ido binyamini july 25, 2012 - bar ilan university · final thesis ido binyamini the research thesis...

Adaptive precoder for FEXT cancellation in

multichannel downstream VDSL

Ido Binyamini

July 25, 2012

Adaptive precoder for FEXTcancellation for multichannel

downstream VDSL

Final Thesis

Ido Binyamini

The Research thesis was done under the supervision of dr. Itsik Bergel from the School of

Engineering at Bar-Ilan University.

Submitted as Partial Fulfillment of the Requirements for the DegreeMaster of Science in Electrical Engineering

Ramat Gan, Israel July 25, 2012

Acknowledgement

I wish to express my deep gratitude and appreciation to my supervisors Dr. Itsik Bergel.

This thesis could not have been written without his supervision and encouragement.

iii

Contents

Abstract 1

Notation 3

1 Introduction 5

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Precoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Background 9

2.1 Precoding schemes for DSL . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 The contribution of this research . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Problem Formulation 13

3.1 Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 The research structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Partial precoding 17

4.1 Partial precoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 Convergence analyzes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

v

vi CONTENTS

4.2.1 Convergence conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2.2 Theorem proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.3 Steady State Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Low Bit Rate Feedback 39

5.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2 Low Bit Rate Convergence analyzes . . . . . . . . . . . . . . . . . . . . . . . 41


5.2.2 Steady State Approximation . . . . . . . . . . . . . . . . . . . . . . . 47

5.3 Performance analysis with additional coarse power quantization . . . . . . . 47


5.3.2 theorem proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.4.1 Phase feedback only . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.4.2 Phase feedback plus coarse power feedback . . . . . . . . . . . . . . . 56

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6 Research summary 61

6.1 Research summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Bibliography 63

List of Figures

1.1 Network topology with fiber to the point (FTTB/FTTC/FTTN) topology.

Bold lines are the channels from the ONU to the end users’, thin lines are the

far end crosstalk (FEXT) coupling channels. . . . . . . . . . . . . . . . . . . 6

4.1 Convergence of the adaptive precoder to the ideal precoder in mean squares as

a function of the number of symbols transmitted, at a distance of 300m and a

frequency of 14.25MHz (averaged over 500 systems). Also shown is the upper

bound on the convergence defined by (4.56). . . . . . . . . . . . . . . . . . . 32

4.2 Average SINR (of all 28 users) at a distance of 300m and frequency of 14.25MHz,

using the adaptive precoder and the ideal precoder, as a function of the number

of symbols transmitted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3 Average capacity (of all 28 users) using the adaptive precoder and the ideal

precoder, as a function of the number symbols transmitted. . . . . . . . . . . 34

4.4 Sum of all users’ square error over time, using channel measurements at a dis-

tance of 300m over a frequency of 14.25MHz. Also shown is the approximation

given by (4.74). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

vii

viii LIST OF FIGURES

4.5 Sum of all users’ square error in steady state, derived by using channel mea-

surements at a distance of 300m over a frequency of 14.25MHz and cancelling

20 cross-talkers per user (r=20). Also shown is (dotted line) the approximation

given by (4.74). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.1 Phase quantization with M=4 levels when rotating the axis in a randomly phase. 40

5.2 Performance of the dithered phase quantization and the regular phase quan-

tization, measured by the averaged in time absolute error scaled by σv, as a

function of time. The figure compares three different levels of phase quantization. 52

5.3 The absolute error of one user, the corresponding averaged in time absolute

error, and the bound given by (5.10) as function of time, α√

Ps = 5 · 10−4. . 53

5.4 Maximum over all users, of the averaged in time absolute error, and the bound

given by (5.10), as a function of time, α√

Ps = 0.1 . . . . . . . . . . . . . . . 54

5.5 The averaged in time absolute error in the steady state as a function of the

precoder step size. The empiric figure shows the averaged absolute error,

averaged over 100 simulations, as well as the bound given by (5.10) and the

approximation given by (5.40). . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.6 Average user rate at a distance of 300 meters as a function of time for the non-

quantized error feedback, the phase only quantization and the phase quanti-

zation plus coarse power quantization with K = 10. . . . . . . . . . . . . . . 57

5.7 Sum of the square errors of all users over all frequencies using a precoder that

adapts according to the phase quantization plus coarse power feedback and

K = 10, divided by the sum of the corresponding square errors using the ZF

precoder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Abstract

In this thesis we analyze a simplified, adaptive precoder for FEXT cancellation in the down-

stream of multichannel VDSL which, is based on error signal feedback. This thesis analyzes

this precoder for two low complexity schemes. In the first scheme the system complexity is

reduced by canceling only the FEXT generated by subset of the other users. Unlike previous

works, in which each user experienced either a complete FEXT cancellation or no cancel-

lation at all, we present a precoder which can mitigate the FEXT from any desired subset

of interfering users for each user. We derive sufficient conditions for the convergence of this

adaptive precoder, for any partial cancellation scheme, and provide closed form steady state

error analysis. The second scheme reduces the system complexity by limiting the feedback

to low bit-rate quantization of the error signals measured by the receivers. We derive bounds

on the performance, convergence conditions and steady state approximation. The precoder’s

performance and convergence properties are also demonstrated through simulations.

1

2 LIST OF FIGURES

Notation

x scalarx column vectorA matrixaij the (i, j) element of the matrix A

Aj the j-th row of the matrix A

A−1 matrix inverse

(·)T transpose operation(·)∗ conjugate operation

(·)H transpose-conjugate operation

(·)−H inverse-transpose-conjugate operationdiagx diagonal matrix with the vector x on its diagonal

(·)1

2 for diagonal matrices, a diagonal matrix with the square root of the diagonal‖ · ‖ Euclidian norm operationI identity matrixE (·) expectation operation

3

4 LIST OF FIGURES

Chapter 1

Introduction

1.1 Introduction

Digital Subscriber Line (DSL) systems are rapidly gaining popularity as a broadband access

technology capable of reliably delivering high data rates over telephone subscriber lines. The

successful deployment of asymmetric DSL (ADSL) systems, suggested in [1], has helped

reveal the potential of this technology, and current advanced DSL technology such as VDSL

(very high-bit-rate DSL) is designed to support the wide deployment of triple play services

such as voice, video, data, high definition television (HDTV) and interactive gaming. This

thesis deals with canceling interference between users in the downstream of Very high-bit-

rate Digital Subscriber Line (VDSL) system. This interference is caused by electromagnetic

coupling between adjacent wires serving different users over the same frequency band.

In a typical DSL topology the optical fiber ends at an optical network unit (ONU) and the

data are further distributed to the users over the existing copper infrastructure. Thanks to the

increasing deployment of optical fiber infrastructure, advanced DSL technology such as VDSL

is now required to reliably deliver high data rates over shorter loops of telephone subscriber

lines. Using shorter lines significantly improves the SNR. However, delivering high data

rates over unshielded twisted pair (UTP) of copper wires causes significant electromagnetic

coupling between neighboring twisted-pairs within a binder. This electromagnetic coupling

commonly termed crosstalk limits both the data rate and reach of a VDSL service [2], [3].

The crosstalk can be divided into two types: Near-End Crosstalk (NEXT) which refers to the

signals originated from the same side of the binder (due to the existence of downstream and

upstream transmission) and Far-End crosstalk (FEXT) which refers to the coupling between

the received signal and signals originated in the opposite side of the binder. In VDSL the

5

6 CHAPTER 1. INTRODUCTION

Figure 1.1: Network topology with fiber to the point (FTTB/FTTC/FTTN) topology. Boldlines are the channels from the ONU to the end users’, thin lines are the far end crosstalk(FEXT) coupling channels.

impact of NEXT is avoided by employing Frequency Duplexing Division (FDD) to separate

downstream and upstream transmission and transmission synchronization. Based on Discrete

Multi-Tone (DMT) Transmission for Asymmetric Digital Subscriber Lines (ADSL) [4], [5],

VDSL systems use duplex scheme, called Zipper ([6], [7], [8]) to avoid NEXT by using different

orthogonal subcarriers in the upstream and downstream directions to divide the available

bandwidth. It has high flexibility to divide the capacity between the up and down stream, as

well as good coexistence possibilities with other systems such as ADSL. The FEXT, however,

can significantly reduce the system capacity. In upstream (US) transmission FEXT can be

removed by using crosstalk cancellation techniques [9], [10]. These techniques typically rely

on the fact that receiving (RX) modems are collocated at the (ONU). In the downstream, Rx

modems are scattered at different customer premises and there is no possibility to perform

FEXT cancellation at the end users side. On the other hand, the TX modems are co-located

at the ONU, and hence FEXT cancellation is feasible for the downstream using a proper

precoding at the ONU.

1.2. PRECODING 7

1.2 Precoding

As mentioned above, in the downstream, the receiving modems are located in different cus-

tomer premises and it is not possible to cancel FEXT at the end users’ side. On the other

hand, the transmiting modems are co-located at the ONU, and hence FEXT cancellation is

feasible for the downstream if proper precoding is employed at the ONU. A precoding sys-

tem uses some channel state information in order to approximate the interference the receiver

will pick in addition to the transmitted signal. Based on this approximated interference the

transmitter can distort the transmitted signal in order to Pre-compensate for the interference.

From an implementation perspective, precoding algorithms can be sub-divided into linear and

nonlinear precoding types. The capacity achieving algorithms are nonlinear [11], but linear

precoding approaches usually achieve reasonable performance with much lower complexity.

Optimal linear precoding is known as Minimum Mean Square Error (MMSE) precoding,

[12]. In this method the precoding weights for a given user are selected to maximize the

ratio between the signal power and the interference power (generated at other users) plus

noise. Thus, optimal precoding means finding the optimal balance between achieving strong

signal gain and limiting co-user interference [13]. Finding the optimal MMSE precoding is

often difficult, leading to approximate approaches that concentrate on either the numerator

or denominator of the mentioned ratio; that is, maximum ratio transmission (MRT) [14] and

zero-forcing (ZF) [15] precoding. MRT only maximizes the signal gain at the desired user.

MRT is close-to-optimal in noise-limited systems, where the co-user interference is negligible

compared to the noise. ZF precoding aims at nulling the co-user interference, at the expense

of losing some signal gain. ZF precoding can perform close to the system capacity when the

system is interference-limited, (i.e., the noise is weak compared to the interference) as is the

case in a typical DSL systems.

1.3 Thesis Structure

The Thesis is structured as follows: In Chapter 2 we briefly review some precoding schemes

proposed for FEXT cancelation in DSL systems and describe the motivation for our work.

In chapter 3 we describe the mathematical model for multi-pair DSL systems and describe

the problem we dealt with in our work. In Chapters 4 and 5 we present and analyze two low

complexity adaptive precoders for FEXT cancelation in DSL.

8 CHAPTER 1. INTRODUCTION

Chapter 2

Background

2.1 Precoding schemes for DSL

Most research on downstream precoding is based on the assumption that accurate enough

channel state information (CSI) is available at the ONU, based on channel estimate feedback

from the end-user. Ginis and Cioffi [16], [10], considered the problem of a DSL system with

coordinated transmission and uncoordinated receivers, and proposed a transmitter precoding

scheme based on (generalized) zero-forcing equalization and TomlinsonHarashima precoding

[17], [18], [19], which might be interpreted as a suboptimal implementation of the zero-forcing

dirty-paper precoding scheme proposed in [20] and studied in [21]. Their decision-feedback

structure, based on the Tomlinson-Harashima precoder (THP), was shown to operate close

to the single-user bound [10]. An improvement of the scheme of [16] was proposed in [22],

where TomlinsonHarashima precoding is replaced by more efficient trellis precoding schemes.

But, this structure relies on a nonlinear modulo operation at the receiver side, thus requiring

an upgrade of all CPEs in the network. Cendrillon et. al [23], [24] [25], noted that it

is sufficient to use linear precoding due to the diagonal dominance property of the FEXT

coupling matrix. Still, their zero-forcing (ZF) solution requires matrix inversion at each

tone of the multichannel DSL system (and state of the art DSL systems use thousands of

tones). Leshem and Li [26], [27], proposed a simplified approximate precoder, based on a first

or second order approximation of the the ZF precoding matrix, which significantly reduces

the computational complexity. All the solutions above assume sufficiently good channel

estimates. In [28] the effect of precoder quantization and channel estimation is discussed,

assuming that crosstalk estimation is performed at the Rx modem.

