Principles of Telecommunications

• Communication is the largest sector of the

electronics field, hence knowledge &

understanding is a must for every student

• The field of Electronic Communication changes

so fast

• Need for Firm Grounding in Fundamentals: also

understanding of the real world components,

circuits, equipment & systems in everyday use.

• Balance in Principles of latest techniques

• Study the system level understanding

EEB317 Principles of Telecoms

•Signals & Systems

•Amplitude Modulation

•Angle Modulation

•Detection & Demodulation

•Noise in Communications System

Signals & Systems

•Classification

•Power & Energy Spectral Density

•Correlation

•Fourier Series & Transform

•Convolution

Overview

• Tx of information between 2 distant points

• Dominated by 4 important sources: speech,

television, facsimile & personal computers

• Three basic processes:

• Transmitter, Channel & Receiver

Communication System

Communication System

Classification of signals & Systems• A system is an interacting set of physical

objects or physical conditions called system

components

• A signal: set of information or data. Can be

input, output or internal

• Signals may be functions of independent

variables such as time, distance, force, position,

pressure, temp … for simplicity only time will be

used in this class

• Mathematical models are mathematical

equations that represent signals & systems

• They permit quantitative analysis and design of

signals & systems

• Continuous-time signal x(t), has a value

specified for all points in time, & a continuous-

time system operates on and produces

continuous-time signals

Continuous-time signal

Signals

Signals• Discrete-time signal: Signal specified only at

discrete values

• Analog signal: Signal whose amplitude can

take on any value in a continuous range

• Digital signal: Signal whose amplitude can

take on only finite number of values (M-ary)

• Periodic signal: Signal g(t) is periodic if for

some +ve constant T0 (period):

)()( 0Ttgtg

DT signals

Energy & Power Signals• Energy signal: Signal with finite energy & zero

power

• Power signal: Signal with finite power & infinite

energy

dttgE

2)(

2/

2/

2

2/

2/

2

)(1

lim

)(1

lim00

T

TT

T

TT

dttgT

P

dttgT

P

Are these energy or power signals?

•

Energy & Power Signals examples

• Since in (a) amplitude approaches zero

• It’s an energy signal:

• Since (b) is periodic, it’s a power signal:

t

84422)(0

20

1

22 2/

dtedtdttgEt

3

13

2

1)(

1)(

1lim

32

11

1

1

1

21

1

22/

2/

2

t

dttdttgT

dttgT

PT

TT

Worked example

• Determine the power of the following signals

)cos()( tCtg

2/

2/

22

2/

2/

222/

2/

22/

2/

2

2)]22cos(1[

2

)(cos1

)cos(1

)(1

lim

T

T

T

T

T

T

T

TT

Cdtt

T

CP

dttCT

dttCT

dttgT

P

Work these out

• (b)

• (c)

)cos()cos()( 222111 tCtCtg21

tjDetg 0)(

More examples

• Compute the signal energy & signal power for

the following complex valued signal and

indicate whether the signal is an energy or

power signal

tjAetg 2)(

More examples

• Since g(t) is a periodic signal, it cannot be an energy signal. Therefore compute the signal power first. Signal Period:

• Since the signal has a finite power, it is a power signal & has infinite energy (VERIFY!)

10 T

2/1222

2 1

1

/11

1

/11

1

AtAdtAdtAePt

t

tjt

t

t

t

Deterministic & Random Signals

• Deterministic signal: Physical description is

known in either a mathematical or graphical

form. eg:

• Random signal: Signal known only in terms of

probabilistic description such as mean value,

mean square value rather than its complete

mathematical or graphical description

• eg: noise signal, message signal

)1(tan)( 1 ttg

Deterministic & Random Signals

• Use matlab to plot a deterministic signal:

• & random noise

• To use matlab you must:

• (a) declare the variables

• (b) since it’s first time, use the plot command

• (c) label the plot

)1(tan)( 1 ttg

Signal Operations

• Time shifting: If a signal g(t) is time shifted by

t1 units, it is denoted as f(t) =g(t-t1).

