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S-72.244 Modulation and Coding Methods

Introduction / Overview into linear system analysis

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S-72.244 Modulation and Coding Methods: Practical arrangements (2 cr) Lectures: Timo Korhonen, Otakaari 8, room 209, 2. Floor,

phone 451-2351, timo.korhonen@hut.fi Tutorials: Jahangir Sarker, Otakaari 5, phone 451-2347,

jahangir.sarker@hut.fi Lectures on Mondays, 14-16, hall S1. Tutorials on Thursdays,

12-14, hall S5 (Exercises carry bonus points!) Text books :

– A.B. Carlson: Communication Systems, IV ed

– B.P. Lathi: Digital and Analog Communication Systems

– Reference: J. G. Proakis: Digital Communications Homepage http://www.comlab.hut.fi/opetus/244

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Course outline, fall 2001

Introduction

Linear modulation

Exponential modulation

Carrier wave systems

Noise in carrier wave systems

Revision I

Sampling and pulse coded modulation I

Sampling and pulse coded modulation II

Baseband digital transmission

Error control coding I

Error control coding II

Bandpass digital transmission

Revision II

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Introduction

Electrical systems utilize varying currents, voltages (eg. varying electromagnetic fields) to store and convey information

Telecommunication systems are dedicated to transporting information from point to point using links or networks

Telecommunication messages are transmitted via different media as copper wire, microwave or optical fiber

Signal is adapted to the transmission and media by modulation and coding

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Motivation for applying modulation and coding

Modulation is done to enable to use the media for the intended message. Thus the modulation scheme is selected based on

• Message to be transmitted as

– voice

– data

– continuous / bursty traffic

• Allowed delay

• Media that is to be used; compare

– Note that for instance in wireless networks a different modulation method can be more appetizing than in wire-line local area networks (LANs) or in public switched telephone network (PSTN)

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Motivation for applying modulation and coding (cont.)

Coding is done ...– For detection and/or correction of errors produced by the channel (as

block and convolutional coding) by• noise• interference• distortion

– linear– nonlinear

– To alleviate synchronization problems (as Manchester coding)– To alleviate detection problems (as differential coding)– To enable secrecy (as scrambling or ciphering)

Channel coding principles:– ARQ (Automatic Repeat Request)– FEC (Forward Error Correction)

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A classification of communication systems

Communication systems balance between different efficiencies With respect of markets, systems is a failure if anyone of these

efficiencies criteria is not met for a particular application platform:

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System efficiencies, examples

Microwave links:– good bandwidth efficiency, low error rate required

– power efficiency and cost not so important Wireless mobile systems:

– power, bandwidth and cost efficiency very important

– earlier bandwidth efficiency was easiest to compromise

– nowadays signaling rates increase and bandwidth efficiency is becoming more and more important issue

It is most important that a system designer should recognize communication system relationship to these basic qualifiers:

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Telecommunication research areas 2001

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The sinusoidal wave is parameterized by constant amplitude, frequency and phase:

All parameters known, thus: convoys no information! Mathematically and experimentally convenient basic

formulation whose parameterization by variables enables presenting all the modulation formats:

Unmodulated sinusoidal

( ) ( )cos ( )cx t A t t t

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Practical signals

A set of voice tones:

– Several tones superimposed

– Tones can not be separated from the time domain representation

– Frequency components can be separated from a frequency domain representation

“This is some speech”

– Bursts

– Amplitude varies

– Frequency (phase) varies

– Many other practical sources are bursty as

• video signals

• data packets (for instance in Ethernet)

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Time domain representation can only seldom reveal small superimposed signals

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Frequency domain representation of the same signal reveals more!

