١

Chapter one

Introduction

Analog and Digital Communication

٢

Introduction to Communication Systems

•

What is a communication system?

•

Any means for transmission of information.

•

Examples: Telephone, Telegraph, Mobile phone, TV, Radio, Internet, hard disk in a PC, Radar, Satellite, microwave link,…

٣

Elements of a Communication System

•

Communication involves the transfer of information

or intelligence

from a source

to a recipient

via a channel

or medium.•

Basic block diagram of a communication system:

Source Transmitter Receiver Recipient

٤

Brief Description

•

Source: emits analog or digital data.•

Transmitter: transducer, amplifier, modulator, oscillator, power amp., antenna

•

Channel: e.g. cable, optical fiber, waveguide, radio link (free space)

•

Receiver: antenna, amplifier, demodulator, oscillator, power amplifier, transducer

•

Recipient: e.g. person, speaker, computer

٥

Transmitter•

It may include transducer, amplifier, modulator, oscillator, power amplifier and antenna.

•

It modifies the message or the baseband

signal for efficient transmission by a process called modulation.

•

Other functions: filtering, amplification, radiation

٦

Modulation

Modulation: the process by which the base band signal is used to modify some parameter of a high frequency carrier.

Types of modulation•

Continuous wave (CW) modulation.–

RF sinusoidal carrier wave(30K-300GHz).

•

Pulse modulation.–

RF pulse carrier wave.

٧

Why modulation?

•

For ease of radiation.•

Modulation for multiplexing.

•

For exchange of SNR with B.•

To over come equipment limitation.

•

To match channel characteristics.

٨

Example of analog modulation

٩

Channel•

It is the physical medium between the transmitter and the receiver. It can be guided, as optical fiber cables, waveguide, or unguided as radio link, water, free space.

•

Whatever the medium, the signal is corrupted in a random manner by noise and interference (thermal noise, lightning discharge, automobile ignition noise, interference from other users …)

•

Both additive and nonadditive

signal distortions are usually characterized as random phenomena and described in statistical terms.

١٠

Elements of Communication System

١١

Mathematical Model of Channel

١٢

i/o of a comm. channel

)()()(

)()(*)()(

tndtsh

tnthtstr

+−=

+=

∫∞+

∞−

τττ

١٣

Channel Bandwidth

•

The bandwidth of a channel is the range of frequencies that it can transmit with reasonable fidelity.

•

For example, the bandwidth of– twisted pair: several hundred kHz– coax cable: several hundred MHz– wave guide: few GHz– optic fiber: very wide

١٤

Receiver•

Its main function is to recover the message from the received signal.

•

It includes antenna, amplifier, demodulator, oscillator, power amplifier, transducer

•

Demodulation: inverse of the modulation•

Operates in the presence of noise & interference. Hence, some distortions are unavoidable.

•

Some other functions: filtering, suppression of noise & interference

١٥

Types of Communication Systems

•

guided & unguided (wireless). •

Digital & analog,

•

Point-to-point & broadcasting,

١٦

Types of comm. systemsTypes of comm. systems••

Analog comm. systemAnalog comm. system

Transport analog information using analog modulation techniques (AM,FM,PM).

••

Digital comm. system.Digital comm. system.Transport digital information using digital modulation techniques (ASK,FSK,PSK).

••

Hybrid comm. system.Hybrid comm. system.Transport digitized analog information using one of the following digital techniques:

1.

Analog pulse modulation schemes (PAM,PDM,PPM).

2.

digital modulation schemes (ASK,FSK,PSK).3.

Pulse code modulation schemes (PCM,DPCM, δ).

١٧

Types of Transmission

•

Base-band transmission:–

Short distance.

–

No modulation is needed.•

Band-pass transmission:–

long distance.

–

Modulation is needed.–

Analog or digital.

١٨

Digital versus analog transmission

•

Immunity to noise and distortion.•

Viability of regenerative repeaters.

•

Digital hardware implementation is flexible and scalable.

•

Digital signals can be encoded to yield low BER.•

Multiplexing is easier and more efficient.

•

Realization of exchange SNR and B is more efficient.

•

Digital storage easier and cheaper.•

cost of digital hardware is cheaper.

١٩

Transmission Terminology

•

Simplex transmission–

One direction

•

e.g. Radio and television broadcast.

•

Half duplex transmission–

Either direction, but only one way at a time

•

e.g. police radio(walki-talki)

•

Full duplex transmission–

Both directions at the same time

•

e.g. telephone,

٢٠

Simplex vs. Duplex

٢١

Data and Signals

•

Usually use digital signals for digital data and analog signals for analog data

•

Can use analog signal to carry digital data–

Modem

•

Can use digital signal to carry analog data –

codec

٢٢

Analog Signals Carrying Analog and Digital Data

٢٣

Digital Signals Carrying Analog and Digital Data

٢٤

Analog Transmission

•

Analog signal transmitted without regard to their content (May be analog or digital data)

•

Attenuated over distance •

Use amplifiers to boost signal

•

Also amplifies noise, thus received signal will be distorted.

•

If digital data is encoded then amplifiers will increase BER (bit error rate).

٢٥

Digital Transmission

•

Concerned with content of the signal.•

Integrity endangered by noise, attenuation etc.

•

Repeaters used to achieve greater distance.

•

Repeater receives signal-Extracts bit pattern-Retransmits new signal-Attenuation is overcome-Noise is not amplified

٢٦

Transmission Impairments

•

Signal received may differ from signal transmitted

•

Analog -

degradation of signal quality•

Digital -

bit errors

•

Caused by–

Attenuation and attenuation distortion

–

Delay distortion–

Noise

٢٧

Attenuation

•

Signal strength falls off with distance•

Depends on medium

-guided: attenuation is logarithmic.-unguided: attenuation depends on atmospheric

structure.•

Received signal strength:–

must be enough to be detected

–

must be sufficiently higher than noise to be received without error

٢٨

Atten. Cont.

•

Attenuation is an increasing function of frequency.

-attenuation distortion affects analog signals much more than digital signals.

•

Fading channel.•

Equalizers: reduce attenuation distortion.

٢٩

Delay Distortion

•

Only in guided media•

Caused by: Propagation velocity varies

with frequency.-different frequency components arrive at

the receiver at different times causing phase shifts.

•

for digital data delay distortion introduces inter-symbol interference (ISI).

•

Equalizers : reduce delay distortion.

٣٠

٣١

Noise•

Additional signals inserted between transmitter and receiver

•

Thermal–

Due to thermal agitation of electrons

–

Uniformly distributed–

White noise

•

Intermodulation–

produce signals at frequency that is the sum

and difference of original frequencies sharing a medium.

–

Caused by nonlinearity in Tx, Rx, or channel because of signal strength.

٣٢

Noise cont.

•

Crosstalk–

A signal from one line is picked up by another

•

Impulse–

Irregular pulses or spikes

–

e.g. External electromagnetic interference–

Short duration

–

High amplitude–

Severe effect on digital signal of high data rate.

٣٣

Channel capacityChannel capacity•

Two primary resources : channel bandwidth and transmitted power.

•

Limitation imposed on the transmission rate by channel bandwidth B, and SNR is given by

C: channel capacity (maximum rate at no error)M: #of signaling levels

2

2

log (1 ) bps (Shannon equation)2 log bps (Nyquist equation)

C B SNRC B M= +=

٣٤

Channel capacity cont.•

Transmitted power (SNR) and channel bandwidth are exchangeable.

1

22 1( )

BBS N R S N R≈

٣٥

Radio Communication Channels

٣٦

٣٧

١

Chapter 2

Introduction to Signals and systems

٢

Outlines•

Classification of signals and systems

•

Some useful signal operations•

Some useful signals.

•

Frequency domain representation for periodic signals

•

Fourier Series Coefficients•

Power content of a periodic signal and Parseval’ s theorem for the Fourier series

٣

Classification of Signals•

Continuous-time and discrete-time signals

•

Analog and digital signals•

Deterministic and random signals

•

Periodic and aperiodic

signals•

Power and energy signals

•

Causal and non-causal.•

Time-limited and band-limited.

•

Base-band and band-pass.•

Wide-band and narrow-band.

٤

Continuous-time and discrete-time periodic signals

٥

Continuous-time and discrete-time aperiodic

signals

٦

Analog & digital signals

•

If a continuous-time signal can take on any values in a continuous time interval, then is called an analog signal.

•

If a discrete-time signal can take on only a finite number of distinct values, { }then the signal is called a digital signal.

)(tg)(tg

( )g n

٧

Analog and Digital Signals

0 1 1 1 1 0 1

٨

Deterministic signal

•

A

Deterministic signal is

uniquely described by a mathematical expression.

