Analog Modulation” is the subject concerned · Quadrature null effect of the coherent detector....

Chapter 2 Continuous-Wave

Modulation

“Analog Modulation” is the subject concerned

in this chapter.

2.1 Introduction

Analog communication system

The most common carrier is the sinusoidal wave.

Carrier wave

(Analog)

2.1 Introduction

Modulation

A process by which some characteristic of a carrier

is varied in accordance with a modulating wave

(baseband signal).

Sinusoidal Continuous-Wave (CW) modulation

Amplitude modulation

Angle modulation

2.1 Introduction

Amplitude Modulation

Frequency Modulation

Baseband signal

Sinusoidal carrier

2.2 Double-Sideband with Carrier or simply

Amplitude Modulation

Two required conditions on amplitude sensitivity

1 + ka m(t) 0, which is ensured by |ka m(t)| ≦ 1. The case of |ka m(t)| > 1 is called overmodulation.

The value of |ka m(t)| is sometimes represented by “percentage” (because it is limited by 1), and is named (|ka m(t)|100)% modulation.

fc >> W, where W is the message bandwidth. Violation of this condition will cause nonvisualized envelope.

index modulationor y sensitivit amplitude is where

),2cos()](1[)( Signal Modulated

)( Baseband

)2cos()(Carrier

a

cac

cc

k

tftmkAts

tm

fAtc

2.2 Overmodulation

overmodulaion 1|)(| tmka

2.2 Example of Non-Visualized Envelope

2.2 Example of Visualized Envelope

2.2 Transmission Bandwidth

)()(2

)()(2

)(

)2cos()](1[)(

cc

ca

cc

c

cac

ffMffMAk

ffffA

fS

tftmkAts

2W

Transmission bandwidth BT = 2W.


Transmission bandwidth of an AM wave

For positive frequencies, the highest frequency

component of the AM wave equals fc + W, and the

lowest frequency component equals fc – W.

The difference between these two frequencies

defines the transmission bandwidth BT for an AM

wave.


The condition of fc > W ensures that the sidebands

do not overlap.

2.2 Negative Frequency

Operational meaning of “negative frequency”

in spectrum

If time-domain signal is real-valued, the negative

frequency spectrum is simply a mirror of the

positive frequency spectrum.

We may then define a one-sided spectrum as

Hence, if only real-valued signal is considered, it is

unnecessary to introduce “negative frequency.”

.0for )(2)(sided-one

ffSfS


So the introduction of negative frequency part is

due to the need of imaginary signal part.

Signal phase information is embedded in imaginary

signal part of the signal.

phase

mI(t)

mQ(t)


As a result, the following two spectrums contain the

same frequency components but different phases

(90 degree shift in complex plane).

fc

fc

fc

fc

)(c

ff )(c

ff

)(c

ff

)(c

ff

)2cos(2 tfc

)2sin(2 tfjc


Summary

Complex-valued baseband signal consists of

information of amplitude and phase; while real-

valued baseband signal only contains amplitude

information.

One-sided spectrum only bears amplitude

information, while two-sided spectrum (with

negative frequency part) carries also phase

information.

2.2 Virtues of Amplitude Modulation

AM receiver can be implemented in terms of simple

circuit with inexpensive electrical components.

E.g., AM receiver

)]4cos(1[)](1[2

)2(cos)](1[)()(

2

2

2222

1

tftmkA

tftmkAtstv

ca

c

cac


The bandwidth of m2(t) is twice of m(t). (So to

speak, the bandwidth of m(f)*m(f) is twice of m(f).)

Lowpass filter


So if 2fc > 4W,

)(2

mean zero is )( if

)](1[2

)(

)](1[2

)(

block DC

3

22

2

tmkA

tm

tmkA

tv

tmkA

tv

ac

ac

ac

By means of a squarer, the receiver can recover the information-

bearing signal without the need of a local carrier.

2.2 Limitations of Amplitude Modulation

(DSB-C)

Wasteful of power and bandwidth

Waste of power in the information-less “with-carrier” part.

)2cos()()2cos(

)2cos()](1[)(

carrier with

tftmktfA

tftmkAts

cacc

cac

2.2 Limitations of Amplitude Modulation

Wasteful of power and bandwidth

Only requires half of bandwidth after modulation

2.3 Linear Modulation

Definition

Both sI(t) and sQ(t) in

s(t) = sI(t)cos(2fct) – sQ(t) sin(2fct)

are linear function of m(t).

2.3 Linear Modulation

For a single real-valued m(t), three types of modulations can be identified according to how sQ(t) are linearly related to m(t), at the case that sI(t) is exactly m(t): (Some modulation may have mI(t) and mQ(t) that

respectively bear independent information.)

1. Double SideBand-Suppressed Carrier modulation (DSB-SC)

2. Single SideBand (SSB) modulation

3. Vestigial SideBand (VSB) modulation

2.3 DSB-SC and SSB

Type of modulation sI(t) sQ(t)

DSB-SC m(t) 0

SSB m(t) Upper side band transmission

SSB m(t) Lower side band transmission

)(ˆ tm

)(ˆ tm

DSB-SC

SSB

SSB

)( of ansformHilbert tr)(ˆ* tmtm

usb

lsb

, which is used to completely “suppress” the other sideband.

2.3 DSB-SC

Different from DSB-C, DSB-SC s(t) undergoes

a phase reversal whenever m(t) crosses zero.

)2cos()()( tftmtsc

Require a receiver that can

recognize the phase reversal!

2.3 Coherent Detection for DSB-SC

For DSB-SC, we can no longer use the “envelope

detector” (as used for DSB-C), in which no local

carrier is required at the receiver.

The coherent

detection or

synchronous

demodulation

becomes necessary.


. provided

),()cos(2

1 '

Wf

tmAA

c

cc

LowPass

)()cos(2

1)()4cos(

2

1

)()2cos()2cos(

)()2cos()(

''

'

'

tmAAtmtfAA

tmtftfAA

tstfAtv

ccccc

cccc

cc


Quadrature null effect of the coherent detector.

