BLIND CHANNEL ESTIMATION IN CDMA SYSTEMS
by
Ajit Pal Singh Jassal
B.S., Rajiv Gandhi Technical University, India, 2001
A thesis submitted to the
University of Colorado at Denver & Health Sciences Center
in partial fulfillment
of the requirements for the degree of
Master of Science in Electrical Engineering
2006
This thesis for the Master of Science
degree by
Ajit Pal Singh Jassal
has been approved
by
o
Jassal, Ajit Pal Singh (M.S., Electrical Engineering)
Blind Channel Estimation in CDMA Systems
Thesis directed by Associate Professor Dr. Mike Radenkovic
ABSTRACT
Adaptive blind channel estimation with applications in CDMA communication
systems are presented. Algorithm development and simulation results are given for
adaptive blind channel estimation in case of multipath. Additionally, the Multiuser
Detection (MUD) and RAKE receiver structure, in the context of orthogonal and non-
orthogonal, is given with theoretical background, robustness consideration and
simulation verification. The theory are primarily verification of results in the recent
literature, however some intermediate steps and additional derivations are presented.
This abstract accurately represents the content of the candidates thesis. I recommend
its publication.
Signed
jV^-Mikc Radenkovic
DEDICATION
To my parents,
For their great love and endless support
AjitPal Singh Jassal
ACKNOWLEDGEMENT
I would like to express my deep esteem and sincere thanks to Dr. Mike Radenkovic,
at University of Colorado at Denver and Health Sciences Center, U.S.A. It is true
without his guidance, endless help, proper remarks and continuous availability; the
present work would have been by no means the same.
I am deeply indebted to my family for their incredible support and encouragement
from starting line to kickoff, my family never failed to show.
Thanks to my friends who has seen my best and worst you know who you are.
TABLE OF CONTENTS
Figures........................................................
Tables.........................................................
Chapter
1. Introduction to Blind Estimation........................
2. Overview.....................................
3. The Development of Wireless Mobile Communication Systems
3.1 Modem Communication Systems..................
3.2 Cellular Communication Concepts..............
3.3 Mobile Cellular Environment.............................
3.4 Generations of Cellular Systems.........................
3.4.1 First Generation (1 g) Cellular Systems.................
3.4.2 Second Generation (2g) Cellular Systems.................
3.4.3 Third Generation (3g) Cellular Systems
4. Introduction to Multiple Access.........................
4.1 Frequency Division Multiple Access (FDMA)...............
4.1.1 Advantages of FDMA......................................
4.1.2 Disadvantages of FDMA...................................
4.2 Time Division Multiple Access (TDMA)....................
4.2.1 Advantages of TDMA......................................
4.2.2 Disadvantages of TDMA..................................
4.3 Code Division Multiple Access (CDMA)...................
4.3.1 Time Hopping (TH-CDMA).................................
4.3.2 Frequency Hopping CDMA (FH-CDMA).......................
4.3.3 Direct Sequence Code Division Multiple Access (DS-CDMA)
4.3.4 Advantages of CDMA.....................................
4.3.5 Disadvantages of CDMA..................................
5. Power Control in CDMA Cellular
Communication Systems..................................
5.1 Introduction...........................................
5.2 Uplink Versus Downlink Power Control...................
5.3 Open Loop, Closed Loop and Outer Loop Power Control....
5.4 Quality Measures for Power Control.....................
5.5 Practical Aspects on Power Control.....................
6. Models for communication channels......................
6.1 Additive Noise Channel.................................
6.2 Multipath Channel......................................
6.2.1 Characteristics of Multipath Propagation...............
6.3 Time Variant Multipath (Fading) Channel................
7. Subspace Tracking, The Unknown Rank Case...............
7.1 Introduction...........................................
VII
I
7.2 Projection Estimation.....................
7.3 Adaptive Projection Tracking Algorithm....
7.3.1 Channel Estimation Via
Recursive Least Square (RLS)..............
7.3.2 Channel Estimation Via
Leakage Least Mean Square (LMS)...........
7.3.2.1 Mean Behavior of Recursion for LMS.......
7.4 Important Remarks
8. Channel Estimation in CDMA Systems........
8.1 System Model.............................
8.2 Motivation...............................
8.3 Channel Estimation Key Idea............
8.4 Consistency..............................
8.5 Adaptive Implementation..................
8.6 Summary of the RLS Scheme.................
8.7 Summary of the LMS Scheme................
9. Simulations of Adaptive Schemes.............
10. Simulation Code............................
Appendix
A. Subspace Tracking Method Known Rank Case
A.l Singular Value Decomposition.................
viii
A.2 Iterative Subspace Computation Techniques
A. 3 Power Method............................
B. Jakes Model for Fading Channels.........
B. 1 General Description......................
B.2 Multiple Complex Gain Generators..........
Bibliography..................................
I
I
IX
LIST OF FIGURES
Figure
3.1 Structure of cellular communication system........
3.2 Frequency reuse pattern in cellular system........
3.3 Radio channels used in the transmission of information
inside the cell..................................
4.1 Frequency division multiple access................
4.2 Time division multiple access.....................
4.3 Code division multiple access.....................
4.4 Time hopping CDMA system (TH-CDMA)................
4.5 Block diagram of time-hopping spread spectrum.....
4.6 Frequency hopping CDMA system (FH-CDMA)...........
4.7 Block diagram of frequency hopping spread spectrum..
4.8 Direct Sequence CDMA system.......................
5.1 Near Far Effect...................................
5.2 General Model For Uplink Power Control
in CDMA System...................................
5.3 Uplink Open Loop Power Control....................
5.4 Closed Loop Power Control Principles..............
6.1 Additive noise channel............................
IX
6.2 Multipath fading channel...................................
6.3 Multipath effects..........................................
6.4 General system model for multipath fading channel..........
6.5 ...Characteristics function of multipath fading channel (a).
6.6 Characteristics function of multipath fading channel (b)...
6.7 Multipath time variant channel.............................
8.1 Representation of the vectors composing the signal subspace
9.1 Transmitter and receiver structures for CDMA assuming
known flat channels with orthogonal PN code................
9.2 BER curves for ideal receiver for orthogonal and
non-orthogonal PN codes for 2 users........................
9.3 Transmitter and receiver structures for CDMA assuming
Known flat channels with correlated PN codes...............
9.4 BER curves for the least-square and ideal detector for the
case of non-orthogonal PN sequence with 2 user in case
of flat channels............................................
9.5 Inter-symbol interference caused by adjacent transmissions
on the received data vector r. The figure assumes r, > r2
for illustration purposes..................................
9.6 Rake receiver structure for CDMA transmissions over
frequency-selective (multipath) channels with orthogonal
PN codes....................................................
9.7 BER for the RAKE receiver for both cases of orthogonal
and non-orthogonal PN codes over multipath channels
9.8 BER for RAKE receiver using estimated and known
channel taps
IX
B.l Equispaced scatters around a moving mobile receiver
x
TABLES
Table
6.1 Pseudo code of the power method...................
8.1 Proposed RLS algorithm for blind channel estimation
in CDMA system....................................
8.2 Proposed Leakage LMS algorithm for blind channel
estimation in CDMA system.........................
XI
1. Introduction to Blind Estimation
Channel propagation effect is one of the major factors that limit system performance.
These effects can result in many different type of fading such as frequency selective
and frequency non-selective fading. So, linear equalizers are used in digital
communication systems at the receiver to help maximize the information throughput.
In general, digital communication involves the transmission of analog pulses, also
known as symbols, over a channel. Typically the channel is a dispersive medium that
introduces memory and spreads the signal over time. This spreading of the symbols
can corrupt the precise time spacing and make the symbols spill over and corrupt
adjacent symbols. This phenomenon is referred to as inter-symbol interference (ISI).
As ISI grows, the probability of detecting a bit or symbol error at the receiver also
grows. At high enough data rates all physical channels, such as coax, fiber optic,
microwave, and twisted pair, exhibit ISI.
Equalization methods estimate the linear filtering dispersion characteristics of the
channel and insert an inverse filter at the receiver. The combined channel-equalizer
linear filter pair then acts as an ideal channel. In the time domain this would be a
Dirac delta impulse response. In the frequency domain this would be flat line passing
all frequencies (i.e. information) without distortion.
Usually the non-ideal characteristics of the communication channel are not known a
priori, especially in high speed, high capacity channels. The term a priori is used
1
because some equalizations methods do eventually figure out (and use) the channel
characteristics.
There has been and still intensive research on channel estimation. Some are training
based methods, while others are blind or semi-blind methods. However, more
attention is being paid to blind and semi-blind channel estimation techniques since
they reduce the complexity of the system, increase its capacity and minimize the
amount of data checking between the transmitters and the receivers.
The presence of multipath delays, unfortunately, destroys the assumed Orthogonality
between the users spreading codes. As a result, the accuracy of training based
estimators is severely limited by the cross interface between data and the pilot
symbols. Certain applications where a training sequence is either too costly in usable
bandwidth or where a training sequence is impractical require a receiver design which
operates on the received signal and possibly some statistics of the source. Such an
approach is termed blind.
2
2. Overview
Research and development of digital communications systems is undergoing a
revolution fueled by rapid advances in technology. With the ever growing
sophistication of signal processing and computation, advances in communication
theory have an increasing potential to bridge the gap between practically feasible
channel utilization and the fundamental information theoretic limits on channel
capacity. If conquering channel capacity is the manifest destiny of communications
technology, the need for efficient use of channel bandwidth and transmission power is
felt most acutely in wireless communications communication, where the
exponentially growing demand for data rate must be accommodated in a finite
segment of the radio spectrum. To add to the challenge, information is transmitted not
by a single source but several uncoordinated, bursty, and geographically separated
sources.
This thesis begins by presenting some introduction to the general problem of channel
estimation in communication systems and gives theory and simulation verification of
the channel estimation techniques. It presents theory and simulation related to the
Multiuser Detection (MUD) from Sergio Verdu and Rake receiver techniques for
orthogonal and non-orthogonal sequences from Sayed Ali and explains different
channel models and performance measures.
Chapter 3 gives introduction to Communication systems and Generations of cellular
systems.
Chapter 4 gives a brief introduction to the main approaches in multi-access
communication like Frequency Division Multiple Access (FDMA), Time Division
Multiple Access (TDMA), and Code Division Multiple Access (CDMA) along with
their advantages and disadvantages are discussed.
