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THROUGHPUT OPTIMIZATION FOR WIRELESS DATA
TRANSMISSION
THESIS
Submitted in Partial Fulfillment
of the REQUIREMENTS for the
Degree of
MASTER OF SCIENCE (Telecommunications Networks)
at the
POLYTECHNIC UNIVERSITY
by
Saket Sinha
June 2001
________________________________ ________________________________ ________________________________ ________________________________
Advisor
Date
Department Head
Date
Copy No.________
ii
Vita
I was born in Patna, Bihar, India. I completed my high school education till 11th
grade from Delhi Public School, R.K.Puram, New Delhi. In 1996 my family moved to
the country of opportunities, United States and I completed the rest of my high school
education in New York.
I completed my 12th grade from Bayard Rustin High School for Humanities and
was accepted into Polytechnic University. I was accepted into the Accelerated BS/MS
Honor Program after a successful completion of which I would be simultaneously
awarded Bachelor of Science and Master of Science degrees. I chose Computer
Engineering for the Bachelor of Science degree and Telecommunication Networks for the
Mater of Science degree.
During my educational carrier at Polytechnic University I had a number of
professional experiences. During the freshman year I worked as a tutor in the Office of
Special Services and Higher Education Opportunity program. I used to teach the students
the fundamental concepts of Programming in C++ and Chemistry. During my
sophomore year, I worked as grader for the department of Computer and information
science. The following summer during my Junior Year, I worked as an Intern with
Pitney Bowes where I was involved in the development of Netscape Plug-in programs
and security issues in the field of wireless data transmission. During the past one and half
years, I have been substantially involved in my thesis in which I have looked at the
Optimization of the Throughput of wireless data. The research has been done under the
able guidance of Dr. David J. Goodman, Department Head of ECE Department at
iii
Polytechnic University. I worked as a Research Assistant under him and have done
substantial amount of work in the field of wireless data transmission.
iv
For my family and friends…
for their love and support throughout my life
v
Acknowledgment
First, I would like to give my deep thanks to Dr. David J. Goodman for his constant and
generous guidance, help and encouragement for the research study at Polytechnic
University. Working with him has been a great pleasure to me. Dr. Goodman’s patience
and guidance have made him not only an excellent advisor, but also a friend.
I would like to express my deep gratitude to Dr. Elza Erkip, whose wisdom, intuition,
encouragement and generous advising helped me a lot during the study of wireless data
transmission. She was of help to me always whenever I needed her and helped in moving
along with my thesis from time to time. Dr. Elza Erkip’s knowledge of information
theory was an invaluable asset to the technical merit of my work.
I would also like to thank Dr. Philip Balaban for his enormous debt, for his kindness and
inspiration and giving me the opportunity to work with him. He was very helpful in
teaching me the basics of wireless data transmissions and furnished me with a lot of
useful information, which helped me in moving ahead with my research and completing
my thesis successfully. His mentoring and guidance is deeply appreciated.
Last but not the least, I would like to thank all my colleagues at Integrated Information
Systems Laboratory, Richard Lavery, Michael Fainberg, Yelena Gelzard, Zinan Lin,
Virgilio Rodriguez and Seong-Gu Kim for creating an enjoyable and friendly atmosphere
for many useful discussions. I would especially like to thank Richard Lavery, Michael
Fainberg, Yelena Gelzard who have been with me since my freshman year and who know
better than anyone about the rigors of graduate study life. They were always there to
answer a question and to get their comments on new ideas I might have had.
vi
AN ABSTRACT
Maximizing Throughput by Way of Power Control for Wireless Data
by
Saket Sinha
Advisor: Dr. David J. Goodman
Submitted in Partial Fulfillment of the Requirements
for the Degree of Master of Science (Telecommunications Networks)
June 2001
In this thesis, we introduce the concept of maximizing the Throughput of the
system while maintaining optimum signal-to-interference ratios (SIR) by means of
optimizing the powers between the terminals inside the cellular system. We have looked
at two kinds of cellular networks: network in which all the terminals operate with equal
priorities and a network in which different terminals are assigned unequal transmission
priorities and tried to optimize the overall throughput while maintaining equal signal-to-
interference ratio by way of power control of the transceivers.
Power control is essential to the operation of wireless networks, because each user’s
power output contributes to the interference experienced by others. Generally, it is
desirable to identify a choice of power levels, which optimize certain network metrics
such as the throughput of the CDMA system being studied. Throughput is highly
dependent on the product of each transceiver’s information rate by its frame success
probability. This probability can be reasonable modeled as strictly depending on the
product of two key variables in a CDMA system: Processing Gain and the signal to
interference ratio (SIR). The maximum effective throughput data on a wireless
transmission is directly related to the channel characteristics. The throughput of a
wireless channel can be maximized by maintaining optimum level of signal to
vii
interference ratio between the transmitted powers in the system. One lesson of cellular
telephone network operation is that effective power control is essential in order to
promote system quality and efficiency [4]. The operating points in a wireless data
communication system results in an unfair equilibrium in that users operate with unequal
signal-to-interference ratios. Further, the power control required to achieve such
operating points are more complex than the simple signal-to-interference ratio balancing
algorithms for voice.
viii
Table of Contents
Abstract ……………………………………………………………………………… vi List of Figures ……………………………………………………………………….. x List of Tables …………………………………………………………………………. xi 1. Introduction ……………………………………………………………………… 1
1.1 Abstract …..……………………………………………………………… 1
1.2 Introduction…………………………….………………………………….. 3
1.2.1 Background of CDMA systems………………………………….. 3
2. Motivation and Description of Utility Function ………………………………. 8
2.1 Motivation for this Research...……………………………………………. 8
2.2 Approach ………. ……………………………………………………….. 9
2.2.1 A model of data transmission over a wireless CDMA network…. 9
2.2.2 The Data Utility Function ……………………………………….. 11
2.2.3 Power Control for Maximum Utility/ Distributed Power Control.. 14
2.3 Network Assisted Power Control …………………………………...……. 16 3. Throughput Optimization using Power Control and SIR balancing in a Non-Fading Channel …………………..………………………………………………….. 19
3.1 Assumptions and Definitions……………………………………………… 19
3.2 Definition of Terminal Throughput ………………………………………. 20
3.3 Literature used in the Derivation of Bit Error Rate ………………………. 20
3.3.1 Different Modulation Schemes ………………………………… 21
3.4 Throughput Optimization with No White Gaussian Noise in the channel… 24
3.4.1 Analysis of the system with no Gaussian Noise ………………... 29
3.5 Conclusions ……………………………………………………………… 31
4. Throughput Optimization in a CDMA network via Power Control in the presence
of White Gaussian Noise ..……………………………………………………………. 35
4.1 Introduction to Signal to Noise Ratio .……………………………………... 35
ix
4.2 Significance of SNR in Communication Channels.…………………….… 36
4.3 Analysis of the Plots ……………………………………………………… 41
4.4 Analysis of the graph of V versus SNR …………………………………. 46
4.5 Conclusions ………………………………………………………………. 47 5. Throughput Maximization in a CDMA network via Power Control of Tranceives
with Different Priorities ……………………………...…………………………….. 49
5.1 Introduction ……………………………………………………………… 49
5.2 Throughput Optimization in a Priority based system……………………... 50
5.2.1 Performance Analysis ………………………………………….. 53
5.3 Relationship of Information Priority, β to Processing Gain, G …………. 59
5.4 Relationship between information priority, β and α ……………………. 63
6. Summary, Conclusions and Future Work ……………………………………. 68
6.1 Concluding Remarks ..…………………………………………………… 68
6.2 Future Work ……………………………………………………………… 70
7. Works Cited ………………………………………………………………………. 71
x
List of Figures
1. DS-CDMA Transmitter Bock Diagram ………………………………………. 4
2. DS-CDMA Receiver Block Diagram …………………………………………... 5
3. Plot of Receiver Power levels versus Distance ………………………………… 13
4. Non-Coherent Detection of binary FSK ……………………………………….. 22
5. Plot of ( )αGf ' ,
αG
f ' , ( )αT vs α when G = 10 and 1=β ……………….. 28
6. Plot of ( )αGf ' ,
αG
f ' , ( )αT vs α when G =16 and 1=β ………………… 28
7. Optimization of base station throughput versus α )1( =β …………..……….. 29
8. Plot of normalized throughput versus α (SNR=1) ………………... ………... 38
9. Plot of normalized throughput versus α (SNR=2) …………………….………. 39 10. Plot of normalized throughput versus α (SNR=5) …………………….………. 39
11. Plot of normalized throughput versus α (SNR=10) …………………….…….. 40
12. Plot of normalized throughput versus α (SNR=50) …………………………… 40
13. Plot of normalized throughput versus α (SNR=100) ………………………….. 41
14. Plot of normalized throughput versus SNR ………………………………… 44
15. Plot of normalized versus α ( )2=β …………….…………………………… 53
16. Plot of normalized throughput vs α )8,2( == Gβ …………………………. 57
17. Plot of normalized throughput vs α )9,2( == Gβ ………………………… 57
18. Plot of normalized throughput vs α )16,2( == Gβ …….………………….. 58
19. Relationship between Processing Gain and β ………………………………… 62
20. Plot of optα versus β …………………………………………………………… 65
xi
21. Throughput optimization with respect to α when 4=β ………………………..66 22. Plot of ( )αT versus α for β =1, 2, 4, 8 …………………………………………67
List of Tables
1. Different Modulation Schemes …………………………………………………... 21
2. Comparison of G and ( )γf ……………………………………………………… 33
3. Maximum values of Overall Throughput and throughput of individual terminals . 58
4. Relationship of criticalG and β …………………………………………………… 62
5. Relationship between β and optα for corresponding Gcritical …………………… 64
1
Maximizing Throughput by Way of Power Control for Wireless Data
Saket Sinha, David J. Goodman
Polytechnic University Brooklyn, NY 11201
Chapter 1
Introduction 1.1 Abstract
Power control is essential to the operation of wireless networks, because each
user’s power output contributes to the interference experienced by others. Generally, it is
desirable to identify a choice of power levels, which optimize certain network metrics
such as the throughput of the CDMA system being studied. Throughput is highly
dependent on the product of each transceiver’s information rate by its frame success
probability. This probability can be reasonable modeled as strictly depending on the
product of two key variables in a CDMA system: Processing Gain and the signal to
interference ratio (SIR). The maximum effective throughput data on a wireless
transmission is directly related to the channel characteristics. The throughput of a
wireless channel can be maximized by maintaining optimum level of signal to
interference ratio between the transmitted powers in the system. One lesson of cellular
telephone network operation is that effective power control is essential in order to
promote system quality and efficiency [4]. The operating points in a wireless data
communication system results in an unfair equilibrium in that users operate with unequal
signal-to-interference ratios [4]. Further, the power control required to achieve such
operating points are more complex than the simple signal-to-interference ratio balancing
2
algorithms for voice. In this paper, we introduce the concept of maximizing the
throughput of the system while maintaining optimum signal-to-interference ratios (SIR)
by means of optimizing the powers levels between the terminals inside the cellular
system. Chapter 1 gives you general introduction about what a CDMA system is and
describes the functionality and the role that a CDMA system plays in today’s cellular
environments. Chapter 2 begins by describing what motivated us to work on the research
presented in this paper and also provides a brief overview of the utility function that we
are trying to maximize in my work by maximizing the throughout of the base station
under the constraints of the power levels of the transmitters in the system. Chapter looks
into optimizing the throughput of a CDMA system using Power Control and SIR
balancing in a non-fading channel. Chapter 4 looks at a system with the presence of
White Gaussian Noise in the channel. Chapter 5 provides a brief overview of optimizing
the Throughput by way for Power Control in a CDMA system in which the transceivers
are given different priorities. Chapter 6 ends the thesis with conclusion and a word on
future work that can be done on the material studies in this thesis.