• If t1>0, the shift is to the right (time delay)

• If t1<0, the shift is to the left (time advance)

• To demonstrate time shifting plot the signals:

)1(tan)(

)1(tan)(

)(tan)(

1

1

1

ttY

ttG

ttg

Time Shift matlab code

• close all; % close graghs

• clear all; % clear all the variables & functions from memory

• t = -5:.3:5; % declear variable "t"

• g = atan(t);

• G = atan(t-1);

• Y = atan(t+1);

• plot(t,g); % plots the function "g"

• hold;

• plot(t,G,'r'); % plots “G”

• plot(t,Y,'k');

• hold off;

• grid on;

• xlabel('t');

• ylabel('g(t),G(t) & Y(t)');

• title('time shifting demonstration');

• legend('g-original','G-delay','Y-advance');

Time Shift gragh

Signal Operations

• Time Scaling: Compression or expansion of a signal

• Signal f(t) is g(t) compressed by a factor of ‘a’ if f(t) = g(at), therefore f(t/a) = g(t) for a>1

• Similarly f(t) is g(t) expanded (slowed down) by a factor of ‘a’ if f(t) = g(t/a), therefore f(at) = g(t) for a<1

• To time-scale a signal by a factor of ‘a’, replace t with at.

• If a > 1 the scaling is compressed & if a < 1, the scaling is expanded.

Time Scaling demonstration

• close all;

• clear all;

• t = -5:.3:5;

• f = sawtooth(t);

• G = sawtooth(2*t);

• F = sawtooth(t/2);

• plot(t,f); hold;

• plot(t,G,'r');

• grid on;

• xlabel('t');

• ylabel('f(t) & G(t)');

• title('time scaling demonstration');

• legend('f','G');

Time Scale gragh

Signal Operations

• Time Reversal/Inversion/Folding:

• To time reverse a signal, replace t with –t

• If f(t) is a time resersal of g(t) then

• f(t)=g(-t)

• See the matlab code of g(-t)

Time Reversal Demo-code

• close all; clear all;

• t = -5:.3:5;

• g = atan(t);

• G = atan(-t);

• plot(t,g); % hold;

• plot(t,G,'r'); % hold off;

• grid on;

• xlabel('t');

• ylabel('g(t) & G(t)');

• title('time Reversal Demonstration');

• legend('f-original', 'G-timeReversed');

• Continous time signal: g(t)

• Samples of continuous-time signal: g(nT)

• Discrete-time signal: g(n)

Samples of Continuous-time signal

• close all; clear all;

• t=-5:.5:5;

• g=atan(t);

• plot(t,g);

• ylabel('g(t)');

• grid on;

• title('Continuous-time signal Demonstration');

• figure

• stem(t,g);

• grid on;

• xlabel('nT');

• ylabel('g(nT)');

• title('Samples of Continuous-time signal');

Delta Function

• Delta/Dirac/Unit Impulse function:

Rectangular pulse with an infinitesimally small

width & infinitely large height & an overall area

of unity.

1)(

0)(

dtt

t

0t

Multiplying a function by delta

• Sampling/Sifting property of delta

)()(()

)(*)0()()(

TtTfTttf

tfttf

)()()()()(

)0()()0()()(

TfdtTtTfdtTttf

fdttfdtttf

Sampling/sifting property

• The area under the product of a function with

delta is equal to the value of that function at the

instant where delta is located

• Function f(t) must be continuous where the

delta is located

Unit Step Function

t

d1

0)(

0

0

t

t

t

tud )()(

)(tdt

du

Time Shifted, scaled, reversed step

• Causal function: It’s zero before t = 0 otherwise

is non-causal

)(

)(

)(

a

btu

a

btu

batu0

0

t

t

Correlation

• Correlation coefficient cn:

• Cross-correlation:

• Autocorrelation:

11 nc

dttztg

EEc

zg

n )()(1

dttztggz )()()(

dttgtgg )()()(

Fourier Series

• Fourier analysis considers signals to be

constructed from a sum of complex

exponentials with appropriate frequencies,

amplitude & phases

• Frequency components are the complex

exponentials (sines & cosines) which, when

added together, make up the signal

• Orthogonality of signal set: ntxtxtx )(),...(),( 21

Generalized Fourier Series

nm

nm

1

2211

)()(

...)(...)()()(

n

nn

nn

txctg

txctxctxctg

21 ttt

dtxtgEdtx

dtxtgc

t

tn

n

t

tn

t

tn

n

2

12

1

2

1 )(1)(

2

n

t

tnm

Edtxtx

0)(

2

1

Fourier coefficients

nm xx , Are othorgonal

Exponential FS

• Orthogonality:

• Expon. FS

0

)(0

)(0

00

0

0

Tdtedtee

T

tnmjtjn

T

tjm

nm

nm

0

0

0

)(1

)(

0T

tjn

n

n

tjn

n

dtetgT

D

eDtg

Parseval’s Theorem

• Energy of the sum of orthogonal signals is

equal to the sum of their energies:

• Parseval’s theorem:

1

2

1

2

1

2

1

2

1

2

1

222

111

2

1

2

1

)()(

)....()(

);()(

EcdttxcdttxcE

txctg

txctg

t

t

t

t

1g

n

n

ng

g

EcE

EcEcE

2

2

2

21

2

1 ...

Trigonometric Fourier Series

• Trig. FS

01

1

01

1

01

101

1

01

1

0

0

0

0

0

00

2

0

0

1

00

2sin)(2

)(1

2cos)(2

2cos

2cos)(

2sin2cosag(t)

Tt

tn

Tt

t

Tt

tTt

t

Tt

t

n

n

n

n

tdtfntgT

b

dttgT

a

tdtfntgTtdtfn

tdtfntga

tfnbtfna

Fourier Transform & Spectra of

Aperiodic Signals• The spectrum of a periodic signal is found from

FS of a signal over one period

• Since the FS is a periodic function of time, it is equal to an aperiodic signal only over the FS expansion interval, outside this interval it repeats

• FS is used to produce the spectrum of the periodic extension but not the spectrum of aperiodic signal

• To find the spectrum of an aperiodic signal we use FT

Fourier Transform (FT)

• To develop the FT, let’s start with the exponential FS

representation of a periodic signal over the interval

-T/2<t<T/2

• Let the interval increase until the entire time axis is

encompassed

• Since FT is developed from the FS the conditions for

the existence follow from those of the Dirichlet

conditions

dttg )(

Fourier Transform

)()(

)]([)()(

)]([)()(

)(1

)(

12

2

2/

2/0

0

0

fGtg

fGdfefGtg

tgdtetgfG

dtetgT

D

eDtg

ftj

ftj

T

T

tjn

n

n

tjn

n

Fourier Transform (Spectrum of g(t))

Inverse Fourier Transform

Fourier Transform pair

Fourier Transform Theorems

• FT characteristics are expressed in the form of

theorems

• The theorems are useful in computing FT of

complicated signals

• Linearity:

• If x(t) X(f) & y(t) Y(f)

• Then

• ax(t)+by(t) aX(f)+bY(f)

• Integral in a linear operation

Fourier Transform Theorems

• Time Shift

• If

Then

Prove:

02

0 )()(

)()(

ftjefXttx

fXtx

)(22)(2

0

)(22)(2

0

00

2

00

00

00

)(])([)]([

)()()]([

)()]([

tfjfjtfj

tfjfjtfj

ftj

efXdexettx

deexdexttx

ttddttt

dtettxttx

Fourier Transform Theorems

• Time Reversal

)()(

)()(

fXtx

fXtx

Fourier Transform Theorems

• Time scale (Scale change)

)/(1

)(

)()(

afXa

atx

fXtx

Frequency Translation

• If

• Then

• Modulation• If

• then

)()(

)()(

0

2 0 ffXetx

fXtx

tfj

)(2

1)(

2

12cos)(

)()(

000 ffXffXtftx

fXtx

Inverse FT

• Find the inverse FT of the delta function

)(2

)(2

2

1)(

2

1)(

2

1)]([

)]([)()(

0

0

0

00

1

12

0

0

0

0

tj

tj

tj

tjtj

ftj

e

e

e

ede

fGdfefGtg

FT of cosine

• Find the FT of the sinusoid (cosine)

)()(][cos

)](2)(2[2

1)](

2

1[][cos

)(2

1cos

000

000

0

00

00

t

eet

eet

tjtj

tjtj

FT of rectangle

• Find the FT of a rectangle of width tau

/22/0)2/(sin

)2/(sin)(

)2/(sin)2/sin(2/

)2/sin(2

)(

2

)(2)(

1)(

)()(

2/2/2/2/

2/

2/

nnc

ct

rect

cX

j

eeee

jX

dtedtet

rectX

jjjj

tjtj