apacity

No. of realizations = 50000

K=10

2 x 2

2 x 3

3 x 3

3 x 4

4 x 4

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Rate (bps/Hz)

CD

F o

f C

apacity

No. of realizations = 50000

K=10

2 x 2

2 x 3

3 x 3

3 x 4

4 x 4

51

observed that channel capacity increases with increase in SNR. The magnitude

of capacity at each SNR value is higher in the case where CSIT is available

(waterfilling). The presence of rich scattering environment increases the

achievable data rate at the receiver as shown in Figure 3.3(a). This is because

the transmitter pumps in relatively more power in those channels with better

SNR. However, the presence of correlation at both the transmitter and receiver

reduces the capacity increase as the number of independent channels reduces

due to spatial correlation as shown in Figure 3.3(b). The uncorrelated capacity

has been plotted for different configurations, viz., 2 x 2, 2 x 3, 3 x 3 and 4 x 4.

However, to reduce correlation, the antenna should be appropriately designed

with optimum antenna spacing between each element. It can be observed that

as diversity increases, the rate of increase of capacity becomes linear.

Figure 3.3. Spectrum Efficiency vs SNR in dB for system using Waterfilling

algorithm. (a) System with no correlation (b) System with double sided correlation.

Figure 3.4(a) shows the asymptotic plots of capacity of a system using

VMF and Rician fading vs its analytical value. As expected, the analytical

5 10 152

4

6

8

10

12

14

16

18

20

SNR (dB)

a

Spectr

um

Eff

icie

ncy (

bps/H

z)

K=7

5 10 152

4

6

8

10

12

14

16

SNR (dB)

b

Spectr

um

Eff

icie

ncy (

bps/H

z)

Nt=2 N

r=2

Nt=3 N

r=3

Nt=2 N

r=3

Nt=4 N

r=4

Nt=4 N

r=4

Nt=3 N

r=3

Nt=2 N

r=3

Nt=2 N

r=2

52

curve lies beneath the asymptote, which is the tangent at infinity. It gives the

system performance at high SNRs as characterized by the asymptote of the

achievable rate over the logarithmic SNR. Figure 3.4(b) shows the upper bound

of capacity of a system using VMF and Rician fading. The analytical value of

capacity for both doubly correlated and uncorrelated signals have been plotted.

It can be observed that both plots lie below the upper bound of capacity. The

upper bound gives the theoretical limit of the data rate that can be achieved in a

channel undergoing Rician fading. The presence of VMF which directionally

distributes the data, increases the data rate but lies within the constraints set by

the SNR at the receiver.

(a) (b)

Figure 3.4 (a) Plot for Asymptotic tightness of Spectrum Efficiency with Rician

Fading and VMF distribution. (b) Plot for upper bound of spectrum efficiency with

Rician Fading and VMF for correlated and uncorrelated signals.

0 5 10 1530

40

50

60

70

80

90

100

110

120

130

SNR (dB)

Spectr

um

Eff

icie

ncy (

bits/s

ec/H

z)

K=2

Conc. Parameter=100

E=0.577215

0 5 10 15 200

10

20

30

40

50

60

SNR (dB)

Spectr

um

Eff

icie

ncy (

bits/s

ec/H

z)

Nt=4 N

r=4

K=2

Upper Bound

Uncorrelated

Correlated

Asymptote

Analytical Capacity

53

3.3 INTERFERENCE IN CELLULAR NETWORKS

The fundamental principle of cellular systems is that limited radio

bandwidth has the potential to support a large number of users by means of

frequency reuse. Two limiting factors relating to this subject are Co-Channel

Interference (CCI) and Adjacent Channel Interference (ACI). The former is

produced by simultaneous use of the same frequency channel in different

spatial cells. The latter is caused due to spill over of emissions in immediately

adjacent channels as well as non-immediately adjacent channels. Tighter

filtering transition characteristics and receiver filtering are important in high

performance cellular systems.

The Signal-to-interference ratio considering CCI and ACI was

investigated by many researchers [61-69]. Furthermore, mathematical analysis

for calculation of the ratio of signal power to the sum of interference powers

was presented by Rappaport [70].