If = /2 or –/2, the output of coherent detector for DSB-

SC is nullified.

If is not equal to either /2 or –/2, the output of coherent

detector for DSB-SC is simply attenuated by a factor of

cos(), if is a constant, independent of time.

However, in practice, often varies with time; therefore, it is

necessary to have an additional mechanism to maintain the

local carrier in the receiver in perfect synchronization with

the local carrier in the transmitter.

Such an additional mechanism adds the system complexity of

the receiver.

2.3 Costas Receiver for DSB-SC

An exemplified

design of

synchronization

mechanism is the

Costas receiver,

where two

coherent

detectors are

used.

In-phase coherent detector

Quadrature-phase coherent detector

2.3 Costas Receiver for DSB-SC

Conceptually, the Costas receiver adjusts the

phase so that it is close to 0.

When drifts away from 0, the Q-channel output

will have the same polarity as the I-channel output

for one direction of phase drift, and opposite

polarity for another direction of phase drift.

The phase discriminator then adjusts through the

voltage controlled oscillator.

-pi/2 0 pi/2

)sin(

2.3 Single-Sideband Modulation

How to generate SSB signal?

1. Product modulator to generate DSB-SC signal

2. Band-pass filter to pass only one of the sideband

and suppress the other.

The above technique may not be applicable to a

DSB-SC signal like below. Why?

SSB

filter filter


For the generation of a SSB modulated signal

to be possible, the message spectrum must have

an energy gap centered at the origin.


Example of signal with 300 Hz ~ 300 Hz energy gap

Voice : A band of 300 Hz to 3100 Hz gives good articulation

Also required for SSB modulation is a highly selective filter


Phase synchronization is also an important

issue for SSB demodulation. This can be

achieved by:

Either a separate low-power pilot carrier

Or a highly stable local oscillator (for voice

transmission)

Phase distortion that gives rise to a Donald Duck voice

effect is relatively insensitive to human ear.

2.3 Vestigial Sideband Modulation

Instead of transmitting only one sideband as

SSB, VSB modulation transmits a partially

suppressed sideband and a vestige of the other

sideband.

SSB

VSB

Still, no information loss in principle.

2.3 Requirements for VSB filter

1. The sum of values of the magnitude response |H(f)| at any

two frequencies equally displaced above and below fc is

unity. I.e., |H(fc f)| + |H(fc f)| = 1 for fv < f < fv.

2. H(f fc) + H(f + fc) = 1 for W < f < W.

fc fc

So the transmission band of VSB filter is BT = W + fv.

f f+fc ffc

2.3 Generation of VSB Signal

Analysis of VSB

Give a real baseband signal m(t) of bandwidth W.

Then,

Let where

.||for 0)( and )()( * WffMfMfM

,2/]/)(1)[()( jfHfMfMQVSB

).()( and ,

0,0

0),1,0(

,1

)(1 * fHfH

f

ff

ff

fHj

QQv

v

Q

<<

The filter is denoted by HQ because it is used to generate sQ(t) (cf. Slide 2-23)

2.3 Generation of VSB Signal

1.0

W

)(1

)( fHj

fLQQ

2/)(1

12/)](1[

fH

jfL

QQ

).(

)()(1

)(1

)(1

)(real. is )(1

)( *

*

*

fL

fLfHj

fHj

fHj

fLfHj

fL

Q

QQQQQQQ

2.3 How to recover from VSB signal?

).()( because ),(

)]()(2)[(2

1

)()( and real, is )](1[ since

,)](1)[()](1)[(2

1

)](1)[()](1)[(2

1

)()(

*

*

fLfLfM

fLfLfM

fMfMfL

fLfMfLfM

fLfMfLfM

fMfM

QQ

QQ

Q

QQ

QQ

VSBVSB

2.3 VSB upper sideband transmission

)(tmVSB

)(tsVSB

)()(22

1)(

}{)](1[}{)](1[2

1)(

}{2

)](1[}{

2

)](1[)(

}{2

)](1[}{

2

)](1[

)()(2

1

}{2

)](1[}{

2

)](1[

}{)(}{)(2

1

*

*cont.

cQcQDSB

ccQccQDSB

c

cQ

c

cQ

DSB

c

cQ

c

cQ

cc

c

cQ

c

cQ

cccccc

ffLffLfs

WffffLWffffLfs

WffffL

WffffL

fs

WffffL

WffffL

ffMffM

WffffL

WffffL

WffWfffMWffWfffM

11

11

11

11

11

(See next slide.)

.0)()()()( if

)]()()][()([)()()()(

fFfMfFfM

fFfFfMfMfFfMfFfM

LRRL

RLRLRRLL

}{)](1[ WffffL ccQ 1

}{)](1[ WffffL ccQ 12

2

2

}{)](1[

}{)](1[

WffffL

WffffL

ccQ

ccQ

1

1

)( cQ ffL

)( cQ ffL

1

1

2 2)()( cQcQ ffLffL

)()()(

2)()(2

1)()(

fHfsfs

ffLffLfsfs

DSBVSB

cQcQDSBVSB

2)()(2

1)(

cQcQffLffLfH

Consequently,

1 )( fH

1 )(

cffH

1

)(c

ffH

1

)()(cc

ffHffH

1

)()(cc

ffHffH

WfffHffHjfH

WfffHffHfL

ccQ

ccQ

||for )]()([)(

||for )()()(

2.3 Mathematical Representation of VSB

signal

)('2

1)(

2

1

)()(2

1)(

2

1

2/)](1)[()(

fjMfM

fHfjMfM

fjHfMfM

Q

QVSB

).()()()()(' where fLfjMfHfMfM QQ

.)]('[)]()([)()()(

)()()()()('

*****

*

fMfLfjMfLfMj

fLfjMfLfjMfM

QQ

QQ

Transform. Hilbert of extension an is This real. is )(' Notably, tm

2.3 Application of VSB Modulation

Television Signals

1. The video signal exhibits a large bandwidth and

significant low-frequency content.