•

They are reproducible, predictable and well-behaved mathematically.

•

Thus, everything is known about the signal for all time.

٩

A deterministic signal

١٠

Deterministic signal

١١

Random signal

•

Random signals are unpredictable. •

They are generated by systems that contain randomness.

•

At any particular time, the signal is a random variable, which may have well defined average and variance, but is not completely defined in value.

١٢

A random signal

١٣

Periodic and aperiodic Signals•

A signal is a periodic signal if

•

Otherwise, it is aperiodic signal.

0( ) ( ), , is integer.x t x t nT t n= + ∀

( )x t

0

00

: period(second)1 ( ), fundamental frequency

2 (rad/sec), angulr(radian) frequency

T

f HzT

fω π

=

=

١٤-2 0 2-3

-2

-1

0

1

2

3

Time (s)

Square signal

١٥-1 0 1

-2

-1

0

1

2

Time (s)

Square signal

١٦-1 0 1

-2

-1

0

1

2

Time (s)

Sawtooth signal

١٧

• A simple harmonic oscillation is mathematically described by

x (t)= A cos (ω

t+ θ), for - ∞

< t < ∞• This signal is completely characterized by three

parameters:A: is the amplitude (peak value) of x(t).ω:

is the radial frequency in (rad/s),

θ: is the phase in radians (rad)

١٨

Example:Determine whether the following signals are

periodic. In case a signal is periodic, specify its fundamental period.

a) x1

(t)= 3 cos(3π

t+π/6), b) x2

(t)= 2 sin(100π

t), c) x3

(t)= x1(t)+ x2(t)d) x4

(t)= 3 cos(3π

t+π/6) + 2 sin(10 t), e) x5

(t)= 2 exp(-j 20 π

t)

١٩

Power and Energy signals

•

A signal with finite energy is an energy signal

•

A signal with finite power is a power signal

∞<= ∫+∞

∞−

dttgEg2)(

∞<= ∫+

−∞→

2/

2/

2)(1limT

TTg dttg

TP

٢٠

Power of a Periodic Signal

•

The power of a periodic signal x(t) with period T0

is defined as the mean-

square value over a period

0

0

/ 22

0 / 2

1 ( )T

xT

P x t dtT

+

−

= ∫

٢١

Example•

Determine whether the signal g(t) is power or energy signals or neither

0 2 4 6 80

1

2g(t)

2 exp(-t/2)

٢٢

Exercise•

Determine whether the signals are power or energy signals or neither

1) x(t)= u(t)2) y(t)= A sin t3) s(t)= t u(t)4)z(t)= 5) 6)

)(tδ( ) cos(10 ) ( )v t t u tπ=( ) sin 2 [ ( ) ( 2 )]w t t u t u tπ π= − −

٢٣

Exercise

•

Determine whether the signals are power or energy signals or neither

1)

2)

3)

1 1 2 2( ) cos( ) cos( )x t a t b tω θ ω θ= + + +

1 1 1 2( ) cos( ) cos( )x t a t b tω θ ω θ= + + +

1( ) co s( )n n n

ny t c tω θ

∞

=

= +∑

٢٤-1 0 1

-2

-1

0

1

2

Time (s)

Sawtooth signalDetermine the suitable measures for the signal x(t)

٢٥

Some Useful Functions•

Unit impulse function

•

Unit step function •

Rectangular function

•

Triangular function•

Sampling function

•

Sinc

function•

Sinusoidal, exponential and logarithmic functions

٢٦

Unit impulse function•

The unit impulse function, also known as the dirac delta function, δ(t), is defined by

⎩⎨⎧

≠=∞

=0,00,

)(tt

tδ 1)( =∫+∞

∞−

dttand δ

٢٧

Δ 0

٢٨

•

Multiplication of a function by δ(t)

•

We can also prove that

)0()()( sdttts =∫+∞

∞−

δ

)()0()()( tgttg δδ =)()()()( τδττδ −=− tgttg

)()()( ττδ sdttts =−∫+∞

∞−

٢٩

Unit step function•

The unit step function u(t) is

•

u(t) is related to δ(t) by

⎩⎨⎧

<≥

=0,00,1

)(tt

tu

∫∞−

=t

dtu ττδ )()( )(tdtdu δ=

٣٠

Unit step

٣١

Rectangular function

•

A single rectangular pulse is denoted by

⎪⎩

⎪⎨

⎧

>

=

<

=⎟⎠⎞

⎜⎝⎛

2/,0

2/,5.0

2/,1

τ

τ

τ

τt

t

ttrect

٣٢-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

3

Time (s)

Rectangular signal

٣٣

Triangular function

•

A triangular function is denoted by

⎪⎪⎩

⎪⎪⎨

⎧

>

<−=⎟

⎠⎞

⎜⎝⎛Δ

21,0

21,21

τ

τττ t

ttt

٣٤

•

Sinc

function

•

Sampling function

sin( )sinc( ) xxxπ

π=

( ) ( ), : samplig intervalsT s s

nt t nT Tδ δ

∞

=−∞

= −∑

٣٥-5 0 5

-0.5

0

0.5

1

1.5

2

2.5

3

Time (s)

Sinc signal

٣٦

Some Useful Signal Operations•

Time shifting

(shift right or delay)

(shift left or advance)•

Time scaling

( )g t τ−

( )g t τ+

ta

ta

( ) , 1 is co m p ress io n

( ), 1 is ex p an s io n

g ( ), 1 is ex p an s io n

g ( ), 1 is co m p ress io n

g at a

g at a

a

a

≺

≺

٣٧

Signal operations cont.

•

Time inversion

( ) : mirror image of ( ) about Y-axisg t g t−

( ) : shift right of ( )( ) : shift left of ( )

g t g tg t g t

ττ

− + −− − −

٣٨-10 -5 0 5

0

1

2

3

Time (s)

g(t)g(t-5)g(t)g(t-5)g(t)g(t-5)

٣٩-10 -5 0 5

0

1

2

3

Time (s)

g(t+5)

٤٠

Scaling

-5 0 5

024

-5 0 5

024

-5 0 5

024

g(t)

g(2t)

g(t/2)

٤١

Time Inversion

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.5

1

1.5

2

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.5

1

1.5

2

g(t)

g(-t)

٤٢

Inner product of signals

•

Inner product of two complex signals x(t), y(t) over the interval [t1,t2] is

If inner product=0, x(t), y(t) are orthogonal.

2

1

( ( ), ( )) ( ) ( )t

t

x t y t x t y t dt∗= ∫

٤٣

Inner product cont.

•

The approximation of x(t) by y(t) over the interval is given by

•

The optimum value of the constant C that minimize the energy of the error signal

is given by

( ) ( ) ( )e t x t cy t= −2

1

1 ( ) ( )t

y t

C x t y t dtE

= ∫

1 2[ , ]t t

( ) ( )x t cy t=

٤٤

Power and energy of orthogonal signals

•

The power/energy of the sum of mutually orthogonal signals is sum of their individual powers/energies. i.e

if

Such that are mutually orthogonal, then

1

( ) ( )n

ii

x t g t=

= ∑

( ), 1,....ig t i n=

1i

n

x gi

p p=

= ∑

٤٥

Time and Frequency Domains representations of signals

•

Time domain: an oscilloscope displays the amplitude versus time

•

Frequency domain: a spectrum analyzer displays the amplitude or power versus frequency

•

Frequency-domain display provides information on bandwidth and harmonic components of a signal

٤٦

Benefit of Frequency Domain Representation

•

Distinguishing a signal from noisex(t) = sin(2π 50t)+sin(2π 120t);y(t) = x(t) + noise;

•

Selecting frequency bands in Telecommunication system

٤٧0 10 20 30 40 50-5

0

5Signal Corrupted with Zero-Mean Random Noise

Time (seconds)

٤٨0 200 400 600 800 10000

20

40

60

80Frequency content of y

Frequency (Hz)

٤٩

Fourier Series Coefficients

•

The frequency domain representation of a periodic signal is obtained from the Fourier series expansion.

•

The frequency domain representation of a non-periodic signal is obtained from the Fourier transform.

٥٠

•

The Fourier series is an effective technique for describing periodic functions. It provides a method for expressing a periodic function as a linear combination of sinusoidal functions.

•

Trigonometric Fourier Series

•

Compact trigonometric Fourier Series

•

Complex Fourier Series

٥١

Trigonometric Fourier Series

0

00

2 ( ) cos(2 )nT

a x t n f t dtT

π= ∫

( )0 0 01

( ) cos 2 sin 2n kn

x t a a n f t b n f tπ π∞

=

= + +∑

0

00

2 ( ) sin(2 )nT

b x t n f t dtT

π= ∫

٥٢

Trigonometric Fourier Series cont.

