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dissertation entitled

BLIND SIGNAL DETECTION FOR DS—CDMA OVER
FREQUENCY-SELECTIVE FADING CHANNELS

presented by

Weiguo Liang

has been accepted towards fulfillment
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BLIND SIGNAL DETECTION FOR DS-CDMA
OVER FREQUENCY-SELECTIVE FADIN G
CHANNELS

By

Weiguo Liang

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Electrical and Computer Engineering

2004

ABSTRACT

BLIND SIGNAL DETECTION FOR DS-CDMA
OVER FREQUENCY-SELECTIVE FADING
CHANNELS

By

Weiguo Liang

In conventional wireless communication systems, training sequences are transmit-
ted periodically in order to track the time-varying channel environments. This is
neither power efficient, nor spectrally efficient. As an effort to improve the spec-
tral efficiency by reducing the overhead, this dissertation is focused on blind channel
estimation and signal detection for direct sequence spread spectrum systems.

In literature, if the spreading sequences are periodic and repeat with every infor-
mation symbol, the system is referred to as short-code CDMA, and if the spread-
ing sequences are aperiodic or essentially pseudo-random, it is known as long-code
CDMA. The time-varying nature of long-code CDMA significantly complicates the
development of blind detectors, as needed statistics can no longer be obtained through
time averaging of the observed data. For this reason, research on blind multiuser de-
tection has largely been limited to short-code CDMA. On the other hand, long—code
is widely used in virtually all operational and commercially proposed CDMA systems
due to its inherent security features and performance stability in frequency fading
environment.

In this dissertation, statistics based algorithms are developed for both short-code
and long-code DS-CDMA systems, with emphasis on the long-code systems. The
major contributions of the dissertation can be briefly summarized as: (i) Blind de-
tectors based on the code-constrained super-exponential algorithm have been devel-
oped for multi-rate short-code CDMA systems, while only the spreading code of the.

desired user is assumed to be known. (ii) For long-code CDMA systems, consider-

ing the time variant. nature of the Spreading code, the chiprate scrambled signal is
taken as the system input, and the long-code CDMA is characterized using a time
invariant model. (iii) In downlink long-code systems, after chip-level equalization,
the descrambled signal is treated as the received signal of a short-code CDMA sys-
tem, and super—exponential algorithm is applied to recover the information symbols.
(iv) For uplink, twcystage approaches have been developed in this research. In the
first stage, multi-step linear-prediction—based methods are developed to eliminate the
inter-symbol interference. In the second stage, if the spreading codes are of noncon-
stant modulus, blind channel estimation is performed by exploiting the second—order
statistics; if the spreading codes are of constant modulus, higher-order statistics based
algorithms need to be applied to estimate the channels. (v) To further improve the
transmission quality without increasing the transmission power or bandwidth, time-
reversal space-time block coding (TR-STBC) scheme is applied at the base-station
side in downlink CDMA, and blind signal detection algorithm based on the principal
component analysis has been developed.

Throughout this research, computer simulations are carried out to illustrate the

proposed approaches

Dedicated to my family

ACKNOWLEDGMENTS

I wish to express my gratitude to my advisor Professor Tongtong Li for her guid-
ance, support and patience throughout the research work. It is Professor Li who gave
me the key to the world of communications, taught me how to be a researcher, and
made my studies in Michigan State University an unforgettable experience.

I would like to thank the members of my guidance committee: Professor John R.
Deller, Professor Hayder Radha, Professor Ning Xi from the Department of Electrical
and Computer Engineering, and Professor Chichia Chiu from the Department of
Mathematics. Special thanks go to Professor Hayder Radha who gave me the chance
to pursue my Doctorate Degree in Michigan State University.

Thanks also go to my lab-mates: Qi Ling and Huahui Wang. The discussions with
them enriched my knowledge in the field of wireless communications.

I am grateful to all the teachers and friends who believe in my ability, and give
me confidence during my life.

I wish to express my deepest thanks to my parents, who give me the support,
teach me to be independent, and always let me make my own decision. They shape
me into the person who I am today. I also thank my sister, who give me moral support
throughout my study.

Finally, I especially thank my girlfriend Ying Yan, who has been sharing sweetness
and bitterness with me in these years. It is her care, belief and love help me to

overcome all the difficulties in my research work.

TABLE OF CONTENTS

 

 

 

 

LIST OF TABLES ix
LIST OF FIGURES x
1. Introduction - -1
1.1 Evolution of Wireless Communications ............................................................... l
1.2 CDMA Systems .................................................................................................... 2
1.3 Blind Equalization ................................................................................................ 5
1.3.1 Second-Order Statistics Based Methods ................................................... 6
1.3.2 Higher-Order Statistics Based Methods .................................................... 7
1.3.3 Blind Channel Estimation and Equalization in CDMA Systems ............. 9
1.4 Space-Time Coding ............................................................................................ 11
1.5 Objectives ........................................................................................................... 13
1.6 Contributions ....................................................................................................... 14
1.7 Outline of the Dissertation .................................................................................. 15
2. DS-CDMA System Models 17
2.1 Discrete-Time Uplink CDMA System Models ................................................... 17
2.1.1 Short-Code CDMA ................................................................................. 17
2.1.2 Long-Code CDMA ................................................................................. 20
2.2 Discrete-Time Downlink Long-Code CDMA System Models .......................... 22
2.3 General Assumptions .......................................................................................... 24
2.4 Summary ............................................................................................................. 25
3. Blind Signal Detection for Short-Code DS-CDMA Systems .................................. 26
3.1 Introduction to Super-Exponential Algorithms .................................................. 26
3.2 Code Constrained Super-Exponential Methods for Blind Detection of
Single-Rate Short-code DS-CDMA Signals ....................................................... 28
3.2.1 Matrix Representation of the Combined Channel-equalizer Impulse
Response ................................................................................................. 28
3.2.2 Super-Exponential Approach .................................................................. 30
3.2.3 Code-Constrained Super-Exponential Algorithm ................................... 33
3.3 CSEA Based Blind Equalization of Multirate Asynchronous CDMA
Systems ............................................................................................................... 38
3.3.1 System Model for Multirate DS-CDMA Systems .................................. 39
3.3.2 Code-Constrained Super-Exponential Algorithms for Multirate
CDMA Systems ...................................................................................... 41

vi

3.3.3 Summary of the Multirate CSEA Approach ........................................... 47

 

 

 

 

 

 

3.3.4 Simulation Examples .............................................................................. 48
3.4 Summary ............................................................................................................. 53
. Blind Equalization for Long-Code CDMA Systems - - 54
4.1 Blind Equalization For Downlink Long-Code CDMA Systems ........................ 54
4.1.1 Multistep Linear Prediction Based Blind Channel Estimation ............... 55
4.1.2 MMSE Equalization and SEA Enhancement ......................................... 57
4.1.3 Simulation Examples .............................................................................. 60

4.2 Blind Equalization for Uplink Lon g-Code CDMA with Non-Constant Modulus
Spreading Sequences .......................................................................................... 64
4.2.1 System Model ......................................................................................... 65
4.2.2 181 Reduction and Separation Based on Multistep Linear Predictors ..... 67
4.2.3 Channel Estimation through the Fourier Analysis .................................. 70
4.2.4 Channel Estimation Based on the Matrix Pencil Approach .................... 71
4.2.5 Channel Equalization using the Cyclic Wiener Filter ............................. 72
4.2.6 Extension to Multirate CDMA Systems ................................................. 75
4.2.7 Simulation Examples (Nonconstant Modulus) ....................................... 75

4.3 Blind Equalization for Uplink Long-Code CDMA Spreading Sequences with

Constant Modulus ............................................................................................... 80
4.3.1 Blind Channel Identification using JADE Algorithm ............................. 80
4.3.2 Simulation Example (Constant Modulus) .............................................. 82
4.4 Summary ............................................................................................................. 85
. Blind Equalization of Space-time Block Coded DS-CDMA 86
5.1 Transmission Scheme ......................................................................................... 86
5.2 Blind Channel Estimation ................................................................................... 91
5.3 Signal Detection .................................................................................................. 94
5.3.1 Space-Time Decoding ............................................................................. 94
5.3.2 MMSE ................................................................................................... 97
5.4 Simulation Examples .......................................................................................... 98
5.5 Summary ........................................................................................................... 100
. Conclusions and Future Works 101
6.1 Conclusions ....................................................................................................... 101
6.2 Related Future Works ....................................................................................... 104
APPENDICES 106
A. List of Abbreviations and Acronyms 107
B. Definitions of Higher-Order Statistics 109

vii

C. Convergence Analysis of the CSEA Approach 111

 

C.l Equivalent Representation in the Combined Response Domain ...................... 1 11
C2 Convergence Analysis ...................................................................................... 113
BIBLIOGRAPHY 118

 

viii

3.1

4.1
4.2

LIST OF TABLES

Example 1 - VSL system, average time per run, equal power case, same

condition as in Fig.3.1 ........................... 49
Average time per run ........................... 79
Average time per run using MSLP+J ADE approach. ......... 83

1.1
1.2
1.3

2.1

2.2

2.3

3.1

3.2
3.3

3.4
3.5

3.6

4.1

4.2

LIST OF FIGURES

Evolution of modern wireless communication networks ........ 2
Basic multiple access schemes ...................... 3
Principle of DS-CDMA .......................... 3
Block diagram of uplink short-code DS-CDMA system with single re-

ceive antenna ............................... 18
Block diagram of uplink long-code DS-CDMA system with multiple

receive antennas .............................. 21

Illustration of downlink long-code CDMA with multiple receive antennas 23

VSL system - Normalized MSE for the high rate user: 8 chips/ symbol
for the high rate (HR) user, 32 chips / symbol for basic rate (BR) users,
1 HR and 2 BR users, record length = 1024 BR symbols, evaluation
length = 3072 BR symbols, 100 Monte Carlo runs. (“without enhance-
ment” means that the subequalizers are calculated from (3.95)-(3.96)
directly.) .................................. 50
VSL system - BER of the high rate user, same condition as Figure 3.1 50
MC system - Normalized MSE for the high rate user: 32 chips/ symbol
for each symbol stream of the high rate (HR) user, 32 chips / symbol for
basic rate (BR) users, 1 HR and 2 BR users, record length = 1024 BR
symbols, evaluation length = 3072 BR symbols, 100 Monte Carlo runs.
(“without enhancement” means that the subequalizers are calculated
from (3.91) directly) ........................... 51
MC system - BER of the high rate user, same condition as Figure 3.3 51
Load test of VSL systems. Bit error rate of different loads at SN R =
20dB. Totally 100 Monte Carlo runs. The sequence (5, 6, 7, 8, 9) of
virtual users corresponds to the pair sequence (number of HR users,
number of BR users) = (1,1), (1,2), (1,3), (1,4), (2,1) .......... 52
Load test of MC systems. Bit error rate of different loads at SNR =
20dB. Totally 100 Monte Carlo runs. The sequence (8, 10, 12, 14, 16,
18) corresponds to the pair sequence (1,4), (2,2), (2,4), (3,2), (3,4), (4,2). 53

Example 1. MSE of channel estimation for 8 users, N=16, channel
estimation was based on 256 symbols, MSE was averaged over 100

Monte Carlo runs .............................. 61
Example 1. MSE of equalization for 8 users, N=16, 100 Monte Carlo
runs and 1024 symbols per run. ..................... 61

4.3
4.4

4.5

4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

5.1
5.2
5.3
5.4

5.5

Example 1. BER performance, same condition as that in Figure 4.2. .
Example 2. Symbol MSE under different loads, SNR = 15dB, 2 receive
antennas, N =16, 100 Monte Carlo runs and 1024 symbols per run. . .
Example 2. BER under different loads, SNR = 15dB, 2 receive anten-
nas, N =16,100 Monte Carlo runs and 1024 symbols per run. .....
Two equivalent representation of the spreading and scrambling proce-
dures ....................................
Normalized MSE of channel estimation versus SNR, single rate cases
with N=16 and N=8 respectively ....................
Comparison of BER versus SNR, single rate cases with N =16 and N =8
respectively ................................
Normalized MSE of channel estimation versus SNR for matrix pencil
based approach with different codes, multirate configuration with N =
8 for the high rate user, and N = 16 for the low rate user. ......
Comparison of BER versus SNR for matrix pencil based approach with
different codes, multirate configuration with N = 8 for the high rate
user, and N = 16 for the low rate user. .................
MSE of channel estimation, two users, N = 16 for the low rate user
and N = 8 for the high rate user, 100 Monte Carlo runs and 1024
symbols per run. .............................
MSE of symbol estimation, two users, N = 16 for the low rate user and
N = 8 for the high rate user, 100 Monte Carlo runs and 1024 symbols
per run. ..................................
Bit error rate, N = 16 for the low rate user and N = 8 for the high
rate user, 100 Monte Carlo runs and 1024 symbols per run .......

Downlink Transmission Scheme .....................
Data Structure of the Transmitted Signal ................
Data Structure of the Transmitted Signal using Shortened Prefix (Post-
fix) .....................................
MSE of channel estimation. Spreading gain N = 16, and number of
users M = 8. ...............................
Bit error rate. Spreading gain N = 16, and number of users M = 8. .

xi

62

62

63

66

77

78

78

79

83

84

84

87
88

89

99
99

CHAPTER 1

Introduction

1.1 Evolution of Wireless Communications

The evolution of modern wireless communication networks is shown in Figure 1.1. The
first generation (1G) cellular networks emerged in late 19708 and early 19808, such
as the US. Advanced Mobile Phone System (AMPS) and the European Total Access
Cellular Systems (ETACS). They all used analog modulation, large cells and omnidi-
rectional base station antennas. The number of users supported by these systems is
very limited. To increase the system capacity and to improve call capabilities, second
generation (2G) wireless systems were deployed in mid 19903. 2G systems, such as the
US. Digital Cellular Standard IS-136, 13-95 and the Pan-European Global System
Mobile (GSM), use digital voice coding and digital modulation, and could provide at
least a three-fold increase in the overall system capacity. While frequency modulation
(FM) is used for radio transmission in AMPS standard, TDMA-based technologies
are applied in most second generation standards.

Since 2G systems were designed before widespread utilization of the Internet,
they mainly supported voice-centric services with very limited data services, like
short messages, FAX, etc. In an effort to support modern Internet applications, new
data-centric standards, known as 2.5G standards that can be overlaid upon exist-
ing 2G technologies were developed. With new base station add-ons and subscriber
unit upgrades, these new standards (such as Enhanced Data Rate for GSM (EDGE),
CDMAone) support higher data rate transmission for web browsing, e-mail traffic,
mobile commerce (m-commerce) and location based mobile services. The third gen-

eration (3G) systems aim to support multi-megabit Internet access, and simultaneous

  

AMPS -— lS-41

 

TD-SCDMA

UMTS WCDMA

 

  

ETACS

 

(3GPP)

CDMA 2000

1X, 3X (3GPP2)

 

1G 26 2.56 so

Figure 1.1. Evolution of modern wireless communication networks

voice and data access with multiple parties at the same time using a single mobile
handset. 3G systems also allow seamless global roaming. Representative 3G stan-
dards include 3GPP (3rd generation partnership project) UMTS Wideband CDMA,
3GPP2 CDMA2000 3X, and 3G TD-SCDMA.

To support high-speed multimedia services, 3G networks use large bandwidths and
rely on spread spectrum technologies. Code Division Multiple Access (CDMA) is the
most popular spread spectrum technique applied in cellular systems. Compared with
TDMA and FDMA, CDMA has a soft capacity limit. Besides, the inherent frequency
diversity of CDMA can mitigate the effects of small scale fading, and make it robust

to malicious narrow-band jamming.

1 .2 CDMA Systems

Three basic multiple access schemes, FDMA, TDMA and CDMA, are illustrated in
Figure 1.2. In TDMA and FDMA systems, the channels are separated in time domain
and frequency domain respectively. In CDMA systems, the signals of different users
can be transmitted simultaneously through the same frequency band. This is achieved
by assigning each user a unique code sequence (known as spreading code or signature
sequence), which is used to spread its information signal. The receiver recovers the

symbols of the desired user by despreading the received signal with the its spreading

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A Frequency A Frequency 1 Frequency
Ll—

Channel 4 g g g g / ,
Channel 3 g g g 3
Channel 2 3, :3; i 3:. _'

d N (a) b
Channel 1 Time

Tim; Tim; ‘
Code
FDMA TDMA CDMA

Figure 1.2. Basic multiple access schemes

code. To mitigate multiuser interference, spreading codes are generally orthogonal to
each other, or have low cross-correlation. Because the bandwidth of the spread signal
is chosen to be much larger than the original signal, the bandwidth of the signal is
enlarged significantly by the spreading process.

In DS-CDMA (direct sequence CDMA), the data signal is directly multiplied by
the spreading code. The principle of DS-CDMA is illustrated in Figure 1.3. The

Information bits

 

 

 

l5

 

 

 

 

Spreading sequence

ll

 

 

 

1Tlfll'lll HHH [TlllI'll—I'll'llllll'la
-1 UUJL lJLJLlJLL J ULUU ULUL JUJ

llnnnn n IT nmmnnnn ,
_1|ULlJL .lJL J llLUULlJUEUL JUJ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1.3. Principle of DS-CDMA

spreading code changes at chip rate, which is much higher than the symbol rate. The

spreading gain (or processing gain) is equal to the ratio of chip rate to symbol rate.

In Figure 1.3, there are 16 chips per symbol period, that is the processing gain equals
to 16.

Consider a DS-CDMA systems with M users. Let u,(n) denote the nth symbol of
user 2', and let the N-vector cm = [r:,-‘,,(0), c,,,,(1), - - - ,c,-,,,(N — 1)] denote the spreading
sequence for symbol u,(n), where N is the spreading gain. The spreading result of
u,(n) is simply given by

s,-(n) = u,(n)c,-’n. (1.1)

When the channels are ideal, and in the absence of noise, the received signal can be

presented as

M M
y(n) .-. 25,-(72) = Zu,(n)c,,,. (1.2)

If the spreading sequences are chosen to be orthogonal, i.e. cmcfn = 0,Vz' # j,
and satisfy omega 2 N, then the nth symbol of user m can be recovered by the
despreading process

mm = iy<n>cT . (1.3)

To perform the despreading operation, in addition to the knowledge of the spreading
sequence of the desired user , the codes of the received signal and the locally generated
codes have to be synchronized.

Because of the enlarged bandwidth and the simultaneous transmission through
the same radio channel, CDMA has some properties that differ from other multiple

access schemes. The major advantages of DS-CDMA systems are:

0 Soft capacity limit. There is no absolute limit on the number of users in CDMA

systems.

0 Privacy. The transmitted signal can be recovered only if the the spreading code

is known to the receiver.

0 Resistance to narrow band jamming. In CDMA, the signal is spread over a large
bandwidth. After despreading, the narrow band jamming is spread, making it

appear as background noise compared with the despread signal.

0 Low probability of interception. Because the signal is spread over a larger band-
width. the power spectral density is low. Thus the signals can easily be con-

cealed within the noise floor.

While CDMA systems have the noted advantages, at the same time they also face

the following challenges:

0 Chip—level synchronization. The synchronization of the locally generated code

signal and the received signal has to be kept within a fraction of the chip period.

0 Frequency selective fading. High chip rate implies that the signal bandwidth
is larger than the coherent bandwidth of the channel, so different frequency

components will experience different fading characteristics.

0 Multiuser interference. Because multiple users transmit through the same fre-
quency band simultaneously, multiuser interference occurs due to asynchronous

transmission and multipath fading.

o Near-far efiect. The power received by the base station from users close to it
could be much higher than that received from users further away. As a result,
the strongest user may capture the receiver, while the weaker users experience
severe performance loss. To deal with this, tight power control is performed in

practical CDMA systems.

1.3 Blind Equalization

Due to the time-varying nature of wireless environment, training sequences need to be
sent periodically and frequently to obtain accurate channel estimation. Currently, all
operational wireless communication systems are training based, and the channels are
estimated based on the response of the known signals. For example, in GSM systems,
each time slot has 156.25 hits, including a 26bit training sequence [1]. In IS-95
systems, typically about 20% of the radiated power on the downlink is dedicated
to the pilot signal [2]. Pilot channels are defined in W-CDMA and CDMA2000

standards to aid the channel estimation at the mobile devices [3]. It can be seen that
training signals consume a lot of system resources. It is even worse in MIMO wireless
systems, in which spatial diversity is exploited by employing multiple antennas at
both the transmitter and receiver ends, and channels need to be estimated for each
transmitter receiver antenna pair.

In the last few decades, wireless communications have seen explosive growth, and
the number of worldwide subscribers has grown from less than 100 million to more
than one billion. Since the frequency resource is limited, to increase the system
capacity, the spectral efficiency has to be improved. By reducing or eliminating the
training signal, blind signal detection has emerged as a promising technique, and has
attracted more and more research attention.