Hence, no energy gap exists (SSB becomes

impractical).

VSB modulation is adopted to save bandwidth.

Notably, since a rigid control of the transmission VSB

filter at the very high-power transmitter is expensive, a

“not-quite” VSB modulation is used instead (a little

waste of bandwidth to save cost).


VSB Filter for Television Signal Transmissions

55.25 MHz 59.75 MHz


As the transmission signal is not quite VSB modulated,

the receiver needs to “re-shape” the received signal

before feeding it to a VSB demodulator.


2. In order to save the cost of the receiver (i.e., in

order to use envelope detector at the receiver), an

additional carrier is added.

Notably, additional carrier does not increase

bandwidth, but just add transmission power.

)2sin()('2

1)2cos()(

2

11

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

1)2cos()(

tftmAktftmkA

tftmtftmkAtfAts

ccacac

ccaccc


Distortion of envelope detector

Distortion

22

2

2LowP ass

2

22222

2

22

))('(4

1)(

2

11

2

1

)2cos()2sin()(2

11)('

2

1

)2(sin))('(4

1)2(cos)(

2

11)(

tmktmkA

tftftmktmAk

tftmAktftmkAts

aac

ccaca

ccacac

The distortion can be compensated by reducing the amplitude

sensitivity ka or increasing the width of the vestigial sideband. Both

methods are used in the design of Television broadcasting system.

2.3 Extension Usage of DSB-SC

Quadrature-Carrier Multiplexing or Quadrature

Amplitude Modulation (QAM)

Synchronization is critical in QAM modulation, which is often achieved by a

separate low-power pilot tone outside the passband of the modulated signal.

2.4 Frequency Translation

The basic operation of SSB modulation is simply a special case of frequency translation.

So SSB modulation is sometimes referred to as frequency changing, mixing, or heterodyning.

The mixer is a device that consists of a product modulator followed by a band-pass filter, which is exactly what SSB modulation does.

2.4 Frequency Translation

The process is named upconversion, if f1 + f is the wanted signal, and f1 – f is the unwanted image signal.

The process is named downconversion, if f1 f is the wanted signal, and f1 + f is the unwanted image signal.

2.5 Frequency-Division Multiplexing

Multiplexing is a technique to combine a

number of independent signals into a composite

signal suitable for transmission.

Two conventional multiplexing techniques

Frequency-Division Multiplexing (FDM)

Time-Division Multiplexing (TDM)

Will be discussed in Chapter 3.

2.5 Frequency-Division Multiplexing

Example 2.1

2.6 Angle Modulation

Angle modulation

The angle of the carrier is varied in accordance with the baseband signal.

Angle modulation provides us with a practical means of exchanging channel bandwidth for improved noise performance.

So to speak, angle modulation can provide better discrimination against noise and interference than the amplitude modulation, at the expense of increased transmission bandwidth.


Commonly used angle modulation

Phase modulation (PM)

Frequency modulation (FM)

y.sensitivit phase is where)],(2cos[)(ppcc

ktmktfAts

y.sensitivitfrequency is re whe

,)(22cos

))((2cos)(

0

0

f

t

fcc

t

fcc

k

dmktfA

dmkfAts


Main differences between Amplitude

Modulation and Angle Modulation

1. Zero crossing spacing of angle modulation no

longer has a perfect regularity as amplitude

modulation does.

2. Angle modulated signal has constant envelope; yet,

the envelope of amplitude modulated signal is

dependent on the message signal.


Similarity between PM and FM

PM is simply an FM with in place of m(t).

Hence, the text only discusses FM in this chapter.

t

dm0

)(

t

fccFMdmktfAts

0)(22cos)(

)](2cos[)( tmktfAtspccPM

2.7 Frequency Modulation

s(t) of FM modulation is a non-linear function of m(t).

So its general analysis is hard.

To simplify the analysis, we may assume a single-tone

transmission, where

)2cos()( tfAtmmm

t

fcc

t

fcc

t

ic

dmktfA

dmkfAdfAts

0

00

)(22cos

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

From the formula in the previous slide,

deviation.frequency theis where

)2cos(

)2cos(

)()(

mf

mc

mmfc

fci

Akf

tfff

tfAkf

tmkftf

)2sin(2cos

)]2cos([2cos

)(2cos)(

0

0

tff

ftfA

dfffA

dfAts

m

m

cc

t

mcc

t

ic

signal. FMof

index modulation thecalled often is / wherem

ff

Modulation index is the largest deviation from 2fct in

FM system.

As a result,

1. A small corresponds to a narrowband FM.

2. A large corresponds to a wideband FM.

)2sin(2cos)( tftfAtsmcc

mccmcicmcfffftffftfffff )2cos()(

2.7 Narrowband Frequency Modulation

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

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

)2sin(2cos)(

tftfAtfA

tftfAtftfA

tftfAts

mcccc

mccmcc

mcc

(Often, < 0.3.)


Comparison between approximate narrowband

FM modulation and AM (DSB-C) modulation

))(2cos(2

))(2cos(2

)2cos(

)2cos()]2cos(1[

)2cos()](1[)(

tffAk

tffAk

tfA

tftfAkA

tftmkAts

m

ma

m

ma

cc

cmmac

cacAM

))(2cos(2

))(2cos(2

)2cos(

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

tffA

tffA

tfA

tftfAtfAts

mc

c

mc

c

cc

mccccFM


Represent them in terms of their low-pass

isomorphism.