Chapter 5 presents some information regarding Power control in CDMA with an
introduction to various power control methods and how they affect system
performance.
Chapter 6 introduces different models for communication channels where again a
brief introduction to Additive noise channel, Multipath channel and Characteristics of
Multipath propagation are given.
Chapter 7 focuses in estimating the subspace tracking problem. The motivation for
developing the tracking scheme is that in CDMA the noise subspace has dimension
which is unknown and varies with time. Proceeding with this theory at hand adaptive
projection estimation algorithm for Recursive Least Square (RLS) Estimation and
Least Mean Square (LMS) Estimation are also presented.
Chapter 8 presents the theory behind channel estimation with problems that are key to
the thesis along with adaptive implementation of those schemes.
Chapter 9 introduces the channel models used in simulation.
Chapter 10 gives description of simulations, including graphs and results.
Chapter 11 is a listing of the Matlab code used in the simulations
3. Communication Systems
3.1 Modern Communication Systems
Advancement in communication and information processing technologies creates
more new applications and products. In particular, the demand for wireless
communication services has increased rapidly and the trend is expected to continue.
There are already numerous modem wireless systems, as: mobile telephones
transmitting to a base station, ground stations communicating with a satellite, local
area networks, packet-radio networks, interactive cable television networks, etc, and
the area is in complete growth always promising new and exciting services to the end-
user. Therefore, stringent requirements on the capacity of communication systems are
needed in terms of the number of users a system can serve simultaneously. In other or
more appropriate words, as much information as possible should be transferred. Also
in the design of systems employing such technologies will be their ability to perform
with adequate margin over a channel perturbed by a host of impairments not the least
of which is multipath fading.
3.2 Cellular Communications Concepts
In the early mobile radio systems a large coverage was attained by placing an antenna
with a high-power transmitter in one of the highest point of the coverage area for
instance, on top of a hill or a high building. Nevertheless, it meant that only a small
number of users could be allocated in a large area due to the few available radio
frequencies. So any attempt to reuse the same frequencies throughout the system
would result in interference. Thus, the need of higher capacity with limited radio
channel brought into the cellular concept. A cellular mobile communications system
uses a large number of low-power wireless transmitters to create cells which are the
basic geographic service area of a wireless communications system. Figure 1.1
illustrates the structure of a cellular communication system. Each cell consists of a
base station (BS) transmitting over a small geographic area usually depicted as an
hexagon (the true shape of a cell is not a perfect hexagon due to constraints impose by
the terrain). According to the density and demand of mobile users (Mobile Station)
within a certain region, the cell size is determined. The base stations in turn are
connected to a centre called the mobile switching centre (Mobile Switching Center)
which provides connectivity between the public switched telephone network (PSTN)
and the base stations. Thus a global communication network is formed with PTSNs
which connects the conventional telephone switching centers with MSCs all over the
world.
An obstacle in the cellular network arises when a mobile user travel from one cell to
another during a call. To ensure that mutual interferences between users remain low,
adjacent cells do not use the same radio frequency channel, then when a user moves out from
a cell the call has to be transferred to another stronger frequency channel (which becomes the
new cell where the user is moving in). This process is known as hand-off or hand-over,
changing a call from one cell to another without being notice by the users. Another important
concept inside the established cellular systems is the frequency planning or frequency reuse.
Due to the reduce number of radio frequency channels available for mobile systems, a reuse
of the frequency channels had to be implemented into the cellular concept.
Figure 3.1 Structure of cellular communication system
This reuse process means that the radio frequency channels used in one cell can also be
reused in another cell some distance away. Usually clusters of cells (no frequency channels
are reused in a cluster) are reused in a regular pattern during the entire coverage area as it
shown in Figure 3.2. Hence, the frequency reuse factor in a system is determined by the
available frequency channels, i.e. for the particular case depicted in Figure 3.2 the frequency
reuse factor of the system is 1/7.
Cell 2
Figure 3.2 Frequency reuse pattern in cellular system
3.3 Mobile cellular environment
A cellular communication system provides with a full duplex communication
between the mobile user and the base stations to carry through a normal conversation
talk (back and forth). To achieve this type of radio transmission the mobile users and
the base station both need circuitry to transmit on one frequency while receiving on
another. The radio link from the base station to the mobile phone (BS-MS) is usually
referred as downlink (or forward link) and the inverse process (MS-BS) is called
uplink (or reverse link). In the downlink, all the users signals are transmitted by the
same single source, base station; therefore the signals received at each mobile
terminal are synchronous. On the other hand, in the uplink the signals received in the
base station are asynchronous as now the transmissions are yielded by several
uncoordinated and geographically separated sources (mobile users).
3.4 Generations of Cellular Systems
3.4.1 First Generation (lg) Cellular Systems
The cellular concept was developed in Bell Laboratories in 1947. Instead of
transmitting signals from one location with high power, the system capacity could be
dramatically increased by limiting the range of the transmission, which enables the
same frequencies to be reused at much shorter distances. However, the concept was
not implemented until 1979. It was not until that time that the technological
developments such as integrated circuits, microprocessors, frequency synthesizers,
etc. have made it possible. Soon after this came the first generation of commercial
cellular radio systems, such as The Nippon Telephone and Telegraph (NTT) system
in 1979, the Nordic Mobile Telephone (NMT) system in 1981, the Advanced Mobile
Phone Service (AMPS) in 1983 and the British Total Access Communications
System (TACS) in 1985 (a modified version of AMPS). TACS is also used in Japan,
where the system is called JTACS. These systems are/were analog, and where
designed for wireless speech service.
3.4.2 Second Generation (2g) Cellular Systems
The developments of digital signal processing methods along with the rapid
development of integrated circuits and microprocessors led to the replacement of the
analog 1G cellular system by the digital 2G cellular systems. The first of these, the
Global System for Mobile Communication (GSM), was realized in 1992 in Europe. It
operates in 900 MHz band, and is based on FDMA/TDMA. Variants of GSM have
been developed for higher frequency bands, such as the Cellular System 1800 (DCS
1800) in Europe and PCS 1900 in North America. The GSM system became a huge
success. As of December 2004, there were 626 GSM networks on air in 198 countries
or territories around the world.
The major improvements offered by the digital transmission of the 2G systems over
1G systems were better speech quality, increased capacity, global roaming, and data
services like the Short Message Service (SMS), which gained tremendous popularity
in the 1990s. Major improvements in the data services were also the introduction of
packet switched services such as the General Packet Radio Services (GPRS) and
higher data rate circuit switched services such as the High Speed Circuit Switched
Data Service (HSCSD).
3.4.3 Third Generation (3g) Cellular Systems
Although the 2G systems could already provide some basic data services, the possible
data rates were still relatively low, and could not satisfy the needs of future services
like mobile web browsing, file transfer, real-time video, digital TV etc. The 3G
cellular systems are known with the name International Mobile Communications for
the year 2000 (IMT-2000), and are being implemented in many countries around the
world. The 3G systems introduce wireless wideband packet-switched data services
for wireless access to the internet with speeds up to 2 Mb/s. The 2G systems have
been ( and still are) evolving towards the next generation with the introduction of new
technology enhancements, such as GPRS and HSCSD in GSM, Cellular Digital
Packet Data (CDPD, that operates over AMPS, and High Speed Data (HSD) in IS-
95. A step further towards the 3G networks is the Enhanced Data Rates for GSM
Evolution (EDGE) technology, which enables three times higher data rates than
those possible with the ordinary GSM/GPRS network.
Several standardization bodies joined their forces in 1998 in the 3rd Generation
Partnership Project (3GPP) agreement with the joint goal of producing globally
applicable technical specifications and technical reports for a 3rd generation mobile
system. It has a sister project, the 3rd Generation Partnership Project 2 (3GPP2),
which compromises North America and Asian interests developing the 3G mobile
systems. The 3GPP projects produces the radio interference standard for the 3G
networks, wideband CDMA (WCDMA), which is the main 3rd generation air
interface in the world and will be deployed in Europe and Asia, including Japan and
Korea. The 3G systems within the scope of 3GPP are generally known with the name
Universal Mobile Telecommunication Services (UMTS), and WCDMA is called
Universal Terrestrial Radio Access (UTRA) Frequency Division Duplex (FDD) and
Time Division Duplex (TDD), the name WCDMA being used to cover both FDD and
TDD operation.
The air interface to be developed by 3GPP2 is referred to as cdma 2000, which is
based partly on IS-95 principles. It is further divided in two standards, namely
cdma2000 lx and cdma2000 3x. cdma2000 lx is considered a 2.5G system, and it has
the same bandwidth (1.25 MHz) as IS-95.
Voice quality comparable to Public Switched Telephone Network (PSTN).
Support for high data rate.
Support of both packet-switched and circuit-switched data services.
More efficient usage of the available radio spectrum.
Backward Compatibility with the pre-existing networks and flexible
introduction of new services and technology,
An adaptive radio interface suited to the highly asymmetric nature of most
internet communications, a much greater bandwidth for the downlink than the
uplink.
4. Introduction to Multiple Access
Multi-access technologies are in the heart of this modem era of communication
explosion. Multiple Accesses schemes are used to allow many mobile users to share
simultaneously a finite amount of radio spectrum. The sharing of spectrum is required
to achieve high capacity by simultaneously allocating the available bandwidth (or the
available amount of channels) to multiple users. For high quality communications,
this must be done without severe degradation in the performance of the system. The
freedom in use when designing multi-user communication systems include space,
time and frequency. The time and frequency domains are dual of each other via the
Fourier transform so that the actual options to use are space domain and time-
frequency domain designs. In the space domain users can be separated by making
their distance large enough or in other words by assigning different channel for
different user. An example is to use different cables for each user to separate signals
in wire-lines communication. More advanced techniques include Polarization
Division Multiple Access (PDMA) and Space Division Multiple Access (SDMA). In
PDMA two users can be separated by using electromagnetic waves with different
polarization but in SDMA sectorized antennas are usually applied to separate users at
the same frequency. In TDMA, different time slot can be assigned to different user
whereas in FDMA different frequency band can be assigned to different user and then
all users are linearly combined to form one signal. Spread spectrum multiple access
communication is a driving technology behind the rapidly advancing personal
communications industry each user transmitted data is modulated by a unique
signature waveform then all users signals are linearly combined to form one signal.
