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
Recommended Reading
Course text

book
H. L. Van Trees,
Detection, Estimation, and modulation
theory
Additional references
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

Link, and Physical Layers.
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
do use physical wires to control access to the medium.
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
radio link
.
Source
: http://en.wikipedia.org/wiki/OSI_model
Three

part PHY

layer system model
Tx: Transmitter
Rx: Receiver
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
Receiver performs the
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
Adaptivity
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
Adaptive Metric
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
fading
channel
Question
: Under what circumstances is the two

ray channel
distortion

less
Answer
: 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
receiver structures
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
Radar example:
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
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