Multirate Transceivers for Block wireless Transmissions...
Transcript of Multirate Transceivers for Block wireless Transmissions...
Multirate Transceivers for Block wireless
Transmissions and Multicarrier CDMAIrrespective of Unknown Multipath Channels
A Thesis Presented to
the Facuty of the School of Engineering and Applied Science
University of Virginia
In Partial Fulfillment
of the Requirements for the Degree
Master of Science Electrical Engieering
by
Zhengdao Wang
January 1999
Abstract
Suppression of multiuser interference (MUI) and mitigation of multipath effects
constitute major challenges in the design of third-generation wireless mobile systems.
Most wideband and multicarrier uplink CDMA schemes suppress MUI statistically in
the presence of unknown multipath. For fading resistance, they all rely on transmit-
or receive-diversity and multichannel equalization based on bandwidth-consuming
training, or, alternatively, blind equalization techniques. Either way, they impose
restrictive and difficult to check conditions on the FIR channel nulls. Relying on
symbol blocking, we design A Mutually-Orthogonal Usercode-Receiver (AMOUR)
system for quasi-synchronous blind CDMA that eliminates MUI deterministically
and mitigates fading irrespective of the unknown multipath and the adopted signal
constellation. AMOUR converts a multiuser CDMA system into parallel single-user
systems irrespective of multipath and guarantees identifiability of users’ symbols with-
out restrictive conditions on channel nulls in both blind and non-blind setups. The
designed system is flexible to accommodate users with different rates through judi-
cious allocation of what we term signature roots. An alternative AMOUR design
called Vandermonde Lagrange (VL) AMOUR is derived to add flexibility in the code
assignment procedure. Analytic evaluation and preliminary simulations reveal the
generality, flexibility, and superior performance of AMOUR over competing alterna-
tives.
i
Contents
1 Introduction 1
1.1 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Contributions of the thesis . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Modeling and Problem Statement 9
2.1 Discrete time baseband filterbank model . . . . . . . . . . . . . . . . 9
2.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Deterministic MUI Elimination 17
3.1 Transceiver design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 MUI elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 An illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Modulo-Orthogonality in AMOUR . . . . . . . . . . . . . . . . . . . 25
3.4.1 The modulo interpretation . . . . . . . . . . . . . . . . . . . . 25
3.4.2 Re-designing the codes . . . . . . . . . . . . . . . . . . . . . . 27
3.4.3 Reducing bandwidth . . . . . . . . . . . . . . . . . . . . . . . 28
3.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Blind Equalization 35
4.1 Indirect method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2 Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
ii CONTENTS
5 Performance and Comparisons 43
5.1 AMOUR performance with ZF equalization . . . . . . . . . . . . . . 43
5.2 Testing AMOUR’s performance . . . . . . . . . . . . . . . . . . . . . 44
6 Extensions of the AMOUR design 51
6.1 Maximization of single-user information rate . . . . . . . . . . . . . . 51
6.2 VL-AMOUR for code assignment flexibility . . . . . . . . . . . . . . . 52
6.3 Accommodating different bit rates . . . . . . . . . . . . . . . . . . . . 57
6.3.1 Multirate services via root allocation . . . . . . . . . . . . . . 58
6.3.2 Multirate services via virtual users . . . . . . . . . . . . . . . 58
7 Conclusions and future topics 61
A Lagrange Interpolating Polynomial 63
B Proof of Lemmas 3.1 and 3.4 65
B.1 Proof of Lemma 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
B.2 Proof of Lemma 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
C Existence and uniqueness of Cm(z) in (3.18) 67
iii
List of Figures
2.1 Discrete-time baseband (VL-)AMOUR system . . . . . . . . . . . . . 10
2.2 Actual channel setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Spreading Gain of a system with M = 16 users . . . . . . . . . . . . . 14
2.4 No IBI is present, due to trailing zeros . . . . . . . . . . . . . . . . . 15
2.5 Z-domain representation of the system in Fig. 2.1 . . . . . . . . . . . 15
3.1 Equivalent parallel AMOUR-CDMA system . . . . . . . . . . . . . . 21
3.2 Equivalent single user block diagram . . . . . . . . . . . . . . . . . . 22
3.3 An example: the users’ signature roots . . . . . . . . . . . . . . . . . 24
3.4 An example: the users’ transmit-filters in the frequency domain . . . 24
3.5 An example: channels’ nulls . . . . . . . . . . . . . . . . . . . . . . . 24
3.6 An example: Fourier Transform amplitudes of the received noise-free
block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.7 The modulo-equivalent of Fig. 2.1 . . . . . . . . . . . . . . . . . . . . 25
3.8 Modify the transmitter to reduce bandwidth expansion . . . . . . . . 29
3.9 Bandwidth overexpansion . . . . . . . . . . . . . . . . . . . . . . . . 30
5.1 AMOUR vs. OFDMA, 16 users, L = 1 . . . . . . . . . . . . . . . . . 45
5.2 AMOUR with L < L = 6 (Rayleigh fading) . . . . . . . . . . . . . . . 46
5.3 Blindly equalized constellations (Es/N0 = 15dB) . . . . . . . . . . . . 47
5.4 Blind equalization: I = 34, indirect method . . . . . . . . . . . . . . 48
5.5 Blind equalization: I = 100, indirect method . . . . . . . . . . . . . . 49
iv LIST OF FIGURES
5.6 Blind equalization: I = 34, direct method . . . . . . . . . . . . . . . 50
5.7 Blind equalization: I = 100, direct method . . . . . . . . . . . . . . . 50
6.1 Modification of AMOUR transmitter to maximize information rate . 53
6.2 Equivalent single user block diagram of Fig. 6.1 . . . . . . . . . . . . 53
6.3 Variable rates in ADSL applications . . . . . . . . . . . . . . . . . . . 57
6.4 Using virtual users to incorporate multiple rates . . . . . . . . . . . . 59
v
List of Abbreviations
3G Third Generation (wireless communication systems)
ADSL Asymmetric DSL
AGN Additive Gaussian Noise
AMOUR A Multually-Orthogonal Usercode-Receiver
BER Bit Error Rate
BPSK Binary Phase-Shift-Keying
CDMA Code Division Multiple Access
DF Decision Feedback
DS Direct Sequence
DSL Digital Subscriber Line
FFT Fast Fourier Transform
FIR Finite Impluse Response
IBI Inter Block Interference
ICI Inter Chip Interference
IFFT Inverse Fast Fourier Transform
IMT-2000 International Mobile Telecommunications in the year 2000
ISI Inter-Symbol Interference
LV Lagrange Vandermonde
LMS Least Mean Square
MC Multicarrier
ML Maximum Likelihood
MMSE Minimum Mean Square Error
MOE Minimum Output Energy
vi LIST OF ABBREVIATIONS
MUI Multiuser Interference
OFDM Orthogonal Frequency Division Multiplexing
OFDMA Orthogonal Frequency-Division Multiple Access
QS Quasi-Synchronous
RLS Recursive Least Squares
SNR Signal-to-Noise Ratio
SNIR Signal-to-Noise and Interference Ratio
TDMA Time Division Multiple Access
UMTS Universal Mobile Telecommunication System
VL Vandermonde Lagrange
ZF Zero Forcing
1
Chapter 1
Introduction
Code Division Multiple Access (CDMA) has gained widespread international accep-
tance by cellular radio system operators as an upgrade that will dramatically in-
crease both their system capacity and their service quality. Wideband direct se-
quence (DS) CDMA is emerging as the predominant radio access technology for the
third-generation (3G) regional and global wireless standard [14]. DS-CDMA is a very
common one in the CDMA family. In contrast with TDMA and FDMA, in DS-CDMA
systems all the users share the same temporal and spectral resources and user sepa-
ration is made possible by spreading each user’s information bits by a unique code.
There are usually two categories of CDMA receivers depending on how they sepa-
rate the received CDMA signals: single-user and multiuser receivers. In a single user
receiver, signals from interfering users are modeled as noise, whereas in a multiuser
receiver, all the users are included in the signal model.
2 Chapter1
Current CDMA systems use the so-called RAKE single-user receiver (see e.g., [25,
p.798], [26]), which does not attempt to suppress multiuser interference (MUI), but
correlates the received signal with the code of the desired user at time instants cor-
responding to the time-delays introduced by the multipath channel. RAKE receiver
outputs are combined to maximize the signal-to-noise ratio (SNR) at the receiver.
Equivalently, a RAKE receiver can be viewed as performing a matched filtering op-
eration, with respect to both the channel and the user code. Although the RAKE
receiver only requires the signature waveform and the timing (bit-epoch and carrier
phase) of the desired user, it is well known that due to imperfect cross correlation
properties of the spreading codes, the RAKE receiver is sensitive to multipath and
near-far problems.
Significant performance gains can be achieved if a multiuser detection approach
is employed at the receiver (e.g., [42]). The optimum maximum-likelihood (ML)
receiver of [41] assumes knowledge of number of users, all the users’ signature wave-
forms, timing, and amplitudes. Although ML receivers yield optimum performance
and optimum resistance to near-far effects, they have high computational complexity
that is exponential in the number of users. Therefore, many efforts have focused
on designing simple linear multiuser detectors (e.g., [19, 21, 36]). Among the linear
multiuser detectors, the most commonly used are the decorrelating detectors [19, 20],
and MMSE detectors (e.g., [43]). The minimum output energy (MOE) criterion has
been proposed in [12] to asymptotically (in the length of the observation interval) ap-
Introduction 3
proach the MMSE performance without requiring knowledge of the interfering users’
signature waveform, timing, and amplitude. In [12] however, absence of multipath
was assumed. In an attempt to generalize [12] to multipath situations, [36] (and later
on [38] for unknown multipath) proposed a CDMA receiver that has a performance
close to MMSE at high SNR.
As critical performance limiting factors, multiuser interference (MUI) and mul-
tipath induced interchip interference (ICI) define the system’s capabilities in handling
high data rates and interactive multimedia services. MUI and ICI suppression is thus
of paramount importance in mobile wideband CDMA standards such as UMTS and
IMT-2000 [22]. Multipath causes frequency-selective fading, destroys orthogonality
of user codes, and when unknown, it precludes usage of linear zero-forcing (ZF),
MMSE [12, 21], or nonlinear (Decision-Feedback (DF), Maximum Likelihood (ML))
multiuser detectors for MUI suppression [42]. But even when multipath channel es-
timates are available (e.g., using bandwidth-consuming training sequences) it is well
known that especially for multichannel uplink CDMA systems multiuser equalization
is only possible under certain rank conditions on channel matrices that are difficult
to check at the receiver [34, 10]. Thanks to relaxed requirements for power control,
and minimal co-operation among users, self recovering (blind) CDMA receivers are
appealing for mobile radio and digital broadcasting systems. However, even for the
constrained class of equalizable channels blind receivers require subspace decomposi-
tions (see e.g., [35, 18, 1]), or, suppress MUI statistically (and thus asymptotically)
4 Chapter1
when reduced complexity adaptive receivers are sought [34, 38].
Rather than relying on receiver design to suppress MUI and ICI, recent research
turned to transmitter design to improve performance of CDMA systems [32]. Antenna
diversity was also used to trade off improved performance and fading mitigation for
receiver complexity and statistical MUI suppression [10].