)]2sin()2[cos(2

)]2sin()2[cos(2

)0()(~

tfjtfAk

tfjtfAk

jAts

mm

ma

mm

ma

cAM

)]2sin()2[cos(2

)]2sin()2[cos(2

)0()(~

tfjtfA

tfjtfA

jAts

mm

c

mm

c

cFM


Phaser diagram

)]2sin()2[cos(2

tfjtfA

mm

c

)0( jAc

))2sin(

)2(cos(2

tfj

tfA

m

m

c

)(~ tsFM

)]2sin()2[cos(2

tfjtfA

mm

c

)(~ tsAM

.Let mac

AkA

2.7 Spectrum of Single-Tone FM

Modulation

)2exp()(~Re

)2sin(2expRe

)2sin(2cos)(

tfjts

tftfjA

tftfAts

c

mcc

mcc

)2sin(exp)(~ tfjAtsmc

n

ntfj

ncmeJAts

2)()(~ )sin()( jx

n

jn

n eexJ

kind.first theof function elorder Bess nth theis )( where nJ

(See Slide 1-247)


Modulation

n

mnc

n

tnffj

nc

ftj

n

ntfj

nc

ftj

nffJA

dteJA

dteeJA

dtetsfS

m

m

)()(

)(

)(

)(~)(~

)(2

22

2


Modulation

n

mcmcnc

n

mcmcnc

cc

nfffnfffJA

nfffnfffJA

ffSffSfS

)()()(2

)()()(2

)(~

)(~

2

1)( *

Consequently,


Modulation

The power of s(t)

By definition, the time-average autocorrelation function

is given by:

Hence, the power of s(t) is equal to:

T

TT

T

TTs

dttstsT

dttstsET

R )()(2

1lim)]()([

2

1lim)(

22

)2sin(24cos1

2

1lim

)2sin(2cos2

1lim)(

2

1lim)0(

2

2

222

cT

T

mc

cT

T

Tmcc

T

T

TTs

Adt

tftfA

T

dttftfAT

dttsT

R


Modulation

The time-average power spectral density of a

deterministic signal s(t) is given by (cf. Slide 1-117)

From

we obtain:

)()(2

1lim)( *

2fSfS

TfPSD

TT

.||)( of ransform Fourier t theis )( where2

TttsfST

1

n

mcmcncTnfffTnfffTJTAfS ))(sinc(2))(2(sinc)()(

2

n

mcncTTttnffJAts ||))(2cos()()(

21

n

mmcmmcn

k

mcmck

c

p

TT

fnfffpfnfffpJ

kfffkfffJA

fSfST

fPSD

)/)(sinc()/)((sinc)(

)()()(4

lim

)()(2

1lim

)(

2

*

2

For simplicity, assume that 2T increases along the multiple of

1/fm., i.e., 2T = p/fm, where p is an integer. Also assume that fc is

a multiple of fm, i.e., fc = qfm., where q is an integer. Then

mcnff

mcfnf )1(

mcfnf )1(

p

fnff m

mc

)]()([)(4

)()()()()(

)()()()()(4

)()()/)(sinc()(

)()()/)(sinc()(

)()()/)((sinc)(

)()()/)((sinc)(lim4

)(

2

2

2

2

2

2

2

2

mcmc

n

n

c

mc

n

nmcqn

n

n

mcqn

n

nmc

n

n

c

k

mckmmc

n

n

k

mckmmc

n

n

k

mckmmc

n

n

k

mckmmc

n

np

c

nfffnfffJA

nfffJnfffJJ

nfffJJnfffJA

kfffJfnfffpJ

kfffJfnfffpJ

kfffJfnfffpJ

kfffJfnfffpJA

fPSD

2.7 Average Power of Single-Tone FM

Signal

Hence, the power of a single-tone FM signal is given by:

Question: Can we use 2f to be the bandwidth of a single-tone FM signal?

2

)()()1(12

)()()(2

)(

2

2

2

2

2

2

c

n

qnn

nc

n

qnn

n

n

c

A

JJA

JJJA

dffPSD

Example 2.2

Fix fm and kf,

but vary = f/fm =

kf Am/fm.

Example 2.2

Fix Am and kf,

but vary = f/fm

= kf Am/fm.

2.7 Spectrum of Narrowband Single-Tone

FM Modulation

.

2for 0)(

2)(

1)(

small, is When1

0

nJ

J

J

n

2.7 Spectrum of Narrowband Single-Tone

FM Modulation

This results in an approximate spectrum for narrowband single-tone FM signal spectrum as

)()()(4

)()()(4

)()()(4

)()()(4

)(

2

1

2

2

0

2

2

1

2

2

2

mcmc

c

cc

c

mcmc

c

n

mcmcn

c

ffffffJA

ffffJA

ffffffJA

nfffnfffJA

fPSD

)()(16

)()(4

)()(16

)()()(4

)()()(4

)()()(4

)(

22

2

22

2

1

2

2

0

2

2

1

2

mcmcc

ccc

mcmcc

mcmcc

ccc

mcmcc

ffffffA

ffffA

ffffffA

ffffffJA

ffffJA

ffffffJA

fPSD

)()()()1()( 22 nnn

n

nJJJJ

4

2

cA

16

22

cA

4

2

cA

16

22

cA

cf

cf

mf

16

22

cA

16

22

cA

2.7 Transmission Bandwidth of FM signals

Carson’s rule – An empirical bandwidth

An empirical rule for Transmission Bandwidth of

FM signals

For large , the bandwidth is essentially 2f.

For small , the bandwidth is effectively 2fm.

So Carson proposes that:

11222 fffB

mT

2.7 Transmission Bandwidth of FM signals

“Universal-curve” transmission bandwidth

The transmission bandwidth of an FM wave is the

minimum separation between two frequencies

beyond which none of the side frequencies is

greater than 1% of the carrier amplitude obtained

when the modulation is removed.

n

mcmcn

c nfffnfffJA

fS )()()(2

)(

)()(2

)2cos(cc

c

ccffff

AtfA

0.1

0.3

0.5

1.0

2.0

5.0

10.0

20.0

30.0

2

4

4

6

8

16

28

50

70

max

2n

maxmax

22 n

f

fn

f

B

m

mT

.larger a causes smaller a , fixedFor T

Bf

20.0

13.3

8.0

6.0

4.0

3.2

2.8

2.5

2.3

fBT

/

2.7 Bandwidth of a General FM wave

Now suppose m(t) is no longer a single tone but a

general message signal of bandwidth W.