Based on whether the channels are estimated explicitly, blind equalization meth-
ods can be divided into two groups: direct blind equalization and indirect blind equal-
ization. For indirect blind equalization techniques, the channel is estimated first, and
then equalizer is designed based on the estimated channel. A review of multichan-
nel blind identification is presented in [4]. For direct equalization, the equalizer’s
parameters are extracted from the received data without explicit channel estimation.

Based on whether the statistical properties are exploited, the blind equalization
methods can also be classified as statistical methods or deterministic methods. Our

research is focused on the statistical methods, which again are generally split into two

classes: SOS based and HOS based approaches.

1.3.1 Second-Order Statistics Based Methods

As presented in [5] and [6], when the channels are driven by cyclostationary processes,
or can be modelled as single input and multiple outputs (SIMO) systems, it is possible
to identify the channels blindly by exploiting only second-order statistics (SOS). In
[7,8], the cyclostationarity was introduced by modulating the input stream with a
periodic sequence at the transmitter side. The SIMO structure can be obtained by
applying multiple receive antennas, or by oversampling the received signal to generate

multiple virtual channels (subchannels) [5,9]. While most SOS based approaches need

to identify the channel first, some direct. blind equalization methods were presented
in [10—13].

Lots of existing second-order moment-based algorithms can be classified into one
of the following categories: eigenstructure-based algorithms, correlation / spectrum fit-
ting methods and linear prediction based approaches. Among eigenstructure—based
algorithms, subspace methods [14,15] are a class of most popular blind channel esti-
mation techniques in recent years. The main idea is that the orthogonality between
the signal space and the noise space can be exploited to estimate the channel vector
up to a scalar factor. The major advantage of the subspace methods is that they can
provide a closed-form solution, however, they could be sensitive to modelling errors.

The optimal weighted correlation fitting approach was proposed in [16], which
achieved the performance bound of asymptotic normalized mean square error (AN-
MSE). A more practical suboptimal approach has been presented in [17]. Compared
with eigenstructure-based approaches, correlation/ spectrum fitting methods are ro-
bust to channel order selection and ill conditioning of the channels, but they are not
easy to implement because of the local minima in the optimization. An approach
combines the strength of both correlation fitting and subspace methods was pro-
posed in [18], which is referred to as the joint optimization with subspace constraints
(JOSC).

Multi-step linear prediction based method is an SOS based approach that elimi-
nates the inter-symbol interference before channel identification, which was proposed
in [19], and was extended to IIR channels with common zeros in [20]. Compared to
subspace methods, the multi-step approach is more robust to order mismatches of

the underlying FIR channel [19,20].

1.3.2 Higher-Order Statistics Based Methods

Compared with second—order statistics, higher-order statistics can provide more in-
formation about random signals. While phase information is lost in the second-
order statistics (SOS) of symbol rate sampled channel outputs, higher-order statistics

(HOS) have the ability to preserve both magnitude and non-minimum phase infor—

mation. Therefore, by applying higher order statistical methods, a much larger class
of channel models can be identified. Another advantage is that for Gaussian sig-
nals, all cumulants of order greater than two are identically zero. If a non-Gaussian
signal is received with additive Gaussian noise, higher-order cumulants can suppress
the effect of Gaussian noise effectively. A review of the properties and applications
of higher-order statistics can be found in [21]. Most higher-order statistical blind
equalization methods are inverse filter criteria (IFC) based algorithms, which were
initially proposed in [22-24], and were extended to MIMO systems in [25]. This kind
of approaches find the optimal equalizer by searching for the maximum of a class of
cumulant based cost functions either explicitly or implicitly. Two well known IF C
based approaches are super-exponential algorithm (SBA) and constant modulus al-
gorithm (CMA). Usually they have better performance than the SOS based methods,
but require higher computational complexity. A short review can be found in [26].

Constant modulus algorithm (CMA) was first proposed by Godard in [27]. It is
one of the earliest blind receiver designs, and is also one of most widely used blind
equalization methods. A CMA receiver is obtained by minimizing a cost function
which is defined by a constant modulus criterion. As in most IF C based methods,
gradient search method is usually applied to find the minimum of the CM cost func-
tion. The receivers designed with CMA methods have similar MSE performance to
the non-blind Wiener receivers [27]. Not only can the CMA approach recover source
symbols possessing a constant modulus, but also can equalize signals characterized
by source alphabets not possessing a constant modulus, like 16-QAM [28].

The SEA (super-exponential algorithm) is a class of iterative algorithms for solving
the blind deconvolution problem, which was first proposed in [29], and has been
extended for multichannel deconvolution in [30]. In [31], it is shown that SEA method
is equivalent to a gradient search algorithm which minimizes an inverse filter criteria
with dynamic step—size.

CMA and SEA algorithms both are special cases of IFC based algorithms, and
they are closely related to each other. Actually, under certain circumstances, they

are equivalent to each other [26, 31-34].

1.3.3 Blind Channel Estimation and Equalization in CDMA

Systems

In CDMA systems, besides inter-symbol interference, the receivers have to combat
multiuser interference (MUI). While the signals of the interfering users are treated
as Gaussian noise in most commercial systems, significant gain can be achieved by
modelling the MUI as part of the system explicitly. Some initial work of multiuser
detection can be found in [35]. In this section, existing blind multiuser detection

methods for both long-code and short-code CDMA systems are briefly reviewed.

A. Blind Signal Detection for Short-Code CDMA Systems

In short-code CDMA systems, a time-invariant MIMO model can be obtained when
taking the symbol-rate signals as the system inputs. Therefore, by exploiting the
code structure, the algorithms developed for single user systems can be applied to
short-code CDMA systems. For this reason, most research works of blind equalization
for DS-CDMA systems have been focused on short-code systems.

SOS based blind signal detection methods for DS-CDMA can be found in the
following research works [36-49]. In [36], a blind multiuser detection method, which
minimizes the output energy (MOE) of the receiver subject to a special constraint,
was proposed for CDMA systems without multipath distortion. An extension to the
multipath case was presented in [37]. This approach coincides with the MMSE solu-
tion in the constrained space, while the knowledge of the desired user’s transmission
delay is supposed to be known at the receiver. A general framework and the perfor-
mance analysis of the MOE based approach were given in [38]. Subspace based blind
channel estimation methods for DS—CDMA systems were proposed in [39—44], and a
direct equalizer design approach was presented in [45]. The blind detection method
developed in [44] is claimed to have lower computational complexity compared with
the MOE detector, assuming the prior knowledge of the timing of the desired user.
The research work of [41] has been extended to multi-rate DS—CDMA systems in [46].

In [47] and [48], moment matching methods was developed to estimate the chan-

nels blindly for single-rate and multirate DS-CDMA systems, respectively. In [49],
mnltistep linear predictor based (MSLP) methods are presented for CDMA systems.

Higher-order inverse filter criterion based algorithms have been proposed in [50],
[51] and [52]. The approaches presented in [51] and [52] have no control that which
user is extracted first, and a user identification algorithm (UIA) was developed in [52]
to identify the extracted signals. In [50], code-constrained IFC methods was developed
so that only the code sequence of the desired user need to be known. CMA based
approaches can be found in [53—56], and SEA based approaches have been proposed
in [57—59].

B. Blind Signal Detection for Long-Code CDMA Systems

In long-code CDMA, the time-varying nature of the received signal model severely
complicates the equalizer development approaches, since consistent estimation of the
needed signal statistics can not be achieved by time-averaging over the received data
record. More recently, both training based (e.g. [60—62]) and blind (e.g. [63-71])
multiuser detection methods targeted at the long code CDMA systems have been
proposed. In this dissertation, we will focus on blind channel estimation and user
separation for long code CDMA systems. Based on the channel model, most existing

blind algorithms can roughly be divided into three classes:

0 Symbol-by-symbol approaches As in long code systems, each user’s spread-
ing code changes for every information symbol, symbol-by-symbol approaches
(see [68—71] for example) process each received symbol individually based on
the assumption that channel is invariant in each symbol. In [68,69,71], channel
estimation and equalization is carried out for each individual received symbol
by taking instantaneous estimates of signal statistics based on the sample val-
ues of each symbol. In [70], based on the BCJ R algorithm, an iterative Turbo

multiuser detector was proposed.

0 Frame-by-frame approaches Algorithms in this category (see [66,72] for

example) stack the total received signal corresponding to a whole frame or slot

10

/"

 

into a long vector, and formulate a deterministic channel model. In [66], com—
putational complexity is reduced by breaking the big matrix into small blocks
and implementing the inversion “locally". As can be seen, the “localization”
is similar to the process of the symbol-li)y-symbol approach. And the work is

extended to fast fading channels in [72].

o Chip-level equalization By taking chip rate information as input, the time-
varying effect of the pseudo-random sequence is absorbed into the input se-
quence. With the observation that channels remain approximately stationary
over each time slot, the underlying channel, therefore, can be modelled as a
time-invariant system, and at the receiver, chip-level equalization is performed.

Please refer to [65, 73—75] and references therein.

In all these three categories, one way or another, the time-varying channel is “con-

verted” or “decomposed” into time-invariant channels.

1 .4 Space-Time Coding

To support multimedia services, high transmission data rates are required in the next
generation wireless communication systems. Although the capacity can be improved
by increasing the transmission power and bandwidth, they are not practical due to the
limitations of devices, systems and available bandwidth. In addition, increasing the
transmission power will introduce more co—channel interference. For these reasons,
exploiting the spacial diversity has attracted a lot of research interest, and intensive
research has been undertaken in this area. To apply spacial diversity, multiple anten-
nas are equipped in either the transmitter, or the receiver side, and sometimes in both
sides, which results in a so called MIMO (multiple-input multiple-output) system. A
review of the MIMO systems is presented in [76].

In a cellular system, because of the limitation of the size and power of the mo-
bile devices, it is more feasible to equip multiple antennas in the base stations than

in the mobile devices. In the uplink, receiver diversity can be achieved by comb-

ll

ing the output of each receive antenna, while in the downlink, space-time coding
(STC) can be applied to provide transmission spacial diversity. In [77], space-time
trellis codes (STTC) were developed to improve the reliability of communications.
Although STTC can provide both spacial diversity gain and coding gain, the trellis
complexity increases exponentially as a function of the spectrum efficiency and the
diversity order [77]. Meanwhile, space-time block coding (STBC), first introduced
by Alamouti in [78], has become a popular space—time coding scheme for its simplic-
ity in decoding. The original Alamouti scheme was developed for systems with two
transmission antennas. It is generalized to support arbitrary number of transmission
antennas in [7 9]

The STBC schemes were originally designed for flat fading channels. While in the
third generation (3G) communication systems, because of the high chip-rate of the
transmitted signal, the systems suffer from frequency selective fading. The structure
of the coding blocks is corrupted by the inter-symbol interference in this situation. To
overcome this obstacle, MIMO equalizer (MIMO-EQ) was designed to mitigate the
inter-symbol interference before space—time decoding in [80]. Although the diversity
gain can be observed from the simulation results, the theoretical analysis is not ad-
dressed in the paper. Time-reversal space-time block coding (TR-STBC), which can
provide full diversity gain, was designed specifically for frequency selective channels
in [81]. The detailed coding structure and simulation examples were presented in [82].
A general coding scheme was proposed in [83], which achieves the maximum diversity
gain in frequency fading channels, and includes the TR—STBC as a special case.

As CDMA has been identified as the major multiple access technique in 3G stan-
dards, researchers have been investigating space-time coding schemes for CDMA sys-
tems. In fact, the Alamouti scheme has been adopted in the W—CDMA standard.
Recently, following [81] and [83], schemes combining TR—STBC and CDMA were pro-
posed in [84] [85] to provide maximum diversity in downlink CDMA systems. In [84],
equalization is carried out in time domain, while in [85], equalization is performed in
frequency domain.

Existing blind and semiblind equalization approaches for space-time coded CDMA

systems are mainly designed for Alamouti based coding schemes. Code constrained
inverse filter criterion [86] and subspace based approach [87] have been proposed for
ST coded short-code CDMA systems, and a semiblind approach is presented in [88]
for STBC CDMA systems with aperiodic spreading codes. In [86] and [87], each user
is assigned two short-codes, while only one spreading code is required for each user

in [88].

1.5 Objectives

This dissertation is focused on blind channel estimation and signal detection for
CDMA systems. Upon reviewing the research works in literature, we aim to achieve

the following objectives:

0 Develop HOS based fast blind equalization approach for short-code
multi-rate CDMA systems. For short-code CDMA, most existing HOS
based blind detection methods search for a global minimum / maximum of a cost
function explicitly, or implicitly. Therefore the bottleneck is the convergence
speed of the iteration procedure. In multi-rate short-code systems, a high-rate
user is usually modelled as several basic-rate virtual users. Instead of extracting
the signals of each virtual user independently, the correlations between different
virtual users need to be exploited to speed up the iteration procedure. The
first objective of this dissertation is to solve these problems, and to develop an

efficient blind equalization algorithm for multi-rate systems.

0 Design novel chip-level blind equalization approaches for long-code
CDMA systems. As mentioned previously, research works about blind equal-
ization are mainly focused on short-code CDMA systems. Blind detection of
long-code CDMA signals remains a challenging topic due to its time-varying
nature caused by aperiodic scrambling. As mentioned in Section 1.3.3 that, by
taking the chip-rate signals as inputs, the system model can be characterized

using a time-invariant model. Our second objective is to design novel chip-level

13

blind equalizers for long-code systems.

a Design blind equalizer for space-time coded long-code CDMA sys-
tems. It is a trend to incorporate transmission diversity at base station to im-
prove the downlink transmission quality. Notice that the main research works
in this area consider systems combing short-code CDMA with STBC scheme
which was originally designed for flat fading channels, in this dissertation, we
aim to design a blind equalizer for space-time coded long-code CDMA systems

with full transmission diversity in frequency selective environment.

1.6 Contributions

In this dissertation, statistics based blind equalization methods are developed for
both short-code and long-code CDMA systems. Long-code CDMA systems with
transmission space-time diversity are also considered. The main contributions of this

dissertations are:

0 Blind equalization algorithm design for multi-rate shortscode CDMA

A fast HOS based blind equalization approach — code-constrained super expo-
nential algorithm (CSEA) is designed for multi-rate short-code systems [89].
Compared with existing IFC based algorithms, the proposed method has sig-

nificantly faster convergence speed due to its dynamic step size.

0 Chip-rate blind equalizer design for long-code CDMA

Multistep linear prediction based two-step approaches are developed for both
downlink and uplink long-code CDMA. In the first step, the convolutional mix-
ture models are transformed to 181 free instantaneous mixture models. In the
second step, the channels are identified using either SOS or HOS based methods,

and the chip-rate equalizers are designed based on the estimated channels.

— In the downlink, the novelty of this research resides in that, after descram-

bling, the chip sequence is treated as the output of a short-code system,

14

and SEA can be applied to extract the signal of the desired user. By using
this approach, the performance is improved significantly, and the spread-
ing sequences are no longer required to be orthogonal to each other, which

is a necessary condition for most downlink chip-rate equalizers.

— In the uplink, for systems whose spreading codes are nonconstant modulus,
transmitter induced cyclostationarity is exploited to identify the channels
blindly [90]. The codes with special property in frequency domain are
designed in order to apply Fourier analysis to identify the channels. By
using matrix pencil method, requirements on the spreading codes can be
relaxed. For systems whose spreading codes are constant modulus, JADE
algorithm —- a higher order statistics method is applied to solve the instan-

taneous mixture problem [91].

0 Blind equalizer design for space-time coded long-code CDMA systems

Downlink CDMA combining time-reversal space-time coding (TR-STBC)
scheme is considered in this part. The major contribution is that the principal
component analysis method is applied to perform blind channel estimation. The
advantages of the proposed approach are that the ISI, MUI, and noise are com-
pletely suppressed in the channel estimation procedure, and only one spreading

code is needed for each user.

1.7 Outline of the Dissertation

This dissertation is organized as follows. In Chapter 2, discrete system models for
both short-code and long-code DS—CDMA are presented, along with the assumptions
used through the dissertation. The difference of the symbol-rate model and chip-rate
model are also discussed for long-code CDMA in this chapter. In Chapter 3, super-
exponential algorithms (SEA) are introduced, and code-constrained SEA (CSEA)
based approach is designed for multi-rate short-code CDMA. Both multi-code (MC)

and variable sequence length (VSL) multi-rate schemes are discussed. Convergence

15

 

analysis and identifiability aspects are provided, while the detailed proof is given in
Appendix C. In Chapter 4, multi-step linear prediction based methods are developed
for both downlink and uplink long-code CDMA systems. In the uplink, spreading
codes with both constant and nonconstant modulus are considered. The methods
developed for uplink can be used for downlink directly, since the downlink is a special
case of the uplink. In Chapter 5, blind equalizer is designed for space-time coded long-
code CDMA. Principal component analysis method is developed to perform channel
estimation. Both zero padding and cyclic prefix/postfix are exploited to remove
the inter-block interference. Conclusions and related future works are presented in

Chapter 6.

16

CHAPTER 2

DS-CDMA System Models

In this Chapter, system models are presented for both uplink and downlink DS—
CDMA systems, and the differences between short-code and long-code CDMA are
explained. Without special notification, the discussions in the remaining part of the

dissertation will follow the notations and models defined in this chapter.

2.1 Discrete-Time Uplink CDMA System Models

In short-code CDMA systems, spreading codes repeat symbol by symbol, while in
long-code systems, every symbol in a given frame has a distinct spreading codes.
This radical difference results in different characteristics of the system model and
different statistical properties of the received signal. In this section, uplink system

models of both short-code and long-code CDMA are introduced.

2.1.1 Short-Code CDMA

Consider an up—link DS-CDMA system with M users. For m = 1, 2, . - . , M, let um(k)
denote the kth symbol transmitted by user m, and cm(n), n = 0,1,...,Nm - 1,
denote the spreading code of user m, where Nm is the spreading gain for user m.
As illustrated in Figure 1.3, the information sequence is spread with the signature

sequence before transmission. Thus the spreading result of user m is given by

my.) = Z um(k)c,,,(n-kNm), (2.1)

17

C. 01)]

 

 

 

 

 

 

 

 

 

 

 

 

 

u,(k) ‘ . s,(n) )’m(n) w(nl
spreading g. ('1)
. W!)
(k C” (")1 (M)
—u—M-—)-> spreading M 8M (n) y (n)

 

 

 

 

 

 

Figure 2.1. Block diagram of uplink short-code DS-CDMA system with single receive
antenna

and the transmitted continuous time signal has the format
00

82.0) = Z smelled-kn), (2.2)

kz—oo

where p(t) is the chip pulse shaping filter, and Tc is the chip duration. Let ij and
am, denote the delay and the complex gain of the lth ray respectively, then the linear

time-invariant multipath channel for user m is given by
9;,(t) = Z am,;6(t — W). (2.3)
1
Therefore the component of the received signal due to user m can be obtained as

y(’")(t) = 82(t)*91n(t)

= Z 3m(k)gm(t _ chli (2'4)

k=—oo
where * denotes the convolution operation, and gm(t) = g;n(t) *p(t). Sampling y("‘)(t)
at the chip rate, it can be obtained that
00

Mo) = Z sm(k)gm(n—k> (2.5)

kz—oo

18

where _q,,, is the effective channel impulse response of user In sampled at chip rate.
The discrete time system diagram is shown in Figure 2.1. From (2.1) and (2.5), it

can be obtained that

y(’")(n) - Z um(k)hm(n —kN,,,), (2 6)
k=—oo
where
hm(n) = Z gm(l)cm(n—l) (2.7)
l=—oo

represents the effective signature sequence of user m. The overall received signal at

chip rate observed in additive white Gaussian noise w(n) is given by

Wt) = 31“") (n) + 100%)

00

MsiMs

um(k)hm(n — kNm) + w(n). (2.8)
k=—oo

3
ll

In a single rate DS-CDMA system, all users have the same spreading gain, that
is Nm = N. The system can be represented with a multiple inputs multiple outputs

(MIMO) model. Collect N samples of y(n) into an N -vector
y(k) == [y(kN).y(kN +1)...-.y(kN + N -1)lT, (2.9)
and define
hm(l) := [hm(lN), hm(lN +1),...,h,,,(lN + N — 1)]T, (2.10)

then the symbol-rate MIMO model is given by
1

M L",—
y(k) = Z Z hm(l)um(k — r) + w(k), (2.11)
i=0

m=l

where w(k) = ['w(kN), w(kN + 1), . . . ,w(kN + N — 1)]T. Because all symbols of a

19

given user share the same spreading code, this is a time-invariant model.

Define the N X M matrix
1:10) :2 lhl(l)1h2(l)1' ° ' 7 hAl(l)lv (2'12)

and .M x 1 vector

“(kl 1: 011(k), U203)» ' ' ' , UnlrfkllTa (2-13)

then the system model can be represented by

b.
H

y(k) = H(l)u(k -— r) + w(k), (2.14)
I

II
o

where L is the maximum possible value of Lm.