0

00

1 ( )T

a x t dtT

= ∫

٥٣

Compact trigonometric Fourier series

0 01

2 20 0

1

( ) cos(2 )

,

tan

n nk

n n n

nn

n

x t c c n f t

c a b c a

ba

π θ

θ

∞

=

−

= + +

= + =

⎛ ⎞−= ⎜ ⎟

⎝ ⎠

∑

٥٤

Complex Fourier Series

•

If x(t) is a periodic signal with a fundamental period T0

=1/f0

• are called the Fourier coefficients

2( ) oj n f tn

nx t D e π

∞

=−∞

= ∑

0

0

2

0

1 ( ) j n f tn

T

D x t e dtT

π−= ∫

nD

٥٥

Complex Fourier Series cont.

1212

n

n

n n

jn n

jn n

j jn n n n

D c e

D c e

D D e and D D e

θ

θ

θ θ

−−

−−

=

=

= =

٥٦

Frequency Spectra

•

A plot of |Dn

| versus the frequency is called the amplitude spectrum of x(t).

•

A plot of the phase versus the frequency is called the phase spectrum of x(t).

•

The frequency spectra of x(t) refers to the amplitude spectrum and phase spectrum.

nθ

٥٧

Example•

Find the exponential Fourier series and sketch the corresponding spectra for the sawtooth

signal with period 2 π

- 1 0 - 5 0 5 1 00

0 . 5

1

1 . 5

2

٥٨

•

Dn

= j/(π

n);

for n≠0•

D0

= 1;

02

0

1 ( )o

j n f tn

T

D x t e dtT

π−= ∫

( )12 −=∫ taaedtet

tata

٥٩

-5 0 500.5

11.5 Amplitude Spectrum

-5 0 5-100

0

100 Phase spectrum

٦٠

Power Content of a Periodic Signal

•

The power content of a periodic signal x(t) with period T0

is defined as the mean-

square value over a period

∫+

−

=2/

2/

2

0

0

0

)(1 T

T

dttxT

P

٦١

Parseval’s

Power Theorem

•

Parseval’

s power theorem series states that if x(t) is a periodic signal with period T0

, then

0

0

2

/ 2 22 2

010 / 2

2 220

1 1

1 ( )2

2 2

nn

Tn

nT

n n

n n

D

cx t dt cT

a ba

∞

=−∞+ ∞

=−∞ ∞

= =

⎧⎪⎪⎪

= +⎨⎪⎪

+ +⎪⎩

∑

∑∫

∑ ∑

٦٢

Example 1

•

Compute the complex Fourier series coefficients for the first ten positive harmonic frequencies of the periodic signal f(t) which has a period of 2π

and

defined as

( ) 5 ,0 2tf t e t π−= ≤ ≤

٦٣

Example 2

•

Plot the amplitude spectrum of x(t) if T1

= T/4

٦٤

Example 3

•

Plot the spectra of x(t).

0( ) ( )n

x t t nTδ∞

=−∞

= −∑

٦٥

Suggested problems

•

2.1.1,2.1.2,2.1.4,2.1.8•

2.3.1,2.3.3,2.3.4

•

2.4.2,2.4.3•

2.5.2, 2.5.5

•

2.8.1,2.8.4,2.8.5•

2.9.2,2.9.3

Chapter 3

Analysis and Transmission of Signals

Outline

• Introduction• Fourier transform and its inverse• Fourier transform of some useful functions• Properties of Fourier transform

Introduction

• Fourier series works for periodic signals only. What’s about aperiodic signals? This is very large & important class of signals

Introduction (cont.)• Aperiodic signal can be considered as

periodic for T ∞• Fourier series changes to Fourier transform,

complex exponents are infinitesimally close in frequency

• Discrete spectrum becomes a continuous one, also known as spectral density

Fourier Transform and Its Inverse

• Fourier transform: if g(t) is aperiodic signal then

∫∞

∞−

−= dtetgfG tfj π2)()(

2( ) ( ) j f tg t G f e dfπ∞

−∞

= ∫

( ) ( )g t G f⇔

• For real g(t),

:amplitude spectrum:phase spectrum

( )G f

( )( ) ( ) j fG f G f e θ=

* ( )( ) ( ) ( ) j fG f G f G f e θ−− = =

( )fθ

Example

• Find the Fourier transform and plot amplitude and phase spectra.

1)

2)

( ) ( )atg t e u t−=

( ) ( )atx t e u t= −

Fourier Transform of Some Useful Functions

1)2) 3)4) 5) 6)

( ) 1tδ ⇔

1 ( )fδ⇔2

2

( )

( )

c

c

j f tc

j f tc

e f f

e f f

π

π

δ

δ−

⇔ −

⇔ +

( ) sin ( )tTrect T c fT⇔

2( ) sin ( )tT T c fT∆ ⇔

121

2

cos(2 ) [ ( ) ( )]sin(2 ) [ ( ) ( )]

c c c

c c cj

f t f f f ff t f f f f

π δ δπ δ δ

⇔ − + +

⇔ − − +

Spectra of 0( ) sin 2v t A f tπ=

rect

Properties of Fourier Transform

• Linearity:

• Time shifting:

• Time reversal:

• Time scaling:

1 1 2 2 1 1 2 2( ) ( ) ( ) ( )ax t a x t aX f a X f+ ↔ +

20( ) ( ) j f tx t t X f e π−− ↔

( ) ( )x t X f− ↔ −1( ) ( )f

ax at Xa

↔

• Frequency shift (modulation):

• Time differentiation:

• Time integration:

( ) 2 ( )d x t j f X fdt

π↔

12

1( ) ( ) (0) ( )2

t

x d X f X fj f

τ τ δπ−∞

↔ +∫

2( ) ( )cj f tcx t e X f fπ ↔ −

2( ) ( )cj f tcx t e X f fπ− ↔ +

• Time Convolution:

• Time Multiplication (Frequency convolution)

( ) ( ) ( )* ( )x t y t X f Y f↔

( )* ( ) ( ) ( )x t h t X f H f↔

∫∞

∞−

−= τττ dthxthtx )()()(*)(

• Duality

• Differentiation in frequency

If ( ) ( )then ( ) ( )

x t X fX t x f

↔↔ −

( )( 2 ) ( ) dX fj t x tdf

π− ↔

Examples

• Use the Fourier transform properties to find the Fourier transform of the following:

1) 2) 3)4)5)6)

( ) tx t e −=( ) sin (2 )g t c Bt=

/ 2( ) ( )t TTy t rect −=

( ) sin(2 ) ( )tcv t e f t u tπ−=

( ) sgn( )x t t=

( ) ( )g t u t=

Fourier transform of periodic signal

• If is periodic signal of period then

Then the Fourier transform of is

( )px t0T

0

0

210 0( ) ( ) ( ) j nf t

p Tm m

x t x t mT X nf e π∞ ∞

=−∞ =−∞

= − =∑ ∑

0

10 0 0( ) ( ) ( )T

m n

x t mT X nf f nfδ∞ ∞

=−∞ =−∞

− ↔ −∑ ∑( )px t

Signal Transmission Through a Linear Time Invariant System

• System representation• Impulse response and transfer

function• Distortionless transmission

System Representation• A system is defined mathematically as a

transformation or operator that maps an input x(t) into an output y(t).

( )x tSystem

T[ ]( )y t

Impulse Response of an LTI system

• The impulse response of an LTI system is defined as the response of the system when the input is δ(t). i.e

• For any arbitrary input signal x(t), the response

( ) ( )* ( ) ( ) ( )y t x t h t x h t dτ τ τ+∞

−∞

= = −∫

( ) ( )( ) ( ) x t th t y t δ== ↓

Frequency response of an LTI system

• The transfer function of an LTI system is

• The response to an input x(t) is

2( )

( )( )( ) j ftx t e

y tH fx t π=

= ↓

( )( ) ( ) j fH f H f e θ=

( ) ( ) ( )Y f X f H f=

Signal Distortion during Transmission• The transmission of an input signal x(t) through

a system changes it into the output signal y(t).

• During transmission through the system, some frequency components may be boosted in amplitude while others may be attenuated.

• The relative phases of the various components also change due to different delays.

Distortionless Transmission

• Transmission is said to be distortionless if

( ) ( )dy t k x t t= −

2( ) dj f tH f k e π−→ =

2( ) ( ) ( ) ( ) dj ftY f X f H f kX f e π−= =

Dispersive channel

• Channel which adds distortion is dispersive channel.

-Amplitude distortion: when ,channel is a fading channel.

-phase distortion: when , channel is a jittering channel.

