ERG2310A-I p. I-1

Introduction

Information: Voice, data, image, video, music, etc.

Communications systems

Deliver or exchange information among various distant parties

Examples: telegraph, telephony, facsimile, radio, satellite, optical fiber systems, cellular mobile, data networks, etc.

ERG2310A-I p. I-2

Introduction

Block Diagram of a Communication System

Transmitter- transform the message signal produced by the source of information into a form

suitable for transmission over channel

Channel

- transmission media- may distort the transmitted signal - may add noise and interfering signals to the received signal

Receiver- reconstruct a recognizable form of the original message signal- deliver it to the user destination

Transmitter

Channel

ReceiverEstimateofmessagesignal

Receivedsignal

Transmittedsignal

Messagesignal

Source of information

Recipient of information

ERG2310A-I p. I-3

Analog and Digital Communications

Analog Communication Systems- information is from an analog source- the signal waveform changes according to the information

content - Sensitive to noise

Digital Communication Systems- information is from a digital source or an analog is digitized before

transmission - information is carried in form of bit sequence or pattern- Distorted or noise-corrupted digital signal can be recovered by

digital processing techniques Less sensitive to noise

ERG2310A-I p. I-4

Forms of Communications

Simplex Communication- one way communication, in one direction only

Transmitter Receiver Information OUTInformation IN

ChannelA B

Half Duplex Communication- one way communication at any time, but in both directions

TransmitterInformation IN

ChannelReceiverInformation OUT

ReceiverInformation OUT

TransmitterInformation IN

A B

ERG2310A-I p. I-5

Forms of Communications

Full Duplex Communication- simultaneous two-way communication

TransmitterInformation IN

ChannelReceiverInformation OUT

ReceiverInformation OUT

TransmitterInformation IN

A B

ERG2310A-I p. I-6

Signal Representation

s(t) = A sin(2π fo t +φo ) or A sin(ωo t +φo ) Time-domain: waveform

A: Amplitude

f : Frequency (Hz) (ω=2πf)

φ : Phase (radian or degrees)

Time (seconds)

Period (seconds)S(f)

Frequency-domain: spectrum

fo Frequency (Hz)

ERG2310A-I p. I-7

Energy and Power of Signals

For an arbitrary signal f(t), the total energy normalized to unit resistance is defined as

joules, )(lim 2 dttfET

TT ∫−∞→

∆

=

and the average power normalized to unit resistance is defined as

, watts )(21lim 2 dttfT

PT

TT ∫−∞→

∆

=

• Note: if 0 < E < ∞ (finite) P = 0.• When will 0 < P < ∞ happen?

ERG2310A-I p. I-8

Periodic Signal

A signal f(t) is periodic if and only if

ttfTtf allfor )()( 0 =+ (*)

where the constant T0 is the period.

The smallest value of T0 such that equation (*) is satisfied is referred to as the fundamental period, and is hereafter simply referred to as the period.

Any signal not satisfying equation (*) is called aperiodic.

ERG2310A-I p. I-9

Deterministic & Random Signals

Deterministic signal can be modeled as a completely specified function of time.

Example)cos()( 0 θ+ω= tAtf

Random signal cannot be completely specified as a function of time and must be modeled probabilistically.

ERG2310A-I p. I-10

System

Mathematically, a system is a rule used for assigning a function g(t)(the output) to a function f(t) (the input); that is,

g(t) = h{ f(t) }where h{•} is the rule or we call the impulse function.

h(t)f(t) g(t)

For two systems connected in cascade, the output of the first system forms the input to second, thus forming a new overall system:

g(t) = h2 { h1 [ f(t) ] } = h{ f(t) }

ERG2310A-I p. I-11

Linear System

If a system is linear then superposition applies; that is, if

g1(t) = h{ f1(t) }, and g2(t) = h{ f2(t) }then

h{ a1 f1(t) + a2 f2(t) } = a1 g1(t) + a2 g2(t) (*)

where a1, a2 are constants. A system is linear if it satisfiesEq. (*); any system not meeting these requirement is nonlinear.