Hence, the “worst-case” tone is fm = W.

For nonsinusoidal modulation, the deviation ratio D = f / W is used instead of the modulation index .

The derivation ratio D plays the same role for nonsinusoidal

modulation as the modulation index for the case of

sinusoidal modulation.

We can then use Carson’s rule or “universal curve” to

determine the transmission bandwidth BT.

2.7 Bandwidth of a General FM wave

Final notes

Carson’s rule usually underestimates the

transmission bandwidth.

Universal curve is too conservative in bandwidth

estimation.

So, a choice of a transmission bandwidth in-

between is acceptable for most practical purposes.

Example 2.3

FM radio in North America requires the

maximum frequency derivation f = 75 kHz.

If some message signal has bandwidth W = 15 kHz,

then the deviation ratio D = f / W = 75/15 = 5.

Then

kHz 180 1

17521

12Carson,

DDfB

T

kHz 240755

162max

Curve Universal, f

D

nB

T

Example 2.3

In practice, a bandwidth of 200 kHz is allocated to

each FM transmitter.

So Carson’s rule underestimates BT, while

“Universal Curve” overestimates BT.

2.7 Generation of FM Signals

Direct FM

Carrier frequency is directly varied in accordance with the message signal as accomplished using a voltage-controlled oscillator.

Indirect FM

The message is first integrated and sent to a phase modulator.

So, the carrier frequency is not directly varied in accordance to the message signal.


dm )( (See next slide.)


Frequency multiplier

))cos(10)3cos(5)5(cos(16

1)(cos

))cos(3)2cos(4)4(cos(8

1)(cos

))cos(3)3(cos(4

1)(cos

)1)2(cos(2

1)(cos

5

4

3

2

xxxx

xxxx

xxx

xx

2.7 Demodulation of FM Signals

Indirect Demodulation – Phase-locked loop

Will be introduced in Section 2.14

Direct Demodulation

Balanced frequency discriminator

)(ts

)(1

ts

)(2

ts

|)(~|1

ts

|)(~|2

ts

)(~ tso

differentiation filters

elsewhere,02

||,2

2

2||,

22

)(1

T

c

T

c

T

c

T

c

Bff

Bffaj

Bff

Bffaj

fH

elsewhere,02

||,2

2

2||,

22

)(2

T

c

T

c

T

c

T

c

Bff

Bffaj

Bff

Bffaj

fH

2.7 Analysis of Direct Demodulation in

terms of Low-Pass Equivalences

otherwise0,2

||,2

4

otherwise0,2

||),(2)(

~1

1

TT

T

c

Bf

Bfaj

BfffH

fH

elsewhere,02

||),(~

22

)(~

)(~

2

1)(

~11

TTB

ffSB

fajfSfHfS

)(~)(~

)(~1

tsBjdt

tsdats

T

t

fccdmktfAts

0)(22cos)(

t

fcdmkjAts

0)(2exp)(~

t

f

T

f

cT

t

fcT

t

ffc

T

dmkjtmB

kaABj

dmkjABj

dmkjtmkjAa

tsBjdt

tsdats

0

0

0

1

)(2exp)(2

1

)(2exp

)(2exp)(2

)(~)(~)(~

t

fc

T

f

cT

t

fc

T

f

cT

c

t

f

T

f

cT

c

dmktftmB

kaAB

dmktftmB

kaAB

tfjdmkjtmB

kaABj

tfjtsts

0

0

0

11

2)(22cos)(

21

)(22sin)(2

1

)2exp()(2exp)(2

1Re

)2exp()(~Re)(

message. equivalent lowpass theof amplitude theobtain to

used be candetector envelope then, and 1)(2

If WftmB

kc

T

f<

)(

21|)(~|

1tm

B

kaABts

T

f

cT

otherwise0,2

||,2

4)(

~2

TTB

fB

fajfH

elsewhere,02

||),(~

22

)(~

)(~

2

1)(

~22

TTB

ffSB

fajfSfHfS

2 Similarly,

t

f

T

f

cT

T

dmkjtmB

kaABj

tsBjdt

tsdats

0

2

)(2exp)(2

1

)(~)(~)(~

t

fc

T

f

cT

c

dmktftmB

kaAB

tfjtsts

0

22

2)(22cos)(

21

)2exp()(~Re)(

)(

21|)(~|

2tm

B

kaABts

T

f

cT

)(4|)(~||)(~|)(~21

tmaAktststscfo

Final Note: a is a parameter of the filters, which can be used to

adjust the amplitude of the resultant output.

2.7 FM Stereo Multiplexing

How to do Stereo Transmission in FM radio?

Two requirements:

Backward compatible with monophonic radio receivers

Operate within the allocated FM broadcast channels

To fulfill these requirements, the baseband message

signal has to be re-made.

fm = 19 kHz

detectioncoherent For reception monophonicFor

)2cos()4cos()]()([)]()([)( tfKtftmtmtmtmtm mmrlrl

m

m

m

Demultiplexer in receiver of FM stereo.

2.8 Nonlinear Effects in FM Systems

The channel (including background noise, interference and circuit imperfection) may introduce non-linear effects on the transmission signals.

For example, non-linearity due to amplifiers.