Louveaux and var der veen [29] suggested to use the sign of the symbol error measured

9

10 CHAPTER 2. BACKGROUND

by the users as feedback to the transmitter in order to allow the estimation of the crosstalk

channels at the transmitter (where the precoder needs to be computed). In [30] they sug-

gested a precoder that adapts according to the symbol error measured by the users and

transmitted back as feedback from the end-users to the ONU. Bergel and Leshem, [31], an-

alyzed a simplified version of this adaptive precoder and presented convergence conditions

and a steady state performance analysis. In their analysis they have showed that the conver-

gence conditions can be more relaxed when not performing FEXT cancellation from all users.

Moreover, they have showed that the convergence performance depends on the users signal

to noise ratio (SNR) and is actually dominated by the users’ with the worst SNR. Thus, If

the system includes some users with poor SNR, not performing FEXT cancellation for them,

can significantly accelerate the precoder convergence. On the other hand, the performance

of users with poor SNR are limited by the received noise and hence such users cannot benefit

much from FEXT cancellation and therefore the operation of FEXT cancellation for such

users will waste system resources with almost no gain.

In addition to not canceling FEXT for users with poor SNR in practical systems partial

FEXT cancelation is important also because of complexity limitations. Currently DSL system

serves dozens of users and state of the art DSL systems use thousands of tones, therefore

multiplying the data before each transmission by a precoder matrix, significantly increases

the computational complexity. Cendrillon et al. [32] noted that crosstalk channels in DSL

are space and frequency selective and that the majority of crosstalk comes from a few users

and its effect is limited to a subset of tones. Partial FEXT cancelation exploits this by

limiting FEXT cancelation to the tones and lines where it gives the maximum benefit. Their

model results with a precoder matrix which have zeros in the places that corresponds to non

canceled FEXT from other users. As a result, these schemes can achieve the majority of the

gains of full crosstalk FEXT cancellation at a fraction of the run-time complexity. They also

presented a method to calculate the partial FEXT cancellation matrix for given subsets of

canceled FEXT terms.

2.2 The contribution of this research

In this thesis research we analyzed two adaptive precoders. In chapter 4 of this thesis we

present and analyze an adaptive partial percoder, based on the precoder analyzed in [31],

which can reach the performance of the partial FEXT cancellation matrix presented at [32].

2.2. THE CONTRIBUTION OF THIS RESEARCH 11

Moreover, we analyze the general case of partial FEXT cancellation, where we show that

the proposed adaptive partial percoder can reach the performance of any required partial

FEXT cancellation matrix. For this adaptive partial percoder we consider also a variable

step size used for the adaptation. We use the variable step size to prove that the partial

adaptive precoder converges to the required partial FEXT cancellation matrix when the step

size decreases in the correct rate. We also give an approximation steady state analysis of this

precoder. All results are demonstrated through simulations. This work has been accepted

for publication as a regular paper in the IEEE Transactions on Signal Processing under

the name: ”Arbitrary partial FEXT cancellation in adaptive precoding for multichannel

downstream VDSL”.

However, all analysis of adaptive precoding for DSL has assumed a rich enough quanti-

zation. This is obvoiusly not the case in practical systems, where the adaptive precoder is

expected to rely on a low rate feedback. In chapter 5 we analyze the performance of the

precoder analyzed in [31] with low bit-rate quantization of the error signals measured by the

receivers. We give a sufficient condition on the channel that allows the precoder convergence

for any step size used for adapting the precoder. We also give a good steady state error

approximation. Yet, in many cases faster convergence is desirable. We show that a coarse

power quantization, in addition to the phase quantization can significantly accelerate the

precoder convergence, with a negligible cost in feedback rate. All results are demonstrated

through simulations. This work has been submitted to the IEEE Transactions on Signal Pro-

cessing under the name ”Adaptive precoder using sign error feedback for FEXT cancellation

in multichannel downstream VDSL”.

12 CHAPTER 2. BACKGROUND

Chapter 3

Problem Formulation

3.1 Signal Model

Following VDSL2 conventions, We consider a discrete multi-tone (DMT) system where trans-

mission takes place independently over many narrow sub-bands. Also, we assume that the

system operates in a frequency division duplex mode (FDD), where upstream and down-

stream transmissions operate at separate frequency bands, and all transmissions in the binder

are synchronized. Hence, near end cross talk (NEXT) is eliminated. To reduce the FEXT,

the system jointly coordinates the transmissions on all u twisted pairs in a binder. Focusing

on a single frequency bin, the n-th received symbol for all pairs can be written in vector form

as:

xn = HFnsn + vn, (3.1)

where

H =

h11 · · · h1u...

. . ....

hu1 · · · huu

(3.2)

is the channel matrix, sn = [s1n . . . su

n]T

and vn = [v1n . . . vu

n]T, are the vectors containing the

transmitted symbols and sampled noise for all pairs respectively, and Fn is the precoding

matrix for the n-th symbol. We denote the power transmitted by each user at each frequency

bin by Ps (and assume that all users have identical PSD and hence transmit in the same

power). To simlify the derivation we assume that the transmitted symbols and the noise

samples are independent identically distributed (iid) complex Gaussian random variables

with:

E[sn] = 0, E[

snsHn

]

= Ps · I, (3.3)

13

14 CHAPTER 3. PROBLEM FORMULATION

and

E[vn] = 0, E[vnvHn ] = σ2

v · I. (3.4)

The precoder aims to minimize the FEXT measured by all users. For example, the popular

ZF linear Precoder achieves a FEXT free channel using: FZF = H−1D, where D is the

diagonal matrix containing the diagonal elements of the matrix H. As was mentioned above,

calculation of the desired precoder requires good channel estimation and a matrix inversion

operation each time the precoder updates. An efficient precoder update scheme, which was

suggested in [30], uses the error signal feedback from the users to the ONU for updating the

precoder. A simplified version of [30] was analyzed by Bergel and Leshem [31], where the

precoder update equation is given by:

Fn+1 = Fn − α

εnsHn , (3.5)

where

εn is a quantized version of the scaled error signal measured by the receivers, which

was sent as a feedback to the ONU. In [31] the authors derived sufficient conditions that

guarantee that this updated precoder, (3.5), converges to the ZF solution.

3.2 The research structure

In the next two chapters we analyze the adaptive precoder, (3.5), for two cases. The first

is partial precoding and the second is adaptation of the precoder with low bit-rate feedback

of the error signals measured by the receivers. It was shown in [31] that cancelling the

FEXT for fewer lines can help channels to satisfy the convergence conditions, and makes it

possible to increase the step size for faster convergence. They have suggested implementing

partial FEXT cancellation in which users with low SNR do not apply FEXT cancellation. In

practical systems partial FEXT cancellation is important also because of complexity limita-

tions. Cendrillon et al. [32] presented a method to calculate the partial FEXT cancellation

matrix for given subsets of cancelled FEXT terms. In this paper we analyze the general

case of partial FEXT cancellation, which require lower implementation complexity. In the

next chapter we describe the method to calculate the partial FEXT cancellation matrix for

given subsets of cancelled FEXT terms presented by Cendrillon et al. [32]. Then we present

an adaptive partial precoder for arbitrary partial FEXT cancellation based on the adaptive

precoder (3.5) and give convergence analyzes of the presented partial precoder to the partial

FEXT cancellation matrix presented by Cendrillon et al. [32]. In this work we also concern a

3.2. THE RESEARCH STRUCTURE 15

variable step size, αn (used for adapting the precoder based on the n-th transmitted symbol).

We use the variable step size to prove that the partial adaptive precoder converges to the

required partial FEXT cancellation matrix when the step size decreases in the correct rate.

In chapter 5 we analyze the performance of the precoder (3.5) with low bit-rate quanti-

zation of the error signals measured by the receivers. We analyze the precoder performance

with an M-point phase quantization scheme of the error signals measured by the receivers.

We give a sufficient condition on the channel that allows the precoder convergence for any

phase quantization, and for any step size used for adapting the precoder. We also give a good

steady state error approximation. Yet, in many cases faster convergence is desirable. We

show that a coarse power quantization, in addition to the phase quantization can significantly

accelerate the precoder convergence, with a negligible cost in feedback rate. All results are

demonstrated through simulations.

16 CHAPTER 3. PROBLEM FORMULATION

Chapter 4

Partial precoding

4.1 Partial precoding

A useful precoder for partial FEXT cancellation was presented by Cendrillon et al. [32]. With

a minor modification we describe this precoder as follows: Let Gj = g1j . . . g

rj

j ⊆ 1, 2 . . . ube the set of receivers that need FEXT cancellation from the j-th user. In the j-th column of

the precoding matrix only the elements indicated by the set Gj are not set to zero. Denote

the set size by 0 ≤ rj = |Gj | ≤ u (i.e., Gj can also be an empty set). We define the partial

precoder FP as:

FP = FP + I, (4.1)

where I is the identity matrix and the matrix FP satisfies (FP)ij = 0 for any i /∈ Gj . The

elements with non-zero values in FP are constructed according to a generalization of the ZF

principle. In this way we completely cancel the FEXT generated by the j-th user to all users

i ∈ Gj.1 To simplify the precoder analysis we define the diagonal j-th selection matrix:

(Γj)m,i =

1 i = m ∈ Gj

0 otherwise, (4.2)

and the minimized j-th (rj × u) selection matrix:

(

Γj

)

m,i=

1 i = gmj

0 otherwise(4.3)

(Note that ΓTj Γj = Γj). We also define the global selection matrix, which selects all rows

that were selected by at least one of the Γj ’s. Setting: Gmax =u⋃

j=1

Gj , the u × u diagonal

1This precoder is identical to the precoder presented in [32], but, we also allow the case in which Gj doesnot necessarily include j.

17

18 CHAPTER 4. PARTIAL PRECODING

global selection matrix is given by:

(Γmax)m,i =

1 i = m ∈ Gmax

0 otherwise. (4.4)

The j-th column of the precoder is constructed to satisfy:

ΓjHF(j)P = ΓjD

(j), (4.5)

where A(j) is the j-th column of the matrix A and the non zero elements of the j-th column

of the precoder can be calculated by solving:

ΓjHF(j)P = ΓjD

(j). (4.6)

Note that F(j)P has zeros in the rows that do not belong to the set Gj , therefore:

F(j)P = ΓjF

(j)P = ΓT

j ΓjF(j)P . (4.7)

Using (4.1) and (4.7), the left side of (4.6) is:

ΓjHF(j)P = ΓjH

(

F(j)P + I(j)

)

= ΓjH(

ΓTj ΓjF

(j)P + I(j)

)

= HjΓjF(j)P + ΓjH

(j) (4.8)

where Hj = ΓjHΓT

j is an rj × rj non singular matrix formed by the elements of the channel

matrix that are located in the rows and columns that belong to Gj (i.e. by (H)m,i : m, i ∈ Gj).