Various signature waveform designs result in multiple access techniques, but to
receive each users data, the signature waveforms must be known at the receiver. In
the following sections we review the most famous multiple access systems followed
by the evolution of communication systems.
4.1 Frequency Division Multiple Access (FDMA)
The oldest multiple access technique is Frequency Division Multiple Access
(FDMA). Each user is allocated a unique frequency band or channel. These channels
are assigned on demand to users who request service. During the period of the call, no
other user can share the same channel. In FDD systems, the users are assigned a
channel as a pair of frequencies; one frequency is used for the forward channel, while
the other frequency is used for the reverse channel.
4.1.1 Advantages of FDMA
Reducing the information bit rate and using efficient digital codes can
increase the capacity.
As FDMA systems use low bit rates (large symbol time) compared to average
delay spread, they reduce the cost, and there is low Inter Symbol Interference
(ISI).
Technological advances required for implementation are simple. A system can
be configured so that improvements in terms of speech coder bit-rate
reduction could be readily incorporated.
Since the transmission is continuous, less number of bits are needed for
synchronization and framing.
The symbol time of a narrowband signal is large as compared to the average
delay spread. This implies that the amount of inter symbol interference is low
and, thus, little or no equalization is required in FDMA narrowband systems.
After assignment of a voice channel, the base station and the mobile station
transmit simultaneously and continuously.
4.1.2 Disadvantages of FDMA
It does not differ significantly from analog systems; capacity improvement
depends on reducing signal to interference ratio, or signal to noise ratio (SNR)
The maximum bit rate per channel is fixed and small.
The guard band between two users results in wastage of capacity.
Hardware involves narrow band filters, which cannot be realized in VLSI and
thus increase cost.
FDMA requires tight RF filtering to minimize adjacent channel interference.
Non-Linear Effects in FDMA In a FDMA system, many channels share the
same antenna at the base station. The power amplifier or the power combiners, when
operated at or near saturation for maximum power efficiency, are nonlinear. The
nonlinearities cause signal spreading in the frequency domain and generate
intermodulation (IM) frequencies. IM is undesired RF radiation which can interfere
with other channels in the FDMA systems. Spreading of the spectrum results in
adjacent-channel interference. Intermodulation is the generation of undesirable
harmonics. Harmonics generated outside the mobile radio band cause interference to
adjacent services, while those present inside the band cause interference to other users
in the wireless system.
4.2 Time Division Multiple Access (TDMA)
The introduction of digital modulation enabled the appearance of Time Division
Multiple Access (TDMA). TDMA is a digital transmission technology that allows a
number of users to access a single Radio-Frequency (RF) channel without
interference by allocating unique time slots to each user within each channel as
shown in Figure 1.2. TDMA system transmit data in a buffer-and-burst method, thus
the transmission for any user is noncontinuous. This implies that, unlike in FDMA
systems which accommodate analog FM, digital data and digital modulation must be
used with TDMA.
^ Power
Time
30 kHz
Figure 4.2 Time division multiple access
TDMA is relatively simple to implement and is very flexible for providing variable
bit rates. Increasing the bit rate can be implemented by assigning to a user more
transmission intervals. However, the transmission of all users must be exactly
synchronized to each other.
4.2.1 Advantages of (TDMA)
Permits flexible bit rates by assigning more slots per frame to a certain user.
TDMA shares a single carrier frequency with several users, where each user
makes use of non overlapping time slots. The number of time slots per frame
depends on several factors such as modulation technique, available
bandwidth, etc.
Data transmission for users of a TDMA system is not continuous, but occurs
in bursts. This result in low battery consumption, since the subscriber
transmitter can be turned off when not in use (which is most of the time).
Because of discontinuous transmissions in TDMA, the handoff process is
much simpler for a subscriber unit, since it is able to listen for other base
stations during idle time slots. An enhanced link control, such as that provided
by mobile assisted handoff (MAHO) can be carried out by a subscriber by
listening on an idle slot in the TDMA frame.
TDMA uses different time slots for transmission and reception, thus duplexer
are not required.
No narrowband filters required for wideband system.
TDMA has an advantage in that it is possible to allocate different numbers of
time slots per frame to different users. Thus, bandwidth can be supplied on
demand to different users by concatenating or reassigning time slots based on
priority.
4.2.2 Disadvantages of (TDMA)
The high bit rates of wideband systems require complex equalization.
Because of burst mode of operation, a large number of overhead bits for
synchronization and framing are required.
Guard time between each user is required in each slot to accommodate time
inaccuracies because of clock instability.
Electronics operating at high bit rates increase power consumption.
Complex signal processing is required for synchronizing within a short slot
time.
4.3 Code Division Multiple Access (CDMA)
In CDMA each user is assigned a unique code sequence (spreading code) to encode
its information-bearing signal. The receiver, knowing the code sequence of the user,
decodes a received signal after reception and recovers the original data. This is
possible since the cross-correlations between the code of the desired user and the
codes of the other users are small. Since the bandwidth of the code signal is chosen to
be much larger than the bandwidth of the information-bearing signal, the encoding
process enlarges (spreads) the spectrum of the signal and is therefore also known as
spread-spectrum modulation. The resulting signal is also called a spread-spectrum
signal, and CDMA is often denoted as spread-spectrum multiple access (SSMA).
Figure 4.3 Code division multiple access
In CDMA, the power of multiple users at a receiver determines the noise floor after
decorrelation. If the power of each user within a cell is not controlled such that they
do not appear equal the base station receiver, then the near-far problem occurs.
The near-far problem occurs when many mobile users share the same channel. In
general, the strongest received mobile signal will capture the demodulator at a base
station. In CDMA, stronger received signal levels raise the noise floor at the base
station demodulators for the weaker signals, thereby decreasing the probability that
weaker signals will be received. To combat the near-far problem, power control is
used in most CDMA implementations. Power control is provided by each base station
in a cellular system and assures that each mobile within the base station coverage area
provides the same signal level to the base station receiver. This solves the problem of
a nearby subscriber overpowering the base station receiver and drowning out the
signals of far away subscribers.
Power control is implemented at the base station by rapidly sampling the radio signal
strength indicator (RSSI) levels of each mobile and then sending a power change
command over the forward radio link.
CDMA systems can be implemented in numerous ways including Frequency Hopping
(FH-CDMA) spread spectrum, Time Hopping (TH-CDMA), Direct Sequence
(DSCDMA) spread spectrum techniques.
4.3.1 Time Hopping (TH-CDMA) is a spread spectrum system in which, time
interval, is selected to be much larger than the reciprocal of the information rate, is
subdivided into a large number of time slots. The coded information symbols are
transmitted in a pseudo randomly selected time slots as a block of one or more code
words. PSK modulation may be used to transmit the coded bits. Time hopping is
often used effectively with frequency hopping to form a hybrid multiple access
(TDMA) spread spectrum system.
A
\7
f V Programmable Delay k y
riMZB FTI
y <
/ PN Generator v V
i i
Time
Interval
Figure 4.4 Time hopping CDMA system (TH-CDMA)
Block diagram of a transmitter and a receiver for TH-CDMA system is shown in
Figure 4.5. Because of the burst characteristics of the transmitted signal, buffer
storage must be provided at the transmitter in a TH system. A buffer may also be used
at the receiver to provide a uniform data stream to the user.
Information
Sequence
FnrnHpr Buffer and PSK ^ Channel . ^ . Gate . ^ PSK
-> Interleaver modulator S' Demodulator
PN
Sequence
Generator
Figure 4.5 Block diagram of time-hopping spread spectrum
4.3.2 Frequency Hopping CDMA (FH-CDMA) Frequency hopped Multiple
access is a digital multiple access system in which the carrier frequencies of the
individual users are varied in a pseudorandom fashion within a wideband channel.
Figure 4.6 illustrates how FHMA allows multiple users to simultaneously occupy the
same spectrum at the same time, where each user dwells at a specific narrowband
channel at a particular instance of time, based on particular PN code of the user. The
digital data of each user is broken into uniform sized bursts which are transmitted on
different channels within the allocated spectrum band. The instantaneous bandwidth
of any one transmission burst is much is smaller than the total spectrum bandwidth.
Time Interval
>
Figure 4.6 Frequency hopping CDMA system (FH-CDMA)
A block diagram of the transmitter and receiver for an FH spread spectrum is shown
in Figure 4.7. The modulation scheme usually applied is either binary or M-ary FSK.
For example, if binary FSK is employed, the modulator selects one of the two
frequencies corresponding to the transmission of either a 1 or a 0. The resulting FSK
signal is translated in frequency by an amount that is determined by the output
sequence from the PN generator, which, in turn is used to select a frequency that is
synthesized by the frequency synthesizer. This frequency is mixed with the output of
the modulator and the resultant frequency-translated signal is transmitted over the
channel. For example, m bits from the PN generator may be used to specify 2m -1
possible frequency translations.
At the receiver, we have an identical PN generator, synchronized with the receiver
signal, which is used to control the output of the frequency synthesizer. Thus, the
pseudo random frequency translation introduced at the transmitter is removed at the
receiver by mixing the synthesizer output with the received signal. The resultant
signal is demodulated by means of an FSK demodulator. A signal for maintaining
synchronism of the PN generator with the frequency- translated received signal is
usually extracted from the received signal.
Phase Shift Keying modulation gives better performance than FSK in an Additive
White Gaussian Noise channel, it is difficult to maintain phase coherence in the
synthesis of the frequencies used in the hopping pattern and, also, in the propagation
of the channel as the signal is hopped from one frequency to another over a wide
bandwidth. Consequently, FSK modulation with non-coherent detection is usually
employed with FH spread spectrum signals.
Advantages of FHMA
A frequency hopped system provides a level of security, especially when a
large number of channels are used, since an unintended (or an intercepting)
receiver that does not know the pseudo-random sequence of frequency slots
must retune rapidly to search for the signal it wishes to intercept.
In addition, the FH signal is somewhat immune to fading, since error control
coding and interleaving can be used to protect the frequency hopped signal
against deep fades which may occasionally occur during the hopping
sequence.
Error control coding and interleaving can also be combined to guard against
erasures which can occur when two or more users transmit on the same
channel at the same time.
