Chapter 5

AM, FM and Digital Modulated Systems

In this chapter, we will study the techniques of bandpass communications.

1. Analog baseband signal (AM, FM, etc)

2. Digital baseband signals (OOK, BPSK, etc.)

3. Multiplexing (TDM, FDM, CDM)

Basic Model for Bandpass Communication

Source Destination

m(t)tm

g(t)tg

tr

ts

tg

tm

ingapproximat signal tedreconstruc :)(~ ingapproximat signal ddemodulate :)(~

frequency) (shifted channel in the corrupted signal modulated :)(

frequency) (shifted ed transmittbe tosignal modulated :)(

(baseband) modulated be n toinformatio processed :)(

(baseband) sourceby sent be n toinformatio :)(

* *

( ) Re ( ) where ( ) . The goal is to fine a function

• that satisfies the objectives of the communication.

1 1

2 4

cjw t

c c s g c g c

s t g t e g m t

g

S(f) G f f G f f (f) f f f f

P P P

Amplitude Modulation

max

min

max min

( ) 1 ( )

s( ) 1 ( ) cos

Definitions.

% positive modulation 100 max ( ) 100

% negative modulation 100 min ( ) 100

max ( ) min ( )% modulation 100 100

2 2

c

c c

c

c

c

c

c

g t A m t

t A m t w t

A Am t

A

A Am t

A

m t m tA A

A

max min max 1 ( ) , min 1 ( )c cA A m t A A m t

Almost always, -1 min ( ) max ( ) 1. Otherwise, the

transmitter and receivers become very complicated.

In general, to simplify the receiver and transmitter complexity,

1 ( ) cos if ( ) 1 s( ) c c

m t m t

A m t w t m tt

0 if ( ) 1

This will increase the bandwidth for ( ).

m t

s t

max

max

1.5

0.5

Positive modulation % = 50%

Negative modulation % = 50%

Modulation % = 50%

c

c

A A

A A

22 2

2 2

2 2

1( ) 1 ( )

21

1 2 ( ) ( )21

1 ( ) if ( ) 0.2

Definition:

Modulation efficiency = % of total power of ( ) containing information

c

c

c

s t A m t

A m t m t

A m t m t

E s t

2

2

2

( ) 100 (for AM signals)

1 ( )

If an AM signal is 100% modulated with sinusoidal modulating signal,

1i.e., max ( ) 1, ( ) 0.5. Thus, the modulation efficiency = .

3The highe

m tE

m t

m t m t

st possible modulation efficiency is achieved when ( ) 1 for

all . Then, 50%.

m t

t E

22

1 max ( )2

( )2

Note. AM signals use twice the bandwidth of the original baseband signals.

Example.

Suppose an AM transmitter has the average carrier power of 5,0

cPEP

cc c c c

AP m t

AS f f f M f f f f M f f

2

2

00 .

The AM transmitter is connected to a 50 resistor.

15,000 707

2 50Suppose ( ) cos 2 . << . Thus the AM signal is 100% modulated.

( ) 0.5 and max ( ) 1.

1Total transmission power =

2

cc

m m c

W

AA

m t f t f f

m t m t

2 22

22

2

1( ) 7,500

50 2 50

11 max ( ) 20,000

2 501

33.33%1 ( )

c c

cPEP

A Am t W

AP m t W

Em t

Generation of High-Power AM by Pulse Width Modulation

The transmitter does not send the carrier signal to reduce power.

( ) ( ) cos

( )2

Since there is no carrier signal, % modulat

Double Sideband Suppressed Carrier (DSB-SC)

c c

cc c

s t A m t w t

AS f M f f M f f

ion = and 100%.

The envelop detector cannot be used. The product detector is needed.

By not sending the carrier signal, the transmitter power can be increared for the sideband.

E

Coherent detection produces high quality demodulated signals.

It is difficult to do so for DSB-SC, because it has no carrier signal.

DSB-SC has a spectrum

Coherent Detection of DSB-SC

symmetric arout the center frequncy.

Two technology solutions:

- Costas Loop

- Squaring Loop

Costas Loop

22

3

1 1( ) ( )sin 2

2 2 o c ev t A A m t

22

4

1 1( ) ( ) sin 2

2 2 o c ev t A A m t

Act like an integrator.

Squaring Loop

Single sideband (SSB) signals have only half of the sideband of the

ordinary AM signals.

Upper single sideband (USSB) has no spectrum for .

Lower single si

Single Sideband

cf f

deband (LSSB) has no spectrum for > .

Theorem. For SSB signals,

( ) ( ) ( ) ( ) ( ) cos ( )sin for USSB

( ) ( ) ( ) ( ) ( ) cos ( )sin for LSSB

where

c

c c c c

c c c c

f f

g t A m t jm t s t A m t w t jm t w t

g t A m t jm t s t A m t w t jm t w t

o

1( ) ( ) ( ) and ( ) .