Substituting (4.8) back into (4.6) the non-zero elements of F(j)P are then calculated by:

ΓjF(j)P = H−1

j Γj

(

D(j) − H(j))

, (4.9)

and the actual precoder is given by (4.1). Note that for full FEXT cancellation Γj = I, and

FP is equal to the ZF precoder (FP = FZF). Partial precoder results with residual FEXT

whose powers (for all users) are given by the diagonal of:

Z = Ps

u∑

j=1

(

HF(j)P − D(j)

)(

HF(j)P − D(j)

)H

. (4.10)

As was mentioned above, calculation of the desired precoder requires a good channel estima-

tion and a matrix inversion operation each time the precoder updates. An efficient precoder

update scheme, which was suggested in [30], uses the error signal vector transferred back

from the users to the ONU to update the precoder. Assuming that each user knows its direct

channel coefficient, the error signal measured by the i-th user in the n-th received symbol is

given by:

εin = xi

n − hiisin. (4.11)

4.1. PARTIAL PRECODING 19

Representing the measured error signal of all users in vector form gives:

εn = xn − Dsn = (HFn − D)sn + vn. (4.12)

The users send back to the ONU a quantized version of the error signal vector which can be

written as:

εn = D−1εn + wn, (4.13)

where wn is the error due to quantization which in the following will be assumed to be a

statistically independent random variable with zero mean and covariance matrix E[

wnwHn

]

=

σ2wI. A simplified version of the precoder suggested in [30] was analyzed by Bergel and

Leshem [31], where the precoder update equation is given by:

Fn+1 = Fn − αnεnsHn . (4.14)

We distinguish between two kinds of partial FEXT cancellations. The first is full FEXT

cancellation for selected users as was analyzed in [31], where the FEXT due to all cross-

talkers of those selected users is cancelled, while for the unselected users there is no FEXT

cancellation at all. This type of partial FEXT cancellation was suggested when not applying

FEXT cancellation for users with low SNR. The equation for this case is given by setting

Γj = Γmax for all 1 ≤ j ≤ u:

Fn+1 = Fn − αΓmaxεnsHn . (4.15)

In [31] the authors derived sufficient conditions that guarantee that the adaptive precoder

(4.15) converges to the ideal partial precoder FP. It was shown in [31] that cancelling the

FEXT for fewer lines can help channels to satisfy the convergence conditions, and makes it

possible to increase the step size for faster convergence. In this work we analyze the general

case of partial FEXT cancellation, which require lower implementation complexity. Here,

each of the users has its own set of receivers that require FEXT cancellation. The structure

of the partial precoding matrix is determined by the set of u precoder update sets G1 . . .Gu.In this case the precoder update equation cannot be written in a matrix form, and we write

an update equation per column:

F(j)n+1 = F(j)

n − αnΓjεnsj∗n , (4.16)

where sj∗n is the complex conjugate of sj

n. Note that in this work we consider also a variable

step size, αn. In practical systems, the use of a variable step size allows a better control


on the tradeoff between better FEXT cancellation and faster convergence. In particular,

the step size can be increased in system transition times to allow faster convergence, and

decreased in steady state to allow better FEXT cancellation. In the following subsection we

use the variable step size to prove that the partial adaptive precoder, (4.16), converges to

the partial precoder, FP, when the step size decreases in the correct rate. We also give an

approximation steady state analysis of this precoder.2

4.2 Convergence analyzes

In this section we analyze the performance of the precoder defined in (4.16). In the first

sub-subsection we present our main results for the convergence of (4.16) in Theorem 1, and

in the next two sub-sections we give the proof.

4.2.1 Convergence conditions

We define the distance between the adaptive precoder and FP by:

∆n = Fn − FP (4.17)

and:

Wn = E[∆n∆Hn ], δn = trWn. (4.18)

Note that δn is the Frobinous norm of ∆n and therefore convergence of δn to zero indicates

that every element in Fn converges in mean squares (MS) to the corresponding element in

FP. Our main result is given by the following theorem, which provides sufficient conditions

for the precoder convergence:

Theorem 1 If:

βmax =

max

m∈Gmax

∑

i∈Gmax

i6=m

∣

∣

∣

∣

hm,i

hm,m

∣

∣

∣

∣

, max

i∈Gmax

∑

m∈Gmax

m6=i

∣

∣

∣

∣

hm,i

hm,m

∣

∣

∣

∣

< 1, (4.19)

then:

2In this work we focus on the adaptive precoder update, given the precoder structure. For an analysis ofdifferent methods to select the precoder structure, see [32].

4.2. CONVERGENCE ANALYZES 21

1. a sufficient condition for the convergence of the precoder to FP (i.e., limn→∞ δn = 0) is

that

αn = µ · n−θ, (4.20)

for some 0.5 < θ < 1 and µ > 0 .

2. For any constant step size the asymptotic difference between Fn and FP is bounded if:

αn = α <2

KPs(1 + βmax), (4.21)

where K = u + L− 1 and in that case the Forbinous norm of the difference between FP

and the actual precoder Fn after sufficiently long time is bounded by:

limn→∞

δn ≤α · tr

u∑

j=1

(

(

D−HD−1)

Γj (Z + σ2vI) + σ2

wΓj

)

2 − αKPs (1 + β2max) − 2βmax |1 − αKPs|

, (4.22)

where Z is the residual FEXT given in (4.10).

Note that the conditions that ensure that the distance between Fn and FP is bounded, are

the same as the sufficient conditions, derived in [31], for bounding the distance between the

precoder given in (4.15) and FZF. Also note that (4.22) shows that the convergence error,

δ∞, can be made as small as desired using a small enough α.

4.2.2 Theorem proof

Following definition (4.17) the difference between each column of the precoder and the cor-

responding column in FP is given by:

∆(j)n = F(j)

n − F(j)P . (4.23)

The received error signal term, given in (4.12), can be written using (4.23) as:

εn = (HFn − D)sn + vn

=

u∑

i=1

[

(HF(i)n −D(i))si

n

]

+ vn

=u∑

i=1

[(

H(

∆(i)n + F

(i)P

)

−D(i))

sin

]

+ vn

=

u∑

i=1

[

H∆(i)n si

n +(

HF(i)P −D(i)

)

sin

]

+ vn. (4.24)


We define Γj to be the complementary matrix of Γj (i.e., Γj + Γj = I). Using (4.5), equation

(4.24) becomes:

εn =

u∑

i=1

[

H∆(i)n si

n +(

Γi + Γi

)

·(

HF(i)P − D(i)

)

sin

]

+ vn

=u∑

i=1

[

H∆(i)n si

n + Γi ·(

HF(i)P − D(i)

)

sin

]

+ vn. (4.25)

Subtracting F(j)P from both sides of (4.16) and using (4.13) we get:

∆(j)n+1 = ∆(j)

n − αnΓj

(

D−1εn + wn

)

sj∗n . (4.26)

Substituting (4.25) in (4.26) we get:

∆(j)n+1 = ∆(j)

n − αnΓj

(

D−1u∑

i=1

[

H∆(i)n si

n + Γi

(

HF(i)P −D(i)

)

sin

]

)

sj∗n

− αnΓj

(

D−1vn + wn

)

sj∗n . (4.27)

Taking the j-th element out of the summation in (4.27) using Γj · Γj = 0 (note that the

multiplication of two diagonal matrices is commutative, therefore: ΓjD−1 = D−1Γj) , We

get:

∆(j)n+1 = ∆(j)

n − αn

∣

∣sjn

∣

∣

2ΓjD

−1H∆(j)n

− αnΓj

(

D−1u∑

i=1,i6=j

[

H∆(i)n si

n + Γi

(

HF(i)P − D(i)

)

sin

]

)

sj∗n

− αnΓj

(

D−1vn + wn

)

sj∗n . (4.28)

Next we show that (4.28) converges in expectation to zero. Then, we complete the proof and

show that ∆n converges to zero in the mean square sense as well.

Convergence in expectation

We evaluate the expectation of both sides of (4.28). Noting that ∆n is statistically indepen-

dent of sn (since Fn is calculated based on sn−1), and that the expectation of the third and

fourth terms on the right hand side of (4.28) is 0, we have:

E[

∆(j)n+1

]

= E[

∆(j)n

]

− αnPsΓjD−1HE

[

∆(j)n

]

. (4.29)

Similarly to (4.1) we define Fn by:

Fn = Fn + I, (4.30)


where F(j)n has zeros in the elements that do not belong to the set Gj , and the precoder error,

(4.23), is equal to the error in the update elements:

∆(j)n = F(j)

n − F(j)P = F(j)

n − F(j)P . (4.31)

Note that ∆(j)n has zeros in the rows that do not belong to the set Gj . Therefore:

∆(j)n = Γj∆

(j)n . (4.32)

Defining Hj = ΓjHΓj, noting that ΓjΓj = Γj and D−1Γj = ΓjD−1, we measure the conver-

gence of (4.29) to zero by the value of:

E[

∆(j)n+1

]H

E[

∆(j)n+1

]

= E[

∆(j)n

]H(

Γj − αnPsD−1Hj

)H

·(

Γj − αnPsD−1Hj

)

E[

∆(j)n

]

. (4.33)

Defining: Qj,n = (Γj − αnPsD−1Hj) we bound (4.33):

E[

∆(j)n+1

]H

E[

∆(j)n+1

]

≤ E[

∆(j)n

]HE[

∆(j)n

]

· ρ(

QHj,nQj,n

)

≤ E[

∆(j)1

]H

E[

∆(j)1

]

n∏

k=1

ρ(

QHj,kQj,k

)

, (4.34)

where ρ(

AHA)

is the spectral radius (or maximal eigenvalue) of the matrix(

AHA)

, which

is bounded by ([35] page 223):

ρ(

AHA)

≤∥

∥AH∥

∥

1· ‖A‖1 =

[

maxi

u∑

j=1

|Aij|][

maxj

u∑

i=1

|Aij|]

, (4.35)

therefore:

n∏

k=1

ρ(

QHj,kQj,k

)

≤ maxj

(

n∏

k=1

ρ(

QHj,kQj,k

)

)

≤ maxj

(

n∏

k=1

∥

∥QHj,k

∥

∥

1· ‖Qj,k‖1

)

. (4.36)

Testing the elements of the matrix Qj,n we bound (4.36) by:

maxj

(

n∏

k=1

∥

∥QHj,k

∥

∥

1· ‖Qj,k‖1

)

≤n∏

k=1

(|1 − αkPs| + αkPsβE)2, (4.37)

where:

βE = maxj

maxm∈Gj

∑

i∈Gj

i6=m

∣

∣

∣

∣

hm,i

hm,m

∣

∣

∣

∣

, maxi∈Gj

∑

m∈Gj

m6=i

∣

∣

∣

∣

hm,i

hm,m

∣

∣

∣

∣

. (4.38)


Inspecting (4.37) we can conclude that if there exists ξ < 1 so that:

(|1 − αnPs| + αnPsβE) < ξ < 1, (4.39)

for all n, then the expectation converges to zero as time tends to infinity. The condition in

(4.39) is satisfied if:

βE < 1, (4.40)

and the step size parameter for all n satisfies:

ζ < αn <2

Ps(1 + βE), (4.41)

for some ζ > 0. From the definition of βE, (4.38), one can conclude that the convergence

conditions depend on the ”worst” cancelling set. Thus, if FEXT cancellation is applied to

fewer users, then the channel condition and the condition on αn can be more relaxed.

Mean square error convergence

We now show that the precoder Fn in (4.16), converges in mean squares to FP (which

completes the proof of theorem 1). Following the definitions of Wn and δn, given in (4.18),

we now define:

Wj,n = E[∆(j)n ∆(j)H

n ] (4.42)

and

δj,n = trWj,n, (4.43)

noting that Wn =u∑

j=1

Wj,n and δn =u∑

j=1

δj,n. Rewriting (4.28) as:

∆(j)n+1 = ∆(j)

n − αn

∣

∣sjn

∣

∣

2ΓjD

−1H∆(j)n

− αnΓjD−1

u∑

i=1,i6=j

H∆(i)n si

nsj∗n

− αnΓjD−1

u∑

i=1,i6=j

Γi

(

HF(i)P −D(i)

)

sinsj∗

n

− αnΓj

(

D−1vn + wn

)

sj∗n , (4.44)


and substituting it into (4.42) we get:

Wj,n+1 = Wj,n − αnPsΓjD−1HWj,n − αnPsWj,nH

HD−HΓj

+ α2nLP 2

s ΓjD−1HWj,nH

HD−HΓj

+ α2nP 2

s ΓjD−1

u∑

i=1,i6=j

HWi,nHHD−HΓj

+ Φj,n + α2nΩj, (4.45)

Where the four terms in the first two lines of (4.45) are the outcome of the multiplication of

the two terms in the first line of (4.44) by their conjugates, using L = E[

|sjn|

4]/

P 2s which

ranges from 1 in QPSK modulation to 2 in Gaussian modulation. The term in the third line

of (4.45) is the outcome of the multiplication of the term in the second line of (4.44) by its

conjugate. The term Φj,n, defined as:

Φj,n = α2nD

−1Γj

u∑

i=1

P 2s HE

[

∆(i)n

]

(

HF(i)P −D(i)

)H

ΓiΓjD−H

+ α2nP 2

s D−1Γj

u∑

i=1

Γi

(

HF(i)P −D(i)

)

E[

(

∆(i)n

)H]

HHΓjD−H (4.46)

is a matrix that includes the term E[

∆(j)n

]

in each of its terms, which results from the

multiplications of the second line of (4.44) by the conjugate of the third line and vice versa.