On the other hand, combining code division and OFDM, multicarrier CDMA
schemes, were proposed in 1993 and have attracted much attention since then [11].
Three types of systems belong to this category, namely, “multicarrier (MC-) CDMA”
[45, 9, 4], “multicarrier DS-CDMA” [6], and “multitone (MT-) CDMA” [39]. Per-
formance of such systems was evaluated over a frequency selective Rayleigh channel
and compared with that of a single-carrier RAKE system in [17]. Although all these
MC CDMA schemes mitigate fading over a frequency selective channel, none of these
systems has the ability to eliminate MUI and to guarantee identifiability of the users’
information symbols at the same time, irrespective of the multipath channels. User
code hopping and maximal ratio combining diversities are often used to combat fading
in the multicarrier (MC) CDMA systems [3, 33, 11, 8].
A significant step in tackling the multipath problem was made in [32], which
introduced redundancy at the transmitter using filterbank precoders and judiciously
chosen user codes for MUI elimination at the receiver. In the forward link, where all
users share one common multipath channel, it has been shown in [32] that carefully
designed filterbank precoding can guarantee identifiability of the users’ symbols irre-
Introduction 5
spective of the common multipath channel. Various equalizers have also been derived
in [32] (see also [40] for fractionally-spaced detectors).
In the uplink, users can be distributed arbitrarily in space and the multipath
channels experienced by different users are generally different. Therefore, the results
in [32] are not applicable. Relying on transmitter introduced redundancy and gener-
alizing the Orthogonal Frequency-Division Multiple Access (OFDMA), the recent, so
called Lagrange - Vandermonde (LV) CDMA transceivers [30] have low complexity
and offer blind MUI elimination by judicious design of user codes. But similar to
OFDMA and depending on the multipath channel, LV-CDMA transceivers require
extra diversity to ameliorate (but not eliminate) fading effects caused by channel nulls
[28]. In other words, there exist non-equalizable channels (loss of identifiability).
Relying on symbol blocking, we develop in this thesis A Mutually-Orthogonal
Usercode-Receiver (AMOUR) structure for quasi-synchronous blind uplink CDMA
that eliminates MUI deterministically and mitigates fading irrespective of the un-
known multipath. The system encompasses LV-CDMA and MC-CDMA systems as
special cases, it can have low FFT-based complexity, and appears to offer consid-
erable design flexibility. Although our design targets the uplink CDMA channel, it
can be also applied to the downlink channel as well as to other wireline applications
such as DSL (Digital Subscriber Line). Our focus will be on the blind scenario but
AMOUR is also attractive when the multipath channel has been estimated (e.g., via
pilot signals).
6 Chapter1
1.1 Thesis Organization
Based on the multirate filterbank model of Chapter 2, we develop the AMOUR-
CDMA system in Chapter 3. The AMOUR transceivers designed are shown to be
capable of eliminating MUI completely, which is illustrated by a simple example. It
turns out that the system also has a modulo interpretation which is reminiscent of
channel coding ideas. In Chapter 4, we derive blind equalizers for the AMOUR re-
ceiver. Two categories of such equalizers are considered, namely indirect methods and
direct methods. Adaptive implementations of them are also discussed. The perfor-
mance of the designed AMOUR system is evaluated in Chapter 5 and compared with
competing schemes such as OFDM in both deterministic channels and slow Rayleigh
fading channels. The performance of the proposed blind equalization methods is also
tested. Chapter 6 is devoted to possible extensions of the AMOUR design, including
maximizing single-user information rate, adding flexibility to the code assignment pro-
cedure, and incorporating different rates. Conclusions and future topics are discussed
in Chapter 7
1.2 Contributions of the thesis
The major contribution of this thesis is to propose a novel multiuser communication
system (AMOUR) that is capable of eliminating MUI deterministically and guar-
anteeing identifiability of the users’ symbols, irrespective of the multipath channel.
Introduction 7
Most importantly, these claims are achieved without bandwidth over expansion (if we
relax this constraint such systems can be easily designed). In addition, the AMOUR
design is shown to convert multipath multiuser access channel into parallel single
user (decoupled) multipath channels. Therefore techniques developed for single user
systems can be easily applied in the AMOUR system.
The designed system turns out to include many existing systems such as DS-
CDMA, OFDM, TDMA, Multi-Carrier CDMA, as special cases. Thus our approach
unifies many existing systems within a common framework.
Other contributions of this work include:
• An alternative design (VL-AMOUR) which in addition to the AMOUR’s MUI
separating ability, offers added flexibility in the code assignment procedure. The
VL-AMOUR is a direct generalization of the MC-CDMA system proposed in
[17]. Note that in [17], identifiability is not assured in presence of multipath.
• A modulo interpretation of the AMOUR system, which we have used to modify
the AMOUR transmitter and possibly reduce unnecessary bandwidth overex-
pansion.
• Blind indirect equalizers within the AMOUR framework motivated by the meth-
ods proposed in [31].
• A criterion based direct equalizer design, which is readily implementable using
adaptive methods of the LMS or RLS type.
8 Chapter1
• Performance analysis of the system with zero-forcing equalization and calcu-
lation of the bit error rate for the BPSK transmissions. The performance for
Rayleigh fading channels and for blind (both direct and indirect) methods is
simulated and the results compared with each other and also with theoretical
MMSE bounds derived when the channels are known.
• Generalization of the basic AMOUR transmitter to show that AMOUR has
the flexibility (through specification of filter bank precoders ) to further convert
each of the single user equivalent multipath channel into a number of flat fading
channels while maximizing the mutual information rate for each user.
• Incorporation of users with different rate requirements by judicious allocation
of what we term signature roots. This is a desirable feature when the system
is expected to support multimedia services, which are inherently multirate and
packet-oriented (burst transmissions). Thanks to the block transmission nature
of the AMOUR system, such incorporation is made easy.
9
Chapter 2
Modeling and Problem Statement
In this chapter, we propose the model of our designed system in a discrete time
baseband equivalent filterbank form. Relying on blocking and guard zeros, the system
can be written in a very compact form in the Z-domain, which eventually leads to
the AMOUR transceivers we design in Chapter 3.
2.1 Discrete time baseband filterbank model
Adopting the multirate framework for CDMA proposed in [35, 36], the block diagram
in Fig. 2.1 represents the uplink channel of a CDMA system, described in terms
of its discrete-time equivalent baseband model, where signals, codes, and channels
are represented by samples of their complex envelopes taken at the chip rate (only
transmitter and receiver filters for the mth, user are shown). Advance/delay elements
and down/up-samplers (D/U) serve the purpose of blocking and inserting zeros, so
10 Chapter2
P
PK
K
P
Pz z�1
.
.
.
z�1z
sm(n)Cm(z) Hm(z)
.
.
....
mth -transmitter
Sm(i; z).
.
.
mth-receiver
�V�1m
ym;L+K�1(i)
�m
Gm;0(z)
sm(0)
sm(K � 1)Gm;L+K�1
ym;0(i)
(z)
mth-channel
.
.
.
~xm(i)
� � �
MUI
�(n)
X(i; z)
.
.
.
Figure 2.1: Discrete-time baseband (VL-)AMOUR system
that each of the M users maps successive blocks of K symbols of the information
sequence sm(n) to blocks of length P > K, each containing P − K trailing zeros
(guard chips). The ith block is depicted in Fig. 2.1 with its Z transform
Sm(i; z) :=K−1∑k=0
sm(iK + k)z−k. (2.1)
Each of these P -long blocks is encoded with a code cm(p) of length M(L + K) < P
denoted by its Z-transform
Cm(z) :=
M(L+K)−1∑p=0
cm(p)z−p, (2.2)
In the Z-domain, the transmitted discrete-time signal at the chip rate is
Um(i; z) = Sm(i; z)Cm(z), (2.3)
or, in the time domain
um(iP + p) =K−1∑k=0
sm(iK + k)cm(p− k − iP ). (2.4)
Let the chip sequence um(n) be transmitted at a rate 1/Tc using a spectral shaping
pulse g(tr)c (t) modulated at the carrier frequency (see Fig. 2.2). The transmitted
Modeling and Problem Statement 11
Spectral Pulse ChannelReceive-filter
Chip
g(tr)c (t) g
(ch)c;m (t) g
(rec)c (t)
��c(t)
um(n)
xc;m(t)
xm(n)
t = nTc
Figure 2.2: Actual channel setup
signal propagates through a dispersive channel g(ch)c,m (t) and through the (chip rate)
receiver filter g(rec)c (t). We define gc,m(t) to be the overall impulse response of the
channel including the transmitter and receiver filters, i.e., with ? denoting convolution,
gc,m(t) = g(tr)c (t) ? g
(ch)c,m (t) ? g
(rec)c (t). The received signal from user m is thus given by
xc,m(t) =∞∑
n=−∞
um(n)gc,m(t− nTc − τm), (2.5)
where τm is the propagation delay. The overall received signal from all users is there-
fore
xc(t) =M−1∑m=0
xc,m(t) + ηc(t), (2.6)
where ηc(t) = ηc(t) ? g(rec)c (t) and ηc(t) is the received additive noise. If the received
signal is sampled at the chip rate (every Tc seconds), then the discrete-time received
signal is
x(n) = xc(t)|t=nTc =M−1∑m=0
xm(n) + η(n), (2.7)
where xm(n) = xc,m(t)|t=nTc =∑∞
j=−∞ um(j)hm(n − j), and hm(n) is a sampled
version of the continuous-time impulse response: hm(n) = gc,j(nTc − τm).
12 Chapter2
The channels hm(n) or their Z-transform Hm(z), m = 0, . . . ,M − 1 are as-
sumed to be of order ≤ L. In addition to multipath, hm(n) includes the spectral-
shaping pulse and the mth user’s asynchronism in the form of delay factors. In
quasi-synchronous (QS) CDMA systems (e.g. [7, 13]), mobile users attempt to syn-
chronize with the base-station’s pilot waveform. But exact synchronization is difficult
to implement in the reverse channel due to multipath and Doppler effects arising from
relative motion, especially when the chip period is small. In general, mobile users’
timing maybe off by a delay spread of 2− 3 chips.
The ith received block x(i) := [x(iP ) · · ·x(iP + P − 1)]T (T denotes transpose)
is represented by its Z transform
X(i; z) :=P−1∑p=0
x(iP + p)z−p, (2.8)
and consists of chips from the mth user of interest along with MUI chips from
other users and AGN η(i) := [η(iP ) · · ·η(iP + P − 1)]T . The receive-filterbank,
gm(p) := [gm,0(p) · · · gm,L+K−1(p)]T , performs vector filtering of the block x(i) and
after downsampling (to get back to the symbol rate) and multiplication by a ma-
trix V−1m (to be specified in Chapter 3) we obtain the (L + K) × 1 vector ym(i) :=
[ym,0(i) · · · ym,L+K−1(i)]T .
Modeling and Problem Statement 13
2.2 Problem statement
Our goal is to design user code polynomials {Cm(z)}M−1m=0 and receive-filters {Gm,l(z)}
L+K−1l=0
capable of eliminating MUI deterministically, and thus enabling usage of single user
equalizers, denoted by the K × (L + K) matrix Γm, to eliminate multipath effects,
suppress noise, and recover the ith symbol block sm(i) := [sm(0) · · · sm(K−1)]T from
ym(i).