2.1.2 Long-Code CDMA

Due to its better performance stability and information security, long-code CDMA
systems are used in virtually all operational and commercially proposed DS-CDMA
systems. As illustrated in Figure 2.2, in a commercial long code CDMA system, the
user’s symbols is first spread by user specified short code, then a pseudorandom long
sequence is used to scramble the spread sequence. Therefore the overall spreading
codes are aperiodic in each frame. This results in a time-variant symbol-rate system
model. However, it can be observed that, if the channels remain unchanged in one
frame, and the chip-rate scrambled signal is taken as the input, the system can be
characterized using a time-invariant model. In this subsection, we consider a system
with K receive antennas as shown in Figure 2.2.

Let rm(n) denote the spreading result of user m, which is given by

Tad") ‘2 Z'un1(klcrn(n—kNm)- (2.15)

20

 

 

F i gt
ten

W

and

“(1‘) C|(n)l r(n) d|(") ( ) VA")

6 (-n)]y ‘13”)
u (k) r, (n) 3 ('7) yd")

. CM 01)] (15,07) . .
k - . é ]
"1A ) rd") N 5., (n) y. (N)

Figure 2.2. Block diagram of uplink long-code DS-CDMA system with multiple re-
ceive antennas

The chip-rate scrambled signal can be expressed as

sm(n) = r,,,(n)dm(n), (2.16)

where dm(n ) IS the pseudorandom scrambling sequence of user m. Let 9(1)) (l ) represent
the channel impulse response between user m and the pth receive antenna of the base

station, then the received signal at antenna p can be represented as

M
=2: g()(l) )sm (n—l)+'wp(n ), (2.17)

m=l l

1‘

II
o

where wp(n) is the additive noise at pth antenna. Define
s(n) := [sl(n),sg(n), . . .,sM(n)]T, (2.18)

and the received signal vector

y(n) :-—- [91(71): y,(..), . . . .yximlT, (2.19)

21

then the chip-rate time-invariant MIMO model can be obtained as

L—l
y(n) —_- ZH(l)s(n—l)+w(n)
l=0

 

 

;= y,(n)+w(n), (2.20)
where
"9%) 99(1) git-la)"
Ha) = 9% gm 99(1) , (2.2,)
.glKla) 995(1) 951550)-
and we) == [w1(n).w2(n). . . . ,wanni‘.

Note that Nm (m = 1,. . . , M) could be different, therefore this model is suitable
for both single-rate and multi—rate CDMA systems.

When the symbol-rate signals are taken as the system input, the spreading codes
are implicitly part of the channel impulse response, resulting in a time-varying channel
model. On the other hand, when taking the chip-rate signals as input, the under-
lying channel becomes time—invariant. For channel estimation and signal detection
methods which are based on the cyclostationarity of the received signals, the time-
variant model introduces significant complexity, as required statistics can not be
estimated through time-averaging of the observed signals. This makes the chip—rate

time-invariant model be more attractive for long-code CDMA systems.

2.2 Discrete—Time Downlink Long-Code CDMA
System Models

In downlink CDMA, signals are transmitted from the base station to every user in
the same cell as shown in Figure 2.3. Signal destined for each user is first spread

using the user-specific channelization code, then signals from every users are added

22

lW,(k)
ulin) ””“fl r|(k) ._.

_ Spreafiing or , .
channelization / ’ g‘ ' ’(k) y | (k)
._ _ :w2(k)
_‘:.~( It) .' spreading 07—] s(k) (2) $ , k
._. —." channelization Scrambling g (k) i )3( )

 

  
 
 

 

MEET“ Spreading or

channelization

E 'wP(k) 3
_.- +4»

 

Base station

Figure 2.3. Illustration of downlink long-code CDMA with multiple receive antennas

together and scrambled using the same pseudorandom sequence.

When inter-cell interference is negligible, system model of downlink CDMA is just
a special case of the uplink system. Consider a downlink long-code CDMA system
with one transmission antenna at the base station, and P receive antennas at each
mobile device. If we take the scrambled chip-rate signal s(k) as the system input, the
system model will be reduced to a single input multiple outputs (SIMO) model.

As shown in Figure 2.3, the transmitted chip-rate signal is given by

s(n) := r(n)d(n), (2.22)
where
M
r(k) := Z rm(k) (2.23)

is the sum of all spread signals, and d(n) is the scrambling sequence. Let glp)(l)
represent the channel between the base station and the pth antenna of the desired
user. The received signal at the antenna p of the desired user is given by

L—l

yp(n) = 29(p)(l)s(n — l) + wp(n), (2.24)
(:0

23

Define

EU) I: lg‘”(l>.g‘2‘(l), - - - ,ywlUilT (225)
w(n) = [w1(n), ur-2(n), . . . ,wp(n)]T, (2.26)
then the received signal vector y(n) = [y1(n), w(n), - - - ,yp(n)]T can be expressed as

L—l
y(n) = Zg(l)8(n-l)+w(n)
1:0

= y3(n) + w(n), (2.27)

which is a time-invariant SIMO model.

2.3 General Assumptions

Without special indication, the algorithms presented in this dissertation are based on

following assumptions:

(A1) The information sequences {um(k)},m = 1, . . . , M, are zero mean, mutually

independent i.i.d, and are drawn from a finite alphabet with E{[um(k)|2} = 1;

(A2) The scrambling sequences {dm(k)} are i.i.d. BPSK sequences, independent of

the information sequences;

(A3) The noise sequence W( k) is zero mean Gaussian, independent of the information

sequences, with E{w(n -l- k)w(n)} = 02 16(k).

It)

( A4) 77(2), 71(2) and G(z) are FIR, and have full column rank for every 2, including
2 = 00 but excluding z = 0. (77(z), ’H(z) and 0(2) denote the z-transform
of H(l), H(l) and g(l) respectively.) This last assumption is to ensure the

existence of an FIR inverse filter (please refer to [25]).

Obviously, (A4) implies that we need N 2 M. That is, the processing gain is larger

than or equal to the number of active users.

24

2.4 Summary

System models of both short-code and long—code CDMA systems are presented in
this chapter. When the symbol-rate signals are taken as the input, short-code CDMA
systems can be characterized using a time-invariant model, while long-code CDMA
system results in a time variant model due to the chip-level pseudo-random scram-
bling. However, when the chip-rate signals are taken as the input, both short-code
and long-code systems can be modelled as time invariant systems.

General assumptions which are used throughout the dissertation are also presented

in this chapter.

10
U1

CHAPTER 3

Blind Signal Detection for
Short-Code DS-CDMA Systems

In this chapter, the super-exponential algorithm (SEA) is applied to short-code sys-
tems for its fast convergence speed. Code constrained SEA (CSEA) approach of blind
detection of short-code DS-CDMA signals is presented first, then the CSEA approach
is developed for multi-rate DS-CDMA systems.

3.1 Introduction to Super-Exponential Algo-
rithms

The super-exponential algorithm (SEA), first proposed in [29], is a class of itera-
tive algorithms for solving the blind deconvolution problem. In this section, we use
a single-input single-output (8130) system to introduce the basic idea of the SEA

method. Consider a noise free 8180 system

S”
H

y(n) = h(l)$(n - 1), (3-1)
I

II
o

where h(l) is the channel impulse response, and w(n) is the channel input. Let f (n)
denote the equalizer with length Le, and z(n) denote the equalizer output, then the

overall response of the system is given by

d(n) = 2 (1(1) f(n — r). (3.2)

26

We want to find an equalizer f (n) such that
d(n) = ej®6(n — k), (3.3)

where It stands for the equalization delay, and 3 represents the phase shift. Consider

the following two—step iterative procedure:

9'(n) = 9"(n)(9*(n))", (3-4)

6"(n) = W") (3.5)

V2. l9’(n)|2o

Compared to other higher-order-statistics based blind deconvolution algorithms, the

 

major advantage of SEA is that the above iteration forces d(n) to converge ex-
ponentially to the desired response. More specifically, the leading tap converges
to one, while all other taps converge to zero. Define the equalizer vector f =
[f(O),f(1), . . . ,f(Le — 1)]T. It was shown in [29] that the above algorithm can be

realized as:

f’ = R#d,

~r
r" = —.5———, (3.6)
(fr)HRf‘r

where R is the Le x L, matrix whose (i, j)th element is

R _ E{y(n - j)y*(n - 2)}

 

 

1.. _ , .7
’ E{x<n>x*(n)} ‘3 )
and d is the Le x 1 vector whose kth element is given by

d(n) = cum{z(k) :p, z (k) : q,y (k -— n} (3.8)

cum{a:(k) : p,:r*(k) : q +1} ’

where “cum” stands for “cumulant” (Please refer to Appendix B for the definition of
cumulants). It has been proven in [31] that SEA is equivalent to a gradient search

algorithm for cumulant maximization with an optimal time-varying step-size, which

27

ensures the fast convergence of SEA.

3.2 Code Constrained Super-Exponential Meth-
ods for Blind Detection of Single-Rate Short-
code DS-CDMA Signals

In this section, equation of combined channel-equalizer impulse response is derived
first. Next, the unconstrained SEA approach is presented. Finally, code-constrained
SEA approach is developed, and the convergence and identifiability aspect are ana-

lyzed.

3.2.1 Matrix Representation of the Combined Channel-

equalizer Impulse Response

Recall that (please refer to Section 2.1.1) the chip-rate sampled channel output of an
uplink short-code CDMA can be represented as

M Lm—l
=2 2 11,,(1) — r) + w(k) (3.9)
m=1 l=0

where y k()— [y(kN ) y(kN +1) (kN +N — 1)]T contains the chip signal observed

in the kth symbol period.
Let {f(i)}Nx1,i= 0,1,. .,Le — 1 denote the N x 1 vector equalizer of length Le

symbols, then the equalizer output is given by

Le—l
k) = Z fH(i)y(/c — r). (3.10)
i=0
It follows from (3.9) and (3.10) that
M [xi-Le"
=2 2 0:,(1)u,,,((1.:—r)+ Z f"(-Z) ' (3.11)
m=1

where 0;,(1) :2 2:04 f"(i)h,,,(l — r). Let L = L -+- LC -— 2, and define

6 :2 (morale).---.0.(£>.o.<o).o.<1>.---,0-.<L>.

"‘10M(0)16M(1)1"‘9011/l(I—4)]ZI‘I(L+1)XI’ (3'12)
“(kl 3: [u1(/€),u1(k -1)."°.u1(l€— Llsu'szlluflk ‘1),"°,U2(k " L),
~ ° ° , Um“), Uri/(k —-1),---,uM(/€ " [ZULU-aunt (3'13)

then (3.11) can be rewritten as

e(k) = @Hu(k) + flied), (3.14)

where
i” := [f”(0),f”(1),--- ,f”(L, — 1)]”, (3.15)
w(k) := [w”(k),w”(k —1),~-,w”(k — L, +1)]”. (3.16)

Define the block—Toeplitz matrix

115,1(0) 0 0
11.110) 113(0) 0
film-3- 11:3,(L—1) h,’,’,(L—2) 0 , (3.17)

0 11,1;(L— 1)

 

 

0 0 hfn’(L—1) -
- - [L+l]X[NLc]

and

H2: [H{,H2T,...HE]T, (318)

then the combined channel-equalizer impulse response can be represented as

e = Hf. (3.19)

If the moth user is the desired user, the output of the ideal equalizer should satisfy
e(n) = aum_0(n — d), (3.20)

where d is the equalization delay, and a is a complex scaling factor. In other words,

the combined impulse response should satisfy

(-)=[ 0,0,~-,0 ,a*id,0,0,.-.,0]T, (3.21)

(mo-1)(I.+1) zeros
with
L: [0,0,.-.,0,1,0,0,-~,0],,,[,;,,]. (3.22)
d ZBI’OS

3.2.2 Super-Exponential Approach

In this section, following [29] [92] [93], (unconstrained) SEA approach is presented
for CDMA systems. Consider the following two—step iterations (m = 1, 2, - - - , M and

k=0,1,---,L)

65.93) = snowman)", (3.23)
6.2%) = 95.?(k)/|l9“’ll2, (3.24)

where 9‘” is obtained by substituting 6m(k) in (3.12) with 653(k), and p, q are
positive integers that satisfy p, q > 0, p+q Z 2. According to [29] and [93], as long as
the “leading” (maximum magnitude) tap of the initial value of 6 is unique, the above
iterations converge at a “super—exponential” rate to the combined channel-equalizer

response
e=[ 0,0,---,0 ,a‘Id,0,0,-~,0]T, (3.25)

‘15P“
(m—1)(L+1) zeros

for some m and d, where 1 S m _<_ M and 0 S d g L. For simplicity, we choose
= 2 and q = 1 through out this chapter. Since G is not available, an algorithm in

terms of the equalizer f and data is need to be developed.

30

Define

1901) :: (0.7:.(A‘))2(07n(k)), (3-26)
(n=(m—1)(L+1)+k, m=1,2,---,M, k=0,1,---,L.)

and construct an M(L + 1)-column vector 9 = [19(1),79(2), - - - ,i9(M(L + 1))]T. Ac-
cording to (3.23), we seek for f’ to minimize lle’ — 9]]. This is a linear least square

problem whose solution is given by
i’ = (11” HWH” g. (3.27)

The normalization operation in (3.24) can be carried out as

fr

\/ f'H HHHI'.

Projecting the f-domain (equalizer domain) algorithm back to 9-domain (combined

fl! _

 

(3.28)

channel-equalizer domain) following (3.19), we obtain

9’ = H(H”H)#H”g, (3.29)
9!

” = —, 3.30
”9’“ ( )

As H and g are unknown, the algorithm given by (3.27) and (3.28) is still cannot
be implemented directly. Next, this algorithm will be converted into a realizable

algorithm in terms of joint cumulants of the input and the output of the equalizer.

Define

Yf(k) 2= [y:(k)sy3(k —1)»' ' ° .YZUC — Le + :1)th (331)

where M L -1
yak) = Z hmeiume — z) (3.32)

m=l [=0

is the noise—free channel output. Under assumption (A1), it follows that the correla-

tion matrix of the imise-free channel output is given by
72,, :z E{Y,(1.~)Y;’ (1.)) = 1:1”131. (3.33)

Consider the noise-free equalizer output
M I.
as) = Z [annuals — I), (3.34)
m=1 (=0
then the iterations (3.23)(3.24) can be realized as

i" = Rid, (3.35)

~ I
f” = ——f—— (3.36)

where

cum{e;(k), e;(k), €306), Y,(k)}.

 

d := 3.37
cumaumen ( )
(3.38)
In the presence of the additive noise, define
YT(k) := [yT(k), yT(k — 1), - - . ,yT(k — L, + 1)]T, (3.39)

if follows (3.9) and (3.32) that Y(k) = Y8(k) + W(k), where W(k) is the [NLC] x 1
noise vector defined in the same manner as Y(k) and Y3(k). The correlation matrix

of the channel output is given by
73,, := E{Y(k)Y”(k)} = 72,, + 0,3le,. (3.40)

Since all cumulants of order greater than two are zero for Gaussian random variables,

p
r.L
b

I

 

It.

the SEA in the presence of noise can be obtained as

~

’ __ #
f — Ryyd, (3.41)
~I
, ’f’HR-yyf’
In the limit of iterations, the equalizer obtained from the super-exponential algorithm
blindly converges to a solution which is approximately the non-blind Wiener filter

[34,92, 94, 95] which extracts user m with delay d for some m 6 {1,2, - - - , M} The

problem is: there is no control over which user the system will converge to.

3.2.3 Code-Constrained Super-Exponential Algorithm

In this section, to ensure that the system will extract the desired user, a constraint
based on the knowledge of the desired user’s spreading code is formulated first, and
then the code-constrained super-exponential approach is briefly presented.

From Section 3.2.1, it can be observed that to extract the desired user mg with

equalization delay d, the inverse filter equalizer should satisfy
Hi =[ 0,0,---,0 ,a*id,0,0,-~,0]T
\—:"-—/
(mo-l)(L+1) zeros
4:» f”Y,,(k) = e,(k) = aum,(k - d). (3.43)

Multiplying both sides by H” , we obtain

HHHi = a*[h,’,,’0(d)h”(d-1),...,hf,’,0(0),0,---,0]"

7 7710

any). (3.44)
It then follows from (3.33) that

7238f = whiff]. (3.45)

33

According to the orthogonal projection theorem [96], (3.45
cient condition for f to satisfy (3.43).

For any desired equalization delay d, define

hi3) == [h,7,;(0),h§,(1), - --

 

11137101)? : Chabgma

 

where
- c..<0) 0 0 -
cm(1) cm(0)
' 613(1) 0
Cm(N - 1) . 6171(0)
Giff) == 0 Cm(N - 1) cm(1)
0 .
0 cm(N - 1)
_ 0 0 0 _
and

gm == [9m(0).9m(1). '-

Define an [N(d + 1)] x [N(d + 1)] matrix ’1; as

p

 

b

then from (3.46), we have

hlffl, = 71115.0“)

0
0

IN

hH

7 m0

0 IN
IN 0
0 0
(d_ 1):

.-

 

[(d+1)N] x [LN]

-,gm(LN — 1)]T.

~ . 115.302”-

) is a necessary and suffi-

(3.46)

(3.47)

(3.48)

(3.49)

(3.50)

Further define an [NLe] x [NLC] matrix Tl“) as

T 0
Tld) :2 d . (3.51)

0 Irv(r.,—1—d)

Thus, from (3.45)-(3.51), we have

7952.3 = a*r<d>iim, = wefgggmo, (3.52)
where
cfd)
d ,_ 7""
65,3 ._ (3.53)
[L,N)x[r:rv)

Let the columns of H.332 denote an orthonormal basis for the orthogonal complement
of (35:3. Because C55,] is of full column rank, 375,12 is an [N Le] x [N L, — N L] matrix.
This leads to

uggmfldlnflf = 0. (3.54)

In the absence of noise, we have Ryy = 7235, therefore it can be obtained that
ufiffldlnyyi := AW)? = 0, (3.55)

where .A‘m") is an [N(Le - L)] x [N Le] matrix. Thus, in order to extract the desired
user, a necessary condition is that f belongs to the null space of A(’"°). Assume the

effective rank of Am") is r, and carry out an SVD of Am")

A‘m‘” = [ U. U. ] , (3.56)
0 2 v,”

C
b

where Z, is the diagonal r x r matrix containing the effective non-zero singular values

of Am"), and )3,- contains the insignificant singular values of Am"). Therefore, in the

absence of noise. (3.55) is equivalent to
v,”f = 0. (3.57)

With the existence of noise, V1 is determined by the effective rank of A0“).

Recall that f" was chosen to minimize ||Hf’—Q II. This requirement is now modified
as: choose f’ to minimize ||I:If’ --9 || subject to V1” f’ = O. The minimum-norm solution
to this problem is given by

f’ = (Human. (3.58)

where

Hfihno) 3: V2V2H (3.59)

is the [N Le] x [N L] projection matrix onto the null space of Am"). Note that
ni (fini )# = (fini )#, from (3.58), we have IIi f" = P. Then the

A(mo) Abno) A(mo) Abno)

combined response is given by
e = (Hnjumogf. (3.60)

Therefore, mimicking the development in Section 3.2.2, the code-constrained algo-

rithm is given by

= (Hfifmo)Ryij-1(mo))#Hj(mo)d1 (3'61)

1*" = —f———- (3.62)

(hwnwr,

which are the counterparts to (3.35) and (3.36).

Convergence and Identifiability Aspects

Proposition 3.1 Define space 80A 2 range(fiHL

Am»), and let 7%,; represent the

orthogonal projection operator on to the space 80A, then iteration procedure given in

36

(3.61) and (3.62) is equimlent to the gradient search algorithm (10912111 by

I

1
_ (n) __ . . (n)
e _ e +2F4(e(n))PCAV(—) F4(G )

GI

@(n-H) __,___,
“9 H2

which maximizes the cost function

3.3433(1),

 

subject to G E SCA.

Please refer to Appendix C for the proof of Proposition 3.1.
For unconstrained case, the restriction is in a larger subspace S, which is the range
space of 1:1. According to the assumption (A1) in Chapter 2, we have E { lum(k) I2} = 1.

In absence of noise, we have
me) = J42(9)/|cum4(um(k))l, (3.63)

where
__ lcum4(e(k))|
“ [Ele(k)|2]2 ' (“’4’

J42(e) := .1426) ;
Consider the unconstrained case under the sufficient order condition of ’PA = I . In
this case, 6 is unrestricted (except for G 76 0) and the global maxima of lnF4(9)
and F4(@) coincide. Let f be a subequalizer for which J42(f) achieves the global
maxima |ry4s|, corresponding to maxelnF4(€-)) = 0, where |74s| := cum4{um(k)}.
The corresponding output is given by e(n) = aumd(n — do). Thus f leads to the
extraction of user md with delay do. Then, by construction, A<mdlf = 0. Therefore,
for code-constrained SEA, maxeescAlnF4(O) == 0. If J42(f) aé Ins], (3.43) cannot
hold true. Therefore, constrained global maxima of J42(f) are given by those f’s for
which J42(f) = 743 and Af == 0, equivalently, for which (3.43) and (3.55) hold true.

