# Channel Equalization in Digital Communications

Networking and Communications

Oct 23, 2013 (4 years and 8 months ago)

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Digital Modems

Lecture 1

Fall 2008

Course “mechanics”

Schedule & names for this semester

Every Tuesday, 12 pm
-
2:15 pm

Lecturers

Andreas Polydoros

Costas Aidinis

Stelios Stefanatos

{polydoros}, {caidinis}, {sstefanatos}@phys.uoa.gr

Offices: Building V, Second floor.

Course Outline

Fundamentals of detection theory

Detection problem formulation

Cost functions

Likelihood ratio

Optimal detection rules (Bayes/Neyman
-
Pearosn)

Handling of nuisance parameters

Discrete representation of stochastic processes

Signal space/basis

Orthonormal/Karhunen
-
Loeve expansion

Likelihood functionals

Application in communications

Binary/M
-
ary systems

Coherent/Non
-
coherent detection in AWGN

Error probability

Course text
-
book

H. L. Van Trees,
Detection, Estimation, and modulation
theory

J. G. Proakis,
Digital Communications

S. M. Kay,
Fundamentals of statistical signal processing:
Detection theory

A Systems View

ISO
-
OSI Protocol stack

Terminology

The '
Open Systems Interconnection Basic Reference Model'

(OSI Reference Model or
OSI Model
)

is an abstract description for layered communications and computer
network protocol

design.

It was developed as part of the
Open Systems Interconnection

(OSI) initiative
[1]
.

In its most basic form, it divides network architecture into seven layers which, from top to bottom, are the

Application, Presentation, Session, Transport, Network, Data
-

It is therefore often referred to as the
OSI Seven Layer Model
.

The
Physical Layer

defines the electrical and physical specifications for devices.

In particular, it defines the relationship between a device and a physical medium.

To understand the function of the Physical Layer in contrast to the functions of the Data Link Layer,

think of the Physical Layer as concerned primarily with the interaction of a single device with a medium,

where the Data Link Layer is concerned more with the interactions of multiple devices (i.e., at least two)

with a shared medium. The Physical Layer will tell one device how to transmit to the medium, and another device

how to receive from it (in most cases it does not tell the device how to connect to the medium).

Obsolescent Physical Layer standards such as
RS
-
232

The major functions and services performed by the Physical Layer are:

Establishment and termination of a
connection

to a
communications

medium
.

Participation in the process whereby the communication resources are effectively shared among multiple users.

For example,
contention

resolution and
flow control
.

Modulation
, or conversion between the representation of
digital data

in user equipment and the corresponding signals

transmitted over a communications
channel
. These are signals operating over the physical cabling

(such as copper and
optical fiber
) or over a
.

Source
: http://en.wikipedia.org/wiki/OSI_model

Three
-
part PHY
-
layer system model

Tx: Transmitter

Channel: Models the physical distortion

Noise: Thermal noise, interference, …

Block
-
Diagram Functions of Tx

Source

Discrete or analog

Source coding

Redundancy removal (entropy coding)

Data compression (introducing distortion)

Channel coding

Introduces redundancy to compensate for channel/noise

Data format

Mapping bits to symbols, create packets, frames, e.t.c.

Modulator

Convert the discrete
-
time input to the continuous
-
time transmitted
waveform

inverse operations

Tx
-
Rx diagram for different AI’s

Scrambler

Puncturing

Interleaver

Reed
-
Solomon

Convolutional
Encoder

Turbo Encoder

Puncturing

Constellation
Encoder

Pilot
Generator

Pilot & Data
Multiplexer

ST Encoder (TSD)

Mapping

Mapping

IFFT

IFFT

Cyclic Prefix
Insertion

PAPR
Scaling

Output
Logic

Control

From

Rx

Preambles
Generator

Output
Logic

PAPR
Scaling

Cyclic Prefix
Insertion

Input Logic

Data

Command

A modern Tx: MIMO/OFDM

A modern Rx: MIMO/OFDM

Input
Logic

Input
Logic

PAPR
Scaling

PAPR
Scaling

Synchronization

Frame Acquisition

Symbol Offset Estimation

Frequency Offset Estimation

Frame Acquisition

Symbol Offset Estimation

Frequency Offset Estimation

Joint Symbol Synch

Joint Frequency Offset Synch

Sync
Preamble
Extraction

Cyclic Prefix
Extraction

Cyclic Prefix
Extraction

FFT

FFT

Sync
Preamble
Extraction

Demapper /
DC Extraction

Demapper /
DC Extraction

Channel
Acquisition

ST Decoder (TSD)

Maximum Ratio
Combiner

Phase
Tracking

Phase
Correction

Soft Decision
Constellation
Decoder

LLR
Constellation
Decoder

De
-
Puncturing

De
-
Interleaver

Reed
-
Solomon

Convolutional
Decoder (Viterbi)

Output
Logic

Turbo Decoder

De
-
Puncturing

Noise variance /
SNR estimation

Calculation

Preambles

Data Rx

Pilots

Data

Preambles

Data Rx

Theory

Physical Channel

Distortion
-
less (LOS) channel:

Two
-
ray channel:

: channel gain

: delay

Physical Channel

The two
-
ray channel is the simplest example of a
multipath

channel

Question
: Under what circumstances is the two
-
ray channel
distortion
-
less

: It depends on the pulse shape

If the channel is (approximately) distortion
-
less

If the channel inevitably introduces severe distortion

Inference in general

Inference is the task of learning (e.g., making
estimations/decisions) based on given data

Examples of inference:

Estimate

the path loss introduced by a fading channel

Estimate

the range of an enemy aircraft

Predict

the stock market’s gain/loss

Decide

on which product is best

Decide

on which model best fits the observations

In this course we concentrate on a single sub
-
topic of
inference theory:
Hypothesis testing (Detection theory)

Emphasis will be given on how the theory is applied to
design
optimal

Decision criteria

A
cost function

must be defined in order to obtain a
detection rule

This function quantifies the cost of taking erroneous
decisions

What is the cost of “detecting” an aircraft when it is
actually not there?

What is the cost of missing the presence of the aircraft?

After construction of the cost function an optimal
decision rule can be obtained that results in minimum
cost

The appropriate cost function depends upon the
context of the specific problem and is not unique

Hypothesis testing @ Rx side

Problem formulation:

We are given a set of data (observations)

This set could have been generated as the outcome of one of
M

possible hypothesis

Given the data, and any other
statistical

information, we
want to
decide

on the correct hypothesis

Examples:

Decide if the data provided by a radar indicate the presence
of an aircraft

From a noisy received signal, decide on the transmitted
digital sequence

Rx Problem formulation

Binary transmission example:

: observed signal

: signal generated by the aircraft (if present)

: AWGN of power

Rx Problem formulation

In this class, only distortion
-
less

channels will be
considered, including AWGN

The observed signals are of the form:

In case the observation is discrete we have

: Observation interval

where now we use vectors instead of continuous
-
time
functions