Our system design parameters are as follows:
d1) Each input block is designed to contain K � L symbols, and each transmit-
block size (at the chip rate) is chosen equal to P = (M + 1)(L+K)− 1.
d2) Codes {Cm(z)}M−1m=0 are selected to have order M(L + K)− 1, which creates L
trailing zeros per block (guard chips).
d3) Receive-filters {Gm,l(z)}L+K−1l=0 will turn out to have order P , ∀m.
Note that except for an upper bound L on their orders, all uplink channels are al-
lowed to be unknown. In [7], multicarrier signaling was proposed as a means of reduc-
ing the synchronization offsets. Even without such techniques in quasi-synchronous
channels, L is generally no more than 2 or 3. Thus, our choice K � L does not entail
very large K’s and hence it does not entail excessive decoding delays.
With each user transmitting K symbols per block, our system’s spreading gain
14 Chapter2
is [c.f. d1)]:
R :=P
K=
(M + 1)(L+K)− 1
K, (2.9)
which for sufficiently large K � L is approximately M (= to the number of users);
hence, bandwidth is not over expanded (see Fig. 2.3).
2 4 6 8 10 12 14 160.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
K
L=3
L=1L=2
M/R
Figure 2.3: Spreading Gain of a system with M = 16 users
Thanks to the L trailing zeros [c.f. d2)], no interblock interference (IBI) is present
in our received P -long blocks (see also Fig. 2.4). Therefore, despite the presence of
MUI and ICI that is allowed in our QS setup, one can focus on each block X(i; z)
separately and express it in the Z-domain as (see also Fig. 2.5):
X(i; z) =M−1∑µ=0
Sµ(i; z)Cµ(z)Hµ(z) +N(i; z) , (2.10)
Modeling and Problem Statement 15
guard chips
������������������������������������������������������������
������������������������������������������������������������
������������������������������������������������������������
������������������������������������������������������������
����������������
������������������������������
������������������������������
������������������������������
������������������������������
L
P
P-L
hm(l)
um(i� 1) um(i)
L+1
Figure 2.4: No IBI is present, due to trailing zeros
N(i; z)
...
CM�1(i; z) HM�1(i; z)
C0(i; z) H0(i; z)S0(i; z)
X(i; z)
SM�1(i; z)
......
Figure 2.5: Z-domain representation of the system in Fig. 2.1
where N(i; z) :=∑P−1
p=0 η(iP + p)z−p denotes the Z-transform of the noise in the ith
received block.
17
Chapter 3
Deterministic MUI Elimination
Using the model we developed in Chapter 2, we fully specify the AMOUR transceiver
design in this chapter. In addition to MUI elimination, the system is shown to be
capable of converting the multiple input multiple output CDMA channel into M
parallel single-user channels. Following a simple example, we give the system another
interpretation using polynomial modulo operations.
3.1 Transceiver design
Suppose we start with M(L+K) distinct points ρm,l on the complex plane and assign
L+K of them to be roots common to all Cµ(z) polynomials in (2.10), except the mth
one. Our assignment amounts to
Cµ(ρm,l) = 0 , ∀ µ 6= m , l ∈ [0, L+K − 1] , (3.1)
18 Chapter3
which shows that evaluating X(i; z) in (2.10) at z = ρm,l eliminates MUI determinis-
tically from user m and yields
X(i; ρm,l) = Sm(i; ρm,l)Cm(ρm,l)Hm(ρm,l) +N(i; ρm,l) . (3.2)
Note that thanks to (3.1) at most L of the L + K roots {ρm,l}L+K−1l=0 can be roots
of Hm(z); thus, we guarantee that X(i; ρm,l) 6= 0 on at least K points, which is
precisely the minimum number of values we need to know about the (K − 1)st-order
polynomial Sm(i; z) in order to identify uniquely the mth user’s ith block sm(i) from
the ith received data block. Roots {ρm,l}L+K−1l=0 are “signature roots” of user m but
are not roots of Cm(z). To achieve MUI elimination as in (3.2) for every user, Cm(z)
must contain signature roots {ρµ,l, µ ∈ [0,M − 1], µ 6= m}L+K−1l=0 of the remaining
M − 1 users; hence,
Cm(z) = KmQm(z)M−1∏
µ=0,µ6=m
L+K−1∏λ=0
(1− ρµ,λz−1), (3.3)
where Km is a constant controlling the mth user’s transmit power, and Qm(z) is a
code-normalizing polynomial of order L + K − 1 whose coefficients are chosen to
satisfy:
Qm(ρm,l)M−1∏
µ=0,µ6=m
L+K−1∏λ=0
(1− ρµ,λρ−1m,l) = 1 , ∀l ∈ [0, L+K − 1]. (3.4)
Specifically,
Qm(z) =L+K−1∑l=0
L+K−1∏j=0j 6=l
M−1∏µ=0µ6=m
L+K−1∏λ=0
1− ρm,jz−1
(1− ρm,jρ−1m,l)(1− ρµ,λρ
−1m,l)
. (3.5)
Deterministic MUI Elimination 19
Specification of the receive-filters gm(p) follows if we observe that
X(i; ρm,l) = [1 ρ−1m,l · · · ρ
−P+1m,l ] x(i)
:= vT (ρm,l) x(i) , (3.6)
which amounts to convolving x(i) with the (time-reversed) Vandermonde vector
[ρ−P+1m,l · · · ρ−1
m,l 1]; hence, the receive-filter impulse responses in Fig. 2.1 are gm,l(p) =
ρ−P+1+pm,l .
With the normalization in (3.4), eq. (3.3) yields Cm(ρm,l) = Km, which along
with definition1
xm(i) :=
[Xm(i; ρm,0) · · · Xm(i; ρm,L+K−1)
]T(3.7)
allows one to use (3.6) and write the concatenation of (3.2) for l = 0, 1, . . . , L+K−1,
as
xm(i) = Vmx(i) = Km
Sm(i; ρm,0)Hm(ρm,0)
...
Sm(i; ρm,L+K−1)Hm(ρm,L+K−1)
+
N(i; ρm,0)
...
N(i; ρm,L+K−1)
,
(3.8)
where Vm := [v(ρm,0) · · · v(ρm,L+K−1)]T is an (L+K)× P Vandermonde matrix.
So far we have achieved user separation at certain Z-transform values of the ith
received data. To establish separability of the entire Z-transform X(i; z) and hence
1Boldface vectors with tilde that we use henceforth, denote vectors having Z-transforms values
as entries (no tilde stands for time-domain vectors as usual).
20 Chapter3
of x(i), let us consider the polynomial Ym(i; z) of maximum degree L+ K − 1 given
by
Ym(i; z) := KmSm(i; z)Hm(z) +Nm(i; z), (3.9)
where Nm(i; z) is the unique polynomial that satisfies: i) deg(Nm(i; z) ) ≤ L+K−1;
and ii) Nm(i; ρm,l) = N(i; ρm,l), ∀l ∈ [0, L+K − 1].
Let us now express Ym(i; z) in the time-domain as the coefficient vector ym(i) :=
[ym,0(i) · · · ym,L+K−1(i)]T , starting with the Z-domain vector
ym(i) :=
[Ym(i; ρm,0) · · · Ym(i; ρm,L+K−1)
]T, (3.10)
and using the (L+K)× (L+K) Vandermonde matrix
Vm =
1 ρ−1
m,0 · · · ρ−(L+K−1)m,0
......
...
1 ρ−1m,L+K−1 · · · ρ
−(L+K−1)m,L+K−1
(L+K)×(L+K)
. (3.11)
Matrix Vm captures the Z-transform operation at the points z = ρm,l,m ∈ [0,M −
1], l ∈ [0, L+K − 1], and is full rank because the ρm,l’s were chosen to be distinct. It
now follows that (see Appendix A)
ym(i) = Vmym(i) = xm(i) =⇒ ym(i) = V−1m xm(i), (3.12)
where the last equality explains the role of the multivariate operation V−1m in the block
diagram of Fig. 2.1. Matrix V−1m can be cascaded with Vm and then the combination
Vm := V−1m Vm can be applied on x(i) to blindly isolate user m as: ym(i) = Vmxm(i).
Deterministic MUI Elimination 21
We have thus shown that our judicious code design assures perfect separation and
recovery of Ym(i; z) in (3.9) ∀z.
3.2 MUI elimination
Note that eq. (3.9) reveals another important feature of our AMOUR design; namely
how AMOUR design converts the multiuser CDMA system into M parallel single-user
systems irrespective of the multipath (see also Fig. 3.1).
In vector form, (3.9) can be written as (see also Fig. 3.2):
N0(i; z)
S0(i; z) Y0(i; z)H0(z)
SM�1(i; z)
NM�1(i; z).
.
.
YM�1(i; z)HM�1(z)
A0
AM�1
Figure 3.1: Equivalent parallel AMOUR-CDMA system
ym(i) = Hmsm(i) + ηm(i), (3.13)
where ηm(i) = V−1m Vmη(i) is the coefficient vector of the polynomial Nm(i; z), and
22 Chapter3
Hm is the Toeplitz convolution matrix built from the coefficients of Hm(z) as:
Hm :=
hm(i; 0) · · · 0
.... . .
...
hm(i;K − 1) hm(i; 0)
.... . .
...
0 · · · hm(i;K − 1)
(L+K)×L
. (3.14)
�m
sm(i) sm(i)Hm(i) �m(i)
Figure 3.2: Equivalent single user block diagram
Due to the trailing zeros introduced at the transmitter (c.f. d3)), Hm is always
full rank. This property will be exploited later on to find the blind equalizers. To
illustrate the basic idea of AMOUR design and its MUI eliminating ability, we give
an simple example.
3.3 An illustrative example
Suppose L = 1 and M = 3. For simplicity, we choose K = 2 (recall that K should
be much greater than L to avoid bandwidth overexpansion). The signature roots
of the three users are shown in Fig. 3.3 equally spaced and interleaved along the
Deterministic MUI Elimination 23
unit circle. Assuming Km = 1, the user codes can be computed from (3.3) and
(3.4) to be C0(z) = 13(1 + z−3 + z−6), C1(z) = 1
3(1 + ej
2π3 z−3 + ej
4π3 z−6), C2(z) =
13(1 + e−j
2π3 z−3 + e−j
4π3 z−6). In Fig. 3.4, we show the Fourier transform magnitude
of the codes. In close accordance with the signature root allocation, the spectra are
interleaved and overlapped. We assume the ith blocks (each of length K = 2) sent
by the three users are S0(i; z) = 1, S1(i; z) = −1 + 0.3z−1, S0(i; z) = 1 − z−1, and
the discrete-time equivalent baseband channels are chosen to be H0(i; z) = 1 + jz−1,
H1(i; z) = 1 + 0.5z−1, H2(i; z) = 1 + 0.7z−1, whose nulls are plotted in Fig. 3.5. The
ith noiseless received block then turns out to be x(i) = [0.33, 0.17+0.33j, 0.18, 0.33−
0.58j, 0.08 + 0.36j, 0.09 + 0.25j, 0.33 + 0.58j, 0.08 + 0.30j, 0.09 − 0.25j, 0, 0], whose
Fourier transform magnitude is depicted in Fig. 3.6 and the components due to the
three respective users are also shown in the same figure with thin lines. We use circles
on the curve to denote frequencies that correspond to the first user’s signature roots
and hence are free of MUI.
