S-72.244 Modulation and Coding Methods
Introduction / Overview
into linear system analysis
1
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
2
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
3
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
4
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 wireline local area networks (LANs) or in public switched
telephone network (PSTN)
5
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)
6
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:
7
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:
8
Telecommunication research areas 2001
9
Unmodulated sinusoidal

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:

x(t )  A(t ) cos  ct   (t ) 
10
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)
11
Time domain representation can only seldom
reveal small superimposed signals
12
Frequency domain representation of the same
signal reveals more!
13
Examples of other signals’ spectra

All finite signals have spectra that can be computed via Fourier
transformations or Fourier series
14
Noise and interference

In practical communication systems signals are blurred by noise
and interference:
Time Domain
Frequency Domain
15
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
16
Signal classification








Deterministic signals
Random signals; purely and pseudo-random
Energy signals; pulses
Power signal; periodic
Also:
Continuous time - discrete time:
x(t ), x[n]  x(nTS ), X ( f ), X [k ]
Analog - digital
Real - complex
17
Deterministic signals

Deterministic signals are signals which are completely specified
as a function of time.
Examples of deterministic signals:
A cos 2 f C t exp( at )  (t / T )  (t / T )
 (t )  u '(t )  du (t ) / dt

Some properties for delta function  (t ) :

 (t )   (t )
x(t ) (t  t )  x(t ) (t  t )
0
0
0
  (t  t0 ) (t )dt   (t0 )
 (at )   (t ) / a

(n)
n
(n)
  (t  t0 ) (t )dt  ( 1)  (t0 )

18
Example


Verify that  (at )   (t ) / a
Proof is based on equivalence property, e.g. two functions f1 and
f2 are equal only if


f
(
t
)

(
t
)
dt

 1
 f 2 (t ) (t )dt
where (t) is any function. Consider thus now an integral
1 


(
at
)

(
t
)
dt


  ( ) ( / a) d
a
1
1
  (0), a  0; if a  0 equals   (0)
a
a
and thus
1
a
  (at ) (t )dt   (0)  




 (t )
a
 (t )dt
19
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
K
 x(n)  mx   x[i] p[i] mx   xpx ( x)dx
i 1
A
where A is the span where p(x) exists.
The variance (AC power) is defined by
K
mx   ( xk  mx )
k 1
2
2
mx2   x 2 p( x)dx
A
T /2

The RMS value is
 x (t )dt / T  mx  mx
2
2
2
T / 2
20
Using MathCad for calculating averages and
variances
21
22
Energy and power signals

The normalized energy of a signal (pulse) is
E   x(t ) dt
Examples of energy signals
(t /  ) (t /  )
For periodic signals average power is defined instead of energy



1
P

T
P  T 1 lim
x
(
t
)
dt
T / 2 x(t ) dt  A cn

T   T / 2
Note that some signals are not energy or power signals as:
y(t )  u(t ), y(t )  t 2 , t  0
2
T /2


2
T /2
2
2
Wiener-Kinchine theorem defines an important relationship
between PSD P() and autocorrelation Rxx

P( )  X ( )  F  Rxx    Rxx ( )exp( j )d

2
23
Correlation and convolution

Correlation between signals x and y is defined by

Rxy   T x(t ) y (t   )dt
Convolution is defined by
T
x  y(t )  T x(t ) y(  t )dt
T

The Fourier transform of an autocorrelation yields the PSD
2
F x(t )  x(t )  X ( f ) X *( f )  X ( f )

Signal energy (or power) can be calculated in time domain or in
frequency domain


2
2
E   X ( f ) df   x(t ) dt




 Rxx (0)
Correlation is often used to find out PSD for random signals
Convolution is used for determining output of linear systems
24
Fourier transform and Fourier series


F-transforms are used to analyze pulses, F-series for analyzing
periodical signals
Definitions, Fourier transform:

F [ y (t )]  Y ( f )   y (t )exp(2 ft )dt


F [Y ( f )]  y(t )   Y ( f )exp(2 ft )df
1


Fourier series:

y (t )   cn exp( jn 0t )
n 
cn  T0

1
T0 / 2
 y (t ) exp( jn 0 t )dt
 T0 / 2
Total signal power (Parseval’s theory):
Ptot  To
1
T0 / 2

cn
 x(t ) dt  n

 T0 / 2
2
2
25
Some Fourier transforms
26
Some Fourier transforms (cont.)
27
Inspecting Fourier series using Mathcad
T0 / 2
compare: cn  T0  y (t )exp( jn0t )dt
T / 2
1
0
28
Inspecting Fourier series by Mathcad (cont.)
29
Inspecting Fourier series by Mathcad (cont.)

compare : y(t )   cn exp( jn0t )
n
30