( )H f k≠

( )f fθ α≠

The Nature of Distortion in Audio and Video Signals

• The human ear can perceive amplitude distortion but it is relatively insensitive to phase distortion.

• The human eye is sensitive to phase distortion but is relatively insensitive to amplitude distortion.

3.5 Ideal and Practical Filters

• A filter is a system whose transfer function takes significant values only in certain frequency bands. Filter are usually classified as– Low-pass, – high-pass, – Band-pass, or – Band-stop

Ideal Low-Pass Filter( )H f

cf

f

0

0cf−

( )fθ

f

f

Transfer function of an ideal LPF

2( )2

( ) 2 sin (2 ( ))unrealizable

dj f tLPF

c

c c d

fH f rect ef

h t f c f t t

π−⎛ ⎞= ⎜ ⎟

⎝ ⎠→ = −

→

Ideal High-Pass Filter( )H f

cfcf− 0 ω

0 ω

( )fθ

Ideal Band-Pass Filter

0 ωω1−ω1

|H(ω)|

ω2−ω2

ω

θh (ω)

0ω1

−ω1−ω2

ω2

Ideal Band-Stop Filter

0 ωω1−ω1

|H(ω)|

−ω2 ω2

0 ωω1

−ω2

θh (ω)

ω2−ω1

2

21 2

,( )

0,

,( )

0,

d

d

j f tc

HPF

j f t

BPF

e for f fH f

otherwise

e for f f fH f

otherwise

π

π

−

−

⎧ ≥⎪= ⎨⎪⎩

⎧ ≤ ≤⎪= ⎨⎪⎩

Ideal versus Practical Low-Pass Filter

( )x t ( ) ( )dy t x t t= −

LPF

h(t)

[ ]( ) 2 sinc 2 ( )c c dh t f f t t= −

td t

h(t)

0

• For a physical realizable system, h(t) must be causal; that is,

• One practical approach is to cut off the tail of h(t) for t<0

• If td is sufficient large

)()()(ˆ tuthth =

)()(ˆ thth ≈

( ) 0, 0h t t= ∀ ≺

Filter or System Bandwidth

• The bandwidth of an ideal low-pass filter

• The bandwidth of an ideal band-pass filter

• No bandwidth for high-pass and band-stop filters.

• For practical filters, a common definition of filter bandwidth is the 3-dB bandwidth (HPBW).

cBw f=

h lBw f f= −

Signal Bandwidth

• The bandwidth of a signal can be defined as the range of frequencies in which most of the energy or power lies.

• It can also be defined in terms of the 3-dB bandwidth.

• The signal bandwidth is also called the essential bandwidth of the signal

Signal distortion over a communication channel

• Linear distortion– Non ideal characteristics of either the magnitude, the

phase or both• Distortion caused by channel nonlinearities-interference with signals on same channel

• Distortion caused by multipath effects– Causes non-idealities in the magnitude and phase of

• Fading channels– Channels characteristics changes with time

( )H f

Distortion caused by multipath effects

Delay td

Delay td+∆tα

s(t) r(t)

Signal Energy and Energy Spectral Density

• The signal energy can be determined from its Fourier transform using Parseval’s theorem

2 2( ) ( )gE g t dt G f df+∞ ∞

−∞ −∞

= =∫ ∫

Example

• Verify Parseval’s theorem for the signal

1)

2)

( ) ( )atg t e u t−=

( ) sin (2 )x t c wt=

Energy Spectral Density (ESD)

is called the energy spectral density (ESD)

2

( )

where ( ) ( )

g g

g

E f df

f G f

∞

−∞

= Ψ

Ψ =

∫

( )g fΨ

Example

• Estimate the essential bandwidth B of the signal

if the essential bandwidth is required to contain 95% of the signal energy.

( ) ( )atg t e u t−=

Correlation of Energy Signals• There are applications where it is

necessary to compare one reference signal with one or more signals to determine the similarity between the pair from which some information will be extracted.

• This comparison can be done by computing the correlation between these signals.

Cross-correlation

• A measure of similarity between a pair of energy signals, is given by the cross-correlation function expressed as

( ) ( ) ( )xy x t y t dtψ τ τ∞

−∞

= −∫

( ) and ( ) x t y t

Cross-correlation cont.

• If we wish to make the reference signal, then the corresponding cross-correlation function is given by

( ) ( ) ( )y x y t x t d tψ τ τ∞

−∞

= −∫

( )y t

Autocorrelation function

• In the special case where , we have

the autocorrelation of which is defined as

( ) ( ) y t x t=

( ) ( ) ( )x x t x t d tψ τ τ∞

−∞

= −∫

( )x t

Properties of Crosscorrelation and

Autocorrelation functions

( ) ( )

( ) ( )(0)

xy yx

x x

x xE

ψ τ ψ τ

ψ τ ψ τψ

= −

= −=

• Time autocorrelation function and the energy spectral density

• ESD of the Input and the Output

2( ) ( )x t X fψ ↔

2( ) ( ) ( )y xf H f fΨ = Ψ

Signal Power and Power Spectral Density

• For a real power signal g(t)

• The time-averaged autocorrelation function of g(t) is defined as

∫+

−∞→

=2/

2/

2 )(1limT

TTg dttg

TP

/ 2

/ 2

1( ) lim ( ) ( )T

g TT

g t g t dtT

τ τ+

→∞−

ℜ = −∫

Autocorrelation of periodic signal

• If g(t) is periodic with period T

/ 2

/ 2

1( ) ( ) ( )T

gT

g t g t dtT

τ τ+

−

ℜ = −∫

• The power spectral density (PSD) of g(t), Sg(ω), is the Fourier transform of ℜg(τ)

2

2

( ) ( )

( ) ( )

j fg g

j fg g

S f e d

S f e df

π τ

π τ

τ τ

τ

∞−

−∞

∞

−∞

= ℜ

ℜ =

∫

∫

/ 22

/ 2

1(0) lim ( ) ( )T

g g gTT

g t dt S f df PT

+ ∞

→∞− −∞

ℜ = = =∫ ∫

• Input and output spectral densities

2( ) ( ) ( )y xS f H f S f=

Example

Find the autocorrelation function and spectral density of

1)

2)

( ) ( )tx t e u tα−=

( ) sin(2 )cg t f tπ=

Example

• Find the crosscorrelation function and ESD .check for orthogonality.

( ) ( )( ) ( )

t

t

x t e u ty t e u t

α

α

−=

= −

Chapter 4

Amplitude (Linear) Modulation

٢

Outlines•

Introduction

•

Base-band and Carrier Communication•

Amplitude Modulation (AM):DSB-Large Carrier

•

Amplitude Modulation: Double sideband-

Suppressed Carrier (DSBSC)

•

Quadrature

amplitude Modulation (QAM)•

Single Sideband Modulation (SSB)

•

Vestigial Sideband (VSB)•

Frequency mixing

•

Superhetrodyne

AM radio.•

Frequency division multilplexing

(FDM).

٣

Introduction•

Modulation is a process that causes a shift in the range of frequencies of a message signal.

•

A communication that does not use modulation is called baseband communication

•

A communication that uses modulation is called Carrier communication

٤

Example of AM transmitter

٥

Example of AM (radio) Receiver

٦

Baseband

and Carrier Communication

•

Baseband

signal: is message signal (information bearing signal) delivered by the information source or the input transducer .it is usually low frequency signal.

•

Communication that uses modulation to shift the frequency spectrum of message signal is known as carrier communication.– Amplitude modulation (AM), – Frequency modulation (FM)– Phase modulation (PM)

٧

Amplitude Modulation (AM) Double Sideband Large Carrier (DSB-LC)

( ) ( ) cos 2 cos 2AM c cs t m t f t A f tπ π= +

٨

٩

Another example of AM Waveform

( ) sin 2( ) sin 2

c

m

c t Ec f tm t Em f t

ππ

==

( ) ( ( ))sin 2 cS t Ec m t f tπ= +

( ) sin 2( ) sin 2

c

m

c t Ec f tm t Em f t

ππ

=

=( ) ( ( ))sin 2 cs t Ec m t f tπ= +

١٠

Modulation Index

•

The amount of modulation in AM signal is given by its modulation index:

max min

max min

, min ( )pp

m E Eor m m tA E E

μ −= =

+

When mp

= A ,

μ

=1 or 100% modulation.OverOver--modulation, i.e. mmodulation, i.e. mpp

>A>A

, should be avoided, should be avoided

because it will create distortions.because it will create distortions.

max min,p pE A m E A m= + = −

١١

Effect of Modulation Index

μ

<1

١٢

μ

>1

μ

=1

١٣

Effects of Modulation Index

μ

= 1 μ

> 1

١٤

Sideband and Carrier Power

•

Carrier Power

•

Sideband Power •

Total power

•

Power efficiency

•

For single tone modulation

2

2cAP =

sc

s

PPP+

=η

22

22 100%, [1 ]2 tot cP P μμη

μ= = +

+

2m

sPP =

tot c sp P P= +

١٥Modulation index

١٦

Example•

Conventional AM signal with a sinusoidal message has the following parameters:

A=10, μ=0.5, fc

= 1MHz, and fm

= 1kHz1.