0( ) ( ) ( ( ) shifts the signal phase by -90 .)

0

m t m t h t h tt

j fh t H f H f

j f

Spectrum for USSB Signal

( ) ( ) ( )

( ) ( ) ( )

2 ( ) 0( )

0 0

( ) 0( )

0 ( )

Proof of Theorem (for USSB)

c

c

c

c c cc c

c c c

G f A M f j m t

G f A M f jH f

A M f fG f

f

M f f f f f fS f A A

f f M f f f f

F

Generation of SSB Signals

222 2 2

2 2 2 2 2

22 2 2

22 2

1

1 1( ) ( ) ( ) ( )

2 2

( ) ( ) because ( ) ( )

1 1max ( ) max ( ) ( )

2 2

For SSB-AM: ( ) ( ) ( ) ( )

( )For SSB-PM: ( ) ( ) tan for USSB

( )

c

c

PEP c

c

s t g t A m t m t

s t A m t m t m t

P g t A m t m t

R t g t A m t m t

m tt g t

m t

1 ( )

= tan( )

( ) ( )out c

m t

m t

v t KA m t

DSB takes up too much bandwidth, and SSB is difficult to implement.

VSB is a compromise between DSB and SSB.

VSB partially suppresses one of the side

Vestigial Sideband (VSB)

bands.

( ) ( ) ( )

( ) : DSB signal ( ) : impulse response for VSB filter

( ) ( ) ( ) ( )

Requirement for VSB filter: ( ) ( ) , for

( ) ( ) ( ) (2

VSB v

v

VSB VSB v

v c v c

cVSB c v

s t s t h t

s t h t

s t S f S f H f

H f f H f f C f B

AS f M f f H f M f

) ( )c vf H f

VSB Spectra

( ) ( ) cos ( )

1 1( ) ( ) ( )

2 2

( ) 1 for

( ) ( ) for 4

( ) ( ) for due to the requirement for

out o VSB c

out o VSB c c

c oout v c v c

out

v t A s t w t h t

V f A S f f f f f H f

H f f B

A AV f M f H f f H f f f B

V f KM f f B

vH f

Angle Modulation( ) ( )

where ( ) ( ) = and ( ) ( ).

Phase Modulation (PM) and Frequency Modulation (FM) are special

cases of angle modulation.

j tc

c

g t A e

R t g t A t m t

( ) cos ( )

For PM, ( ) ( ) : phase sensitivity factor

For FM, ( ) ( ) : frequency deviation factor

Note: ( ) is a nonlinear function of ( ), and ( ) is a linear f

c c

p p

t

f f

s t A w t t

t D m t D

t D m s ds D

g t m t t

*

( )

unction of ( ).

1 ( )

2

( ) ( )

In general we don't know how to find ( ).

c c

j tc

m t

S f G f f G f f

G f g t A e

G f

F F

Let ( ) : modulating message for PM, ( ) : modulating message for FM.

( )( ) ( ) ( )

p f

tp p ff p f

f p

m t m t

D dm t Dm t m t m s ds

D dt D

Angle Modulator Circuits

RFC : Radio Frequency Choke

Definition.

Bandpass signal: ( ) ( ) cos ( ) where ( ) ( ).

Instantaneous frequency:

1 1 ( ) ( ) ( )

2 2

1 ( )For PM: ( )

2

For FM:

c

i i

i c

s t R t t t w t t

d tf t w t

dt

d tf t f

dt

1

( ) ( )2

Note: For angle modulated signals, the instantaneous frequency varies in

time around the center frequency .

i c f

c

f t f D m t

f

FM with m(t) = Sinusoidal Signal

1 ( )Frequency deviation: ( ) = ( )

2

1 ( )Peak frequency deviation: = max 0, necessarily.

2

Peak-to-peak frequency deviation:

1 ( ) = max

2

d i c

PP

d tf t f t f

dt

d tF

dt

d tF

d

1 ( )min 0, necessarily.

2

1For FM: = where max ( ) .

2 Note. The bandwidth requirement varies by the amplitude of ( ) and

, the fr

f p p

f

d t

t dt

F D V V m t

m t

D

2

equency deviation factor.

The power of FM signal is influence only by the amplitude of the

modulated signal, not ( ). power = 2cAm t

Peak phase deviation: = max ( )

For PM, where max ( ) .

Definition.

Phase modulation index

Frequency modulation index

If ( ) is a single sinusoidal wave,

p p p

p

f

t

D V V m t

F

B

m t

then .p f

(

Example.

For PM, let ( ) sin . ( ) sin sin

Thus, .