Next we consider the identifiability aspect of the algorithm. Consider the following

condition:

37

C 3.1 The [N136] >< [ZN + 1] matrix [C533 3 7<d>i1§,‘{“’] has full column rank for every
772. 7f mo and every do 6 {d — L + 1,d — Z + 2, . . . ,d} where L _>_ d +1 and d 2 2.

Proposition 3.2 Under Condition 3.], any solution that satisfies (3. 55) and (3.52)
corresponds to m = mo and do 6 {d - L +1,d — L + 2, - - - , d}, where Le _<_ d +1 and
d g L.

Please refer to [53] for the proof. From the above discussion, under the condition
C31, all constrained global maxima of lnF4(9) in 80A are specified by the solutions

given in Proposition 3.2.

3.3 CSEA Based Blind Equalization of Multirate
Asynchronous CDMA Systems

As is well known, the third generation wireless system aims to provide multimedia
services with high data rate and variable quality of service. Multirate design is there-
fore required to map different data rates into the allocated bandwidth. For CDMA
systems, two basic multirate schemes are multicode (MC) transmission and variable
sequence length (VSL) (also known as variable spreading factor). In MC systems,
the information sequence of a high-rate user are subsampled to obtain several sym-
bol streams, and each stream is spread using a distinct signature sequence. In VSL
systems, users requiring different rates are assigned signature sequences of different
lengths. For both schemes, the signal of a high-rate user can be treated as the super—
position of several basic-rate virtual users’ signals.

In this section, motivated by the previous works in [97] and [57], we consider
blind detection of the desired user’s signal in multirate CDMA systems using super-
exponential algorithm. Compared with the single-rate case, the major challenge of the
multirate case lies in that the signals of different virtual users corresponding to the 3

same high—rate user have to be extracted synchronously and in correct order.

38

3.3.1 System Model for Multirate DS-CDMA Systems

In this section, an equivalent baseband system model is presented for both VSL and
MC systems. In the following discussion, R denotes the basic symbol rate, and the
pair (i, j) is used to denote user j at rate R,, whereJL- = p,R (p,- is an integer). K
stands for the number of different user rates, and assume there are Q,- users at rate
R,. N represents the processing gain of the basic rate users. In MC systems, all users
have the same processing gain, while in VSL systems, the processing gain of users at

rate R,- is given by N,- = N /p,.

A. VSL Systems

In VSL systems, users at different rates have different processing gains. Let the
signature sequence of user j at rate R,- be denoted by cij := [cg-(0), c,,-(1), - - - ,Cij(Ni —

1)]T. Then the effective signature waveform of user (i, j) is given by
N,—1
n) = Z cave-An —- k), (3.65)
k=0

where g,j(n) is the effective channel impulse response with respect to user (i, j). A
high rate user at rate p,- times basic rate can be converted to p.- basic rate virtual
users. The symbols of the mth (m = 0,1,...,p,—- 1) virtual user uf-J m’(l) can be
extracted by subsampling:

u‘r’u) := new +m), (m = o. 1. . . . .p. - 1). (3.66)

U")

where {712303)} is the symbol sequence of user (i, j) drawn from a finite alphabet,
and is independently and identically distributed (i.i.d). Let 12,5 denote the received
signal component due to user (i, j). If we stack 13-]- into an N x 1 column vector
xij(n) := [x,j(nN), . . . ,x,j(nN + N —1)]T, then we have

Pi-‘l Liz" I

x,,-(n =2 2 hfim’u) , -—1), (3.67)

m=OI=O

39

where hff’u) :2 [h,j(lN— d”. — an ) ..... 1,.,(1..V— d,,- —mN + N — 1)]?“ (1,,- is the
transmission delay of user (i. j), and L), is the maximum length of the channel impulse
response of each virtual user of user (i, j) in terms of basic rate symbols. Thus, x”-
can be considered as the superposition of p,- virtual users derived from user (i, j). The

spreading code for the mth virtual user is given by [97]
C371) i: [lehnNJ CZ; le[(p,—1—m)N,-]]T- (368)

B. MC Systems

In MC systems, high rate user (i, 3') can also be converted to p,- basic rate virtual
users. The symbol stream of each virtual user is derived by the same way as (3.66).
Those virtual users have their own spreading codes cf]. ), (m = O, 1, . . . , p, — 1). The

effective signature waveform of the mth virtual user of user (i, j) is given by

N— 1
h(m’(n =$1(Zcm(l’l)g,~]( n — l). (3.69)
z=o
If we stack 53")(71) into an N x 1 vector, and define 6);")(1) := [hgn’uN —
d,,-),. . ., hgn’UN— d.,- + N — 1)]T, then the received signal from user (i,j) can also

be represented by (3.67).

C. Unified Model

Based on the above derivation and take the received signals from all users into con-

sideration, we can obtain an MIMO model for both VSL and MC systems:

K Qi pi—l Lh- “'1

)—Z Z Z Z hgvu — l) + w(k), (3.70)

i=1j=lm=01=0

where y(k): =[:I:(kN), f(kN +1),. . . ,ii7(kN + N —1)]T, and 513(n) is the total received
signal. w(k) is the Gaussian noise defined in a manner similar to y(k). In other words,

the total received signal can be represented by the superposition of total 2:, Qip,

41f)

 

 

virtue

then.

ha

virtual users plus the additive white Gaussian noise. Define

hfi’" ‘:"’ _[h"."‘”(0) 0,1153")”(1), . . . , hlt”"(d)]”, (3.71)

U

g] : [gtj(_dij)i ' - 7 igtj(—dij — 1+ NI-‘gHTv (372)

then, for both VSL and MC systems, we have hgn‘d’ -- an'd)g,-j, where

cfif’w) 0 0
4;”)(1) <0)
0
4;?”(N — neg-"10>
GET”: 0 e§;">(N—1) 2 , (3.73)
(m)
0 0 CU (N 1)
0 0 0 -
. ~ [(d+1)N]><[LgNl

 

 

d is the desired equalizer delay, and L9 is the maximum length of {gij} in terms of

basic rate symbols. According to the definition of Lh, L, = 59 + 1.

3.3.2 Code-Constrained Super-Exponential Algorithms for
Multirate CDMA Systems

A. The Combined Channel-Equalizer System

Let {f(mo’Uc) )},c _31 denote the N x 1 vector subequalizer for virtual user (i0, jo, mg),
the moth virtual user of the desired user (in, jg). Then its output is given by
1.6—1

e‘m0’tn) = Z tt‘m°’(k)t”ytn — k) == [i‘m°)t”Y(n). (3.74)

k=O

41

where

W“ := [(Fm°’(0))”, . . . . (WWI... — 1))”1”.

Y(n) := [yT(n), . . . ,yT(n — Le +1)]T.

Defining ugn’m) = [ugn’(n), - - - ,ugn’m — Le — L), + 2)]T, and collect all {ugn’(n)}s

to a vector u(n). Following (3.70), we get

~

Y(n) = H”u(n) + e(n), (3.75)
where PI is the transpose of the signature matrix, and 17v(n) :2 [w” (n), - ,w” (n —
Le + 1)]” . Therefore, (3.74) can be rewritten as

e<mo>(n) = eHu(n) + [i<mo>]”\7v(n), (3.76)
where
e = Him) (3.77)

is the combined channel-equalizer impulse response.
Assuming that user (in, jg) is the desired user, we wish to design subequalizer f(m")
such that
01‘5“ “ d) if (2.1.7.1771) = (ioJoflnol

(m)
613‘ (l) = . (3.78)
0 otherw1se.

In the absence of noise, this leads to

e(m°’(n) = ozulm")(n — d). (3.79)

lojo

That is, the equalizer output is a scaled and shifted version of the signal of the virtual
user (i0, jo, me). After we get all the subequalizers, their outputs are interleaved to

get the equalized output of the high rate user (i0, jg)

..,c(0’(n), e(l’(n), . . . , emit—”(72), e(0’(n +1),

42

(1)011.1)H_,(J(l"o—]’(n+ 1),... (380)

B. Multirate CSEA (MR-CSEA)

In this subsection, the CSEA method presented in [57] is extended to multirate sys-
tems.

Following [57], we define an [N(d + 1)] X [N(d + 1)] matrix ’1; as

- q

0 0 [N
0 I 0

712% . . f" . , (3.81)
[N 0 0

 

 

and an [N Le] x [N Le] matrix T“) as

:r 0
’1'”) 2 d . (3.82)

0 IN(L¢——l—d)

In the absence of noise, following [97] and [57], if the equalizer output has the for-

mat given in (3.79), it can be shown that the equalizer should satisfy the following

equation:
~ m’ )= a; a; m
T<d>7zyye ° ar<d>h§3§ d) _ —a cfng ”gm, (3.83)
where aw -i‘—. E{Y(k)Y”(k)}, and
C(mo d)
(m ,d) A i
Ciojg' 03° , (3.84)
[I.,N]x[LgN]
m ,d A meH m H
11$ng ) —[h[0Ji0’ (d ),. .,thJg) (0),0, . . . ,0)". (3.85)
Let U531: ‘0 denote an orthonormal basis for the orthogonal complement of 6533‘”.
Because (35ng d" is of full column rank, L153: "1’ is an [N Le] x [N Le — N Lg] matrix.
This leads to
L153? MT “”72 1W):A AM it“) = 0. (3.86)

43

Thus, in order to extract the desired virtual user. a. necessary condition is that f(m“)
belongs to the null space of Am”). Let Ilium, denote the [N Le] x [N Le] projection

matrix onto the null Space of 34"“). Thus the combined response is given by
e = (Hui )i<"'°> (3.87)
A(mo) '

Then the subequalizer f(m") can be calculated using the following two-step iterative

procedure similar to that of [57]:

f = (Hi(mo)Ryy1-I:(mo))#Hfi(mO)d (3'88)

f

W

57+” (3.89)

 

where
*(m0) *(m0) (7710)
d = CUE{€ (k)? e (k),€ (k7),Y(k)} ’ (3.90)
cum4luiojo(k)}
with
CHECK], 1132,1133, 1134) = E{$1$2$3$4} — E{$1£E2}E{$3IE4}

-E{$1$3}E{$2$4} - E{$1$4}E{$2$3}

(2:1, (to, 1:3 and x4 are zero-mean random variables), and e(m°)(k) is the equalization
output of the nth iteration. As in [57], the code-constrained SEA is followed by
unconstrained SEA (without the projection operator) to enhance the system perfor-

mance .

MC System

Consider the following condition (the counterpart to the condition C31):

C 3.2 The [NLC] x [Lg/v +1] mat1i:r[CJ-((:'Ji§‘d’ 2 T‘d’fiiin’d03 has full column rank for

every (i,j, m) aé (io,jo,m0) and every do 6 {d — Lg + 1,d — £9 + 2, . . . ,d} where
Le Zd+1andd22

As proved in [50], under condition C32, any solution that satisfies (3.79) with the
constrain (3.86) corresponds to (213'. m) = (io.jo. mo) and do 6 {d — Z9 + 1. d — £9 +
2, . . . ,d}. In MC systems, condition C32 is easy to be satisfied, since each virtual
user is assigned a distinct spreading code.

Because all virtual users corresponding to a high rate user share the same channel,
the equalizer f(mO), which is obtained from the iteration (3.88)-(3.89), can be used to
set the initial values of subequalizers for all other virtual users. To ensure different
virtual users which correspond to one high-rate user have the same equalizer delay
and scale factor, the initialization method of [97] is adopted. For MC systems, we

then get the initial guess for f(m), m % mo, from (3.83) as

:(m) 2(m0)
r := [7(leyy]#C(m’d)[C(m‘d)]#T(d)Ryyf . (3.91)

iojo 1’010

This initial guess is used to initialize the SEA procedure to extract virtual user m,

Vm aé mo.

VSL System

In VSL systems, the spreading codes of virtual users corresponding to the same user
are simply shifted version of each other, so the condition C32 is not necessarily

satisfied. An example is given in [97]. Therefore, C32 has to be modified as

C 3.3 For VSL systems, the [N Le] x [EQN + 1] matrix [Cfsgfl 5 T(d)fig"’d°)] has
full column rank for every (2',j) ¢ (io,jo), every m 6 {0,1, . . . ,p, — 1}, every mo 6
{0,1,...,p,-0 — 1}, and every do 6 {d— 139+ 1,d— Zg+2,...,d} where L8 2 d+1
and d 2 2.

Under condition C33, the solution that yields (3.79) and satisfies (3.86) corresponds
to (z',j) = (z'o,jo), m e {0,1,....p,-, — 1}, and do 6 {d— E, +1,d— IZ,+2,...,d}.
This indicates that the two step iteration (3.88)-(3.89) will converge to an equalizer
corresponding to a virtual user of high rate user (to, jo), but it is not guaranteed that

(to, jo, mo) will be extracted.

Due to the fact that the VSL spreading codes CE; for m— - O 1, . . . .1), - — 1 share
the same CU, special care needs to be taken in order to extract the virtual users in

correct order. Define

x. := ONioXIN'L‘3-N‘Ol 0N..,x~., (3.92)
I[NL¢- ,0] 0[NLC— iolXNio
then it can be proved that [97]
(Pi-‘7") d”(m,do) d (Odo- 1)
x. 0 ° T< )h,o,.g =T< >h,0,.0 . (3.93)

Let fg‘) denote the subequalizer of virtual user (to, jo, m) with equalization delay do.

In the absence of noise, it follows from (3.83) and (3.93) that
T<d>7zyyr§§>_, _—_ Xép‘0'm°’7<d>R,,i§g‘°). (3.94)

Therefore, instead of (3.91), once the system converges to a virtual user (to, jo, mo)
:(m )
with delay do, i.e. f do is obtained, we use following method to initialize the sube-

qualizers for virtual user (to, jo, m):

:(m) 20'” )
fdO z: (T‘leyy)#Xé"“"‘°)T(d)Ryyfdoo (for mo < m < 1),), (3.95)
2(770

fdo—I

:(m )
(T‘dmyy)#X5Pi+m—mo>7<d>1zwfdo° (for 0 g m < mo). (3.96)

H

The interleaved subequalizer output will be

m ( 11)
,aufojg)(n - d0) Wig: (n - d0),
CXU £11,030“), + 1 — do)” WOU£33_1)(TL +1— do),
aulmlm + 1 - d 0,) au§$3+1)(n + 1 -— do), . .. (3.97)

From (3.97), we can see that, even it is not sure which virtual user is extracted first,

the interleaved output is still in correct order.

46

3.3.3 Summary of the Multirate CSEA Approach

The Multirate CSEA algorithm for both MC and VSL systems can be summarized

as:

1) Take mo 2 0, and use multirate CSEA method to extract virtual user
(io, jo, mo). The iteration is considered converged if ||f<n+1l — f(n) H < e (where e
is a small positive number, such as 10‘3). For MC system, when the algorithm
converges, it will converge to user (io, jo,0). For VSL system, the algorithm
may converge to (io, jo, mo), with mo sé 0, since the spreading codes of all the
virtual users corresponding to one high rate user are highly correlated (please
refer to equation (3.68). However, in this case, even if mo 3£ 0, the resulted
output (see equation (3.97)) will only be a shift version compared to that of

m0=0.

2) For MC systems, use (3.91) to initialize the subequalizers for all other virtual
users (io,jo,m), m E {0,1,...,p,0 — 1}, and m yé mo. For VSL systems, use
the initialization (3.95) and (3.96). Then use SEA procedure to extract virtual

users (io,jo,m), m E {0, 1, . . . ,p,o - 1}, and m 75 mo.

3) Interleave the outputs of the subequalizers to obtain the output of user (io, jo)

according to (3.80).

Remark 3.1 Please refer to Section 3.2.3 and Appendix C for the convergence and
identifiability analysis.

Remark 3.2 Another possibility is to use 79,; = nglTwmfgg‘d)[T(d)C,($:'d)]T'R;J to
project the equalizer onto the weighted null space of Am" as suggested in [.98]. In
the absence of noise or when SNR is high, Rm, may not be full rank or tend to be
ill-conditioned, therefore 72;; has to be replaced with the pseudoinverse of Ryy- In
this case, ’PA is no longer a projection matrix onto the null space of Am". Simulation
results indicate that when ”PA is used, the system still converges, but loses control of

which user the system is extracting, so we stick to the projection matrix Him), in this

research.

47

3.3.4 Simulation Examples

In this section, simulation examples are explored to illustrate the proposed approaches
and compare the performance with that of MR-CC-IFC in [97] and MMSE method
with known channels. For both VSL and MC schemes, we consider DS-CDMA sys-
tems with two different symbol rates. Each user transmits 4—QAM signals. The
symbol rate of a high rate user is 4 times that of a low rate user. The basic rate
processing gain is 32. Thus, each high rate user can be converted to 4 virtual users.
The spreading codes were randomly generated binary (i1 with equal probability)
sequences. The multi-path channels for each users have 4 paths. The transmission
delays are uniformly distributed over one basic rate symbol (32 chips), and the re—
maining 3 paths have delays (w.r.t. the first arrival) uniformly distributed over one
basic rate symbol. The multi—path amplitudes are mutually independent, complex
Gaussian with zero mean and unit variance. The channels for each user were gener-
ated randomly for each Monte Carlo run. The subequalizers of length L; = 5 and
delay d = 3 were designed based on a record length of 1024 symbols. Then they
were applied to an independent record of 3072 symbols for calculating the normalized
equalization mean-square error (NEMSE) (normalized by the desired user’s informa-
tion sequence power). N EMSE was averaged over 100 Monte Carlo runs to evaluate
the performance of the equalizers. In the equal power case, a symbol of a high rate
user is transmitted with the same energy as a symbol of a low rate user. In the near
far case, the power of the desired user is 10dB below that of other users at the same
rate. SNR refers to the chip signal-noise—ratio with respect to the desired user. For
MR—CC-IFC approach, we set [1,4 = pg = M) = 2, where M4, #3, and up are the
scaling factors for certain penalty functions. Please refer to [97] for the meaning of
those coefficients. For both MR—CSEA and MR—CC-IFC methods, 8 = 10‘3 was used
in the simulations, and the iterative procedure is ended either when this criterion is

met, or 30 iterations have been performed.

48

 

 

 

 

 

 

 

 

 

 

 

SNR (dB) 0 5 10 15 20 25
MR—CC-IFC(s) 20.2 14.5 14.0 19.8 20.4 19.8
MR-CSEA(s) 28.5 24.08 17.4 3.7 3.9 3.7

MR—CSEA (without enhancement) (5) 21.3 17.4 13.0 2.4 2.3 2.3

 

 

Table 3.1. Example 1 - VSL system, average time per run, equal power case, same
condition as in Fig.3.1.

A. Example 1

In this simulation example, we consider a DS-CDMA system with 3 users (two basic
rate users and a high rate user). The high rate user is the desired user. The results
for VSL systems are shown in Fig. 3.1 and Fig. 3.2, and the results for MC systems
are shown in Fig. 3.3 and Fig. 3.4. It can be observed that, when both methods
converge, the MR-CSEA approach has better performance than that of the MR—CC-
IFC approach. At high SNRs, we can get comparable results with MR—CC-IFC by
calculating the subequalizers using (3.91) for MC systems and (3.95)-(3.96) for VSL
systems directly. In the meantime, due to the existence of noise, we can get better
results if we enhance each subequalizer individually as in this research.

Table 3.1 shows the average time in seconds per Monte Carlo run for both algo-
rithms in the equal power case in the VSL system. The model of the computer used
in this simulation is Dell Dimension 4550, P4 2.8GHz, 1G RAM. From the table, we
can see that, when SNRZ 15dB, the MR-CSEA method converges much faster than
the MR—CC-IFC method.

49

 

—&— CSEk equal power

A CSEA: near—tar

+ CSEA: equal power. w/o enhancement
-0 CSEA: near—far, w/o enhancement

 

 

 

 

Figure 3.1. VSL system - Normalized MSE for the high rate user: 8 chips/symbol
for the high rate (HR) user, 32 chips/ symbol for basic rate (BR) users, 1 HR and 2
BR users, record length = 1024 BR symbols, evaluation length = 3072 BR symbols,
100 Monte Carlo runs. (“without enhancemen ” means that the subequalizers are
calculated from (3.95)-(3.96) directly.)

 

   
 
   

hr
+ CSEA: equal power. wlo owlncamom
a CSEA near-hr. w/o cinnamon!
—9- MR-CC-IFC: equal power
-9- MR—CC-IFC: near-hr
-9- MMSE: "‘ -‘ '

 

equal pan!