3
1.2 Introduction 1.2.1 Background on CDMA systems:
In 1989 Code Division Multiple Access (CDMA) was a radically new concept in
cellular communications. Since then it has gained widespread international acceptance by
cellular radio system operators who are attracted by high system capacity and service
quality. CDMA is a form of spread-spectrum, a family of digital communication
techniques that have been used in military applications for many years. The core principle
of spread spectrum is the use of noise-like carrier waves, as was suggested decades ago
by Claude Shannon [1]. Instead of partitioning either spectrum or time into disjoint
“slots” each user is assigned a different instance of the noise carrier. While those
waveforms are not rigorously orthogonal, they are nearly so [2]. And, as the name,
spread spectrum implies, bandwidths are much wider than that required for simple point-
to-point communication at the same data rate. Originally there were two motivations:
a. Either to resist enemy efforts to jam the communications (anti-jam, or AJ),
or
b. To hide the fact that communication was even taking place, sometimes
called low probability of intercept (LPI).
A basic property of the spread spectrum is a substantial increase in bandwidth of an
information-bearing signal, far beyond that needed for basic communication. The
bandwidth increase, while not necessary for communication, can mitigate the harmful
effects of interference, either deliberate, like a military jammer, or inadvertent, like co-
4
channel users. The interference mitigation is a well-known property of all spread
spectrum systems. However the cooperative use of these techniques in a commercial,
non-military, environment, to optimize spectral efficiency was a major conceptual
advance [2]. Spread Spectrum systems generally fall into one of two categories:
frequency hopping (FH) or direct sequence (DS). In both cases synchronization of
transmitter and receiver is required. Both forms can be regarded as using a pseudo-
random carrier, but they create the carrier of the signals in different ways.
CDMA cellular systems use a form of direct sequence. Direct sequence is, in essence,
multiplication of a more conventional communication waveform by a pseudonoise (PN)
±1 binary sequence in the transmitter. Figure 1 below represents the frequencies
occupied by the information signal and the transmitted signal.
Figure 1: A DS-CDMA Transmitter block diagram A second multiplication by a replica of the same ±1 sequence in the receiver recovers the
original signal. The figure below represents decoding the signal at the receiver end.
5
Figure 2: DS-CDMA Receiver Block Diagram The noise and interference, being uncorrelated with the PN sequence, become noise-like
and increase the bandwidth when they reach the detector. The signal-to-noise ratio can be
enhanced by narrowband filtering that rejects most of the interference power. It is often
said that the SNR is enhanced by the processing gain of the channel W/R, where W is the
spread bandwidth and R is the data rate. This is a partial truth. A careful analysis is
needed to accurately determine the performance. In IS-95A CDMA W/R = 10 log(1.2288
MHz/9600Hz) = 21 dB for the 9600 bps rate set [1]. There are two CDMA common air
interface standards:
c. Forward CDMA channel
The forward CDMA channel is the cell-to-mobile direction of communication. It carries
traffic, a pilot signal, and overhead information. The pilot is a spread, but otherwise
unmodulated Direct Sequence Spread Spectrum (DSSS) signal [1]. The pilot and
overhead channels establish the system timing and station identity. The pilot channel also
is used in the mobile-assisted handoff (MAHO) process as a signal strength reference.
d. Reverse CDMA channel
6
The REVERSE CDMA CHANNEL is the mobile-to-cell direction of communication. It
carries traffic and signaling. Any particular reverse channel is active only during calls to
the associated mobile station, or when access channel signaling is taking place to the
associated base station.
Under ideal conditions, in a CDMA network users should not interfere with one another.
In such a network, each user transmits digital information by modulating a waveform
“signature” which uniquely identifies the user. Signatures are either orthogonal or
minimally crosscorrelated. Hence, a correlation detector tuned to the intended
transmitter’s signature allows a receiver to separate the desired signal from those of other
simultaneous transmitters. Thus, under ideal conditions, multi-user interference does not
exist.
However, a typical wireless channel is far from ideal. In such a channel, much
impairment affects the transmitted signal. In particular, lack of user’s coordination
(asynchronous transmission) and multi-path disrupt the orthogonality of the users
signature. This stops the correlator from separating the desired signal from those of
simultaneous users. Under these conditions, multi-user interference becomes a highly
detrimental factor to the operation of a CDMA network. In particular, it gives rise to the
so-called “near-far” problem: a sufficiently powerful interferer could degrade the
receiver’s performance to an arbitrary degree.
7
The above implies that power control is an issue of paramount importance to the efficient
operation of CDMA wireless networks, since, under realistic conditions, each user’s
power output contributes significantly to the interference experienced by others. But
power is not the only factor that needs to be optimized in order to get greater network
efficiency. In general, one would like to determine a power level for each active user in
the network in such a manner that a suitable measure of network performance, such as the
throughput, be optimized. The system’s throughput can suitably be defined as a (possibly
weighted) sum of the contribution of each transceiver. Each transceiver’s contribution to
the throughout can be defined as the product of the transceiver’s information rate by its
frame success probability. The probability can be reasonably modeled as strictly
depending on the product of two key variables: a transceiver’s processing gain (the ratio
of the available bandwidth to the transceiver’s information rate) and its signal to
interference ratio, SIR, (the ratio of the transceiver’s own transmit power to the sum of
the interfering transceiver’s power plus applicable noise power that exists in the
environment). In this thesis, we have tried to maximize the overall throughput of the
wireless system by maximizing the optimizing the power levels of the transceivers in the
system. The throughput of the individual transceiver is maximized based upon the power
levels that are operating at in the system.
8
Chapter 2
Motivation and Description of Utility Function 2.1 Motivation for the Research In today’s world, the success of cellular phones prompts the wireless communications
community to turn its attention to other information services, many of them in the
category of “Wireless data” communications. The quality and bandwidth efficiency of
wireless communication systems depend of effective power control algorithm. A
terminal and base station need to transmit enough power to deliver a useful signal to the
receiver. However, excessive power causes unnecessary interference to other receivers,
and in the case of transmission from a portable terminal, it drains battery energy faster
than necessary. An effective power control is essential to promote system quality and
efficiency. A Network Assisted Power Control (NAPC) techniques is used to maximize
utilities for users while maintaining equal signal-to-interference ratios for all users [4].
The optimization is based upon the properties of the utility function for wireless data
systems defined as the number of information bits delivered accurately to a receiver for
each joule of energy expended by the transmitter. A power control system that
maximizes the utility function maximizes the amount of information that can be
transmitted by a terminal to the base station in a cellular system. The goal of the work is
to provide a means of achieving a fairer (or more equitable) operating point and also
allow implementation of distributed power control using signal-to-interference ratio non-
balancing. The network keeps on broadcasting a common signal-to-interference ratio as
the target. In a CDMA system, the target signal-to-interference ratio depends on the
number of users simultaneously transmitting information to a base station using the same
9
carrier frequency [8]. The number of users present in a system determines the throughput
of the base station. We find that there is a user population size that maximizes the
throughput of the base station. This population size can be viewed as the capacity of a
wireless data system. It corresponds to the capacity of a wireless telephone system,
defined as the maximum number of conversations that a base station can handle within a
signal-to-interference ratio constraint. The goal of the work is to provide a fairer
operation point and also to implement distributed power control using SIR balancing
between the transmitters. The availability of variable transmission rates in a cellular
network raises the problem of controlling them in the most spectrally efficient way [8].
In a cellular environment, the transmission rates are closely related to the signal-to-
interference (SIRs) and the SIRs can be effectively controlled by means of power control,
which is addressed in this paper.
2.2 Approach
2.2.1 A model of data transmission over a wireless CDMA network In a somewhat general situation, a simple model of data transmission over a CDMA
network could be described as follows: In a single cell, non-orthogonal codes carry data
packets. Packet errors are caused by interference and noise. A selective-repeat scheme
based on error detecting codes and acknowledgments allows retransmission of those
packets not successfully detected by the base station. I assume a perfect error detection at
the base station and error-free transmission of acknowledgments from the base station to
the transceivers.
In this simple model, the following quantities and/or concepts are of interest:
10
• N is the number of transceivers transmitting data simultaneously to the base
station.
• Rs is the source rate being used by the transceiver to transmit the data. We
assume in our model that the transmission rates being used by all the transceivers
are constant.
• Rc chips per second is the chip rate of the channel.
• G =S
C
RR
and is identified as the processing gain of the channel.
• The data is transmitted as “packets”, each of which contains L information bits
and a total of M>L bits, which accounts for bits added for error
correction/detection, as well as other overhead.
Of fundamental importance is the probability of correct reception of a packet.
This probability depends on the physical attributes of the system, including the
binary modulation technique being used, the forward error correction scheme, the
nature of the channel, and the details of the receiver, including its demodulator,
forward error correction scheme, the decoder and antenna diversity, if any. We
assume that these properties of the physical layer can be captured via a single
real-valued function, which gives the packet/frame error probability as a function
of the product of the transceiver’s processing gain to its signal-to-interference
ratio.
11
2.2.2 The Data Utility Function
A utility function is a measure of the satisfaction experienced by a person using a product
or service. In the wireless communication literature, QoS is closely related to utility. The
main objectives of the QoS are: low delay and low probability of error [4]. The utility
function of wireless data systems is defined as the ratio of throughput of the system
(number of information bits delivered accurately) to the power transmitted by the
transceiver. The number of bits delivered accurately to the base station = ( )γfM
LRs ,
where M is the size of the packet, L is the number of information bits being transmitted,
Rs is the transmission rate of each transceiver transmitting in the cellular system being
considered here and f(γ) is the probability of successful transmission or in other words
the frame success rate. With channel coding the total size of each packet is M>L bits.
The information is transmitted over the network in packets each containing L bits of
information. The utility (U) of a packet transmission can be viewed as the ratio of the
number of bits transferred, to the energy consumed in the transmission [4]. The number
of bits transferred is given by: )(γLf and the energy consumed in the transmission is give
by: R
PM. Our goal is to maximize the utility function by maximizing the throughput of
the system and by minimizing the power level received by the base station of each
transceiver in the system.