(if conditions (4.40) and (4.41) are valid, then limn→∞

Φj,n = 0). The last term in (4.45) results

from multiplying (separately) each element in the third and fourth lines of (4.44) by its

conjugate value, where:

Ωj = PsD−1Γj

(

Z + σ2vI + Dσ2

wDH)

ΓjD−H . (4.47)

This matrix includes the effect of noise, quantization, and the power of the residual FEXT,

Z (given in (4.10)). This term is constant in time and negatively affects the performance

even in the absence of noise. Note that summations (4.46) and (4.47) also include the case

i=j because: Γj · Γj = 0. Also note that the terms Φj,n and Z contain the residual FEXT

which is not cancelled even in the ideal partial precoder, FP. This FEXT appears in the

unselected terms and therefore: HF(i)P − D(i) = Γi

(

HF(i)P − D(i)

)

. If Γj = Γmax for all j, as

was analyzed in [31], Φj,n and the term ΓmaxZΓmax in Ωj , are zeroed, using Γmax · Γmax = 0.


Summing (4.45) over j = 1, .., u, using Wn =u∑

j=1

Wj,n , the trace of Wn is given by:

δn+1 = tr

Wn − αnPs

u∑

j=1

D−1ΓjHWj,n − αnPs

u∑

j=1

Wj,nHHΓjD

−H

+ tr

α2n (L − 1)P 2

s

u∑

j=1

D−1ΓjHWj,nHHΓjD

−H

+ tr

α2nP 2

s

u∑

j=1

D−1ΓjHWnHHΓjD

−H

+ tr

Φn + α2nΩ

, (4.48)

where Φn =u∑

j=1

Φj,n and Ω =u∑

j=1

Ωj . We now upper bound (4.48) by replacing Γj throughout

(4.48) with Γmax, using the fact that Wj,n is semi positive definite for all j:

δn+1 ≤ tr

Wn − αnPs

u∑

j=1

D−1ΓmaxHWj,n − αnPs

u∑

j=1

Wj,nHHΓmaxD

−H

+ tr

α2n (L − 1)P 2

s

u∑

j=1

D−1ΓmaxHWj,nHHΓmaxD

−H

+ tr

α2nP 2

s

u∑

j=1

D−1ΓmaxHWnHHΓmaxD

−H

+ tr

Φn + α2nΩ

. (4.49)

Note that the trace of the first line of (4.48) did not change because Wj,n has zeros throughout

the columns and rows that do not belong to set Gj (i.e., Wj,n = ΓjWj,n = Wj,nΓj). Using

again Wn =u∑

j=1

Wj,n we merge rows two and three in (4.49), defining K = u + L− 1, to get:

δn+1 ≤ tr

Wn − αnPsD−1ΓmaxHWn − αnPsWnH

HΓmaxD−H

+ tr

α2nKP 2

s D−1ΓmaxHWnHHΓmaxD

−H

+ tr

Φn + α2nΩ

. (4.50)

Noting that Wn = ΓmaxWn = WnΓmax (because all rows that were selected by any Γj are

also selected by Γmax), we Define Hmax = ΓmaxHΓmax, and (4.50) can be written as:

δn+1 ≤ tr

Wn − αnPsD−1HmaxWn − αnPsWnH

HmaxD

−H

+ tr

α2nKP 2

s D−1HmaxWnHHmaxD

−H

+ tr

Φn + α2nΩ

. (4.51)


The first and second rows of (4.51) have the same structure as the bound on the Frobinous

norm of the distance between the full updated precoder for selected users, given by (4.15),

and the ZF solution presented in [31]. Repeating the steps presented in [31] we define:

Qn = (Γmax − αnKPsD−1Hmax), (4.52)

and rearrange the elements in (4.51) to get:

δn+1 ≤ tr

(

1

KQnWnQ

Hn +

K − 1

KWn

)

+ tr

Φn + α2nΩ

= tr

Wn

(

1

KQH

n Qn +K − 1

KI

)

+ tr

Φn + α2nΩ

. (4.53)

Using the spectral radius, (4.53) is bounded by:

δn+1 ≤ δn

(

1

Kρ(

QHn Qn

)

+K − 1

K

)

+ tr

Φn + α2nΩ

, (4.54)

and the spectral radius is upper bounded by (using (4.35) again):

ρ(

QHn Qn

)

≤∥

∥QHn

∥

∥

1· ‖Qn‖1 ≤

(

|1 − αnKPs| + αnKPsβmax

)2

. (4.55)

Substituting (4.55) into (4.54) the convergence error is bounded by:

δn+1 ≤ δn ·(

1

K

(

|1 − αnKPs| + αnKPsβmax

)2

+K − 1

K

)

+ tr

Φn + α2nΩ

. (4.56)

In the constant step size case (αn = α for all n), conditions (4.19) and (4.21) ensures that:

(

|1 − αKPs| + αKPsβmax

)

< 1, (4.57)

which, together with the requirement of the convergence of Φn guaranty the convergence

of (4.56). Note that conditions (4.19) and (4.21) are stricter than the conditions for the

convergence in expectation given in (4.40) and (4.41); Therefore, when these conditions are

met, after long enough convergence time, the term Φn (that includes the term E[∆n]), fades

(limn→∞ tr Φn → 0). Substituting the definition of Ω from (4.47), the bound on the MS

convergence is given by:

δ∞ ≤α2KPs · tr

u∑

j=1

((

D−HD−1)

Γj (Z + σ2vI) + Γjσ

2w

)

1 −(

|1 − αKPs| + αKPsβmax

)2 . (4.58)


Extending the denominator gives (4.22), and hence completes the proof of the second part

of theorem 1.

As mentioned, we expect that the partial precoder (4.16) will allow convergence even in

channels in which the full update precoder (4.15), does not converge. Such better convergence

conditions can indeed be derived, by continuing from equation (4.48) without replacing Γj

with Γmax. But, the resulting condition, although less restrictive, is much more complicated,

and hence not useful for practical scenarios. In this paper we restricted ourselves to the more

convenient convergence conditions given in the theorem, which are satisfied in most practical

channels.

To complete the proof of theorem 1 we restrict the channel and set the step size as required

by the first part of the theorem: βmax < 1 and αn = µ · n−θ, where µ > 0 and 0.5 < θ < 1

are constants. As the step size decreases in time we can find ℓ > 0 such that for all n ≥ ℓ:

αnKPs < 1, (4.59)

and

µ · Ps (1 − βmax)

nθ>

1

n(4.60)

(note that θ < 1). From (4.55) we have for n ≥ ℓ:

ρ(

QHn Qn

)

≤ (1 − αnKPs (1 − βmax))2 ≤ 1 − αnKPs (1 − βmax) , (4.61)

and substituting into (4.54) gives:

δn+1 ≤ δn (1 − αn · c) + α2nγ, (4.62)

where: c = Ps (1 − βmax) and γ = tr |Φℓ| /α2ℓ + Ω are both constant in time (note that

tr|Φℓ|α2

ℓ

≥ tr|Φn|α2

nfor n ≥ ℓ). Writing (4.62) recursively starting at n = ℓ we get:

δn+1 ≤ δℓ

n∏

k=ℓ

(1 − αk · c) + γ ·n∑

k=ℓ

α2k

n∏

m=k+1

(1 − αm · c). (4.63)

Substituting the step size, (4.20), gives:

δn+1 ≤ δℓ

n∏

k=ℓ

(

1 − µ · ckθ

)

+ µ2 · γ ·n∑

k=ℓ

1

k2θ

n∏

m=k+1

(

1 − µ · cmθ

)

. (4.64)

Using (4.60):

δn+1 ≤ δℓ

n∏

k=ℓ

(

1 − 1

k

)

+ µ2 · γ ·n∑

k=ℓ

1

k2θ

n∏

m=k+1

(

1 − 1

m

)

. (4.65)

4.3. STEADY STATE ANALYSIS 29

Noticing thatn∏

k=ℓ

(

1 − 1k

)

= ℓ−1n

andn∏

m=k+1

(

1 − 1m

)

= kn, we write (4.65) as:

δn+1 ≤ δℓℓ − 1

n+ µ2 · γ · 1

n·

n∑

k=ℓ

k

k2θ. (4.66)

For θ > 0.5 : limn→∞

δn → 0, which completes the proof of theorem 1.

4.3 Steady State Analysis

After ensuring the precoder convergence, we test the steady state error for the case of a fixed

step size α. Testing the covariance matrix of the received error signal:

Rεn= E

[

εnεHn

]

, (4.67)

we evaluate the steady state error as time goes to infinity. Substituting (4.25) into (4.67),

using E[∆(j)n→∞] → 0 and the statistical independence of ∆n and sn (since Fn is calculated

based on sn−1) we have:

Rε∞ =

[

u∑

i=1

PsHWi,∞HH + PsΓi

(

HF(i)P −D(i)

)(

HF(i)P − D(i)

)H

Γi

]

+ σ2vI

= PsHWHH + Z + σ2vI, (4.68)

where Z is defined in (4.10), Wi,∞ = limn→∞

Wi,n, and W =u∑

i

Wi,∞ is the steady state value

of the precoder error covariance matrix (note that the multiplication cross term disappeared

because Γi∆(i)n = 0). To evaluate (4.68) we first need to evaluate W = lim

n→∞Wn. Inspecting

(4.45), taking time to infinity (so limn→∞

Φi,n → 0) the steady state error covariance matrix per

column must satisfy:

α2Ωj = αPsΓjD−1HWj,∞ + αPsWj,∞HHD−HΓj

− α2 (L − 1) P 2s ΓjD

−1HWj,∞HHD−HΓj

− α2P 2s ΓjD

−1HWHHD−HΓj . (4.69)

We make two approximations in the right side of (4.69). The first is removing Γj from the

last term, the second (and more significant one) is assuming D−1H ≈ I. Summing (4.69)

over j = 1, .., u , using Wj = ΓjWj = WjΓj and K = u + L − 1 we get:

W ·(

2αPs − α2KP 2s

)

≈ α2Ω. (4.70)


Substituting the definition of Ω from (4.47), W in the steady state can be approximated by:

W ≈ α

(2 − αKPS)

u∑

j=1

D−1Γj

(

Z + σ2vI + Dσ2

wDH)

ΓjD−H . (4.71)

Substituting (4.71) into (4.68) we get:

Rε∞ ≈ αPS

(2 − αKPS)

(

HD−1

u∑

j=1

(

Γj

(

Z + σ2v + Dσ2

wDH)

Γj

)

D−HHH

)

+ Z + σ2vI. (4.72)

Using again the approximation HD−1 ≈ I:

Rε∞ ≈ αPS

(2 − αKPS)·(

u∑

j=1

Γj

(

Z + σ2v + Dσ2

wDH)

Γj

)

+ Z + σ2v · I. (4.73)

In order to obtain a more insightful expression, we consider the trace of (4.73), and making

one last approximation, we assume that the sets Gj = g1j . . . g

rj

j for each j are selected

independently. Therefore, for a large u, defining the empiric average set size r = 1u

u∑

j=1

|Gj |,

we have:u∑

j=1

Γj ≈ r · I3. Using trAB = trBA and Γj · Γj = Γj , the sum of the steady

state error of all users is approximated by:

tr Rε∞ ≈(

αPsr

(2 − αKPs)+ 1

)

· tr

Z + σ2vI

+αPsrσ

2w

(2 − αKPs)tr

DDH

. (4.74)

In an ideal partial FEXT cancellation, the error signal covariance is given by Z + σ2vI,

where Z contains the error due to the non-cancelled element FEXT terms. The error increase

is due to the use of the iterative precoder and can be determined by α and by the number

of users to be cancelled. Note that for full cancellation, r = u and Z = 0, so that (4.74) is

the same as the result derived in [31]. Although the approximations made in the derivation

of (4.74) are fairly heuristic, the resulting approximation is surprisingly accurate as shown

in the following section.

4.4 Simulation Results

To demonstrate the results derived in the previous sections we conducted several simulations

using 28 channels measured by France Telecom4. In the following simulations we chose the

3Alternatively, one can consider a “fair” partial FEXT cancellation scheme, which cancels the same number

of interferers to each receiver. For such a scheme the approximation is replaced by an equality:u∑

j=1

Γj = r · I.4We would like to thank M. Ouzzif, R. Tarafi, H. Marriott and F. Gauthier of France Telecom R&D, who

conducted the VDSL channel measurements as partners in the U-BROAD project.

4.4. SIMULATION RESULTS 31

active elements to be the 28 · r strongest elements in the channel matrix. The transmitted

PSD was set to -60dBm/Hz, the noise PSD is -140dBm/Hz and the step size α was chosen

to be half of its maximal value calculated by (4.21) for all cases.5 These simulations do not

apply error signal quantization (σw = 0).

