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UNIVERSITY OF CALGARY
Jacobi-SVD Channel Estimation for Amplify and Forward MIMO Relay
by
Ayo-Bello Olubunmi Ayobami
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
ELECTRICAL AND COMPUTER ENGINEERING
CALGARY, ALBERTA
DECEMBER 2013
© Ayo-Bello Olubunmi Ayobami 2013
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Abstract
In this thesis we present a least squares based channel estimation scheme for amplify and
forward relaying systems. Firstly, a composite channel estimate is obtained at the destination by
least squares method. This is followed by a sequence of least squares minimizations and Jacobi
rotation to obtain individual channel estimates of source-relay and relay-destination channels.
Theoretical method of obtaining singular value decomposition by computing the Eigen
decomposition of gram matrix have proven to be very difficult and impractical to implement due
to associated numerical difficulties. Therefore Jacobi Singular Value Decomposition parallel
algorithm for channel decomposition is proposed. Performance evaluation is carried out to
determine the accuracy of individual channel estimates using normalized mean square error
(NMSE). It was also shown that using the channel state information (CSI) of source-relay
channel to determine the relay gain improves the BER performance of the system.
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Acknowledgements
I would like to express my profound gratitude to my supervisor Professor Abu B. Sesay for his
support, encouragement, valuable and timely suggestions throughout my research work.
Professor Sesay is a great teacher and I am privileged to have been under his supervision. This
thesis would not have been possible without his help and guidance. I also wish to extend my
sincere gratitude to Dr. Yang Gao, Mr. Norman Bartley and Dr. Edwin Nowicki for serving as
members of my thesis defense committee.
I also wish to thank the entire Faculty and Staff of Electrical Engineering department at the
University of Calgary. I want to specially thank Dr. Abraham Fapojuwo for his help and support
during the course of my studies.
Next, I would like to acknowledge my Dad and Mom for their love, care and support. I am most
grateful to them for playing a vital role in my life right from childhood. I also want to sincerely
thank my siblings Seun, Folusho and Gbenga for their support and encouragement. I am grateful
to Dasola Oluge and Virali Shah for their friendship. I also want to thank Xiaobin Yang for his
insightful discussions and suggestions. I truly appreciate everyone that has shown acts of
kindness towards me in the past years.
Finally, I want to deeply appreciate my best friend Abidemi Akinsete for his love, care and
support he has shown to me.
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Dedication
To my beloved parents,
Ayo and Bosede
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Table of Contents
Abstract ............................................................................................................................... ii Acknowledgements ............................................................................................................ iii Dedication .......................................................................................................................... iv
Table of Contents .................................................................................................................v List of Tables .................................................................................................................... vii List of Figures .................................................................................................................. viii List of Abbreviations ...........................................................................................................x List of Symbols ................................................................................................................. xii
CHAPTER 1: INTRODUCTION ........................................................................................1
1.1 Background ................................................................................................................4
1.2 Motivation ..................................................................................................................5 1.3 Thesis Contributions and Objectives .........................................................................6 1.4 Thesis Outline ............................................................................................................6
CHAPTER 2: AMPLIFY AND FORWARD MIMO RELAYING SYSTEM ...................8
2.1 Characteristics of Wireless Communication Channels ..............................................8 2.1.1 Fading Effect .....................................................................................................8
2.1.2 Multipath Effects ...............................................................................................9 2.1.3 Doppler Effect .................................................................................................10
2.2 Introduction to Cooperative MIMO Communications ............................................10
2.3 Diversity Techniques ...............................................................................................11 2.3.1 Frequency Diversity ........................................................................................11
2.3.2 Temporal Diversity ..........................................................................................12 2.3.3 Spatial Diversity ..............................................................................................13
Space-time trellis coding ..........................................................................................13 Space time block coding ...........................................................................................13
2.4 Types of Cooperative Relay Communication ..........................................................13
2.4.1 Amplify and Forward ......................................................................................13 2.4.2 Decode and Forward ........................................................................................14
2.4.3 Demodulate and Forward ................................................................................14 2.5 Matrix Decomposition .............................................................................................14
2.5.1 Singular value decomposition .........................................................................15
2.5.2 QR decomposition ...........................................................................................16 2.6 System Model ..........................................................................................................17
CHAPTER 3: CHANNEL ESTIMATION FOR AMPLIFY AND FORWARD MIMO
RELAYS ...................................................................................................................20
3.1 Introduction ..............................................................................................................20 3.2 Composite Least Squares Channel estimation. ........................................................21 3.3 Two-by-two Jacobi Singular Value Decomposition ................................................25
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3.4 Performance Evaluation ...........................................................................................31
3.5 Chapter Summary ....................................................................................................36
CHAPTER 4: JACOBI SVD CHANNEL ESTIMATION USING THE SYSTOLIC ARRAY
...................................................................................................................................38 4.1 Introduction ..............................................................................................................38
4.2 The systolic arrays for Jacobi SVD .........................................................................38 4.2.1 Serial ordering of Jacobi-SVD ........................................................................45 4.2.2 Parallel ordering of Jacobi-SVD .....................................................................49 4.2.3 Extension of BLV systolic array to complex matrices ....................................51
4.3 Least Squares based channel estimation for amplify and forward using systolic arrays
................................................................................................................................53 4.4 Performance evaluation ...........................................................................................59
Chapter Summary ..........................................................................................................63
CHAPTER 5: PERFORMANCE ANALYSIS OF ZERO FORCING DETECTION FOR
MIMO SYSTEMS ....................................................................................................64 5.1 Introduction ..............................................................................................................64
5.2 Effect of Channel Estimation Errors on Zero Forcing MIMO Receivers ................64 5.3 Semi-Analytical Performance Analysis ...................................................................67
5.4 Numerical Results ....................................................................................................70 5.5 Chapter Summary ....................................................................................................73
CHAPTER 6: CONCLUSIONS AND FUTURE WORK .................................................75
6.1 Summary and conclusions .......................................................................................75
REFERENCES ..................................................................................................................79
vii
List of Tables
Table 3.1 Summary of Jacobi-SVD for 2-by-2 complex matrix ...................................................30
Table 4.1 Serial ordering algorithm on systolic array ....................................................................47
viii
List of Figures
Figure 2.1: Schematic Diagram of Amplify and Forward MIMO Relay System ......................... 18
Figure 3.1: Transmission model for Amplify and forward MIMO relaying system .................... 22
Figure 3.2 Channel estimation performance for H1 with no spatial correlation at the relays ....... 33
Figure 3.3 Channel estimation performance for H2 with no spatial correlation at the relays ....... 34
Figure 3.4 Channel estimation performance for H1 with spatial correlation ρ = 0.9 .................... 35
Figure 3.5 Channel estimation performance for H2 with spatial correlation ρ = 0.9 .................... 36
Figure 4.1 Propagation of rotation angles for the Brent-Luk-Van Loan systolic array ................ 39
Figure 4.2 Input and output communication links for diagonal processor ................................... 41
Figure 4.3 Sub-diagonal Processor for BLV array ....................................................................... 41
Figure 4.4 Super-diagonal Processor for BLV array .................................................................... 42
Figure 4.5 The Brent-Luk-Van Loan systolic arrays for diagonal and off diagonal processor
connections for n = 8 ............................................................................................................. 44
Figure 4.6 Illustration of 4x4 Jacobi SVD serial ordering for iteration and sweep process ......... 48
Figure 4.7 Staggering computations of BLV Arrays .................................................................... 50
Figure 4.8 Extension of BLV staggering computations on the complex SVD array .................... 52
Figure 4.9 Schematic diagram of Amplify and Forward MIMO relaying .................................... 54
Figure 4.10 Channel estimation performance for H1 with no spatial correlation (ρ = 0). ............ 61
Figure 4.11 Channel estimation performance for H1 with spatial correlation (ρ = 0.9). .............. 61
Figure 4.12 Channel estimation performance for H2 with no spatial correlation (ρ = 0). ............ 62
Figure 4.13 Channel estimation performance for H2 with relay spatial correlation (ρ = 0.9). ..... 62
Figure 5.1 BER performance of QPSK MIMO zero forcing receiver N = 2, R = 2 and M = 2 ... 71
Figure 5.2 BER performance of QPSK MIMO zero forcing receiver N = 4, R = 4, M = 4 ......... 72
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Figure 5.3 BER Performance comparison of amplify and forward MIMO relays with relay
gain chosen using knowledge of CSI and choice of relay gain without knowledge of
CSI. ....................................................................................................................................... 73
x
List of Abbreviations
Abbreviation Meaning
1G The first generation analog cellular systems
2G The second generation digital cellular systems
3G The third generation digital cellular systems
3GPP 3G partnership projects
4G The fourth generation network
AMPS Advanced mobile phone system
BER Bit Error Rate
BLV Brent-Luk-VanLoan
CDMA Code Division Multiple Access
CSI Channel State Information
EV-DO Enhanced Version Data only
FM Frequency Modulation
FDMA Frequency Division Multiple Access
GSM Global Systems for Mobile communications
HDR High Data Rate
HSDPA High Speed Downlink Packet Access
HSUPA High SpeedUplink Packet Access
IMT International Mobile Telecommunication
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ITU International Telecommunications union
JSVD Jacobi Singular Value Decomposition
LAPACK Linear Algebra Package
NMSE Normalized Mean Square Error
PSK Phase-Shift Keying
QPSK Quadrature Phase-Shift Keying
SVD Singular Value Decomposition
xii
List of Symbols
Symbol Meaning
h i thi element of vector h
H i ,th
i k element of matrix H
1 : , 1 :H N M A sub-matrix consisting of the first N rows and
first M columns
: 1H i thi element of row of matrix H
T
Transpose operator
H
Hermitian operator
Complex conjugate
T
Tr Trace operator
Khatri-Rao product
Kronecker product operator
E Statistical Expectation
N by NI Identity matrix with N-by-N dimension
, ,J a b Jacobi rotation matrix
Avec Vectorization of matrix A
Avecd Vector formed from diagonal matrix A 2
F
Frobenius norm operator
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CHAPTER 1: INTRODUCTION
There has been a remarkable evolution in communication between people over the past decades
since 1897 when Guglielmo Marconi first pioneered communication over long distance using
radio telegraph [1]. This invention has drawn a lot of interest into the field of radio transmission
providing better and efficient methods of transmitting information. To provide this service to a
wider range of population, Bell laboratories developed the concept of cellular networks in the
1960s. The early generation of the cellular network can be traced in successive generation
starting with the AMPS (Advanced Mobile Phone System) which was the first generation (1G)
of analog cellular telephone system. This system uses separate frequencies for transmission and
therefore requires a considerable amount of bandwidth for large number of users. Analogue
frequency modulation (FM) and frequency division multiple access (FDMA) technologies were
employed in transmission. Incompatibility of devices between different geographic areas which
makes roaming impossible and low data rate of the system necessitated the need for the next
generation. In the second generation (2G), digital transmission techniques such as code division
multiple access (CDMA) and time division multiple access (TDMA) were employed [2]. The
GSM (global system for mobile) and CDMA standards was widely used by the European and the
United States respectively in the second generation. The GSM service packages has new data
and Teleservices packages which includes short messaging systems (SMS), facsimile, videotext
and emergency calling. Although the 2G system provides a better and efficient voice and data
transmission, but low transmission rate for internet applications made it give way for the third
generation (3G) networks. During the early stages of development of 3G, CDMA 2000X1 and
GPRS (generalized packet radio service) standard was developed for 2.5G [2]. Although 2.5G
2
networks was an extension of 2G networks but could still not satisfy the increasing demand for
higher data rate to support a wide range of media services.
The ITU (International telecommunications union) set up IMT-2000 (international mobile
telecommunications) standard requirements for peak data rate to support high mobility rate
which necessitated the advent of the next generation.