U = i
i
PMRL )f(γ
(1)
where the throughput of the packet transmitted by the transceiver, i, is given by
12
T = MRL
f(γi) (2)
where (RL/M) is the payload transmission rate and ( )γf is the frame success rate defined
as the probability of receiving the packet correctly. The frame success rate depends on γ,
the target signal-to-interference ratio (SIR) at the receiver. The properties of ( )if γ that
make it interesting are: ( ) 1=∞f and ( )
0≈i
i
Pf γ
for 0=iP . The product of the number of
information bits present in the packet (L) and the probability of successful transmission
represents the expected number of bits received accurately by the base station. In a
system of N users, each terminal transmits data to a single base station. The receiver for
terminal i receives energy transmitted by all other terminals in the system. The target
SIR γi, depends on the power level that each of the transmitters in the system is operating
at and their distances from the base station [7]. In real systems, the signal strength at the
receiver depends on the distance from the transmitter. In reality, the received signal is
influenced by myriad details of the physical environment of the transmitter, receiver and
the space between them. Some of these factors are terrain, buildings, people and vehicles
in the signal path, antenna characteristics and the motion of the transmitter and the
receiver. For a system with N terminals simultaneously transmitting to a single base
station, there is a lot of interference experienced by a terminal as a result of the other
transmissions. A terminal, located at the edge of a cell, experiences more interference
from other terminals because it is far away from the base station and the signal strength at
the receiver of that terminal decreases significantly. To achieve a target signal-to-
interference ratio, mobile users at the cell border have to use the highest transmit powers.
On the other hand, the terminals, which are close to the base station, do not have to use
13
much of their battery power. Figure 3 represents the graph of the signal strength vs
distance, at various locations equidistant from the transmitter. Once could observe that
the received signal strength exhibits a wide range of values. Figure 3 also illustrates that
as the distance decreases, the signal strength of the transmitter increases and this is the
exact reasoning for why do the terminals close to the base station have to transmit at a
lower power as compared to terminal that are far away from the base station. Thus,
system capacity becomes a very important issue in maximizing the overall performance
of the system. Thus, power control is needed to overcome the near far problem. This is
how power control works in speech analysis but for digital data it is quite different.
Figure 3: Plot of received power versus distance (m)
14
2.2.3 Power Control for Maximum Utility/ Distributed Power Control
In our research, we consider a single cell of a CDMA wireless data system with N
terminals transmitting their data to the same base station. The path gain of terminal i to
the base station is defined as hi, i = 1, 2,…, N. The SIR experienced by terminal i is
defined as:
γi =
∑=≠
+N
jij1
2jj
ii
hP
hGP
σ=
∑≠=
+N
ijj
j
i
Q
GQ
1
2σ (3)
where G is the CDMA processing gain and is defined as source
channel
RR
. Pi is the transmitted
power of terminal i, and σ2 is the noise power in the base station receiver. The distributed
power control problem seeks an algorithm in which each terminal uses its local
information about its transmission to choose power levels that maximizes the utility
function of each terminal in the CDMA system. Thus, each terminal in the system tries
to achieve the target SIR by periodically learning about its current SIR and then adjusting
its power to reach the equilibrium, assuming that all other terminals in the system
maintain their power levels constant. The maximum utility occurs at a power level for
which the partial derivative of U with respect to Pi is zero:
0=∂∂
iPU
(4)
We observe in Equation (4) that in order to differentiate U with respect to Pi, we need to
know the partial derivative of iγ with respect to Pi. Taking the partial derivative of
iγ with respect to Pi:
15
∑≠=
+=
∂∂
N
ijj
jj
i
i
i
hP
GhP
1
2σ
γ=
i
i
pγ
(5)
Referring to Equation (1) and (5), we can express the derivative of utility with respect to
power as:
( ) ( )
−
∂∂
=∂∂
ii
ii
ii
ff
MPLR
PU
γγγ
γ2 (6)
Therefore, with Pi>0, the necessary condition for terminal i to maximize the utility is
( ) ( ) 0=−
∂∂
ii
ii f
fγ
γγ
γ (7)
We adopt a notation *γγ =i for a signal to interference ratio that satisfies equation (7)
and we call this the equilibrium SIR. Once this equilibrium is reached, all the transceivers
in the wireless system operate with the same SIR, *γ , the solution to Equation (7). What
needs to be examined is whether equal SIR is the best thing to do and whether the base
station throughout be optimized under such conditions. In the case of powers that
represent the solution to Equation (7), we can show that there are power reduction factors
1<α such that all terminals can increase their utility to *'ii UU > by simultaneously
reducing their transmitted powers from *iP to *'
ii PP α= [4]. The power reduction causes
all the terminals to operate at a common SIR *' γγ <i . This in turn results in lower value
of f(γ) in Equation 1. However, with respect to utility, the advantage of a lower power
outweighs the disadvantage of a lower value of f(γ).
16
2.3 Network Assisted Power Control In the Network Assisted Power Control, a terminal involved in the cellular environment
periodically learns about the current signal-to-interference ratio iγ and adjusts its power
to aim for Tγ [4]. Thus if the power of the terminal is Pi, the adjusted power level is
i
TiPγγ
. This affects the SIR of other terminals in the system and causes them to change
their power levels [4]. When all the terminals operate with the same SIR, their signals
arrive at the base station with the same power level, Prec. Thus, for balanced signal-to-
interference ratio:
recii PhP = f or i = 1,2,…,N (8) In our study of wireless data transmission, we seek a value of ?T that produces the
optimum results with respect to the utility function given in Equation (1).
Thus with ?i=?T for i = 1,2,…,N,
Tγ = 2)1( σ+− rec
rec
PNGP
(9)
recP = T
T
NG γσγ
)1(
2
−− for all i (10)
By referring to Equation (1) and substituting Equation (10) in it, we can derive an
expression for the utility function in terms of the common signal-to-interference ratio Tγ :
( )
−−= )1(2 N
Gf
hMLR
UT
Ti
i γγ
σ (11)
17
It could be seen from the equation above that the utility of terminal i is proportional to the
path gain, hi. If we consider, the proportionality factor to be a constant, the target SIR
defined as γT affects the terminals in the same way for all terminals. Therefore, the
utility function of each terminal is maximized once they have achieved the target SIR, γT.
If we adopt the notation, γopt, as the maximizing value of ?T , we can find the value of γopt
by differentiating Equation (11) and setting the derivative equal to zero. When we
differentiate Equation (11) with respect to γT , we obtain the following result:
( ) [ ] ( )T
TTTT d
dfNGGf
γγ
γγγ )1( −−= (12)
Thus, the optimum signal-to-interference ratio, γopt is a solution to Equation (12). Just
like γ*, the equilibrium SIR, γopt is dependent on the function f(γ), which describes the
dependence of frame success rate on signal to interference ratio. Note that γopt, unlike γ*
depends on the number of the users in the system and also the processing gain of the
CDMA system. Therefore, in contrast to the distributed power control scheme with a
target γT=γ*, the Network Assisted Power Control algorithm aims for γT=γopt, the solution
to Equation (12). Also note that γopt, changes constantly as the users come and leave the
system. To keep the terminals informed about the changes of the optimum SIR, the base
station communicates with the terminal on the slow associated control channel that exists
in the wireless systems, and transmits the optimum value of SIR periodically so that the
terminals are informed about the changing levels and they could change their power
levels accordingly [1]. According to the literature, the maximum number of users a
18
CDMA system can support simultaneously under the conditions described above is given
by:
N =T
Gγ
+ 1 (13)
Note that Equation (13) represents the channel capacity of a CDMA system. With G
being constant, we can see that as N grows, the target SIR, Tγ , must decrease in order to
compensate for the increase in N. Also, note that when N = 1, there is only one terminal
in the cellular system and Equation (12) reduces to Equation (7). Therefore, the lone
terminal in the system acts to maximize the utility function, by achieving the optimum
SIR. γopt=γ*, the solution to Equation (7). When two or more terminals transmit to the
same base station, all terminals in the system aim for the common SIR, γopt which is
easier to implement [4].
In speech, the distributed power control leads t globally optimum solution. This is not
the case in data systems. In a data system we can show that if all terminals operate with
the power levels that satisfy Equation (7), they can all increase their utilities by
simultaneously reducing their power by a small amount. The result is formally proven in
[7].
19
Chapter 3
Throughput Optimization using Power Control and SIR balancing in a Non-Fading Channel
In this chapter, I have taken into consideration the Network Assisted Power Control
algorithm described in the previous section in which the optimum SIR is dependent upon
the number of users operating in the system. The capacity of the CDMA system plays a
very crucial rule on the throughput at the base station receiver. The paper on NAPC
began by looking at the throughput of the collection of terminals using a common base
station. In this section, I have taken a general look at throughput of the base station
receiver and how it is maximized based upon the power levels of the transceivers
operating in the wireless system. It could be shown that a good power control mechanism
preserves the equal signal-to-interference requirement and it achieves good results in
maximizing the overall throughput of the base station.
3.1 Assumptions and Definitions:
The factors described in section 2.2.1 apply to this section too. The following
assumptions are made in this section:
(i) R b/s is the transmission rate of the transceiver and is fixed.
(ii) We assume that there are 16 redundant bits being used for channel coding.
We also assume that all of these bits appear in the frame check sequence for
error detection.
(iii) The number of undetected errors is negligible.
(iv) The total packet length, M = 80 bits.
20
3.2 Definition of Terminal Throughput In this section, we are considering a two-user scenario transmitting their data
simultaneously to the base station. This is the most primitive case of a CDMA system.
According to assumptions (iii) and (iv) described in the previous paragraph, if binary
errors affecting the number of bits in a frame are mutually independent, then the
probability of successful transmission of each terminal, ( )if γ and is a consequence of
the independence assumption:
( ) ( )( )Mii BERf γγ −= 1 (14)
where M represents number of bits present in a transmitted frame and γi represents the
SIR experienced by terminal i in the system. Equation (14) takes into account the amount
of information bits being delivered accurately to the base station, which depends on the
SIR level of one transmitter which in turns depends on the received power levels of all of
the signals but only on iγ , the SIR of terminal i in the CDMA system [4].
3.3 Literature used in the derivation of the Bit Error Rate John J. Proakis in his book has proven that the probability of correct decision at the
receiver in the transmission of M-ary (not to be confused with the total number of bits in
a frame) orthogonal equal energy signals over an AWGN channel, which are envelope
detected at the receiver is given by [2]:
( )
+−
+
−−= ∑
= o
snM
nC Nn
nnn
MP
)1(exp
111
10
ξ (15)
21
where o
s
Nξ
is the SNR per symbol. Then, the probability of a symbol error, which is
PM =1-PC, becomes
( )
+
−+
−−=
+−
=∑
o
bnM
nM Nn
nnn
MP
)1(exp
111
111
1
ξ (16)
where o
b
Nξ
is the SNR per bit [2]. For binary orthogonal signals (M = 2), Equation (16)
reduces to a very simple form and the probability of correct decision is given by
o
b
Nb eP 2
21
ξ−
= (17)
Relating Equation (17) to the ( )iBER γ in section 3.2, we will assume that the
interference has the same effect as White Gaussian Noise (WGN) and therefore,
o
ii N
ξγ = where iξ is the received energy per bit at the receiver.