Figure 4.1 depicts the mean square error convergence of the adaptive precoder Fn to the

ideal partial precoder FP as a function of the number of symbols transmitted for a distance

of 300 meters in a frequency of 14.25 MHZ (βmax = 0.3576). The convergence is measured

by δn defined in (4.18). The figure shows the empirical evaluation of δn based on a Monte

Carlo simulation of 500 systems for various values of r. The figure also shows the bound

defined in (4.56). It can be seen that as r decreases, the precoder convergence is faster, but

the precoder converges further away from FP due to the higher residual FEXT. One can see

that although the convergence bound is not tight it well describes the precoder convergence.

5In practice the value of the channel parameter βmax is often unknown. In such case, a practical approachcan take the worst case value βmax = 1, and use: α = const

KPs

.


0 50 100 150 200 250 300 350 400 450 500 55010

−4

10−3

10−2

10−1

100

Time (symbols)

Converg

ence E

ror

(δn)

aver sim r=15

bound r=15

aver sim r=20

bound r=20

aver sim r=28

bound r=28

Figure 4.1: Convergence of the adaptive precoder to the ideal precoder in mean squares asa function of the number of symbols transmitted, at a distance of 300m and a frequency of14.25MHz (averaged over 500 systems). Also shown is the upper bound on the convergencedefined by (4.56).


Figure 4.2 shows the signal to noise plus interference ratio (SINR) averaged over all users

as a function of time for the same case (300 meters, 14.25 MHz). For reference, the figure

also shows the average SINR using the ideal partial precoder Fp. It can be seen that for all

values of r, the SINR converges to a value which is very close to the SINR obtained with an

ideal partial precoder. Note that using smaller values of α will allow the SINR to be as close

as desired to the ideal value at the price of slower convergence.

0 100 200 300 400 500 60020

25

30

35

40

45

50

Time (symbols)

SIN

R [dB

]

ideal r=28

adaptive r=28

ideal r=20

adaptive r=20

ideal r=15

adaptive r=15

ideal r=5

adaptive r=5

Figure 4.2: Average SINR (of all 28 users) at a distance of 300m and frequency of 14.25MHz,using the adaptive precoder and the ideal precoder, as a function of the number of symbolstransmitted.


Extending the simulation to the whole bandwidth (30MHz), Figure 4.3 shows the average

capacity achieved from all users as a function of time at a distance of 300 meters. The figure

shows the curves for different values of r and the capacity achieved using Fp for each case.

The results show similar behavior as the results in figure 4.2 for the SINR convergence.

0 100 200 300 400 500 600200

250

300

350

400

450

500

Time (symbols)

Mean C

apacity [M

bps]

ideal r=28

adaptive r=28

ideal r=20

adaptive r=20

ideal r=15

adaptive r=15

ideal r=5

adaptive r=5

Figure 4.3: Average capacity (of all 28 users) using the adaptive precoder and the idealprecoder, as a function of the number symbols transmitted.


figure 4.4 demonstrates the accuracy of the approximation given in (4.74) for the sum

of all users’ steady state square errors. Again we use channel measurements corresponding

to a distance of 300 meters at a frequency of 14.25 MHz. The figure shows the curves

of the sum of all users’ square errors as function of time for different values of r. The

figure also shows by markers the corresponding approximation for each case. It can be seen

that the approximation gives an almost exact prediction in all cases. Note that (4.74) was

derived using the assumption that the FEXT cancellation sets are independent and uniformly

distributed. The accuracy of (4.74) indicates that this assumption gives a good representation

of system performance.

0 100 200 300 400 500 600−65

−60

−55

−50

−45

−40

−35

Time (symbols)

Cu

mu

lative

sq

ua

re e

rro

r [d

B]

empirical r=5

approx steady state r=5

empirical r=15


empirical r=20


empirical r=28


Figure 4.4: Sum of all users’ square error over time, using channel measurements at a distanceof 300m over a frequency of 14.25MHz. Also shown is the approximation given by (4.74).


To study the amount of increase in noise and FEXT due to the iterative scheme and its

distribution among the users, figure 4.5 depicts the histogram of all users’ steady state SINR

loss, using the same channel measurements as before (300m, 14.25MHz) when cancelling 20

cross-talkers per user. The dotted line indicates the approximation derived in (4.74). One

can see that the maximum deviation from the users’ performance is about 0.5 dB. This is

also a typical result for different frequencies and different numbers of cross-talkers canceled

per user. Thus, this simple steady state approximation provides a useful and robust tool for

system design and analysis.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.20

1

2

3

4

5

6

7

8

9

10

Steady state squre error increase ratio [dB]

Nu

mb

er

of

use

rs

0

1

2

3

4

5

6

7

8

9

10

Figure 4.5: Sum of all users’ square error in steady state, derived by using channel measure-ments at a distance of 300m over a frequency of 14.25MHz and cancelling 20 cross-talkersper user (r=20). Also shown is (dotted line) the approximation given by (4.74).

4.5. CONCLUSION 37

4.5 Conclusion

In this chapter we presented and analyzed an adaptive precoder for partial FEXT cancella-

tion. Partial FEXT cancellation is very important because it can overcome major practical

constraints (e.g., availability of appropriate end-user equipment or limited system complex-

ity). We derived sufficient conditions that guarantee the convergence of the precoder to the

ideal partial FEXT precoder presented by Cendrillon et al. [32]. The analysis also provides

bounds that enable the right choice of parameters and provides prediction on the expected

performance. The results are supported by numerical simulations which show that the pre-

coder converges very closely to the ideal precoder presented in [32] and that partial FEXT

cancellation also accelerates the convergence.

Chapter 5

Low Bit Rate Feedback

In this chapter we analyze the performance of the precoder suggested by bergel and leshem

[31] with low bit-rate quantization of the error signals measured by the receivers. We analyze

the precoder performance with an M-point phase quantization scheme (where M can even be

2). We give an upper bound on the precoder convergence for any step size used for adapting

the precoder and for any number of allowed values used for the phase quantization. We give

sufficient conditions to derive this upper bound and also give steady state approximation.

We also suggest to use coarse power quantization of the error signals, in addition to the phase

quantization, for adapting the precoder. We give steady state approximation for this scheme

and show through simulations that with minor increase in feedback rate, it significantly

improves the performance.

5.1 System Model

As was described in Section 3.1 Bergel and Leshem [31] have analyzed the precoder update

equation given by (see (3.5)):

Fn+1 = Fn − αεnsHn , (5.1)

where εn is the scaled error signal in which the i-th element is given by:

εin =

xin − hiis

in

hii. (5.2)

In [31] the authors derived sufficient conditions that guarantee that this updated precoder,

(5.1), converges to the ZF solution. The analysis in [31] considered also feedback quantization,

but, under the assumption that the quantization error is independent of the quantized signal.

In practice this assumption only holds for high rate quantization.

39

40 CHAPTER 5. LOW BIT RATE FEEDBACK

In this chapter we analyze the precoder convergence when the feedback contains only

quantized phase of the error signals. We consider a dithered quantization scheme (e.g. , [33])

in which the error signal, εin, is first rotated by multiplying it with ejφi

n where φin is a random

variable uniformly distributed over [0, 2π]. The rotated error signal is then quantized to one

of the values ej k2πM , where k = 0, 1, ..., M − 1, and M is the number of quantization values,

and then rotated back by e−j k2πM as is depicted in Figure 5.1. For quantization of M levels

Figure 5.1: Phase quantization with M=4 levels when rotating the axis in a randomly phase.

we have:

E[

ejθin

]

=1

2 (π/M)

π/M∫

−π/M

cos θdθ =M

πsin

π

M≡ q(M), (5.3)

which is constant in time for all users (e.g it is not dependent of i or n). This scheme doesn’t

change the performance, as it is demonstrated in the simulation in section 5.4, however the

quantized feedback error phase becomes statistically independent of the transmitted signal,

sn, and of the additive noise vn.

5.2. LOW BIT RATE CONVERGENCE ANALYZES 41

The quantized error signal of the i-th user in the n-th received symbol can be written as:

εin =

ejθin εi

n

|εin|

, (5.4)

where, θin, the phase difference between the actual error signal and the quantized error signal

is uniformly distributed over [−π/M, π/M ] (note that the dithered quantizer is required

mostly for analysis purposes. In Section 5.4 we show that the performance of the simple

quantizer are nearly identical). Representing by εn the error signal, measured by all users,

after quantization in vector form, we analyze in the next section the precoder which adapts

according to:

Fn+1 = Fn − αεnsHn . (5.5)

In the following sections we derive an upper bound on the difference between the adaptive

precoder, (5.5), and the zero forcing solution. In Section 5.3 we also present a scheme that

use coarse power of the error signals in addition to the phase quantization for adapting the

precoder.

5.2 Low Bit Rate Convergence analyzes

In this section we analyze the performance of the precoder defined in (5.5). We first give our

main results for the convergence in a theorem form, and in the next subsection we will give

the full proof.


In this section we derive an upper bound on the absolute value of the error signal using the

adaptive precoder presented in (5.5). We define the difference between the precoder and the

zero forcing solution FZF by:

∆n = Fn − FZF = Fn − H−1D, (5.6)

and define:

Wn = E[

∆n∆Hn

]

. (5.7)

We also define the averaged in time expectation, (ATE), of the absolute error, measured by

the j-th user by:

εjN =

1

N

N∑

n=1

E[∣

∣εjn

∣

∣

]

(5.8)


where N is the number of symbols transmitted. Defining hij to be the element in the i-th

row and the j-th column of the channel matrix, H, we prove the following theorem:

Theorem 2 if:

β = max

(

maxj

∑

i6=j

∣

∣

∣

∣

hij

hii

∣

∣

∣

∣

, maxi

∑

j 6=i

∣

∣

∣

∣

hij

hii

∣

∣

∣

∣

)

< 1, (5.9)

then the ATE of the absolute error, measured by the j-th user is bounded by:

εjN ≤ (1 + β2) tr W1

αMπ

sin πM

· 2 (1 − β)N+

αuPs (1 + β)2

Mπ

sin πM

· 2 (1 − β)+

√

π4

∣

∣h−1jj

∣

∣ σv

(1 − β), (5.10)

where M is the number of levels of quantization for the feedback error.

Note that the downstream VDSL systems are typically considered to be row-wise diagonal

dominant (RWDD) [27]. A more accurate characterization of this property is that the matrix

D−1H is diagonal dominant. Hence condition (5.9) is satisfied in almost all VDSL channels.

Also note that the first term in the right side of (5.10) goes to zero as time tends to infinity.

Hence, in steady state (N → ∞) the bound depends linearly with α.

Theorem proof

There is much resemblance between the analysis of the adaptive precoder and the analysis

of the LMS filters. Yet the well studied results on LMS filtering cannot be directly applied

to the precoder analysis. The following proof follows the general concept of the proof in [34],

with appropriate adjustments. We start our proof by subtracting FZF from both sides of

(5.5) and using the definition in (4.6) to get:

∆n+1 = ∆n − αεnsHn . (5.11)

Following the definitions of Wn given in (4.8) we get:

Wn+1 = Wn − αE[

εnsHn ∆H

n

]

− αE[

∆nsnεHn

]

+ α2E[

εnsHn snεH

n

]

, (5.12)

and using (5.12) recursively we get:

Wn+1 = W1 − α

N∑

n=1

E[

εnsHn ∆H

n

]

− α

N∑

n=1

E[

∆nsnεHn

]

+ α2

N∑

n=1

E[

εnsHn snεH

n

]

. (5.13)


Defining:

A = α1

N

N∑

n=1

E[

εnsHn ∆H

n

]

HHD−H , (5.14)

and

Cn = E[

εnsHn snεH

n

]

, (5.15)

we write (5.13) as:

ADHH−H + H−1DAH =1

NW1 −

1

NWn+1 + α2 1

N

N∑

n=1

Cn. (5.16)

Multiplying (5.16) by D−1H from the left and with HHD−H from the right we have:

D−1HA + AHHHD−H = D−1H

(

1

NW1 −

1

NWn+1 + α2 1

N

N∑

n=1

Cn

)

HHD−H

≤ 1

ND−1HW1H

HD−H + α2D−1H1

N

N∑

n=1

CnHHD−H , (5.17)

where the inequality is in the semi-positive definite sense and in particular:

zHj

(

D−1HA + AHHHD−H)

zj ≤ zHj

1

ND−1HW1H

HD−Hzj

+ α2zHj D−1H

1

N

N∑

n=1

CnHHD−Hzj , (5.18)

where zj is the vector in which all elements are zero except for the j-th which equals 1.