The third generation (3G) networks are equipped to handle higher data rate for high speed web
browsing applications like video conferencing, multimedia and gaming service. This generation
provided backward compatibility of the 2.5G and 2G networks making it cost effective for
system upgrade. 3GPP (3G partnership project for wideband CDMA) and 3GPP-2 (3G
partnership project for wideband CDMA-2000 3X) were the standard used in the third
generation. The 3GPP standard involves wideband code division multiple access (WCDMA) also
known as Universal Mobile Telecommunications System (UMTS). In order to achieve higher
data rate for high speed internet data, 3GPP-2 which employed CDMA high data rate (HDR) was
developed by Qualcomm. The CDMA-HDR also referred to as 3G 1X EV-DO (3G 1X Enhanced
Version Data only) has a great improvement in downlink structure with data burst rate as high as
2.4Mbps with downlink data rate of 0.5 - 1Mbps [2]. High mobile data rates for these 3G
systems enabled it to support high data rate required for video conferencing applications which
has been found very useful in the business industry. Further extension to the 3G gave way to the
3.5G with HSDPA (High Speed Downlink Packet Access) and HSUPA (High Speed uplink
Packet Access) standards. This system has a data rate up to 20Mbps which is higher than the data
rate for 3G systems. The increasing demand of data and video service to be available at anytime
and everywhere (indoor and hot spot locations) paved the way to the fourth generation systems
(4G). The main objective of 4G systems was to integrate wide area networks (e.g cellular) with
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short range wireless technologies like personal area networks (PANs) with Bluetooth
technologies, Metropolitan area networks (MAN) using WiMAX technology and WLANs
(wireless local area networks) with IEEE802.11 and its variants [3]. The fourth generation (4G)
cellular networks developed by 3GPP is known as LTE (long term evolution) [4] and the 3GPP-2
version was called UMB (ultra mobile broadband) [5]. In order to improve spectra efficiency
whilst achieving higher data rate, 4G cellular networks were based on orthogonal FDM
(frequency division multiplexing) technology and single-carrier FDM (SC-FDM) [6] for
downlink and uplink respectively. Further improvements done by 3GPP in LTE networks was
published in LTE Rel-10 version termed as LTE-Advanced (LTE-A) [7] to meet the IMT-
advanced standard. Enabling technologies were introduced in LTE-A networks to achieve
superior performance, among these are the multiple input multiple output (MIMO) which
incorporated use of multiple antennas at the transmitter and receiver. Relay nodes were also
introduced in LTE-A to improve coverage area. The function of the relay node is to forward
message signal from base station to the mobile station. The two types of relays identified in
3GPP LTE-A are type-I and type-II [8]. Type-I relays help the mobile station located in a remote
location access the base station. Its main function is to extend service coverage to the mobile
station, hence improving overall system capacity. On the other hand, the type-II relays assist
mobile stations located within the coverage area of the base station improves its quality of
service. There are different types of relay transmission techniques used to establish two hops
communication between the base station and mobile station. The commonly used ones are
amplify and forward, decode and forward, detect and forward and demodulate and forward
methods [9-10]. The full duplexing method was introduced in [11]-[12] where relay node was
equipped with multiple antennas which are partitioned into transmit and receive antenna sets. In
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half duplexing techniques relay nodes are equipped with single antenna with transmit and receive
operations separated in time [13]. Therefore, transmit and receive cannot occur simultaneous as
additional time slot is required to complete both operation.
1.1 Background
Due to the challenging nature of wireless networks which causes degradation in the transmitted
signal, processing techniques are required at the receiver in order to correctly receive the
transmitted signal at the destination. One of the signal processing is channel estimation which
involves obtaining the channel state information (CSI) to correctly detect transmitted data at the
receiver. Various methods of obtaining CSI have been studied. Some of the most commonly used
methods are the least squares (LS) and minimum mean square error (MMSE) estimation methods
[14]-[18]. A known training sequence (over- head signals) is transmitted between the source and
destination over the wireless channel in order to obtain channel estimate. Two main types of
signalling techniques used in OFDM include the comb type and block type [19] and [20]. These
classifications are based on the pilot arrangement. In block type, the whole OFDM symbol is
dedicated to carry pilot signal which makes this method suitable when the wireless channel is
static or slowly varying channel. Comb type signalling involves spreading the pilots over
selected subcarriers which make this method useful when the channel is time varying.
Channel estimation in wireless communication has taken a new approach since the incorporation
of multiple antennas (MIMO systems) as well relaying techniques. Use of multiple antennas at
the transmitter and receiver helps to achieve diversity which enhances decoding of transmitted
signals since multiple copies of signals are received over independent fading channels. In MIMO
relaying systems, two time slots are usually required for signal transmission to the destination. In
the first time slot, the source transmits signal to the relay and the relay processes and forwards
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the signal to the destination. Relays are classified into different categories based on the type of
processing they perform. Most commonly used are amplify and forward and decode and forward
techniques. Decode and forward relays also known as regenerative relays receive the signal in
the first time slot, decode and re-transmit the decoded signal in the second time slot. In this
relaying method channel estimators are incorporated in the relays for efficient decoding of signal
hence increasing the design complexity of the relay node. Amplify and forward relays receive
the signal in the first time slot, amplify and forward the received signal to the destination. There
are two types of pilot assisted channel estimation for amplify and forward method namely
cascaded and disintegrated. In cascaded estimation, the relays are not equipped with channel
estimators, therefore, the source-relay and relay-destination channel estimation are performed at
the destination. Disintegrated estimation involves disintegrating and separately estimating the
source-relay and relay-destination path at both relay and destination respectively. In this method,
the relay nodes are equipped with estimators, quantised version of source-relay channel estimate
are forwarded to the destination [21]. This introduces more complexity at the relay nodes making
the cascaded method more attractive in reducing system complexity.
1.2 Motivation
Despite the advantages involved in two hop communications, there are still challenges that need
to be overcome to achieve efficient design of MIMO relaying systems. One of the major
challenges posed is power allocation, several optimization schemes have been proposed in [22]-
[26] to minimize the total energy consumption without impacting the quality of signal. Most of
these schemes require individual channel knowledge of source-relay and relay-destination
channels which are mostly assumed to be known at destination. Also, designing an efficient
system, amplification factor and capacity bounds need to be investigated to obtain achievable
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rate of information that can be reliably transmitted over the communication channel [27] and
[28]. Since the choice of cascaded channel estimation is preferred for reduced complexity
systems, it is therefore necessary to further decompose the channel to obtain individual channel
estimates. Existing method for channel decomposition in cascaded (composite) channel
estimation method use singular value decomposition (SVD) to obtain individual channel
estimates [29]-[31]. However, numerical difficulties associated with the computation of SVD
make it difficult and impractical for implementation. Hence, the need to obtain a more stable
algorithm which lends itself to practicable implementation is required. In real life applications, it
is desirable to obtain individual channel estimates in a recursive manner. Hence, we propose an
adaptive method of obtaining individual estimates using the Jacobi-SVD algorithm.
1.3 Thesis Contributions and Objectives
The main contribution of this thesis is as follows:
Individual channel estimation using Jacobi Singular Value Decomposition. Jacobi-
SVD algorithm offers better stability and is more suitable for parallel implementation.
Adaptive implementation of individual channel estimate of amplify and forward using
systolic arrays for Jacobi-SVD
Obtain reduced complexity of channel estimation algorithm for individual channels in
amplify-and-forward MIMO relays.
1.4 Thesis Outline
The rest of this thesis is organized as follows:
Chapter 2 reviews the characteristics of wireless channels and provides an introduction to
the different types of commonly used relays in cooperative communication. This chapter
7
also presents the system model and assumptions used throughout this thesis as well as
mathematical algorithms for matrix diagonalization.
Chapter 3 contains a summary of existing schemes for channel estimation in amplify-and-
forward MIMO relays. It also proposes and discusses a more stable and efficient
algorithm for individual channel estimation. The proposed algorithm applies Jacobi-SVD
to perform decomposition of the composite channel estimate to obtain individual
estimates of the source-relay and relay-destination channel. Lastly, a performance
comparison of the proposed and existing techniques is presented by simulations.
Chapter 4 is an extension to chapter 3 which introduces systolic arrays for complex SVD.
This chapter presents an adaptive implementation of the SVD algorithm using systolic
arrays where individual channel estimate can be obtained in real time. Finally, the
normalized mean square error performance of this algorithm is compared with
conventional methods using Monte-Carlo simulations to obtain the normalized mean
square error.
Chapter 5 evaluates a semi-analytical BER and compares it with a simulation
performance of the proposed techniques.
Chapter 6 concludes this thesis and provides direction to future work.
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CHAPTER 2: AMPLIFY AND FORWARD MIMO RELAYING SYSTEM
This chapter begins with a brief overview of wireless channel characteristics. Section 2.1
provides an overview of wireless communication channels. Section 2.2 gives an introduction to
cooperative communication and the importance of cooperative communication is briefly
discussed. In Section 2.3, various methods of achieving diversity were discussed. Section 2.4
provides an overview of commonly used relaying techniques in cooperative communication.
Some methods of matrix decomposition techniques are discussed in Section 2.5. Lastly, Section
2.6 describes the system model and assumptions used in this work
2.1 Characteristics of Wireless Communication Channels
Radio waves are electromagnetic waves that propagate through space. The waves are subject to
reflection, diffraction and scattering as they travel through space. Reflection occurs when these
waves strike an object causing a change in the direction of propagation. Diffraction occurs when
radio wave bend while approaching an obstacle. Scattering is a phenomenon that occurs when
radio signal hits an object smaller than or of same wavelength as the signal causing dispersion of
signal energy in different direction. The effects of the propagation environment include path loss,
attenuation, small scale and large scale fading. In the absence of reflectors, diffractors and
scatterers, loss of signal power (attenuation) can still occur in free space. This is referred to as
free space attenuation.
2.1.1 Fading Effect
Small scale fading occurs when multiple copies of signal arriving at the receiver with different
delay combine constructively or destructively. This results in rapid variation of signal amplitude
when mobile station moves over a short distance. Large scale fading occurs due to signal path
loss as a function of distance and shadowing by large objects such as building and mountains.
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The probability distribution of shadowing effects is represented by a Log-normal distribution and
is commonly referred to as Lognormal fading. In heavily shadowed locations where there is no
line of sight transmission, which can form a dominant component, the probability distribution of
the received signal is approximated by a Rayleigh distribution. If a dominant path exists the
distribution is approximated by a Rician distribution.
2.1.2 Multipath Effects
Multipath occurs due to reflections and scattering of the transmitter signal. This causes multiple
versions of the signal to arrive at the destination at different time delays with varying amplitude,
phase and angle depending on the path each one travels. Multipath results in time dispersion of
signals and the degree of dispersion is observed in the time interval it takes for the first and the
last signal copies to reach the destination. A signal can also experience frequency dispersion
when different frequency components arrive at the destination at different times. In this case, the
channel exhibits a different behavior for each frequency component of the signal. Such a channel
is said to be frequency selective. The multipath effect can be measured by the coherence time
and coherence bandwidth of the signal. Coherence bandwidth is the frequency range over which
different frequency components exhibit identical behavior in the wireless channel. Attenuation
remains constant in this frequency range [2], therefore no dispersion occurs and signal
components arrive at the destination at the same time. Coherence time is the period over which
the channel is assumed to vary insignificantly. Both the coherence time and the coherence
bandwidth determine the degree of fading experienced by the signals. If channel coherence time
exceeds the signal coherence time the signal experiences a slow fading. In this case the channel
is assumed to vary insignificantly during the duration of a symbol. When signal coherence time
is greater than channel coherence time, the channel changes rapidly causing signal distortion, this
10
case is referred to as fast fading. A channel is assumed to be flat fading if the coherence
bandwidth of the channel is greater than signal coherence bandwidth, otherwise, the channel is
frequency-selective.
2.1.3 Doppler Effect
Relative movement between the transmitter and receiver results in Doppler Shift. The maximum
frequency spread mf due to Doppler shift is approximately inversely proportional to the
coherence time cT
1
2c
m
Tf
(2.1)
The maximum Doppler frequency mf is mathematically expressed as [32]
cosm m
vf
(2.2)
where m denotes the angle of arrival of the m-th wave component, the maximum Doppler
frequency occur when 0m .
v is the speed of the vehicle conveying the receiver or transmitter
denotes the wavelength.
2.2 Introduction to Cooperative MIMO Communications
Cooperative communication requires the cooperation of an intermediate node to send and receive
information between the transmitter and receiver. MIMO systems improve performance of the
network by providing diversity through the use of multiple transmit and receive antennas at the
transmitters and receivers. Cooperation of an intermediate (relay) node further improves system
capacity especially in cases where direct transmission between transmitter and receiver is not
achievable. Two types of relays were identified in [27], based on the role they perform, namely
11
Type 1 and Type 2. The function of Type 1 relay is to extend service coverage to mobile stations
not within the coverage area of the base station. Type 2 relays are used to improve the link
capacity and quality for mobile stations within the coverage area of the base station.
2.3 Diversity Techniques
Diversity involves transmitting the same information over multiple independent paths to obtain
multiple copies of same signal for efficient and reliable detection. This techniques helps to
mitigate the effect of fading and can be achieved by different methods. There are three types of
diversity commonly used to mitigate the effect of channel fading on signal. These include
frequency diversity, temporal diversity and spatial diversity.