3.3.1 Different Modulation Schemes: In the context of this study, the main effect of the modem is on the optimum signal-to-
noise ratio γopt, the solution to Equation (9), with f(γ) given by Equation (11). Table 1
represents data for four modems described in communications textbooks used in a non-
fading Gaussian channel: binary phase shift keying, differential phase shift keying,
coherent frequency shift keying, and non-coherent frequency shift keying.
Table 1: Different Modulation Schemes Binary PSK Differential
PSK Coherent FSK Non-Coherent
FSK ( )iBER γ ( )iQ γ2 ( )iγ− exp
21
( )iQ γ
−2
exp 21 iγ
22
∫∞
−=x
duuxQ )2/exp(21
)( 2
π
For our research we are using a Non-Coherent Frequency Shift-Keying (FSK) channel. The non-coherent receiver for FSK is shown in figure 4 below:
Figure 4: Non-Coherent detection of binary FSK
The filters H0(? ) and H1(? ) are matched to the two RF pulses corresponding to 0 and 1,
respectively. The outputs of the envelope detectors at t = T0 are r0 and r1, respectively.
The noise components of output of filters H0(? ) and H1(? ) are the gaussian r.v.’s n0 and
n1, respectively with nnn σσσ ==10
. An orthogonal FSK is assumed here.
From the practical point of view in communication systems, FSK is preferred over
Amplitude Shift keying (ASK) because FSK has a fixed optimum threshold, whereas the
optimum threshold in ASK depends on the signal level [3]. Hence, ASK is more
susceptible to signal fading than FSK. In FSK, the decision requires comparison between
r0 and r1, the problem of signal fading does not arise here. This is one of the biggest
advantages that a non-coherent FSK receiver have over the non-coherent ASK receiver
( )ω1H
( )ω0H
Envelope detector
Envelope detector
Comparator
Input
t = T0
Decision: Select target
or
1r
t = T0
23
and thus we choose this model for our research. One of the biggest disadvantages of the
FSK is that it requires greater bandwidth than that of ASK [2].
Referring to Equation (14) in section 3.2 in this thesis, we will model the interference
experienced by a terminal from other terminals in the system as white gaussian noise and
hence we make use of the expression in Equation (17) to represent the Bit Error Rate
(BER) of the channel being used in this research:
BER( iγ ) =
−
2exp5.0 iγ
(18)
The throughput achieved by each terminal in the CDMA system is defined as the number
of correct bits received per second by the base station and is expressed in b/s. It is
proportional to the frame success rate given in Equation (14). The throughput of
individual terminals in the system, Ti be stated as follows:
Ti = M
RLf i )(γ (19)
where R is the transmission rate of each terminal in the system, and L is the information
bits contained in a frame. The total throughput of the system, T, is the sum of the
individual throughputs of the terminals operating simultaneously in the system
T = ∑=
N
iiT
1
= ∑=
N
i
i
MRLf
1
)(γ (20)
Since we are considering a CDMA system with only two users, substituting N = 2 in
Equation (20) gives us the overall throughput of the system as:
24
T = ∑=
2
1iiT =∑
=
2
1
)(
i
i
MRLf γ
(21)
We are trying to maximize the overall throughput, T, at the receiver of the base station
with respect iγ for each transceiver in the system. Referring to Equation (3) in section
2.2.3 and using N =2, we can represent the interference experienced by terminal 1, 1γ
and the interference experienced by terminal 2, 2γ , as follows:
22
12
22
111 σσ
γ+
=+
=GQ
GQhP
hGP (22)
211
221 σ
γ+
=hP
hGP=
21
2
σ+GQGQ
(23)
3.4 Throughput Optimization with no Gaussian Noise in the channel Let us first consider a system with only two terminals transmitting to a single base station
located inside the cell. We assume that the channel has no White Gaussian Noise or in
other words 02 =σ in equations (3), (5), (22) and equation (23). The system being
looked at has fixed packet length of M bits and fixed transmission rate of R b/s. We start
our analysis by considering a non-prioritized base system in which all the terminals
operating in the system have equal data priority. Then, according to Equation (21), the
throughput of the system with only two terminals transmitting simultaneously can be
expressed as:
T = MLR ( )
+
αα
GfGf (24)
25
where γ1 is the SIR experienced by transceiver 1 and γ2 is the SIR experienced by
transceiver 2 are defined as follows:
γ1 = G2
1
(25)
γ2 = G1
2
(26)
Note that the above two equations are derived from the definition of iγ in Equation (3)
with 02 =σ . We adopt a notation α =2
1
and applying this to Equations (25) and (26)
above, reduces them to the following set of equations:
γ1 = Gα (27)
γ2 = αG
(28)
Substituting (27) and (28) in (24), reduces the definition of the throughput of the system
to:
T = MLR ( )
+α
αG
fGf (29)
In this thesis, the factor MLR
in Equation (29) is assumed to be constant and thus we
normalize the throughput of the system in Equation (29) by this factor and hence adopt a
new notation to represent the normalized throughput of the base station as
26
( )
+=α
ααG
fGfV )( (30)
which will be used throughout this thesis.
Notice from Equation (30) that the normalized throughput of the base station is a function
of processing gain, frame success rate, the number of bits in a frame and the ratio of
transmitted powers of the transceivers. Thus, the problem being addresses here is finding
the optimum level of the received powers at the receiver so that the throughput of the
system is maximized. In other words, we were trying to find out the optimum value of α,
which would result in maximum number of successful transmissions. To maximize ( )αV
with respect to α , we consider α to be a continuous variable and differentiate Equation
(30) to obtain
( )
−
+=
∂∂
2
1''
ααα
αG
GfGGfV
(31)
Setting Equation (30) = 0 and solving for optimization with respect to α gives us the
following result
=
ααα
GfGf '
2' 1
)( (32)
Note that α =1 satisfied this condition.
The first point to address in optimizing the normalized throughput with respect to α is
whether 1=α represents a minimum or a maximum. Figure 7 indicates that this depends
on the value of the processing gain, G. The results of the plot will be discussed in the
next section. We can see from figure 7 that the critical value of G needed to support
27
more than one user in the system in a non-prioritized based system is slightly above 8.
When G = 8, the throughput of the system still stays below 1 at α =1 and by the time
G=9, the throughput of the base station atα =1 is above 1, thus indicating to us that the
critical value of G needed to support the services of two users is above 8. Throughout the
paper, we assume that the packet that is being transmitted between the terminals and the
base station has a length of 80 bits, or in other words M = 80. Another way of looking at
the optimum value of α needed to maximize the throughput of the base station is the
point that satisfies equation (32). Figure 5 shows the plot of
αα
GfGf '' ),( , V( )α
versus α when G=10. The value of G was chosen above the critical value as indicated
earlier. One could notice from the graph that the point where ( )αGf ' and
αG
f '
intersect is exactly 1 and the throughput of the base station is maximized at that point
where they meet. Figure 6 shows the plot of the same functions described above but the
processing gain in this case is G=16. Again, it is clearly visible from the plot that the
optimum value of α needed to maximize the normalized throughput is again 1. We could
also observe from Equation (32) that the optimum value of α that maximizes or
minimizes the equation is indeed 1 provided that the processing gain of the channel is
well above the critical value. The results of the value below the critical value of the
processing gain are discussed in the following section. Through the plots in figures 5 and
6, we have confirmed that 1=α is the point where the local maxima of ( )αV lies and
hence the base station throughput is maximized at this point. Two other interesting points
to be looked at would be the two extreme values of α = 0, ∝.
28
0 1 2 30
1
2Optimization of Throughput
alpha
T α( )
g1 α( )
g2 α( )
α α, α,
Figure 5: Plot of ( )αGf ' ,
αG
f ' , ( )αT vs α when G = 10
0 1 2 30
1
2Optimization of Throughput
alpha
2
0
T α( )
g1 α( )
g2 α( )
30 α α, α,
Figure 6: Plot of ( )αGf ' ,
αG
f ' , ( )αT vs α when G =16
29
0 0.1 0 .2 0 .3 0 .4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
G = 1
G = 4
G = 6
G = 8
G = 9
G = 16
G = 64
Graph of Normalized Throughput versus α (M = 80)
α
Nor
mal
ized
Thr
ough
put
Figure 7: Optimization of base station throughput versus α )1( =β
3.4.1 Analysis of the system with no White Gaussian Noise Figure 5 represents the plot of ( )αV vs α for values of α between 0 and 1. The plot
when presented on a logarithmic scale is symmetric for values of α > 1 because
( )
=
αα
1VV . The observations that could be made from the plot are as follows:
• Initially, when α =0, transceiver 1 is the only transmitting terminal in the system
and the throughout of the system starts off at 1 since it is the lone terminal in the
system.
• When G is low, the system does not have sufficient bandwidth to support two
terminals and 0=α leads to higher throughput than 1=α . With 0=α , P1=0 and
30
terminal 2 transmits without interference producing V(0) = 1. In order for the
system to allow two terminals to transmit their data simultaneously, the
processing gain of the channel has to be increased. Increasing the processing
gain increases the bandwidth of the channel. By increasing the bandwidth of the
channel, we increase the capacity of the system and hence more terminals will be
able to operate in the system simultaneously as the value of G keeps on
increasing.
• Figure 7 suggests to us that for values of G < 8, one of the two terminals has to
shut off its power and let the other terminal use the system all the time.
• When the value of G is greater than critical value 8, the figure shows that the
throughput of the system starts rising above 1 and approaches 2 (the maximum
value of )(αV ) asymptotically as G approaches ∝. This gives us an indication
that as the number of chips being used for the channel starts rising above 8, the
bandwidth of the system is able to let two terminals transmit their data
simultaneously to the central base station located inside the cell. Thus, when G is
high, there is sufficient bandwidth to support two terminals. In this case V(1) =
2f(G) > 1. In the limit as G increases without bound →)1(V 2.
• When G > 8, it can be seen that the optimum value of α needed to maximize the
throughput of the system is equal to 1, at which point of time the two terminals in
the system operate at equal powers. Thus, as the bandwidth of the system
increases, more and more users can be incorporated into the system due to more
network resources becoming available.
31
• An interesting issue is the value G = criticalG such that for G < criticalG , 1 = V(0) >
V(1) and for G > criticalG , 1 = V(0) < V(1). At G = criticalG :
( )GfVV 2)1()0(1 ===
Therefore, criticalG satisfies the equation
( ) 12 =Gf or 21
)( =Gf . (33)
For BER given in Equation (18) and M = 80, this processing gain is
criticalG =8.12
The method that was used in calculating the value of criticalG is shown in section 5.3
below in this thesis.