Applying the V−1m Vm operations on x(i) for m = 0, 1, 2 respectively, we get
y0(i; z) = 1+jz−1, y1(i; z) = −1−0.2z−1+0.15z−2, and y2(i; z) = 1−0.3z−1−0.7z−2,
which are precisely the convolutions of transmitted block symbols with the corre-
sponding channel impulse responses of each user, confirming the equivalent model
shown in Fig. 3.1. 2
24 Chapter3
−1.5 −1 −0.5 0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
1
1.5
Re{z}
Im{z
}
user 0user 1user 2
Figure 3.3: The users’ signature roots
−π −0.5π 0 0.5π π0
0.5
1
1.5
θ
Am
plitu
de
user 0user 1user 2
Figure 3.4: The users’ transmit-filters
in the frequency domain
−1.5 −1 −0.5 0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
1
1.5
−> user 0’s channel null −> user 1’s channel null −> user 2’s channel null
Re{z}
Im{z
}
Figure 3.5: Channels’ nulls
−π −0.5π 0 0.5π π0
0.5
1
1.5
2
2.5
− sig. roots of user 0: NO MUI
Am
plitu
de
θ
from user 0 from user 1 from user 2 composite
Figure 3.6: Fourier Transform ampli-
tudes of the received noise-free block
Deterministic MUI Elimination 25
single
equalizeruser
Sm(i; z)
Cm(z)
Ym(i; z)X(i; z)
sm(n)
v(n)
� � �
MUI
modGm(z)Hm(z)
Figure 3.7: The modulo-equivalent of Fig. 2.1
3.4 Modulo-Orthogonality in AMOUR
The AMOUR CDMA system design in Section 3.1 relied heavily on the Z-domain
representation (2.10). It turns out that the system has another interpretation using
the polynomial modulo operations. The system is re-derived here relying on a modulo-
orthogonality condition between user-code and receive-filter polynomials. Such an
interpretation is reminiscent of channel coding for noise cancellation.
3.4.1 The modulo interpretation
Equation (3.12) shows AMOUR’s ability for the user separation. We rewrite it here
as follows:
Ym(i; ρm,l) = X(i; ρm,l), ∀l ∈ [0, L+K − 1] (3.15)
which suggests that Ym(i; z) and X(i; z) are equal at z = ρm,l, ∀l ∈ [0, L+K − 1].
To reveal the intrinsic relationship between the two polynomials Ym(i; z) and
X(i; z), we will need the following Lemmas (see Appendix B for proofs).
26 Chapter3
Lemma 3.1 Two polynomials a(z) and b(z) are equal at n points z = z0, . . . , zn−1,
if and only if
a(z) mod c(z) = b(z) mod c(z), (3.16)
where c(z) :=∏n−1
i=0 (z − zi). 2
Lemma 3.2 Let a(z), c(z) be polynomials in z. If deg(a(z)) < deg(c(z)), then
a(z) mod c(z) = a(z). 2
Because Ym(i; z) is a polynomial of order L+K− 1, it follows from Lemmas 3.1
and 3.2 that (3.15) holds true if and only if
Ym(i; z) = X(i; z) mod Gm(z), (3.17)
where Gm(z) :=∏L+K−1
l=0 (1− ρm,lz−1).
Thus the cascade of receive filter bank, down-sampler, and matrix mapping
V−1m , perform equivalently a modulo operation ( mod Gm(z)) on the received data
polynomial X(i; z) to produce Ym(i; z) that is free of MUI. The system model can
therefore be re-drawn as in Fig. 3.7. It also can been seen from the definition of
Nm(i; z) in (3.9) that Nm(i; z) = N(i; z) mod Gm(z), which provides intuition on how
the noise in the parallel equivalent single-user models of Fig. 3.1 relates to the original
noise polynomial N(i; z).
Deterministic MUI Elimination 27
3.4.2 Re-designing the codes
Interestingly, we can design the codes starting from the modulo-interpretation of
Fig. 3.7.
The precoders {Cm(z)}M−1m=0 and receive-filters {Gm(z)}M−1
m=0 could have been de-
signed as follows:
i) Choose coprime polynomials {Gm(z)}M−1m=0 of order L+K.
ii) Build {Cm(z)}M−1m=0 of order < M(L+K) such that
Cm(z) mod Gµ(z) = δm,µ, ∀m,µ ∈ [0,M − 1]. (3.18)
The existence and uniqueness of Cm(z), ∀m ∈ [0,M − 1] is established in Ap-
pendix C.
To illustrate the deterministic MUI elimination property of the designed {Cm(z)}M−1m=0
and {Gm(z)}M−1m=0 , it will be convenient to use the following additional (easy to prove)
Lemma 3.3 about the modulo operation.
Lemma 3.3 Let a(z), b(z), c(z) be three polynomials.Then
[a(z) + b(z)] mod c(z) = a(z) mod c(z) + b(z) mod c(z).
[a(z)b(z)] mod c(z) ={
[a(z) mod c(z)][b(z) mod c(z)]}
mod c(z). 2
Using Lemmas 3.2 and 3.3, we can write Ym(i; z) in the presence of noise as
follows:
28 Chapter3
Ym(i; z) = X(i; z) mod Gm(z)
= [M−1∑µ=0
Sµ(i; z)Cµ(z)Hµ(z) +N(i; z)] mod Gm(z)
=M−1∑µ=0
{[Sµ(i; z)Hµ(z)] mod Gm(z)[Cµ(z) mod Gm(z)]
}mod Gm(z)
+N(i; z) mod Gm(z)
=M−1∑µ=0
Sµ(i; z)Hµ(z)δm,µ +Nm(i; z)
= Sm(i; z)Hm(z) +Nm(i; z),
where Nm(i; z) := N(i; z) mod Gm(z).
The multiuser CDMA system is therefore converted into M parallel independent
single-user systems irrespective of the multipath channels the same way as in Fig. 3.1.
3.4.3 Reducing bandwidth
It has been shown in (3.17) that Ym(i; z) is the reminder of X(i; z) divided by Gm(z).
Using this fact and the following Lemma (proved in Appendix B) provide us with a
means of reducing the bandwidth required by the system without affecting perfor-
mance by reducing unused redundancy.
Lemma 3.4 Let a(z), c(z), and d(z) be three polynomials. If d(z) divides c(z), then
[a(z) mod c(z)] mod d(z) = a(z) mod d(z). 2
Deterministic MUI Elimination 29
single
equalizeruser
Sm(i; z)
Cm(z) modGm(z)
Ym(i; z)X(i; z)
sm(n)
v(n)
� � �
MUI
Hm(z)modG(z)
Figure 3.8: Modify the transmitter to reduce bandwidth expansion
Lemma 3.4 reveals that the receiver output will not change if we perform a
modulo operation ( mod G(z)) at the transmitter (see Fig. 3.8), using
G(z) =M−1∏m=0
Gm(z). (3.19)
Note that G(z) is a polynomial of order M(L + K) divisible by every Gm(z), m =
0, . . . ,M − 1.
With the mod G(z) operation at the transmitter, the new spreading gain of the
AMOUR system becomes (c.f. Eq. (2.10))
R′ =M(L +K) + L
K< R . (3.20)
The ratio RR′ increases as the number of users M decreases, or, as K increases
(see Fig. 3.9). In practical systems, if we want to use large K to reduce R, we should
consider using the modification in Fig. 3.8 to further reduce bandwidth.
3.5 Remarks
Having specified the AMOUR design, several remarks are now in order.
30 Chapter3
5 10 15 201.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
1.18
M
K=2 K=16
R/R′
Figure 3.9: Bandwidth overexpansion
Remark 1 AMOUR has low complexity if ρm,l’s are chosen regularly around the
unit circle; e.g., with l ∈ [0, L+K − 1] and using complex exponentials as signature
roots for user m,
ρm,l = ej
2π(m+lM)M(L+K) , m ∈ [0,M − 1] . (3.21)
With such signature roots, the codes can be found to be
cm(n) =
Km
M(L+K)e
2πmM
q if n = q(L+K), q = 0 . . .M − 1,
0 otherwise.
(3.22)
Notice that for every user m, the code cm(n) is non-zero only at multiples of L+K,
which means that at the transmitter, the convolution of sm(i) with cm(z) entails
Deterministic MUI Elimination 31
shift and scaling only. At the receiver, matrix multiplications (by Vm) and inversion
(V−1m ) can be replaced by FFTs. Algorithmcally, the receiver computes a P -point
(Zoom) FFT of the received data x(i) at the M(L + K) frequencies given in (3.21).
Then each of the M users computes an (L+K)-point IFFT at the L+K frequency
samples corresponding to his/her signature roots to obtain ym(i), on which a single
user equalizer can be applied. In addition to computational simplicity, such a “user-
balanced” root selection in (3.21) possesses additional optimality in terms of SINR
improvement. Additional research is needed in this direction but it will not be pursued
in this thesis.
Remark 2 If channel estimates are available, MUI elimination is possible even with
L+K < L+K signature roots which decreases the spreading in (2.9) and allows usage
of smaller K’s to reduce decoding delays. In this case, the block and code lengths in
d1), d2) are M(L + K) + L + K − 1 and M(L + K), respectively. In fact, we show
in Chapter 5 that considerable gains in Bit Error Rate (BER) are achieved with L
as small as 1 or 2. This feature makes the AMOUR system attractive in applications
such as digital subscriber line (DSL), where channels may be long but can be retrieved
via the feedback channels. However, in applications where no feedback channel is
available to provide information about forward channels, or, when multipath causes
rapid fading that renders channel adaptation at the transmitter too cumbersome, it
is convenient to guarantee (FIR) ZF channel-independent blind equalization. In this
case, we need L = L (at least L+K values of Ym(i; z) are needed in (3.2)) to recover
32 Chapter3
ym(i) from which sm(i) can be found as we will see in Chapter 4.
Remark 3 AMOUR resembles MC CDMA although the latter does not guarantee
channel-independent demodulation (especially in the uplink), and resorts to hopping
in order to ameliorate performance in the presence of deep fades [3]. Instead of using
harmonic subcarriers as in MC CDMA [24], the ‘subcarriers’ in AMOUR system can
be general complex exponentials. This feature can be exploited to optimize BER
performance with respect to signature root locations on the complex plane when
dealing with frequency and time-selective channels. Since average user performance
depends on the choice of signature roots, root allocation is also potentially useful
when the system is desired to be flexible enough to accommodate different priority
users. Furthermore, given that our AMOUR design guarantees channel independent
demodulation without resorting to frequency hopping, signature root hopping along
the lines of [3] can still be implemented to improve average performance further in
frequency- and/or time-selective channels. Root hopping and the resulting perfor-
mance improvement deserve further investigation and will reported elsewhere.
The AMOUR system also generalizes the LV/VL-CDMA systems of [28], which
correspond to no input blocking (K = 1) and one signature root per user (as opposed
to L + K roots used herein). The LV/VL-CDMA systems also guarantees MUI
elimination irrespective of channel nulls. But when the channel nulls happen to
coincide with one user’s single signature root, this user will suffer consistently (BER=
1/2). Even when one of the nulls is close to the signature root, the performance
Deterministic MUI Elimination 33
degrades considerably. To ameliorate this situation, root hopping for such systems
was developed in [28] and is still applicable within the AMOUR framework.