2.3.1 Frequency Diversity
In frequency diversity, the same information bearing signal is transmitted on different frequency
subcarriers with a separation distance greater than the coherence bandwidth of the channel.
Hence, the signal experiences different channel fading effect. Orthogonal frequency division
multiplexing (OFDM) is one of the methods used to achieve frequency diversity. In OFDM
technique, the transmission bandwidth is divided into narrowband subcarrier [32]. Each
subcarrier is orthogonal to each other such that the transmission bandwidth can be efficiently
utilized. The bandwidth of each subcarrier is chosen to be less than the coherent bandwidth,
which makes it suitable in frequency selective channels. Therefore the signal carried on the
subcarrier experiences frequency flat fading which can be easily retrieved at the destination by
using one-tap equalization. Insertion of guard band (cyclic prefix) between the subcarriers helps
to mitigate the effect of inter-symbol interference (ISI). The duration of the guard band is
designed to be greater than the maximum delay caused by multipath.
12
2.3.2 Temporal Diversity
Temporal diversity is another method of achieving diversity by sending the same information
bearing signal in different time slots. The interval between the time slots should be equal to or
exceed the channel coherence time so that the signal copies can undergo independent fading.
Temporal diversity can be achieved in different ways [33] these include repetition code,
Automatic Repeat reQuest (ARQ) and Combination of interleaving and coding
Repetition Coding: This is the simplest form of temporal diversity and it involves repetition of
signal over an interval long enough to make the signals uncorrelated. This method is does not
utilize available bandwidth efficiently.
Automatic Repeat reQuest (ARQ): In this method, the receiver sends a request for transmission
to the transmitter when the quality of the received signal is low. The transmitter then wait for a
period of time greater than the coherence time to retransmit the signal. Retransmission is only
done when received signal quality is poor, hence this method has a better bandwidth utilization
compared to repetition coding method.
Combination of interleaving and coding: In this method symbols to be transmitted are coded
with error correcting code to handle error during transmission. In noisy channels burst errors
(group of consecutive errors) are likely to occur on one code word which may exceed the
capability of error correcting code and could result to failure in recovery of transmitted code
word. To prevent these burst errors, interleaving technique is carried out where different code
words are arranged in a way that burst errors are spread across multiple code word. At the
receiver code words can be deinterleaved to obtain their original order.
13
2.3.3 Spatial Diversity
Spatial diversity explores the principle of transmitting using multiple antennas or receiving with
multiple antennas at the destination. In order to achieve spatial diversity antennas are placed at a
minimum distance of half the wavelength.
Diversity in MIMO systems can also be achieved by space time coding (STC) technique. STC
involves splitting encoded data into multiple streams which are simultaneously transmitted using
multiple antennas over multiple time slots. The two types of commonly used STC’s are space-
time trellis codes and space-time block codes.
Space-time trellis coding first introduced in [34] employs Trellis coding scheme to encode data.
The encoded data is divided amongst multiple transmit antennas, which simultaneously transmit
the data.
Space time block coding provides diversity by transmitting streams of data encoded in blocks
over multiple antennas at different time slot. Alamouti code was the first well known Space time
block codes. In this scheme a complex orthogonal code is designed using two consecutive
symbols which are transmitted over two time slots [35].
2.4 Types of Cooperative Relay Communication
2.4.1 Amplify and Forward
In the Amplify and forward (AF) scheme the relay station receives transmitted signal from the
source station in the first time slot, amplifies and forwards signal to the destination in the second
time slot. This scheme offers reduced complexity [36] and overall cost of the system as the relay
is not over burdened with complex processing. The drawback of this technique is that noise is
amplified alongside the signal.
14
2.4.2 Decode and Forward
The signal transmitted from source is decoded and re-encoded at the relay station prior to
forwarding to destination. In order to accurately decode the signal received from the source, the
relay has to be equipped with a channel estimator, which introduces additional complexity at the
relays. The overall performance of this technique is dependent on how the relay can reliably
decode the transmitted information [37]-[38]. Due to the processing at the relay, delay is quite
long but error propagation at destination can be minimized.
2.4.3 Demodulate and Forward
In this scheme the relay receives signal transmitted from the source, demodulates and re-
modulates it before forwarding it to the destination. Channel estimation, decoding and re-
encoding is not required in this scheme, hence it offers more simplicity when compared with
decode and forward relaying method. This scheme also offers reduction in power consumption,
complexity and overall system delay when compared to decode and forward method [39]. The
drawback of this scheme is the error propagation to destination. Since the relays are not equipped
with error correction codes, relaying nodes can forward erroneous information to the destination
leading to detection errors at the destination.
2.5 Matrix Decomposition
Matrix decomposition has been found useful in signal processing application. Matrix
decomposition techniques have been used to decompose multiuser MIMO channels into single
user MIMO sub channels [40]-[43]. Multiuser channel decomposition is required for easy design
of modulation, demodulation, coding and decoding schemes when channel state information
(CSI) is known, [41] and [42]. Matrix decomposition algorithms are also useful in channel
estimation where the channel state information (CSI) is required for efficient decoding of
15
transmitted signal at the destination. QR decomposition and singular value decomposition (SVD)
algorithms are the most commonly used matrix decomposition algorithm for channel estimation
in MIMO systems [44]-[47]. In recent times matrix decomposition algorithms have been
extended to channel estimation of amplify and forward cooperative communication. Most of the
existing channel estimation methods for amplify and forward estimates the composite channel
which is the combination of source-relay and relay-destination channels. In order to achieve
optimal system design and efficient power allocation individual channel estimates of source-
relay and relay-destination are required [48]-[50].
2.5.1 Singular value decomposition
In Singular value decomposition an m-by-n m n matrix A is decomposed as
A = U V (2.3)
where U is an m-by-m unitary matrix, V is an n-by-n unitary matrix and Σ m by nR is a
diagonal matrix with real non-negative elements. Let
1 m 1 m 1 2(u ,......u ), (v ,......v ), ( , ....... )nU V (2.4)
The parameter i is referred to as the i -th singular value of A, while iu and iv are the left and
right singular vectors respectively. The diagonal singular values can be arranged in any order,
although if it is required, permutation matrices can be used to arrange them in descending order.
Let Pm m and Q
n n be permutation matrices such that P Q is diagonal, then
A=(UP )(P Q)(QV )T H (2.5)
In this case matrix A is referred to as an SVD if P and Q matrices are chosen such that the
singular values are obtained in descending order:
1 2....... 0r (2.6)
16
Theoretically, Eigen value decomposition can be used to solve the Singular Value
Decomposition problem as follows [51]:
2 2
1( ) ( ...... )V A A VH H
rdiag (2.7)
where 1 2, ........V nv v v , 1 2 1....... ..... 0r r n and r = rank(A).
The next step is to obtain vectors iu
ii
i
Avu
, 1,........i r (2.8)
Therefore, we obtain 1 2,u ........uU mu
In order words, r columns of U with corresponding none zero singular values span the column
space while r columns of V span the row space. If matrix A is real then the singular vectors U
and V are also real, likewise if matrix A is complex then the singular vectors are also complex
matrices. Theoretically, the SVD of matrix A can be computed through decomposition of the
Gram matrix ( )A AH . Unfortunately, formation of the gram matrix has well known numerical
difficulties [51].
2.5.2 QR decomposition
QR decomposition of an m-by –n matrix A is defined as [49]
R
A Q0
(2.9)
Where Q is an m-by-m orthogonal matrix and R is an n-by-n upper triangular matrix and 0 is a
null matrix.
There are various methods for computing the QR, which include Householder transformations,
Givens rotations, fast Givens rotation and Gram-Schmidt orthogonalization. For the purpose of
17
this work more emphasis will be on the Givens rotations because of its parallel architecture
which has been found suitable for hardware implementation.
The Givens rotation method of decomposition for matrix A involves plane rotations to
selectively annihilate off diagonal elements to obtain a lower or upper triangular matrix. Givens
rotations can be represented in matrix form as [52]
1 0 0 0
0 0( , , )
0 0
0 0 0 1
Gc s
a bs c
a
b (2.10)
a b
Where is the rotation angle, cosc and sins . Givens rotations are orthogonal rotations.
Pre-multiplication of matrix A by ( , , )G a b annihilates (a,b)th
element of matrix A, which
translates to a clockwise rotation through angle in the (a,b)th
coordinate plane. For counter
clockwise rotation, matrix A is post-multiplied by T( , , )G a b through angle in the (a,b)
coordinate plane.
2.6 System Model
The Basic block diagram of MIMO relay system with two relays is shown in Figure 2.1. The
source and destination are equipped with N transmit and M receive antenna respectively.
Transmission between the source and destination is assisted by R relay stations. Each relay
station is equipped with a single antenna that operates in half duplex mode. Therefore, a relay
station cannot simultaneously transmit and receive in one time slot. Transmission of signal from
the source to destination is done in two time slots. In the first time slot, transmitter with N
transmit antennas sends data packets to the relay stations which amplify the received signal.
18
Transmission of amplified data packets between the relay and destination occupies the second
time slot.
There is no direct transmission between the source and destination station since the destination
station is assumed not to be within the coverage area of the transmitter.
1 H1 H2 1
N M
R
Figure 2.1: Schematic Diagram of Amplify and Forward MIMO Relay System
For simplicity we shall assume frequency flat fading channels between the source-relay channels
and the relay-destination channels. The received signal at the relay station can be modeled as
1 1( ) ( ) ( )r H s ws n n n (2.11)
1( )rRx
s n is the received signal vector at the relay station
1( )s Nxn denotes signal transmitted from the source to the relay station
1HRxN is the channel matrix between the source and relay station
1( )w n is the additive white Gaussian noise (AWGN) vector at the relay stations
TX
RX
19
The received signal at the relay is amplified and forwarded to the destination during the second
time slot ( 1)n . The signal model at the destination can be expressed as:
2 2 21( 1) ( ) ( ) ( 1)1y H FH s H Fw wn n n n (2.12)
2HMxR is the channel matrix between the relay and destination.
FRxR represents the amplification matrix which is diagonal.
where vectors 2
1( ) (0, )w R Rn I and 2( ) (0, )
2w y Mn I represent the noise contribution at
the relay and destination, respectively, and are assumed to be independent white complex
Gaussian noise. Both the source-relay ( 1H ) and relay-destination 2( ) H channels are also
assumed to be frequency flat fading. Similarly, the received signal vector in the second time slot
can be expressed as
( 1) ( ) ( 1)y Hs wn n n (2.13)
where 2 1 2( 1) ( ) ( 1)w H Fw wn n n
The matrix 2 1H H FH , HM N denotes the composite channel matrix between source-relay
and relay-destination, while, 1( 1)w Mn is the overall noise vector. No channel estimation
is performed at the relay so as to reduce the complexity of the relay stations. In practice it is cost
effective to minimize the complexity at the relays, therefore, the only processing performed is
amplification of transmitted signals. The composite channel is estimated at the destination.
20
CHAPTER 3: CHANNEL ESTIMATION FOR AMPLIFY AND FORWARD MIMO
RELAYS
3.1 Introduction
There are two channel estimation methods for amplify and forward relaying scheme namely; the
cascaded (composite) channel estimation method and disintegrated channel estimation method.
In cascaded channel estimation, the source-relay and relay-destination channel estimation is
performed at the destination. In disintegrated channel estimation, the source-relay and relay-
destination estimation is separately done at the relay station and destination, respectively. The
disintegrated method offers more complexity than the composite method because the relay
station is equipped with a channel estimator. In order to perform channel estimation, a known
training sequence is transmitted from source to destination to obtain channel state information.
The receiver, therefore, uses the channel knowledge obtained during training to equalize the
channel effect for accurate decoding of received signal and also for efficient power allocation.
Channel estimate can still be obtained without using a known training sequence; this method is
known as blind channel estimation. In blind channel estimation, the receiver exploits the
statistical properties of the received signal and channel structure to obtain channel estimates. One
of the widely studied blind estimation is the second order statistics [SOS] based blind estimation
[53]-[55]. This estimation method has an advantage of bandwidth efficiency since no overhead
training sequence is required. However, the computational complexity of this method makes it
unattractive in real life implementation. Therefore, the training based method will be used to
estimate the composite channel in this thesis. The main objective of this chapter is to propose the
Jacobi-SVD method to obtain source-relay and relay-destination channel estimate separately. In
Section 3.1 least squares estimation is performed to obtain the composite channel estimate of
21
source-relay and relay-destination channels. In Section 3.3 Jacobi rotations are used to
decompose the composite channel estimate in order obtain the individual channel estimate.