• Confining our attention to integer values of G, we find that for 8≤G , V(0) >
V(1) and the system throughput is highest when terminal 1 turns off its
transmitter. However, for 9≥G , )(αV is a maximum at 1=α . In this case it is
best to have 21 QQ = in which case both the signals arrive at the base station with
equal power. Because ( )
=
αα
1VV , we could have performed the same analysis
by comparing ( )∞V with ( )1V . With ∞=α , P2 = 0 and thus terminal 1 uses the
system with no interference.
3.5 Conclusions One could derive from the above analysis that as long as the value of G>8, the system is
able to support more than one user and as long as the received powers of the terminals at
the base station receiver are equal, the overall throughout of the system is maximized. At
32
this point, the two terminals utilize the network resources properly, hence maximizing the
performance of the cellular system. This is a very important aspect in designing cellular
communication models since the mobile service providers promise their customers to
offer them with the best quality of service and hence the network engineers have to
design such networks very carefully since there are a lot of changing parameters in
cellular communications. A change in one parameter can adversely affect the
performance of the cellular system and hence a lot of research needs to be put in before
designing an optimal cellular network. A cellular network design consists of all the
parameters that are needed to provider a customer with a low blocking probability, higher
QoS, less usage of battery power, lower call dropping probability, lower signal-to-
interference ratio and much more. All these objectives have to be kept in mind in
designing a cost-effective cellular network. In this section we have addressed the
problem of adjusting the received power levels at the base station receiver in order to
minimize the signal-to-interference ratio among the terminals operating in the system.
By having a lower signal-to-interference ratio, the probability of successful transmission
increases, as fewer packets will be dropped because there is not too much interference
from the interfering terminal. The probability of successful transmission depends
heavily on the processing gain (G) of the channel. The probability that the packets will be
successfully delivered to the base station for different values of G when the two users are
transmitting at equal powers is presented in the Table 2:
33
Table 2: Comparison of G and ( )γf
Processing Gain(G), number of chips used Probability of Successful transmission
( )γf
4 3.68e-3
6 0.12
8 0.47
10 0.76
16 0.98
64 1.0
Thus, once could see from Table 2 that as the number of chips used for channel coding
increases, the probability of successful transmission also increases. In other words, the
greater the bandwidth of the channel, the greater is the probability of information being
delivered to the base station accurately. The overall throughput of the system increases
with increasing G and approaches a maximum value of 2 as ∞→G . Thus as long as the
terminals adjust their powers accordingly and the received powers of the terminals are
equal, the maximum amount of data is delivered accurately to the base station with
minimum errors. Note that in this section we have considered a closed loop power
control algorithm in the reverse direction in which the base station calculates the received
power levels and transmits them back to the terminal telling the terminals in the system to
adjust their power levels accordingly such that the terminals operates with minimal power
and at the same time maximize the signal to interference ratio. The next chapter looks at
34
optimization of throughput of the system in the presence of Gaussian Noise. We will still
consider a system with non-prioritized terminals operating simultaneously.
35
Chapter 4
Throughput Optimization in a CDMA network via Power Control in the Presence of White Gaussian Noise
4.1 Introduction to Signal to Noise Ratio (SNR)
The above results were based upon the assumption that there is no Gaussian noise
present in the channel. But in practical systems, all channels and receiver circuits contain
noise of some sort, which affects the performance of the transmission of data from the
transmitter to the receiver. The ratio of the magnitude of the wanted signal to that of
unwanted noise can be expressed in simple arithmetic ratio called the Signal-to-Noise
Ratio (SNR). In order to understand the effects of noise in communication networks let
us take a simple example. Suppose we take a microphone and connect it to an
oscilloscope tuned appropriately. The emitter is put a few centimeters away from the
microphone. When the emitter is switched off, the oscilloscope will show a straight line,
no signal. When the emitter is switched on, the oscilloscope will show a sine wave.
Suppose now that we put the emitter four times (inverse square law) farther from the
microphone. When the emitter is on, the signal shown is two times weaker and hence by
increasing the amplification of the oscilloscope we can view the signal properly. As we
keep on moving the emitter away from the microphone, the noise present in the system
starts chipping in and this could be clearly observed when the emitter is turned off.
When the emitter is at a considerable distance from the microphone and is turned off, we
will still some waves on the oscilloscope, which is caused due to the noise in the system.
When the emitter is turned on, the sine wave simply adds itself to noise. The noise in the
system remains the same whether the emitter is turned off or on.
36
4.2 Significance of SNR in Communication Channel
In wireless system where we have a transmitter and a base station, the transmitter plays
the role of the emitter and the base station plays the role of the microphone. As the
terminals in the system move away from the base station, the received power of the
terminal at the base station decreases along with the distance. Wireless systems are
particularly difficult to design due to the signals high vulnerability to noise interference,
and changing channel conditions [9]. Under such circumstances the throughput of the
system becomes an important factor. There are many factors that influence the
throughput and hence are there are many different approaches that can be taken to
maximize it [9]. Choosing an optimum power level to maximize the throughput in
presence of noise has been investigated here. As described earlier in this paper, the SIR
experienced by transceiver1 and transceiver2 with no noise power in the base station
receiver is given by equations (25) and (26).
When there is noise in the base station of the receiver the signal-to-interference noise
ratio (SINR) experienced by transceiver 1 is defined as follows:
2
2
12
22
111 σσ
γ+
=+
=Q
QG
hPhP
G (34)
Dividing (34) by 2Q simplifies the equation to
2
22
1
1
1Q
G
σγ
+= (35)
37
Let us adopt the notation that 2
2
Qσ
= SNR
1. Thus the SINR experienced by transceiver1
from the transmission of transceiver2 and noise power in the channel reduces (35) to the
following form
1γ =
SNR
G
11
2
1
+ (36)
Using the earlier definition of2
1
=α , simplifies equation (36) to
11
11 +
⋅⋅=
+
⋅⋅=
SNRSNRG
SNR
SNRG ααγ (37)
In a similar manner the SINR experienced by transceiver2 in presence of noise can be
shown as follows:
112 +⋅
⋅=
+=
SNRSNRG
SNR
Gαα
γ (38)
Notice that when SNR -> ∝, equations (37) and (38) reduces to equations (27) and (28)
which complies with our results when we considered a system with no Gaussian Noise in
the channel. Thus, the normalized throughput of the base station can then be expressed
as:
+⋅⋅
+
+⋅⋅
=11
)(SNR
SNRGf
SNRSNRG
fVα
αα , max10 QQ ≤≤ ; 10 ≤≤ α (39)
It would be interesting to find the optimum value of α that is needed to maximize the
overall throughput of the base station in the presence of White Gaussian Noise.
Depending on what the value of SNR is, the throughput of the base station will vary
38
accordingly and the critical value of processing gain, Gcritical, needed to support more than
one user in the system has to be high where there is too much noise in the channel (lower
SNR) and low when there is less noise in the channel (high SNR). Figure 8 below starts
off by showing the plot of normalized throughput of the base station versus α when
SNR=1.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
α
Nor
mal
ized
Thr
ough
put
Graph of Normalized Throughput versus α in the present of AWGN (SNR=1)
G=8 G=10G=11G=12G-14G=16G=20G=32
Figure 8: Plot of normalized throughput versus α (SNR=1)
39
Figures 9 through 13 represents the same plot as above but for different values of SNR.
0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 10
0 . 2
0 . 4
0 . 6
0 . 8
1
1 . 2
1 . 4
1 . 6
1 . 8
2
α
Nor
mal
ized
Thr
ough
put
G r a p h o f N o r m a l i z e d T h r o u g h p u t v e r s u s α i n t h e p r e s e n t o f A W G N ( S N R = 2 )
G = 4 G = 8 G = 1 0G = 1 1G = 1 2G = 1 4G = 1 6G = 2 0
Figure 9: Plot of normalized throughput versus α (SNR=2)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
α
Nor
mal
ized
Thr
ough
put
Graph of Normalized Throughput versus α in the present of AWGN (SNR=5)
G=4 G=8 G=9 G=10G=12G=16G=20
Figure 10: Plot of normalized throughput versus α (SNR=5)
40
0 0.1 0.2 0 . 3 0 . 4 0.5 0.6 0.7 0.8 0 . 9 10
0 . 2
0 . 4
0 . 6
0 . 8
1
1 . 2
1 . 4
1 . 6
1 . 8
2
α
Nor
mal
ized
Thr
ough
put
G raph o f Norma l i zed Throughpu t ve rsus α i n t h e p r e s e n t o f A W G N ( S N R = 1 0 )
G = 1 G = 4 G = 8 G = 9 G = 1 0G = 1 6
Figure 11: Plot of normalized throughput versus α (SNR=10)
0 0 .1 0 .2 0 . 3 0 . 4 0 .5 0 .6 0 .7 0 .8 0 . 9 10
0 . 2
0 . 4
0 . 6
0 . 8
1
1 . 2
1 . 4
1 . 6
1 . 8
2
α
Nor
mal
ized
Thr
ough
put
G r a p h o f N o r m a l i z e d T h r o u g h p u t v e r s u s α i n t h e p r e s e n t o f A W G N ( S N R = 5 0 )
G = 1 G = 4 G = 8 G = 9 G = 1 0G = 1 6
Figure 12: Plot of normalized throughput versus α (SNR=50)
41
0 0.1 0.2 0 .3 0.4 0.5 0.6 0.7 0.8 0 .9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
α
Nor
mal
ized
Thr
ough
put
Graph of Normalized Throughput versus α in the present of AWGN (SNR=100)
G=1 G=4 G=8 G=9 G=10G=16
Figure 13: Plot of normalized throughput versus α (SNR=100)
4.3 Analysis of the Plots
• Figure 8 represents the plot of the base station throughput versus α when SNR=1,
thus signifying that the noise power in the channel is equal to the received power
of the transmitter and hence it would greatly effect the throughput of the base
station. We can see from the plot above, for values of 20≤G , the base station
throughput remains below 1. This tells us that as long as the value of the
processing gain of the channel is less than 20, only one terminal in the system can
operate while the other turns off its transmitter. Infact the plot conveys to us that
the performance of the lone terminal under heavy noise conditions is not very
significant and it does contribute much to the base station throughput. One could
42
observe from the plot than when G is small, the contribution of the lone terminal
to the throughput is not that significant and the throughput starts off at a value
near 0.5 when G = 8 and very quickly approaches 0. Thus in order for the lone
terminal to perform better under heavy noise conditions, the processing gain of
the channel need to be increased so that enough resources can be allocated for the
terminal to transmit its data accurately. The plot tells us that in order for the lone
terminal to perform better the minimum value of processing gain needed is G=20
which is fairly high. Hence in this case, criticalG =20. If both the terminals try to
transmit their data simultaneously to the base station, both will interfere with each
other’s transmission and there will be a lot of bit errors caused and hence the
performance of the network will degrade very drastically. Under such conditions,
the information bits of both the terminals will not be delivered accurately to the
base station. As the value of criticalGG > , the throughput of the base station
receiver starts increasing and rises above 1. Thus, the processing gain of the
channel needs to be increased in order for both the terminals to operate with equal
powers and minimum interference from each other. One could see from the plot
that when G =32, reasonably high, the base station throughput is maximized when
1=α . Hence, we arrive at the conclusion from the plot that when there is a lot of
noise present in the channel and only one terminal is transmitting, the noise in the
channel prevents the terminal to operate with maximum efficiency for lower
values of G. Hence, to maximize the throughput of the base station, the value of
G needs to be increased to a fairly high level (20) in order to provide the terminal
with enough network resources and bandwidth. For values of 8≤G , the
43
throughout of the base station is 0 since G is very low and hence are not presented
in the plot. When critcialGG > , the system is able to incorporate more than one
terminal and as the value of G keeps on growing, both the terminals are able to
operate with maximum efficiency, thus maximizing the throughput of the base
station at 1=α .