Find time-domain expression

2. Find its Fourier transform3. Sketch its spectrum 4. Find the signal power, carrier power and the

power efficiency5. Find the AM signal bandwidth

( )Ams t

١٧

Example•

A given AM (DSB-LC) broadcast station transmits an average carrier power output of 40kW and uses a modulation index of 0.707 for sine-wave modulation. Calculate

a) the total output powerb) the power efficiencyc) the peak amplitude of the output if the

antenna is represented by a 50-Ω resistance load.

١٨

Generation of AM Signals diode as NLE or as switch

Square-law modulator2( ) ( ) ( )o i iv t av t bv t= +

' ( ) [ 2 ( )]cos 2o cv t aA Abm t f tπ= +

3cf B≥ To avoid overlap the spectrum of

2 ( ) and ( )cm t M f f−

Switching modulator

•

Assume

1

1

'

( ) ( ) ( ),

1 2 ( 1)( ) cos 2 (2 1)2 2 1

2( ) [ ( )]cos 22

o in

cn

o c

v t v t w t where

w t f n tn

Av t m t f t

ππ

ππ

−∞

=

=

−= + −

−

⇒ = +

∑

( ) ,and diode an ideal switchm t A

٢١

Demodulation of AM signals

•

AM signals can be demodulated by– Envelope detector– Rectifier detector– Coherent (synchronous) detector.

٢٢

Envelope Detector

٢٣

Envelope Detector (Cont.)

٢٤

Rectifier Detector

٢٥

Coherent detector

cos(2 )cA f tπ θ+

LPFV(t)

Local oscillator

2 ( ) cosA m t θ

( )Ams t

٢٧

Advantages/Disadvantages of Conventional AM (DSB-LC)

•

Advantages–

Very simple demodulation (envelope detector)

–

“Linear”

modulation

•

Disadvantages–

Low power efficiency

–

Transmission bandwidth twice the message bandwidth.

٢٨

Double-sideband suppressed carrier DSBSC

٢٩

The modulating signal m(t)

٣٠

Modulated signal m(t) cos(ωc

t)

٣١

Modulated signal m(t) cos(ωc

t)

٣٢

Example.

٣٣

٣٤

DSBSC Modulators

•

DSBSC signal can be generated using several types of modulators:– Multiplier Modulators– Nonlinear Modulators– Switching Modulators

Multiplier modulator

٣٦

Nonlinear Modulators

٣٧

Switching Modulators

( ) cos 2 ckm t f tπ

a b

BPFM(t)

+

-

v2

٣٨

Switching Modulators

٣٩

٤٠

Diode-bridge electronic switch

٤١

Series-bridge diode modulator

٤٢

Shunt-bridge diode modulator

٤٣

Ring Modulator

٤٤

Ring modulator

٤٥

Demodulation of DSBSC

( ) cos(2 )cm t f tπ

cos(2 )cA f tπ θ+

LPFV(t)

Local oscillator

2 ( ) cosA m t θ

٤٦

Quadrature

Amplitude Modulation (QAM)

٤٧

Transmitter

٤٨

Receiver

٤٩

QAM cont.

•

Quadrature

multiplexing is used in color television to multiplex the signals which carry the information about colors.

٥٠

Single Sideband (SSB)

SSB time representation

ˆ( ) ( ) cos 2 ( )sin 2 ,::

1ˆ ( ) ( ) Hilpert transform of ( )

SSB c cS t m t f t m t f tUSBLSB

m t m t m tt

π π

π

=−+

= ∗

∓

٥٢

Selective filtering method

٥٣

Selective filtering method (Cont.)

٥٤

Phase–Shift Method

٥٥

Phase–Shift Method

٥٦

Hilbert transform

٥٧

Phase–Shift Method (Cont.)

•

Advantages:– Does not deploy bandpass

filter.

– Suitable for message signals with frequency content down to dc.

•

Disadvantage:– Practical realization of a wideband 90o

phase shift circuit is difficult.

٥٨

Demodulation of SSB Signals•

Demodulation of SSB signals can be accomplished by using a synchronous detector as used in the demodulation of normal AM and DSBSC signals.

•

If we want to use an envelope detector, it can be shown that we must insert a pilot carrier signal Acos(2 πfc t) to the SSB signal, where A >> m(t) and A >> m^(t).

•

The pilot signal carries most of the transmission power which becomes inefficient.

٥٩

Example

•

A DSB-LC signal is generated using a 1-kHz carrier and the input is m(t)= cos(200πt). The modulation index is 80%. The lower sideband is attenuated (assume ideal filter). Find an expression for the resulting SSB-LC signal if it develops 0.58 W across a one-Ohm resistive load.

٦٠

Vestigial-Sideband Modulation (VSB)

٦١

VSB modulator

٦٢

Demodulation of VSB•

Demodulation of VSB signals can be accomplished by using a synchronous detector.

٦٣

Vestigial-Sideband Modulation (VSB)

٦٤

VSB modulator

٦٥

Demodulation of VSB•

Demodulation of VSB signals can be accomplished by using a synchronous detector.

Transfer function of LPF in VSB receiver

1( ) ,( ) ( )LPF

BPF c BPF c

H f f BH f f H f f

= ≤− + +

٦٧

٦٨

٦٩

VSB+C

•

VSB modulated signals can also be detected by an envelope detector.

•

As in the demodulation of a SSB signal, we need to send a pilot carrier signal, resulting an inefficient use of available transmitted power.

٧٠

Comparison of conventional AM, DSB-SC, SSB and VSB.

•

Conventional AM: simple to modulate and to demodulate, but low power efficiency (50% max) and double the bandwidth

•

DSB-SC: high power efficiency, more complex to modulate & demodulate, double the bandwidth

•

SSB: high power efficiency, the same (message) bandwidth, more difficult to modulate & demodulate.

•

VSB: lower power efficiency & larger bandwidth but easier to implement.

٧١

Multiplexing•

Multiplexing: combining a number of message signals into a composite signal to transmit them simultaneously over a wideband channel.

•

Two commonly-used types: time-division multiplexing (TDM) and frequency division multiplexing (FDM).

•

TDM: transmit different message signals in different time slots (mostly digital).

•

FDM: transmit different message signals in different frequency slots (bands) using different carrier frequencies.

٧٢

FDM

٧٣

٧٤

٧٥

Time Division Multiplexing

٧٦

TDM

٧٧

AM receiver for many radio stations ?

Frequency mixing

•

It is desired in communication system to translate the spectrum of the modulated signal up word or down word in frequency to be centered around desired frequency

0

0

: up conversio: down conversio

l c

l c

c l

f f f

f f nf

f f n

= −

−⎧⇒ = ⎨ −⎩

٧٩

Superheterodyne

AM Receiver

٨٠

•

The RF amplifier amplifies the incoming signal and start the process of selecting the wanted station and rejecting the unwanted ones.

٨١

The Mixer and the IF Amplifier

٨٢

٨٣

٨٤

٨٥

Introduction to Carrier Acquisition

•

Consider a DSB-SC demodulator where a received signal is m(t) cos(ωc

t) and the local carrier is 2 cos[(ωc

+Δω) t+δ

] . Find the LPF output ifa) Δω=0, and b) δ=0

٨٦

Carrier Acquisition•

To ensure identical carrier frequencies at the transmitter and the receiver, we can use quartz crystal oscillators, which are generally very stable.

•

At very high carrier frequencies, the quartz- crystal performance may not be adequate, we

can use the phased-locked loop (PLL)

٨٧

Phased-Locked Loop (PLL)•

Phase-locked loop is one of the most commonly used circuit in both telecommunication and measurement engineering.

•

PLL can be used to track the phase and the frequency of the carrier component of an incoming signal.

٨٨

•

A PLL has three basic components:1.

A voltage controlled oscillator

2.

A multiplier3.

A loop filter H(s)

recovered carrier signal

vout

(t)

vin

(t) e0

(t)x(t) Loop Filter

H(s)

Voltage-Controlled Oscillator (VCO)

٨٩

•

In every application, the PLL tracks the frequency and the phase of the input signal. However, before a PLL can track, it must first reach the phase-locked condition.