Performance evaluation is carried out in Section 3.4 to determine the accuracy of the individual
channel estimate obtained.
3.2 Composite Least Squares Channel Estimation.
There are generally two approaches used in estimation theory; these are the classical and
Bayesian approach. In the Classical approach, the parameter to be estimated is considered to be
deterministic but unknown. Although a signal model is assumed, no optimality is claimed. On
the other hand, the Bayesian approach considers the parameter to be estimated as random and
requires prior knowledge of the probability density function (apriori PDF) [56]. Using the apriori
information contained in the PDF improves estimation accuracy. Estimates from both classical
and Bayesian converge asymptotically when the same model is assumed [57]-[58]. The classical
approach has less complexity and easy to use in many applications since no probabilistic
assumptions are made on the parameter to be estimated. A family of classical approach known as
the least squares will be used for the end-to-end (composite) channel estimation as well as
obtaining individual channel estimates. We illustrate the transmission model for amplify and
forward MIMO relaying system used for estimating the channels in Figure 3.1. The source-relay
and relay-destination channels are denoted as H1 and H2 respectively. The objective is to obtain
estimates of H1 and H2 individually. First, we estimate the composite channel using a least
squares method. We then decompose the composite channel to obtained H1 and H2 individually
by another least squares method using kronecker products.
22
H2
h1,i h2,i
Source
h2,R
Figure 3.1: Transmission model for Amplify and forward MIMO relaying system
Using the system model described in Chapter 2 and further illustrated in Figure 3.1, with N
transmit and M receive antennas at the source and destination respectively. Transmission of
signal from source antennas to destination antennas is assisted by two relay stations. These relay
stations are both equipped with a single antenna that operates in half-duplex mode. A known
training sequence s = 1 .. Ls s of length L ( L N ) is transmitted from the source to the relay
stations in the first time slot. The received signal at the relay is amplified at the relay station and
then forwarded to destination in the second time slot. Each relay station amplifies the received
signal with amplification factor F(i)
(1........R)i therefore, we obtain R channel pairs of
1
1, 1,
N
i ih h and 1
2, 2,
M
i ih h . The rows of H1 channel matrix (source-relay channel) are
derived from vector 1
1, 1,
N
i ih h and the columns of H2 channel matrix (relay-destination
channel) are derived from 1
2, 2,
M
i ih h . At the destination, the composite channel estimate is
Destination
1● ● ●
h2,1
ith
Relay
h1,R
SSour
ce
1●
● ●
FR
F1
Fi
Relay 1
=1
●
● ●
Relay R
H1 h1,1
23
obtained using the least squares method and this estimate is then decomposed to obtain
individual channel estimates of H1 and H2. The first step is to obtain the lest squares estimate of
the i-th composite channel as follows [56]:
( )†
H YSi
(3.1)
where † 1( )S S SSH H , 1 2 3 L(y , y , y .....y )Y represents the received signal vector at the
destination and ( )
Hi
is the estimate of composite channel matrix.
In order to decompose the composite channel estimate, further least squares minimization is
performed using the estimate obtained in Equation (3.1) as follows [31]:
21
2( )
( )
2 1
1
min ,H H H H F HR i
i
i F
(3.2)
where 2
F denotes the Frobenius norm operator.
The next step is to obtain a new set of matrices by vectorizing each term in Equation (3.2). The
vectorized terms are arranged such that they form the columns of the new matrix. We define the
new matrix HMNxR obtained by arranging the vector
( )
, 1.......Hi
i R into a matrix as
(1) (R)
( )........vec( )H H Hvec
, where (1)
( )Hvec denotes the vector form of (i)
H . The diagonal
amplification factor is also arranged in a vector form, therefore the new matrix F is expressed as
(1) (R)( ),...... (F )F= vecd F vecd , FR R . We also obtain matrix Ξ
MNxR as 1 2)Ξ= (H HT
where . the Khatri-Rao product (column wise kronecker product). Therefore, the new set of
matrices obtained is expressed as
21
2
min ,H H H ΞFF
(3.3)
24
Using least squares estimation, the estimate of Ξ is obtained from Equation (3.3) as
†Ξ=HF (3.4)
In order to obtain individual channel estimates of H1 and H2, further minimizations are then
performed on the estimate of Ξ obtained in Equation (3.4) as follows:
21
2
1 2min ,ˆ )H H Ξ- (H HT
F (3.5)
Using the property of kronecker product, the minimization in Equation (3.5) can be carried out
column by column. Denoting the i-th column by фi , a column-wise kronecker product
relationship exist between h1 and h2 that is,
1 2ф h hT
i (3.6)
where operator represents the kronecker product and 1.........ф = R .
The minimization problem for the i-th column becomes
21 2
2
1h ,hmin ф - h hT
i F (3.7)
From [60] it follows that solving the minimization on Equation (3.7) is equivalent to obtaining an
approximate rank-1 matrix for фi . The Singular Value Decomposition (SVD) is a well known
method for solving the approximate rank-1 matrix problem.
The SVD of фi can be expressed as
*ф UΣVi (3.8)
where U and V are the singular vectors and Σ is a diagonal matrix representing singular values
of фi .Van Loan in [60] defines the optimum values of h1 and h2 for matrix Φ as
25
1 1( ) (:,1)h Vopt (3.9)
2 1( ) (:,1)h Uopt (3.10)
1 represents the maximum singular value.
Loan further suggests that an arbitrary scaling 0) can be used to multiply h1 and h2 such
that 1( ).h opt and
2( ).h opt are also optimum. Thus performing the minimizations for each i-th
column 1........i R , we obtain the estimates of the channels H1 and H2.
Due to implementation difficulty associated with constructing the gram matrix required for
singular value decomposition, we propose a more practical approach based on rotation. The
Jacobi-SVD algorithm can be used to diagonalize a matrix to obtain its singular values and
singular vectors. This algorithm is suitable for practical implementation and is numerically
stable.
3.3 Two-by-two Jacobi Singular Value Decomposition
The main principle of Jacobi methods is to diagonalize a given matrix via a sequence of
orthogonal rotations in order to annihilate the off diagonal elements of the matrix. The Jacobi
rotations were first used for real symmetric matrices and latter extended to real non-symmetric
matrices. A symmetric matrix A is a square matrix whose transpose is equal to the matrix, that
is, a matrix A that satisfies the condition A = AT. In solving for eigenvalues and eigenvectors of
an arbitrary real symmetric matrix Anxn , rotations are performed using the Jacobi rotation
matrix abP [61]. The Jacobi rotation matrix is given by
26
1
1
Pab
c s
s c
a
b
(3.11)
a b
The rotation matrix has all ones in its diagonal except for c in rows a and b, the off diagonal
elements are zero with the exception of s and –s. The non-zero off diagonal elements s and - s
are the sine of a rotation angle while the non-unity diagonal elements c are the cosine of a
rotation angle , such that 2 2 1+ c s The Jacobi rotation matrix is used to transform matrix A
to obtain a new matrix A which can be expressed as
'A P APT
ab ab (3.12)
The rotation matrix Pab is used to transform the rows a and b of matrix A, while PT
ab transforms
the columns of a and b. Each rotation annihilates one of the off diagonal element, therefore
multiple rotations are required to diagonalize the matrix. The diagonalization process requires
successive rotations to selectively annihilate off diagonal elements. Subsequent rotations can
usually undo the annihilated off diagonal elements in previous rotations, nevertheless, the off
diagonals are reduced. Therefore sets of rotations (sweeps) are required for achieving a diagonal
matrix. The idea of these basic rotations has been extended to non-symmetric real matrices.
A real non-symmetric square matrix Anxn can be diagonalized by computing the sine and
cosine of rotation angles which annihilate the off diagonal elements [51] such that
1 1 2 2 1
1 1 2 2 2
0
0
Tc s c s dw x
s c s c dy z
(3.13)
27
where Aw x
y z
When the rotations are not able to achieve ( 1 2 0)diag d d this is referred to as unnormalized
SVD. In order to obtain normalized SVD, diagonal elements need to be sorted in descending
order. The diagonal elements can either be sorted by permutation or BLV normalization (Brent-
Luk-VanLoan) algorithm. The BLV normalization is given by
1 1 2 2 1
1 1 2 2 2
0
0
Tc s k c s dw x
s c k s c dy z
(3.14)
If the value of k is -1, then the left rotation is a reflection and can be written as
1 1
1 1
c s
s c
An alternative method is to use permutation matrices so that the singular values d1 and d2 are
arranged in descending order. Let Pnxn and Q nxn be permutation matrices, U, V are the
singular vectors and Σ is the singular value then the normalized SVD obtained by permutation is
expressed as
T T H( ) ( )( )A UP P Q Q V (3.15)
The permutation matrices are chosen to achieve ( 1 2 0)diag d d .
Singular value decomposition has been found to be very useful in signal processing applications.
Complex matrices are often encountered in signal processing, hence the need to extend the
Jacobi-SVD to complex matrices. Each element of a complex matrix is a complex number with
real and imaginary part.
28
Any complex number r iA A iA (rA and
iA are the real and imaginary component of A
respectively) can be expressed in exponential form as
i a
AA R e (3.16)
where 2 2
r iR A A and 1tan i
a
r
A
A
The Jacobi-SVD algorithm performs a two-step transformation to diagonalize a complex 2-by-2
matrix. The two-step transformation is illustrated as follows [62]:
Consider a complex matrix A
1 1 2 2
3 3 4 4
Aa +ia b +ib
c +ic d +id
(3.17)
Expressing complex matrix A in exponential form, we obtain
11 12
21 22
A
i a i b
i c i d
A AAe Be
A ACe De
(3.18)
Step 1 annihilates matrix element A21. This is done by using a unitary transformation to convert
the matrix elements in the second row to real. QR decomposition is then performed to annihilate
A21. Transformations performed in Step 1 are illustrated as follows:
0 0
0 0
i i a i b i i y i z
i i c i d i
e Ae Be e Ae Be
e Ce De e C D
(3.19)
where,
( )
2
d c+
and
( )
2
d c
29
In Equation (3.19), we perform a unitary transformation which converts the second row of the
matrix to real. This step is followed by QR decomposition to annihilate matrix element A21.
0
y zi i w i xic s c s We XeAe Be
s c s c ZC D
(3.20)
where, 0 and arctanC
D
The first step of diagonalization can be written more compactly as
0
a b
c d
ii i ii i i w i x
i i i iii
c e s e c e s e We XeAe Be
s e c e ZCe Des e c e
(3.21)
Step 2 completes the diagonalization of matrix A by annihilating matrix element A12 as follows:
0 0
00 00
i ii w i x
ii
e W XWe Xe e
ZZ ee
(3.22)
,2
X W
2
X W
,
2
W X
In Equation (3.22), we perform a unitary transformation to convert the matrix elements in the
first row to real. QR decomposition is then performed to annihilate A12 as follows:
0
0 0
c sc s W X P
s c Z s c Q
(3.23)
where tanX
Z W
and tan
X
Z W
The second step of the transformation is expressed as
1
2
0
00
w x
i ii ii i
ii i i
c e s ec e s e We Xe
Zs e c e s e c e
(3.24)
30
The resulting diagonal matrix in Equation (3.24) is the singular value of matrix of matrix A. The
left singular vectors U are obtained from the product of all the left hand side rotation matrices,
similarly we obtain the right singular vectors V by multiplying all the right hand side rotation
matrices.
Therefore, we can express U and V as
00
0 0U Q
ii
H L
i i
c s c see
s c s ce e
(3.25)
0 0
0 0V Q
ii
R
i i
c s c se e
s c s ce e
(3.26)
The Jacobi-SVD for diagonalizing a complex matrices can be summarized as follows:
Table 3.1 Summary of Jacobi-SVD for 2-by-2 complex matrix
Step Transformation performed Objective
1 Unitary Transformation Transforms the second row of matrix to real
QR decomposition Annihilates A21 matrix element
2 Unitary Transformation Converts the first row of step 1 output to real
QR decomposition Annihilates A12 matrix element
Applying the Jacobi-SVD algorithm, we can decompose channel matrix , 1.......фi i R in
Equation (3.7) to obtain the individual channel estimates. Using the two-step transformation, we
can obtain the singular vectors and singular values of фi as follows:
1 2,ф QQR
i i
L
i diag (3.27)
31
QL
iand Q
R
idenote the product of left and right rotation matrices respectively and are referred to
as the singular vectors while 1 and 2 denote the singular values. Therefore, we can rewrite
Equation (3.9) and (3.10) as Equation (3.28) and (3.29) respectively,
RH
1 1( ) (:,1)h Qopt (3.28)
2 1( ) (:,1)h QLopt (3.29)
Where 1 is the maximum singular value.