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Examples of other signals’ spectra

All finite signals have spectra that can be computed via Fourier transformations or Fourier series

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Noise and interference

In practical communication systems signals are blurred by noise and interference:

Time Domain Frequency Domain

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A short review to signals and systems

(A) Signals and spectra

– Line spectra, Fourier series

• Phasors and line spectra

• Periodic signals - average power

• Parseval’s power theorem

– Fourier transform and continuous spectra

• Fourier transforms

• Rayleigh’s energy theorem

• Duality

– Time and frequency transformations

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Signal classification

Deterministic signals Random signals; purely and pseudo-random Energy signals; pulses Power signal; periodic Also: Continuous time - discrete time:

Analog - digital Real - complex

( ), [ ] ( ), ( ), [ ]S

x t x n x nT X f X k

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Deterministic signals

Deterministic signals are signals which are completely specified as a function of time.

Examples of deterministic signals:

Some properties for delta function

cos2C

A f t exp( )at( ) '( ) ( ) /t u t du t dt

( ) :t

0 0 0

0 0

( ) ( )

0 0

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) /

( ) ( ) ( 1) ( )n n n

t t

x t t t x t t t

t t t dt t

at t a

t t t dt t

( / )t T ( / )t T

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Example

Verify that Proof is based on equivalence property, e.g. two functions f1 and

f2 are equal only if

where (t) is any function. Consider thus now an integral

and thus

( ) ( ) /at t a

1 2( ) ( ) ( ) ( )f t t dt f t t dt

1( ) ( ) ( ) ( / )

1 1(0), 0; if 0 equals (0)

at t dt a da

a aa a

1 ( )( ) ( ) (0) ( )

tat t dt t dt

a a

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Random signals

Random signals are signals that take a random value at any given instance of time

These signals must be modeled probabilistically (e.g. using distribution functions as PDF and CDF)

The discrete and continuous mean (e. g. mean for a discrete variable) is defined by

where A is the span where p(x) exists. The variance (AC power) is defined by

The RMS value is

1( ) [ ] [ ]

K

xi

x n m x i p i ( )

x xA

m xp x dx

22

1( )

K

x k xk

m x m 2 2 ( )

xA

m x p x dx/ 2 22 2

/ 2

( ) /T

x xT

x t dt T m m

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Using MathCad for calculating averages and variances

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Energy and power signals

The normalized energy of a signal (pulse) is

Examples of energy signals

For periodic signals average power is defined instead of energy

Note that some signals are not energy or power signals as:

Wiener-Kinchine theorem defines an important relationship between PSD P() and autocorrelation Rxx

2/ 21

/ 2lim ( )T

TTP T x t dt

( / ) ( / )t t

2 2/ 21

/ 2( )T

T nA

P T x t dt c

2

( ) ( ) ( )exp( )xx xx

P X R R j d

F

2( ) ( ), ( ) , 0y t u t y t t t

2

( )E x t dt

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Correlation and convolution

Correlation between signals x and y is defined by

Convolution is defined by

The Fourier transform of an autocorrelation yields the PSD

Signal energy (or power) can be calculated in time domain or in frequency domain

Correlation is often used to find out PSD for random signals Convolution is used for determining output of linear systems

( ) ( )T

TxyR x t y t dt

( ) ( ) ( )T

Tx y t x t y t dt

2

( ) ( ) ( ) *( ) ( )x t x t X f X f X f F

2 2

( ) ( )

(0)xx

E X f df x t dt

R

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Fourier transform and Fourier series

F-transforms are used to analyze pulses, F-series for analyzing periodical signals

Definitions, Fourier transform:

Fourier series:

Total signal power (Parseval’s theory):

1

[ ( )] ( ) ( )exp( 2 )

[ ( )] ( ) ( )exp(2 )

y t Y f y t ft dt

Y f y t Y f ft df

F

F

0

0

0

/ 21

0 0/ 2

( ) exp( )

( )exp( )

nn

T

nT

y t c jn t

c T y t jn t dt

0

0

/ 22 21

/ 2

( )T

tot o nnT

P T x t dt c

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Some Fourier transforms

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Some Fourier transforms (cont.)

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Inspecting Fourier series using Mathcad

0

0

/ 21

0 0/ 2

compare: ( )exp( )T

nT

c T y t jn t dt

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Inspecting Fourier series by Mathcad (cont.)

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Inspecting Fourier series by Mathcad (cont.)

0compare : ( ) exp( )

nn

y t c jn t