Remark 4 There is a tradeoff in selecting the number of symbols K transmitted in
each block. On the one hand, one wants to have K as large as possible to reduce
redundant overhead needed per block, or equivalently to avoid bandwidth over expan-
sion (c.f. (2.9)). On the other hand, as K increases, the block becomes longer and the
decoding delay increases proportionally. The channels are also implicitly assumed to
remain the time-invariant over a longer period. Moreover, when one transmits more
symbols per block, the peak-to-average power increases and power amplifiers with
larger back-off may become necessary.
35
Chapter 4
Blind Equalization
Depending on complexity vs. performance tradeoffs, our channel-independent MUI-
free receiver can be followed by any single user equalizer, be it linear (e.g., ZF,
MMSE) or nonlinear (e.g., DF or ML), in order to recover the block signal esti-
mates sm(i) from ym(i). As we mentioned in Chapter 3, AMOUR’s built-in ability
for channel-irrespective blind user separation is very attractive even when channel
estimates Hm(z) are available.
However, to maintain an overall blind and computationally simple demodulator,
we recommend the single-user blind filterbank approach of [31] that capitalizes on
input redundancy which is also present in our transmitter design in the form of L
trailing zeros.
36 Chapter4
4.1 Indirect method
To show how [31] applies in our single-user setup, let Rym := E{ym(i)yHm(i)} be the
correlation matrix of ym(i) = Hmsm(i) + ηm(i) , which is given by
Rym = HmRsmHHm +Rηm , (4.1)
whereRsm := E{sm(i)sHm(i)}, andRηm := E{ηm(i)ηHm(i)} = E{V−1m Vmη(i)ηH(i)V
H
m
V−Hm } = V
−1m VmRηV
H
mV−Hm , with Rη denoting the autocorrelation matrix of η(i).
Minimum persistence of excitation guarantees that Rsm is full rank. Because the
Toeplitz matrix Hm in (3.14) is also full column rank irrespective of the FIR channel,
the nullity of HmRsmHHm is L and Range(HmRsmH
Hm) = Range(Hm). Assuming
that Rη is known, the noise ηm(i) can be whitened by pre-multiplying ym(i) by any
matrix Φ−1ηm
such that Rηm = ΦηmΦHηm . The covariance matrix of the prewhitened
ym is
Φ−1ηmRymΦ−Hηm = Φ−1
ηmHmRsmH
HmΦ−Hηm + I. (4.2)
Focusing on the signal and noise subspace of the matrix in (4.2) reveals that Φ−Hηm U(Φ)
ym
is a basis for the null space of HmRsmHHm, where U
(Φ)
ymcontains the L eigen-vectors
associated to the smallest eigenvalues of Φ−1ηmRymΦ−Hηm . It follows that (see also [31])
Φ−Hηm U(Φ)
ymHm = 0 ⇒ uHl Hm = 0H , l = 1, . . . , L (4.3)
where ul denotes the lth column of Φ−Hηm U(Φ)
ym. Because convolution is a commutative
operation, (4.3) can also be written in terms of the unknown channel coefficients hm
Blind Equalization 37
as follows:
hHmU := hHm(U1 · · ·UL) = 0H, (4.4)
where each U l is an (L + 1) × K Hankel matrix formed by ul with first column
[ul(0), . . . , ul(L)]T and last row [ul(L), . . . , ul(L + K − 1)]. From (4.4), the channel
vector hm can be obtained as the unique (within a scale factor) null eigen-vector of
U [31].
In practice, the method is implemented by replacing ensemble averages, i.e., the
covariance matrices, by sample averages. In short, the mth user collects I blocks of
ym(i) in an (L+K)× I matrix Y m := [ym(0) · · ·ym(I − 1)] and forms
1
IY mY
Hm =
1
I(HmSmS
HmH
Hm +VmV
Hm)
=1
I(HmSmS
HmH
Hm + VmVmVV
HVH
mVHm)
(4.5)
where Sm := [sm(0) · · ·sm(I−1)]K×I , Vm := [ηm(0) · · ·ηm(I−1)]P×I , and V := [η(0)
· · · η(I−1)]P×I . The sample averages in (4.5) will converge in the mean square sense
to correlation matrices (ym(i) in (3.13) is mixing because sm(n) has finite moments
and hm(n) is FIR). Finally, because Φ−Hηm U(Φ)
ym is a continuous function ofRym and the
sample estimate of Rym is consistent, it follows that the channel estimation algorithm
is also consistent.
Once Hm is found, a zero forcing equalizer Γ(zf)m = H†m, or, an MMSE equalizer
Γ(mmse)m = RsmH
Hm(Rηm +HmRsmH
Hm)−1, (4.6)
38 Chapter4
can be applied to the vector ym(i) to obtain estimates of the symbols sm(i). Direct
and adaptive equalizers are also possible (see [31] for details). Detailed derivation
of the equalizer, its adaptive implementation, and its performance analysis within
the AMOUR framework is beyond the scope of this thesis. However, we want to
underscore that all existing blind CDMA schemes [35, 18, 1, 34, 38, 10, 44] do not
guarantee channel-irrespective identifiability in the uplink; e.g., reducible transfer
function matrices offer counterexamples as recognized in [34]. Some also rely on
whiteness assumptions which may fail for coded inputs, or, require additional (e.g.,
antenna) diversity to ameliorate (but not eliminate) channel fades.
4.2 Direct Method
In this section, we will derive a direct criterion based blind equalizer, which is readily
implementable in an adaptive fashion.
By direct we mean that the equalizer can be constructed without explicit esti-
mation of the channel. Note that in the indirect method we outlined in the previous
section, the blind equalization problem was accomplished in two steps: channel es-
timation first using subspace method (4.4), and ZF or MMSE equalization second
based on the channel estimates. Such methods may lose optimality even if each of
the two steps is optimal [37]. So we want to derive a blind criterion based method
that seeks the equalizer directly according to a certain optimality criterion.
Blind Equalization 39
We can write (3.13) as follows:
ym(i) = Hmsm(i) + ηm(i), (4.7)
where,
Hm :=
hm(i; 0) · · · 0 · · · 0
.... . .
......
hm(i;K − 1) hm(i; 0) 0
.... . .
.... . .
...
0 · · · hm(i;K − 1) · · · hm(i; 0)
(L+K)×(L+K)
(4.8)
is a lower triangular Toeplitz matrix and sm(i) := [sm(i; 0), . . . , sm(i;K−1), 0, . . . , 0]T
has L trailing zeros appended to sm(i). The vector model in (4.7) expresses explicitly
the fact that trailing zeros were introduced at the transmitter.
Since we know that the last L symbols of sm(i) are zeros, a good criterion
for selecting the equalizer will be to minimize the total energy (sample averaged
or ensemble averaged) in the last L symbols of the equalized data. Notice that
decorrelating solution Γ(zf) = H−1
m forces the last L symbols to be zeros when there
is no noise. Since Hm is a lower triangular Toeplitz matrix, so is Γ(zf). But we
know that at high SNR, MMSE solutions boil down to zero-forcing ones, so it is
reasonable if we pose a similar condition on the direct equalizer Γm; i.e., to constrain
the equalizer to be lower triangular Toeplitz matrix. Subject to this constraint, we
40 Chapter4
propose the following criterion
Γm = arg min(εL(Γm)), (4.9)
where εL(Γm) is the total energy in the equalized last L symbols. The energy can
be defined either as a sample average or as an ensemble average. Here we adopt the
former one for simplicity.
Let Γm = [γ0,γ1, . . . ,γL+K−1]T . Then the energy in (4.9) is given by
εL(Γm) =L∑l=0
E{||γHL+K−1−jx(i)||2}. (4.10)
Since Γ is a lower triangular Toeplitz matrix, all rows {γi}L+K−1i=0 can be obtained
from the last row γTL+K−1. So the design of the direct equalizer in (4.9) entails finding
the vector γTL+K−1 only. To exclude the trivial solution of (4.9), we further constrain
||γL+K−1|| = 1.
Let J denote a P × P shift matrix having all ones in the first sub-diagonal:
J :=
0 0 · · · 0
1 0...
. . . . . ....
0 · · · 1 0
(4.11)
Using J , the rows of Γm can be obtained from γTL+K−1 as
γTl = γTL+K−1JL+K−1−l. (4.12)
Blind Equalization 41
Using (4.12), we can write (4.10) as
εL(Γm) =L∑l=0
E{||γHL+K−1−jx(i)||2}
=L∑l=0
γHL+K−1JlRx(J
l)TγL+K−1
= γHL+K−1
(L∑l=0
J lRx(Jl)H
)γL+K−1
= γHL+K−1RxγL+K−1,
(4.13)
where Rx is the covariance matrix of x(i).
The vector γL+K−1 can be found to be the eigenvector that corresponds to the
minimum eigenvalue of matrixRx. Our result resembles (but is different from) that of
[31, 38], although [31] does not adopt an optimality criterion. Using similar arguments
as in [31, Theorem 2], we can establish that as the SNR increases, the solution will
converge to the zero-forcing one as the MMSE equalizer does.
We must admit that although this method is criterion based, it does pose an
unnecessary condition on the structure of equalizer Γm, which may result in perfor-
mance degradation. Actually, it has been shown in [27] that a more general equalizer
with optimum delay can be considered and the optimum delay is determined by the
number of maximum phase channel zeros. If the noise covariance were known, it can
also be taken in to account to potentially improve performance. However, the method
we proposed in this section has one major advantage: it can be implemented online
with adaptive algorithms like LMS or RLS following standard procedures. We com-
ment that other methods (direct or indirect) proposed in [31] can also be implemented
42 Chapter4
adaptively, although not so immediately because eigen-decomposing of expectation of
a random matrix (in our case, the covariance matrix) adaptively requires stochastic
approximation algorithms (see [2], and [23] for details).
43
Chapter 5
Performance and Comparisons
In this chapter, we will analyze performance of the AMOUR system and test it in
several experimental systems with BER calculation and Monte-Carlo simulations.
5.1 AMOUR performance with ZF equalization
Because η(n) is AGN theoretical BER evaluation is possible for a given constellation
when we adopt a ZF receiver. For simplicity, we focus first on BPSK sm(n)’s and
the downlink setup (same Hm(z) = H(z) ∀m). The mth user’s ZF receiver can be
described by the matrix
Gm := H†mV−1m Vm, (5.1)
whose kth row is denoted as gHmk. Our figure of merit is the average BER
Pe := (MK)−1M−1∑m=0
K−1∑k=0
Pe,mk, (5.2)
44 Chapter5
where Pe,mk denotes BER for the kth symbol of user m. Because
sm(iK + k) = Kmsm(iK + k) + gHmkη(i), (5.3)
our SNR will be K2m/(N0g
Hmkgmk/2) and with 2Eb/N0 denoting bit SNR, we have
K2m = Eb/Ec,m where
Ec,m :=
M(L+K)−1∑p=0
|cm(p)|2 (5.4)
is the energy of the mth user’s code (note that Ec,m is same ∀m for the ρm,l’s chosen
as in (3.21)); hence,
Pe,mk = Q
(√1
gHmkgmkEc,m
√2EbN0
), (5.5)
where Q(·) denotes the Q-function.