Therefore, performing the Jacobi-SVD for , 1.......фi i R we can obtain all the matrix elements
of individual channel estimate of H1 and H2.
3.4 Performance Evaluation
In this section, we use Monte Carlo simulations to evaluate the performance of our channel
estimations algorithm. The MIMO channels with Nt and Nr antennas at the source and
destination station respectively is modeled using the kronecker model. The transmission of a
signal between source and destination is aided by single antenna relay stations R. For simulations
purpose we use two transmit and receive antennas at the source and destination nodes
respectively and two single antenna relay nodes. The source-relay channel matrix is denoted as
H1 and relay-destination channel matrix by H2. We generate 10,000 realizations of H1 and H2
using the kronecker model for the MIMO channel. The spatial correlation is modeled as
1 12 2
1 2 1H R H Rwsr and 1 1
2 2
2 3 2H R H Rwrd (3.30)
HRxNt
wsr and HNrxR
wrd are complex Gaussian random variables, independent and
identically distributed with zero mean and a variance of 1. R1, R2 and R3 denote the transmit
antenna spatial correlation, relay antenna spatial correlation and receive antenna spatial
32
correlation, respectively. The spatial correlation matrix R is an identity matrix when no spatial
correlation exists between the antennas at the source, relay and destination station. In this case
the correlations matrix has diagonal elements equal to one and off diagonals elements are zero.
When spatial correlation exists at the antennas, the correlation matrix has diagonal elements
equal to one and non zero off diagonal elements denoted by ρ. The off diagonal elements ρ
represent the degree of antenna correlation. In this simulation model, two antennas were used at
the source and destination nodes and two relay stations with single antenna. An orthogonal
training sequence is transmitted in frames from the source to amplify and forward relay node.
We use a short training sequence of length L = Nt. A complex orthogonal matrix is generated
which serves as the amplification matrix during the training period.
The accuracy of the individual channel estimates of H1 and H2 is obtained by computing the
NMSE (normalized mean square error) as follows:
2
11
1 2
1
H DH
H
F
F
NMSE
(3.31)
21
22
2 2
2
H H D
H
F
F
NMSE
(3.32)
To obtain an optimum value of 1H and 2H an arbitrary scaling matrix is required to resolve the
SVD ambiguity of the channel estimate as earlier suggested in Section 3.3. The matrix D is
obtained by dividing the maximum element in each row of 1H by the maximum element in each
row of 1H .
We compare the performance of individual channel estimate obtained by SVD LAPACK (Linear
algebra PACKage) traditional standard software in Matlab with those obtained with Jacobi SVD
33
algorithm. The results are shown in Figures 3.2, 3.3, 3.4, and 3.5. We observe that both methods
have similar performance when applied to the channel estimation algorithm. However, for
practical implementation, the Jacobi-SVD is preferred because of its numerical stability and
suitability for hardware implementation because they are amendable to parallel architecture.
In Figure 3.2 and 3.3, we compare the performance of channel estimates of H1 and H2 when
there is Spatial no correlation between the relay station antennas. In Figure 3.2, the normalized
mean square error of H1 channel estimate (NMSE1) is high at low SNR. This is because for
amplify and forward relaying systems, the relay amplifies the noise along with the transmitted
signal from the source. Therefore, we observe performance degradation in the channel estimate
of H1 channel at low SNR.
Figure 3.2 Channel estimation performance for H1 with no spatial correlation at the relays
NMSESVD- Normalized Mean Square error for LAPACK SVD
NMSEJ- Normalized mean Square Error for Jacobi SVD
0 5 10 1510
-2
10-1
100
101
SNR [dB]
NM
SE
1
NMSESVD1
NMSEJ1
34
Figure 3.3 shows the normalized mean square error (NMSE2) performance of H2 channel
estimate. It is observed that H2 channel estimate performs better at low SNR compared with H1
channel estimate. Therefore, at low SNR the noise contribution effect is more evident in channel
H1 due to the noise amplification at the relay stations.
Figure 3.3 Channel estimation performance for H2 with no spatial correlation at the relays
Next, we compare the NMSE performance of H1 and H2 channel estimates when spatial
correlation exists between relay station antennas. Figures 3.4 and 3.5 show the NMSE
performance of H1 and H2 channel estimates in the presence of spatial correlation at the relay
stations. The correlation coefficient ρ at the relays is chosen to be equal to 0.9 which indicates a
strong correlation between antennas at the relaying nodes. It is seen that the NMSE (normalized
mean square error) for the case of relay antenna correlation is higher compared to when no
0 5 10 1510
-2
10-1
100
101
SNR [dB]
NM
SE
2
NMSESVD2
NMSEJ2
35
correlation exist. This is because when spatial correlation exists, full antenna diversity gain is not
achievable, which leads to increased estimation errors. Therefore, we observe that antenna
correlation increases the channel estimation errors for H1 and H2 channels as seen in Figures 3.4
and 3.5.
Figure 3.4 Channel estimation performance for H1 with spatial correlation ρ = 0.9
0 5 10 1510
-2
10-1
100
101
SNR [dB]
NM
SE
1
NMSESVD1
NMSEJ1
36
Figure 3.5 Channel estimation performance for H2 with spatial correlation ρ = 0.9
3.5 Chapter Summary
In this chapter we proposed the Jacobi-SVD for decomposing the composite channel estimate for
amplify and forward MIMO relay system. Firstly, we obtain the channel estimate of the
composite channel which includes the source-relay and relay-destination channels by least
squares estimation method. Secondly, we perform a sequence of minimizations to obtain a
matrix that reveals the relationship between individual channel estimates. The Jacobi-SVD
decomposition method is then used to decompose this matrix to obtain individual channel
estimate of source-relay and relay-destination channels. The performance evaluation of the
proposed Jacobi-SVD method for channel decomposition was carried out. Results show that
Jacobi-SVD decomposition method for obtaining individual channel estimates performs better
when the there is no spatial correlation between antennas. The Jacobi-SVD algorithm and the
0 5 10 1510
-2
10-1
100
101
SNR[dB]
NM
SE
2
NMSESVD2
NMSEJ2
37
theoretical LAPACK SVD have the same performance when applied to our channel estimation
algorithm. The LAPACK SVD has associated numerical difficulties in constructing the Gram
matrix which makes it impractical to implementation. In hardware implementation, Jacobi-SVD
algorithm is generally preferred because they offer stability and good numerical properties. In
the next chapter, we use the Jacobi-SVD on systolic arrays to obtain individual channel
estimates. The systolic arrays have suitable characteristics required to perform real time
computations.
38
CHAPTER 4: JACOBI SVD CHANNEL ESTIMATION USING THE SYSTOLIC
ARRAY
4.1 Introduction
Systolic arrays employ the principle of parallel computing which involves a network of
processing units that compute and exchange data with other units. In parallel computing, large
and complex computations are divided into smaller units and each unit is computed
simultaneously. Systolic arrays are very useful because they possess characteristic features that
enable real time signal processing [63]. The 2-by-2 Jacobi singular value decomposition (JSVD)
described in Chapter 3 provides the frame work to decompose a larger matrix using systolic
arrays. Each processing unit on the systolic array contains a 2-by-2 matrix and the units are
interconnected to exchange data elements and rotation angles. Brent and Luk [64] described the
Jacobi algorithm as the most efficient SVD algorithm because they can be easily implemented in
parallel computing. In Section 4.2 we give a brief description of Jacobi rotations on systolic
arrays. In Section 4.3 the least squares method was first used to obtain the composite channel
estimate and then, the individual channel estimate was obtained using Jacobi rotations on systolic
arrays. We evaluate the performance of the individual channel estimation method in Section 4.4.
Lastly, we provide a brief summary of the chapter.
4.2 The systolic arrays for Jacobi SVD
This section gives an overview of the BLV (Brent-Luk-VanLoan) systolic arrays used for
decomposition. These systolic arrays were primarily used for singular value decomposition of a
real square matrix and were adapted for complex n-by-n square matrix in [51]. The array contain
2n -by
2n processing units, each processing unit contains a 2-by-2 matrix as illustrated in
39
Figure 4.1. This implies that an arbitrary n-by-n square matrix can be decomposed by breaking
the large matrix into smaller matrices in each processor and computing rotation angles
simultaneously for each unit.
Figure 4.1 Propagation of rotation angles for the Brent-Luk-Van Loan systolic array
40
The processors that contain the diagonal element of the matrix are called the diagonal processors
while the processors containing the off diagonal elements are known as the off diagonal
processor. Diagonal and off diagonal processors perform different roles. The diagonal processor,
also referred to as boundary processor, is used to compute the rotation parameters which are
transferred to the nearest adjacent processor for zeroing off diagonal elements. They perform
very important role in that they compute the rotation angles and which are propagated along
the rows and columns to annihilate the off diagonal elements. Generally, for BLV systolic array,
the off diagonal processor 2( , 1, )nijP i j i j contain real matrix element M as represented
in [51].
Mij ij
ij ij
(4.1)
The diagonal processors ( 1,...... )2ii
nP i compute the left and right rotation pairs of angle
,SL L
i iC and ,SR R
i iC respectively. Figure 4.2 shows a single diagonal processor and the links for its
inputs and outputs. The Position of the off diagonal processor on the systolic array would
determine the flow of the rotation angles as shown in Figures 4.3 and 4.4.
41
Figure 4.2 Input and output communication links for diagonal processor
Figure 4.3 Sub-diagonal Processor for BLV array
In δ
●
● ●
Out δ
●
● ●
Out α
●
● ●
In
● ●
●
Out
●
● ●
Out
β
●
●
●
Out
●
● ●
In
●
● ●
In γ
●
●
●
Out
γ γ γ
●
●
●
In α
●
● ●
In β
●
●
●
In β
●
●
●
In δ
●
● ●
In α
●
● ●
Out α
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●
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Out β
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●
Out
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Out δ
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Out
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● ●
Out
●
● ●
Out
●
● ●
In γ
●
●
●
Out
γ
●
●
●
42
Figure 4.4 Super-diagonal Processor for BLV array
During the process of diagonalizing the matrix, processor iiP computes the left and right rotation
elements for zeroing off diagonal elements ii and ii as below:
'
'
0
0
TL L R L
ii iii i i i ii
L L L Rii iii i i i ii
c s c s
s c s c
(4.2)
The rotations in Equation (4.2) can also be expressed as
'
'
cos sin cos sin 0
sin cos sin cos 0
T
ii ii ii
ii ii ii
(4.3)
Once the rotation angle has been generated by the diagonal processor, the matrix elements are
interchanged and the rotation angles are propagated to the row and columns. Propagation of
In α
●
● ●
Out α
●
● ●
In γ
●
● ●
Out γ
●
● ●
Out
●
● ●
In
●
● ●
Out β
● ●
●
In β
●
●
●
Out
●
● ●
Out
●
● ●
Out δ
●
● ●
In δ
●
● ●
43
rotation angles are illustrated in Figure 4.1, which shows that only the diagonal processor
computes rotation angles which are propagated as input to the off diagonal processor.
Angle is propagated along the columns while is propagated along the rows of the diagonal
processor. Figure 4.5 shows the connection of the diagonal and off diagonal processor to
facilitate exchange of data matrix. Both Figures 4.1 and 4.5 represent the conventional BLV
systolic array for 8-by-8 matrices.
44
Figure 4.5 The Brent-Luk-Van Loan systolic arrays for diagonal and off diagonal
processor connections for n = 8
45
The IEEE 802.11 MIMO system with 4x4 antenna constellations would require breaking down
the complex matrix into a 2-by-2 SVD on the systolic arrays. Each 2-by-2 SVD processor
computes the left and right rotation angles to pre-multiply and post-multiply a square matrix
n by nM R . In the systolic array each the 2-by-2 processor performs one iteration. For a larger
matrix, multiple iterations performed on the matrix constitute one sweep. Therefore, multiple
sweeps are required to completely annihilate off diagonal elements for Jacobi-SVD convergence.
The Jacobi rotation matrix is given by
, ,
0 0 0 01 1
0 0
0 0
0 0 0 0 0 01 1
Q QH
i j i j
ic s c s
js c s c
i
j
(4.4)
i j i j
Where c = cos θ, s = sin θ , θ is the rotation angle, ,Qi j and ,QH
i j are the left and right Jacobi
rotation matrix respectively. The rotation matrix ,Qi j transforms i-th and j-th rows of matrix M
and ,QH
i j transforms i-th and j-th columns. Since the i-th and j-th rows and columns are affected,
subsequent iterations are likely to undo the zeroing done by previous iteration, hence the need for
multiple sweeps to ensure convergence. The iterations on systolic arrays required for each sweep
can be done in serial or parallel order depending on the computational complexity required.