 

Bit error rate

 

 

 

 

10 15 20 25

Figure 3.2. VSL system - BER of the high rate user, same condition as Figure 3.1

50

 

 

 

 

 

 

 

 

 

10 1 I L j
+ CSEA: equal power
5 «A- CSEA: near—far _
- + CSEA: equal power. w/o enhancement
at - CSEA: near-far, w/o enhancement
.4” —e— MR-CC-IFC: equal power -
"xx 0- MR-CC-IFC: near-far
_5_ 1‘». ,, , ,. , _ + MMSE: known channel, equal power H
'fiL,o_ ;
m -10» \\."';:\'>...V‘ u
g ] x \ ‘ ‘1 ————— -e —————— a
Z -15- \. -F$
. f \\
_20_M..M.....,. .. h. . H H.‘ H . . , ,. ,7 . H ,_
. \'\. .
-25“' ..\., -
_3o_.wu..HWVH.,.H.H.W_6.. “a..n.sw ... iii a
_35 I l L 1 ‘
O 5 10 15 20 25
SNR

Figure 3.3. MC system - Normalized MSE for the high rate user: 32 chips/ symbol for
each symbol stream of the high rate (HR) user, 32 chips/ symbol for basic rate (BR)
users, 1 HR and 2 BR users, record length = 1024 BR symbols, evaluation length =
3072 BR symbols, 100 Monte Carlo runs. (“without enhancement” means that the
subequalizers are calculated from (3.91) directly)

 

 

 

 

 

 

 

10° 4
' + CSEA: equal power
..,... CSEA: near-tar
+ CSEA: equal power. w/o enhancement
..,. CSEA: near-far. w/o enhancement
-e- MR-CC-IFC: equal power
0 - MR-CC-IFC: near—tar
+ MMSE: known channel, equal power
\
X. .
2 :
s g
B \ \ \\ 1
g 4 _‘h'_ _. _ .. 6. — ------ '. ----- '1‘.
,4; 10- ‘ - G ----- 1)
fl) , :
\
-5 \
1 O \ +1—
\.
\. e
1 0— ‘ ‘~, - >
\
\
-7 ,
1 o l 1 A L L
O 5 1O 1 5 20 25
SNR

B. Example 2

In this example, we study the performance of both approaches for different loads in
both VSL and MC systems. We fixed the desired user’s SN R at 20dB and vary the
number of active users. In VSL systems. let the pair (HR users. LR users)=(1,1),
(1,2), (1,3), (1,4), (2,1), which corresponds to 5 ~ 9 virtual users. In MC systems, the
number of users (8, 10, 12, 14, 16, 18) corresponds to the pair sequence (HR users,
LR users)=(1,4), (2,2), (2,4), (3,2), (3,4), (4,2). The desired user is the first high rate
user. The simulation results of MC and VSL systems are shown in Fig. 3.5 and Fig.

3.6, respectively.

 

 

+ CSEA: equal power - .

45- CSEA2near-far ’ ..,. 31737
-1 + MR-CC-IFC: equal power ‘

10 -c— MFl—CC—IFC: near—tar -

 

Bit error rate

 

 

 

 

7
Number of virtual users

Figure 3.5. Load test of VSL systems. Bit error rate of different loads at SNR =
20dB. Totally 100 Monte Carlo runs. The sequence (5, 6, 7, 8, 9) of virtual users
corresponds to the pair sequence (number of HR users, number of BR users) = (1,1),
(1,2), (1,3), (1,4), (2,1).

52

 

 

3 + CSEA: equal power , _ - .. _; ,0
’ .4- CSEA: near-tar ' ' ' ' , , ’1,
104,, + MR-CC-IFC: equal power . : A?)

._._ MR-CC-IFC: near tar ‘ _ '

 

   

 

 

  

Bit error rate
a

 

 

 

 

10
... ._l

.. ...,t'.
10-55 ,-Ij
: ,‘ "git
CTT'T If" 'f' . , , I ':f_

—6 1 l , 1 I
10 8 10 12 14 16

Number of virtual users

Figure 3.6. Load test of MC systems. Bit error rate of different loads at SNR = 20dB.
Totally 100 Monte Carlo runs. The sequence (8, 10, 12, 14, 16, 18) corresponds to
the pair sequence (1,4), (2,2). (2,4), (3,2). (3,4), (4.2).

3.4 Summary

In this chapter, code-constrained super-exponential approach is developed for blind
equalization of multirate DS-CDMA signals. Both multicode and variable sequence
length are considered. Simulation examples demonstrate that, when the algorithm
actually converges (SN R2 15dB), MR-CSEA delivers better results with faster con-

vergence speed compared with existing cumulant maximization blind detectors.

53

CHAPTER 4

Blind Equalization for Long-Code
CDMA Systems

In this chapter, we consider blind detection of long-code CDMA signals. Multistep
linear prediction based blind channel estimation methods are developed for both
downlink and uplink DS-CDMA, and the equalizers can readily be designed based
on the estimation results. For downlink CDMA, the scrambled chip-rate signal is
recovered first. The descrambled signal is then regarded as the output of a short-
code CDMA system, and SEA approach is applied to extract the symbol-rate signal.
For uplink CDMA, first, based on multistep linear prediction methods, convolutive
mixtures are reduced to instantaneous mixtures. Secondly, for systems with noncon-
stant modulus spreading codes, the channels can be identified by exploiting only the
second-order statistics of the signal. While if the spreading codes are constant mod-
ulus, higher-order statistics approaches should be utilized to perform blind channel

identification.

4.1 Blind Equalization For Downlink Long-Code
CDMA Systems

In downlink long-code CDMA, the signals destined for all users are transmitted syn-
chronously. Therefore, at each mobile device, the received signal components corre-
sponding to different users share the same channel impulse response. For this reason,
chip-level equalization can be performed to restore the signals transmitted from the

base station, and then the orthogonality between different data streams can be ex-

54

ploited to recover the signals of the desired user (see [99103]). In this section, inspired
by [20] and [104]. multistep linear predictor based (MSLP) algorithms are applied to
perform blind channel estimation, then MMSE equalizer is designed to extract the

chip-rate signal.

4.1.1 Multistep Linear Prediction Based Blind Channel Es-

timation

The system model presented in Section 2.2 is applied in this section. Consider a
downlink CDMA system with M users, and each user has P receive antennas, then

the received signal vector is given by (2.27):

L-l
y(n) = Zga>e(n—l>+w(n)

(:0
:= y5(n) + w(n). (4.1)

Following [20], y3(n) can be decomposed in the following format:
y,(n) = y3(n|n — l) + e(nln - l), (l = 1,2,...) (4.2)

where ys(n|n—l) is the l-step (ahead) linear predictor of y3(n) given {y3(k), k S n—l},

and has the following presentation
Lt
y3(n|n — l) = ZAPan — i). (4.3)

i=1

for some L, S L + l — 2. The prediction error e(nln — l) is given by

1—1
e(nln — z) = Z g(i)s(n — 2'). (4.4)
satisfying
E{e(n|n — l)yf(n — m)} = 0, Vm 2 l. (4.5)

From (4.2)-(4.5), we can get

R,,J3(m) :2 E{y3(n)y I{(71—171)}
L1

._. Z A§’>E{y.(n — oyftn — m)}

i:

 

 

 

L1
= ZAE”R...<m—e>
i=l
_ Balm—z) l
= [A(’)A]21,...,A)])] Brim—1‘1) . (4.6)
_ Rum—Lt) _
Define
' Rare) 134.0) Reta—z) ‘
72...: Rat—1) Rate) Hamel-+1) . (4.7)
_Re.(l—Lz) Beau—Ll) 0 ,

 

It follows from (4.6)-(4.7) that

M)”, A") . . .,A3f317z,, = [11,,(1),R,,,(z+ 1), . . . , 19414)]. (4.8)

(+11

A minimum norm solution to (4.8) can be obtained as [105]
1A5” Atom-AB]:lR..(l),R...(z+1)....,R...<Lz>17zi.. (4.9)
From (4.2) and (4.3), we have

L
n(In—l) )AI’zZ y3(n—i). Vl >1, (4.10)
i=0

56

where L = L, - l + 1 + d, and

I), i: 0
Ai = . (4.11)
—A(') lg i g L,
0 LI 3 i g L
Define
é1(n) := e(nln — l) — e(nln — l + 1), (4.12)
and
E(n) = [é5+1(n + d), éfln + d —1),...,é§(n +1),eT(nIn — 1)]T, (4.13)

then it can be derived from (4.2)-(4.4) that

- s(d) l
d — 1
E(n) = g( Z I s(n) ;= gem). (4.14)
_ 8(0) ,

 

 

According to (A1), we have E{s(n)s(n)*} = 1, then it can be obtained that
REE := E{E(n)E”(n)} = gg”. (4.15)

Note that REE is a rank one matrix, in the noise free case, the channel impulse
response vector g (which contains the channel impulse response from the signal source

to every receive antenna) can be obtained up to a scalar from (4.15).

4.1.2 MMSE Equalization and SEA Enhancement

In this section, MMSE equalizer is designed based on the channel estimation result to

recover the chip-rate signal transmitted from the base station. Next, the descrambled

chip sequence is treated as the channel output of an unknown short-code CDMA
system, and SEA approach is applied to extract the symbol-rate information sequence
of the desired user.

Let {fd(k)k_1’:—l denote the K X 1 vector equalizer with equalization delay d. The
equalizer output is then given by

Lc-‘l

(n. -— d): 2 fd(i y(n — 2'). (4.16)

i=0

The mean square error between {s(n)} and its estimate value is

E{le(n)|2} = E {|§(n) - 8(n)l2}- (4-17)

Define
1.1;: [f3"(0),f}‘(1),. . . ,f}‘(Le —1)]T, (4.18)
Y(n) == [1’T(n),yT(n -1),---,yT(n - Le +1)lT, (4-19)

then, the MMSE solution which minimizes (4.17) is given by [106]
f4- — Rygd (4.20)

where Ry = E{Y(n)Y(n)”}, and gd = [gT(d),gT(d — 1),gT(0),0, . . .,0]T.
In downlink, signals of all users are synchronous. According to (2.22), the de-
scrambling process r(n) = s(n)d(n) results in an estimate of r(n). At this stage, the

desired user’s signal can be extracted directly using correlators. That is
amue) = —c,T,,(k)i~(k). (4.21)
where

f(k) := [f(kN). r(kN +1),-~,r(KN + N —1)]T. (4.22)

58

Instead of (:lepreading the descrambled signal r(n) directly, here we choose to

model it as an hr‘lllVlO short-code CDMA system

M Lo— 1
:2 Z hm - —l) (4.23)
m=l l: 0

where the channel responses {hm( l),} m=1,. . , M, are unknown. Therefore, exist-

ing blind multiuser detection methods for short-code CDMA system can be applied
to eliminate the multiuser interference. In this work, super exponential algorithm

(SEA) is chosen to extract the signal of the desired user from {r(k)}. Suppose user

1 is the desired user. Our goal is to design an N x 1 vector equalizer {f’(i)= .1’8 0’1
L’e—l
u1(n - d’) = Z f’H(i)i'(n — i). (4.24)
i=0

Comparing (4.24) with (4.21), CI can be used to initialize {7(0) to carry out the fol-

lowing two—stage iteration [58]:

{fin-H) ___ Rfv,
fl(n+1)
1‘4"“) = 4.25
J?!(n+l)HRTi‘I(n-l-l), ( )

 

 

where

R. = E{i‘(n)f‘H(n)},
cum{e*(k), e*(k), e(k), f(k)}
cum4{ui(k)} ’

 

and {e(k )} is the equalization result using f’(")

Lg—i

= Z [f’(")(i)]”i~(k — 2'). (4.26)
i=0

The iteration is quit once If’ln‘”) — f’WI < e, where e is a small positive number.

Please refer to Chapter 3 for more details.

59

4.1.3 Simulation Examples

In the simulations, it is assumed that the base station transmits QPSK signals to
each mobile user with equal power. The spreading gain is N = 16, and the spreading
sequences were randomly generated. Gold sequence with period (218 — 1) is used
to scramble the spread signals. The downlink channel corresponding to each receive
antenna has 4 paths, and the first path is the dominant path. The initial transmission
delay and the delay spread are unknown, only assumed to be uniformly distributed
over one symbol period. The noise is white Gaussian with zero mean, and SNR refer
to the chip rate signal to noise ratio with respect to the desired user. The channels
were randomly generated. Channel estimation and equalizer design were based only
on a blocksize of 256 symbols. MSE of channel estimation (CHMSE) and MSE of
input-output symbols (SMSE) were calculated over 100 Monte Carlo runs. In the
simulations, by “w/o SEA enhancement”, we mean using c1 as the correlator to
extract user 1 directly after we get {f(n)}.

In Example 1, the number of users is assumed to be 8. Both one and two receive
antennas were considered. The proposed approach is compared with MMSE equal-
izer with known channel, which is not enhanced with SEA. The CHMSE is shown
in Fig.4.1, while the performance of the equalizer is shown in Fig.4.2 and Fig.4.3. It
can be seen that, compared with direct correlation based user extraction, SEA en-
hancement delivers much better results. Meanwhile, as expected, the space diversity
provided by applying multiple receive antennas (2 in our case) results in significant
performance improvement. Because of the superior performance of the channel esti-
mation algorithm, the equalization results (without enhancement) using the estimated
channels are very close to those of using known channel parameters.

In Example 2, SN R was fixed at 15dB, and the proposed approach was tested
under different loads. The results are shown in Fig.4.4 and Fig. 4.5. It can be seen

that SEA enhancement can increase the system capacity significantly.

60

 

I l
G- 2 receive antannas
E —Ei- 1 receive antenna

1. l.
to,N m

MSE of Channel Estimation (dB)
1
N
O

 

 

 

an I;

 

1O 15 20 25
SNR (dB)

Figure 4.1. Example 1. MSE of channel estimation for 8 users, N=16, channel
estimation was based on 256 symbols, MSE was averaged over 100 Monte Carlo runs.

 

 

 

l “4!:
I “<1

 

‘8’ j' W
t

 

 

 

   

 

 

 

7:; -e— 2 antennas. with SEA enhancement
7: -B- 2 antennas. w/o SEA enhancement
E +2 antennas. MMSE with known channel
E +1 antenna. with SEA enhancement
>
to +1 antenna. w/o SEA enhancement
3 +1 antenna. MMSE with known channel
a) 4
2
-18 -
—20 -
“‘0 5 20 25

1o 15
SNR (dB)

Figure 4.2. Example 1. MSE of equalization for 8 users, N=16, 100 Monte Carlo runs
and 1024 symbols per run.

61

 

 

 

 

   

Bit Error Rate

 

—e—2 antennas, with SEA enhancement
+2 antennas, w/o SEA enhancement
: - +2 antennas, MMSE with known channel
+1 antenna, with SEA enhancement
—v-—1 antenna, w/o SEA enhancement
+1 antenna, MMSE with known channel

20 25

   
 

 

 

 

 

10 15
SNR (dB)

Figure 4.3. Example 1. BER performance, same condition as that in Figure 4.2.

 

—a— with SEA enhancement
+ w/o SEA enhancement

  

MSE of Symbols (dB)

 

 

 

‘4 6 8 1o 12 14 16
Number of Users

Figure 4.4. Example 2. Symbol MSE under different loads, SNR = 15dB, 2 receive
antennas. N =16, 100 Monte Carlo runs and 1024 symbols per run.

62

 

10

 

E+ wi‘hSEAenhancemm 7333f
. + w/o SEA enhancement . - g .

 

............................................

 

 

 

10 . ' ~'

-2

........................ .. .. . . . .
............ .. . ,
................ . . . . .

...........................................

.........................

Bit Error Rate
8

.........................................................................................

................................................................................................

..................

 

...........................................................................................

 

4 6 8 10 12 14 16
Number of Users

 

 

 

Figure 4.5. Example 2. BER under different loads, SNR = 15dB, 2 receive antennas,
N=16,100 Monte Carlo runs and 1024 symbols per run.

63

4.2 Blind Equalization for Uplink Long-Code
CDMA with Non-Constant Modulus Spread-
ing Sequences

In this section, the long-code CDMA system is characterized as a time-invariant
MIMO system as in Section 2.1.2. Actually, the received signals and MUIs can
be modeled as cyclostationary processes with modulation induced cyclostationar-
ity, and we consider blind channel estimation and signal separation for long code
CDMA systems using multistep linear predictors. Linear prediction-based approach
for MIMO model was first proposed by Slock in [107], and developed by others
in [20, 104,108-110]. It has been reported [104, 109] that compared with subspace
methods, linear prediction methods can deliver more accurate channel estimates
and are more robust to overmodeling in channel order estimate. In this section,
multistep linear prediction method is used to separate the intersymbol interference
introduced by multipath channel, and channel estimation is then performed using
non-constant modulus preceding technique both with and without the matrix pencil
approach [111,112]. The channel estimation algorithm without the matrix pencil ap-
proach relies on the Fourier transform, and requires additional constraint on the code
sequences other than being non-constant modulus. It is found that by introducing a
random linear transform, the matrix pencil approach can remove (with probability
one) the extra constraint on the code sequences. After channel estimation, equaliza-
tion is carried out using a cyclic Wiener filter. Finally, since chip—level equalization is
performed, the proposed approach can readily be extended to multirate cases, either
with multicode or variable spreading factor. Simulation results show that compared
with the approach using the Fourier transform, the matrix pencil based approach can
significantly improve the accuracy of channel estimation, therefore the overall system

performance.

64

4.2.1 System Model

Consider a DS-CDMA system with M users and K receive antennas, and the same
notations as in section 2.1.2 are adopted in this section. Assume the processing gain
is N, and the channelization code sequence extends over LC symbols. The Spreading

code for user m is denoted by
cm 2: [cm(0),cm(1), . - -,cm(LCN —1)]. (4.27)
The spreading process can be carried out as

[rm(kN), rm(lcN +1),---,rm(kN + LON — 1)]

= [um(k)cm(0),um(klcm(1)w-',%(k)cm(N —1),---,
um(k + LC —1)c,.,,((LC —1)N),um(k + LC —1)c.,,((Lc — 1)N +1),-~,
um(k + L. —1)c,,,(LcN —1)], (4.28)

where k is equal to an integer times of LC. The successive scrambling processing is

achieved by

[sm(kN), sm(kN + 1), ~ - . ,sm((lc + LC)N — 1)]
= [rm((kN)dm(kN), rm((kN + 1)dm(kN +1),...,
rm((k + LC)N — 1)dm((le + LC)N —1)], (4.29)

where dm(n) is the scrambling sequence of user 172. Define

(..,,(1N),...,vm((k + L.)N — 1)]

[-u,,,(k)dm(kN), - - - ,um(k)d,,,(kN + N — 1),

...,um(k + LC —1)d,,,((le + LC —1)N),

~-,u,.,,(k+LC — 1)dm((K+LC)N—1)], (4.30)

65

Channchzation Scrambling

 

 

 

 

 

 

 

 

 

 

code c l“(11) code (1 min)
um(k) A Symbo, rm(n) sm(n)
Symbol rate Repetition Chip rate
input signal
(a)
Scrambling Channelization
coded l_(n) code c III(n)
"w(k) ‘ Symbol
V R tition
Symbol rate epe
input signal

(b)

Figure 4.6. Two equivalent representation of the spreading and scrambling procedures

then it can be obtained that

[sm(kN),sm(kN +1),---,sm((k + LC)N — 1)]
= [vm(kN)cm(0),vm(kN +1)cm(1),

' ° '1vm((k + Lc)N _ 1)CTn(LCN _ 1)]1 (4°31)

or in brief
sm(n) = vm(n)cm(n). (4.32)
Here cm(n) = cm(n + LCN) serves as a periodic precoding sequence with period

LCN. This is illustrated in Fig. 4.6. After symbol repetition, the chip-rate sequence
can be multiplied with either the channelization code cm(n) or the scrambling code
dm(n) first. They are just two equivalent procedures. Please note that, this form

of periodic precoding (4.32) has been proposed in [7] to introduce cyclostationarity

66

in the transmitted signal, therefore making blind channel estimation based 011 the
second-order statistics of the channel outputs possible for SISO systems.

Following section 2.1.2, the channel output still can be represented as

y(n) = 20ml) s()n—l +)w(n)
[2
= ys(n)+w(n), (4.33)

where H(l) is defined in (2.21), and y,(n) is the noiseless channel output.