<

1)(,3

2

1)(,3

2

1|)(|),(3

1)(

)(

3

tv

tv

tvtvtv

tv

i

i

iii

o

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1.5 -1 -0.5 0 0.5 1 1.5

2.8 Nonlinear Effects in FM Systems

Suppose

t

f

cci

iiio

dmkt

ttfAtv

tvatvatvatv

0

3

3

2

21

)(2)(

)(2cos)(

)()()()(

Then

)](2[cos

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

)()()()(

33

1

22

11

3

3

2

21

ttfAa

ttfAattfAa

tvatvatvatv

cc

cccc

iiio

WfB

cc

WfB

cc

WfB

cccc

ccc

cccco

TT

T

ttfAattfAa

ttfAaAaAa

ttfttfAa

ttfAattfAatv

26

3

1

24

2

2

22

3

31

2

1

3

3

2

21

Carson,Carson,

Carson,

)](36cos[4

1)](24cos[

2

1

)](2cos[4

3

2

1

)](36cos[)](2cos[34

1

)](24cos[12

1)](2cos[)(

Thus, in order to recover s(t) from vo(t) using band-pass filter

(i.e., to remove 2fc and 3fc terms from vo(t)), it requires:

2/)22(2/)24(2 WffWffcc

.23 ly,equivalentor Wffc

The filtered output is therefore:

)](2cos[4

3)( 3

31filtered,ttfAaAatv

ccco

Observations

Unlike AM modulation, FM modulation is not affected

by distortion produced by transmission through a channel

with amplitude nonlinearities.

So the FM modulation allows the usage of highly non-

linear amplifiers and power transmitters.

2.8 AM-to-PM Conversion

Although FM modulation is insensitive to amplitude nonlinearity, it is indeed very sensitive to phase nonlinearity.

A common type of phase nonlinearity encountered in microwave radio transmission is the AM-to-PM conversion.

The AM-to-PM conversion is owing to that the phase characteristic of amplifiers (or repeaters) also depends on the instantaneous amplitude of the input signal.

Notably, the nonlinear amplifiers discussed previously will leave the phase of the input unchanged.

Often, it requires that the peak phase change for a 1-dB change in input envelope is less than 2%.

2.9 Superheterodyne Receiver

A commercial radio communication system

contains not only the “transmission” but also

some other functions, such as:

Carrier-frequency tuning, to select the desired

signals

Filtering, to separate the desired signal from other

unwanted signals

Amplifying, to compensate for the loss of signal

power incurred in the course of transmission


A superheterodyne receiver or superhet is designed to facilitate the fulfillment of these functions, especially the first two. It overcomes the difficulty of having to build a tunable highly

selective and variable filter (rather a fixed filter is applied on IF section).

heterodyning function

First detector


Example AM Radio FM Radio

RF carrier range 0.535-1.605 MHz 88-108 MHz

Midband frequency of IF section 0.455 MHz 10.7 MHz

IF bandwidth 10 kHz 200 kHz

Second detector

2.9 Image Interference

Fix fIF and fLO at the receiver end. What is the fRF that will survive at the IF section output?

Answer:

Example. Suppose the receiver use 1.105MHz local oscillator, and receives two RF signals respectively centered at 0.65MHz and 1.56MHz.

||IFLORF

fff

MHz455.0

IFf

MHz455.0

IFf

MHz65.0

RFf

MHz56.1

ceInterferenf

MHz105.1

LOf

MHz.4550

RFLOIFfff MHz.7551

ageIm

RFLOfff

MHz.4550ceInterferen Image

LOceInterferen

fff MHz665.2LOceInterferen

ff


Then at the output of the IF section, we respectively obtain:


A cure of image interference is to employ a highly

selective stages in the RF session in order to favor the

desired signal (at fRF) and discriminate the undesired

signal (at fRF + 2fIF or fRF – 2fIF).

2.9 Advantage of Constant Envelope for FM

modulation

Observations

For FM modulation, any variation in amplitude is

caused by noise or interference.

For FM modulation, the information is resided on

the variations of the instantaneous frequency.

So we can use an amplitude limiter to remove the

amplitude variation, but to retain the frequency

variation after the IF section.

2.9 Advantage of Constant Envelope for FM

modulation

Amplitude limiter

Clipping the modulated wave at the IF section

output (almost to the zero axis) to result in a near-

rectangular wave.

Pass the rectangular wave through a bandpass filter

centered at fIF to suppress harmonics (due to

clipping).

Then the filter output retains the frequency

variation with constant amplitude.

2.10 Noise in CW Modulation Systems

To simplify the system analysis, we assume: ideal band-pass filter (that is just wide enough to pass the

modulated signal s(t) without distortion),

ideal demodulator,

Gaussian distributed white noise process.

So the only source of imperfection is from the noise.

TB


),()()( tntstx

So after passing through the ideal bandpass filter, s(t) is

unchanged but w(t) becomes a narrowband noise n(t).

Hence,

).2sin()()2cos()()( where tftntftntncQcI


Input signal-to-noise ratio (SNRI)

The ratio of the average power of the modulated

signal s(t) to the average power of the filtered noise

n(t).

Output signal-to-noise ratio (SNRO)

The ratio of the average power of the demodulated

message signal to the average power of the noise,

measured at the receiver output.


It is sometimes advantageous to look at the lowpass equivalent model.

Channel signal-to-noise ratio (SNRC)

The ratio of the average power of the modulated signal s(t) to the average power of the channel noise in the message bandwidth, measured at the receiver input (as illustrated below).


Notes

SNRC is nothing to do with the receiver structure, but depends on the channel characteristic and modulation approach.

SNRO is however receiver-structure dependent.

Finally, define the figure of merit for the receiver as:

C

O

SNR

SNRmerit of figure

2.11 Noise in Linear Receivers Using

Coherent Detection

Recall that for AM demodulation

when the carrier is suppressed, linear coherent

detection is used. (Section 2.11)

when the carrier is additionally transmitted,

nonlinear envelope detection is used. (Section 2.12)

The noise analysis of the above two cases are

respectively addressed in Sections 2.11 and

2.12.