4.2.1 Serial ordering of Jacobi-SVD
This section illustrates the iterations and sweeps required for serial ordering. Let matrix A in
Equation (4.5) be 4-by-4 matrix to be diagonalized using the serial ordering
46
1,1 1,2 1,3 1,4
2,1 2,2 2,3 2,4
3,1 3,2 3,3 3,4
4,1 4,2 4,3 4,4
A
A A A A
A A A A
A A A A
A A A A
(4.5)
Each processor in the systolic array contains a 2-by-2 matrix, therefore matrix A is divided into a
2-by-2 matrix which can be handled by the processor. During the first iteration matrix elements
1,1 1,4 4,1 4,4, , ,A A A A are chosen to form a 2-by-2 sub-matrix. These matrix elements are located on
the 1st and 4th rows and 1st and 4th columns of matrix A. The purpose of this iteration is to use
the Jacobi rotations in Equation (4.4) to annihilate elements 1,4A and
4,1A of matrix A. The Jacobi
rotation matrix Qij transforms the i-th and j-th row of matrix A and ,QH
i j transforms its i-th and j-
th columns. Therefore, all matrix elements on the same row and column of the sub-matrix are
affected by the rotation. In this case the i-th and j-th rows correspond to the 1st and 4th rows and
the i-th and j-th columns correspond to the 1st and 4th columns. The resulting matrix '
1A is given
by
' ' '
1,1 1,2 1,3
' '
2,1 2,2 2,3 2,4'
1 ' '
3,1 3,2 3,3 3,4
' ' '
4,2 4,3 4,4
0
0
A
A A A
A A A A
A A A A
A A A
(4.6)
The matrix elements with apostrophe indicate the elements affected by the Jacobi rotation while
those without marking are not affected. This step is followed by the second iteration. During the
second iteration, matrix elements 1,1 1,3 3,1 3,3' , ' , ' ,A A A A are chosen to form the 2-by-2 sub-matrix.
The purpose of this iteration is to annihilate 1,3'A and 3,1'A matrix element of '
1A using the Jacobi
rotation matrix in Equation (4.4). The rotation matrix Qij transforms i -th and j -th row of
47
matrix A and ,QH
i j transforms it i-th and j-th columns. The sub-matrix elements are located on
the 1st and 3rd rows and the 1st and 3rd columns of '
1A . Therefore all the elements on the same
rows and columns with the sub-matrix will be transformed by the rotation, and we obtain a new
matrix '
2A which is expressed as
'' '' ''
1,1 1,2 1,4
'' ''
2,1 2,2 2,3 2,4'
2 '' '' ''
3,2 3,3 3,4
'' ''
4,1 4,2 4,3 4,4
0
0A
A A A
A A A A
A A A
A A A A
(4.7)
The double apostrophes on the matrix elements indicate the elements that have been affected
during the second iteration. All the six iterations required to complete one sweep are illustrated
in Figure 4.6. The result of previous iterations need to be available before subsequent ones can
be carried out, therefore, more time is required for convergence. Convergence rate of serial
algorithm are discussed in [65]. Table 4.1 gives a summary of the serial ordering algorithm
Table 4.1 Serial ordering algorithm on systolic array
While A does not converge do
o Chose the sub-matix element
o perform 2-by-2 Jacobi rotation on the submatrix
o multiply matrix A ,
H
i jQ and ,i jQ to transform i-th and j-th rows and column location of
submatrix.
o update matrix A with the new values after rotation.
if all i-th and j-th row and column pairs have been rotated
then Multiply ,
H
i jQ and ,i jQ by the previous rotations
loop until A = diag(a1.....n)
end if
end while
48
A A A A
A A A A
A A A A
A A A A
2 3 0 3
3 3 3 0
0 3 2 3
3 0 3 3
A A A
A A A
A A A
A A A
2 2 1 0
2 1
0 1
2 1 1 1
A A A
A A A A
A A A
A A A A
1 1 1 0
1 1
1 1
0 1 1 1
A A A
A A A A
A A A A
A A A
.
4 0 4 4
0 4 4 4
4 4 2 3
4 4 3 3
A A A
A A A
A A A A
A A A A
4 5 5 4
5 5 0 4
5 0 5 5
4 5 5 3
A A A A
A A A
A A A
A A A A
4 5 6 6
5 5 6 6
6 6 6 0
6 6 0 6
A A A A
A A A A
A A A
A A A
' 0 0 0
0 ' 0 0
0 0 ' 0
0 0 0 '
A
A
A
A
Figure 4.6 Illustration of 4x4 Jacobi SVD serial ordering for iteration and sweep process
4th
Iteration 5th
Iteration 6th
Iteration
Sweeps until
convergence
ITERATION
NUMBER
MATRIX ELEMENT
ELEMENTS
AFFECTED BY
ITERATION
1 A1
2 A2
3 A3
4 A4
5 A5
6 A6
2nd
Iteration 3rd
Iteration
Convergence
sweep ends
1st Iteration
49
4.2.2 Parallel ordering of Jacobi-SVD
The parallel ordering has been used in BLV systolic array [51] where multiple iterations can be
performed simultaneously. This makes the parallel ordering more desirable compared to the
serial ordering where the previous iteration has to be completed before proceeding to subsequent
ones. An illustration of parallel ordering used in the BLV systolic array is shown in Figure 4.7.
The off diagonal processors connected to the nearest diagonal make it possible for the exchange
of data matrix. Each diagonal processor generates the rotation angles which are propagated at a
constant time to off diagonal processor in the same row and column. The location of the matrix
element determines its new input after exchange of matrix element as shown in Figure 4.2. The
staggering computation used by the Brent-Luk-Vanloan illustrated in Figure 4.7 shows how the
10-by-10 matrix is computed to control the broad cast of rotation angle computed. The number
in each square box represents the iteration number and each processor with the same iteration
number can be done simultaneously with three iterations representing one sweep. During each
iteration process all matrix elements on the same row and columns with the diagonal processor
are affected by the rotation. Each iteration step is completed when all rows and columns have
interchanged their matrix elements and the new set of matrix elements are ready to be computed.
50
Figure 4.7 Staggering computations of BLV Arrays
1
1
1
1
1
1 1
1 1
1 1
1
1
1
1 1
1
1
2
2
2
2
2
2
2 1
2 2
2 2
2 2
1 2
2 1
2
1
2
1 2
1 2
3 2
3
2
3
2
3
2
3
3
2
3
3
3
3
3
3
2
3
3
3
3
3
3
3
51
4.2.3 Extension of BLV systolic array to complex matrices
In applications, such as wireless communication the channel matrices are mostly represented by
complex matrices Hemkumar et al in [62] have extended the BLV systolic arrays for complex
matrices. In this case, each diagonal processor is required to compute four unitary angles to
convert the complex matrix to real in addition to two rotational angles to annihilate the off
diagonal element. Figure 4.8 illustrates the computation of complex SVD on the BLV systolic
array. The BLV staggered computation has been extended to complex SVD computation. A two-
step transformation is performed on the complex matrix for diagonalization, which involves
inner rotation and outer rotations. The inner rotations are responsible for converting the complex
matrix to real while the outer rotation zeros off diagonal elements. In the computation of the
complex Jacobi-SVD, the off diagonal processor applies the first Q transformation for
conversion of complex matrix to real. The diagonal processors can also compute the second Q
transformation angles simultaneously during the first iteration. Therefore, both computation of
the two-step transformation can sequentially follow each other on the array. Complex matrix
diagonalization requires more rotation angles than the real matrices hence SVD for complex
matrices requires more number of sweeps and time for convergence.
52
X - Represents first Q transformation of X iteration step
X' - represents second Q transformation of X iteration step
Figure 4.8 Extension of BLV staggering computations on the complex SVD array
1
1
1
1
1
1
1'
1
1 1'
1
1 1'
1
1 1'
1
1 1'
1'
1
1'
1'
1
1
'
1'
1'
1
1 1'
1'
1 1'
2'
2'
2'
2
2 2'
2'
2
2
2'
2'
2
2'
2'
2
2'
2
2'
2'
2
2'
2
2'
3 2'
2
3
2'
3
2'
3
2 2'
3
2'
'
2
1'
2
2'
2
2
2'
2
2
2'
2
1'
2
2'
2 1'
1
2
1'
2
1'
2
1
'
1'
2
1'
1
1'
1
1'
'
1'
1
'
1'
1
1'
53
4.3 Least Squares based channel estimation for amplify and forward using systolic arrays
Consider the system model described in Section 2.6 for amplify and forward MIMO system
which employs N and M number of antennas at the source and destination respectively with R
relay stations. In the first time slot n, the source transmit signal vector s to the relay stations.
The received signal at the relays is written as
1 1( ) ( ) ( )y H s wn n n (4.8)
where
1HRxN denotes the channel matrix between the transmitter and the relays stations
denotes the transmitted signal vector
1( )w n denotes the noise component in the first time slot
The signal received at the relay station is amplified and transmitted to destination in the second
time slot 1n .
2 2( 1) ( ( )) ( 1)y + H F y w +n n n (4.9)
The composite signal received at the destination in the second time slot in Equation (4.9) can be
expressed as
2 2 21( 1) ( ) ( ) ( 1)1y H FH s H Fw wn n n n (4.10)
( 1) ( ) ( 1)y Hs wn n n (4.11)
where
2HMxR denotes the channel matrix between the relay station and the destination
FRxR denotes the diagonal amplification matrix
( 1)w n1Mx denote the composite noise vector
54
Let 2 1H = H FH denote the composite channel matrix.
Transmission model used for amplify and forward MIMO relays channel estimation is illustrated
in Figure 4.9.
h1,i h2,i
Source
h1,R h2,R
Figure 4.9 Schematic diagram of Amplify and Forward MIMO relaying
A training sequence S of length L ( )L N is transmitted from source to relay stations. The relay
stations amplify the received signal and forward the amplified signal to the destination. There are
R number of relay stations, with Fi amplification factors ( 1....... )i R , therefore, we obtain R
channel pairs of 1
1, 1,( )xN
i ih h and 1
2, 2,( )Mx
i ih h . Vectors 1,ih form the i-th row of H1 and
vectors 2,ih form the i-th column of H2 as illustrated in Figure 4.9. The first step is to estimate i-
th composite channel matrix H(i)
using the least squares estimation method and then decompose
it to obtain individual channel estimates of H1 and H2.
Destination
1● ● ●
H2
h2,1
i-th Relay
FR
F1
Fi
Fi
Relay 1
=1●
● ●
Relay R
1●
● ●
H1
h1,1
55
The least squares composite channel estimate of H(i)
is expressed as [56]
( )†
H YSi
i (4.12)
where
† 1( )S S SSH H
The composite channel defined in Section 4.3 is expressed as
2 1H H FH (4.13)
To obtain the individual channel estimates of H1 and H2 from the composite channel, the
following minimization problem need to be solved
21
2( )
( )
2 1
1
min ,H H H H F HR i
i
i F
(4.14)
This problem may be solved by vectorizing each term in Equation (4.14) as [59]. The vectorized
terms are arranged such that they form the columns of the new set of matrices. We define the
new matrix HMNxR obtained by arranging the vector of
( )
Hi
, 1.......i R into a matrix form as
(1) (R)
( )........vec( )H H Hvec
, where (i)
( )Hvec denotes the vector form of (i)
H . The diagonal
amplification factor ( )F
i is also arranged in a vector form, therefore the new matrix is expressed
as (1) (R)( ),...... (F )F= vecd F vecd FR R . Also matrix Ξ
MNxR is expressed as
1 2)Ξ = (H HT where, operator is the Khatri-Rao product (column-wise kronecker product).
Therefore, after vectorization Equation (4.14) is expressed as
21
2
min ,H H H ΞFF
(4.15)
The next step is to obtain the estimate of Ξ from Equation 4.15.
56
Using least squares estimation the estimate of Ξ is expressed as
†Ξ=HF (4.16)
Further minimization needs to be performed in order to obtain individual estimates of H1 and H2
channels as follows [31]:
21
2
1 2min ,ˆ )H H Ξ- (H HT
F (4.17)
Using the property of the kronecker product the minimization in Equation (4.17) can be carried
out column by column. Denoting the i-th column of Ξ by 1 2, ,.......ф ф ф фR , the
minimization problem for the i-th column becomes
2
1 2min )h ,h21
- (hф hT
F (4.18)
The minimization in Equation (4.18) is equivalent to obtaining an approximate rank-1 matrix
[60]. The singular value decomposition (SVD) is a well known method for solving an
approximate rank-1 problem.