• Figure 9 represents the plot of throughput of the base station versus α but now the
value of SNR=2, which is not a very significant improvement over the previous
value. The noise present in the channel is still very high and hence we expect the
value of G needed to incorporate the transmissions of two terminals to be fairly
high. It could be observed from the plot that as long as G 14≤ , the system is
much better off by letting only one of the terminals transmit in the system. When
G=14, the throughput of the base station shows an improvement and the system is
able to incorporate the transmission data of the second terminal. Observe that
when G = 4 (not shown in figure 9), the throughput of the base station starts off at
a value near 0.5. Hence in order for the one terminal to perform better, the value
of G needs to be increased thus providing the terminal with enough network
resources. When G=8, terminal 2 operates with maximum efficiency
( ( ) 0;1 == ααV ) but when terminal 1 turns on its transmitter power, because of
unavailability of bandwidth, it causes a lot of interference to the second terminal
causing the throughput of the base station to fall down to 0. Hence, we could say
that criticalG =14 in this case.
• When SNR=5, it could be seen from the plot in figure 10, Gcritical is a bit below 11
because at G=11, the throughput of the base station stabilizes and does not fall off
44
the threshold value of 1. We could see from the plot that when G=10, the
throughput of the base station initially stays below 1 but is barely able to make it
above 1 for values of α near 1. As the value of G increases, the base station
throughput also increases and gets maximized when 1=α . Thus the system
performs better as G increases.
• In a similar manner, when SNR=10, one could expect the value of criticalG to
reduce since the noise in the channel is decreasing and hence there will be less
errors made in the transmission of information bits of the terminals to the base
station.
• When SNR is very high such as SNR=50, 100 in figures 12 and 13 respectively,
we observe that Gcritical reduces by a big amount and the system is already better
off when G = 9 and the base station throughput is maximized when 1=α . This is
very obvious because higher values of SNR signify lower noise power in the
channel and hence we do not need a higher degree of coding to protect the
information bits from being corrupted or damaged. The bandwidth necessary to
allow the successful transmission of two terminals in the system is achieved when
G is quite low.
• Thus the general conclusion that can be drawn from the plots presented above is
that when SNR is very high, the noise in the channel is high and hence the
processing gain of the channel needed to allow two terminals to simultaneously
transmit their data is very high since we need better coding and protection. When
SNR is low, it is close to staying that there is no noise in the channel and hence
the system is able to allocate enough network resources to both the terminals for
45
lower values of G and the system performs with greater efficiency. In every
practical network, the engineer of the system tries to maintain the SNR level as
low as possible and the reason to do that is presented above in the plots.
Since we already know that the optimum value of α , optα , needed to maximize the
throughput of the base station is 1, it would be interesting to plot the throughput of the
base station with respect to SNR for 1=α different G. Figure 14 below presents such a
plot when SNR = 14.
0 10 20 30 4 0 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
S N R
Nor
mal
ized
Thr
ough
put o
f the
sys
tem
Graph of throughput versus SNR (M=80)
α =1 G =6
α =1 G =8
α =1 G =10
α =1 G =16
α =1 G =64
Figure 14: Plot of Normalized throughput versus SNR
46
4.4 Analysis of the Graph of V versus SNR
In the earlier chapter, based upon the results we came to a conclusion that in order to
maximize the throughput of the base station, the received power levels of all the
terminals must be equal and hence in this plot we have assumed 1=α and the results are
based upon that. The following conclusions could be drawn from the plot of the
throughput of the system versus SNR:
• It could be seen that as the SNR increases for values of G < 8, the throughput of
the system never quite actually rises above 1 since the network resources are not
sufficient enough to support a system with two users since they require a larger
bandwidth. We could also observe that for lower values of SNR close to zero, the
systems throughput is almost zero since the network resources are under-utilized
suggesting that the network is better off by letting only one terminal operate in the
system. Thus, we could conclude that for lower values of processing gain, only
one user can use the system until there is enough bandwidth available for the
second user to transmit its data to the base station.
• For values of G 8≥ , the throughput of the system increases as the signal to
interference noise ratio of the system increases and hence the network is able to
incorporate the transmission bits of two terminals in the system and let them
transmit at equal powers but the throughout does not quite reach the maximum
value, until the value of G gets closer to 16.
• It could be observed that in order to allow more and more terminals to share the
system and at the same time maximize the throughput of the system, the number
of chips being used should be increased above the critical value of G=8.
47
• The throughput of the system starts reaching the maximum value and is
approximately flat near the highest value at which point more and more users can
begin to share the system and make efficient use of it. Thus, the results agree
with our previous results that we obtained earlier in the paper. The level of signal
to noise ratio for values of G ≥ 16 should be high enough in order to let the two
terminals transmit at equal powers and at the same time maximize the throughput
of the system. More number of are users are allowed to use the system at this
point.
• The results we obtained here are very similar to the results we obtained in section
4.3 earlier. The reader should be able to see that when SNR is too low, the noise
power is high in the system and hence the value of G needed to support more than
one terminal should be very high. G = 8 is the critical value when we have a high
SNR rather than a low SNR.
When both the terminals in the system operate at equal powers, the normalized
throughput of the base station reduces to the following equation:
( )
+⋅
⋅=
+⋅
+
+⋅
=SNRSNRG
fSNRSNRG
fSNR
SNRGfV
12
11)(α (40)
Note that when ∞→SNR , Equation (40) reduces to (33). In a similar manner,
( ) ( )SNRGfVV ⋅==∞ )0( (41)
4.5 Conclusions
In this chapter, we have shown the effects of SNR on the throughput of the base station
and looked at the system for different values of SNR. Some general conclusions can be
48
drawn from the plots that we have seen in this chapter. We have seen that when the SNR
of the system is very low, the noise in the channel is very high and hence in order for the
system to protect the information bits of the two terminals and transmit their information
successfully to the base station, the processing gain of the channel should be high. In a
similar manner, when there is less noise in the channel, SNR is high, the value of
processing gain needed to support the services of more than one user is not very high and
the network is able to provide the terminals with enough resources to transmit their data
at the same time. However, the optimum value of α , optα , needed to optimize the
throughput base station is still 1 provided that the value of SNR is high. The received
power levels of the transceivers in the presence of white gaussian noise are not affected
significantly provided the network has enough resources available to support more than
one user in the system. The noise plays a very important role in communication networks
and may affect the system’s performance. Note that the upper bound on the number of
user in the system is still bounded by Equation (13). If the number of terminals operating
in the system crosses the boundary given by Equation (13), the network performance is
drastically degraded and the network resources will be over utilized. This would result in
an unstable network. Transmission of packets successfully to the base station becomes a
very important issue when there is noise present in the channel. If there is too much
white gaussian noise available in the channel, the packets of information might not be
delivered accurately which results in retransmissions of packets and this might affect the
performance of the network significantly. The next chapter looks into a CDMA system
in which different transceivers are given unequal information priority.
49
Chapter 5
Throughput Maximization in a CDMA network via Power Control of Transceiver with Different Priorities
5.1 Introduction In the previous chapters a fixed information rate has been typically been assumed, and an
optimal power vector has been sought. A transceiver’s processing gain and SIR are equal
partners in determining the probability of success of a packet transmitted from or to the
transceiver. Furthermore, a transceiver’s contribution to the system’s throughput
increases monotonically with the frame success probability to the processing gain [2].
Hence, both transmit power levels and processing gain should be optimized
simultaneously. In the work below we show that even if the contribution of each
transceiver to the network throughput is weighted differently, through a linear
combination (which would reflect a situation in which bits from different users are valued
dissimilarly by the system administrator) still they should aim for the same effective SIR.
Thus, an interesting area of research in the field of wireless communications is how the
power should be allocated between the different transceivers in a system where each
transceiver has a different information priority than other. In practical systems, every
user wants to have his/her data to be delivered accurately and faster to the base station
and the Quality of service offered by the network is very important. A user certainly
wants to have a certain degree of priority given to their mobile terminal while
transmitting data to the base station. The user might be willing to invest more money in
order to get a higher degree of performance. Different cellular subscriber companies
offer different rate plans, which provide the user with a better degree of transmission
efficiency. If a particular mobile subscriber is willing to pay more, his/her terminal will
50
be given a higher information priority as compared to other terminals in the system.
Hence, different telephone companies offer different competitive rate plans to their
subscribers promising them to offer a better quality of service as compared to other
telephone companies. This keeps the competition going in the market and day by day
there has been a growing popularity among the people trying to own a cell phone. If a
customer is provided with a better quality of transmission, then he/she will be definitely
willing to invest their money in that particular rate plan and hence we got the motivation
to explore the system in which different terminals are given unequal information
priorities. It is obvious that if one terminal has a higher information priority than the
other, then it will have better access to the base station as compared to others. It would
be interesting from an engineering point of view to explore how do the terminals adjust
their power in order to maximize the overall weighted throughput of the system, hence
increasing the system efficiency.
5.2 Throughput Optimization in a Priority Based System The following section looks at the throughput optimization in a CDMA system where one
terminal is given a higher information priority compared to other terminals information.
The assumptions are the same as made in the previous chapters. An ideal transmission of
corrupted packets is assumed in this section. We are considering a CDMA system in
which every terminal is transmitting to the base station at the same transmission rate, R.
The number of bits that are transmitted in one packet of data is again considered to be
constant, M = 80 bits. Hence, referring back to Equation (19), the weighted throughput
of terminal i having an information priority, iβ is given by
51
( )ii
i fMRL
T γβ
= (41)
where iβ is the weighting coefficient, reflecting the possible fact that the system’s
administrator may make a “value” of a transceiver’s contribution to the network
throughput dependent upon the specific user and iγ is the signal-to-interference ratio
experienced by the particular terminal from other transmitting terminals in the system.
