Multirate Transceivers for Block wireless Transmissions...

81
Multirate Transceivers for Block wireless Transmissions and Multicarrier CDMA Irrespective 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

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

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������������������������������������������������������������

������������������������������������������������������������

������������������������������������������������������������

����������������

������������������������������

������������������������������

������������������������������

������������������������������

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.

16 Chapter2

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.

34 Chapter3

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

L+K−1∑λ=0

Gm,l(ρµ,λ)Sµ(i; ρµ,λ)Hµ(ρµ,λ) + gTm,lη(i)

=M−1∑µ=0

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

60 Chapter6

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

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