The SVD of ф can be expressed as
ф UΣVH (4.19)
Where U and V are the singular vectors and elements of the diagonal matrix Σ represent the
singular values of ф . Van Loan [60] defined the optimum values of h1 and h2 as
1 1( ) (:,1)h Vopt (4.20)
2 1( ) (:,1)h Uopt (4.21)
where 1 represents the maximum singular value
57
Van loan further suggested that using an arbitrary scaling α ≠ 0 to multiply the 1( )h opt and
2 ( )h opt such that 1( )h opt and
2( )1 h opt also gives an optimum value of h1 and h2.
Performing the Jacobi rotations on systolic array for complex matrix фi we obtain the singular
value and singular vectors used to derive the estimates of individual component in Equation
(4.20) and (4.21) Since the channel matrices H1 and H2 are complex, the Jacobi-SVD for
complex matrix on systolic array can be used to diagonalize фi when M = 4 and N = 4. Since
each processor on the systolic array contains a 2-by-2 matrix, the matrix фi is divided into a 2-
by-2 sub-matrices. A two step Q transformation is performed as described in Section 4.2 to
diagonalize each 2-by-2 sub-matrix. In the first step the Q transformation annihilates one of the
off diagonal and the second step completes the diagonalization. We illustrate Jacobi-SVD on
systolic arrays using parallel ordering for ф MxN
i , when M= 4 and N=4.
Let фi be represented as
1,1 1,2 1,3 1,4
2,1 2,2 2,3 2,4
3,1 3,2 3,3 3,4
4,1 4,2 4,3 4,4
фi
A A A A
A A A A
A A A A
A A A A
(4.22)
Since each processor in the array contains a 2-by-2 matrix, we divide фi into four 2-by-2 sub-
matrices as follows:
11 12
21 22
P P
ф
P P
i
(4.23)
58
The diagonal processor 11P and 22P contains matrix elements 1,1 2,1 1,2 2,2, , ,A A A A and
3,3 3,4 4,3 4,4, , ,A A A A respectively. These processors compute the rotation angles θ and which are
propagated along the rows and columns, respectively. Multiple rotation angles are required to
annihilate all the off diagonal elements of фi. We denote the product of all the rotations
propagated along the columns as LQ and those propagated along the rows as RQ .
Then
1
2
0
0Q Qфi
L R
(4.24)
Performing the rotations for фi, . obtain all the matrix elements of the channel estimate of H1
and H2 as
1, ,1h QRi i (4.25)
2, ,1h QLi i (4.26)
where
LQ and RQ denote the product of all the left and right rotations respectively.
The diagonal elements of Equation (4.24) are the singular values of фi while ,1i denotes the
maximum singular value. Van Loan in [60] suggests that multiplying 1,h i and 2,h i by an arbitrary
scaling 0) still gives an optimal value that is 1,.h i and 2,ˆ1 h i are also optimum.
Arranging the values of 1,i 2,i and h h obtained into matrix form for 1,.......,i R we obtain the
estimate of H1 (source-relay) and H2 (relay-destination) channels.
59
4.4 Performance evaluation
For the performance evaluation we assume a flat fading Rayleigh channel for multiple transmit
and receive antennas which can be statistically modeled using the kronecker spatial correlation
model as in [31]. Using this model, the H1 and H2 channel matrices can be expressed as
1 12 2
1 2 w,1 1H R H R and 1 1
2 2
2 3 w,2 2H R H R (4.27)
The channel spatial correlation matrices at the source relay and destination are represented as R1,
R2 and R3 respectively. w,1H and
w,2H are independent and identically distributed complex
Gaussian random variables with zero mean and unit variance. The source and destination are
equipped with N transmit and M receive antennas respectively. There are R relay stations which
are equipped with single antenna to assist in forwarding transmitted information from the source
to the destination. Using Matlab simulations, 10000 random realizations of H1 and H2 are
generated. The spatial correlation matrix between the antennas is an identity matrix when there is
no spatial correlation. In the case of spatially correlated antennas the off diagonal elements are
non zero and are represented as ρ which denotes the degree of antenna correlation. A training
sequence S with length L=N is with each frame in the sequence chosen to be orthogonal. The
amplification matrix at the relay station is chosen to be a diagonal matrix. The noise variance at
the relay and destination are chosen to be equal. Setting the convergence rate for the complex
Jacobi-SVD function to 10-9
and performing three sweeps. The singular value and singular
vectors of the channel matrix are obtained. For the purpose of simulation we use an arbitrary
scaling as suggested in [60] to obtain the optimum value of 1H and 2H . The elements of scaling
matrix is chosen as
60
1
1
( , ),
( , )
HD
H
i
i
i k
i k 1......i R (4.28)
where 1H denotes the estimate of 1H .
The elements of the diagonal ambiguity matrix D are obtained by dividing the maximum element
in each row of 1H by 1H . We compute the normalized mean square error (NMSE) to evaluate
the performance of the channel estimates. The NMSE can be expressed as [31]
2
11
1 2
1
H DH
H
F
F
NMSE
(4.29)
21
22
2 2
2
H H D
H
F
F
NMSE
(4.30)
In Figure 4.10 and 4.12, we evaluate the NMSE performance of H1and H2 channel estimate
when there is no relay antenna correlation. We observe that at low SNR, H2 channel estimate has
a better performance when compared with the H2 in Figure 4.12. This is because at the relay
stations the noise is amplified along with the signal prior to transmission through H2 channel,
thus making the noise contribution effect more evident in the H1 (source-relay) channel estimate.
Therefore the channel estimator performs much better for source-relay channels at high SNR.
We also evaluate the NMSE peformance of the channel estimator in the presence of spatial
correlations at the relay in Figure 4.11 and 4.13. It is seen that the the channel estimation error
are higher for individaul channel estimate of H1 and H2. Since diversity benefits are not fully
exploited in the presence of antenna correlation, hence the degradation in estimator antenna for
spatial correlation at the relay station. For simulation purpose, we evaluate the estimator
61
performance for strong relay antenna correlation, ρ = 0.9. It can also be seen that estimator
performance is very similar for Jacobi-SVD and the theoretical SVD.
Figure 4.10 Channel estimation performance for H1 with no spatial correlation (ρ = 0).
Figure 4.11 Channel estimation performance for H1 with spatial correlation (ρ = 0.9).
0 5 10 1510
-2
10-1
100
SNR [dB]
NM
SE
1
NMSESVD1
NMSEJ1
0 5 10 1510
-2
10-1
100
101
SNR [dB]
NM
SE
1
NMSESVD1
NMSEJ1
62
Figure 4.12 Channel estimation performance for H2 with no spatial correlation (ρ = 0).
Figure 4.13 Channel estimation performance for H2 with relay spatial correlation (ρ = 0.9).
0 5 10 1510
-3
10-2
10-1
100
SNR [dB]
NM
SE
2
NMSESVD2
NMSEJ2
0 5 10 1510
-2
10-1
100
SNR [dB]
NM
SE
2
NMSESVD2
NMSEJ2
63
Chapter Summary
In this chapter we proposed obtaining the individual estimate of source-relay (H1) and relay-
destination channel (H2) by using the Jacobi-SVD on systolic arrays. The systolic arrays are of
significant importance in real time signal processing applications. Firstly, we obtain the
composite estimate of source-relay and relay-destination channels. Then a sequence of least
squares minimization was performed to obtain the channel matrix that reveals the column-wise
Kronecker relationship between the individual channel estimates. Decomposing the channel
matrix using a sequence of Jacobi rotations we obtain the individual channel estimate H1 and H2.
The performance of individual channel estimate was evaluated and it was observed that the H2
channel estimation errors are less than H1 estimate due to the noise amplification at the relay
station. It can also be seen diversity gain is not fully achieved when antennas are spatially
correlated, hence leading to a degradation in performance.
64
CHAPTER 5: PERFORMANCE ANALYSIS OF ZERO FORCING DETECTION FOR
MIMO SYSTEMS
5.1 Introduction
In this chapter, we analyze the performance of channel estimation algorithm proposed in this
work. Monte-Carlo simulations are used to illustrate the effect of imperfect channel knowledge
on the BER (Bit-Error-Rate) performance of MIMO zero forcing receivers and semi-analytical
simulations are used to verify results. Zero forcing receivers are used to retrieve data streams
transmitted from source antenna to the destination antenna. These receivers require perfect
channel knowledge to accurately decode streams of transmitted data. In practice, perfect channel
knowledge are not available at the receiver, hence the need to evaluate the performance of these
receivers in the presence of channel estimation errors. Although Zero forcing receivers provide
ease of implementation because they offer significantly low computational complexity, their
performance is sub-optimal compared to Maximum likelihood receivers. The effects of imperfect
channel knowledge on zero forcing MIMO receiver is investigated in Section 5.2. We derive the
end-to-end SNR expression for amplify and forward MIMO relays to obtain the BER
performance in Section 5.3. Simulation and semi-analytical results were obtained in Section 5.4
to compare the BER performance of zero forcing MIMO receivers in the presence of perfect and
imperfect channel estimates.
5.2 Effect of Channel Estimation Errors on Zero Forcing MIMO Receivers
We consider a MIMO system with N transmit and M receive antennas at the source and
destination respectively as shown in Figure 2.1. Transmission of signal between the source and
destination is achieved in two hops using single antenna relay stations placed between the source
and destination station. In the first hop the source transmits signal to the relay station, and the
65
received signal at the relay is amplified and forwarded to destination. The MIMO channel is
assumed to be frequency flat fading with independent identically distributed (i.i.d) Gaussian
channel matrix. Also, the antennas are assumed to be sufficiently separated such that there is no
spatial correlation between the antennas. The received signal vector at the relay station in the
first time slot can be expressed as
1 1( ) ( ) ( )r H s ws n n n (5.1)
where,
1( )rRx
s n is the received signal vector at the relay,
1( )s Nxn denotes signal transmitted from the source to the relays,
1HRxN is the channel matrix between the source and relays and 1( )w n is the additive white
Gaussian noise (AWGN) vector at the relay stations
In the second time slot, the received signal at the relay is amplified and then forwarded to the
destination antenna. The signal model at the destination can be expressed as:
2 2 21( 1) ( ) ( ) ( 1)1y H FH s H Fw wn n n n (5.2)
where
2HMxR denotes the channel matrix between the relay and destination,
FRxR is the diagonal amplification matrix and
2
1( ) (0, )w R Rn I and 2( ) (0, )
2w y Mn I represent the noise contribution at the relay and
destination, respectively. The noise components are assumed to be independent white complex
Gaussian noise. The received signal for both time slots at the destination in Equation (5.2) can
further be simplified as
66
( 1) ( ) ( 1)y Hs wn n n (5.3)
where
2 1H H FH denotes the composite channel matrix between source-relay and relay-destination.
The overall noise vector at the destination is given by
2 1 2( 1) ( ) ( 1)w H Fw wn n n (5.4)
The transmitted signal from the source can still be decoded using the composite channel
estimate. In order to enhance the performance of the cooperative relaying system, an optimal
power allocation scheme is required. For most power allocation schemes in amplify and forward
relaying systems, the channel state information of the source-relay and relay-destination channels
is required [66] and [67].
When perfect channel knowledge is available at the receiver, zero forcing estimate of the
transmitted signal vector s n at a given time instant n is given by [68]
( ) ( ( ) ( )) (n) (n)s G Hs w s Gwn n n (5.5)
where,
† 1( )G H H H HH H is the pseudo-inverse of the composite channel matrix H.
2 1H H FH represents the composite channel matrix,
(n)w represents the noise vector.
In the presence of imperfect channel knowledge at the receiver, zero forcing estimate of the
transmitted signal vector ( )s n is given by
( ) ( ( ) ( )) (n) (n)s G Hs w s Gwn n n (5.6)
67
where
†1( )G H H H H
H H is the pseudo-inverse of the composite channel estimate.
2 1H H FH is the estimate of the composite channel matrix H.