ERG2310A-I p. I-12

Time-Invariant and Time-Varying

A system is time-invariant if a time shift in the input resultsin a corresponding time shift in the output so that

.any for )}({)( 000 tttfhttg −=−

The output of a time-invariant system depends on time differences and not on absolute values of time.

Any system not meeting this requirement is said to be time-varying.

ERG2310A-I p. I-13

Fourier Series

A periodic function of time s(t) with a fundamental period of T0 can be represented as an infinite sum of sinusoidal waveforms. Such summation, a Fourier series, may be written as:

∑∑∞

=

∞

=

π+

π+=

1 01 00 ,2sin2cos)(

nn

nn T

ntBTntAAts (1)

where the average value of s(t), A0 is given by

∫−= 2

0

20

,)(1

00

T

T dttsT

A (2)

while

∫−

π= 2

0

20

,2cos)(2

00

T

T dtTntts

TAn (3)

and.2sin)(2 2

0

20

00∫−

π=

T

T dtTntts

TBn (4)

ERG2310A-I p. I-14

Fourier Series

An alternative form of representing the Fourier series is

∑∞

=

φ−

π+=

1 00

2cos)(n

nn TntCCts (5)

where (6),00 AC =

,22nnn BAC +=

.tan 1

n

nn A

B−=φ

(7)

(8)

The Fourier series of a periodic function is thus seen to consist of a summation of harmonics of a fundamental frequency f0 = 1/T0.

The coefficients Cn are called spectral amplitudes, which represent the amplitude of the spectral component Cn cos(2πnf0t − φn) at frequency nf0.

ERG2310A-I p. I-15

Fourier Series

The exponential form of the Fourier series is used extensively in communication theory. This form is given by

∑∞

−∞=

π

=n

nTntj

eSts ,)( 02

where

∫−

− π

= 20

20

02

.)(1

0

T

TTntj

dtetsT

Sn

Note that Sn and S−n are complex conjugate of one another, that is

(9)

(10)

(11).*nn SS −=

These are related to the Cn by

.2

njnn eCS φ−= (12),00 CS =

ERG2310A-I p. I-16

Fourier Series

Amplitude Spectra (Line Spectra)Cn

Fig.(a)

Note that except S0 = C0, each spectral line in Fig. (a) at frequency fis replaced by the two spectral lines in Fig. (b), each with half amplitude, one at frequency f and one at frequency - f.

0 fo 2fo 3fo 4fo 5fo 6fo (n-1) fo nfo

|Sn|

-nfo -(n-1)fo ••• - 6fo0-5fo -4fo -3fo -2fo -fo 0 fo 2fo 3fo 4fo 5fo 6fo ••• (n-1) fo nfo

••• •••

Fig.(b)

ERG2310A-I p. I-17

Fourier Series : Example

The Bn coefficients are given byConsider a unitary square wave defined by

( )

( ) ( )

( ) ( )

( )π−π

=

ππ+

ππ−=

π−+π=

π=

π=

∫∫∫

∫−

nn

nnt

nnt

dtntdtnt

dtnttx

dtTnttx

TB

T

Tn

cos12

22cos

22cos2

2sin22sin2

2sin)(2

2sin)(2

1

5.0

5.0

0

1

5.0

5.0

0

1

0

00

20

20

<<−<<

= 150 ,15.00 ,1

)(t.t

tx

and periodically extended outside this interval. The average value is zero, so

.00 =A

Recall that

( )

( ) ( )

( ) ( )

0

22sin

22sin2

2cos22cos2

2cos)(2

2cos)(2

1

5.0

5.0

0

1

5.0

5.0

0

1

0

00

20

20

=

ππ−

ππ=

π−+π=

π=

π=

∫∫∫

∫−

nnt

nnt

dtntdtnt

dtnttx

dtTnttx

TA

T

Tn

Thus all An coefficients are zero.

which results in

π=even is ,0

odd is ,4

n

nnBn

ERG2310A-I p. I-18

Fourier Series : Example

The Fourier series of a square wave of unitary amplitude with odd symmetry is therefore

)10sin516sin

312(sin4)( K+π+π+π

π= ttttx

1st term 1st + 2nd terms 1st + 2nd + 3rd terms

Sum up to the 6th term

ERG2310A-I p. I-19

Fourier Transform

Representation of an Aperiodic Function

)()(lim tftfTT=

∞→

Consider an aperiodic function f(t)

To represent this function as a sum of exponential functions overthe entire interval (-∞, ∞), we construct a new periodic functionfT(t) with period T.