Coherent Detection

. todbandlimite )( PSDand meam zero withstationary:)( WfStmM

)2cos()()( tftmCAts cc

)2cos( tfc


Coherent Detection

Average signal power

2

)2(cos2

1lim

)()2(cos2

1lim

)()2(cos2

1lim)]([

2

1lim

22

222

2222

22222

PAC

dttfT

PAC

dttmEtfACT

dttmtfACET

dttsET

c

T

Tc

Tc

T

Tcc

T

T

Tcc

T

T

TT

power. message theis )()]([ where 2

W

WM dffStmEP


Coherent Detection

Noise power in the message bandwidth

00

2)( WNdf

NdffS

W

W

W

Ww

)2cos( tfc


Coherent Detection

Channel SNR for DSB-SC and coherent detection (observed at x(t))

Next, we calculate output SNR (observed at y(t)) under the condition

that the transmitter and the receiver are perfectly synchronized.

o

c

o

c

CWN

PAC

WN

PACSNR

2

2 2222

SC-DSB,

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

)()()(

tftwtftwtftmCA

twtstx

cQcIcc

)(2

1)(

2

1

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

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

)2cos()()(

LowPass

22

twtmCA

tftftwtftwtftmCA

tftftwtftwtftmCA

tftxtv

Ic

ccQcIcc

ccQcIcc

c

)(2

1)(

2

1)( twtmCAty Ic

0

22

2

22

2

222

SC-DSB,2)]([4/)(

4/)(

WN

PAC

twE

PAC

twE

tmACESNR cc

I

cO

)].)([)]([)]([ (Recall 222 twEtwEtwE QI

(See Slide 1-219.)

.1detectioncoherent and SC-for DSBmerit of Figure

Similar derivation on SSB and coherent detection yields the

same figure of merit.

Conclusions

Coherent detection for SSB performs the same as

coherent detection for DSB-SC.

There is no SNR degradation for SSB and DSB-SC

coherent receivers. The only effect of these modulation

and demodulation processes is to translate the message

signal to a different frequency band to facilitate its

transmission over a band-pass channel.

No trade-off between noise performance and bandwidth.

This may become a problem when high quality

transceiving is required.

2.12 Noise in AM Receivers Using Envelope

Detection

)2cos()](1[)( tftmkAtscac

mean.) zero )( (Assume )1(2

)2(cos2

1lim))(1()]([

2

1lim

2

2

2222

tmPkA

dttfT

tmkEAdttsET

a

c

T

Tc

Tac

T

TT

Hence, channel SNR for DSB-C is equal to:

Next, we calculate output SNR (observed at y(t)) under the condition that the transmitter and the receiver are perfectly synchronized.

0

0

2)( Also, WNdf

NdffS

W

W

W

Ww

o

ac

CWN

PkASNR

2

)1( 22

AM,

)2sin()()2cos()]())(1([

)2sin()()2cos()()2cos()](1[

)()()(

tftntftntmkA

tftntftntftmkA

tntstx

cQcIac

cQcIcac

)]()([2

1

)]())(1([2

1

)()]())(1([2

1

)()(

block DC

22

LowP ass

2

tntmkA

tntmkA

tntntmkA

txty

Iac

Iac

QIac

envelop detector

x

x

)(~)](1[ if tntmkAac

(Refer to Slide 2-136 and 2-138.)

0

22

2

22

2

222

AM,2)]([2/)(

2/)(

WN

PkA

tnE

PkA

tnE

tmkAESNR acac

I

ac

O

11)2()1(

)2(2

2

22

0

22

AM,

AM, <

Pk

Pk

WNPkA

WNPkA

SNR

SNR

a

a

oac

ac

C

O

Conclusion

Even if the noise power is small compared to the

average carrier power at the envelope detector output,

the noise performance of a full AM receiver is

inferior to that of a DSB-SC receiver due to the

wastage of transmitter power.

Example 2.4 Single-Tone Modulation

Assume

)2cos()]2cos(1[)(

)2cos()(

tftfAkAts

tfAtm

cmmac

mm

)1(24

02

1

)2(cos)2(cos

)2(cos)2cos(2)2(cos2

1lim

)2(cos)2cos(12

1lim)]([

2

1lim

2

222

2

2222

222

2222

PkAAk

A

dttftfAk

tftfAktfT

A

dttftfAkT

AdttsET

a

cma

c

cmma

T

Tcmmac

Tc

T

Tcmma

Tc

T

TT

Hence,

.2/1

2/

1

,discussion previous as proceduresimilar Following

.2

)2(cos1

lim where

22

22

2

2

AM,

AM,

2

0

22

ma

ma

a

a

C

O

mT

mmT

Ak

Ak

Pk

Pk

SNR

SNR

AdttfA

TP

So even if for 100% percent modulation (kaAm = 1), the figure of merit

= 1/3. This means that an AM system with envelope detection must

transmit three times as much average power as DSB-SC with coherent

detector to achieve the same quality of noise performance.

2.12 Threshold Effect

What if is violated (in AM modulation

with envelope detection)?

)2sin()()2cos()]())(1([

)2sin()()2cos()()2cos()](1[

)()()(

tftntftntmkA

tftntftntftmkA

tntstx

cQcIac

cQcIcac

)(~)](1[ tntmkAac

|~||~||~|2|~|2)(|~|

22)(|~|

222222

2222222

nBnBnBnBnBnnBnB

nBnBnBnnBnBnnBnB

IQI

IIIQIIQI

<<

|))(~|))(1((2

1

|))(~|))(1((2

1

|)(~|))(1(|)(~|2))(1(2

1

|)(~|))(1()(2))(1(2

1

)()]())(1([2

1

)()(

2

222

222

22

LowP ass

2

tntmkA

tntmkA

tntmkAtntmkA

tntmkAtntmkA

tntntmkA

txty

ac

ac

acac

acIac

QIac

|])(~|)([2

1block DC

tntmkAac

.)(~)](1[ Assume tntmkAac

<<

<<

|)(~|)](1[|,)(~|)(

|)(~|)](1[),()()(2

tntmkAtntmkA

tntmkAtntmkAty

acac

acIac

0

22

22

22

2

222

AM,4)]([)]([|)(~|

)(

WN

PkA

tnEtnE

PkA

tnE

tmkAESNR ac

QI

acac

O

2

1

)1(2)2()1(

)4(2

2

22

0

22

AM,

AM, <

Pk

Pk

WNPkA

WNPkA

SNR

SNR

a

a

oac

ac

C

O

2.12 Threshold Effect

Threshold effect

For AM with envelope detection, there exists a

carrier-to-noise ratio r (namely, the power ratio

between unmodulated carrier Ac cos(2fct) and the

passband noise n(t) ) below which the noise

performance of a detector deteriorates rapidly.