5.3 Semi-Analytical Performance Analysis
In this section, we derive an expression for the end to end equivalent SNR (signal-to-noise
ration) for amplify and forward MIMO system. From Equation (5.2), the signal component is
represented by 2 1 ( )H FH s n while the composite noise component is denoted as
2 1 2( ) ( 1)H Fw wn n . Therefore, we express the overall SNR as [69]
H
2 1 2 1
H
2 2
E[( )( ) ]
1 1E[( ) )( ) ) ]
H H s H H s
H w H wF F
TrSNR
Tr
(5.7)
where
1H denotes the source to relay channel matrix
2H denotes the relay to destination channel matrix
F represents the amplification matrix
We denote the transmitted signal vector as s and the noise vector as .w Where
( ) .ssH
sE E I and ( ) .wwH
NE E I
After expansion Equation 5.7 can be simplified using the property of the trace.
1
( )AN
ii
i
Tr a
(5.8)
For a 2-by-2 matrix H1 with independent and uncorrelated components we obtain
68
2 21 1
11 12
1 1 2 21 1
21 22
0
0
H HHh h
E E
h h
(5.9)
1 1
11 12
1 1 1
21 22
Hh h
h h
1 1 .H HH
sE N I
Similarly, we express the average channel power of H2 as
2 22 2
11 12
2 2 2 22 2
21 22
0
0
H HHh h
E E
h h
(5.10)
where,
2 2
11 12
2 2 2
21 22
Hh h
h h
2 2 .H HH
dE N I
For convenience we normalize the diagonal elements to 1.
Therefore, we can write the signal energy component as
H
2 1 2 1E[( )( ) ]H H s H H s s s dTr E N N (5.11)
where, Ns and Nd denote the number of source and destination antennas
Similarly, the energy of the noise component can also be obtained as follows:
H
2 2
1 1E[( )( )) ]H H
F FNE Tr
(5.12)
69
Using the Choice of the relay amplification factor suggested in [71] which depends on the
channel state information (CSI) between the source and relay station H1 channel. Therefore, we
express the relay gain as
1 1
1[ ]
[H H ] [ ]
H
H HE FF
E E ww
(5.13)
Therefore, substituting Equation (5.13) into (5.12), the noise energy component can be written as
H
2 2 d s
1 1E[( )( )) ] (N N 1)H H
F FN NE Tr E
(5.14)
The equivalent SNR at the receiver can be expressed as
1( 1)
s s dR
N s d
E N N
E N N
= .
( 1)
s dT
s d
N N
N N
(5.15)
where T denotes the transmit SNR and 1R denotes the SNR at the receiver.
Equation (5.15) is not mathematically tractable due to complexity of obtaining the statistics such
as the probability density function (PDF) and cumulative distribution function (CDF) [69].
Therefore, we use the suggested upper bound in [69].
1 2 .( )
s dR R T
s d
N N
N N
(5.16)
Equation (5.16) has the advantage of mathematical tractability over Equation (5.15) and also
gives a tight upper bound at medium to high average SNR [69].
Therefore, we obtain the BER performance by using the approximate BER expression for MIMO
zero forcing [68]
2
1min(2, /4 )
2max(log ,2)
1 0
1 11 1
2 2
D jM D
MPSK i iM
i j
D jBER
j
(5.17)
70
2
2
sin (2 1)
1 sin (2 1)i
i
i
,
s dD N N and .( )
s dT
s d
N N
N N
5.4 Numerical Results
In this section, Monte Carlo simulations are used to illustrate BER performance of zero forcing
receivers in the presence of channel estimation errors. We investigate the performance of QPSK
modulation for amplify and forward MIMO system which employs least squares channel
estimation as described in earlier chapters. Considering a system with the source and destination
equipped with N transmit and M receive antennas respectively. Transmission between source
and destination is assisted by R relay stations. Each relay station operates in half duplex mode
and are equipped with a single antenna. Generating 10,000 realizations of source- relay channel
H1 and relay-destination channel H2 using the kronecker model as described in Section 4.4. An
orthogonal training sequence S is transmitted from the source to destination with the cooperation
of the relay stations. The length L of training sequence is chosen such that L = N, (N is the
number of transmit antennas) in order to use smallest possible length of training. The
amplification matrix during training is chosen to be orthogonal. Assuming equal noise variance
at the relays and destination antenna, we set the training SNR to compare with results obtained in
[29]. The composite channel estimate is decomposed using the Jacobi-SVD as described in
earlier chapters to obtain individual channel knowledge which is used to design an optimum
relay amplification matrix. During data transmission, an optimum choice of relay gain is such
that the output power of relay is dependent on the fading effect of source-relay channel. In
Figures 5.1 and 5.2, the plots of simulation results are shown.
71
Figure 5.1 BER performance of QPSK MIMO zero forcing receiver N = 2, R = 2 and M = 2
In Figure 5.1 and 5.2, we observe that when perfect channel knowledge is available at the
destination BER obtained from simulations agree with the semi-analytical results. As the SNR
values increases, the BER decreases. In the presence of channel estimation errors, the BER
reaches an error floor at high SNR. For example, at BER 5.5 × 10-3
it is observed that an error
floor function is reached for the case of two transmit and received antennas at source and
destination, respectively. For the case of four transmit and receive antennas at the source and
destination respectively with four relay stations, the error floor function is reached at 2.5 × 10-3
.
Therefore, it is seen that the BER performance of zero forcing receivers is sensitive to channel
estimation errors. For zero forcing detection as observed in [68], at high SNR value the BER rate
does not decrease to zero but an error floor function is reached. The value of the error floor
0 5 10 15 20 25 30 3510
-4
10-3
10-2
10-1
100
SNR [dB]
BE
R
Semi Analytical
Perfect Channel Knowledge
Estimated Channel
72
function attained depends on the number of transmit and receive antennas as well as the channel
estimation error.
Figure 5.2 BER performance of QPSK MIMO zero forcing receiver N = 4, R = 4, M = 4
In Figure 5.3, we compare the BER performance of our Jacobi-SVD and theoretical SVD for a
zero forcing MIMO receivers. We include the direct path in our simulations to enhance the
system performance. Using four transmit and receive antennas at the source and destination
respectively with four single antenna half-duplex relay stations. In Figure 5.3, at high SNR we
observe there is a constant dB gain of about 3dB when channel state information of source relay
channel is used to determine the relay gain.. When relay gain is chosen without using the source-
relay CSI, we observe performance degradation. An error floor is attained at BER value of 8 ×
10-4
when the choice of relay gain is not according to CSI of source-relay channels. When the
0 5 10 15 20 25 30 3510
-5
10-4
10-3
10-2
10-1
100
SNR [dB]
BE
R
Semi Analytical
Perfect Channel Knowledge
Estimated Channel
73
relay gain is chosen with the knowledge of channel state information, then we achieve an error
floor at BER value of 4.5 × 10-4
. Under the same condition of the choice of relay gain, the BER
of the receivers for Jacobi-SVD and theoretical SVD have identical performance. Once again we
observe that with perfect channel BER decreases as the SNR value gets higher. But in the case of
imperfect channel estimate an error floor value is attained such that increasing the SNR has no
significant effect on the BER.
Figure 5.3 BER Performance comparison of amplify and forward MIMO relays with relay
gain chosen using knowledge of CSI and choice of relay gain without knowledge of CSI.
5.5 Chapter Summary
In this chapter, BER performance of MIMO zero forcing receiver in uncorrelated Rayleigh flat
fading was investigated. We derived an equivalent SNR expression for the end to end
0 5 10 15 20 25 3010
-5
10-4
10-3
10-2
10-1
SNR [dB]
BE
R
Perfect Channel Knowledge
JSVD relay gain with CSI
SVD relay gain with CSI
JSVD relay gain without CSI
SVD relay gain without CSI
74
performance of amplify and forward MIMO channel in order to obtain the BER performance.
When perfect channel knowledge is available at the receiver, BER performance improves as the
SNR increases. It was also shown that in the presence of perfect channel estimates our
simulations results show a very good agreement with the semi-analytical performance. In the
presence of channel estimation error, at high SNR the BER does not tend to zero, instead an error
floor is attained. The value of the error floor is dependent on the number of transmit and receive
antennas at the source and destination station respectively. Therefore, at high SNR estimation
errors do not have significant impact on the BER performance of the receiver. Also the choice of
relay gain using the channel state information (CSI) improves the system performance.
.
75
CHAPTER 6: CONCLUSIONS AND FUTURE WORK
This chapter concludes the thesis and provides an overview of the dissertation and future
research direction. Section 6.1 gives a brief summary of work done in this dissertation as well as
conclusions drawn. Future work directions are presented in Section 6.2.
6.1 Summary and conclusions
The main objective of this work is to estimate the channel state information (CSI) of individual
source-relay and relay-destination channels using the Jacobi rotation. The knowledge of source-
relay CSI is then used to obtain an optimum amplification factor to enhance the overall
performance of MIMO amplify and forward relay system. The work done in this thesis can be
divided into three parts. The first part (Chapter 3) proposes, using the 2-by-2 Jacobi rotations, to
decompose the composite channel estimate of amplify and forward relay. A sequence of Jacobi
rotations is accumulated to obtain the singular values and singular vectors used to derive the
individual channel estimate of source-relay and relay-destination channels. In the second part
(Chapter 4), the Jacobi algorithm for decomposing channel estimates was extended to a systolic
array in order to obtain the individual channel estimate for larger matrices. The normalized mean
square error was used to evaluate the accuracy of the individual channel estimate obtained.
Finally, we evaluate the BER performance of a zero forcing reciever. Also we compare the
overall system performance when relay gain is chosen with or without the knowledge CSI of the
source-relay channel. The summary of each part is as follows:
In Chapter 3, we propose using the 2-by-2 Jacobi rotation to obtain individual channel
estimate of amplify and forward MIMO relays. Firstly, a sequence of least squares
minimization was performed on the composite channel estimate and this was followed by
sequence of Jacobi rotations to obtain the individual channel estimates. The normalized
76
mean square error was used to determine the accuracy of the channel individual channel
estimate. At low SNR the relay-destination channel estimate has better performance
compared to the source-relay channel estimate. In amplify and forward relying systems,
the noise is amplified along with the signal at the relay station, which results in
performance degradation of source-relay channel estimate. Also, the performance of
individual channel estimates was compared, in the presence of spatial correlations at the
relay stations and when no spatial correlation exists. The diversity gains are not fully
exploited in the presence of spatial correlation, hence the reason for performance
degradation.
In Chapter 4, we propose using the Jacobi-rotations on systolic arrays to obtain individual
channel estimate of source-relay and relay-destination channels. Systolic arrays employ
parallel computing techniques where large computations are divided into small ones and
done simultaneously. Systolic arrays have been found to be very useful where real time
singular value decomposition (SVD) is required. Therefore, we used Jacobi-SVD on
systolic arrays to obtain the individual channel estimates of source-relay and relay-
destination channels. The normalized mean square error of source-relay and relay-
destination channel estimates obtained using Jacobi-SVD on systolic arrays was
compared with the theoretical SVD. Although both methods have a similar performance,
the numerical difficulties associated with the theoretical SVD make them impractical for
real time implementations [51].
In Chapter 5, we use knowledge of individual source-relay channel estimates to improve
the overall performance of amplify and forward MIMO relays. This is achieved by using
the source relay CSI to obtain the amplification factor required at the relay stations. Most
77
optimal power allocation schemes [66] and [67] require knowledge of individual channel
estimates of source-relay and relay destination channels. Hence, the need to decompose
the estimated composite channel to obtain the individual estimates.
The main findings of this thesis are as follows;
Individual channel estimates of source-relay and relay-destination channels can be
obtained from the composite channel estimates using the Jacobi rotations.
Simulation results in Chapter 3 show that the effect of noise contribution is more
evident at the source-relay channels at low SNR. Normalized mean square error
performance improves when antennas are not spatially correlated as diversity gain
is fully achieved under no spatial correlation between antennas.
Using systolic arrays individual channel estimates of amplify and forward MIMO
relays can be obtained recursively. In Chapter 4, simulation results show that
performance of channel estimation is improved with increase in the number of
antennas at the source, relays and destination.
Lastly, the individual channel estimates obtained for amplify and forward MIMO
relay can be used to determine the relay gain. Semi-analytical and Simulation
results in Chapter 5 show an improvement in BER performance when the relay
gain is chosen using the CSI of source-relay channel. Hence, decomposing the
composite channel estimates is necessary to improve overall system performance.
78
6.2 Future Work
In this section, we suggest the future research direction for this work as follows:
In Chapters 3 and 4, other optimal channel estimations algorithms like the Minimum Mean
Square Error can be employed to obtain the composite channel estimate which can be
decomposed to obtain individual channel estimate. Recursive channel estimate methods can also
be extended to obtain the composite channel estimate. In Chapter 5, more complex receiver
design structure like the maximum likelihood receiver can be used to evaluate the BER
performance of the overall amplify and forward MIMO relay.
79
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