By letting T→∞,

(13)

ERG2310A-I p. I-20

Fourier Transform

The new function fT(t) can be represented by an exponential Fourier series, which is written as

∑∞

−∞=

ω=n

tjnnT eFtf ,)( 0 (14)

where

∫−

ω−=2/

2/0)(1 T

T

tjnTn dtetf

TF (15)

and ./20 Tπ=ω

ERG2310A-I p. I-21

Fourier Transform

For the sake of clear presentation, we set

,0ω=ω∆

nn ,)( nn TFF∆

=ω

Thus, Eq.(14) and (15) become

∑∞

−∞=

ωω=n

tjnT

neFT

tf ,)(1)(

∫−

ω−=ω2/

2/.)()(

T

T

tjTn dtetfF n

The spacing between adjacent lines in the line stream of fT(t)is

./2 Tπ=ω∆

(16)

(17)

(18)

(19)

ERG2310A-I p. I-22

Fourier Transform

Using this relation for T, we get

∑∞

−∞=

ω

πω∆

ω=n

tjnT

neFtf .2

)()(

As T becomes very large, ∆ω becomes smaller and the spectrumbecomes denser.

(20)

In the limit T → ∞, the discrete lines in the spectrum of fT(t) mergeand the frequency spectrum becomes continuous.

Therefore,∑

∞

−∞=

ω

∞→∞→ω∆ω

π=

n

tjnTTT

neFtf )(21lim)(lim (21)

∫∞

∞−

ω ωωπ

= deFtf tj)(21)(becomes (22)

ERG2310A-I p. I-23

Fourier Transform

In a similar way, Eq. (18) becomes

.)()( ∫∞

∞−

ω−=ω dtetfF tj (23)

Eq. (22) and (23) are commonly referred to as the Fourier transform pair.

Fourier Transform

.)()( ∫∞

∞−

ω−=ω dtetfF tj

Inverse Fourier Transform

∫∞

∞−

ω ωωπ

= deFtf tj)(21)(

ERG2310A-I p. I-24

Spectral Density Function

F(ω): The spectral density function of f(t).

Fig. 3.2

A unit gate function Its spectral density graph

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

ωω

=ω

ERG2310A-I p. I-25

Parseval’s Theorem

The energy delivered to a 1-ohm resistor is

∫∫∞

∞−

∞

∞−== .)()()( *2 dttftfdttfE (24)

Using Eq. (22) in (24), we get

.)()(21

)()(21

)(21)(

*

*

*

∫

∫ ∫

∫ ∫

∞

∞−

∞

∞−

∞

∞−

ω−

∞

∞−

∞

∞−

ω−

ωωωπ

=

ω

ω

π=

ωω

π=

dFF

ddtetfF

dtdeFtfE

tj

tj ∫∞

∞−

ω ωωπ

= deFtf tj)(21)(

(25)

Parseval’s Theorem:

∫∫∞

∞−

∞

∞−ωω

π= .)(

21)( 22 dFdttf (26)

ERG2310A-I p. I-26

Fourier Transform: Impulse Function

The unit impulse function satisfies

(27),1)( =δ∫∞

∞−dxx

≠=∞

=.0 0,0

)(xx

xδ (28)

Using the integral properties of the impulse function, the Fourier transform of a unit impulse, δ(t), is

{ } .1)()( 0 ==δ=δℑ ∫∞

∞−

ω− jtj edtett (29)If the impulse is time-shifted, we have

{ } .)()( 000

tjtj edtetttt ω−∞

∞−

ω− =−δ=−δℑ ∫ (30)

ERG2310A-I p. I-27

Fourier Transform: Complex Exponential Function

The spectral density of will be concentrated at ±ω0.tje 0ω±

{ }

,21

)(21)(

0

001

tj

tj

e

de

ω±

∞

∞−

ω−

π=

ωωωδπ

=ωωδℑ ∫ mm

(31)