•

In general, the VCO center frequency differs from the frequency of the input signal.

•

First the VCO frequency has to be tuned to the input frequency by the loop. This process is called frequency pull-in.

•

Then the VCO phase has to be adjusted according to the input phase. This process is known as phase lock-in.

٩٠

How the PLL works?

)sin()( icin tAtv θω +=

)cos()( ocout tBtv θω +=

vout

(t)

vin

(t) e0

(t)x(t) Loop Filter

H(s)

Voltage-Controlled Oscillator (VCO)

٩١

Signal Squaring Method

( )2BPF

@ 2 ωc

PLL

2:1 Frequency divider

m(t) cos(ωc

t)

k cos(ωc

t)

c cos(2ωc

t)

٩٢

Suggested Problems

•

4.2-1

4.2-2

4.2-3

4.2-4

4.2-6, 4.2-8•

4.3-1 4.3-2

4.3-3

4.3-4

4.3-7 4.3-8

•

4.5-1

4.5-2

4.5-3

4.5-5, 4.5-6 •

4.6-1

•

4.8-1

4.8-2

•

Read Section 4.9 (Television)

Chapter 5

ANGLE MODULATION:FREQUENCY and PHASEMODULATIONS(FM,PM)

Outlines

• Introduction• Concepts of instantaneous frequency• Bandwidth of angle modulated signals• Narrow-band and wide-band frequency

modulations• Generation of FM signals• Demodulation of FM signals• superhetrodyne FM radio

Introduction• Angle modulation: either frequency modulation

(FM) or phase modulation (PM).

• Basic idea: vary the carrier frequency (FM) orphase (PM) according to the message signal.

• While AM is linear process, FM and PM arehighly nonlinear.

• FM/PM provide many advantages (main –noise immunity, interference, exchange ofpower with bandwidth ) over AM, at a cost oflarger transmission bandwidth.

• Demodulation may be complex, but modernICs allow cost-effective implementation.Example: FM radio (high quality, notexpensive receivers).

Concepts of InstantaneousFrequency

• A general form of an angle modulated signal is givenby

is the instantaneous angleis the instantaneous phase deviation.

• The instantaneous angular frequency of

( ) cos ( ) cos(2 ( ))EM i c iS t A t A f t t

( ) ( )( ) i i

i c

d t d tt

dt dt

( )i t

( )i t( )EMS t

• The instantaneous frequency of

• The instantaneous frequency deviation

( ) ( )1 1( )

2 2i i

i c

d t d tf t f

dt dt

( )1( )

2i

i

d tf t

dt

( )EMS t

Example

• for the signal below find the instantaneousfrequency and maximum frequencydeviation.

2( ) cos(10 )x t A t t

• For phase modulation (PM), the instantaneousphase deviation is

•kp is the phase sensitivity of the PM modulator

expressed in (rad/ V) if m(t) is in Volts• The instantaneous frequency of

( )( )i c p

dm tf t f k

dt

Phase modulation (PM)

( ) ( )i t kp m t

( ) cos [2 ( )]PM c pS t A f t k m t

( )PMS t

• For Frequency Modulation (FM), theinstantaneous phase deviation is

• kf is the frequency sensitivity of the FMmodulator expressed in rad/ V s if m(t) in Volts.

• The instantaneous frequency of

( ) cos 2 ( )t

FM c fS t A f t k m d

Frequency Modulation (FM)

( ) ( )t

i ft k m d

( )FMS t

( ) ( )2

fi c

kf t f m t

Angle modulation viewed as FM orPM

PhaseModulator

FrequencyModulator

PhaseModulator

FrequencyModulator

( )m t ( )PMS t

( )m t( )FMS t

( )m t ( )FMS t

( )PMS t( )m t d

d t

• A PM/FM modulator may be used togenerate an FM/PM waveform

• FM is much more frequently used than PM• All the properties of a PM signal may be

deduced from that of an FM signal• In the remaining part of the chapter we

deal mainly with FM signals.

Example 5.1• Sketch FM and PM waves for the modulating

signal m(t) shown in Fig. 5.4a. The constants kfand kp are 2πx105 and 10π ,respectively, andthe carrier frequency fc is 100 MHz..

Example

Bandwidth of Angle ModulatedSignals

1) FM signals

2 3

2 3

( ) cos(2 ) ( )sin(2 )

( ) cos(2 ) ( )sin(2 ) ...2! 3!

FM c f c

f fc c

S t A f t k a t f t

k kA a t f t a t f t

where ( ) ( )t

a t m d

• Narrow-Band Frequency Modulation (NBFM):

• Narrow-Band Phase Modulation (NBPM):

( ) cos(2 ) ( )sin(2 )NBFM c f cS t A f t k a t f t

( ) cos(2 ) ( )sin(2 )NBPM c p cS t A f t k m t f t

BBNBFM 2

| ( ) | 1fk a t

2N BPMB B

| ( ) | 1Pk m t

Generation of NBFM

m(t)

Generation of NBPM

m(t)

• If

∆f: maximum carrier frequency deviationβ: deviation ratio or modulation index

• Wide- Band Frequency Modulation (WBFM)|kf a(t)|>>1 or β>100 fBWBFM 2

2pf mk

f

)1(2)(2 BBfBFM

B

f

| ( ) | 1fk a t

max ( )Pm m t

• For phase modulation: if

2

'ppmk

f

| ( ) | 1Pk m t

2( ) 2 ( 1)PMB f B B

' 'max ( )Pm m t

2WBPMB f

Single tone modulation

• Let

)2sin(2cos)( tftfAtx mcFM

n

mcnFM tfnfJAtx )(2cos)()(

( ) cos 2 mm t f t

• The results is valid only for sinusoidal signal

• The single tone method can be used forfinding the spectrum of an FM wave whenm(t) is any periodic signal.

2 ( 1)

2

FM m

f

m

B f

kf

f

f

Example 1• A single tone FM signal is

Determinea) the carrier frequency fc

b) the modulation index βc) the peak frequency deviationd) the bandwidth of xFM(t)

6 3FMx (t)=10 cos[ 2 (10 )t+ 8 sin(2 (10 )t)]

Example 2

• A 10 MHz carrier is frequency modulated bya sinusoidal signal such that the peakfrequency deviation is ∆f=50 KHz. Determinethe approximate bandwidth of the FM signal ifthe frequency of the modulating sinusoid fm isa) 500 kHz, b) 500 Hz, c) 10 kHz.

Example 3• An angle modulated signal with carrier

frequency 100kHz is

Finda) the power of xFM(t)b) the frequency deviation ∆fc) The deviation ratio βd) the phase deviation ∆ φe) the bandwidth of xFM(t).

EM cx (t)=10 cos[ 2 f t+ 5 sin(3000 t)+10 sin(2000 t) ]

Example 5.3 (Txt book)

a) Estimate BFM and BPM for m(t) whenkf= 2πx105 rad/sV and kp= 5π rad/V

b) Repeat the problem if the amplitude of m(t)is doubled.

Features of Angle Modulation• Channel bandwidth may be exchanged for

improved noise performance. Such trade-offis not possible with AM

• Angle modulation is less vulnerable than AMto small signal interference from adjacentchannels and more resistant to noise.

• Immunity of angle modulation tononlinearities thus used for high powersystems as microwave radio.

• FM is used for: radio broadcasting, soundsignal in TV, two-way fixed and mobileradio systems, cellular telephone systems,and satellite communications.

• PM is used extensively in datacommunications and for indirect FM.

• WBFM is used widely in space andsatellite communication systems.

• WBFM is also used for high fidelity radiotransmission over rather limited areas.

Generation of FM Signals

• There are two ways of generating FMwaves:

–Indirect generation

–Direct generation

Indirect Generation of NBFM

m(t)

Indirect Generation ofWideband FM

• In this method, a narrowband frequency-modulated signal is first generated and then afrequency multiplier is used to increase themodulation index.

m(t)NBFM

xFM(t)FrequencyMultiplier

m(t)

N fcNBFM Frequency

Multiplier

BPF

Local Oscillator(fLo)

xFM(t)

fc

Frequency Converter

m(t)NBFM

FrequencyMultiplier

x64

PowerAmplifier

CrystalOscillator10.9 MHz

fc1=200 kHz∆f1= 25 Hz

FrequencyMultiplier

x48

fc2=12.8MHz∆f2= 1.6 kHz

fc3=1.9 MHz∆f3= 1.6 kHz

fc4= 91.2MHz∆f4= 76.8 kHz

Armstrong Indirect FM Transmitter

BPF

Direct Generation• The modulating signal m(t) directly controls

the carrier frequency. [ ]• A common method is to vary the inductance or

capacitance of a voltage controlled oscillator.