Inspecting the term in the second line of (5.18) we can write:

α2zHj D−1HCnH

HD−Hzj = α2∑

ℓ

∑

i

(

D−1H)

ji· (Cn)iℓ ·

(

(

D−1H)H)

ℓj

≤ α2∑

ℓ

∑

i

∣

∣

∣

(

D−1H)

ji

∣

∣

∣· |(Cn)iℓ| ·

∣

∣

∣

∣

(

(

D−1H)H)

ℓj

∣

∣

∣

∣

,(5.19)

where the notation (B)iℓ denotes the element in the matrix B, that is placed in the i-th row

and in the ℓ-th column. Using εj∗n to denote the complex conjugate of εj

n, we note that for

all i and ℓ (see (5.15)):

(Cn)iℓ = E[

sHn snεi

nεℓ∗n

]

≤ E[

sHn sn

∣

∣εin

∣

∣

∣

∣εℓn

∣

∣

]

= uPs, (5.20)


which is real and constant in time. We also note that:

max

(

maxj

∑

i

∣

∣

∣

(

D−1H)

ij

∣

∣

∣, max

i

∑

j

∣

∣

∣

(

D−1H)

ij

∣

∣

∣

)

= 1 + β, (5.21)

where β is defined in (5.9). Substituting (5.20) and (5.21) in (5.18) we get:

zHj

(


zj ≤ zHj

1

ND−1HW1H

HD−Hzj + α2uPs (1 + β)2 . (5.22)

On the other hand, we also have:

zHj

(


zj = 2 · Re(

zHj

(

D−1HA)

zj

)

= 2 · Re

(

(A)jj +∑

i6=j

(

D−1H)

ji(A)ij

)

≥ 2 · Re

(

(A)jj −∑

i6=j

∣

∣

∣

(

D−1H)

ji

∣

∣

∣

∣

∣

∣(A)ij

∣

∣

∣

)

. (5.23)

Evaluating the matrix A (see (5.14)), we can write:

A = α1

N

N∑

n=1

E[

εn

(

sHn ∆H

n HHD−H + vHn D−H

)]

− α1

N

N∑

n=1

E[

εnvHn D−H

]

= α1

N

N∑

n=1

E[

εnεHn

]

− α1

N

N∑

n=1

E[

εnvHn D−H

]

, (5.24)

and writing each element separately:

(A)ij = α1

N

N∑

n=1

E[

εinεj∗

n

]

− α1

N

N∑

n=1

E[

εinv

j∗n h−1∗

jj

]

. (5.25)

The dithered quantization scheme guarantees uniformly distributed phase error, for quan-

tization of M levels we have:

E[

ejθin

]

=1

2 (π/M)

π/M∫

−π/M

cos θdθ =M

πsin

π

M≡ q(M), (5.26)

which is constant in time for all users (e.g it is not dependent of i or n). This scheme doesn’t

change the performance, as it is demonstrated in the simulation in section 5.4, however the

quantized feedback error phase becomes statistically independent of the transmitted signal,

sn, and of the additive noise vn and obviously it is statistically independent of ∆n (since ∆n


is calculated based on sn−1 (5.11)). Therefore we can separate the expectation on the phase

error due to the quantization:

E[

εinεj∗

n

]

= q(M)E

[

εin

|εin|

εj∗n

]

, and : E[

εinv

j∗n h−1∗

jj

]

= q(M)E

[

εin

|εin|

vj∗n h−1∗

jj

]

. (5.27)

We next note that in the off diagonal element of the matrix A, the right hand term in (5.25),

vanishes due to the statistical independence between εin and vj

n. Thus, for i 6= j we have:

∣

∣

∣(A)ij

∣

∣

∣= αq(M) · 1

N

∣

∣

∣

∣

∣

N∑

n=1

E

[

εin

|εin|

εj∗n

]

∣

∣

∣

∣

∣

≤ αq(M) · 1

N

N∑

n=1

E

[∣

∣

∣

∣

εin

|εin|

∣

∣

∣

∣

∣

∣εj∗n

∣

∣

]

= αq(M) · εjN . (5.28)

The scaled error signal of all users in vector form is given by:

εn = D−1 (xn − Dsn) = D−1(HFn − D)sn + D−1vn. (5.29)

Using the definition

∆n = Fn − FZF , (5.30)

where FZF = H−1D, we have:

εn = D−1 (H∆nsn) + D−1vn. (5.31)

Defining hj to be the j-th row of the channel matrix H each element in the vector εn can

be written as:

εjn = h−1

jj hj∆nsn + h−1jj vj

n. (5.32)

Considering a diagonal element of A, and substituting (5.27) and (5.32) in (5.25) we have:

(A)jj = αq(M) · εjN − αq(M) · 1

N

N∑

n=1

E

[

(

h−1jj hj∆nsn + h−1

jj vjn

)

∣

∣h−1jj hj∆nsn + h−1

jj vjn

∣

∣

(

h−1jj vj

n

)H

]

= αq(M) · εjN

− αq(M) · 1

N

N∑

n=1

E

[

E

[

(

h−1jj hj∆nsn + h−1

jj vjn

)

∣

∣h−1jj hj∆nsn + h−1

jj vjn

∣

∣

(

h−1jj vj

n

)H |∆n

]]

= αq(M) · εjN

− αq(M) · 1

N

N∑

n=1

√

π

4E

|hjj|−2 σ2v

√

Ps |hjj|−2hj∆n∆H

n hHj + |hjj|−2 σ2

v

(5.33)


where the second equality applied the chain rule for expectations, and the third equality

used:

E

[

x∗

|x|z]

=

√

π

4

E [x∗z]√

E [x∗x](5.34)

which holds for any jointly Gaussian random variables x and z.1 Using the positive definite

property of ∆n∆Hn we get:

(A)jj ≥ αq(M) · εjN − αq(M)

√

π

4

∣

∣h−1jj

∣

∣σv. (5.35)

(Note that the right most term in the right hand side of (5.35) does not depend of n.)

Substituting (5.28) and (5.35) into (5.23) we get:

zHj

(


zj ≥ 2αq(M) · εjN − 2αq(M)

√

π

4

∣

∣h−1jj

∣

∣ σv

− 2αq(M) · εjN

∑

i6=j

∣

∣

∣

(

D−1H)

ji

∣

∣

∣

≥ 2αq(M) · εjN (1 − β)

− 2αq(M)

√

π

4

∣

∣h−1jj

∣

∣ σv, (5.36)

where β was defined in (5.6). Combining (5.36) with (5.22) we get:

2αq(M) · εjN (1 − β) − 2αq(M)

√

π

4

∣

∣h−1jj

∣

∣ σv ≤ zHj

1

ND−1HW1H

HD−Hzj

+ α2uPs (1 + β)2 , (5.37)

which can be rephrased as:

εjN ≤

zHj D−1HW1H

HD−Hzj

2αq(M) (1 − β) N

+αuPs (1 + β)2

2q(M) (1 − β)+

√

π4

∣

∣h−1jj

∣

∣ σv

(1 − β). (5.38)

Noting that:

zHj D−1HW1H

HD−Hzj ≤(

1 + β2)

tr W1, (5.39)

and substituting the definition of q(M) from (5.26) leads to (5.10) and completes the proof

of the theorem.

1If x and z are jointly Gaussian, then z can be written as: z = γx + v, where E [x∗v] = 0 and γ =

E [x∗z] /E [x∗x]. Thus E[

x∗

|x|z]

= γE [|x|] =√

π4

E[x∗z]√E[x∗x]

.

5.3. PERFORMANCE ANALYSIS WITH ADDITIONAL COARSE POWER QUANTIZATION47

5.2.2 Steady State Approximation

In order to derive a steady state approximation of the ATE of the absolute error, measured

by the j-th user, εjN , we set β in (4.10) to zero to get:

εjN ≈ tr W1

αMπ

sin πM

· 2N +αuPs

2Mπ

sin πM

+

√

π

4

∣

∣h−1jj

∣

∣σv. (5.40)

Although this approximations is fairly heuristic, the result is surprisingly accurate as shown

in section 5.4.

5.3 Performance analysis with additional coarse power

quantization

The phase quantization feedback scheme, analyzed above, allows a guaranteed convergence

with a low rate feedback. Yet, in many cases one would prefer to achieve a faster convergence.

In this section we show that a coarse power quantization can significantly accelerate the

precoder convergence with a negligible cost in feedback rate. We first give our main results

in a theorem form, and in subsection 5.3.2 we will give the full proof.


Denote by gn the instantaneous error magnitude:

gjn =

√

E[

|εjn|2∣

∣∆n

]

, (5.41)

and let gjn = bj

ngjn be the coarse quantization of gj

n, where bjn is the multiplicative quantization

error. We envision a very coarse feedback which is updated once for many symbols, or is

calculated as the average of many error signals in time or in frequency (one example is

presented in detail in the simulations results section). For such a feedback, it is reasonable

to assume that the quantization error, bjn, is statistically independent of gj

n.

Using a matrix notation, we write

Bn = diag([b1n, ..., bu

n]), (5.42)

Gn = diag([g1n, ..., gu

n]) =

√

E[diag2(|εn|)|∆n], (5.43)

and

Gn = BnGn, (5.44)


where diag(x) denotes the diagonal matrix with the elements of x on its diagonal, and the

notation |x| for a vector means taking the absolute value of each element in it. In the

following we analyze the performance of a precoder which is updated according to:

Fn+1 = Fn − αGnεnsHn . (5.45)

Theorem 3 A sufficient condition for the precoder to converge is that:

βmax = max

max1≤i≤u

u∑

j=1j 6=i

|hij||hii|

, max1≤j≤u

u∑

i=1i6=j

|hij||hii|

< 1 (5.46)

and

α <2q(M)E[b1

n]√

π/4

uE [(b11)

2] PS

· 1

1 + βmax

. (5.47)

If these conditions are met, after sufficiently long convergence time, the Frobenius norm of

the difference between the ZF solution and the actual precoder, (5.45), is upper bounded by:

limn→∞

δn ≤αu · E

[

(b1n)2]

q(M)E[b1n]√

π/4·

u∑

i=1

σ2v

|hii|2

2 − 2β ·∣

∣

∣

∣

1 − αuPsE[(b1n)2]

q(M)E[b1n]√

π/4

∣

∣

∣

∣

− αu (1 + β2) · PsE[(b1n)2]q(M)E[b1n]

√π/4

(5.48)

where δn = tr

E[

(Fn − FZF ) (Fn − FZF )H]

.

5.3.2 theorem proof

Starting by subtracting FZF from both sides of (5.45), defining ∆n = Fn − FZF and Wn =

E[

∆n∆Hn

]

, therefore δn = tr Wn and we get:

δn+1 = tr

Wn − αE[

GnεnsHn ∆H

n

]

− αE[

∆nsnεHn Gn

]

+ α2E[

GnεnsHn snεH

n Gn

]

.(5.49)

We next introduce also a matrix notation of the phase quantization error,

Jn = diag(

[ejθ1n, ..., ejθu

n ])

, (5.50)

so that εn = Jn ·diag−1 (|εn|)·εn, and inspect the second term within the trace in (5.49). Not-

ing that both the phase quantization error and the power quantization error are independent

of the rest of the elements, and using the chain rule for expectations we can write:

E[

GnεnsHn ∆H

n

]

= q(M)E[B] · E[

E[

Gndiag−1(

|εn|)

εnsHn ∆H

n |∆n

]]

= q(M)E[b1n] · E

[

GnE[

diag−1(

|εn|)

εnsHn |∆n

]

∆Hn

]

, (5.51)

5.3. PERFORMANCE ANALYSIS WITH ADDITIONAL COARSE POWER QUANTIZATION49

where the second line also used the assumption that the power quantization error distribution

is identical for all users.