In comparison, M-user OFDMA will exhibit mth equalizer output
SNR = |H(2πm
M)|2Eb/(N0/2), (5.6)
and thus average BER:
Pe =1
M
M−1∑m=0
Q
(|H(ej
2πmM )|
√2EbN0
), (5.7)
where H(z) is for OFDMA, a channel common to all users (downlink setup).
5.2 Testing AMOUR’s performance
Test case 1 We compared (5.5) with (5.7) on a system with M = 16 users sharing
a common multipath channel of order L = 1 with a single root located at ρ =
Performance and Comparisons 45
0 2 4 6 8 10 1210
−6
10−5
10−4
10−3
10−2
10−1
100
Eb/N
0 (dB)
aver
age
BE
R p
er u
ser’s
sym
bol
OFDMA ρ=0 OFDMA ρ=0.5OFDMA ρ=0.7OFDMA ρ=1 AMOUR ρ=0 AMOUR ρ=0.5AMOUR ρ=0.7AMOUR ρ=1
Figure 5.1: AMOUR vs. OFDMA, 16 users, L = 1
0, 0.5, 0.7, 1. Because OFDMA’s spreading gain is (M +L)/M , for a fair comparison
with AMOUR, we chose K = M = 16 (c.f. (1)). Fig. 5.1 shows BER gains of
AMOUR over OFDMA by 2-3 orders of magnitude as ρ approaches the unit circle.
Test case 2 To avoid channel dependent performance, we averaged (5.5) over 100
Monte Carlo realizations of 6th order Rayleigh faded channels (simulated with com-
plex Gaussian coefficients) and obtained AMOUR’s Pe vs Eb/N0 curves parameterized
by the number of signature roots L assigned to each the M = 16 users (Fig. 5.2). K
values were chosen according to (2.9) to maintain the same rate. Even small values
of the diversity factor L offer considerable BER gains over OFDMA (L = 0). Further
BER improvement is possible by precoding sm(i) blocks as um(i) = Fmsm(i), as we
46 Chapter5
−3 −1 1 3 5 7 9 11 13 15 1710
−14
10−12
10−10
10−8
10−6
10−4
10−2
100
Eb/N0 (dB)
Ave
rage
BE
R
~L=0
~L=1
~L=2
~L=3
Figure 5.2: AMOUR with L < L = 6 (Rayleigh fading)
suggested in Fig. 6.1.
Test case 3 To test AMOUR’s ability for channel independent blind demodulation
in uplink systems, we simulated M = 4 users each transmitting blocks of K = 16
QPSK symbols with L+K = 17 signature roots, through a two-ray channel (L = 1).
The signature roots are equally spaced and interleaved along the unit circle. Relying
only on I = L+K = 34 blocks received in AWGN (Es/N0 = 15dB), we recovered each
user’s constellation using the FIR-ZF blind equalizer described in Chapter 4 (equalized
scatter diagrams of the first four users are depicted in Fig. 5.3). Decision feedback
(DF) schemes (see e.g., [42]) can improve performance further. We stress however,
that our basic result does not rely on finite alphabet assumptions; thus, it applies
Performance and Comparisons 47
to general deterministic blind separation and equalization of convolutive mixtures
involving even continuous amplitude precoded (e.g., radar or speech) sources.
−2 0 2−3
−2
−1
0
1
2
3
−2 0 2−3
−2
−1
0
1
2
3
−2 0 2−3
−2
−1
0
1
2
3
−2 0 2−3
−2
−1
0
1
2
3
Figure 5.3: Blindly equalized constellations (Es/N0 = 15dB)
Test case 4 To quantify AMOUR’s performance for blind equalization in uplink
CDMA systems, we used the indirect blind method outlined in Chapter 4, which
first estimates the channels and then applies zero-forcing or MMSE equalization. We
simulated 100 Monte-Carlo realizations of Rayleigh channels of order 1, and tested
performance when I = 34 or I = 100 blocks were received. It follows from (4.5)
that the minimum number of blocks required in order to implement the method is
I = L + K = 17, [31]. Blind performance is shown against the theoretical one
48 Chapter5
0 2 4 6 8 10 12 14 16 18 2010
−4
10−3
10−2
10−1
100
Es/N
0
aver
age
BE
R
AMOUR: M−Carlo. BPSK in−dir eq. M=4, K=16, I=34, L=1, 100 Rayleigh realization
BlindTheoretical
Figure 5.4: Blind equalization: I = 34, indirect method
computed using formula (5.5) with perfect channel information (see Figs. 5.4 and
5.5). We observe that with I = 34 blocks (Fig. 5.4) the blind method undergoes only
a small (2dB) penalty compared with the theoretical bound (5.5), and as I increases
to 100, the blind method can virtually achieve the bound for all simulated SNRs
(> 0). The latter supports substantially the blind equalization of users’ symbols as
promised by our system’s design and speaks for AMOUR’s potential in a wireless
multipath environment.
Test case 5 In this test, we compare the indirect method and the direct method we
described in Chapter 4. We use the same system setup as in Test case 4, except that
instead of using indirect method in equalization, we use the direct method.
In Figs. 5.6 and 5.7, we draw the performance of the direct equalization method
Performance and Comparisons 49
0 2 4 6 8 10 12 14 16 18 2010
−4
10−3
10−2
10−1
100
Es/N
0
aver
age
BE
R
AMOUR: M−Carlo. BPSK non−dir eq. M=4, K=16, I=100, L=1, 100 Rayleigh realization
Monte−Carlotheoretical
Figure 5.5: Blind equalization: I = 100, indirect method
for I = 34 and I = 100, respectively. We can see that the direct equalizer performed
inferior to the indirect method. But the differences are not major, and as I increases
from 34 to 100, we observe that the performance of direct equalizer approaches that
of the MMSE. Although we have not established a proof that the direct method will
converge to the MMSE as I →∞, we have a certain confidence that the direct blind
method performs close to MMSE for most channels, because the performance shown
in Figs. 5.6 and 5.7 were averaged over many (100) random channels.
50 Chapter5
0 2 4 6 8 10 12 14 16 18 2010
−4
10−3
10−2
10−1
100
Es/N
0
aver
age
BE
R
AMOUR: M−Carlo. BPSK dir−eq. M=4, K=16, I=34, L=1, 100 Rayleigh realization
BlindTheoretical
Figure 5.6: Blind equalization: I = 34, direct method
0 2 4 6 8 10 12 14 16 18 2010
−4
10−3
10−2
10−1
100
Es/N
0
aver
age
BE
R
AMOUR: M−Carlo. BPSK dir−eq. M=4, K=16, I=100, L=1, 100 Rayleigh realization
Monte−Carloformular
Figure 5.7: Blind equalization: I = 100, direct method
51
Chapter 6
Extensions of the AMOUR design
The basic AMOUR design of Chapter 3 can be generalized along several directions
to gain flexibility and improve performance. We give three possible extensions here.
6.1 Maximization of single-user information rate
AMOUR transmitters can be generalized as in Fig. 6.1, where fm,k(n), k = 0, . . . , K−
1 are FIR precoding filters of length K. The equivalent block diagram model of
Fig. 6.1 is shown in Fig. 6.2, where Fm is a matrix whose kth column holds coefficients
of the FIR filter fm,k(n):
Fm :=
fm,0(0) · · · fm,K−1(0)
......
...
fm,0(K − 1) · · · fm,K−1(0)
K×K
. (6.1)
52 Chapter6
The motivation behind such a modification is added flexibility to optimize Fm and
Γm when channel estimates of Hm(z) are available, e.g., via a feedback channel as in
CATV and DSL applications. Desirable criteria include minimization of mean-square
error, or, maximization of information rate (throughput) under finite transmit-power
constraints (see [16, 28, 29] and references therein). Using as columns of Fm the eigen-
vectors of the channel matrix Hm obeys the “waterfilling” principle and maximizes
the information rate [29]. OFDM precoders with fm,k(n) = ej2πmnk are also desirable
because they are approximately optimal and computationally attractive since filter-
ing operations are performed via FFTs. With channel eigenvector precoders Fm (or
OFDM with block size K � 1) and corresponding decoding filterbanks denoted by
Γm, each of the parallel frequency-selective single user channels of Fig. 3.1 can be
further split into K parallel subchannels; thus, the AMOUR transceivers are capable
of converting the dispersive multi-input multi-output CDMA system into MK paral-
lel independent flat fading subchannels. Power loading and bit allocation among the
subchannels can be easily incorporated into the AMOUR design thereafter to improve
performance in terms of average bit error rate as in [29].
6.2 VL-AMOUR for code assignment flexibility
In the AMOUR design so far, one user’s signature roots are roots common to all other
users (c.f. eq. (3.1)). Therefore, whenever one user changes code (to avoid channel
nulls, for example), or, when a new user enters the cell, all other users must change
Extensions of the AMOUR design 53
P
PK
K
z
z
sm(n)
.
.
.
.
.
.
mth -transmitter
fm;K�1(n)
fm;0(n)
z�1
z�1
Cm(z)
.
.
.
~Sm(i; z)
Figure 6.1: Modification of AMOUR transmitter to maximize information rate
�m
sm(i) sm(i)Hm(i)Fm(i) �m(i)
Figure 6.2: Equivalent single user block diagram of Fig. 6.1
their codes as well in order to maintain transceiver-orthogonality. Our goal here is
to design the transmit code polynomials {Cm(z)}M−1m=0 and the corresponding receive
polynomials {Gm,l(z)}L+K−1l=0 , m = 0 . . .M − 1, so that they possess the same MUI
eliminating property as AMOUR, but contrary to AMOUR, only the receiver (base
station) needs to change code assignments whenever users change codes, or, whenever
new users enter. The dual Vandermonde Lagrange AMOUR system we will develop
in this section offers such flexibility in the code assignment procedure and generalizes
related results from [28] to the AMOUR framework.
Consider the same system setup as in Fig. 2.1). Instead of using Lagrange
polynomials {Cm(z)}M−1m=0 and Vandermonde receive-filters {Gm,l(z)}
M−1m=0 , ∀l ∈ [0, L+
54 Chapter6
K − 1], we use Vandermonde transmit- and Lagrange receive- filters, resulting in the
dual VL-AMOUR system. In this case, d2) and d3) are replaced by:
d2′) Codes {Cm(z)}M−1m=0 are selected to have order P−1 (no trailing zeros are needed
in VL-AMOUR.)
d3′) Receive-filters {Gm,l(z)}L+K−1l=0 have order M(L+K)− 1, ∀m.