Information
Sequence
Encoder
FSK
Modulator
PN
Sequence
Generator
Mixer
Channel
Mixer
FSK
Demodulator
Decoder
Output
>
Frequency
Synthesizer
Frequency
Synthesizer
Time
Synchronization
PN
Sequence
Generator
Figure 4.7 Block diagram of a frequency hopping spread spectrum
4.3.3 Direct Sequence Code Division
Multiple Access (DS-CDMA)
Direct Sequence Code Division Multiple Access (DS-CDMA) communication
systems have attracted a considerable attention for third generation (3G) mobile
systems. They have the ability to suppress a wide variety of interfering signals
including Narrow-Band Interference (NBI), Multiple-Access Interference (MAI), and
Multipath Interference (MPI).
Compensation of multipath is possible through the use of RAKE receivers that use
several baseband correlators to coherently process the multipath components and
exploits their diversity advantages. In the presence of fading, the capacity of the
system can be improved through multipath diversity gained by utilizing an
appropriately designed RAKE receiver. RAKE receiver correlators can be
implemented as Tapped Delay Lines (TDL) or FIR filter-like which has complex
structures if the number of taps necessary to accurately model the channel is high. For
example in 3G UMTS system the delay spread can be up to 75 chips in a 3.84 Mega
chips per second (Mcps) chip-rate system, thus requiring a RAKE receiver covering a
delay of 75 chip duration.
However, several distinct paths typically dominate the channel, hence only a few
taps, or, in other words low order models, are necessary to capture most of the signal
energy. If all the taps are used in a RAKE structure, even though most do not
correspond to an actual signal path, channel noise would ensure any linear estimate of
their tap weights would produce non-zero weights. This leads not only to undue
complexity, but also reduces system performance. By discarding those taps with a
non-significant contribution to the channel model, it is possible to simultaneously
reduce system complexity and improve the performance.
Figure 4.8 Direct sequence CDMA system
4.3.4 Advantages of CDMA
CDMA can support many users in the same channel i.e. a high capacity.
Lower mobile transmit power, i.e. longer battery life and better power control.
Improved performance in multipath environments,
Soft handoffs can be used. Mobiles can switch base stations without switching
carriers. Two base stations receive the mobile signal and the mobile is
receiving from two base stations.
High peak data rates can be accommodated.
Burst transmission reduces interference.
The generation of the coded signal is easy. It can be performed by a simple
multiplication.
Coherent demodulation of the DS signal is possible.
No synchronization among the users is necessary.
Disadvantages of CDMA
The code length has to be carefully selected. A large code length can induce
delay or even cause interference.
Time synchronization is necessary.
Soft handoff increases use of radio resources and hence can reduce capacity.
As the sum of power received at and transmitted from a base station has to be
constant, a tight power control is needed. This can result in more handoffs.
Synchronization has to be kept within a fraction of the chip time.
For correct reception, the synchronization error of locally generated code
sequence and the received code sequence must be very small, a fraction of the
chip time. This combined with the non-availability of large contiguous
frequency bands practically limits the bandwidth to 10 MHz to 20 MHz.
5. Power Control in CDMA Cellular
Communication Systems
The aim in this chapter is to give a somewhat detailed overview of power control in
CDMA communication systems and to present the relevant problems.
5.1 Introduction
Transmission power control (TPC) is vital for capacity and performance in cellular
communication systems, where high interference is always present due to frequency
reuse. The basic intent is to control the transmission power in such a way that the
interference power from each transmitter to other co-channel users (users that share
the same radio resource simultaneously) is minimized while preserving sufficient
quality of service (QoS) among all users. Co-channel interference management is
important in any system employing frequency reuse. However, in CDMA there are
interfering users inside and outside a cell, which makes CDMA interference limited.
Thus efficient TPC is essential in CDMA1, especially in the uplink (from mobile to
base station communication).
Consider the situation depicted in Figure 5.1. Mobile stations MS 1 and MS2 share
the same frequency band and their signals are separable at the base station BS by
their unique spreading codes.
1 This applies to direct-sequence (DS-CDMA). In frequency-hopping CDMA (FH-CDMA) the intra-cell
interference can be made very small.
Figure 5.1 Near Far Effect
The link attenuation of MS2aX a particular time instant might be greater than that
from MS 1 to BS If power control is not applied, the signal of MSI will overpower
the signal of MSldX the base station. The received signal power attenuation is a
random variable. Thus when power control is applied, it must adapt to changing
attenuation of the desired signals, as well as the changing interference conditions,
since the attenuation of the co-channel users signals are also changing, and those
signals are power-controlled as well.
5.2 Uplink versus Downlink Power Control
In CDMA the uplink transmission creates a near-far situation if power control is not
used. This occurs because the signals of the different mobile stations propagate
through different radio channels before reaching their serving base station. The task
of power control is thus to vary the transmission power in order to compensate for the
varying channel attenuations, so that the signals from the different mobile stations are
received with equal powers at the base station. Uplink power control is critical for the
capacity of CDMA systems. The requirement of the dynamic range of uplink power
control can be of the order of 80 dB.
In downlink the situation is different; since all signals are transmitted by a base
station propagate through the same radio channel before reaching a mobile station.
Therefore, since they undergo the same attenuation, power control is not needed for
near-far problem. Instead, power control is used to provide more power to users
located near the cell borders, suffering from high interference from nearby cells and,
on the other hand, to use only sufficient transmission powers in order to minimize the
interference produced to nearby cells. In principle, the downlink signals to different
users could be made orthogonal by using proper spreading codes.
Unfortunately, the Orthogonality of the downlink signals is lost in practice due to
multipath propagation. Thus, allocating different powers for different users in
downlink could cause a near-far situation at the mobile stations. For this reason, the
dynamic range of downlink power control is usually much smaller than in uplink,
typically of the order of 20-30dB.
Figure 5.2 General Model for Uplink Power Control in CDMA System
5.3 Open Loop, Closed Loop and Outer Loop
Power Control
An intuitive way to compensate for the channel attenuation in the uplink would be to
measure the strength of a pilot signal from the downlink, and adjust the transmission
power proportionally to the inverse of this measurement. Since the pilot signal is
transmitted at constant power, the variation of its strength gives the information of the
downlink link attenuation. This is called open loop power control.
The open loop power control principle is described in Figure 5.3. Since the distance
d\ of the mobile station 1 to the base station (BTS) is shorter than the distance of
mobile station 2 d2 to the base station (BTS), the signal received by the mobile
station 1 has a smaller propagation loss than mobile station 2 that has a longer
propagation path. Assume the mean input power of the mobile station 1 is -70 dBm
(100pW) and the mean input power of the mobile station 2 is -90 dBm (lpW).
Open loop power control is usually used for initial power settings, when two-way
communication link is not yet established and closed loop is not possible. The center
frequencies allocated to up-and-downlink transmissions are usually widely separated,
and thus the correlation between up-and-downlink attenuation is generally weak.
Therefore, the transmission power update of mobile must be based on feedback 2
2 1 dBm means ldB over 1 mW. For example -70dBm is 70 dB (10 million times) less than 1 mW (i.e.
1E-12W = 1 Pico Watt).
d 1
d 2
<-----> <------------->
Figure 5.3 Uplink Open Loop Power Control
information of the received SIR at the base station, forming a closed loop between
them. This closed loop power control (or inner loop power control) aims to keep the
received uplink signal power at a specified target. Moreover, the target must also be
varied, because the SIR requirement for a given Bit Error Rate (BER) is not constant,
but depends on the radio propagation conditions. This is the task of the outer loop
power control.
However, in practice only one bit is used to signal the received SIR information at a
fast rate to track the channel variations. In WCDMA there are some more degrees of
freedom, for instance, the possibility to signal a no change command when the
received SIR is reasonably close to the target, thus reducing a ping-pong affect
around the target.
Since the IS-95 uplink and downlink have a frequency separation of 20 MHz, their
fading processes are not strongly correlated. Even though the average power is
approximately the same, the short term power is different, and therefore, the open
loop power control cannot compensate for the uplink fading. To account for the
independence of the Rayleigh fading in the uplink and the downlink, the base station
also controls the mobile station transmission power.
The base station measures the received SIR 3 over a 1,25-ms, period, equivalent to six
modulation symbols, compares that to the target SIR, and decides whether the mobile
station transmission power needs to be increased or decreased. The power control bits
are transmitted on the downlink fundamental code channel every 1.25-ms (i.e. with a
transmission rate of 800 Hz) by puncturing the data symbols. The placement of a
power control bit is randomized within the 1.25-ms power control group. The
transmission occurs in the second power control group following the corresponding
uplink traffic channel power control group commands in which the SIR was
estimated4.
3 The IS-95 standard suggests that the received signal strength should be measured. However, in
practice usually the SIR or the received bit energy to noise density (Ebj IQ) is used, since they
have a direct impact on the bit error rate (BER).
4 For instance, if the signal is received on the reverse traffic channel in power control group number
5, and the corresponding power control bit is transmitted on the forward traffic channel during power
control group number 5+2=7.
Base station closed loop
power control functions
Mobile station closed loop
power control functions
User Data
H tn i>
% <=
O * C/5
Measure Quality
MUX
Power Control
Commands
SNR
Measurement
>
DEMUX
>
Decoding
Power
Control
<
Decoding
<
Despreader
User Data
Power Amplifier
Figure 5.3 Closed Loop Power Control Principles
Since the power control commands are transmitted uncoded, their error ratio is fairly
high, on the order of 5%. However, since the loop is the delta modulation type (i.e.
power is adjusted continuously up or down) this is tolerable. The mobile station
extracts the power control bits command and adjusts its transmission power
accordingly. The adjustment step is a system parameter and can be 0.25, 0.5 or 1.0
dB. The dynamic range for the closed loop power control is 24 dB. The composite
dynamic range for open and closed loop power control is 32 dB for mobile stations
operating in band class 0, and 40 dB for mobile stations operating in band class 1.
The typical standard deviation of the power control error due to closed loop is of the
order of 1.1 to 1.5 dB.
The SIR required to produce a certain bit error rate varies according to radio
environment and depends on the amount and type of multipath.