3.3.1 Cellular Concept

A cellular mobile communications system uses a large number of low-

power wireless transmitters to create cells: the basic geographical service area

of a wireless communication system. Variable power levels allow cells to be

sized according to subscriber density and demand, within a particular region.

As mobile users travel from one cell to another, their conservations are handed

off between cells in order to maintain seamless service. Channels (frequencies)

used in one cell can be reused in another cell some distance away. Cells can be

added to accommodate growth, creating new cells in unserved areas or

overlaying cells in existing areas.

The ultimate objective of a wireless communication system is to host

large number of users in a wide coverage. This limits coverage at the expense

54

of the number of users or vice versa. Initial deployment of wireless networks

dates back to 1924 with one base station providing city-wide coverage.

Although achieving very good coverage, the network can only host a few users

simultaneously. Another base station using the same spectrum and serving the

same area cannot be placed since that would result in interference. Increase in

demand and poor quality of existing service led mobile service providers to

research ways to improve QoS and to support more users in their systems.

Because the amount of frequency spectrum available for mobile cellular use is

limited, efficient use of required frequencies was needed for mobile cellular

coverage. In modern cellular telephony, rural and urban regions are divided

into areas according to specific provisioning guidelines. Deployment

parameters such as amount of cell-splitting and cell sizes, are determined by

engineers experienced in cellular system architecture. Provisioning for each

region is planned according to an engineering plan that includes cells, clusters,

frequency reuse, and handovers [71].

The cellular concept has introduced smaller cells operating with a

channel, which is a split of the allocated spectrum. The number of base stations

is increased to achieve larger coverage and to reduce interference using the

same channel is not allowed in adjacent base stations, but the same channel is

reused in other base stations that are spatially separated. Hence, the degree of

spatial separation directly affects capacity and interference.

A cell can host limited number of users; to increase capacity, if there is

more demand, more number of base stations can be deployed with reduced

coverage. Channels can be allocated with distributed fashion with spatial

separation in mind for the same channels. For instance, if the allocated

spectrum is F Hz, it can be split into n channels, and distributed into N Base

Stations (BS). This is called a cluster, and a cluster is replicated m times to

cover the area. Total capacity, C, then equals to mF. For instance, in a

55

precellular concept, total capacity is F since m = 1 and n = 1. It is clear that,

with an increasing number of neighboring cells, the undesired effect of

interference becomes more problematic, affecting network performance.

Usually, two types of interference are distinguished: CCI, which is caused by

undesired transmissions carried out on the same frequency channel; and ACI,

produced by transmissions on adjacent or partially overlapped channels [71].

Of course, the above analysis gives theoretical capacity, since in real

deployment; cells operating with the same channel cause CCI to each other. To

reduce CCI, cells operating in the same channel should be separated by a

distance to provide ample protection. Co-channel reuse ratio is given by D/R,

where D is the distance between the centers of two co-channel cells and R is

the cell radius.

There is also ACI, which is basically a leak from adjacent channels in

the spectrum due to imperfection in the devices. ACI can be minimized by

keeping frequency separation between each channel in a given cell as large as

possible. Interference is further mitigated by controlling the power of mobile

subscribers. Power control maintains mobile transmission power low enough to

maintain good quality link. Mobile subscribers close to the BS are forced to

reduce power, and mobile subscribers away from the BS are forced to increase

transmit power.

3.3.2 Co-Channel Interference

Co-Channel Interference is one of the primary sources of noise in

cellular mobile radio systems; although, it is a mistake to completely ignore the

effects of Additive White Gaussian Noise (AWGN). CCI is the interference

produced by cells using the same frequency. CCI can become a major problem

due to reuse of identical frequency channels in different cells. Theoretically, in

a hexagonal-shaped cellular system, there are six co-channel interfering cells in

56

the first tier, as shown in Figure 3.5. CCI can be experienced both at the cell

site and mobile units in the reference cell. If the interference is much greater,

then SIR at the mobile units caused by six interfering sites is (on the average)

the same as the received SIR at the reference cell site caused by interfering

mobile units in the six cells. According to both, reciprocity theorem and

statistical summation of radio propagation, the two SIR values can be very

close, and can be expressed as [71]

∑=

γ−

γ−

=IK

1k

kD

R

I

S, (3.15)

where, γ is determined by the actual terrain environment, KI is the number of

co-channel interfering cells, and Dk is the distance between the centers of the

kth

cochannel cell and the reference (center) cell.