Thus, the aggregate throughput of the system in which two terminals transmit their data
with unequal information priority is given by:
T = ∑=
2
1iiT =∑
=
2
1
)(
i
ii
MfRL γβ
= ( ) ( )[ ]2211 γβγβ ffMRL
+ (42)
Since MRL
is a constant, it does not affect the optimum power level ratio. Thus we define
the normalized throughput:
( ) ( )211)( γγβα ffV += (43)
Let us consider a very preliminary form of the above equation in which transceiver 1 is
given twice the information priority as compared to the transmission priority given to
transceiver 2. Hence, 21 2ββ = where 1β is the information priority of transceiver 1 and
2β is the information priority of transceiver 2 transmitting their respective data’s to a
single base station located inside the cell as the terminals. If we assume 2β =1, then 1β
being twice as large as 2β gives us the value of 21 =β and hence equation (43) can we
re-written as
52
( ) ( ) ( )[ ] ( )
+⋅=+=
ααγγα
GfGfffV 22 21 (44)
where 21 γγ and are given by Equations (27) and (28). We try to optimize ( )αV with
respect to α , the ratio of the transmitted powers of the two terminals. In order maximize
V with respect to α , we have to take the partial derivative of ( )αV in Equation (44) with
respect to α and set it equal to 0.
0=∂∂αV
(45)
Taking the partial derivative of V with respect to α gives us the following equation
( )2
''2
αα
αα
−=∂∂
GGf
GGfV
(46)
Thus, in order to optimize the throughput, we set the above equation to 0. In doing so we
get
( )αα
αGf
Gf
'2'
2=
(47)
Hence we see that unlike Equation (32), 1=α does not guarantee that the first derivative
is 0. Therefore, the normalized throughput will be maximum when 21 QQ ≠ . We expect
that the optimum 1>α , and 21 QQ > because terminal 1 has a higher priority and hence it
should have more powerful received signal than terminal 2. The first derivative does not
give us much information because the solution to the above equation **α could be a local
maximum or local minimum. A sufficient condition for a local maximum throughput at
**αα = is:
53
0**2
2
<=∂∂
αααV
(48)
5.2.1 Performance Analysis Figure 15 represents the plot of throughput of the system versus a for different values of
a between 0 and 10. We can see from the figure that when 0=α and when ∞=α , the
system consists of a lone terminal and that is the only terminal contributing to the
throughput of the base station receiver. We can see from the figure when G is below a
certain threshold value, the CDMA system is able to provide service to only one terminal
in the system since it has not enough bandwidth to support the services of the other
terminals in the system. Initially, when G is below the critical value of G needed to
support two users, the system is not able to provide sufficient bandwidth to let two
terminals transmit simultaneously and hence the throughput of the base station drops
down drastically and approaches zero. The plot tells us that when the processing gain
(chip rate) of the channel is not high enough to support two users, only one terminal be
allowed to use the system without any interference and in this case it would be terminal 1
( 0, 2 =∞= Pα ). Thus ∞=α leads to a higher throughput than 1=α for low values of G
< criticalG . We define criticalG as the minimum value of G needed to satisfy the equation:
V(1) = V(∞ ) = 2. It is easy to achieve ( ) 2=αV by letting ∞== α,02P . The real
question is what processing gain is needed to have ( ) 2>αV ? By achieving ( ) 2>αV , we
can have the two transmitters share the system simultaneously. Figure 15 suggests that
9≥G is necessary. With G=9, 2=β , the system is able to support two terminals with
unequal power because )(2max ∞=> VV . However, with equal power V(1) < 2.
54
However, with G=9, 1=α is no good since the throughput of the base station is below 2
and hence we can get better results for throughput by letting ∞=α . Thus the system
would be better off by letting only one terminal transmit. But one could observe from
figure 15, when G=10, ( ) ( )∞> VV 1 . Thus we can say that criticalG in this case is 10 and
the method of calculating it is shown in the future section. We can argue that the value of
criticalG according to the definition defined above is not exactly 10 but slightly above 9.
We round up criticalG to the nearest integer to make sure that ( ) ( )∞≥ VV 1 . As the
system’s chip rate increases, the capacity also increases and hence the system will be able
to provide services to more than one user. Similarly, the plot tells us when →∝α ,
terminal 2 has to shut off it’s power and terminal 1 be allowed to transmit it data since as
α grows, terminal 1 transmits at a much higher power than terminal 2 and hence it
would cause a lot of interference to terminal 2 and this would affect the throughput of the
base station. Under such circumstances, the probability of successfully transmitting the
packets of terminal 2 declines and it would not be beneficial for terminal 2 to transmit its
data and hence it has to turn off it’s power. Thus the throughput of the system stabilizes
to 2 in the long run and the system efficiency is maintained.
55
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
G = 2
G = 4
G = 6
G = 8
G = 9
G = 1 0
G = 1 6 G = 6 4
α
Nor
mal
ized
Thr
ough
put
Graph of Normalized Throughput versus α (M = 80)
Figure 15: Plot of normalized throughput versus α ( )2=β
We said earlier that as long as the processing gain of the channel is below a certain
threshold, only one of the terminals is allowed to use the system. An engineer is always
interested in knowing the critical value of G above when the two transceivers can
transmit their data successfully to the base station. If one pays a closer attention to the
plot presented in figure 15, we could see that the value of G needed to support more than
one user is 9. As the value of G starts increasing above 9, the network throughout starts
giving us positive results (rising above 2) and hence the system is able to provide
resources to more than one terminal and they are allowed to operate simultaneously.
Notice that the value of α needed to optimize the throughput is still close to 1 and gets
closer and closer to 1 as G increases.
56
One could observe that we arrived at similar conclusions as discussed in chapter 3. Thus,
as long as the value of G < criticalG , in this case 10, the system is able to support one of
the users since there is not enough bandwidth available to let two terminals transmit their
data simultaneously. As the value of the processing gain, G starts increasing and crosses
the critical value, we can observe from the plot that the throughput of the system starts
rising gradually above 2 for increasing values of G and approaches 3, the maximum value
of V(α ). Thus the critical value of G needed to support more than one user in the system
where one of the terminals has twice the information priority as compared to other
terminal is criticalG = 10.
We could see that as the processing gain increases, the maximum value of the throughput
keeps on rising and gets very close to 3. Practically, it will never touch 3 since not all
system are ideal in nature. Due to the increase in G and the bandwidth of the channel, the
capacity of the system keeps on increasing and for higher values of G, the system can
support more and more users The optimal value of α needed to optimize the system
performance is still very close to 1 and as G increases, it moves closer and closer to 1.
Figure 16 through 18 presents the plot of normalized throughput of the base station
versus α for values of G=8, 9 and 16. If one blows up the graphs at points where ( )αV
approaches the maximum value, one could observe that the value of α needed to
maximize the throughput lies in the vicinity of 1 for value of G = 9 and as the processing
gain increases, the point moves closer and closer to 1 and at high values of processing
gain it is exactly 1. Table 3 below lists the optimum values of α and the corresponding
value of ( )1*2 γf and ( )2γf for different values of G.
57
0 1 2 3 4 5 6 7 8 9 1 00
0 . 5
1
1 . 5
2
2 . 5
α
Nor
mal
ized
Thr
ough
put
G r a p h o f N o r m a l i z e d T h r o u g h p u t v e r u s u α (β = 2 G = 8 M = 8 0 )
N . T h p tu s e r 1 u s e r 2
Figure 16: Plot of normalized throughput vs α )8,2( == Gβ
0 1 2 3 4 5 6 7 8 9 1 00
0 . 5
1
1 . 5
2
2 . 5
α
Nor
mal
ized
Thr
ough
put
G r a p h o f N o r m a l i z e d T h r o u g h p u t v e r u s u α (β = 2 G = 9 M = 8 0 )
N . T h p tu s e r 1 u s e r 2
Figure 17: Plot of normalized throughput vs α )9,2( == Gβ
58
0 1 2 3 4 5 6 7 8 9 1 00
0 . 5
1
1 . 5
2
2 . 5
3
α
Nor
mal
ized
Thr
ough
put
G r a p h o f N o r m a l i z e d T h r o u g h p u t v e r u s u α (β = 2 G = 1 6 M = 8 0 )
N . T h p tu s e r 1 u s e r 2
Figure 18: Plot of normalized throughput vs α )16,2( == Gβ
Table 3: Maximum values of Overall Throughput and throughput of individual terminals
G
optα maxT ( )max1*2 γf ( )max2γf
9 1.2 2.048 1.41 0.638
10 1.1 2.2 1.69 0.51
11 1.05 2.627 1.819 0.808
12 1.00 2.717 1.811 0.9055
13 1.00 2.824 1.883 0.941
14 1.00 2.884 1.92 0.964
15 1.00 2.934 1.956 0.978
16 1.00 2.96 1.973 0.9867
59
5.3 Relationship of Information Priority, β to Processing gain, G In the previous section, we saw that the value of processing gain needed to provide
network resources to allow transmission of more than one terminal to a single base
station is dependent on the information priority factor, β given to transceiver 1. The
necessary value of G increases as β increases and this is clearly evident based upon the
plots that we showed you earlier. In this section, we try to establish the relationship that
exists between β and G. In order to so, we follow the following procedure described
below. The conclusions presented below are based upon the observation so far
• As a? 8 , we adopt the notation of representing the throughput of the system by
( )∞V . Substituting the value of a = 8 in (43) by assuming 12 =β and
representing ββ =1 , the throughput of the base station receiver is given by the
equation below
( ) β=∞V (49)
• When a =1, all the transceivers operate with equal signal-to-interference ratio and
hence we presume that Tγγγ == 21 . Plugging in the value of a and
1,1 21 =≥= βββ in (43) reduces the equation of the normalized throughput of
the base station receiver and thus
( ) ( )M
GV
−−+=
2exp5.111 β (50)
60
Thus, we arrive at the conclusion based upon the equations (49) and (50) that for values
of G < 8, ( ) ( )1VV >∞ . In a similar manner as the value of G starts increasing and
approaches ∞ , ( ) ( )∞> VV 1 . The two equations suggest to us that as long as the
processing gain of the channel stays below criticalG and the ratio of the received power of
the two terminals is greater than 1 and approaches ∞ ( 02 =Q ), the overall throughput of
the system approaches the value of the information priority given to terminal 1, β .
An interesting question that arises from the above derivations is what critical value of the
processing gain (Gcritical) is needed so that the throughput of the base terminal receiver of
a system with only one terminal transmitting is equal to the throughput of the base station
receiver with two terminals operating at equal powers ( ) ( )( )∞= VV 1 . Remember the
definition of critcialG from section 5.2.1? The answer to this question would help us in
establishing a direct relationship between the processing gain, G and the information
priority, ß. In other words, our goal is to find the critical value of the processing gain
(Gcritical) of the channel, the solution to the equation ( ) ( )∞= VV 1 . And the solution of the
equation depends on the value of the information priority index, ß. Therefore, in order to
find the value of Gcritical, we have to solve the following equality for Gcritical in terms of β
and M.
( )M
criticalG
−−+=
2exp5.11ββ (51)
61
Solving equation (51) for Gcritical gives us the following result:
+−−= McriticalG
122ln2
ββ
(52)
Equation (52) gives us a lot of interesting insights into the system being looked at. The
right hand side of the equation provides us with some valuable information. One could
see in Equation (52), that the critical value of the processing gain, Gcritical, is independent
of the powers that the terminals are operating at and is dependent on the value of the
information priority given to the terminals, ß, and the packet size, M, being transmitted
by the terminals to the base station. In our priority-based system, we assign terminal 1
twice the priority than terminal 2. Substituting, ß=2 and M = 80 bits in Equation (52) and
rounding up the result to make sure that ( ) ( )∞≥ VV 1 gives the value of critcialG =10, and
hence this complies with the results we obtained through our plots earlier, which showed
that the critical value of the channel processing gain needed to support more than one
user in a priority based system ( 2=β ) was 10. In a similar manner, when we substitute
ß=1 and M= 80 in Equation (52), we obtain Gcritical = 9 which again proves the results we
obtained in figure 7.