4.2.2 ISI Reduction and Separation Based on Multistep Lin-

ear Predictors

In this section, multistep linear prediction method is used to resolve the intersymbol
interference introduced by multipath channel. Following [20] and [104], the noiseless

channel output y3(n) has the following canonical representation:

y(=n) :Afi’,y,(n— )+ e(nIn — l), l= 1,2, - -- (4.34)

i=l

for some L1 5 M (L — 1) + l — 1. The l-step ahead linear prediction error e(nIn — l)

is given by

e()=n|n—l ZH(’£)S( 71—2) (4.35)

which satisfies

E{e(n|n - l)yf(n — m)} = 0, Vm 2 l. (4.36)

From (4.34) and (4.36), it can be obtained that
E{y.(n)y” m>=1 2:34:13..(nE{y.— >y. ” <n—m)},Vm 21. (4.37)

Define
Rm k) =E.{y.<n y"<n — 4)}, (4.38)

67

then we have

Rys(n, 7n) = ZAmiRM (n — i, m. — i)

 

 

"Rate-m4) I
___ [451“ AS)”, ..,/153“] R10" ' l _ I’m — 1“ I) . (4.39)
bRys(n—-L¢,m—L,) .
It follows (4.39) that
[R4 (”1) Begin,” 1l1”'1R'ye(naLl)l
[A5911 1413“," 141531417317. ("6 l, Ll): (4-40)
where
Ry,(n, l, L,) =
'Ryxn—LO) Rate—1,1) Ram—1,144) ‘

R,,(n—l—1,—1) R,,(n—l—1,0) -.. R,,(n—l—1,L,-l—1)

 

 

LRyJ'n—LlJ—Lj) Ry,(n——L1,l—L1+1) R1,(’n— L1,0)
(4.41)

A solution to (4.40) is given by

(l)
[Ah3l1A( nl,)l+1’ . O ' 1 An,Lll

lR’ys(nil)1R3/a(nal + 1), ' ° °1Rya(n1Ll)lRya(n1l1Ll)#' (4'42)

Following (4.33), we have

L—l L—l
Ry,(n, k) = E{ H(l)[s(n — l)s”(n — k — m)]HH(m)}
1.-. If? M
= H(l)E{s(n — l)sH(n — k — m)}HH(m)
l=0 m=0

68

= H(l)Rs(n — l, k + m — l)H”(m), (4.43)
where

R.(n,k) := E{s(n)sH(n—k)}
= diag[I01(n)I2, 1020012,“ ' , ch(n)|2]6(k). (444)

Obviously, Rs(n,k:) = R3(n + LCNJc), since cm(n) = cm(n + LCN). Note that
R3(n, k) = 0,Vk 75 0, define

R,(n) := R8(n, 0), (4.45)

Then L_1
11,,(5, k) = ZH(1)R.(n — l)HH(l -— k). (4.46)

1:0

From the above equation, it can be observed that R1,,(n,k) has the period LCN.
Therefore, according to (4.42), A59, also has a period of LCN. Define E(n) and e(l)
as in Section 4.1.1. It follows (4.35) that

 

 

H(d)
H(d—l) ~
E(n) == . s(n) := Hs(n), (4.47)
_ Hm) .
where _ -
H(d)
Hz: H(df- 1) . (4.48)
. Hm) -

 

 

This is an instantaneouse mixture model. Since E(n) is only determined by the

channel and the current chip-rate channel input, this model is ISI free.

69

4.2.3 Channel Estimation through the Fourier Analysis

In this section, we present that, by designing the precoding sequences properly, the
channels can be determined up to a complex scalar based on only the second order
statistics of the chip—rate sampled channel outputs. First, consider the correlation

matrix of E(n). From (4.47), we have

RE(n) 2: E{E(n)EH(n)} =HRs(n)HH
= fldiagflcl(n)|2e |02(n)|2, ° ° - , lclr('n)|2}fi"- (4-49)

Since cm(n) = cm(n + LCN),j = 1,2,--~,M, RE(n) is has a period of LCN. The
discrete Fourier transform (DFT) of RE(n) is given by

LcN_l

 

33(2) =.- Z RE(n)e-i"2’¢’° =HC,(k)H”, (4.50)
n=0
where
LcN—l . k LcN—l . k
(Mk) = diagf Z [01(n)|2€—'Te“°e Z ICM(n)I2€_l—7"_}
11:0 11:0

= diag{Csi(k)1 ' ' ' 1 Cs.tr(k)}'

Design the precoding sequences {c,-(n) gig-1 in such a way that for a given k = kj,

Csj(k,-) 74 0 and Cs,(kj) = 0,\7’i =,£ 3'. Therefore only one entry in Cs(k) is nonzero,
which is located in the diagonal of Cs(k). Without loss of generality, assume the first
user is the desired user. Based on the design of the code sequences and following

(4.50), it can be obtained that

Seth) = Hdieg{0..(h>.o. - . - .0111”

= aglgfl, (4.51)

where

gt : 191(1)(d)1' ' '1ng)(d)1° ' ° 19:1)(Ola ‘ ° ' vgil\’)(0)lT1 (4'52)

70

and a = C,,,(lcl). Thus the channel of user 1 can be determined up to a complex

scalar from (4.51).

4.2.4 Channel Estimation Based on the Matrix Pencil Ap-
proach
If {(;,,,(n)},",{__.1 are non-constant modulus, the period of RE(n) is LCN. Define random

sequences {,6(n)},~___1,2, which are uniformly distributed in the interval (0,1), then a

matrix pencil {81, S2} can be formed by defining

LcN—l
s,- := Z 6,-(n)RE(n)
n=0
LcN—l LCN l
= 11.114ng nowhere . Z are ))IcM(nI}HN
n=0
:= HRH”, (i=1,2), (4.53)
where
LCN- 1 LcN—l
P~=diag{2jn(n)(lc1n . Zaire )(IcM or} (1:12). (4.54)
n=0 n=0

Consider the generalized eigenvalue problem

Slx = ASQX
e=> H(P1 - 11“,)on = 0. (4.55)

According to assumption (A4), PI is of full column rank, then it can be obtained from

(4.55) that
(r1 — /\I‘2)H”x = 0. (4.56)

Thus the generalized eigenvalue is given by

1. ___ 25“.!" non-(4)12
’ Z£_°_N152(n)lc(n)l2

 

, i=1,2,-~-,M. (4.57)

71

Note that ,l3,(n),=,..2 are generated randomly. /\,-,j = l,- --,1l[ are distinct eigenval-
ues with probability 1. Let x,- denote the generalized eigenvector corresponding to

eigenvalue A,. Because (F1 — A132) is a diagonal matrix, x,- has to satisfy

‘ 11 ,

H X] = 7J1}? (4.58)
where y,- is an unknown scalar, and I,- is an M -vector given by

I,- = [0,0,--.,0,1,0,~-,0]T.
j—l zeros

From (4.53)(4.58), it can be derived that

LrN—l
Sin = HFiHHXj = ’)’j Z fli(n)ch(n)I2gJ-. (4.59)

n=0

Therefore, gj can be determined up to a scalar 'yj 2::ng fl,(n)|cj(n)l2.

Remark 4.1 It should be noticed that the channel estimation algorithm based on
the Fourier analysis requires an additional condition on the coding sequences, which
actually implies that for a given cycle, all but only one antennas are nulled out. More
specifically, this constraint on the code sequences implies that for each user, there
exists at least one narrow frequency band over which no other user is transmitting.
When using the matrix pencil approach, on the other hand, random weights, hence a
random linear transform, is introduced instead of the Fourier transform, resulting in
that the condition on the code sequences can be relaxed to any non-constant modulus

sequences which make /\,- ’s in (4.57) be distinct from each other forj = 1, 2, - - - , M.

4.2.5 Channel Equalization using the Cyclic Wiener Filter

Once the estimated channels are available, MMSE cyclic Wiener filter can be designed
to extract the signal of the desired user. Assuming that user 1 is the desired user,

we want to design a K x 1 MMSE equalizer {fd(n, i)},—I:g of length L, which satisfies

fd(n, i) = fd(n + LCN, i) i = 0. 1, . . . , Le — 1. The output of the equalizer is

Leg-l

4,61 — d) = Z f;’(n,z’)y(n — 7;), (4.66)
i:O

where s1(n) is the estimate of 31 (n), which is given by (4.32), and d is the equalization

delay. The mean square error (MSE) between the input signal and the equalizer

} . (4.61)

Le-l
Z fi’m, 2')y(n — 2') — s1(n — an] y”(n — 10} = 0.
i=0

output is given by

Lc—l

2 tie, z'>y(n — .-) — sun — 4)

i=0

E{|e(n)l2} = E{

 

 

Applying the orthogonality principle, we have

El

 

(k=0,1,...,Le—1). (4.62)
Define
Y(n) : lyT(n)in(n _1))° ° ' ayT(n _ Le +1)lT’ (4'63)
S(n) = [sT(n), sT(n —1),---,sT(n — Le — L + 2)]T, (4.64)
and
P H(O) H(L — 1) o i
H = 3 3 g , (4.65)
_ O H(O) H(L _ 1) . KL.x[(L+L.—1)M]

 

 

then it follows (4.33) that

r(n) = Hsm) + W(n), (4.66)

73

From (A 1 )-(A3). we

where 117(1)) is define in the same manner as Y(n.) and S(n).

have
Ray/(Tl) :2 E{sl(n — d)YH(n)}
= E{Sdn - d)[S(n)”’H” + WHWl}
= |c1(n — d)|21f'HH (4.67)
where Id = [0,...,0, 1,0,...,0 ,...,O]H.
W

(d+l)’s Mxl block

Define fd(n — —,[fd(n 0)”,fd(n,1)H,.. f(,n Le —1)H]”, then (4. 62) can be rewrit-

ten as
(4.68)

Rw(n)?d(n) = l01(n — d)l2HId-
One solution of (4.68) is given by

to) = lam—orRtymmId

= Icl(n — 4))2R,t,,(n)[hl(d)”,hl(d — 1)”, . . . , h1(0)”,0, - - . ,0)” (4.69)

where 111(1) = [95”(1 () 99(1), . . . , g,'°>(z)]T.
Instead of recover the transmitted chip sequence sl(n), the equalizer can also be

designed to recover v1 (n) directly. Actually, the MSE given by (4.61) can be rewritten

E{|€(n )l2} = E{
= |c1(n— d)|2E{ }-(4-70)

)I2} is equivalent to minimize the following error func—

as

Le—l

' (.)y }

Eff ni (n—i)—c1(n—d)vl(n—d)

 

 

Le-

 

If” .
Hahn—mow)

i=0

 

Therefore, to minimize E {|e(n

tion
E{|r*'(7?)|2} = E{

L—l

2
' "’y : A , } (4.71)
i=0

2 (we) (n — 2) — w — 4)

 

 

74

where

fd(n, 2))

mm» = W __ .1)

(4.72)

is the equalizer used to restore the signal v1(n). Define E(n) in the same way as fd(n),

then from (4.69) and (4.78), we have
E(n) = c1(n -— d)R#Y(n)[h1(d)H, h1(d —1)H,...,h1(0)H,0,-~-,O]H, (4.73)

which is the same result as presented in [90].
After the channel equalization, descrambling and despreading process can be car-

ried out as an inverse procedure of (4.28) and (4.29).

4.2.6 Extension to Multirate CDMA Systems

Since chip-level channel modeling and equalization are performed, the proposed ap-
proach can readily be extended to multirate case. As an MC system with high rate
users is equivalent to a single rate system with more users, extension of the proposed
approaches to MC multirate CDMA systems is therefore trivial. For VSL (variable
sequence length) systems, let N be the smallest processing gain and let LcmN denote

the length of the mth user’s spreading code. Define
LC = LCM(LC,13LC,2a ° ° ' 1 LC,M)3

as the least common multiple of {Lc,1, L03, - - - , LQM}, the generalization of the pro-

posed algorithm to VSL systems is then straightforward.

4.2.7 Simulation Examples (Nonconstant Modulus)

We consider the case of two users and four receive antennas. Each user transfers
QPSK signals. The spreading gain is either N = 16 or N = 8. Three cases were
considered: (1) Both users have spreading gain N = 8; (2) Both users have spreading

gain N = 16; (3) Two users have different data rates, the spreading gain for the low

75

rate user is N = 16, and that for the high rate. user is N = 8.

The length of the channelization codes was chosen to be 32 chips, i.e. 2 to 4
symbols depending on the user’s spreading gain. Both randomly generated codes and
codes which satisfy the constrained given in 4.2.3 were considered. For the method

presented in Section 4.2.3, the channelization codes were chosen to be:

c1 = [0.6857, 0.7145, 0.6356, 0.6849, 0.8433, 0.8036, 0.7597, 0.5856,

0.7488, 0.5641, 0.7300, 0.7542, 0.7482, 0.5870, 0.7902, 0.6172,

0.5409, 0.5474, 0.6425, 0.7834, 0.7520, 0.6743, 0.6904, 0.8114,

0.5829, 0.6913, 0.5939, 0.7339, 0.8608, 0.6380, 0.8207, 0.8808], (4.74)
c2 = [0.6670, 0.7275, 0.8540, 0.6100, 0.7518, 0.6363, 0.5545, 0.6887,

0.7092, 0.6143, 0.6313, 0.7625, 0.5210, 0.8036, 0.7582, 0.6979,

0.8136, 0.6944, 0.6902, 0.6660, 0.6536, 0.6908, 0.6010, 0.8078,

0.7622, 0.5486, 0.6005, 0.6395, 0.6176, 0.8070, 0.6382, 0.8265]. (4.75)

The multipath channels have 3 rays, whose amplitudes are Gaussian with zero
mean and identical variance. The transmission delays uniformly spread over 4 chip
intervals. Complex zero mean white Gaussian noise was added to the received signals.
The normalized mean-square—error of channel estimation (CHMSE) for the desired

user is defined as
(p) _ (p) 2
1 g1 ll

 

I K .
1 H g

CHMSE = —— E E
K M n 4» H2

(4.76)
where I stands for the number of Monto Carlo runs, and K is the number of receive
antennas. SNR refers to the chip level signaLto—noise ratio with respect to the desired
user, and is chosen to be the same value at each receive antenna. The normalized mean
square error of symbol estimation and the bit error rate (BER) were used to evaluate
the performance of the equalizer. The result was averaged over I = 100 Monto
Carlo runs. The channel was generated randomly in each run, and was estimated

based on a record of 256 symbols. In the case of multirate, we mean 256 lower

rate symbols. The equalizer with length L. = 6 was constructed according to the

76

 

—a— Without MP. N=16
-8' - —-9—WithoutMP,N=8 ~
_9_ , +wrm MP, N=16
-v-With MP,N=8
-10. ..

 
    

 

 

 

I
_s
.5

MSE of Channel Estimation (dB)

 

 

 

 

5 1o 15 2o
sun (dB)

Figure 4.7. Normalized MSE of channel estimation versus SNR, single rate cases with
N=16 and N=8 respectively

estimated channel, and is applied to a set of 1024 independent symbols in order to
calculate the symbol MSE and BER for each Monto Carlo run. Channel estimation
based on nonconstant modulus precoding was carried out both with and without the
matrix pencil approach. Without matrix pencil approach, the channel estimation
was obtained directly through the second—order statistics of E(n) based on the non-
constant precoding technique. Figure 4.7 and Figure 4.8 correspond to the single rate
cases, where both users have spreading gain N = 8 or N = 16, and the channelization
codes are given by (474-475). Figure 4.9 and Figure 4.10 compared the performance
of the matrix pencil based approach when different codes were used. In the figures,
”codes with constraint” denotes the codes in (4.74—4.75), and we choose N = 8 for
the higher rate user and N = 16 for the low rate user. The simulation results show
that, by introducing a random linear transform, the matrix pencil approach delivers
significantly better results for both single rate and multirate systems.

Table 4.1 shows the average time in seconds per Monte Carlo run for both with
and without matrix pencil approach while the processing gain of both users is 16. The

computer used in this simulation is a Dell Dimension 4550, P4 2.8GHz, 1G RAM.

77

 

' -—a-——Without MP. N=16
—0— Without MP. N=8
‘ +With MP, N=16
+ With MP. N=8

      

 

 

Bit Error Rate

 

 

10-5 A i i

 

10 1 5 20
SNR (dB)

Figure 4.8. Comparison of BER versus SNR, single rate cases with N=16 and N=8
respectively

I
.2
_.

 

 

 

 

 

 

 

4 —9- Codes with constraint. high rate user. N=8
42: ~ —9— Codes with constraint. low rate user, N=16
—-er— Random codes. high rate user, N=8
g '13 ’ rv— Random codes. low rate user. N=16
§ —14 -
E —15 -
.C
o
'5 —17 -
Lu
0)
5 _13 L
-19 i.
_20 . . .
0 5 15 20

1 0
SNR (as)

Figure 4.9. Normalized MSE of channel estimation versus SNR for matrix pencil
based approach with different codes, multirate configuration with N = 8 for the high
rate user, and N = 16 for the low rate user.

78

 

 

.1 'iiffiiffiififffifififf -—B—Codeswithconsiraint.highrateuser.N=8}
It;_:'-‘j“:j:::::j:_'j_‘ —9—-Codeswithoonstraint,Iowrateuser.N=16‘
' —A—Fiandom codes. high rate user. N=8

      

 

 

 

Bit Error Rate
a
a.

 

-4

1O

 

 

 

SNR (dB)

Figure 4.10. Comparison of BER versus SNR for matrix pencil based approach with
different codes, multirate configuration with N = 8 for the high rate user, and N = 16
for the low rate user.

 

 

 

 

 

 

 

Channel Order 5 9
Number of receive antennas 4 3 4 3
With MP (second) 15.59 13.11 34.86 26.86
Without MP (second) 15.57 12.99 34.06 26.79

 

 

 

 

Table 4.1. Average time per run

79

4.3 Blind Equalization for Uplink Long-Code
CDMA Spreading Sequences with Constant
Modulus

The drawback of the approaches presented in section 4.2.3 and section 4.2.4 is that
the spreading codes need to be non-constant modulus, which causes inconvenience
for practical design. To overcome this obstacle, instead of using only second-order
statistics as in [75,113], in this work, higher-order statistics are exploited as in [104]
so that multiuser separation no longer requires the spreading code be non-constant
modulus. In this section, after the system model is transformed to the instantaneous
mixture mode, as in [104, 114], joint approximate diagonalization of eigen-matrices
(JADE) algorithm is used to estimate the channels. MMSE equalizer is designed
to recover the input signals. In the third generation wireless systems, multi-rate
transmission is required to support variable quality of service. Because the chip level
MIMO model is applied in this research, the proposed method supports multi-rate

transmission inherently.

4.3.1 Blind Channel Identification using JADE Algorithm

In this section, {c,,,(n)},",f=1 are assumed to satisfy |cm(n)| = 1. From (4.44), we
have Rs(n, k) = I M606). Therefore, from (4.40) and (4.46), Rya (n, k) and Ag], are
independent of n. It can be obtained from (4.49) that

RE(n) = HH". (4.77)

Since RE(n) is independent of n, we define RE = HH”. Except for the one user
case, H cannot be uniquely determined from (4.77), since Fifi” = HEB” 1:1” for any

unitary B. Thus, higher order statistics has to be exploited. Consider the following

80

instantaneous mixture problem:

.-

E(n) : Hs(n). (4.78)

JADE algorithm [114] is a higher-order-statistics based method used to separate H

and s(n) blindly based on the following assumptions:

(H1) The processes {31(n)}, {32(n)}, ---, {sM(n)} are jointly stationary.

(H2) There is at most one signal source has a zero kurtosis.

(H3) The columns of H are linearly independent.

(H4) The variables 31(n), 32(n), ~ ~ -, sM(n) are statistically independent for each n.

(H1), (H2) and (H4) can be satisfied with the general assumptions (A1)-(A3) given
in Chapter 2, while (H3) requires that the channels corresponding to different users
be independent.

First, find an M x (KL) whitening matrix W which satisfies IM = WREWH =
WHHHWH. Let A.- (i = 1, . . . , M) denote the eigenvalues of RE, and v,-
(i = 1, . . . , M) denote the corresponding orthonormal eigenvectors. We choose
W = P’IVH, where I‘ = diag(\/:\:,...,\/;\E) and V = [v1,...,vM]. Then the

whitening process is carried out as
z(n) := WE(n) = WHs(n) = Us(n), (4.79)

where U := WH. Let QZ(B) denote the M x M cumulant matrix associated with a
M x M matrix B. The (i, j)th entry of QZ(B) is defined by

(16: Z cum(z.-,z;.zk,z;)bz., (4.80)
k,l=1,IW

where 2m is the mth entry of vector z, and buc is the (l, k)th entry of B. Let a,- denote

the M x 1 vector with 1 in the ith position and 0 elsewhere. Define the set of parallel

81

cumulant slice as

S := {Q.(a,a{.)|1 g 14,], _<_ M}. (4.81)

A joint diagonaliser of set S is defined as the a unitary matrix which maximizes the

criterion

0W) = Z Idiag(V”Q.(a.a£)V>I2. (4.82)

131:.ng
As proved in [114], a joint diagonaliser of set S is essentially equal to U (Matrix
A and C' are said to be essentially equal when A = CP, where P is a permutation
matrix).
Let 0 denote a matrix that is essentially equal to U, then the estimate of H can
be obtained as H = W#U. If we select the equalizer length Le = d, the MMSE
equalizer with delay d is given by

F. ;= [Ff(0),FeH(1), . . . ,Fe”(Le —1))H = RchIEI (4.83)

where Ru)“ := E{[yT(n), . . . ,yT(n -— Le +1)]T[y(n),... , y(n -— Le +1)]}. The equal—
ization output is given by

Le—l

s(n — d) = Z F6H(k)y(n — k). (4.84)
k=0

Because U is essentially equal to U, H is also only essentially equal to H. That
means we have no knowledge about which entry of 6 belongs to a specified user. By
calculating the correlation of each component of s with the overall spreading code of

every user, the estimated symbol stream of each user can be identified.