5.4 Quality Measures for Power Control
A great deal of work on power control in CDMA cellular system has focused on how
to set the transmission power so that all users in the system have acceptable bit-
energy-to-interference-spectral-density ratios(Eb/1o}5. This approach is based on a
fairly reasonable assumption that the bit error probability (BEP) at the receiver is a
strictly monotonically decreasing function of(Eb/lo}. For instance, BEP Pb of
5 Usually the term bit-energy-to-noise-spectral-density-ratio (Eb/lo^ is used. However, since the
CDMA cellular systems are in practice interference-limited, we prefer to speak about{^EbjI^ ,
where IQ contains both the interference and background noise.
Binary Phase Shift Keying (BPSK) modulation in an additive white Gaussian noise
(AWGN) channel is given by
Pb = Q
WJ
Where Q(x)is defined by
(5.1)
00
A more relevant case for CDMA cellular systems is the BEP performance of RAKE
receiver in fading channels. Assuming binary modulation, Rayleigh fading channel,
and that the RAKE receiver can coherently combine L paths using maximal ratio
combining (MRC), the BEP performance is
(5-3)
where
and
r-. \-i
1 +
grf
sin2 9
(5.4)
r,n
K
(5.5)
is the average signal-to-noise ratio per bit, corresponding to the Ith channel (or
resolvable path), and Q, is the mean strength of that channel (path). In Equation
(5.3), g = l for the antipodal signals (like in BPSK) and g = 0.5 for orthogonal
signals.
Eb/Io is closely related with another measure, namely the signal-to-interference ratio
(SIR), denoted by y, such that
W
Eblo=y-f
Kb
where W is the transmission bandwidth in Hertz and Rb is the data rate in bits/second.
The quantity W/Rb is called the processing gain. When the data rate is fixed, the SIR
differs from Eb/lo by merely a scaling factor.
In digital cellular communication systems, typically, the information to be transmitted
is arranged in strings of bits called frames, and error correction coding is applied to
each frame to further decrease the bit error rate (BER) after decoding. A frame is
useless if there are still error bits in the frame after decoding, and it must be
discarded. Hence, depending on the service, a sufficiently low frame error rate (FER)
must be guaranteed. However, long time delays are needed to obtain reliable
estimates of BER or FER. Since the channel conditions can change very rapidly,
these delays might be unacceptably long in practice. Hence most of the attention in
the power control field has been on SIR based algorithms. These algorithms typically
adjust the transmission power inversely proportional to the link gain. The frame error
information can be used to adjust the target SIR, which a fast TPC algorithm is trying
to achieve. This increases system capacity, since a worst-case setting of the SIR target
is not required. In addition to the SIR and FER requirements, the delay or latency
requirements must be taken into account. For instance, a voice service tolerates a
certain amount of data loss but is delay-critical, whereas file downloads do not
tolerate bit errors at all (erroneous frames must be retransmitted), but the transmission
need not be continuous and must only satisfy some delay limit on the average.
5.5 Practical Aspects on Power Control:
Many solutions to power control in CDMA systems have been proposed in the
literature. A lot of work has been devoted to find the optimal (in some sense)
algorithms that may be unpractical to implement, but can be used to derive theoretical
bounds to performance of the system. Also, a great deal of work has been done to
seek distributed iterative algorithms that are optimum, and are more practical to
implement. To move closer to real implementation of the TPC algorithms, one must
take into account a number of practical issues that limit the capacity gains achievable
by TPC.
Loop Delay: The loop delay refers to the overall loop delay in closed loop
power control. It greatly affects the performance of a power control algorithm.
This delay is a combination of delays due to the SIR measurement process, the
transmission of the SIR information over the radio channel, the processing
gain of the SIR information to calculate and adjust the transmission power,
and the propagation time after which a new transmission power affects the
next SIR measurement. Therefore the power update is based on outdated
information of the received SIR. This may cause instability in the power
control algorithms, leading to large variations in the interference power at the
receivers and diminished capacity.
SIR Estimation: The received SIR is not known exactly at a receiver, but it
must be estimated, and thus there will always be some estimation error. The
estimation error can be made smaller by increasing the average time of the
measurement, but this might lead to a longer loop delay, which is undesirable.
Feedback Information Accuracy: The information of the SIR at the
receiver should some how be communicated to the transmitter. An accurate
representation of the SIR measurement requires several bits, but this requires
more signaling overhead. This form of feedback is referred as information
feedback (IFB). A usual case in practice is that only one bit is used to inform
the transmitter to either increase or decrease its transmission power by a fixed
account, typically by 1 dB (e.g. in WCDMA). This form of feedback is
referred as decision feedback (DFB).
TPC Update Rate: In CDMA one has to deal with the near-far situation,
and thus the update rate of the power control algorithm must be sufficiently
high so that the variations in the link attenuation can be tracked. Typical
updates rates are from 800 Hz (used in IS-95 system) to 1500 Hz (used in
WCDMA). Note that the update rate cannot be arbitrarily high because of the
inherent delay imposed by the SIR measurement process. Since the available
feedback channel bandwidth for power control signaling is limited in practice,
there is a tradeoff between feedback information accuracy and TPC update
rate.
Errors In The Transmission Of Feedback Information: To
minimize the loop delay, the TPC command bits are sent without error
correction coding. Hence the probability of receiving an erroneous command
can be relatively high, e.g. up to 10%. In the uplink of a DS-CDMA system,
the requirement for power control is the most serious negative point. The
power control problem arises because of multiple access interference. All
users in a DS-CDMA system transmit the message by using the same
bandwidth at the same time and therefore users interfere with one another.
Due to the propagation mechanism, the signal received by the base station
from a user terminal close to the base station will be stronger than the signal
received from another terminal located at the cell boundary. Hence, the distant
users will be dominated by the close user. This is called the near-far effect. To
achieve a considerable capacity, all signals irrespective of distance should
arrive at the base station with the same mean power.
6. Models for Communication Channels
In the design of communication systems for transmitting information through
physical channels, it is necessary to construct mathematical models that reflect the
most important characteristic of transmission medium. The mathematical channel
model can then be used in numerous parts of the communication system, like channel
encoding and modulation in the transmitter, demodulation and channel decoding in
the receiver. We continue with a brief description of the most popular models used in
wireless communications.
6.1 Additive Noise Channel
The simplest mathematical model for a communication channel is the additive noise
channel, which is shown in Figure 6.1. In this model the transmitted signal x(t) is
corrupted by an additive noise process n(t). Physically the additive noise may arise
from electronic components and amplifiers at the receiver of the communication
system or from interference encountered in transmission (as in the case of radio signal
transmission).
If the noise is introduced primarily by electronic components and amplifiers at the
receiver, it may be characterized as thermal noise and characterized statistically as a
Gaussian noise process. In the case where this noise is white, then the mathematical
model is called in the literature as Additive white Gaussian noise (AWGN) channel.
Channel
1 1 1 x(t) r i i i ] y(t) = x(t) + n(t)
1 1 U > K i i i i
n (/)
Figure 6.1 Additive noise channel
Because this channel model applies to a broad class of physical communication
channels and because of its mathematical tractability, this is predominant channel
model used in communication system analysis and design. The effect of channel
attenuation can be easily incorporated into this model. When the signal undergoes
attenuation in transmission through the channel, the received signal becomes
y(t)-ax(t) + n(t)
where a denotes the attenuation factor. Finally, it should be stressed here that due to
the special form of this simple channel, the phenomenon of intersymbol interference
(ISI) does not exist.
6.2 Multipath Channel
Fading and multipath occur in many radio communication systems. In mobile
communication systems, the mobile or the base station is often surrounded by various
objects, such as buildings, trees, etc. These objects produce more than one path over
which the signal can travel between the transmitter and the receiver. In a multipath
situation, the signals arriving along different paths will have different attenuation,
delays and Direction of Arrivals (DOA). They might add at the receiving antenna
either constructively or destructively depending on their phases, resulting in
phenomena known as fading. If the path length add/or the geometry change due to
variations in the transmission medium or due to relative motion between the
transmitter and receiver antennas, the signal level will exhibit some changes to
random fluctuations.
To become familiar with the effects of the multipath propagation, let us consider the
following situations. If we transmit say a short duration pulse to represent a digital
sequence, due to reflections, the received signal may appear as a number of pulses
with different attenuation and delays. Also, if we repeat and transmit another pulse,
attenuation and delays will be different from the previous case and the received
number of pulses does quite often changes see Figure 6.3. Hence the first two
characteristics of the channel are the time spread introduced in the signal and the
channel medium structure is also time varying. Moreover, the time variation of the
channel is unpredictable to the channel user. In some situations and due to the nature
of the reflecting objects, the duration of the received pulses may also change and this
will create other problems such as scattering, dispersion and frequency distortion.
Figure 6.3 Multipath effects
6.2.1 Characteristics of Multipath Propagation
In this section we study the characteristics of the multipath propagation from signal
processing point of view. Hence, let us consider the situation where a signal x(r) is
transmitted over a multipath channel. For continuous multipath propagation, the
equivalent base-band response of the channel g(r,t) can be defined as a function of
two variables, v and t. The variable r is considered to represent the effect of the
channel delay and t is to represent the time variation nature. The received signal
r(t) as shown in Figure 6.4, can then be defined as,
00
r{t) = \s(^,ty(t-z)dT + v{t) (6.1)
-00
Where v(/) represents the channel noise.
v(0
*(')------->
Figure 6.4 General system model for multipath fading channel
(*><)
Due to random nature ofg(r,/), in practice the channel effects can be characterized
by four important factors:
1. T : Multipath delay spread of the channel.
2. Af : The coherence bandwidth of the channel.
J C rp
1 m
3. fid: Doppler spread of the channel.
4. At, : Coherence time of the channel.
Given information about these factors, one can assume the situation over which the
transmitted signal x(/) can be affected then a specific or an approximated channel
model can be considered [4] [5] [l 8].
To become familiar with the meaning of each of the above factors, we will briefly
summarize some important functions of the channel g(r,t) .The channel response,
g(r,t) is characterized in [4] as a wide sense-stationary complex-valued random
process with the autocorrelation function,
^(r1;r2,At) = ^Â£{g*(r1,/)g(r2,t + A/)j (6.2)
In most radio transmission media, the attenuation and phase shift of the channel
associated with the delay r, is uncorrelated with the one presented at delay r2, then it
follows from the above equation that,
^Â£{g*(V)s(r2>' + A0}=^ri~r2A0^(ri~r2) (6-3)
If we replace At = 0, the resulting autocorrelation function, (p{t) = (p{r ,o) is known
as the Multipath intensity profile of the channel.