The precoders {Cm(z)}M−1m=0 and receive-filters {Gm,l(z)}
L+K−1l=0 , m = 0, . . . ,M −
1, are built as follows:
i) Start with M(L+K) distinct points {ρm,l}, on the complex plane and for every
m assign L+K signature roots {ρm,l}L+K−1l=0 , to the mth user.
ii) For everym: Cm(z) has coefficients cm(p) = Km∑L+K−1
l=0 ρm,l−P+1+p, p = 0 · · ·P−
1. Equivalently, the code vector cm := [cm(0) . . . cm(P − 1)]H (H denotes Her-
mitian) is a linear combination of L + K Vandermonde vectors v(ρm,l) :=
[1 ρm,l · · · ρm,lP−1]H, and can be written as
cm = Km
L+K−1∑l=0
ρm,l−P+1v(ρm,l). (6.2)
iii) For every m: {Gm,l(z)}L+K−1l=0 are polynomials of order M(L+K)−1 that satisfy
Gm,l(ρµ,λ) = δm,µδl,λ, ∀µ ∈ [0,M − 1], ∀λ ∈ [0, L+K − 1]. (6.3)
Extensions of the AMOUR design 55
Specifically, Gm,l(z) is a Lagrange polynomial
Gm,l(z) =
M−1∏µ=0µ6=m
L+K−1∏λ=0λ6=l
(1− ρµ,λz−1)
M−1∏µ=0µ6=m
L+K−1∏λ=0λ6=l
(1− ρµ,λρm,l−1)
, (6.4)
and because {Gm,l(z)}L+K−1l=0 are of order M(L+K)− 1, the receive-vectors
gTm,l :=
[0 · · · 0︸ ︷︷ ︸L+K−1
gm,l(M(L+K)− 1) · · · gm,l(0)
]T(6.5)
are equipped with P −M(L + K) = L + K − 1 leading zeros, which cancel
inter-block interference.
To establish the MUI eliminating ability of VL-AMOUR transceivers, let us de-
fine channel vectors {hµ}M−1µ=0 , and Toeplitz matrices Sµ(i) for the µth-user’s symbols
and C0µ, C0
µ code matrices as follows:
hµ := [hµ(0) · · · hµ(L)]T , Sµ(i) :=
sµ(i; 0) · · · 0
.... . .
...
sµ(i;K − 1) sµ(i; 0)
.
... . .
.
..
0 · · · sµ(i;K − 1)
(L+K)×L
,
C0µ :=
cµ(0)
.... . .
cµ(L+K) · · · cµ(0)
..
....
cµ(P − 1) · · · cµ(P − L−K − 1)
P×(L+K)
, C1µ :=
0 cµ(P − 1) · · · cµ(P − L−K)
.... . .
...
0 · · · · · · cµ(P − 1)
..
....
0 · · · · · · 0
P×(L+K)
.
With the aforementioned definitions, the ith received block can be written as:
x(i) =M−1∑µ=0
[C0µSµ(i)hµ + C1
µSµ(i− 1)hµ]. (6.6)
56 Chapter6
From (6.2)–(6.6) it follows that Ym(i; ρm,l) := gTm,lx(i) is given by (c.f. Fig. 2.1) the
output of the lth down-sampler of the mth user’s receiver is
Ym(i; ρm,l) =M−1∑µ=0
Kµ
L+K−1∑λ=0
Gm,l(ρµ,λ)Sµ(i; ρµ,λ)Hµ(ρµ,λ) + gTm,lη(i)
=M−1∑µ=0
Kµ
L+K−1∑λ=0
δm,µδl,λSµ(i; ρµ,λ)Hµ(ρµ,λ) + gTm,lη(i)
= KmSm(i; ρm,l)Hm(ρm,l) + gTm,lη(i).
Mimicking the rationale of Chapter 3, the same linear transformations V−1m and Γm
can thereafter be applied to ym(i) := [Ym(i; ρm,0) · · ·Ym(i; ρm,L+K−1)]T successively,
in order to recover the symbols sm(i; k), ∀k ∈ [0, K − 1].
Therefore, the designed VL-AMOUR transmit- and receive- filters have the same
deterministic MUI elimination property as the LV-AMOUR in Chapter 3 and also
convert the multiuser CDMA system into M parallel single-user systems irrespective
of the multipath (see Fig. 3.1). In addition, since the transmit-filters are linear combi-
nations of the Vandermonde vectors derived from the users’ own signature roots only,
when one user µ changes code (or equivalently signature roots), no other user needs
to change and only the receiver, typically the base station, needs to adjust its fil-
ters {Gµ,l(z)}L+K−1l=0 accordingly. This substantially reduces the need for cooperation
required among users and thus agrees with the CDMA philosophy.
Extensions of the AMOUR design 57
Figure 6.3: Variable rates in ADSL applications
6.3 Accommodating different bit rates
In UMTS/IMT2000, end users are to be accommodated with the necessary service
quality for multimedia communications, mainly Internet access, and video/picture
transfer, and high-bit-rate capabilities [5]. Video, slow-scan video, and picture trans-
fer services require bit rates ranging from a few tens of kilobits per second to roughly
2 Mb/s, depending on quality requirements. Multirate support is also very important
in DSL applications such as ADSL (see also Fig. 6.3). Because many multimedia ap-
plications are packet-oriented, it is essential to optimize 3G techniques to effectively
cater for variable bit rate and packet capabilities. We will consider incorporating
variable rates in the AMOUR design in the following. Provision of packet capabilities
will not be discussed here, thanks to our block transmission nature of AMOUR, such
capabilities can be easily incorporated in the system.
58 Chapter6
6.3.1 Multirate services via root allocation
One good idea is to allocate a different number of signature roots per user, and adjust
the transmitted block size K accordingly. Suppose the system is designed with a total
of Ns = M(K + L) signature roots to be allocated to different users. Among those
roots, one user’s signature root number can range from L + 1 to Ns, instead of the
fixed K+L roots per user allocated in (3.3). But the total number of signature roots
should be no more thanNs in order to guarantee the MUI elimination property. Users
that are given more signature roots can afford to transmit at a higher rate (larger K)
information symbols within one block interval, and vice versa. In addition, Ns can
also take integer values other than M(K + L), which offers added flexibility.
6.3.2 Multirate services via virtual users
Another means of incorporating users with different rates is to use the concept of
‘virtual users’ (see Fig. 6.4). Specifically, let one user split his/her information sym-
bols into several substreams each of which is transmitted as if it were generated by a
different user. However, the user hereby is only able to transmit symbols at multiples
of the designed basic rate.
Extensions of the AMOUR design 59
AMOUR Precoder
AMOUR Precoder
S/P
Virtual users
... Σsm(n)Hm(z)
Figure 6.4: Using virtual users to incorporate multiple rates
61
Chapter 7
Conclusions and future topics
Based on transmitter introduced redundancy, we developed a flexible so-called AMOUR
system for multiuser communications, which is especially suitable for uplink CDMA.
Relying on symbol blocking and redundant precoding at the transmitters of Chap-
ter 2, we specified the AMOUR design in Chapter 3. The deterministic MUI elim-
inating ability of AMOUR was proved and identifiability of input symbols can be
guaranteed under mild conditions, without resorting to multiple channel diversity.
In addition, the system was shown to be equivalent to M single input single output
systems. We also established in Chapter 3 that based on a modulo-interpretation,
we can reduce unnecessary bandwidth expansion at the transmitter. We also showed
that the transceivers can even be designed totally from the modulo interpretation,
and we established this redesign by using Bezout’s result.
In Chapter 4, we applied two kinds of equalizers to the AMOUR design, namely,
62 Chapter7
the indirect methods and the direct equalizers. For the indirect method, we derived
the channel estimator, and based on channel estimates, we formed the Zero-forcing
and MMSE equalizer. For the direct method, we borrowed ideas from [31] and pro-
posed a criterion based equalizer design scheme. We also discussed possibilities of
implementing both the direct and indirect methods adaptively.
In Chapter 5, the performance of zero-forcing solutions was analyzed and com-
pared with OFDM for fixed and slow Rayleigh fading channels. We also simulated
the performance of blind equalizers. Tests and numerical simulations demonstrated
the system’s generality, flexibility, superior performance, and strong potential in mul-
tiuser communications.
In Chapter 6, three extensions to the basic AMOUR design were given. The first
was to generalize the AMOUR precoder so that single user mutual information rate
can be maximized. Then we proposed an alternative AMOUR design by switching
the usercode and receiver filters in order to offer the system extra flexibility in terms
of the code assignment procedure. Finally, we briefly discussed how to incorporate
users of different rates in the system design.
Future possible research topics include (direct) blind adaptive equalizers, design
of optimal root allocation algorithms, root hopping, performance evaluation in time-
selective (Doppler induced) channels, and combination of the AMOUR design with
coding for noise suppression.
63
Appendix A
Lagrange Interpolating Polynomial
The Lagrange interpolating polynomial is the polynomial of degree n−1 which passes
through the n points y0 = f(z0), y1 = f(z1), . . . , yn−1 = f(zn−1), where zi, yi are
complex numbers, i = 0, . . . , n− 1, and zi 6= zj , if i 6= j. It is given by
P (z) =n−1∑j=0
Pj(z), (A.1)
where
Pj(z) =n∏k=1k 6=j
z − zkzj − zk
yj. (A.2)
It can be seen that Pj(zk) = δjkyk; therefore,
P (zk) =n−1∑j=0
Pj(zk) =n−1∑j=0
δjkyk = yk. (A.3)
The Lagrange interpolating polynomial is unique. Suppose there are two polyno-
mials P1(z) and P2(z) that both pass through all the points (zi, yi), i = 0, . . . , n− 1.
64 ChapterA
Then their difference Pd(z) = P1(z) − P2(z) should be zero at n points z = zi,
i = 0, . . . , n − 1. But, Pd(z) is a polynomial of order ≤ n. The only possible case is
Pd(z) ≡ 0.
The coefficients of P (z) = p0 + p1z + · · · + pn−1zn−1 can also be obtained by
solving the linear system of equations:
1 z1 z21 · · · zn−1
1
1 z2 z22 · · · zn−1
2
......
...
1 zn−1 z2n−1 · · · zn−1
n−1
p0
p1
...
pn−1
=
y0
y1
...
yn−1
. (A.4)
Since the Vandermonde matrix in (A.4) is built from n different points z = zi,
i = 0, . . . , n−1, it is non-singular. Thus, it also follows from (A.4) that the Lagrange
interpolating polynomial is unique.
65
Appendix B
Proof of Lemmas 3.1 and 3.4
B.1 Proof of Lemma 3.1
Let a(z), b(z), c(z) be three polynomials, where c(z) =∏n−1
i=0 (z − zi) has n roots at
zi, i = 0, . . . , n− 1. Using the polynomial division identity we can write
a(z) = c(z)qa(z) + ra(z), (B.1)
b(z) = c(z)qb(z) + rb(z), (B.2)
where qa(z) and qb(z) are quotients, and ra(z) and rb(z) are remainders. If deg(ra(z)) ≤
deg(c(z)) and deg(rb(z)) ≤ deg(c(z)), such decompositions are unique and ra(z) =
a(z) mod c(z), rb(z) = b(z) mod c(z).
Suppose a(z) mod c(z) = b(z) mod c(z), i.e., ra(z) = rb(z). Evaluating (B.1)
and (B.2) at roots of c(z), namely, zi, i = 0, . . . , n− 1, reveals that a(zi) = b(zi), i =
0, . . . , n− 1.
66 ChapterB
On the other hand, suppose a(zi) = b(zi), i = 0, . . . , n − 1, it follows from
(B.1) and (B.2) that ra(zi) = rb(zi), i = 0, . . . , n − 1. But both ra(z) and rb(z)
are polynomials of order ≤ n. Therefore, it must hold that ra(z) = rb(z), namely,
a(z) mod c(z) = b(z) mod c(z).
B.2 Proof of Lemma 3.4
Let a(z), c(z), and d(z) be three polynomials and assume that c(z) factors d(z). Since
a(z) mod c(z) = [a(z) mod c(z)] mod c(z) (Lemma 3.2),
a(z) and [a(z) mod c(z)] are equal at roots of c(z), including all roots of d(z) because
c(z) factors d(z). Therefore, a(z) and [a(z) mod c(z)] are equal at all roots of d(z).