4.3.2 Simulation Example (Constant Modulus)

In this section, a simulation example is provided to illustrate the proposed approach.
Uplink CDMA systems with four receive antennas and two users were considered.
Each user transmitted QPSK symbols with equal power. The spreading sequences

and scrambling sequences were randomly generated. The processing gain of the basic

82

 

 

-2 . . L T
i f —e— High rate user, N=16
‘ ' —Ei-- Low rate user, N=8

 

 

 

MSE of Channel Estimation (dB)

 

 

 

 

SNR (dB)

Figure 4.11. MSE of channel estimation, two users, N = 16 for the low rate user and
N = 8 for the high rate user, 100 Monte Carlo runs and 1024 symbols per run.

rate user is N = 16. Each subchannel has four paths, which are uniformly distributed
in one basic-rate symbol period. In this example, the additive noise is white Gaussian.
SN R refers to chip level signal-to—noise ratio with respect to the desired user. The
equalizers were designed based on 256 symbols, and were applied to 1024 symbols to
calculate the normalized symbol mean square error SMSE. SMSE and MSE of chan-
nel estimation (CHMSE) were further averaged over 100 Monte Carlo runs. In this
example, one user transmitted at basic symbol rate, and the other user transmitted
at two times the basic rate. The estimation and equalization results are shown in
Figure 4.11-4.13. Table 4.2 shows the average time in seconds per Monte Carlo run

to extract the low rate user for both with and without matrix pencil approach.

 

Channel Order 15 7
Number of receive antennas 4 3 4 3
Average time per run (5) 3.95 2.18 1.59 1.19

 

 

 

 

 

 

 

 

 

Table 4.2. Average time per run using MSLP+JADE approach.

83

 

 

E . —9— High rate user, N=8
' —a— Low rate user, N=16 ‘

 

MSE of Symbols (dB)

 

 

 

 

5 10 15 20 25
SNR (dB)

Figure 4.12. MSE of symbol estimation, two users, N = 16 for the low rate user and
N = 8 for the high rate user, 100 Monte Carlo runs and 1024 symbols per run.

 

 

—e—
' - —-B—- Low rate user, N=16

 

Bit Error Rate

 

 

 

 

1
SNR (dB)

Figure 4.13. Bit error rate, N = 16 for the low rate user and N = 8 for the high rate
user, 100 Monte Carlo runs and 1024 symbols per run.

84

4.4 Summary

In this chapter, multistep linear prediction based methods are proposed to perform
blind channel estimation and equalization for long-code CDMA. Both uplink and

downlink are considered:

0 Downlink First, chip—rate equalization is performed to recover the scrambled
signal. Secondly, after descrambling, the descrambled signal is regarded as the
channel output of a short-code CDMA system with unknown channel param-
eters, and SEA method is applied to recover the signal of the desired user.
The simulation results show that, the approach with SEA enhancement deliv-
ers much better performance than the approach that perform direct despreading
after the descrambling procedure. It also can be observed that, when the re-
ceiver is equipped with multiple antennas, the performance can be improved

significantly.

0 Uplink In the uplink, if the spreading codes are nonconstant modulus, the
transmission induced cyclostationarity make it possible to conduct blind chan—
nel estimation based on the second-order statistics (SOS) of the received signal.
In this chapter, two SOS approaches are developed. One is based on the Fourier
analysis and strict spreading code design, and the other one is based on the ma-
trix pencil method. It was shown that the matrix pencil approach can relax
the conditions on the spreading codes, and can deliver significantly better per-
formance. If the spreading codes are constant modulus, higher-order statistics
have to be exploited. In this chapter, the JADE (joint approximate diago-
nalization of eigen—matrices) algorithm proposed in [114] was applied for blind
channel estimation. Compared to the methods exploiting transmission induced
cyclostationarity, the MSLP—l-JADE approach developed for the system with
constant modulus spreading codes has better performance, since higher-order

statistics are exploited.

As chip level channel modeling and equalization are performed, the proposed approach

can be extended to multirate CDMA systems in a straight forward manner.

85

CHAPTER 5

Blind Equalization of Space-time

Block Coded DS-CDMA

This chapter considers blind channel estimation and signal detection in space-time
block coded CDMA systems. Because of the size and complexity limitation of the
mobile devices, it is more practical and economic to apply multiple antennas at the
base station. Therefore space-time coding is only considered for downlink systems in
this chapter. First, in order to achieve maximum transmission diversity gain, after
spreading and scrambling, time-reversal space-time block coding (TR-STBC) [81—83]
is applied to formulate a two-branch transmission. Only one spreading code is as-
signed to each user, and the time-reversed blocks can be regarded as spread spectrum
signals whose spreading code is in the reversed order as the regular block in the same
coding block. Secondly, motivated by [65], a blind channel estimation approach is
developed based on the principal component algorithm. The major advantage of this
approach is that it can mitigate all the interference items (including multipath inter-
ference, multiuser interference, and the noise) simultaneously and effectively, and can
achieve good channel estimation even at low SNR levels. Finally, after the channel
estimation, MMSE equalizer was applied to the matched filter output to recover the

transmitted information symbols.

5. 1 Transmission Scheme

In this chapter, TR-STBC technique is applied to achieve maximum transmission

diversity. The transmission scheme is illustrated in Figure 5.1. Similar transmission

86

 

Space‘time
‘ block

coding

 

 

 

S .

32,}. (n)

Spreading

 

Figure 5.1. Downlink Transmission Scheme

scheme can be found in [83] and [84].
Consider a long code DS—CDMA downlink system with M users. The base station
has two transmission antennas, and each mobile station has K (K = 1 or 2) receive

antennas. Let um(k:) denote the kth symbol of user m, then the spreading result is

given by
mm) := Z um(k)cm(n — kN), (5.1)
k=r-oo
where cm(n), n = 0, . . . , N - 1, is the spreading code of user m. The summation of
M

the spread spectrum signals a(n) = Z am(n) is scrambled by a pseudo-random

rn==l

sequence d (n):

v(n) = a(n)d(n). (5.2)

Collect PN consecutive scrambled chips to a 1 x PN block v(b):

v(b) = [vb(0),vb(1), . . .,vb(PN — 1)], (5.3)

where vb(i) = v(bPN + i),i = 0,... ,PN — 1. After prefix and postfix insertion, the

resulting block can be represented as:

V'(b) = [Vp(b),V(b),Vo(b)la (5-4)

where vp(b) and v0(b) are vectors that denote the prefix and postfix of the bth block,

respectively. Let L represent the maximum length of the channels, vp(b) and vo(b)

87

O-

I l
Time (n) l0 PN’h

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I
x PN— 11 A
J 1 r 1_ __ n '
Coding l, l SL1 I l SL2 I
block [g IT 521 I I 523 I
Prefix . [Postfix Prefix] i Postfix
l l I l
Received ] T
signal I l
I) (l) (2) (2)
y].(0) ...... Y,- (PN+L—2) y} (0) ...... y)- (PN+L—2)

Figure 5.2. Data Structure of the Transmitted Signal

are chosen to be 1 x [L — 1] vectors to remove inter—block interference completely.
Next, two consecutive vectors v’ (2b) and v’ (2b + 1) are used to form a space-time

coding block given by:

51,1 81,2 __ V,(2b) -V" (2b + 1)TpN (5 5)
82,1 82,2 V’(2b + 1) V’*(2b)TpN .
where * denotes complex conjugate, and
"' W
0 0 1
0 1 0
T... = . . . . (56>
1 0 -- - 0
. .PNxPN

 

 

is a permutation matrix, which performs a time reversal of the 1 x PN vector v"(b)
through the operation v’*(b)TpN. The two rows of the coding block in (5.5) are trans-
mitted from the two antennas respectively. Here and in the subsequent discussions, b
is omitted from the notation for simplicity. In order to discriminate the information

chip sequence from the prefix and postfix, the entries of SM is denoted by
Si,j : [81.j(_L +1)1Si.j(—~L + 2), - - - 181.1(PN + L — 2)]a (5'7)

where 31'.) (n), 0 _<_ n S PN — 1, is the original information chip sequence before adding

l I 1
Time (n) L0 PN‘ll #0 PN‘li >

 

 

 

 

 

Coding
block '

ifief' [Ref‘ Pnfiix

(Postflfir) i ii’ostfif‘xl ] (PostfiX)

I l I I
Received [ ] 1
signal 1 I /\ 2 2)
w(n) ...... y;>(PN+L—2) yj. )(0) ...... y]. (PN+L-2)

Figure 5.3. Data Structure of the Transmitted Signal using Shortened Prefix (Postfix)

the prefix and the postfix. The data structure of the space—time coded block is further
illustrated in Figure 5.2.

In general cases, the length of both the prefix and the postfix should be no less
than the channel order to avoid inter-block interference. For the channel estimation
method presented in the next section, in order to approximate the correlation matrices
using time average without edge effect, the prefix and postfix should be chosen to have
the same second-order statistical properties as the chips that bearing information bits.
One simple method is to use cyclic prefix and postfix. That is, the prefix vp(b) is
chosen to be the last L — 1 symbols of v(b), and the postfix vo(b) is chosen to be the
first L — 1 symbols of v(b). The total length of the chip sequence inserted between
two information blocks is 2(L — 1) as illustrated in Figure 5.2.

If no special statistical property is required for the prefix and the postfix, shorter
prefix and postfix can be applied. It is mentioned in [115] that, when the prefix and
the postfix possess certain conjugate symmetry properties, they can be merged into
one sequence of length L — 1. The coding structure with totally L - 1 chips inserted
between two information chip sequences is illustrated in Figure 5.3. The coding block

should still satisfy:

81,2 = —S;,1TpN (5.8)

52,2 = Si,1TPN (5-9)

89

Therefore, we have

[81‘2(—L +1),81,2(—L ‘l' 2), ' ' ' , 51,2(—1)]

= —[s;'1(PN+L—2),s;,1(PN+ l),~--,s§.1(PN)] (5.10)
[52,2(-L +1),32,2(—L + 2), - - - , 32,2(—1)]
= [311(PN + L — 2),sI,1(PN +1),-°-,sI‘1(PN)]. (5.11)

From Figure 5.3, [31,2(—L + 1),~ -,sl,2(—1)] and [81'1(PN),H-,81,1(PN + L — 2)]
represent the same sequence, which is the overlapped part of 31,1 and SL2, then it can

be obtained from (5.10) and (5.11) that

[82,1(PN),82’1(PN+1),°'°,82,1(PN+L- 2)]

= —[sz,2(—L +1),32,2(—L + 2):"',82,2(—1)l- (5'12)

Because [32,1(PN), - - - , 32,1(PN+L—2)] and [32,2(—L+1),- - - , 32,2(—1)] also represent
the same sequence, the only choice of the overlapped chips (prefix/postfix) are zeros.
That is , zero padding is the only way to reduce the length of the postfix/ prefix, and
satisfy (5.8-5.9) strictly at the same time.

For a fixed mobile device, let {gfp)(l)}{’=_01,i = 1, 2 denote the channel between
the ith transmit antenna and the pth receive antenna, then the received signal corre-

sponding to block b at the pth antenna is given by

2 L—l
gym) = 22g?)(l)si,j(k—l)+w],j)(k),
i=1 (=0
(ogkgPN+L—zj=12) (am)

where wéj) (k) is the additive white Gaussian noise at the pth receive antenna. Define

the received signal array as:

We) := Millie). 49’s). . . . 4W)?
mgkgPN+L-m, (an)

90

and define the channel vector

4(1):: lgf‘iz). glglll), . . . ,gEK’mlT. (5.15)

then we have

y(j)(k) = Z g,(1)s.-,,(k -— l) + w(j)(k), (5.16)

where w(j)(k) is defined in the same way as y(j)(k).

5.2 Blind Channel Estimation

In this section, based on the principal component algorithm [65], a blind channel
estimation approach is developed for space-time coded long-code DS-CDMA system.
The basic idea can be summarized as: (i) The time reversed chip sequence is still a
spread spectrum signal, which implies that after space-time block coding, the trans-
mitted signal from each antenna can be regarded as a DS—CDMA signal with aperiodic
spreading codes. (ii) For DS—CDMA signals, the auto-covariance matrices of the re-
ceived signal before and after despreading differ only by a rank one matrix, which is
completely determined by the channel impulse response vector. The channel impulse
response, therefore, can be estimated by calculating the principal eigenvector of this
rank one matrix.

For simplicity, in the following discussions, we use c]?)(k,n),n = 0,. . .,N — l
to represent the overall spreading code for the kth symbol of user m carried by the

sequence Si‘j. Despread y(j)(k) at different delays:

22:01 y(j)(kN + n + L —— 1)c[?)(k, n)

N_1 (j) . (m) (5.17)
Z7120 y (kN + n +1)c,-,j (k,n)

_ 22:01 y(jl(kN + n,)(:]?’)(k, n)

 

 

91

Define

yiflbl) == [(y“"l(n))73(y‘~“(n—1))T,---.(y”’(n - L +1))TlT,
(L—lgngPN+L—2) (5.18)
Y‘Wk) := M,” (W + L— 1,) (ij)(kN + L). ..,y‘Lj)(kN + N + L — 2)](519)

then the despreading result can be represented as

 

 

xifyle) = Y<3><k)c.‘-f;')(k). (5.20)
where
c§7>(k) =[c§j")( k ,,0) 653;”(14, 1),...,c§j;?>(k,1v —1))T. (5.21)
Define
f a(n) 4.0) g (L — 1) o 0 ‘
H, := (1 gp(0) s50) - 31201-1) (3 (5.22)
L 0 0 3pm) 310(1) 310(14‘ 1) _
and
S2007.) = [SI-JUL), 83"](7'1. — 1), . . . ,sm-(n — 2L + 2)]T, (5.23)

then it can be obtained that

ng s,,-(n )+ wg)(n ), (5.24)

where w(Lj)(n ) is defined in the same way as y(g)(n ). To further analyze the received
signal, let s(")(n ) ands ’(m ?(n ) denote the contribution of user m to Sid and §,,j(n)

respectively, then y(Lj)(n ) can be decomposed as

92

mu) : sff’.”<n—L+1)g.+

2 35.3%. — k + 1)h,-,,. + Z Z Hqsgffln) + ng>(.n), (5.25)

 

5;”. kgém (#4 4
17;)
where
s.- = [53(L - 1), 53114 - 2), . . . .gf(0)lT, (5.26)

hi), represents the kth column of matrix H,, and I (n) stands for the inter-chip inter-
ference, multiuser interference and the noise.

Calculate the covariance matrix of the received signal before despreading, we get

Ryy = E{Yi?)(n)(yf)(n))”} = as.” + Rn, (5.27)

where R11 = E {I (n)I H (n)}, and ()H represents complex conjugate transpose. From

(5.19), the covariance matrix of YU)(k) is given by

Ryy = E{YU)(k)[Y(”(k)l”}

= NEW
= Nag,” + Nan. (5.28)

From (5.20) and (5.25), it follows that the despread signal can be decomposed as

N—l
x]3')(kt) = Nuffy’(k)g.- + Z I(KN + L — 1 + z)c§j;"(k, z), (5.29)
(=0

where ’tl(?)(k) denotes the kth symbol of user m in block 5):). Therefore, we have

i.

w

R... = E{x‘"-"<k)<xf-T;”(k))”} = N255.” + Nan. (5.30)

93

The difference between R” and Ryy is given by
R... — Ryy = N(N — 1)g,g,”. (5.31)

From (5.31), clearly, the channel impulse response g,- can be estimated up to a scalar

through eigen-value decomposition.

5.3 Signal Detection

In this section, an equivalent system model is derived based on the time-reversal
coding structure, and the MIMO matched filter is used to decouple the two chip
sequences contained in one coding block. After space-time decoding, the MIMO
system is transformed into two SIMO systems, and MMSE equalizer is designed to

recover the signal.

5.3.1 Space-Time Decoding

Since the TR—STBC scheme is applied at the transmitter end, at the receiver, the
signals transmitted from the two antennas can be decoupled by the MIMO matched
filter. Define the received signal block corresponding to coding block b as

I‘d") = y(1)(n)
r2(n) = y*(2)(PN + L — 2 — n)
(n=0,1,...,PN+L—-2). (5.32)

From (5.16), the z—transform of r1(k) is given by

r1(z) = g1(z)sl,1(z) + g2(z)sg,1(z) + w(1)(z). (5.33)

94

For a causal sequence a(k), the z -transform is defined as (1(2) = 2,31,, (1(1))Z—ks “1911

the z—transform of a”(l.‘.) is given by

a*(:) = Bessy—k. (5.34)
Therefore the z—transform of r2(k) can be deduced as

y‘2’(k) -* y‘2’(z)

= y‘2’(-k) -—>y‘2’(2“)
=2 y(2)(PN + L — 2 — k) —_. z~PN-L+2y(2>(z-1)
=> r2(k) = y*(2)(PN + L — 2 — k)

—> r2(z) = z”PN—L+2y‘(2)(z_l). (5.35)
From (5.16) and (5.5), it follows that

y(2)(2) = 510081.20?)+g2(2)82.2(z)+W‘2)(3)

-Z'PN+lgi(Z)8§,1(Z’1)

+

z’PN+1g2(z)sI,1(z_1)+ w(2)(z). (5.36)
Finally, r2(z) has following expression
12(2) = Z7L+1l-gi(2'1)82.1(2) + EEO—981.1(2)] + Z’PN-L+2W(2)’(Z‘l)- (5-37)

Define r(k) = [r{’(k),r£’(k)]”, then we have

r(z)= “(2) 211(2) 81"”) +542), (5.38)
r2(z) 82‘1(Z)
where
H(z) ___ g1(2) g2(z) (5.39)

w“) 2
Me) = H] ( ) )]. (5.40)

__)..'_ _
3 [A L+2w(2)4(: 1

Because u'],j)(k) is white gaussian, from the definition of w(j)(k), j = 1,2, we have

wa(k)- — E{w(n) (n — k)}— - 0216(k ), and wa(z ) = 0:1. From (5.39), the
MIMO matched filter 18 given by
H -—1 L-l gT
H”(z‘1) = 31(2 ) 2 2(2) (5.41)
g§(z“‘)- 2” left?)

It can be observed that H(z) has the following orthogonal properties:

1:1”(Z_1)fi(zl= [31(2 1)g1(z)+g£{(z_l)g2(z)]12, (542)

where 12 denotes the 2 x 2 identical matrix. Thus, the signals can be decoupled using

the MIMO matched filter rim-1), that is

 

q(z)= (“(2) =HH(z_l)r(z)=h(z) 314(2) +w’(z), (5.43)
(12(2) 32.1le
where
5(2) := gi’<z‘1)g()+ge (z ‘)gez() (5.44)
vi/(z) := [103(2)] =HH(z—I)W(z). (5.45)

From the fact wa(z) = 0,2,1, we have

00

R1...<) = Z E{w 'nvv<)l (n— kll”}z"‘

k=-oo

= HH(z—1)R441(Z)H(z)
= 03H”(z“l)H(z)
= 02])(3)I (5.46)

96

Therefore 1113(3) and 111.3(2) are uncorrelated. and the detection 1i)1'ol)len1s are com—

pletely decoupled.

5.3.2 MMSE Equalization

After decoupling, the transmitted signals, 51,1 and 521, have been separated from each
other, while the decoupled results are still distorted by a linear filter h(z). Based on
the channel estimation result, ML detectors can be applied to estimate the original
input. Considering the complexity of the ML approach, in this section, an MMSE
equalizer is designed to recover the information chip sequences. Since 31,1(k) and
32,1(k) have the same statistical properties, from (5.43), both of them can be estimated
using the same equalizer. Let f (k), k = 0,. . . , L, -—1 denote the equalizer coefficients,

the equalization output is then given by:

Le-l

Si,(kl =:f(l)

(2' = 1.2), (5.47)
where r is the equalization delay. Define
f— — [f(O) f(1)fe(L — 1))T (5.48)
and
w(n) = [w(n), (It-(n -1),.--.qt(n — Le +1)]T. (549)
then the MMSE equalizer is given by
_ # ~
f — quh,, (5.50)

where qu- — E{q,(n n)q,” (,n)} h =[h(r), . . . , h(0), 0,... ,0], and R3; represents the
pseudoinverse of matrix qu.
After equalization, the information chip sequence v(2b) and v(2b + 1) can be

recovered, and the symbol sequence of the desired user can be estimated through

97

descrambling and despreading.

