Fundamentals of Communication Theory
Ya Bao
Contact message:
Assessment:
Room:
T700B
Telephone:
020 7815 7588
Email: [email protected]
3-hour written examination – 70%
Lab accessed report – 30% (by lab
tutor)
Website: http://eent3.sbu.ac.uk/staff/baoyb/foct/
Ya Bao
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1
Introduction of the unit

This unit consists of five topics:





Signals and processes.
Fourier analysis and applications
Random signals and processes
Correlation processes.
Electrical noise
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2
Chapter One. Signals and processes
Learning outcomes
You will be expected to know:





the definitions of deterministic and nondeterministic signals;
mathematical representations of deterministic
signals;
the idea of power and energy signal and methods to
calculate these parameters;
processes of multiplication, and convolution of time
signals.
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1.1 Introduction
A Communication System
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1.2 Sinusoidal expressions

Sinusoidal signal
f (t )  A cos(t   )
Or,
f (t )  A sin( 2ft   )
or
v(t )  Vo cos(ot  o )
Note: ω=2πf
cos  sin(    )
2
Where: A is the sinusoid's amplitude
ω is the angular velocity of the sinusoid in radian/s,
θ is an arbitrary phase in radian.
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Time domain graph
v ( t )Vo cos(ot o )
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f (t )  A cos(t   )
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Frequency domain spectra
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1.3 Classification of signals

Energy signals, Power signals
An energy signal is a pulse-like signal that usually exits
for only a finite interval of time or has a major portion of
its energy concentrated in a finite time interval.
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1.3 Classification of signals (cont1)
An energy signal is defined to be one fro which the
t2
E   | f (t ) | dt
2
joules.
t1
Is finite even when the time interval becomes infinite; i.e., when

E   | f (t ) | < 
2

Average power dissipated by the signal f(t)
1 t2
2
p
|
f
(
t
)
|
dt

t2  t1 t1
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1.3 Classification of signals (cont2)

Power signal
1
0 < lim
T  T

T /2
| f (t ) | dt < 
2
T / 2
Then the signal f(t) has finite average power and is
called a power signal.
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1.3 Classification of signals (cont3)

Periodic, Nonperiodic (aperiodic)
A periodic signal is one that repeats itself exactly
after a fixed length of time.
f (t  T )  f (t ) for all t
T – period, it define the duration of one complete cycle of f(t)
If energy/cycle is finite then it is power signal.
Any signal for which there is no value of period
T is said nonperiodic (or aperiodic) signal.
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1.3 Classification of signals (cont4)

Deterministic, non-deterministic (random)
Deterministic signal: no uncertainty in its values.
 an explicit mathematical expression can be
written
Random signal: some degree of uncertainty before it
actual occurs. (discussed later)
 A collection of signals, each of which is different
e.g. uncertain starting phase
 Future values of the signal may not be
predictable. E.g. noise
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1.4 Multiplication and Convolution



Multiplication in frequency domain is convolution in time domain.
Convolution in frequency domain is multiplication in time domain.
Convolution may be defined

g (t )  h(t )   g (u )h(t  u )du


  g (t  u )h(u )du

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1.4 Convolution (con1)-- example

Example: Convolve the two signals g(t) and h(t) in (a)
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1.4 Convolution (con2)– solution of example



Step 1
Introduce a dummy variable to form g(u) and h(u) F(b)
Step 2
Form g(t-u), F(c).
Step 3
F(d). Place g(t-u) and h(u) on a common set of axes.
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1.4 Convolution (con3)– solution of example

Three distinct regimes:
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1.4 Convolution (con4)– solution of example
Step 4
Determine the convolution


g (t )  h(t )   g (t  u )h(t )du

t
 e
-3(t- u)
1

t

2
t -0.5
t -0.5
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2du,
e-3(t-u) 2du,
e
-3(t- u)
2du,
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1  t < 1.5
1.5  t < 2
2  t < 2.5
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1.4 Convolution (con5)– solution of example
3 t
3u t
1
 3t
3u t
t  0. 5
 e (e )
1.5  t < 2
 3t
3u 2
t  0. 5
2  t < 2 .5
g (t )  h (t )  e ( e )
2
3
2
3
1  t < 1.5
 e (e )
2
3
Hence :
g (t )  h(t )  (1  e
2
3
3( t 1)
1  t < 1.5
)
 3t
3t
1.5
( 3t 7.5 )
 e (e  e e )  0.5179
2
3
 e (e
2
3
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3 t 1. 5
 1)
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1.5  t < 2
2  t < 2.5
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1.4 Convolution (con6)– solution of example

The results may be sketched to show pictorially the
effect of convolving g(t) and h(t)
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1.5 Properties of Convolution

Commutative Law
f1 (t )  f 2 (t )  f 2 (t )  f1 (t )
Distributive Law
f1 (t )  [ f 2 (t )  f 3 (t )]  f1 (t )  f 2 (t )  f1 (t )  f 3 (t )
Associative Law
f1 (t )  [ f 2 (t )  f 3 (t )]  [ f1 (t )  f 2 (t )]  f 3 (t )
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exercise

Find the convolution of the rectangular pulse f1(t) and
the triangular pulse f2(t) show in following Fig.
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1.6 Analogue Filters
Filter: a circuit that place a a limit upon the range of frequencies it
will pass, and rejects any frequencies that fall outside this range.
Low pass, high pass, band pass, band stop (notch)
Commonly used in communication systems.
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By Ya Bao
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Bandwidth of a system
Bandwidth W -- the interval of positive
frequencies over which the magnitude H(w)
remains within –3dB.
2
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