Substituting the definition of εn from (5.31) and using identity (5.34) we have:

E[

GnεnsHn ∆H

n

]

= q(M)E[b1n]√

π/4 · E[

Gn

(

E[diag2(|εn|)|∆n])− 1

2 PsD−1H∆n∆

Hn

]

= q(M)E[b1n]√

π/4 · E[

PsD−1H∆n∆

Hn

]

= q(M)E[b1n]√

π/4 · PsD−1HWn. (5.52)

Next, focusing on the right hand side term within the trace in (5.49), we have:

tr

E[

GnεnsHn snεH

n Gn

]

= E[

sHn sn · tr

εnεHn G2

n

]

= E[

sHn sn · tr

G2n

]

= E[

(b1n)2]

E[

sHn sn · tr

E[

εnεHn |∆n

]]

= E[

(b1n)2]

E[

sHn sn · tr

PsD−1H∆n∆

Hn HHD−H + D−1σ2

vD−H]

= uPs · E[

(b1n)2]

tr

PsD−1HWnH

HD−H + D−1σ2vD

−H

, (5.53)

where the equality in the first line used the rotation property of the trace function, the second

line results from the fact that all diagonal elements of εnεHn equal 1, the third line used (5.43)

and the fourth line used (5.31). Substituting (5.52) and (5.53) in equation (4.39), we have:

δn+1 = δn − αq (M) E[b1n]√

π/4 · Pstr

D−1HWn

− αq (M) E[b1n]√

π/4 · Pstr

WnHHD−H

+ α2uPs · E[

(b1n)2]

tr

PsD−1HWnH

HD−H + D−1σ2vD

−H

. (5.54)

This equation, (5.54), have exactly the same structure as equation (20) in [31]. Hence, the

rest of the proof repeats the steps introduced in [31]. For tractability, we present here the

necessary steps with the required adaption of the constants.

Defining:

Qn ,

(

I − αuE [(b1

n)2]

q (M) E[b1n]√

π/4PsD

−1H,

)

(5.55)

we rewrite (5.54) as:

δn+1 = L · tr

QnWnQHn

+ (1 − L) · δn

+α2uPs · E[

(b1n)2]

tr

D−1σ2vD

−H

(5.56)


where

Ln =q2 (M) E2[b1

n] (π/4)

uE [(b1n)2]

. (5.57)

Using the rotation property of the trace function and the fact that Wn is positive definite,

we upper bound (5.56) by:

δn+1 = L · tr

QHn QnWn

+ (1 − L) · δn + α2uPs · E[

(b1n)2]

tr

D−1σ2vD

−H

≤(

L · ρ(

QHn Qn

)

+ (1 − L))

· δn + α2uPs · E[

(b1n)2]

tr

D−1σ2vD

−H

(5.58)

where ρ (Z) is the spectral radius (or maximal eigenvalue) of the matrix Z. Noting that

L < 1, the condition for (5.58) to converge is:

ρ(

QHn Qn

)

< 1, (5.59)

and when time goes to infinity:

limn→∞

δn ≤α2uPs · E [(b1

n)2]u∑

i=1

σ2v

|hii|2

L (1 − ρ (QHn Qn))

. (5.60)

The spectral radius of QHn Qn is upper bounded by ([35] page 223):

ρ(

QHn Qn

)

≤∥

∥QHn

∥

∥

1‖Qn‖1 =

[

maxi

u∑

j=1

∣

∣

∣(Qn)ij

∣

∣

∣

] [

maxj

u∑

i=1

∣

∣

∣(Qn)ij

∣

∣

∣

]

, (5.61)

and testing the elements of the matrix Qn yields:

∥

∥QHn

∥

∥

1· ‖Qn‖1 ≤

∣

∣

∣

∣

∣

∣

1 − αuPsE

[

(b1n)

2]

q (M) E [b1n]√

π/4

∣

∣

∣

∣

∣

∣

+ αuPsE

[

(b1n)

2]

q (M) E [b1n]√

π/4· β

2

. (5.62)

If β < 1 and α <2q(M)E[b1n]

√π/4

uPsE[(b1n)2]· 1

(1+β):

ρ(

QHn Qn

)

≤∥

∥QHn

∥

∥

1· ‖Qn‖1 ≤ 1, (5.63)

which guaranties the convergence of (5.58). The proof is completed by substituting the right

hand side of (5.62) instead of ρ(

QHn Qn

)

in (5.58) to get (5.48).

Note that the steady state approximation, given in [31], can also be adapted to this case.

The covariance matrix of the error signal, is given by:

E[

|εn|2]

= E[

D−1 (HFn − D) snsHn (HFn − D)H

D−H]

+ D−1σ2vD

−H

= PsD−1HE

[

∆n∆Hn

]

HHD−H + D−1σ2vD

−H

= PsD−1HWnH

HD−H + D−1σ2vD

−H . (5.64)


An approximated solution of the sum of all users error power, can be derived by substituting

I instead of D−1H and taking the trace of (5.64) to get:

tr

E[

|εn|2]

≈ Ps · δn + tr

D−1σ2vD

−H

. (5.65)

To evaluate (5.65), we first need to evaluate δ, the steady state value of δn. Inspecting

equation (5.54) and setting δn+1 = δn = δ, and approximating it by substituting again I

instead of D−1H we get:

δn ≈ α2uPs

2αq(M)E [b1n]√

π/4Ps − α2uP 2s E[

(b1n)2] · tr

D−1σ2vD

−H

. (5.66)

Substituting (5.66) into (5.65) we get:

tr

E[

|εn|2]

≈(

α2uP 2s

2αq(M)E [b1n]√

π/4Ps − α2uP 2s E[

(b1n)2] + 1

)

·∑

i

σ2v

|hii|2. (5.67)

Note that the error signal covariance matrix with the ideal FEXT cancelation is

D−1σ2vD

−H

.

Therefore the error increase due to the use of the iterative precoder is described by the term

in brackets in 5.67.

5.4 Simulation Results

5.4.1 Phase feedback only

To demonstrate the results derived in the previous sections we conducted several simulations

using channel measurements, of 28 users in a binder, performed by France Telecom2. The

full setup, as well as statistical characterization of the channels, are described in [36]. The

transmitted PSD is set to -60dBm/Hz, the noise PSD is -140dBm/Hz, and we used channel

measurements at distance of 300 meters. In Figure (5.2) we compare the performance of

the dithered (rotated) phase quantization scheme with a regular (fixed) phase quantization

scheme for three different levels of phase quantization. The figure shows the scaled averaged

in time absolute error, measured by the first user 1σv·N

N∑

n=1

|εjn|, as function of the number of

symbols transmitted, at a frequency of 14.25MHz (β = 0.3576 ), using α√

Ps = 5 · 10−4. One

can see that the quantization scheme has no significant effect on the performance, for any

level of phase quantization.

2The authors would like to thank M. Ouzzif, R. Tarafi, H. Marriott and F. Gauthier of France TelecomR&D, who conducted the VDSL channel measurements as partners in the U-BROAD project.


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

102

Time (symbols)

Norm

alize

dtim

e-aver

aged

abso

lute

erro

r

Fixed quantization M=2

Dithered quantization M=2





Figure 5.2: Performance of the dithered phase quantization and the regular phase quantiza-tion, measured by the averaged in time absolute error scaled by σv, as a function of time.The figure compares three different levels of phase quantization.


100

101

102

103

104

105

10−6

10−5

10−4

10−3

10−2

10−1

100

101

102

Time (symbols)

Norm

aliz

ed a

bsolu

te e

rror

magnitude

∣

∣ǫ1

n

∣

∣

1

N

N∑

n=1

∣

∣ε1

n

∣

∣

bound on ǫ1

N

Figure 5.3: The absolute error of one user, the corresponding averaged in time absolute error,and the bound given by (5.10) as function of time, α

√Ps = 5 · 10−4.

To demonstrate the bound in (5.10), Figure 5.3, depicts, as function of time, the absolute

error of the first user, |ǫ1n|, the corresponding averaged in time absolute error, 1

N

N∑

n=1

|ε1n| ,

and the bound given by (5.10). The figure was depicted for the same frequency (14.25MHz),

using a phase quantization with M = 4 levels and α√

Ps = 5 · 10−4. One can see that (5.10)

indeed bounds the averaged in time absolute error. This bound can be quite loose for small

N , but it improves with time and well represent the performance after conversion.

In Fig. 5.4 we use the same frequency of 14.25MHz (β = 0.3576 ) and phase quantization

level of M = 4, and a much larger step size α√

Ps = 0.1. The figure shows that although the

averaged in time absolute error increases, it still converges and is bounded by (5.10).

This is further demostrated in Fig. 5.5 where for the same channel and same quantization

level, we show the averaged in time absolute error of the first user, after sending 104 symbols,

averaged over 100 simulations vs. the step size. The figure also show the corresponding

bound for each case. The figure shows that even for high step size, although the error can


0 100 200 300 400 500 60010

−3

10−2

10−1

100

Time (symbols)

Norm

aliz

ed a

bsolu

te e

rror

magnitude

Bound on ǫjN

∣

∣ǫjn

∣

∣

ǫjN

Figure 5.4: Maximum over all users, of the averaged in time absolute error, and the boundgiven by (5.10), as a function of time, α

√Ps = 0.1

be high it is still bounded by (5.10). In addition, the figure also depicts the approximation

on the steady state averaged in time absolute error given in (5.40). It can be seen that this

approximation is very good.


10−2

10−1

100

101

102

103

104

103

104

105

106

107

108

109

1010

α√

Ps

Norm

alize

dtim

e-av

eraged

abso

lute

erro

r

Bound

Empirical

Approximation

Figure 5.5: The averaged in time absolute error in the steady state as a function of theprecoder step size. The empiric figure shows the averaged absolute error, averaged over 100simulations, as well as the bound given by (5.10) and the approximation given by (5.40).


5.4.2 Phase feedback plus coarse power feedback

In this subsection we describe a simple scheme for coarse power feedback and illustrate

its performance. This scheme is based on dividing the available bandwidth into K sub-

bandwidths and averaging the squared error signals measured in all subcarriers of each sub-

bandwidth. We assume that the error power measurement and its quantization is performed

as follows: Denote the error signal measured by user i ∈ 1, .., u at the n-th symbol and

the f -th frequency bin as εin,f , where f ∈ 1, .., F, and F is the number of frequency bins

used. The bandwidth is divided into K equal sub-bandwidths, such that: f ∈ K∪

k=1Fk, where

Fk consists of F/K successive frequency bins and k ∈ 1, .., K. Each user calculates the

average error power per sub-bandwidth:

P iεk,n =

K

F

∑

f∈Fk

∣

∣εin,f

∣

∣

2, (5.68)

and the estimate Gk,n, which is the estimate of Gn for all frequencies f ∈ Fk is given by:

Gk,n = diag([√

P 1εk,n, . . . ,

√

P uεk,n

])

. Then Gk,n is obtained by a logarithmic quantization

with a step size of 2dB, and fed back to the transmitter (note that following (5.44) Gk,n =

Bf,n · Gf,n where f ∈ Fk).

Fig. 4.5 displays the performance of this quantization scheme, and compares it to the

performance of the phase-only quantization scheme and a system with no quantization. The

figure shows the mean user rate over the whole system bandwidth (30MHz) for the different

systems as function of time, at a distance of 300 meters. For reference the figure also shows

the performance of the ZF solution (FZF = H−1D). The figure shows the performance of

three different systems. The first is the system using the precoder that adapts with full

error signal feedback (no quantization) given by (5.1), using αPs = 12(u+1)

= 0.0172 for all

frequencies (note that this is half of the largest step size allowed when substituting β = 1 in

the step size condition given in [31]). The lowest curve shows the performance of a system

using the precoder that adapts according to the phase quantization (M=4) given by (5.5),

using α√

Ps = 3.5 · 10−4 for all frequencies. This system can achieve the performance level of

the full error feedback precoder in this case, by using a small enough step size. But in order

to achieve a convergence time that is close to that of the full error feedback precoder, it must

use a much larger step size, which significantly reduces its performance. This highlights the

contribution of coarse power feedback as it can achieve the same rate as the full feedback

at a steady state with only 50% longer convergence time. The coarse power feedback curve


0 500 1000 1500200

250

300

350

400

450

500

Time (symbols)

Mean u

ser

rate

[M

bps]

ZF performance

Full error feedback

Sign plus coarse power

Sign feedback

Figure 5.6: Average user rate at a distance of 300 meters as a function of time for the non-quantized error feedback, the phase only quantization and the phase quantization plus coarsepower quantization with K = 10.

uses K = 10 and an update step size of αPs = 0.0129. As can be seen, the addition of the

coarse power feedback significantly improves system performance compared to phase-only

quantization. This addition of coarse power feedback allows the system to converge much

faster and to a much higher rate. In this simulation, the coarse power quantization required

only 60 bits for the whole frequency band. Since we used F = 6900 frequency bins and

M = 4, this coarse power feedback increased the feedback rate from 2 bits per frequency bin

to 2.01 bits per frequency bin. As can be seen from the figure, this negligible increase in

feedback rate leads to a significant acceleration of precoder convergence.