For FM, ( ) cos . ( ) sin sin

Thus, and .2

For the PM and FM above, ( )

p m m p m m m

p m p

f m m p m m m

f m f mf

m

jc

m t A w t t D A w t w t

D A

m t A w t t D A w t w t

D A D AF

w

g t A e

sin)

th

( ) where is the Bessel function of

the first kind of the order.

mj w ttc

c n m nn

A e

G f A J f nf J

n

Bessel functions: n = 0 ,1, 2, 3, 4, 5, 6

Magnitude spectra for FM or PM

( ) sin

: bandwidth within which

98% of power is contained

2 1 (Carson's rule)

m

T

T

m t w t

B

B B

( )

Narrowband Angle Modulation

Variation of angle is small. ( ) 0.2

( ) 1 ( )

( ) cos ( )sin

( )

j tc c

c c c c

c c c c c

t rad

g t A e A j t

s t A w t A t w t

S f A f f f f j f f f f

( ) for PM

where ( )( ) for FM

2

p

c f

D M f

f f t DM f

j f

F

2

Wideband Frequency Modulation

Theorem. For WBFM, let

max ( )( ) cos ( ) and 1.

2

power spectrum density of ( ),

2

2

t fc c f f

cm

f f

D m ts t A w t D m s ds

B

s t

Af f

D D

P(f) 2 where is p.d.f. of m(t).c m c m

f

f f f f fD

2

Example. WBFM with ( ): triangular function

1for

2 where is the peak of the triangular function.

0 for

1 2 1 2if if

2 22

if all other

ppm p

p

c pcp f p f

f

m t

m VVf m V

m V

f f V fAV D V D

Do f

P(f)

2 2

if all other

if if 8 8

if all other if all other

2

c p

c cc c c c

f p

f V

o f

A Af F f f F f F f f F

F Fo f o f

D VF

P(f)

Spectrum of WBFM Signal

Spectrum of Wideband Binary FSK

Pre-emphasis and De-emphasis

Frequency Division MultiplexingTransmission of multiple messages simultaneously over wideband channel.

Any modulation type (AM, DSB, SSB, PM, FM) can be used.

FM Stereo System

FM Broadcasting

Non commercial: 88.1 - 91.9

Commercial: 92.1 - 107.9

Today, there are analog and digital FM broadcasting with a

channel bandwidth of 200 .

MHz

MHz

KHz

Binary Modulated Bandpass Signals

On Off Keying (OOK): Carrier sinusoid is on/off with a unipolar binary

signals. (e.g., Morse code radio trasmission)

Binary Phase Shift Keying (BPSK): The

o o

phase of the carrier sinusoid is either

0 or 180 based on binary inputs.

Frequency Shift Keying (FSK): Change the frequency of the carr ier sinusoid

by a certain amount.

Digital Bandpass Signals

22

OOK: ( ) ( ) cos where ( ) is a unipolar bainary base band data.

( ) ( )

sin ( ) ( )

2

1 bit rate , null-to-null BW 2 , absolute BW = .

c c

c

c bg b

b

b

s t A m t w t m t

g t A m t

A fTf f T

fT

R BT

P

BPSK: ( ) cos ( ) where ( ) = 1.

( ) cos ( ) cos sin ( ) sin

( ) cos cos sin ( )sin

This can be made into ( ) ( )sin

( )

c c p

c p c c p c

c p c c p c

c c

s t A w t D m t m t

s t A D m t w t A D m t w t

s t A D w t A D m t w t

s t A m t w t

g t j

2

2

( )

sin ( )

1 bit rate , null-to-null BW 2 , absolute BW = .

c

bg c b

b

b

A m t

fTf A T

fT

R BT

P

PSD

Detection of OOK

Detection of BPSK

From Frequency Shift Keying section to the end of Chapter 5, the math is quite complex.

Thus, we will focus on the concepts and understanding how various techniques work.

Section 5.10 will be covered briefly. Section 5.11 will not be covered.

Note that Chapter 5 is concerned with the technology. Do not focus too much about the mathematical details.

Frequency Shift Keying

( )

FSK: message ( ) 0,1 is sent as part of the frequency of bandpass signal.

(1: mark, 0:space)

( ) cos ( )

( ) Re ( ) where ( )

( ) ( )

c

t

c c f

jw t j tc

t

f

m t

s t A w t D m v dv

s t g t e g t A e

t D m v dv

FSK is very popular for telephone modems.

FSK Modem

Example FSK Data

Spectra of FSK Data

PSD for Complex Envelope

of FSK

Detection of FSK

Quadrature Phase Shift Keying (QPSK) (4 Level PSK)

Quadrature Amplitude Modulation (QAM)

(Multi Level AM)

( )

( ) ( ) cos ( )sin

( ) ( ) ( ) ( )

c c

j t

s t x t w t y t w t

g t x t jy t R t e

Generation of QAM

MSK, GMSK, QPSK, and OQPSK

Direct Sequence Spread Spectrum

Autocorrelation and PSD for an m-Sequence PN Waveform

Approximate PSD of BPSK-DS-SS Signal

Frequency-Hopped Spread Spectrum