The six co-channel interfering cells in the second tier cause weaker

interference; hence co-channel interference from the second tier of interfering

cells may be negligible. Considering interference reduction factor, qk, yields

[72]

( )

,

q

1

R

D

1

I

SII

K

1k

k

K

1k

k ∑∑=

γ−

=

γ−=

= (3.16)

where qk is defined as R

Dk , and Dk is the distance between the center of the kth

cochannel cell and the reference cell, where the same frequency may be reused.

The co-channel SIR decreases as the tier coverage increases, because of

interference contributions from more co-channel cells. The decrease in co-

channel SIR becomes negligible when the tier coverage increases to exceed a

threshold value.

57

. . .

. . .

Fig

ure 3

.5.

Co-c

han

nel

cel

ls i

n d

iffe

rent

tier

s.

Refe

ren

ce U

ser

Fir

st t

ier C

o-

ch

an

nel

cell

s

Secon

d t

ier C

o-

ch

an

nel

cell

s

Kth

tie

r C

o-

ch

an

nel

cell

s

58

3.3.3 Signal To Noise Plus Interference Ratio of Co-Channel

Interference

In a high-capacity mobile radio system, the reduction of CCI can be the

most important advantage of diversity. A diversity combiner changes the

probability distribution of the ratio of the desired signal and interfering signal

power presented to the FM detector. The distribution of the Intermediate

Frequency (IF) SIR can be converted to that of the baseband SIR by computing

the average output signal, average output interference, and signal suppression

noise. The baseband interference is dominated by occasions that the FM

detector is captured by the interferer. As a result, suppression of interference by

increasing the modulation index is not successful in the presence of Rayleigh

fading without diversity. With diversity, not only is the IF SIR improved, but

one can achieve some interference suppression with FM demodulation by

increasing the modulation index. The amount of “index-cubed” baseband SIR

improvement that can be achieved in this way increases with the number of

diversity branches [74].

In the case, where there are many co-channel interferers, each with

different modulation signals, the Central Limit Theorem (CLT) can be applied.

The sum of interferers is then approximated by Gaussian noise with power

equal to the sum of average interferer powers.

The generalized expression for Signal to Noise plus Co-channel

Interference ratio is given by [73]

11

CCI

1

0

b

2

I

S

N

E

)IN(

S−

−−

+

µ=

+

, (3.17)

where

µ

0

b

2

N

Erepresents SNR considering AWGN alone, µ represents the

Rayleigh faded random variable, and since signals from other cells’ base

59

stations arrive at the reference user asynchronously, even when the system is

designed to be inter-cell synchronous, multiuser interference due to

transmissions from base stations other than the reference base station, ie; CCI

is approximated as [73]

=

∑∑

==

−i0 k

1k 0

ik

i

1i

1

CCI P

P

N3

2

I

S. (3.18)

Here, N = W/R is the system processing gain (Total channel

bandwidth/Data rate of one user), ki represents the number of users within the

ith

cochannel cell, Pik represents the average transmit power from the ith

cochannel’s base station to the kth

user in that cochannel cell as received by the

reference user in the reference cell and P0 is the average transmitted power.

In practice, only the first-tier co-channel cells (cells adjacent to the

reference cell) significantly affect (S/I)CCI. The effect on (S/I)CCI of the second-

tier co-channel cells (cells adjacent to the first-tier co-channel cells) can be

included in the overall SNR expression, but due to the relatively negligible

effect of second-tier co-channel cells or higher, their corresponding

interference contribution will be omitted.

