Digital Modems

Lecture 1Fall 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

Application Layer

Network Layer

Link Layer

Physical Layer

Transport Layer

Web, FTP, VoIP

TCP, UDP

IP, routing

MAC

Capacity, bits, noise, waveforms

TerminologyThe '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, …

Tx Channel Rx

noise

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

source coding

channel coding

data format

modulatorsource

Transmitted waveform

Receiver performs the inverse operations

Tx-Rx diagram for different AI’s

- BPSK (m=1)- QPSK (m=2)- 16-QAM (m=4)- 64-QAM (m=6) m=log2(Μ)

2log ( )b

W

RR

r M

/bR rbRBitSource

Encoding Interleaving

Modulation

SymbolMapping

IFFTSerial/Parallel

CPInsertion

Spreading Scrabling

Rx

Tx

Channel

Equalization/demodulation

FFTEqualization -Parallel/Serial

CPRemoval

Despreading/Equalization

Descrabling

Channel estimation /Synchronization

Decoding DeinterleavingSymbol

Demapping

WR W CR R

1) Single Carrier2) CDMA3) Multi-carrier

(1)

(2)

(3)

(2)

(1)

(3)

0

bE

N 0

CbE

N

Scrambler Puncturing InterleaverReed-

SolomonConvolutional

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

RxPreambles Generator

Output Logic

PAPR Scaling

Cyclic Prefix Insertion

Input LogicData

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 Sym

bol Synch

Joint Frequency O

ffset Synch

Sync Preamble Extraction

Cyclic Prefix Extraction

Cyclic Prefix Extraction

FFT

FFTSync

Preamble Extraction

Demapper / DC Extraction

Demapper / DC Extraction

Channel Acquisition

ST Decoder (TSD)

Maximum Ratio Combiner

Phase TrackingPhase

Correction

Soft Decision Constellation

Decoder

LLR Constellation

Decoder

De-PuncturingDe-Interleaver Reed-Solomon

Convolutional Decoder (Viterbi)

Output Logic

Turbo DecoderDe-Puncturing Noise variance / SNR estimation

Adaptive Metric Calculation

Preambles

Data RxPilots

Data Preambles

Data Rx

Theory

Physical Channel

Distortion-less (LOS) channel:

Two-ray channel:

2j fh t A t H f Ae F

A

: channel gain: delay

21 j fh t t A t H f Ae F

Tx

Rx H f

f1

21

2

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

2T2T

x t

F

sin 2

2

fTX f

fT

1 T

2 1 2 ( )T T T

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

1.

0

M

m m

H

Rx Problem formulation

Radar example:

Binary transmission example:

1

2

:

:

r t s t w t

r t w t

H

H

r t

s t

w t

: observed signal: signal generated by the aircraft (if present): AWGN of power 0 2N

1

2

:

:

r t s t w t

r t s t w t

H

H

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

; ; , 0,1, , 1mr t s t w t t m M H T

T : Observation interval

; 0,1, , 1m m M r s w

where now we use vectors instead of continuous-time functions