Using Lemma 3.2 again shows that [a(z) mod c(z)] mod d(z) = a(z) mod d(z).
67
Appendix C
Existence and uniqueness of Cm(z)
in (3.18)
We prove that there exist unique Cm(z)’s that satisfy (3.18) ∀m.
For every m, Cµ(z) mod Gm(z) = δµ,m, and {Gm(z)}M−1m=0 are coprime; thus
L+K−1∏µ=0µ6=m
Gµ(z)must divide Cm(z); i.e.,
Cm(z) = Qm(z)L+K−1∏µ=0µ6=m
Gµ(z),
where Qm(z) has order < L+K.
From Cm(z) mod Gm(z) = 1, and the division identity we have that,
Qm(z)L+K−1∏µ=0µ6=m
Gµ(z) = Dm(z)Gm(z) + 1, (C.1)
where Dm(z) denotes the quotient of Cm(z) divided by Gm(z).
68 ChapterC
Since deg(Qm(z)) < deg(Gm(z)), it is clear that deg(Dm(z)) < deg(L+K−1∏µ=0µ6=m
Gµ(z)).
Using Bezout’s result [15, p.141], there exist unique Qm(z) and Dm(z) that satisfy
(C.1). The existence and uniqueness of Cm(z) follow.
69
Bibliography
[1] S. E. Bensley and B. Aazhang, “Subspace-based channel estimation for code
division multiple access communication systems,” IEEE Transactions on Com-
munications, vol. 44, pp. 1009–1020, Aug. 1996.
[2] C. Chatterjee and V. P. Roychowdhury, “An adaptive stochastic approximation
algorithm for simultaneous diagonalization of matrix sequences with applica-
tions,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.
19, no. 3, pp. 282–7, Mar. 1997.
[3] Q. Chen, E. S. Sousa, and S. Pasupathy, “Multicarrier CDMA with adaptive
frequency-hopping for mobile radio systems,” IEEE Journal on Selected Areas
in Communications, pp. 1852–1857, Dec. 1996.
[4] A. Chouly, A. Brajal, and S. Jourdan, “Orthogonal multicarrier techniques
applied to direct sequence spread spectrum CDMA systems,” in Proc. of IEEE
GLOBECOM, Houston, USA, Nov. 1993, pp. 1723–28.
[5] E. Dahlman, B. Gudmundson, M. Nilsson, and J. Skold, “UMTS/IMT-2000
based on wideband CDMA,” IEEE Communications Magazine, vol. 36, no. 9,
pp. 70–80, Sept. 1998.
[6] V. M. DaSilva and E. S. Sousa, “Performance of orthogonal CDMA codes for
quasi-synchronous communication systems,” in Proc. of IEEE ICUPC ’93, Ot-
tawa, Canada, Oct. 1993, pp. 995–99.
[7] V. M. DaSilva and E. S. Sousa, “Multicarrier orthogonal CDMA signals for
quasi-synchronous communication systems,” IEEE Journal on Selected Areas in
Communications, pp. 842–852, June 1994.
70 BIBLIOGRAPHY
[8] K. Fazel and G. P. Fettweis (Eds), Multi-Carrier Spread Spectrum, Kluwer
Academic Publishers, 1997.
[9] K. Fazel and L. Papke, “On the performance of convolutionally-coded
CDMA/OFDM for mobile communication system,” in Proc. of IEEE PIMRC
’93, Yokohama, Japan, Sept. 1993, pp. 468–72.
[10] D. Gesbert, J. Sorelius, and A. Paulraj, “Blind multi-user MMSE detection of
CDMA signals,” in Proc. of Intl. Conf. on ASSP, May 1998, pp. 3161–3164.
[11] S. Hara and R. Prasad, “Overview of multicarrier CDMA,” IEEE Communica-
tions Magazine, pp. 126–133, Dec. 1997.
[12] M. Honig, U. Madhow, and S. Verdu, “Blind adaptive multiuser detection,”
IEEE Transactions on Information Theory, vol. 41, pp. 944–960, July 1995.
[13] R. A. Iltis, “Performance of constrained and unconstrained adaptive multiuser
detectors for quasi-synchronous CDMA,” IEEE Transactions on Communica-
tions, vol. 46, no. 1, pp. 135–431, Jan. 1998.
[14] B. Jabbari, “Wideband CDMA,” IEEE Communications Magazine, vol. 36, no.
9, pp. 46, Sept. 1998.
[15] T. Kailath, Linear Systems, Englewood Cliffs, N.J. , Prentice-Hall, 1980.
[16] S. Kasturia, J. T. Aslanis, and J. M. Cioffi, “Vector coding for partial response
channels,” IEEE Transactions on Information Theory, vol. 36, pp. 741–761, July
1990.
[17] S. Kondo and L. B. Milstein, “Performance of multicarrier DS CDMA systems,”
IEEE Transactions on Communications, vol. 44, no. 2, pp. 238–46, Feb. 1996.
[18] H. Liu and G. Xu, “A subspace method for signature waveform estimation in
synchronous CDMA systems,” IEEE Transactions on Communications, vol. 44,
no. 10, pp. 1346–1354, Oct. 1996.
[19] R. Lupas and S. Verdu, “Linear multiuser detectors for synchronous code-division
multiple-access channels,” IEEE Transactions on Information Theory, vol. 35,
no. 1, pp. 123–136, Jan. 1989.
BIBLIOGRAPHY 71
[20] R. Lupas and S. Verdu, “Near-far resistance of multiuser detectors in asyn-
chronous channels,” IEEE Transactions on Communications, vol. 38, no. 4, pp.
496–508, Apr. 1990.
[21] U. Madhow and M. Honig, “MMSE interference suppression for direct-sequence,
spread-spectrum CDMA,” IEEE Transactions on Communications, vol. 42, pp.
3178–3188, Dec. 1994.
[22] E. Nikula, A. Toskala, E. Dahlman, L. Girard, and A. Klein, “FRAMES multiple
access for UMTS and IMT-2000,” IEEE Personal Communications, pp. 16–24,
April 1998.
[23] E. Oja and J. Karhunen, “On stochastic approximation of eigenvectors and
eigenvalues of expectation of a random matix,” Journal Math. Analysis Appl.,
vol. 106, pp. 69–84, 1985.
[24] T. Ojanpera, “Overview of multiuser detection/interference cancellation for DS-
CDMA,” in IEEE International Conference on Personal Wireless Communica-
tions, New York, NY, 1997, pp. 115–119.
[25] J. Proakis, Digital Communications, McGraw-Hill, New York, 1989.
[26] J. Ramos and M. Zoltowski, “Reduced complexity blind 2-D RAKE receiver for
CDMA,” in Proc. 8th IEEE SP Workshop Statist. Signal Array Process., Corfu,
Greece, June 1996, pp. 502–505.
[27] A. Scaglione, S. Barbarossa, and G. B. Giannakis, “Inverting overdetermined
toeplitz systems with application to block-adaptive equalization,” in Proc. of
the Asilomar Conf., Pacific Grore, CA, Nov. 1998.
[28] A. Scaglione, S. Barbarossa, and G. B. Giannakis, “Fading-resistant and MUI-
free codes for CDMA systems,” in Proc. of Intl. Conf. on ASSP, Phoenix, AZ,
Mar. 1999.
[29] A. Scaglione, S. Barbarossa, and G. B. Giannakis, “Filterbank transceivers
optimizing information rate in block transmissions over dispersive channels,”
IEEE Transactions on Information Theory, vol. 5, no. 3, pp. 1019–1032, Apr.
1999.
72 BIBLIOGRAPHY
[30] A. Scaglione and G. B. Giannakis, “Design of user codes in QS-CDMA systems
for MUI elimination in unknown multipath,” IEEE Communications Letters,
pp. 25–27, Feb. 1999.
[31] A. Scaglione, G. B. Giannakis, and S. Barbarossa, “Self-recovering multirate
equalizers using redundant filterbank precoders,” in Proc. of Intl. Conf. on
ASSP, 1998, pp. 3501–3504.
[32] A. Scaglione, G. B. Giannakis, and S. Barbarossa, “Redundant filterbank pre-
coders and equalizers Part I: Unification and optimal designs,” IEEE Transac-
tions on Signal Processing, vol. 47, pp. 1988–2006, July 1999.
[33] E. Sourour and M. Nakagawa, “Performance of orthogonal multicarrier CDMA
in a multipath fading channel,” IEEE Transactions on Communications, vol. 44,
no. 3, pp. 356–67, Mar. 1996.
[34] M. K. Tsatsanis, “Inverse filtering criteria for CDMA systems,” IEEE Transac-
tions on Signal Processing, vol. 45, no. 1, pp. 102–112, Jan. 1997.
[35] M. K. Tsatsanis and G. B. Giannakis, “Multirate filter banks for code-division
multiple access systems,” in Proc. of Intl. Conf. on ASSP, New York, NY, USA,
1995, vol. 2, pp. 1484–7.
[36] M. K. Tsatsanis and G. B. Giannakis, “Optimal decorrelating receivers for DS-
CDMA systems: A signal processing framework,” IEEE Transactions on Signal
Processing, vol. 44, no. 12, pp. 3044–55, Dec. 1996.
[37] M. K. Tsatsanis and Z. Xu, “Constrained optimization methods for blind equal-
ization of multiple FIR channels,” in Proceedings of the 31st Asilomar Confer-
ence on Signals, Systems & Computers, Pacific Grove, CA, USA, Nov. 1997, pp.
128–132.
[38] M. K. Tsatsanis and Z. Xu, “Performance analysis of minimum variance CDMA
receiver,” IEEE Transactions on Signal Processing, vol. 46, no. 11, pp. 3014–22,
Nov. 1998.
[39] L. Vandendorpe, “Multitone direct sequence CDMA system in an indoor wireless
environment,” in Proc. of IEEE First Symposium of Communications and Ve-
hicular Technology, Benelux, Delft, The Netherlands, Oct. 1993, pp. 4.1–1–4.1.8.
BIBLIOGRAPHY 73
[40] L. Vandendorpe, L. Cuvelier, and O. van de Wiel, “Fractionally spaced linear
and decision-feedback detectors for transmultiplexers,” IEEE Transactions on
Signal Processing, vol. 46, no. 4, pp. 996–1011, Apr. 1998.
[41] S. Verdu, “Minimum probability of error for asynchronous Gaussian multiple-
access channels,” IEEE Transactions on Information Theory, vol. 32, no. 1, pp.
85–96, Jan. 1986.
[42] S. Verdu, Multiuser Detection, Cambridge Press, 1998.
[43] X. Wang and H. V. Poor, “Blind equalization and multiuser detection in disper-
sive CDMA channels,” IEEE Transactions on Communications, vol. 46, no. 1,
pp. 91–103, Jan. 1998.
[44] X. Wang and H. V. Poor, “Blind multiuser detection: A subspace approach,”
IEEE Transactions on Information Theory, vol. 42, pp. 677–690, Mar. 1998.
[45] N. Yee, J-P. Linnartz, and G. Fettweis, “Multicarrier CDMA in indoor wireless
radio networks,” in Proc. of IEEE PIMRC ’93, Yokohama, Japan, Sept. 1993,
pp. 109–13.