The range of values of r over which (p{r ) Ois called the Multipath delay spread
of the channel and denoted by Tm see Figure 6.6. Define the Fourier transform
G(/,t) of the channel response g(r,t) with respect to the delay r,
00
G(f,t)= jg(r,t)e~j2;rfTdT (6.4)
00
Since the Fourier transform will not change the wide-sense-stationary properties, and
then we can also define another function known as the spaced-time spaced-frequency
correlation function of the channel,
(d(A/,A() = iÂ£{G*(/,()G(/ + A/V + A<)}
(6.6)
Figure 6.5 Characteristics function of multipath fading channel (a)
After some manipulations one can show that ^(A/, A/) is the Fourier transform of the
multipath intensity profile function,
Using some practical measurements [4], it has been found that the relation between
the coherence bandwidth Afc and the multipath delay spread Tm can be approximated
by,
Af'y (6-6)
1 m
In order to see the effect of the time variation of the channel, define the Fourier
transform of
(6.7)
s(4M)=
oo
This function gives us the information about the relation between the Channel time
variation and the Channel Doppler effects. With Af = 0 the function
S(T) = S(0,A)is known as the Doppler power spectrum (see Figure 6.6) of the
channel. The range of values of X over which S (T) 0 is called the Doppler spread
of channel,/?,.
Figure 6.6 Characteristics function of multipath fading channel (b)
The reciprocal of [id is a measure of another factor known as the coherence time of
the channel,
(6.8)
According to the above channel factors one can assume the situation over which the
transmitted signal can be modeled then a proper model can be considered.
6.3 Time-Variant Multipath (Fading) Channel
Physical channels, such as underwater acoustic channels or ionosphere radio
channels, resulting in time variant multipath propagation of the transmitted signal,
may be represented mathematically as time variant linear filters. Such linear filters
are characterized by a time variant channel impulse response f{r;t), where
/(r;?),is the response of the channel at time t due to an impulse applied at time
t T. Thus, t represents the age (elapsed time) variable. Figure 6.7 shows such a
time variant channel with additive noise. Such channels in literature are known as
Multipath fading channels.
Channel
*0) i Linear
^ time -variant
filter f(r;t)
A
n(t)
Figure 6.7 Multipath time variant channel
For an input signal x(t) in fading channels, the output becomes
y(t) = x(t)* f(r\t) + n{t)
(6.9)
(r,t)x(t x)d.T + n(t)
(6.10)
A suitable model for multipath signal propagation through physical channels, such as
the ionospehric and mobile cellular radio channels, is a special case of Equation 1.11
in which the time variant impulse response has the form
Substituting Equation (1.12) in Equation (1.11), the received signal takes the form
Where a, aL (t) represent the time-variant attenuation factors for the
L propagation paths and r,,..., the corresponding time delays. The previous
model seems to be the most popular multipath fading model in the literature. A
possible way to generate such channels for discrete time systems is through Jakes
Method. A Description of which is given in Appendix B.
L
(6.11)
L
(6.12)
7. Subspace Tracking,
The Unknown Rank Case
In this is chapter we will focus our attention in estimating the subspace tracking
problem which will help us in Chapter 8 problem 1. We are interested in estimating
the subspace corresponding to the smallest singular value. We will also assume that
this subspace has dimension that can be larger than unity (i.e. the smallest singular
value can be multiple). The motivation for developing tracking schemes for this case
is that in CDMA the noise subspace (that corresponds to the smallest singular value)
has dimension which is unknown and varies with time. This is true since the noise
subspace depends on the number of users occupying the channel and this is variable,
since the numbers of users enters and exit the system continuously.
7.1 Introduction
The covariance matrix R of the observation sequence can be defined as
(7.1)
Applying Singular Vector Decomposition (SVD) then we can write
(7.2)
The orthonormal matrix U as we know spans the noise subspace of the data and
corresponds to the smallest singular value of R (which is equal to cr ). In CDMA,
when following a subspace approach, the solution may turn out to depend on the
noise subspace projection operator UJU1 ^ instead of the basis Ua. So, it is feasible
to develop an algorithm that directly calculates the projection operator instead of the
basic itself.
7.2 Projection Estimation
Let us first see how the product UJUm can be estimated when the data
autocorrelation matrix R is known.
Lemma 1 Let R be as in Equation (7.1) with an SVD as in Equation (7.2) and
p>0a non-negative scalar. We then have
lim
*50
' pI + R
Kp + a2
N-*
)
= uu'
(7.3)
Proof from Equation (7.2)
*=[>. u]
A +ct I. 0
s a
o
a21
lus vay.
N-d
Substituting R in the Equation (7.3) we see
0
lim
Â£>00
' pi+ r'
Kp + cr2 ;
2 t\
pI + A + oLI
V P + G
0
-Â£
f pl + a2!^
p + cr2
\+, y.]'
lim
Â£>00
/ \ -A
pI + R
K P + ^ >
= [^
(p + g2)/ + A5
p +
0
0
{p+rj2)
p + cr2
2 \ r A
lim
Â£*
pI + R
Kp + cr2;
I +
V P + G
0
0
I
[Us Â£/] ' (7.4)
Since
^ p + o-
is a diagonal matrix with diagonal elements strictly greater than
unity.
lim
Â£>oo
' pI + R
kP + cj2
A
-k
t
lim
Â£->oo
f pi+ R
yp + cr2
\
-k
= u u
CO CO
(7.5)
Lemma 1 Suggests a possible way to approximate the product UJU1 co (noise
subspace projection operator). The following remarks can be made for the Equation
derived above.
Remark 1: Approximating the projection operator through Equation (3.6) requires
no knowledge of the subspace rank. This is evident, since we directly utilize the
whole data autocorrelation matrix R.
Remark 2: In Lemma 1 the convergence in Equation (3.6) is exponential and we
observe that the corresponding rate is maximized when p- 0. Regardless of this fact,
the employment of the parameter p > 0 will turn out to be extremely useful for the
adaptive implementation and in particular the LMS version that we are going to use in
our simulations. Indeed, we are going to see, the algorithm will be able to forget past
data much more rapidly with p > 0 than the usual LMS with p = 0.
Remark 3: A subtle and very important remark regarding Lemma 1 concerns the
power &. Notice that the limit is correct, i.e. we obtain the noise subspace
t
projectionU U only when the singular values corresponding to the noise subspace
are exactly equal. But as we know that we only have estimates of R available, this is
rarely the case. This causes a great problem since the corresponding limit instead of
being the desired product will simply become the rank-one matrix uu where u is the
singular vector corresponding to the smallest singular value of the estimate of R.
This in turn will make W a rank-one matrix as well. So, we should not use power of
k much high than desired. As we will notice in our simulation that increasing power
above certain value doesnt make much difference in our results.
7.3 Adaptive Projection Tracking Algorithm
In this part we develop RLS and LMS algorithms, which track the matrix
W = F\UJU1 (0F\ making use of the received data vector y(n) and the known
matrix Fx. Using the approximation proposed in Lemma 1 and according to the
third remark of the section (7.2), we have the following estimate for the matrix W
Wk=F],(pI + Rjt F (7.6)
Where we have discarded the scalar quantity (/r + cr2) since it does not affect any
subspace determination problem. We are interested in producing adaptive estimates
Wk (n) of the matrix Wk defined in (Equation (7.6)). Let us now examine what
possibilities exist for such the estimate Wk(n).
7.3.1 Channel Estimation Via
Recursive Least Square (RLS)
The parameter p was introduced to allow the adaptive schemes to forget past data.
Since the ability is intrinsic in RLS, so we can select p = O.for our scheme. For the
adaptive estimate Wk {n) of the matrix Wk we propose
Wk{n) = F\Pk(n)Fu
k = 1,2,3.
(7.7)
Where
P(n) = R 1 (n)
R(n) is exponentially windowed sample autocorrelation matrix of the data y(n)
i-e., R(n) = Zl QAn r(i)rl (i), with 0
P(n l) is available from the previous time instant when y(n) arrives. We recall
the following well-known RLS adaptation for P(n), we apply
k[n) = P{n-\)y{n)
(7.8)
r(n) =
(a+/(rt)Â£(rt))
(7.9)
P(n) = (p(n -l)-(/(/2)^())A:/ (n)) (7.10)
A,
That has an overall complexity of 5N2 + 3N + 0(\) (Counting together
multiplications and additions). If we compute Wk (n) as
M")=/^(")/=, = /?(A) OWOM'O)-) (7'D
then this part requires 2kLN2 + 2L2N + O(l) operations. It is noticed that the most
computationally intense part is the computation of Wk (n).
7.3.2 Channel Estimation Via Leakage
Least Mean Square (LMS)
An alternate means to generate estimates for Wk consists of writing the matrix
Wk=Ft'(pI+R)-k F
as
Wk=F/Vk (7.12)
where
Vk=(pl + Ryl F (7.13)
and produce adaptive estimates Vk (n) for Vk .
Case 1 Consider first k l,and define the recursion
Vx (n) = A.lVx (n -1) + ju(F{ -r(n)r (n)V^ (n -1)),
(7.14)
where 0 < ZL < 1 is the leakage factor.
By taking expectations on both sides and evoking the Independence Assumption, i.e.
assuming that V] (n -1) is independent from the data vector y(n) we can verify that
1-/1,
lim E[ V,(/i)]= -----Li + R
F
(7.15)
Which is exactly Equation (7.13) with p = 1 XLjp and k = 1.
Case 2 Estimates of higher order / = l,2,...,A:,can be obtained by applying the
following time order recursion,
Vl(n) = AiE/(-l) + ^(FM(/7-l)-r()r'(n)E/(n-l)), (7.16)
where VQ ( n ) = V( (n ) = F,.
7.3.2.1 Mean Behavior of Recursion For LMS
Theorem 7.1 Let Vt(n), l = \,...,k, be as in (7.16) define p = 1-XLj\iand
let Vk {pi + R) k be the expression defined in (7.13) corresponding to the
power l then, under the Independence Assumption, we have for l = We apply
induction in the power l.
/-i fn\
E!>,()] = r, + 2>' (v-A) (^(0)-^).