Taking the Fourier transform of both sides, we have(32)

{ } { }tje 0

21)( 0

1 ω±− ℑπ

=ωωδℑℑ m

which gives{ } )(2 0

0 ωωπδω m=ℑ ± tje (33)

ERG2310A-I p. I-28

Fourier Transform: Sinusoidal Function

The sinusoidal signals and can be written in terms ofthe complex exponentials.

t0cosω t0sin ω

Their Fourier transforms are given by

{ } { }),()(

cos

00

21

21

000

ω+ωπδ+ω−ωπδ=

+ℑ=ωℑ ω−ω tjtj eet

(34)

{ } { }.)()(

sin

00

21

21

000

j

eet tjj

tjj

ω+ωπδ−ω−ωπδ=

−ℑ=ωℑ ω−ω

(35)

ERG2310A-I p. I-29

Fourier Transform: Periodic Functions

We can express a function f(t) that is periodic with period T by itsexponential Fourier series

∑∞

−∞=

ω=n

tjnnT eFtf 0)( (36)where ω0 = 2π/T.

Taking the Fourier transform, we have

e.g.

(37)

{ }

{ }

∑

∑

∑

∞

−∞=

∞

−∞=

ω

∞

−∞=

ω

ω−ωδπ=

ℑ=

ℑ=ℑ

nn

n

tjnn

n

tjnnT

nF

eF

eFtf

).(2

)(

0

0

0

A unit gate function Its Fourier transform

Line spectrum of f(t) with period T

Its spectral density graph

ERG2310A-I p. I-30

Time and Spectral Density Functions

ERG2310A-I p. I-31

Selected Fourier Transform Pairs

ERG2310A-I p. I-32

Properties of Fourier Transform

Linearity (Superposition)

)()()()( 22112211 ω+ω↔+ FaFatfatfa

Time Shifting (Delay)tjeFttf 0)()( 0

ω−ω↔−

Frequency Shifting (Modulation)Complex Conjugate)()( ** ω−↔ Ftf )()( 0

0 ω−ω↔ω Fetf tj

DualityConvolution

).(2)( ω−π↔ ftF)()()()( 2121 ωω↔∗ FFtftf

Scaling

ω

↔a

Fa

atf 1)( Multiplicationfor .0≠a)()()()( 2121 ω∗ω↔ FFtftf

Differentiation

)()()( ωω↔ Fjtfdtd nn

n

ERG2310A-I p. I-33

Properties of Fourier Transform

).(2)( ω−π↔ ftFDuality

ω

↔a

Fa

atf 1)( .0≠aScaling

for

ERG2310A-I p. I-34

Properties of Fourier Transform

Frequency Shifting (Modulation)

)()( 00 ω−ω↔ω Fetf tj

ERG2310A-I p. I-35

Time Convolution

One method of characterizing a system is by its impulse response h(t).

δ(t-τ) h(t−τ)LTI System

where τ is the delay.

An input signal f(t) can be expressed in terms of impulse functions:

.)()()( ∫∞

∞−−= ττδτ dtftf

Thus,f(τ)δ(t-τ) f(τ) h(t−τ)LTI System

By principle of superposition

∫∞

∞−− ττδτ dtf )()(

f(t)

τττ∫∞

∞−− dthf )()(

g(t)

LTI System

ERG2310A-I p. I-36

Time Convolution

For a linear time-invariant system, if an input f(t) pass through a system with impulse response h(t), the output g(t) will be

τττ∫∞

∞−−= dthftg )()()(

Thus,).()()( thtftg ∗= This result is known as convolution integral.

An important property of the Fourier transform is that it reducesthe convolution integral operation to an algebraic product.

).()()( ωω=ω HFG

Thus convolution in the time domain corresponds to multiplication inthe frequency domain.

ERG2310A-I p. I-37

Properties of Convolution

Commutative Law

)()()()( 1221 tftftftf ∗=∗

Distributive Law

[ ] )()()()()()()( 3121321 tftftftftftftf ∗+∗=+∗

Associative Law

[ ] [ ] )()()()()()( 321321 tftftftftftf ∗∗=∗∗

ERG2310A-I p. I-38

Graphical Interpretation: Convolution