5.4 Simulation Examples

In this section, simulation examples are provided to illustrate the proposed approach.
We consider a downlink CDMA system with 8 users. The spreading gain is selected
to be N = 16. The mobile stations with both one and two receive antennas are
considered. QPSK signals are transmitted from the base station. Each space-time
coding block contain 16 information symbols (P = 8). Two kinds of prefix and
postfix are tested. One is simply zero padding. For the other one, the prefix vp(b)
is chosen to be the last L - 1 symbols of v(b), and the postfix vo(b) is chosen to
be the first L — 1 symbols of v(b). 256 symbols are used to perform blind channel
estimation, and the equalization is carried out for 1024 symbols. The MSE of the
channel estimation and the BER are averaged over 100 Monte Carlo runs for different
SNR levels. The channels are generated randomly and independently for each run.
Each multiple channel has 3 rays, which are uniformly distributed over 5 chip periods.
The performance of the proposed approach is compared with the approach without
space-time coding. For the later case, only one transmission antenna is used, and the
same channel estimation method is applied.

The simulation results are shown in Figure 5.4 and Figure 5.5. It can be observed
that, by introducing transmission space diversity, the proposed scheme delivers much
better performance compared with conventional single transmit antenna scheme. Two
kinds of prefix and postfix have similar channel estimation results, but the transmis—
sion scheme using zero padding delivers better BER performance, as less interference

is introduced in this case.

98

 

    

 

 

51‘

E

C

O

'a

«s V 4*-

.§ V

‘15—12- .. . ,. ,. ..
Iii —e— TR—STBC: 1 receive antenna, CP

2 —B— TR-STBC: 2 receive antennas. CP
5 ‘13' ' ' ; +TR-STBC: 1 receive antenna.ZP ‘
o + TR—STBC: 2 receive antenna, 2P

3 _14.. . . . +w/o STBC: 1 receive antenna -
g ‘ —v—— w/o STBC: 2 receive antennas

 

 

    
  

 

 

 

 

1 0
SNR (dB)

Figure 5.4. MSE of channel estimation. Spreading gain N = 16, and number of users
M = 8.

 

 

 

 
   
 

g ........

a ‘ f:

g .....
Lu _4' ' TR-STBC' 1 receive entenna.ZP ”””
E 10 5: - »

 

E:: :1531‘ 1? »w/oSTBC:2receiveamennas§

 

 

 

-5
1o " ....... :7
‘TFl-STBC: 2 receive antennas. CF """"" ,
' 'TR-STBC: 2 receive antenna. ZP")
10.5 I L
0 15 20

10
SNR (dB)
Figure 5.5. Bit error rate. Spreading gain N = 16, and number of users M = 8.

99

5.5 Summary

In this chapter, a blind signal detection approach based 011 the principal component
algorithm is presented for TR-STBC DS-CDMA systems with aperiodic spreading
codes. Compared with the scheme without space-time coding, significant gain can be
obtained by introducing transmission diversity. Both cyclic prefix/postfix and zero
padding were discussed and tested with simulation. While similar performance can
be observed using both kinds of prefix/postfix, it is more spectral efficient to use
zero padding. It is shown that the proposed approach is not only very effective in
mitigating multipath interference and multiuser interference, but is also robust to

additive noise.

100

CHAPTER 6

Conclusions and Future Works

6. 1 Conclusions

The research focus of this dissertation is blind signal detection for DS-CDMA over
frequency selective channels. Statistics based methods are designed for both short-
code and long-code CDMA systems. A fast HOS based approach — MR—CSEA is
proposed for multi-rate short-code systems, and MSLP based chip-rate equalizers are
designed for long-code systems. Blind equalization approaches have been developed
for space-time coded downlink long-code CDMA with maximum transmission diver—
sity. Based on the theoretical analysis and simulation results, we have the following

conclusions:

0 Characterization of CDMA systems

When modelling with symbol-level inputs and chip-level outputs, the short-code
DS-CDMA can be characterized with an MIMO time invariant model, while the
long-code DS-CDMA can only be characterized by time variant model because of
the time variant nature of the spreading code. By taking the chip-rate scrambled
signal as the input, the long-code DS-CDMA can also be modelled as a time

invariant system.
0 Equalizer design for multirate short-code systems

— For both MC and VSL schemes, the signal of a high-rate user can be

regarded as the sum of the signals of several virtual users.

— The code structure can be exploited to recover the symbol—rate signal

101

blindly. and only the spreading code of the desired user is assumed to

be known.

— By using proper initialization, the signals of different virtual users can be
recovered in correct order, which is necessary for restoring the sequence of

the corresponding high-rate user.

— The proposed MR—CSEA approach has fast convergence speed and supe—
rior performance. The simulation results show that, when the chip SNR
is larger than 15dB, the proposed approach has much better performance,
and converges much faster than existing blind approaches. It can be ob—
served that, at high SNR, the proposed method converges to the MMSE

equalizer with known channel information.

o Two-stage blind channel estimation and signal detection for long-code systems

MSLP based blind channel identification methods have been developed for both
uplink and downlink long-code CDMA. First, convolutive mixtures are con-
verted to instantaneous mixtures. Secondly, both second-order-statistics based
and higher-order-statistics based approaches can be developed for blind signal

detection.

— Blind signal detection for downlink long-code systems

After chip—rate equalization, the descrambled signal can be modelled as
the output of a short-code CDMA system. By using this method, the

performance can be improved significantly.
— Blind signal detection for uplink long-code systems

* When the spreading codes are nonconstant modulus, the cyclostation-
arity introduced by the transmitted signal can be exploited to per-

form blind channel estimation based only on the second-order statistics

(SOS) of the observed data.

... By choosing the spreading codes that have special properties in fre-

quency domain, Fourier analysis method can be used to perform blind

102

channel estimation. This method is relatively simple, but additional

constraint is required.

* By introducing a random linear transform, the matrix pencil approach
can remove the extra constraint 011 the code sequences. Simulation re—
sults show that the matrix pencil based approach delivers much better

result than the one relying on the Fourier transform.

* If the spreading codes are constant modulus, higher-order statistics
based methods have to be applied to exploit the phase information.
In this research, JADE algorithm is applied to identify the channels
blindly.

* As chip level channel modeling and equalization are performed, the
proposed approach can be extended to multirate CDMA systems in a

straight forward manner.

0 Blind equalizer design for space-time coded downlink CDMA

— After space-time coding, the two chip sequences assigned to the two trans-
mission antennas can be treated as two long—code CDMA sequence with
prefix/postfix inserted. In the time interval between two consecutive pre-
fixes/pofixes, the spreading sequences of the two transmission antennas
are independent. Based on this observation, statistical method can be

developed to perform blind channel estimation.

— Both cyclic prefix/postfix and zero padding can be used to suppress inter-
block interference, and similar performances have been observed. However,
using zero padding is more spectrally efficient, as the length of the pre-
fix/postfix could be reduced by half.

- Significant improvement can be achieved by introducing transmission di-

versity.

103

6.2 Related Future Works

In this section, possible future research directions are discussed.

0 Further discussions on blind channel estimation of downlink CDMA with TR-
STBC' scheme.

In this thesis, the blind channel estimation is performed for downlink CDMA
with TR—STBC scheme. While the signal components of all active users share
the same channel impulse responses between the base-station and the received
antennas of a given mobile device, only the signal component of the desired
user is analyzed to perform blind channel estimation. The performance can
be improved by exploiting the statistical properties of all signal components.
Because two transmission antennas are applied to the TR-STBC scheme, the
MSLP algorithm developed for conventional downlink systems can not be ap-
plied directly for blind channel estimation, since there are two channels need to
be estimated. Because each antenna can be treated as a virtual user, algorithms
designed for uplink systems may be extended to this case, and simulations need

to be carried out for performance analysis.

0 Blind signal detection for space-time spreading scheme

Applying spacial diversity is a trend in next generation wireless communication
systems. Alamouti space-time block coding scheme has been specified in W-
CDMA standardization, while space-time spreading (STS) scheme [116], which
includes the STBC scheme as a special case, has been adopted in IS-2000 stan-
dard. So far, blind signal detection methods designed for S T S scheme has rarely
been observed. Consider a CDMA system with STS transmission scheme. Let
u(k) denote the kth symbol of the transmitted stream, then the signal trans—

mitted from one antenna is given by

sl(k) = (u(2k)cl + u(2k +1)c2), (6.1)

L
\/'2'

104

and the signal transmitted from the other antenna is given by

52(5) 2‘: —1—(U,(2k +1)C1‘- H(257)C2), (0.2)

t/2

where C1 and c2 are 1 x 2N spreading sequences. It can be observed that
every symbol is spread by both spreading sequences. Therefore, it is difficult
to perform blind signal detection by exploiting the code structure. Blind signal

detection of STS scheme should be considered from a totally new viewpoint.

105

APPENDICES

106

APPENDIX A

List of Abbreviations and

Acronyms

AN MSE
AMPS
CDMA
CM
CMA
CSEA
DS-CDMA
FDMA
FIR
GSM
HOS
IFC
IIR

lSI
JADE
J OSC
LCM
MC
.\’IIMO
ML
MMSE

Asymptotic Normalized Mean Square Error
Advanced Mobile Phone Service

Code Division Multiple Access

Constant Modulus

Constant Modulus Algorithm

Code Constrained Super Exponential Algorithm
Direct Sequence Code Division Multiple Access
Frequency Division Multiple Access

Finite Impulse Response

Global System for Mobile Communications
Higher-Order Statistics

Inverse Filter Criterion

Infinite Impulse Response

Inter-Symbol Interference

Joint Approximate Diagonalization of Eigen-Matrices
Joint Optimization with Subspace Constraints
Least Common Multiple

Multiple Code

Multiple-Input Multiple-Output
Maximum—Likelihood

Minimum Mean Square Error

107

M R-CC-IFC
MR-CSEA
MSE
MSLP
MUI
QPSK
SEA

SIC

SIMO
SISO

SNR

SOS
STBC
STS
STTC
TDMA
TR—STBC
VSL

I\-’lulti-rate Code Constrained Inverse Filter Criterion
Multi-rate Code Constrained Super Exponential Algorithm
Mean Square Error

Multistep Linear Prediction

Multi-User Interference

Quadrature Phase—Shift Keying

Super Exponential Algorithm

Serial Interference Cancellation

Single-Input Multiple-Output

Single-Input Single-Output

Signal-to—Noise Ratio

Second-Order Statistics

Space-Time Block Coding

Space-Time Spreading

Space-Time Trellis Codes

Time Division Multiple Access

Time-Reversal Space-Time Block Coding

Variable Sequence Length

108

APPENDIX B

Definitions of Higher-Order

Statistics

Let x1, x2, . . . ,xn be a set of random variables. Their joint characteristic function is
given by
d(w1,.. . ,n}”) = E{ejZ?=1w‘x‘} (RI)

The joint cumulant of Jimflne. . . . ,xnm, n, E {1, 2, . . . ,n} is defined by

m 8m1n¢(w1, . . . , wn)
0w", - - - Bwnm

 

(13.2)

cum{xn,,:ce2, . . . ,xnm} = (-J')

 

w"! =Wn2 ='°'=wflm :0

The mth order joint cumulant cum{xn,,xn2, . . . , xnm} is related to the moments of

33m, . . . ,xn up to order m. For example, if x1,x2,x3,x4 are zero-mean random

m

variables, then

cum(x1 = O

cum(x1, x2, x3

)

cum(x1,x2) = E{Ill'g}

) = E{1311'21E3}
)

cum(x1, x2, x3, x4

= E{IL'1132I31L'4} — E{$1$2}E{$3$4}
= —E{$1$3}E{$2$4} — E{$1$4}E{$2$3} (8.3)

The kurtosis of x1 is defined as

cum4{x1} é cum{x1,x1,x'{,r]} = E{|x1|4} — 2{E{|x1|2}}2 — |E{x?}|2. (B4)

109

The mth order joint cumulant cun'1{x.,,,,xn2,.. .,xnm} has the following important
property: if xn,.x,,._,. . . . .xnm are jointly Gaussian, then their joint. cumulant is zero

whenever m > 2. For the convenience of notation, we denote

cum{x1 :p1,x2 : p2...) = cum{x1,x1,...,x1,x2,x2,...,x2 . . .} (B5)
W W
P11877718 pzterms

110

APPENDIX C

Convergence Analysis of the CSEA
Approach

In this appendix, the equivalent combined channel-equalizer domain representation
of both CSEA and MR—CSEA is given in Section C.1, then the convergence analysis

of both algorithms is presented in Section 0.2.

C.1 Equivalent Representation in the Combined
Response Domain

Let f ("1“) denote the equalizer to extract the desired user mo in a single-rate system, or
the desired virtual user (i0, jg, mo) in a multirate system, then the combined response

in the code-constrained case is given by (see (3.60) and (3.87))
e = (animoghmo). (C.1)

Let d(n) denote the nth component of O, and 999W) stands for the Hadamard power

in complex case , whose nth component is given by
e®<W>(n) = 0(n)p0*(n)q. (0.2)

For the unconstrained case, 9 = HEW). It can be seen that O E range(H).

111

. A ~ . . . .
Define space SA 2 range(H), the orthogonal projectlon operator onto 8,; is g1ven by

PA = H(H”H)#H”. (<13)

If ’PA = I , then it is called the sufficent order case. Otherwise, it is called undermod-
eled case, in which H is not of full rank.

Compared with (3.29) and (3.30), we have

0 Unconstrained super—exponential algorithm:

 

e’ = H(HHHWHHeW‘I)=7>Ae®<M> ((3.4)

,, e’

e = , 0.5
lie I). ( l

. __ 1
For code constrained case, Define space 80A — range(HIIA(,,,o)

). From (C.1), we

can see that G 6 80A. The orthogonal projection operator on to 80A is given by

790.4 = HE((HE)”H HE) #(fiElH
= Hs(HE)#= H(HE)# (0.6)
(since E(HE)#= (HE)# )

where E: Him). Therefore, the equivalent algorithm of CSEA in the combined

response domain can be obtained as

0 Code Constrained super-exponential algorithm:

6’ = (HE)(HE)#9®(”"’) =r>CAe®<W> ((3.7)

H e,

e = —,— 0.8
He He ( )

Since HE is not of full rank, constrained algorithm is corresponded to a special

undermodel case.

C.2 Convergence Analysis

In this section, we prove that the CSEA approach is equivalent to a gradient search

algorithm which maximizes the cost function

_ nenzp 2”
F2p(e)_{||8||2} ’ (09)

with the constraint 9 E SCA, where p = 2, 3, . . . ,00.

Proposition C.1 The super-exponential algorithm (C. 7) (0.8) is equivalent to the
gradient search algorithm defined by

I 1
9 = 9‘"’+W7’CAVG'F2P<G‘"’> (0.10)
P
e,
9(n+1) _ ..__.__I , C.11
”9H2 ( )

For the convenience of calculation, we prove that the following algorithm is equiv-

alent to ((3.7) (C.8)

I 1
8 z 9%") + WPCAVGF2p((-)(n)) (0.12)
p
e]
#(n+1) = 1
e ”all; (C 3)

Proof:

Define: u = ”9H3; v :2 “QUE”, then

2
||9||2§

2
Ilellz”

 

F2p(e) = = g. ((3.14)

Therefore

Veu = V9 2 WWI?” = V9 2 9p('n)(9”(n))’ = [199p’1(n)(9”(n))‘ 210

= p@®(p—1.p) ((3.15)

113

and

W : Ve(Zl0(n)|2)”=p(Zl9(n)l2)”“9*

 

 

 

 

 

 

——— pIIeIIE‘P“”e* (C16)
From (C.14) to (C.16), we have I
V9F2p(@) = v-Veu—éu Vev
v
z 22- 9900-10) - Maui” —puen§‘”"’e' - Mani: (C 17)
(HE-3H3”)? ,,
For (C.12), HGWIIZ = 1, then we have
, n .. 1 ,,
G = (6( l) +p-F2p(9(")) ‘PCA'V9F2p(e( ))
1 ammo—1,1») _9(n)* e(n) 2»
= ( (71))!!! _W . PCA . p (n) 2p 2” “2p
Pile llzp (ll9 “2 )
pCA . e(n)®(p-1.p)
= 900* _ pCA . 902% + (C.18)
Hemllgfi
If 6“” 6 804, then 90‘) 6 80,; => 770,490“) = 9““). Thus from (C.18), we get
, . (n)®(p-1,p)
= 73"" e (0.19)

 

“9"” “33

Notice that ||o<n>||§g is a scalar, so (C.12) (C.13) are equivalent to (C7) (C8) after
we add the normalization step.

El End of proof

Define an 3 Hrmetnfl‘MP-lr)“2, and A, 3 F2p(o<n>). From ((3.12) (C.13) and
(0.19), we have

a1].@*(n+l) = Anedn) + xt(n), (C.20)

114

where

. 1
x*<"> = EPCAVnga-MW). (0.21)
or equivalently:
an@("+1) —_- An@(")+x(") (0.22)
=>9<n+1> = §em>+ix<ni
an on
= ohmic") (0.23)

where

5'6") ____ ._1_x(") + M

an an

e(n). (0.24)

Proposition C.2 With {an}n20, {An}n20 defined as above, we have

0 < An < an < An“ < 1 ((3.25)
and An“ - An > p(an — An). (C26)

Proof:

(a) orthogonal property: (9*, VF2p(€-))) = O for any ”9H2 = 1.
For ||G||2 = 1, from ((3.17), we have

<e*,VF2p(e>> = p<eze®<rlr>> -pF2p<e><e*,e*>
= pllallii—pF2p(G)=0 (027)
(Here <x.y> é Zx(n>y*<n>)

(b) <@:(n),xt(n)> = 0
Since PM 2 795A, (x,PCAy) = (PCAX, y). Thus, from (C21)

1
(9*("),x‘(")) = Ema/(9%"), VF2p(6)("))) = 0 (0.28)

115

(c) From (a) and (b), it follows from (C22) that

afille‘"+”||§ = AEHGWIE + ”)5"ng-
(Note that ||9(k+”||2 :2 ||G("‘l||2 = 1)

0,2. = 131+ ||X‘"’||§

l

l

an 2 A", Vn. (C29)

on = An if and only if x(") = 0, i.e. e(n) is a stationary point.

lnce unction - lS convex, we ave
(d) S' f ' || “33' h
2 2 n 2 -.
lle‘"“’||2£-IIG‘"’II2£ _>_ |<V||9‘ )llzi, x‘"’)l (C30)
2 2 2 ~..
4:» IIG‘"+”II2‘£ 2 IIG‘"’||2£+I<VIIG‘"’II2£, x ‘”’>|
4:, /\n+1 Z An+|(p6(")®(”‘l'p), i190 A(e(n)®(p—1,p)_ Anew)»

+<pe(n)®(p-l.p), @9471)”

4:) An+1 Z An + Ipan +p)‘k(£‘-lnL — 1) —p§;
4:) An“ 2 An + Ip(an — An)| = An +p(oz,, — An) (Since an 2 An)
4:) An“ 2 an — (p —1)/\n + (p —1)a,,
4:} )‘n+l 2 an + (p —1)(an "' An)
=> ATM-l 2 an (0.31)
[:1 End of proof
The proof of ((3.30):
From [117], we have
2: w(n)?” — Z we)!” 2 2p-Re<e®<P~1~P>,<q -- 6)) (0.32)
i.e.
Ilqlléi — Il9|l§§ 2 2p. amp-LP). (q — 6)) (0.33)

116

 

for any two vectors q, 6) E 12, with restrict inequality whenever q # ('3. Apply this to

GM“) and 9“”, we obtain
”9"”"Il32 — new: 2 2:» He<e<")‘~=><v—‘~P>,(9‘2”) - 9(2)”) (0.34)

Note that
. VHQWHS; = p . 900002—142)
o (VIIG‘nlllgg, km) 2 O, is real

Thus, we have

||9("+”||§§-ll9(")||§§ 2 |(V||@(")l|§5, i‘")>| (C35)

Proposition (3.3 Assume {e(n)};o is defined as (C.7)(C.8) or (C.10)(C.11), then

{e(n) },i',°=0 converges.

Proof:

Since 0 < An < an < An+1< 1,

lim An = lim an = Co > 0 (C36)

for some constant Co. From (C23) and (C28),

An

£1120<e(n+1),e(n)> = nl-iEoQ—<e(n)’ e(n)) = 1
=> 33,, new“) — smug n
.—. "220(e(n+1) _ e(n), 9(n+1) _ @(n))
= giglllem+1>lli+ne<">H§— (as), 9W) — <e<"+1>,e<")>1
= 0
=> {(-9(n)}f,‘?=0 converges (C37)

1:] End of proof

117

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