7=0 \J)
limE [K,(n)] = F)=(p/ + fi)-'f]
rt>00
(7.17)
(7.18)
Step 1 We will first prove that this is true for / = 1. Taking Expectation on both
sides in Equation (7.16), using the independence assumption, and recalling that
V0 (n) = we obtain the following recursion:
E[E/(/i)] = ^E[E/U-l)] + //E[F/_1(/2-l)-r()r,(Az)F/(Az-l)],
E[E/()] = zliE[E/(-l)] + /7E[E/_1(n-l)]-/7E[r(A2)r'(/I)]E[E/(/z-l)]
E[F,()] = ^E[K/(-l)]-//E[r(/z)r/(Az)]E[K/(/2-l)] + //E[FM(-l)]
where
E(r()r/ ()) = R and E(Vl (n -l)) = Fx
substituting this in above Equation we have,
E [Vt (/*)] = (V fiR) E[V, (n 0] + nFx (7.19)
Equation (7.19)can be written in the form
E[v, (w)] = vf. ( V FR)jFx + (V iuR)n vx (0) (7.20)
7=0
using the fact that the sum of powers of a matrix equals
YdAj =(l An)(l A)~\ (7.21)
7=0
using Equation (7.21) in Equation (7.20). The second part of Equation (7.20) would
remain unchanged.
e[>( ()] = (Aj-pR) vx (o)+p((i (V ^)")(
E[V, ()] = (AJ pR)n V, (0) + p
I-ALI + pR) )f,
'(i-(AlI -pR)")
, V-V + mR)
' {i-(\I pR)")
((l-AL)I + pR)
\
7
\
/
*1
1-/L
As we know p - substituting in above Equation we have
E[Vt ()] = (V pR)n Vx (0) + p
E [V, ()] = (V pRf V} (0) + p
{ppI + pR)
[i-(ALI fiR))
p{pI + R)
F
F
E\v{ ()] = (AlI -pR)n Vx (0) + (/ -(AlI -pR)n)(pI + R)~' Fx (7.22)
E[V,()] = (V-pR)n Vx (0) + (/ -(V-pR)n){pI + /?)'* Fx
V\ from 3.8
E[v, ()] = (XLI pR)n vx (0) + (/ (AlI pR)n)vx (7.23)
this is same as Equation (7.17) for / = 1.
Step 2 Let us now assume that Equation (7.17) is true for / = k, then Equation
(7.17) becomes
K-1
fn\
E|/>)] = >; + 2>J . {AJ-nR)"'1 (VK_f 0)-yK.j). (7.24)
j=o \JJ
also
AV =V
K-J K
(7.25)
or
AVi = Vi{Q)-Vi
Substituting this in Equation (7.24) we have
/c1 f
E|>>>] = ^ + 2>7 . (7-26)
7=0 \J)
Step 3 : Let us now again use induction to prove that it is also true for
l = rc +1. Consider the recursion in Equation (7.17) after taking expectation and
applying the independence assumption i.e. V x{n l) is independent from the
received datar(n), we obtain the following recursion.
E[>/l()] = (V-/')E[F,+1(n-l)] + AE |-1)] (7.27)
which yields,
E[^|()] = /ii(V-/'),E[^(-/-l)] + (V-/'*)"^+,<0)- (7-28)
(=0
Substituting E[VK (n i l)J with its equal from Equation (7.26) we have
E[X+1 ()] = V mR)k + (V fiR)" vr+l (o)
/=0
71-1 K-1 (
(7.29)
(=0 7=0
l J )
The sum of first two terms in Equation (7.29), using the fact that
{pi + R) VK = VK+v is equal to
/Z(V-^)'>.+(V-^)^+i(o) = ^+1+(V-^)ae,+1
(7.30)
Furthermore, if we change the order of summation in Equation (7.29) we end up with
the inner sum V '
in=0
n-i-1
1 i )
m
z
, and using the property that
fy+n ' j + m + V
> k J + 1 y
Zn1
/=0
1 1 s: 71-1-/ 'n-i 1N n-\-j ( n \
= l = z / +
v j , ;=0 , J y 1=0 , J,
(7.31)
Moreover, according to Equation (7.31) and by changing the variables p j +1, the
double summation in Equation (7.29) takes the form
AT
I*'
/>=
K+\-p'
(7.32)
By adding Equation (7.32) and Equation (7.30) we have
e[X+1 M]=^+(v - ()
(7.33)
K+l-p
= K+tvP^{lLI-R)n~P (7.34)
Where the second term in Equation (7.33) was included in the sum changing the
initial index to p = 0 which was desired. This concludes the proof.
The computational complexity of the above Least Mean Square (LMS) scheme is as
follows. We need,
1. IkLN + 0(l) for the adaptation in Equation (3.11) i.e. for V^n).
2. 2L2N + O(l) for the computation of Wk Fx Vk
7.4 Important Remarks
1. Observation of Theorem 7.1 reveals that, due to the term {^ALI /jR j there is
an exponential convergence of E[E/(n)J towards the desired quantity
y,=(pi+ry'f,.
2. The speed of convergence is governed by the largest eigenvalue of the matrix
2
A LI juR, which is equal to A L /na .
3. By using Leakage LMS recursion in Equation (3.11) and with 0 < AL < 1, there
would be an exponential convergence in our simulation with a factor that is at
least equal to AL, independently of the SNR level. On the other hand using the
regular Least Mean Square (LMS) with AL = 1 the factor becomes
1 (J.G which in medium to high SNR will result in slow convergence of the
algorithm.
8. Channel Estimation in CDMA Systems
Code-division multiple-access (CDMA) implemented with direct-sequence (DS)
spread spectrum constitute one of the most important emerging technologies in
wireless communications. It is well known that CDMA has been selected as the base
for the 3rd generation mobile telephone systems. In a CDMA system, users are
capable of simultaneously transmitting in time, while occupying the same frequency
band, by using a unique signature waveform assigned to each of them. However, this
important advantage also constitutes the principle weakness of CDMA system as it
results in performance degradation of the system. Indeed for every user, all other
users play the role of multi-user interference.
At the receiver end i.e. the mobile unit, whenever CDMA signal propagate through a
multipath environment (i.e. buildings, trees, mountains etc), the effective signature
are no longer the signature waveforms but rather the convolution of these signals with
the channel impulse response. This combined waveform is also known as composite
signature. Furthermore, we would like to express strong interest towards blind
estimation methods, as this method does not require transmission of any training
sequences.
8.1 System Model
Consider a K user CDMA system with identical chip waveforms and signaling
antipodally through a multipath channel in the presence of additive white (but not
necessarily Gaussian) noise (AWN). Although CDMA systems are continuous in
time, they can be adequately modeled by an equivalent discrete time system.
Specifically, no information is lost if we limit ourselves to the output of a chip
matched filter applied to the received analog signal and sampled at the chip rate. [5] .
Let N be the processing gain of the code and L the length of the channel impulse
response. Moreover, let sj ^ (0) s. (l) s(. (2) 5|. ( Ab l) J be the length
Abnormalized signature waveform or the spreading code of User i ^i.e. |.s; || = 1),
and denote by Â£, () the sequence corresponding to this signature waveform zero-
padded from both ends towards infinity. The transmitted signal due to User i is
given by
zi(n) = aiflsi{n-kN-Ti)bi(k) i = ,K. (8.1)
GO
Where aj is the amplitude of User i
b^n) The corresponding bit sequence
r( The time delay that can take any value in the set {0, ,2V 1}
The signal z, () propagates through a multipath AWN channel with the impulse
response f. = [f. (0) (L l)] and let s. = [ff. (0) si (0) si (N + 1-2)] 1
be the composite signature waveform of User i, i.e. st = si ft where "* denoted
convolution. Then the received signal can be written as
K 00
y(n) = (n-kN- rf )bi (k) + crw(n) (8.2)
/=i -00
Where 5( (/z)is the sequence corresponding to the composite signature waveform of
User i zero-padded from both ends towards infinity and w()is a unit variance
i.i.d. (independently and identically distributed) noise sequence with cr denoting the
power of the AWN.
The model given in equation (4.2) describes fully the uplink (mobile to base station)
scenario of a multipath CDMA system. For the downlink we simply need to select
_/]= = fk f and r, = = Tk = r (since each user receives the information
intended to all users with the corresponding signals propagating through the same
multipath channel and being completely synchronized).
For the presentation of our method it is more convenient to express the received
signal in blocks of data. We are interested in block of size mN + L-1, where m a
positive integer or smoothing factor. Consequently let us consider the block
I
rm(n) = \y(nN)"m y((n~m)N - (8-3)
which is synchronized with the user of interest which is User -1. As downlink
scenario is considered and as synchronization is there, the block rm () contains
m entire copies of the composite signature of user of interest. Specifically () can
be decomposed as
0
(/-1)jVx1
(n)= z
/=1
axbx (-/ + !)
0
(m-l)Nx 1
Entire Composite Signature of User of Interest
(8.4)
K m
+EL
i=2 /=1
0
0,
(l-\)Nx\
Si
(m-I)Nx\
afr (-/ + !) + ISI + crw()
MAI
Observation can be made from the above equation that the sum of the first m terms
involves the entire composite signature of the user of interest, then follows the multi-
access interference (MAI) that contains the terms similar to the first sum but coming
from the interfering users, then the inter symbol interference part coming from all the
user consideration and finally the last term which is nothing but Additive white noise
(but not necessarily Gaussian) vector.
8.2 Motivation
Why the channel estimation problem is important in a CDMA communication
system.
Consider first the case where no multipath is present, i.e. we have a direct link (clear
line of sight.). Then the received signal from Equation (8.4) for the case of only one
data block m = 1, can be written as
We can notice that in the above equation we only have MAI and AWN, while the ISI
interest from rx ().
When no multipath is present, numerous offline as well as adaptive detection scheme
have been proposed and have been analyzed extensively in the literature [l]-[5].
The three most popular class of linear detectors encountered in practice are briefly
presented. The general form of which can be represented as
K
r\ (n) = a\sA () + YuaisPi (n) + (8-5)
part is absent. But we are interested in detecting the bit sequence bx (n) of the user of
(8.6)
and differing only in the way in which the vector c is defined.
Matched Filter
c = s
(8.7)
De-correlator
c =
f k V
Iv/
V=i J
(8.8)
where | denotes pseudo-inverse.
Minimum Mean Square Error (MMSE) detector
c = R~ls, (8.9)
where R is the auto-correlation matrix R = Ejr, {n)r\ ()|.