( ) ( )i c ff t f k m t

• In Hartley or Colpitt oscillator , the frequency isgiven by

• We can show that for k m(t) << C0

LC

1

02

)(1

C

tmkc

0

1

LCc

Varactor Modulator Circuit

• Advantage - Large frequency deviations arepossible and thus less frequency multiplicationis needed.

• Disadvantage - The carrier frequency tends todrift and additional circuitry is required forfrequency stabilization.

To stabilize the carrier frequency, a phase-locked loop can be used.

Example 5.6

• Discuss the nature of distortion inherent in theArmstrong FM generator

– Amplitude distortion– Frequency distortion

Example• A given angle modulated signal has a peak

frequency deviation of 20 Hz for an inputsinusoid of unit amplitude and a frequency of50 Hz. Determine the required frequencymultiplication factor, N, to produce a peakfrequency deviation of 20 kHz when the inputsinusoid has unit amplitude and a frequencyof 100Hz, and the angle-modulation used is(a) FM; (b) PM

Demodulation of FM Signals

• Demodulation of an FM signal requires asystem that produces an output proportional tothe instantaneous frequency deviation of theinput signal.

• Such system is called a frequencydiscriminator.

FM

Demodulator

)(cos)( ttAtx c

dt

tdkty

)()(

• A frequency-selective network with a transferfunction of the form |H(ω)|= a ω+b over theFM band would yield an output proportionalto the instantaneous frequency.

• There are several possible examples forfrequency discriminator, the simplest is theFM demodulator by direct differentiation

FM demodulator by direct differentiation

• The basic idea is to convert FM into AMand then use AM demodulator.

' ( ) 2 ( ) sin 2 ( )t

c c f c fs t A f k m t f t k m d

Bandpass Limiter

• Input-output characteristic of a hard limiter

HardLimiter

BPF

• Any signal which exceeds the preset limits aresimply chopped off

Practical Frequency Demodulators

• There are several possible networks forfrequency discriminator– FM slope detector– Balanced discriminator– Quadrature Demodulator

• Another superior technique for thedemodulation of the FM signal is to use thePhased locked loop (PLL)

FM Slope Detector

FM Slope Detector

FM Slope Detector

Balanced Discriminator

Balanced Discriminator (Cont.)

Balanced Discriminator (Cont.)

Quadrature Demodulator• FM is converted into PM

• PM detector is used to recover messagesignal

Quadrature Demodulator

Transfer function ofQuadrature demodulator

Phase-Locked Loop (PLL)

)sin()( icin tAtv

)cos()( ocout tBtv

vout(t)

vin(t) e0(t)x(t) Loop Filter

H(s)

Voltage-ControlledOscillator (VCO)

dt

tdkte i )(

)(0

Zero-Crossing Detectors• Zero-Crossing Detectors are also used because

of advances in digital integrated circuits.• These are the frequency counters designed to

measure the instantaneous frequency by thenumber of zero crossings.

• The rate of zero crossings is equal to theinstantaneous frequency of the input signal

Summary• Concepts of instantaneous frequency• FM and PM signals• Bandwidth of angle modulated signals

NBFM and WBFM• Generation of FM signals

– Direct and indirect generation• Demodulation of FM signals

– frequency discriminator– PLL

Suggested Problems

• 5.1-1 5.1-2 5.1-3 5.2-1 5.2-2 5.2-3 5.2-4 , 5.2-5 5.2-6 .

• 5.2-7 5.3-1 5.3-2 5.4-1 5.4-2

Chapter 6

Sampling and PulseCode Modulation

Outline• introduction• Sampling and sampling theorem• Practical sampling and pulse amplitude

modulation (PAM)• Pulse code modulation (PCM)• Differential pulse code modulation (DPCM)• Delta modulation.

Introduction• There is an increase use of digital

communication systems• Digital communications offer several

important advantages compared to analogcommunications such as higher performance,higher security and greater flexibility.

• digital transmission of analog signals requireAnalog to digital conversion (AD).

• digital pulse modulation

Analog to Digital converter

PCM system

Sampling• A typical method for obtaining a discrete-

time sequence x(n) from a continuous-time signal x(t) is through periodicsampling.

x(n)= x(nTs), for -∞ < n < ∞• Ts : sampling period.

• fs: sampling frequency

or sampling rates

s

fT

1

x(t)

s(t)

xs (t)

Spectrum of Xs(f)X(f)

k

ss

s fkfXT

fX )(1)(

• Is it possible to reconstruct the analogsignal from the sampled valued?

Samplerx(t) x(nTs)

LPFx(t)

• Given any analog signal, how should we selectthe sampling period Ts (or the samplingfrequency fs) without losing the importantinformation contained in the signal.

Spectrum of Xs(f)

Sampling Theorem• Let m(t) be a real valued band-limited signal

having a bandwidth B, and m(nTs) be thesample values of m(t) where n is an integer.

• The sampling theorem states that the signalm(t) can be reconstructed from m(nTs) with nodistortion if the sampling frequency

fs ≥2B

• The minimum sampling rate 2B is called theNyquist sampling rate.

Typical sampling rates for somecommon applications

Application B fs

Speech 4 kHz 8 kHzAudio 20 kHz 40 kHzVideo 4 MHz 8 MHz

Example

Determine the Nyquist rate of the followinganalog signal and plot the spectrum of thesampled signal for :

1. fs=150Hz 2. fs=300Hz 3. fs=500 Hzx(t) = 3cos(50t) + 10sin(300t) - cos(100t)

To Avoiding aliasing• Band-limiting signals (by filtering) before

sampling.• Sampling at a rate that is greater than the

Nyquist rate.

Anti-Aliasing

Filter

Sampler

fs≥ 2B

x(t) xs(t)

Practical Sampling

• In practice, we multiply a signal x(t) by a train ofpulses of finite width.

• There are two types of practical sampling

– Natural Sampling (Gating)

– Instantaneous Sampling. Also known as flat-top PAM or sample-and-hold.

Natural Sampling

s(t)

Generation of PAM with natural sampling

X(t)

Another example of natural sampling

Sample-and-Hold

τxs(t)

Sample-and-hold (S/H) circuit.

Natural Sampling (Gating)

)()()( tstxtxs

k

skTtrectts

)(

Natural Sampling (Gating): Spectrum

• The spectrum (FT) of the sampled (PAM)signal is

( ) sinc( ) ( )s sk

X f d kd X f kf

sTd Duty cylce of s(t)

Natural Sampling (Gating): Spectrum

d X( f )

d sinc(kd)

Sample-and-Hold( flat-top sampling)

τxs(t)

k

sss

kTtrectTkxtx

)()(

Sample and hold: Spectrum

kss

k

sss

kTtTkxtrect

kTtrectTkxtx

)()(*

)()(

1( ) sin ( ) ( )s sks

X f c f X f kfT

Sample and hold: Spectrum

sinc(τf) X(f)

• we see that by using flat-top sampling wehave introduced amplitude distortion, and theprimary effect is an attenuation of high-frequency components. This effect is knownas the aperture effect.

• If <<Ts, then H( f) represents a LPF.

• Else, we can use a LPF such thatHeq( f)= 1/H(f)

The LPF is called an equalization filter.

Reasons for intentionally lengthening the durationof each sample are:

• Reduce the required transmission bandwidth:B is inversely proportional to pulse duration.

• To get the exact signal value, the transient mustfade away

Pulse Modulation• Pulse modulation results when some

characteristic of a pulse is made to vary in one-to-one correspondence with the messagesignal.

• A pulse is characterized by three qualities:– Amplitude– Width– Position

• Pulse amplitude modulation, Pulse widthmodulation, and Pulse position modulation

Pulse amplitude modulation (PAM)• In Pulse Amplitude Modulation, a pulse is

generated with an amplitude corresponding tothat of the modulating waveform.

• There are two types of PAM sampling

– Natural Sampling (Gating)

– Flat-top or sample-and-hold.

PAM System

• A system transmitting sample values of theanalog signal is called a pulse-amplitudemodulation (PAM) system.

• Like AM, PAM is very sensitive to noise.

• While PAM was deployed in early AT&TDimension PBXs, there are no practicalimplementations in use today. However, PAM isan important first step in a modulation schemeknown as Pulse Code Modulation.

Note

• PBX: Short for private branch exchange, aprivate telephone network used within anenterprise.

• Users of the PBX share a certain numberof outside lines for making telephone callsexternal to the PBX.

Pulse Width Modulation (PWM)

• In PWM, pulses are generated at a regular rate.The length of each pulse is controlled by themodulating signal amplitude at that samplinginstant.

