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Fundamentals of Communication Theory

Ya Bao

Contact message:

Room: T700BTelephone: 020 7815 7588Email: baoyb@sbu.ac.uk

Website: http://eent3.sbu.ac.uk/staff/baoyb/foct/

Assessment:

3-hour written examination – 70%Lab accessed report – 30% (by lab tutor)

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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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Chapter One. Signals and processes Learning outcomesYou will be expected to know: the definitions of deterministic and non-

deterministic 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 IntroductionA Communication System

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1.2 Sinusoidal expressions Sinusoidal signal

)cos()( ooo tVtv

)cos()( tAtf

Or, )2sin()( ftAtf

or

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.

Note: ω=2πf

)2sin(cos

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Time domain graph)cos()( otooVtv )cos()( tAtf

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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

2

1

joules. |)(| 2t

tdttfE

Is finite even when the time interval becomes infinite; i.e., when

<2|)(| tfE

Average power dissipated by the signal f(t)

2

1

2

12

|)(|1 t

tdttf

ttp

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1.3 Classification of signals (cont2) Power signal

<<2/

2/

2|)(|1

lim0T

T

dttfTT

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.

tallfor )()( tfTtf 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

duuhutg

duuthugthtg

)()(

)()()()(

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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

2

0.5-t

u)-3(t-

t

0.5-t

u)-3(t-

t

1

u)-3(t-

5.22 ,2e

25.1 ,2e

5.11 ,2e

)()()()(

<

<

<

tdu

tdu

tdu

duthutgthtg

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1.4 Convolution (con5)– solution of example

5.22 )1(

25.1 0.5179)(

5.11 )1()()(

:Hence

5.22 )(

25.1 )(

5.11 )()()(

)5.73(5.132

5.133332

)1(332

25.0

3332

5.033

32

133

32

<

<

<

<

<

<

tee

teeee

tethtg

tee

tee

teethtg

t

ttt

t

tut

tt

ut

tut

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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

)()()()( 1221 tftftftf

Distributive Law

)()()()()]()([)( 3121321 tftftftftftftf

Associative Law

)()]()([)]()([)( 321321 tftftftftftf

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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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Bandwidth of a systemBandwidth W -- the interval of positive

frequencies over which the magnitude H(w) remains within –3dB.

2