Fig. 5.7 presents the steady state approximation given by (5.67), for three different phase

quantization levels. We used a precoder that adapts according to the phase quantization

plus coarse power feedback with K = 10 for all cases. The figure depicts two curves that

correspond to αPs = 12(u+1)

with a phase quantization level of M=2, and M = 4, and a

curve that corresponds to M=8 using αPs = 14(u+1)

. For each case, the figure also shows the


0 500 1000 15000

5

10

15

20

25

Time (symbols)

Cum

ula

tive

err

or

po

we

r in

cre

ase o

ve

r n

ois

e [d

B]

Empirical M = 2

Empirical M = 4

Empirical M = 8

Approximation M = 2

Approximation M = 4

Approximation M = 8

Figure 5.7: Sum of the square errors of all users over all frequencies using a precoder thatadapts according to the phase quantization plus coarse power feedback and K = 10, dividedby the sum of the corresponding square errors using the ZF precoder.

summation of tr G2n over all frequencies, scaled with the sum of σ2

v · tr

D−1D−H

over all

frequencies (which is the expected square error when using the ZF solution), as function of

time. The figure shows the corresponding steady state approximation for each case given by

(5.67), normalized by the sum of σ2v · tr

D−1D−H

over all frequencies. Note that the 2dB

spacing yields quite a small error, and the approximation is drawn using:

E[b1f,n] ≈ 1, E

[

(b1f,n)2

]

≈ 1, (5.69)

for all f and n (such accuracy requires K to be large enough). It can be seen that (5.67)

provides a good prediction of the steady state error.

5.5. CONCLUSION 59

5.5 Conclusion

In this chapter we presented and analyzed a linear, computationally-efficient, adaptive pre-

coder for FEXT cancellation using sign error feedback. The precoder is based on the precoder

presented in [31] with the proper analysis for very low rate faeedback schemes. This chapter

presents convergence condition for systems that feedback only a quantization of the phase

of the error signal and for systems that use also a coarse quantization of the error power.

The results show good performance of both schemes even with feedback rates of 1-2 bits per

frequency bin. The coarse power quantization significantly accelerates precoder convergence

costing only a negligible increase in the feedback rate.

The presented analysis shows that for phase feedback only, the averaged in time residual

FEXT is bounded, in almost all practical channels and for any precoder step size. In the case

of additional coarse power quantization, the analysis gives an upper bound on the adaptation

step size that guaranties precoder convergence. The analysis also give good approximations

for the steady state performance in both cases.

Chapter 6

Research summary

6.1 Research summary

In this thesis, we analyze an adaptive downstream multichannel VDSL precoder that is based

on error signal feedback. Based on the adaptive precoder presented and analyzed in [31] we

have analyzed two adaptive precoders, each of which reduces the complexity implementation.

In chapter 4 we presented a novel adaptive precoder for FEXT cancellation from only

selected interfering users. Partial FEXT cancellation is very important because it can sig-

nificantly reduce system complexity. We have derived sufficient conditions that guarantee

the convergence of the precoder to the ideal (in ZF sense) partial FEXT precoder presented

by Cendrillon et al. [32]. The analysis also provides bounds that enable the proper choice

of parameters and provides prediction on the expected performance. The results are sup-

ported by numerical simulations which show that the precoder converges very closely to the

ideal precoder presented in [32] and that partial FEXT cancellation also accelerates the con-

vergence. The results show that with proper choice of parameters the precoder converges

exactly to the ideal precoder presented in [32]. We also present steady state approximation

and demonstrated its accuracy through simulations.

In chapter 5 we analyzed an adaptive precoder for FEXT cancellation which adapts

according to a phase quantization of the received error signal used as a feedback. As actual

systems use low bit rate feedback it is important to analyze the performance when the bit rate

for the feedback is limited. We analyze the precoder presented in [31], when the feedback

consists of phase quantization of the received error signal, and even when the feedback is

only a sign bit. The paper presents convergence condition for systems that feedback only a

quantization of the phase of the error signal and for systems that use also a coarse quantization

61

62 CHAPTER 6. RESEARCH SUMMARY

of the error power. The results show good performance of both schemes even with feedback

rates of 1-2 bits per frequency bin. The coarse power quantization significantly accelerates

precoder convergence costing only a negligible increase in the feedback rate.

The presented analysis shows that for the phase feedback only case, the averaged in time

residual FEXT is bounded, in almost all practical channels and for any chosen step size.

In the case of additional coarse power quantization, the analysis gives an upper bound on

the adaptation step size that guaranties precoder convergence. The analysis also give good

approximations for the steady state performance in both cases.

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המחקר: לעבודת בעברית תקציר

Convergence Analysis of Adaptive FEXT

cancellation precoder for multichannel

downstream VDSL.

בנימיני עידו

2012 ביולי 24

DSLב־ משתמשים בין הפרעות של ביטול

המחקר לעבודת בעברית תקציר

בנימיני עידו

להנדסה, מהפקולטה ברגל איציק ד"ר של הנחייתו תחת נעשה המחקר

אילן. בר אוניברסיטת

מהדרישות כחלק הוגש

חשמל בהנדסת שני לתואר

ישראל גן, רמת 2012 ביולי 24

תקציר

המשמשים קיימים, נחושת קווי גבי על תקשורת העברת של מערכת היא DSL מערכת

סיבים ידי על היא שקיימת ביותר המהירה המידע העברת הביתיים. הטלפונים את

פרקטית. ולא מאוד יקרה אופטי לסיב בית כל שתחבר תשתית הכנת אולם אופטיים

מועבר המידע ומשם ראשית, למרכזיה עד אופטיים סיבים דרך מועבר המידע לפיכך

לעשרות המידע להעברת אחראית כזו מרכזיה כל נחושת. קווי דרך הקצה למשתמשי

מהמרכזיה היוצאים שהקווים כך בנין/אזור, באותו הנמצאים משתמשים מאות ולאף

"קולטים" סמוכים קווים אלקטרומגנטית מהשראות וכתוצאה לשני אחד בסמוך יוצאים

כאשר קורות אלו הפרעות הרצוי. המידע בפיענוח הפרעות יוצר וזה סמוכים, שידורים

לתחום מחולק הטלפון קווי של הסרט רוחב תדרים. באותם משתמשים שונים משתמשים

מהמרכזיה נתונים" ל"הורדת שמיועד תדרים תחום בטלפון, לדיבור שמיועד תדרים

אל האישיים מהמחשבים שידור מידע", ו"העלאת האישיים למחשבים ומשם ל"מודם"

חובבנים). לשמש שיכול פנוי תדרים תחום גם (ישנו המרכזיה

שכולם בגלל שכאמור המשתמשים אל מהמרכזיה בשידור עסקנו שלנו בעבודה

המשתמשים הקווים, בין מידע של "זליגה" וישנה תדרים באותם כך לצורך משתמשים

היא זו הפרעה אליהם. מיועד שלא ממידע הפרעה גם הרצוי למידע בנוסף קולטים

בביטול מתמקדים אנו זה ובמחקר DSLב־ הנתונים העברת קצב את המגביל הגורם

"דיבור" אין למשתמשים זו בטופולוגיה .Downstreamב־ או מידע" ב"הורדת הפרעות

הוא המרכזיה בפני שעומד האתגר במרכזיה. להתבצע חייב ההפרעות ביטול ולכן ישיר

לאיזה נגרמה הפרעה איזו המשתמשים לכל ששודרה ידועה אותות סדרת עפ"י להעריך

כלומר סינכרונית, בצורה נעשה מהמרכזיה השידור המפריע. המשתמש ומיהו משתמש

הנגרמת ההפרעה מהי תדע המרכזיה אם לכן זמן. חלון באותו המשתמשים לכל שידור

משתמש לאותו לשדר הבא בשידור תוכל היא אחר, משתמש מכל מסוים למשתמש

יגיע האות וכך הפוך, בסימן בדרך יקלוט שהוא ההפרעה את גם לו הרצוי לאות בנוסף

הכפלת ע"י מתבצע והוא Precoding נקרא זו בצורה שידור מהפרעות. נקי למשתמש

.Precoder שנקראת במטריצה המידע ווקטור

שונה גישה הערוץ. שיערוך על מסתמכות ורובם ההפרעות ללמידת שיטות כמה ישנם

הטעות לגבי למרכזיה הקצה משתמש שנותן במשוב משתמשת הבעיה לפתרון ויעילה

ופשוט יעיל אלגוריתם וניתחו זו בגישה הלכו ברגל וד"ר לשם פרופסור אצלו. שנקלטה

חלק ללמידת שלהם האלגוריתם את ניתחנו זה במחקר המרכזיה. ע"י ההפרעות ללמידת

ובדקנו המערכת מסיבוכיות יוריד ההפרעות של חלקי ביטול משתמש. לכל מההפרעות

המערכת ביצועי את ניתחנו כן כמו .DSL לערוצי נכונותו ואת יכולותיו את במחקר

מוגבל. הוא המשדר לבין המשתמשים בין המשוב של המידע קצב כאשר

DSL מערכת תכונות על סקירה הבאנו שבו מבוא פרק הוא הראשון הפרק

אותות בין הפרעות לביטול רקע הבאנו מכן לאחר אותה. המגבילים הגורמים ועל

במטריצת המשודרים האותות של הכפלה ע"י משדר, מאותו במקביל המשודרים שונים

.precoding

שהוצעו DSL למערכות שונות precoding מטריצות מספר סקרנו השני בפרק

מטריצת את גם הזכרנו .precodingה־ מטריצות לבניית שונות שיטות וכן בספרות,

הפרעות לביטול כלומר ,FEXT של חלקי ביטול של במובן האידיאלית precodingה־

מסוימים. ממשתמשים

המחקר. מטרת את וניסחנו המערכת של המתמטי המודל את הצגנו השלישי בפרק

הפתרון בהצגת פתחנו כאשר הפרעות, של חלקי לביטול כולו הוקדש הרביעי הפרק

את מתמטית בצורה ניסחנו מכן לאחר מתמטית. בצורה וניסוחו זה, במובן האידאלי

איזה בוחר משתמש כל כאשר הכללי, למקרה וביטולם ההפרעות לימוד אלגוריתם

המתארות המשוואות לנו שהיו אחרי מהם. ההפרעה את מבטל הוא מהמשתמשים

לביטול הלומדת המערכת של ההתכנסות את מתמטית הוכחנו הרצויה, המערכת את

אנליזה ע"י הראינו זה. במובן האידאלי precoderל־ הצענו, אותו חלקי הפרעות

ההתכנסות, על חסם מבטיחים נכונה פרמטרים בחירת ידי ועל סטנדרטיים שבערוצים

התוצאות כל ואת המערכת של היציב למצב קירוב הצגנו כן כמו ממש. התכנסות ואף

בסימולציות. הצגנו

בין המשוב של המידע קצב כאשר המערכת ביצועי את ניתחנו החמישי בפרק

את וניתחנו המערכת של מתמטי בניסוח פתחנו מוגבל. הוא המשדר לבין המשתמשים

ואף שידור לכל מידע ביטי של בודד למספר מוגבל היה המשוב כאשר המערכת ביצועי

ורק אך מורכב היה שהמשוב למקרה המערכת את ניתחנו סימן). (ביט אחד לביט רק

והראינו המערכת, להתכנסות חסם פיתחנו שנקלטה. הטעות של פאזה מקוונטיזצית

גם הצענו .DSL במונחי סטנדרטי יהיה שהערוץ בכך ורק אך תלוי ההתכנסות שתנאי

כי בסימולציות והראינו הטעות, אמפליטודת של גס קירוב הפאזה לקוונטיזציית להוסיף

כמשוב. שמשודרים המידע ביטי במספר זניחה עלייה עם דרסטית, משתפרים הביצועים

ido binyamini july 25, 2012 - bar ilan university · final thesis ido binyamini the research thesis...

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