Assuming perfect power control at the base stations, the power ratios in

the SNIR can be replaced with distance ratios. The received power from a co-

channel is inversely proportional to the distance from the appropriate

corresponding co-channel cell transmitter to the reference mobile’s location

raised to the appropriate propagation path loss exponent for that cell, i.e;

,R1P in

iik ∝ (3.19)

where Ri is the distance from the ith

base station transmitter to the reference

user, and ni is the propagation path loss exponent from the ith

cell to the

60

reference user. Likewise, the received power from the reference cell base

station at the reference mobile is inversely proportional to the distance from the

appropriate reference cell transmitter to the reference mobile’s location raised

to the propagation path loss exponent for the reference cell, i.e;

0n

00 R1P ∝ , (3.20)

where R0 is the distance from the appropriate reference cell transmitter to the

reference user, and n0 is the propagation path loss exponent for the reference

cell. Assuming the constant of proportionality is the same for all base stations,

the power ratios can be written as

.R

R

P

Pi

0

n

i

n

0

0

ik= (3.21)

The evaluation of SNIR for an arbitrary location within the reference

cell is both difficult and unnecessary. Systems must be designed for the

smallest expected SNIR, hence evaluation of worst case SNIR is sufficient. For

worst case analysis, the mobile unit is located on its reference cell’s boundary

for omnidirectional and sectoring architectures. Although the cell boundary is

at any point on the perimeter of the cell, the boundary is considered to be at the

farthest location from the center of the cell to truly represent the worst case. As

such, the cell radius R, the distance from the center of the cell to any of the six

vertices of the cell, where each cell is assumed to be hexagonal, is used as the

position of the reference mobile for omnidirectional and sectoring

architectures.

3.3.4 Adjacent Channel Interference

Adjacent-Channel Interference is the interference caused by

extraneous power from a signal in an adjacent channel. ACI may be caused by

inadequate filtering (such as incomplete filtering of unwanted modulation

61

products in FM systems), improper tuning or poor frequency control (in the

reference channel, the interfering channel or both). The problem can be

particularly serious if an adjacent channel user is transmitting at a very close

range to a subscriber’s receiver, while the receiver attempts to receive a base

station on the desired channel. This is referred to as the near-far effect,

whereby a transmitter (which may or may not be of the same type as that used

by the cellular system) captures the receiver of the subscriber.

Alternatively, near-far effect occurs when a mobile close to a base

station transmits on a channel close to one being used by a weak mobile. The

base station may have difficulty in discriminating the desired mobile user from

spill over caused by close adjacent channel mobiles. There is also ACI, which

is basically a leak from the adjacent channel in the spectrum due to

imperfection in the devices.

The presence of ACI reduces the effective SNIR, and therefore, the

number of errors in reception is increased. ACI can result in reduced network

capacity in a multioperator OFDM environment. ACI can be minimized by

keeping frequency separation between each channel in a given cell as large as

possible. Interference is further mitigated by controlling the power of the

mobile subscriber. Power control maintains the mobile transmission power low

enough to maintain a good quality link. Mobile subscriber close to a BS is

forced to reduce the power, and away from the BS, is forced to increase the

transmit power.

62

3.3.5 Signal to Noise plus Interference Ratio of Adjacent Channel

Interference

ACI at the mobile receiver may result from a channel radiated from the

same base station as the desired channel of the mobile shown in Figure 3.6 or

from a different base station.

Figure 3.6. Adjacent Channel Interference caused by the same base station.

The generalized expression for signal to noise plus adjacent channel

interference ratio is given by [73]

11

ACI

1

0

b

2

I

S

N

E

)IN(

S−

−−

+

µ=

+

, (3.22)

where .P

P

NG3

2

I

S 0k

1k 0

k

2

1

ACI

∑=

−

λ=

(3.23)

Here, G represents the power gain of the IF filter for the desired signal

relative to the adjacent channel, λ represents correlation of the received

amplitudes of two signals transmitted from the same base station and received

at the same mobile as a function of their frequency separation, k0 represents the

number of users in the reference cell, Pk represents the average transmitted

power from the reference base station to the kth

user in the reference cell as

received by the reference user in the reference cell, and P0 is the average

transmitted power.

Reference

User

63

3.4 SUMMARY

In this Chapter, closed-form expressions for asymptotic tightness and

upper bound of the system incorporating VMF distribution are obtained and

plotted. Also, the basics and the effects of CCI and ACI are discussed in detail.

The SNIR expressions including interference such as CCI and ACI are shown,

which are used to obtain the simulation results. In Chapter 4, the derivation of

spectrum efficiency of a Rayleigh fading channel in the presence of CCI for

different diversity schemes under various adaptation policies is discussed in

detail.