Pulse Position Modulation (PPM)

• A PPM signal consists of pulsesin which the pulse displacementfrom a specific time reference isproportional to the sample valuesof the information bearing signal.

Pulse Code Modulation(PCM)

Advantages of PCM• Inexpensive digital circuitry may be used in

the system.• All-digital transmission.• Further digital signal processing such as

encryption is possible.• Errors may be minimized by appropriate

coding of the signals.• Signals may be regularly reshaped or

regenerated using repeaters at appropriateintervals.

A single-channel PCM transmission system

Advantages of PCM• Inexpensive digital circuitry may be used in

the system.• All-digital transmission.• Further digital signal processing such as

encryption is possible.• Errors may be minimized by appropriate

coding of the signals.• Signals may be regularly reshaped or

regenerated using repeaters at appropriateintervals.

A single-channel PCM transmission system

Quantization

• Quantizer converts the discrete time signalinto a sampled and quantized signal that isdiscrete in both time and amplitude

m(t) and its sampled value m(kTs)

0 0.002 0.004 0.006 0.008 0.01-8

-6

-4

-2

0

2

4

6

8

D

Input-output characteristics of the quantizer

Output

Input

L=8

mq=8-mq=-8

• Quantization can be uniform andnonuniform

• The quantization discussed so far is said to beuniform since all of the steps are of equalsize.

• Nonuniform quantization uses unequal steps

Uniform Quantization• The amplitude of ms(t) can be confined to the

range [-mq, mq]

• This range can be divided in L zones, each ofstep such that

= 2 mq / L

• The sample amplitude value is approximated bythe midpoint of the interval in which it lies.

Quantization Noise• The difference between the input and output

signals of the quantizer becomes thequantizing error or quantizing noise

mq(t) mq(t)+D/2mq(t)-D/2

ms(t)ms(t)

2)(

2

tq

Quantization Error or Noise

• Assuming that the error is equally likely to lieanywhere in the range (-∆/2, ∆/2), the mean-square quantizing error is given by

121 22/

2/

22

dqqq

2

22

3 Lm

q q2

22 )(3

qo

o

mtmL

NS

Example• For a full-scale sinusoidal modulating signal

m(t)= A cos(ωmt), show that

• or

23 2L

NS

o

o

)()(log2076.1 10 dBLNS

dBo

o

223

q

o

o

o

mSL

NS

Nonuniform Quantization

• For many classes of signals the uniformquantizing is not efficient.

• Example: speech signal has large probability ofsmall values and small probability of large ones.

• Solution: allocate more levels for smallamplitudes and less for large. Thus, totalquantizing noise is greatly reduced

Example of Nonuniform quantization

0 0.005 0.01 0.015 0.02

-6

-4

-2

0

2

4

6

Nonuniform Quantization

• The effect of nonuniform quantizing can beobtained by first passing the analog signalthrough a compression (nonlinear) amplifierand then into the PCM circuit that uses auniform quantizer.

• At the receiver end, demodulate uniform PCMand expand it.

• The technique is called companding.• Two common techniques

1. m-law companding2. A-law companding

m-law Compression Characteristic

• where

qmmmy

1ln

)1ln()sgn(

1qm

m

m-law Compression Characteristic

|m/mq|

y

A-law Compression Characteristic

• where1

qmm

11,ln1ln1

)sgn(

1,ln11

qq

qq

mm

AmmA

Am

Amm

mm

Ay

A-law Compression Characteristic

|m/mq|

y

• The compressed samples must be restoredto their original values at the receiver byusing an expander with a characteristicscomplementary to that of the compressor.

• The combination of compression andexpansion is called companding

• It can be shown that when a µ-lawcompander is used, the output SNR is

• where

22

)1ln(3

LNS

o

o

)(2

22

tmmq

Coding of Quantized Samples• The coding process in an A/D converter

assigns a unique binary number to eachquantization level. For example, we can usebinary and gray coding.

• A word length of n bits can create L= 2n

different binary numbers.• The higher the number of bits, the finer the

quantization and the more expensive thedevice becomes.

Binary and Gray coding of samples.

Output SNR• SNR is controlled by the PCM bandwidth

)(6 dBnNS

dBo

o

c10log10

caseeduncompress

mtm

casecompressedc

q

,)(3

,)1(ln

3

2

2

2

Comments About dB Scale• The decibel can be a measure of power ratio

• It can also be used for measuring power

NS

NS

dB10log10

mWPP

PP

WdBm

WdBW

1log10

log10

10

10

ExamplesGain= (Pout/Pin) = 2 = +3 dB

Pout= 48 mW

Bandwidth of PCM• What is the spectrum of a PCM signal?• The spectrum of the PCM signal depends on

the bit rate, the correlation of the PCM data,and on the PCM waveform pulse shape (usuallyrectangular) used to describe the bits.

• The dimensionality theorem [2] shows that thebandwidth of the PCM waveform is bounded by

BPCM ≥ R/2 = n fs/2where R: bit rate

Transmission Bandwidth

• For L quantization levels and n bitsL= 2n or n= log2L

• The bandwidth of the PCM waveformBPCM ≥ n B Hz

• Minimum channel bandwidth or transmissionbandwidth

BT= n B Hz

Example: PCM for Telephone System• Telephone spectrum: [300 Hz, 3400 Hz]• Min. sampling frequency: fs,min = 2 Fmax=6.8 kHz• Some guard band is required:

fs= 2 Fmax + ∆fg = 8 kHz• n=8-bit codewords are used L=256.• The transmission rate: R=n* fs =64 kbits/s• Minimum PCM bandwidth: ΒPCM = R/2=32 kHz

Example 6.3• A signal m(t) of bandwidth B= 4 kHz is

transmitted using a binary companded PCMwith µ=100. Compare the case of L=64 with thecase of L=256 from the point of view oftransmission bandwidth and the output SNR.

Differential PCM (DPCM)• Samples of a band-limited signal are correlated.• This can be used to improve PCM

performance: to decrease the number of bitsused (and, hence, the bandwidth) or to increasethe quantization SNR for a given bandwidth.

• Main idea: quantize and transmit the differencebetween two adjacent samples rather thansample values.

• Since two adjacent samples are correlated,they difference is small and requires less bits totransmit.

DPCM System-Modulator

mq(k)

d(k)m(k) Quantizer

Predictor

dq(k)

)(ˆ kmq

)(ˆ)()( kmkmkd q )()()( kqkdkdq

)()(ˆ)( kdkmkm qqq

mq(k)= m(k)+ q(k) is the quantized version of m(k)

DPCM System-Demodulator

Predictor

Output mq(k)

)(ˆ kmq

Input dq(k)

)(ˆ)()( kmkmkd qqq )(ˆ)()( kmkdkm qqq

Time Division Multiplexing (TDM)

Ten-Channel PCM System(a) Transmitter (b) Receiver

Signal Shapes

Bandwidth Requirements for TDM• If N band-limited signals are multiplexed

each with bandwidth B• The minimum TDM sampling rate is

fTDM= 2 N B• If each sample is coded with n bits, then the

minimum transmitted data rate isR= 2 n N B

• The minimum transmission bandwidth isBT = n N B

TDM: Concept of Framing and Synchronization

• The time interval TF containing one sample from eachmessage signal is called a frame.

• an extra pulse (called marker) is transmitted forsynchronization

Comparison of Time and FrequencyDivision Multiplexing

• Time division multiplexing: Individual TDMchannels are assigned to distinct time slots butjumbled together in the frequency domain.Channels are separated in the time domain

• Frequency division multiplexing: IndividualFDM channels are assigned to distinctfrequency regions but jumbled together in thetime domain. Channels are separated in thefrequency domain

Comparison of Time and FrequencyDivision Multiplexing

• Many of the TDM advantages are technologydriven. The digital circuits are much cheaperand easier to implement

• In FDM, imperfect bandpass filtering andnonlinear cross-modulation cause cross talk.TDM is not sensitive to these problems.

Example• A binary channel with bit rate Rb=36000 bits/s

is available for PCM transmission. Findappropriate values of the sampling rate fs, thequantizing level, and the binary digits n,assuming the signal bandwidth is B=3.2 kHz.

Example• An analog signal is quantized and transmitted

by using a PCM system. If each sample at thereceiving end of the system must be known towithin 0.5% of the peak- to-peak full-scalevalue, how many binary digits must eachsample contain ?

Example• For a full-scale sinusoidal modulating signal

m(t)= A cos(ωmt), show that

• or

23 2L

NS

o

o

)()(log2076.1 10 dBLNS

dBo

o

223

q

o

o

o

mSL

NS

Suggested problems

• 6.1-1, 6.2-3 and 6.2-5.