MMSE detector exhibits
8.3 Channel Estimation Key Idea
Let y(/t)be the observation sequence and rm(n) the data block defined as
r(n) = y((n- m)N L + 2)J (8.10)
Let us first identify the channel impulse response f assuming the availability of the
data autocorrelation matrix and the initial signature waveform of the user of interest
i.e. 5|. The data autocorrelation matrix satisfies
RU E{r()r'(/7)} = Â£> + cr2/
Observing that all the terms in equation below,
+ l)
^(/-l)iVxl
(m-l)Nx\
afii ( / + l) + ISI + (Tw(fl)
except the last one, are of the form djb^n- _/) where dl are deterministic vectors
corresponding to shifted versions of composite signatures coming from the user of
interest of Multiple Access Interference (MAI) or shifted sections of the composite
signatures coming from Inter symbol interference (ISI). () are the binary data that
are mutually independent from the noise vector. Therefore we have that
e=!
/
0
K m
+ZZ
1=2 /=1
o
(/-l)jVxl
Si
(m-l)Nx\
in) = l!
/=1
is a symmetric, non-negative definite matrix, of dimensions mN + L -1, formed by
the dj vectors introduced in the signal model.
By applying a Singular Value Decomposition on R we can write.
R = [us Â£/]
Aj+<72/
0
0
a2I
\P, Â£/.]'
where U and U are orthonormal1 (orthogonal and normalized) bases for the signal
and noise subspace respectively. We should note that U spans the noise subspace,
2
which corresponds to the smallest singular value of R (being equal to cr ). Due to
the Orthogonality* 2 of the two subspaces, for any vector dj in the signal subspace,
v'A = o
as we can see from signal model
m
rW = Z
1=1
o,
o
(/-l)jVxl
*1
(m-l)Nx. 1
0
axbx (n-l +1)
K m
+ZZ
(=2 /=1
0
(l-\)Nx\
(m-l)Nxl
aibi (-/ + !) + ISI + crw(rt)
that our data block contains m shifted copies of the composite signature of the user of
interest that have the form.
d,=
0
(/-!)*'
0
(m-/)xl
/ = 1,2, m.
In linear algebra, two vectors v and w in an inner product space are said to be orthonormal if they
are both orthogonal (according to a given inner product space! and normalized.
2 Orthogonality means at right angles. It comes from the Greek orthos, meaning "straight", used by
Euclid to mean right; and gonia, meaning angle.
as these vectors lie in signal subspace, and so they satisfy the Orthogonality
conditions
and the same property of Orthogonality is satisfied for their sum also i.e.
<4,14=0
/=!
The composite signature of User -1 i.e. s, can be written as
*1 = SJ
Where 5, is a convolution matrix of size (N + L-\}xL corresponding to the initial
signature of User 1 which is defined as
*,=
(0)
(N-\)
0
0
(0)
0
0
Si(Af-l) \ ,,(0)
0
0
sx{N-\)
Using 5, = S{f the Orthogonality condition introduced in =0can also be
i=\
UFf = 0
CO ] J
expressed as
With
m
/=!
(l-\)NxL
St
0
(l-m)NxL
And has a simple structure same as 5, but of dimensions (wjV + L l) x L, with the
first column containing the signature 5j repeated m times, i.e. of the form
r
s\ s\ lxL-
m times
We should also note that when m = 1 then F, reduces to S,. From UF f = 0 we
1 1 (O
can conclude that
U,aFlf = 0 = (Fl'Ua>U'aFl)f
Which suggests the recovery of f as the singular vector corresponding again to the
smallest singular value (which here is equal is zero) of the matrix F^UJU^F^. This
concludes the channel estimation method.
From the equation U^FJ = 0 = )/ it can be seen that there are two
subspace estimation problems.
Problem 1 Consists in estimating the projectionUJJ[0, where U is an
orthonormal basis for the noise subspace corresponding to the smallest singular
value (72 of R (i.e. projection operator UJJto the noise subspace of the matrix R).
Problem 2 Once the projection operator U JJ1^ is available, we are interested in
estimating the singular vector f corresponding to the smallest singular value of the
matrix.
W =
8.4 Consistency
Let us briefly discuss the consistency of the two problem that were proposed
previously. Denote by rs and rn the signal and noise subspace ranks, respectively;
then, the matrix U^F in equation Ul(0F^f = 0 = iyF^UJU^F^ j/ is of dimension
rn x L If Uw is the exact noise subspace basis then, due to equation U^F f = 0, we
conclude that the column rank of U^F can at most be equal to L -1. In order for
UlFtf = 0 = (RUa,U'F>)/to have a unique solution (modulo a multiplicative
constant-ambiguity), the column rank of U^F must be exactly equal to L 1. It is
also known that the column rank of a matrix is equal to its row rank (and also equal to
the rank of the matrix) in order to have a row rank equal to L 1 a necessary
condition is to have at least L 1 rows, that is,
rn>L-\
Since rs+ rn = mN + L-1
rs < mN
Let us know specify, more precisely, the signal subspace rank. Notice that the number
of columns of U is equal to rs. In fact, U is an orthonormal basis for the subspace
spanned by the vectors dj introduced in Q ^djdj
l
The downlink scenario is considered for the sake of clarity and all K are
synchronized. As seen from the figure above there are m big rectangles of dimensions
(N + L-l)*K, containing the entire composite signature of all K users. The first such
rectangle corresponds to the nth user-bits whereas the last to the (n m + l)i(. The
two smaller rectangles of dimensions (L-l)x K, contain Inter symbol Interference
coming from the (n + l)s/and (n-nif1 user-bits respectively. Each rectangle has a
rank that cannot exceed its smallest dimension.
Making an assumption that the number of users K is smaller than the processing gain
N,than
rs < mK + 2 min {L -1,K}
m
(L-\)xK
(n-m) User
bits
Figure 8.1 Representation of the vectors composing the signal subspace
We therefore conclude that if we select m such that
rs < mK + 2min 1,AT} < mN
than the validity of r < mN is guaranteed.
Calculating the estimate for the number of blocks m
mK + 2 min {Z \,K} < mN
mN-mK>2mm{L-l,K}
(N-K)m>2min{L-\,K}
Equivalently, for a given number of blocks m, we can obtain an upper bound for the
maximum load of the system
mN mK > 2min {L 1,/f}
This bound is for Downlink scenario. Following the same analysis for the Uplink
scenario then, due to lack of synchronization using
rs < mK + 2min{L-l,A^}
we can deduce that
rs < {m + 2) K < mN
mN >mK + 2K
mN-mK>2K
m(N-K)>2K
m >
2 K
N-K
m >
2 K
N-K
.. m
or K < --------- N
(m + 2)
as a possible estimate for m (for given K) or an upper bound for K (for given m).
The bounds introduced in m > and K < N-lare by
(N-K) [m + 2 m J
no means strict and must be used with caution. The above equation ensures only the
validity of the necessary condition rs < mN and are thus not sufficient for
identifiability.
Finally, in situation where the channel length is not available, we have to assume that
L plays the role of a known upper bound for the true parameter. In such a case,
similarly to M. an additional necessary condition for identifiability is needed.
Specifically the difference between the upper bound L and the true filter length
L must be strictly less than the processing gain N, i.e. L L < N Otherwise we
can see that in the opposite case one can easily produce two different solutions for
VF,f = 0 = (F;ujj'c,Fl)f i.e.
1 o . 1 and i o V > O l
lx^L-L j lx^L-Z. -Nj
, where
f is the true channel impulse response and L its corresponding length. Since the
linear combination of
f'0 I and i o < K O 1
lxl L-L J lx^Z.-I -Vj
is also a solution of
r < mN So we can conclude that there is an infinite number of candidates for the
S
role of the channel impulse response.
8.5 Adaptive Implementation
In this section we focus on the development of adaptive implementation for the
offline SVD approach introduced previously. Before proceeding to the adaptive
implementation we would like to recall and explicitly describe the two subspace
problems.
Subspace Problem 1: If R satisfies the decomposition
* = [U, u\
As +cr I 0
0
a21
[". w.]'
we are interested in estimating the projection operator U U1 where U a is an
orthonormal basis for the (noise) subspace corresponding to the smallest singular
2
value cr of R.
Solution To Subspace Problem 1
The Proof of this Subspace Problem can be found in Chapter 7.
Subspace Problem 2 If R and UJU^ are as in Subspace problem 1
and Fx =Â£
/=i
0
(l-\)NxL
Â£
0
(l-m)NxL
we are interested in estimating the singular vector
f corresponding to the smallest singular value of the matrix W = F[U .
8.6 Summary of the RLS Scheme
Let us summarize the steps that constitute our RLS blind adaptive channel estimation
schemes.
CDMA Channel Estimation : RLS Algorithm
Initializations: P(0) = ---/; /(0) = [0 0 1 0 0]
Available for previous instant: f{n l).
New data: r{n)
Apply:
1. &() = .P(tf-l)r(n)
2. y(n) = ---------------r
+ rl ()&(n)j
3- p{n)=-j(p(n-l)-(y{n)k("))k(n))
4. Wk(n) = F;Pk(n)F, =F;P(n)(P(n) (P()F,))
V V J
k times
trace\Wk (n)|
f( x n{n)Wk(n)f(n-\)
\f(n-\)-n(n)Wk(n)f(n-\)\
Table 8.1: Proposed RLS algorithm for blind channel estimation in CDMA system.
The proposed RLS algorithm is shown in Table 8.1. First of all, we compute the
inverse exponentially windowed sample autocorrelation matrix P(n) using RLS.
Then, we form the matrix Wk (n) as shown in Step 4 Table 8.1. Finally, we apply the
power adaptation (4.58) to compute the new channel impulse response
estimate fin). The computational complexity of the RLS scheme for the different
steps is as follows.
1. Steps 1-3, computation of Pin) 5imN + L-\)2 +3imN + L-\) + Oi\).
2. Step 4, computation of Wk in) :
2kLitnN + L-\)2 + 2L2 imN + L-1) + O(l).
3. Steps 5-6, computation of fin) : 2L2 + 5L + Oi\).
It can be seen that the most computational intense part for the RLS algorithm is the
computation of Wk in) i.e. Step 4. Thus the leading complexity of the algorithm is
0(kLimN + L-\)2).