Figure 19 plots the processing gain, G versus the information priority, ß. Notice that as
the value of ß grows, the optimum value of G needed to support more than one terminal
in the system also increases with it. A greater insight into the plot reveals that the
theoretical results obtained earlier for value of ß=1 and 2 agrees with the plot. The graph
also shows that as the priorities of the terminals in the system increases; the processing
gain of the channel also increases along with it since a greater network bandwidth is
62
needed to support more than two terminals operating with minimum interference from
other terminals in the system.
Figure 19: Relationship between Processing Gain and β
Table 4 lists the values of β and the corresponding values of G needed to support the
load of two terminals transmitting simultaneously to the base station.
Table 4: Relationship of Gcritical (rounded up) and β
β criticalG
2 10
4 11
8 12
16 13
32 15
64 16
63
The above table tells us that as long as the channel processing gain is beyond a certain
threshold for a given value of β , the cellular system will be able to provide the
transceivers with sufficient resources and bandwidth to transmit their data.
Thus we have proven that the value of Gcritical depends upon the information priority
index, ß. It increases with increasing value of β . This is exactly what an engineer
would expect to happen under normal circumstances. When a terminal is given a higher
information priority than the other terminal, transmission errors and packet loss are more
costly. The system has to avoid errors in the high priority signals by operating with a high
processing gain or by blocking out low priority transmitters or signals.
5.4 Relationship between information priority, β and α In the previous section, we showed the relationship between β and G. We have seen so
far, that the trade-off in terms of information priority is not that significant and the
optimum value of α needed to optimize the base station throughput lies in the vicinity of
1 as long as the processing gain of the channel is sufficient enough to incorporate the
transmission of two terminals. In the section, we try to establish a relationship between
information priorities β and optα , the optimum value of α needed to maximize the base
station throughput and hence it’s performance. From an engineering point of view one
would expect that as the information priority of terminal 1 increases, the value of
α needed to optimize the throughput of the base station receiver also increases along with
it. We designed a simulation program that found the optimum value of α , optα needed to
optimize the throughput of the base station for the corresponding values of β . Table 5
64
below lists the values of optα , needed to optimize the base station throughput, versus
information priority, β at the critical values of processing gain, criticalG needed to
support more than one user for different β . Keep in mind that values of Gcritical is
rounded off to the nearest integer:
Table 5: Relationship of β and optα for corresponding criticalG
β criticalG optα
2 10 1.120
4 11 1.312
8 12 1.374
16 13 1.400
32 15 1.437
64 16 1.492
As one could see from the table above, the optimum value of a needed to maximize the
overall throughput at the base station receiver increases with increase in the priority index
of terminal 1. Figure 20 represents the plot of the optimum received powers levels at the
base station, optα versus the information priority, β .
65
0 1 0 20 30 4 0 50 60 7 01
1 .05
1 . 1
1 .15
1 . 2
1 .25
1 . 3
1 .35
1 . 4
1 .45
1 . 5
β
Opt
imum
val
ue o
f α
Plo t o f op t imum va lue o f α versus β
Figure 20: Plot of optα versus β
The method of analyzing the throughput of the system based upon different priorities
given to each terminal in the system can be extended to even higher levels such as giving
terminal 1 four times the priority as compared to the second terminal 2. Under such
conditions, the information priority index, ß = 4 and throughput of such as system can be
summarized as follows:
( ) ( ) ( ) ( )
+⋅=+=
ααγγα
GfGfffV 44 21 (53)
where γ1 and γ2 are the same as described in equations (27) and (28). It would be interesting to look at the plot of ( )αV vs α for such a system and find the
optimum value of α that optimizes the base station throughput. According to Table 4,
the critical value of processing gain needed to allow two terminals to transmit their data
66
simultaneously is G = 11 and hence we will be looking at the plot of ( )αV vs α when G
=12. The plot is presented in figure 21. One could notice from the plot that the optimum
value of α needed to maximize the throughput is not exactly 1 (since we chose the value
of G=12). Therefore, as the priority index of a terminal changes, the value of α needed
to optimize the throughput also changes along with it. Hence, as long as the received
power levels of the two terminals transmitting are almost equal, the overall throughput of
the system will be maximized approximately and the system will perform with greater
efficiency and the interference experiences by terminals in the system will be minimized.
In general, all the plots we obtained so far tell us that in order to maximize the throughput
of the system, the received power levels of all the transceivers operating inside the cell be
irrespective of the priorities given to them in the system. Throughout this research we
have concentrated on 1=α because there are known algorithms for achieving equal
received power at the base station receiver and the normalized throughput is always as
good as when optαα = .
0 1 2 3 4 5 6 7 8 9 1 00
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
α
Nor
mal
ized
Thr
ough
put
G r a p h o f N o r m a l i z e d T h r o u g h p u t v e r u s u α ( β = 4 G = 1 2 M = 8 0 )
N . T h p tu s e r 1 u s e r 2
Figure 21: Throughput optimization with respect to α when 4=β
67
In figure 22, we show the plot of the throughput of the base station for values of β =1, 2,
4, 8 when G = 14. This value of the processing was chosen well above the value criticalG
needed for each of the β to support more than one user in the network.
0 1 2 3 4 50
2
4
6
8
Throughput vs alpha
alpha
Thro
ughp
ut o
f th
e sy
stem
9
0
V α( )
V2 α( )
V4 α( )
V8 α( )
xi
50 α α, α, α, s xi( ),
Figure 22: Plot of ( )αT versus α for β =1, 2, 4, 8
The notation used on the y-axis in the figure is of the form )(αβV where β is the
information priority given to terminal 1 and takes on the value described above. The
vertical line evaluates the throughput of the system at 1=α for different β . As you
could see the optimum value of α needed to maximize the throughput of the system
moves away from 1 as β increases and hence complies with the results presented in
Table 5 in section 5.4.
68
Chapter 6
Summary, Conclusions and Future Work
6.1 Concluding Remarks When this work was started, the goal was to provide new insight into the understanding
of the data throughput in a CDMA wireless system. This was not geared towards any
system in particular. No system protocols and/or parameters were used in this work.
Wireless systems engineers, in their design of wireless data systems, should use this
research as a tool.
Some very important results were obtained. We have learned that the data throughput
depends on a wide variety of variables, some of which include the transmission rate, the
received signal power, the received noise power spectral density, the modulation
techniques used and the channel conditions [2]. Given a modulation scheme, and channel
conditions, the optimum received power levels of the transceivers at which they should
operate in the system for maximum throughput could be derived [1]. By optimizing the
power levels of the transceivers, we are able to optimize the signal to interference ratio
between the different transceivers operating in the system Maximizing throughput in a
wireless channel is a very important aspect in the quality of voice or data. In this study,
we have shown that factors such as powers levels of the transmitters and information
priority given to different terminals in the system is a function of processing gain, signal
to interference ratio and the packet length. We have addressed the problem of optimizing
the received power levels of the terminals in the system such that the interference caused
69
due to other terminals is minimized and base station throughput is maximized. The key
concept behind this research is that for each particular channel and transmission scheme
model, there exists an optimum level of received powers at the base station such that the
base station throughput is maximized.
We considered a very simple form of wireless system in which we have assumed that
there are only two terminals operating in the system. We looked at the optimum level of
received powers of different transceivers at the base station that is required to maximize
the throughput of the base station so that the transmission bits are delivered accurately to
the base station without any errors. We have considered two kinds of systems in this
thesis: a system in which all the terminals operate with equal priorities and a system in
which different terminals are given unequal priorities as decided by the system’s
administrator. We have shown through our plots that the optimum ratio of the received
powers of the transceivers required to maximize the throughout of the base station stays
very close to 1 both in a non-prioritized and prioritized networks. We have shown
thatα is quite independent (but not exactly) independent of the priority coefficients
assigned to different terminals in the system.
We can generalize our results to N number of users and obtain similar results.
Throughout this thesis, we have assumed that the terminals involved in wireless data
transmissions are operating within a cell and the co-channel interference is ignored.
However, practical systems are not that simple as one might think but rather complicated.
Co-channel interference plays a very important role in wireless data throughput and
hence cannot be ignored [5]. Successful results could be obtained through simulations.
70
We tried simulating the effects of co-channel interference but were not able to complete
the task because of unavailability of time.
6.2 Future Work
As we mentioned earlier, this work did not take any particular wireless system into
account in different analysis. One possible path for future work is the application of this
research in progressive image coding techniques in which the most significant bits of the
image are given higher priority as compared to the least significant bits of the image. One
could generalize the above results to N number of users and derive some interesting
results. In this thesis, we have assumed constant transmission rate being employed by the
different terminals and hence it would be interesting to analyze the effects on throughput
for variable transmission rates assigned to different terminals. One should definitely not
ignore the effects of co-channel interference and look at a system when a particular
terminal is placed at the boundary of two cells. This research has basically laid the
framework for which future study can be built.
71
Works Cited
[1] David J. Goodman, “Wireless Personal Communications Systems.” Addison-Wesley Wireless Communications Series, 1997. [2]John G. Proakis, “Digital Communications.” McGraw-Hill, Inc., Third Edition, 1995. p. 257-305. [3] Andrew J. Viterbi, “Principles of Spread Spectrum Communication”, Addison-Wesley Wireless Communications Series, 1995. p. 123- 155. [4] David J. Goodman, Narayan Mandayam, “Power Control for Wireless Data,” IEEE Personal Communications, April 2000. [5] J. Zander, “Distributed Co-Channel Interference Control for Cellular Radio Systems,” IEEE Trans. Vehic. Tech. Vol. 41, 1992, pp. 305-311 [6] Jeongrok, Yang, Insoo, Koo, Yeoongyoon Choi, and Kiseon Kim, “ A Dynamic Reaource Allocation Scheme to Maximize Throughput in Multimedia CDMA system,’ Vehicular Technology Conference, 19999. VTC 1999, p. 348-351. [7] V. Shah, “Power Control for Wireless Data Services based on Utility and Pricing,” MS Thesis, Rutgers University, March 1998. [8] Seong-Lyun Kim, Zvi Rosberg and Jens Zander, “Combined Power Control and Transmission Rate Selection in Cellular Networks,” IEEE Personal Communications, May 1998. [9] Shunsuke Seo, Tomohiro Dohi, and Fumiyuki Adachi, “SIR-Based Transmit Power Control of Reverse Link for Coherent DS-CDMA Mobile Radio,”, IEICE Trans. Communications, Vol. E81-B, No. 7, July 1998.