0

2

0

2

42

2/

WN

A

WN

Acc r

2.12 General Formula for SNRO in Envelope

Detection

For envelope detector, the noise is no longer additive; so the

original definition of SNRO (which is based on additive

noise) may not be well-applied.

A new definition should be given:

Definition. The (general) output signal-to-noise ratio

for an output y(t) due to a carrier input is defined as

)]([Var

2

ty

sSNR o

O

alone. noise of presense thein

)( toequal is )( and )],([)]([ where tytytyEtyEsooo

)()( tysty oo


Detection

so is named the mean output signal.

Var[y(t)] is named the mean output noise power.

Example. y(t) = A + nI(t), where nI(t) is zero mean..

)]([)]([Var)]([Var

)]([)]([

2 tnEtnty

AtnEtnAEs

II

IIo

)]([ 2

2

tnE

ASNR

I

O

This somehow shows the backward compatibility of the new

definition.


Detection

Now for an envelope detector, the output due to a carrier

input and additive Gaussian noise channel is given by:

)())(()( 22 tntnAtyQI

pdf withddistribute Rayleighis )()()( 22 tntntyQIo

pdf withddistribute Ricianis )(ty

I0( ) = modified Bessel function

of the first kind of zero order.

Appendix 4 Confluent Hypergeometric

Functions

The Kummer confluent hypergeometric function is a

solution of Kummer’s equation

For b 0, 1, 2, …., the Kummer confluent

hypergeometric function is equal to 1F1(a;b;x).

complex ,for 0)(2

2

baaydx

dyxb

dx

ydx

.1)(

)1()1()( where,

!)()()(

)()()();;(

function trichypergeome dGeneralize

00 21

21

a

kaaaa

k

x

bbb

aaaxbaF

kk

k kqkk

kpkk

qp

rr

./)0(' and 1)0( conditionsboundary with bayy

A4.2 Properties of the Confluent

Hypergeometric Function

);;();;()exp( .5

4

1;1;

22

)2/()()exp( .4

. as 2

22)1(

2exp);1;2/1( .3

.1);1;1( 2.

.0 as 1);;( .1

1111

2110

0

221

2011

11

11

uFuFu

b

mF

b

mduuIubu

xx

xxI

xIx

xxF

xxF

xxb

axbaF

m

m


Detection

Hence,

r

rr

rr

rr

rr

r

;1;2

1

2

;1;2

3)exp(

2

;1;2

3

)4(2

)2/3()exp(

)2(

)(4

exp)exp()2(

)]([

11

11

112/32/3

00

2

2

2/3

F

F

F

duuIu

utyE

N

N

N

N

By Property 4

By Property 5


Detection

As a result,

Similarly, we can obtain:

.12

;1;2

1

2)]([)]([

2

2

11

N

Noo

AFtyEtyEs

2

2

2

112

2

2

2

2

2

112

2

11

2

2;1;

2

1

4212

2;1;

2

1

42;1;12)](Var[

NN

N

NN

N

AF

A

AF

AFty

By Property 2


Detection

This concludes to:

(Continue from the previous slide.)


Detection

rPkSNR aO

2

AM,

es.approximat limiting two theand

;1;2/1)1(4

1;1;2/1 of Curve

2

11

2

11

rr

r

F

FSNR

O

rPkSNR aO

2

AM, 2

0.1

1

10

100

1000

0.1 1 10 100


Detection

Remarks

For large carrier-to-noise ratio r, the envelope detector

behaves like a coherent detector in the sense that the output

SNR is proportional to r.

For small carrier-to-noise ratio r, the (newly defined) output

signal-to-noise ratio of the envelope detector degrades faster

than a linear function of r (decrease at a rate of r2).

From “threshold effect” and “general formula for SNRO,” we

can see that the envelope detector favors a strong signal. This

is sometimes called “weak signal suppression.”

2.13 Noise in FM Receivers

To simplify the system analysis, we assume: ideal band-pass filter (that is just wide enough to pass the

modulated signal s(t) without distortion),

ideal demodulator,

Gaussian distributed white Noise process.

So the only source of imperfection is from the noise.

.)(2)( where)],(2cos[)(0t

fccdmktttfAts

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

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

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

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

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

222ttftttrtttrA

ttftttr

ttftttrA

ttttftrttfA

ttftrttfAtx

cc

c

cc

ccc

ccc

)]()(cos[)(

)]()(sin[)(tan)()( where 1

tttrA

tttrtt

c

))(sin()()(

))(cos()()(

ttrtn

ttrtn

Q

I

)](2cos[

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

Limiter

222

ttfA

ttftttrtttrAtx

c

cc

)('2)(~Output ))(2cos()(Input taAtsttfAtsoc

Recall on Slides 2-94 ~ 2-101, we have talked about the

Balanced Frequency Discriminator, whose input and output

satisfy:

Next, the signal will be passed through a Discriminator.

)(exp)(')(~)(~)(~

1tjBtaAjtsBj

dt

tsdats

TT

)(exp)(')(~)(~)(~

2tjBtaAjtsBj

dt

tsdats

TT

)('2|)(~||)(~|)(~21

taAtststso

Specifically, with :have we)),(exp()(~ tjAts

dt

tttrA

tttrtd

aAtaAtvc

)]()(cos[)(

)]()(sin[)(tan)(

2)('2)(

tor discrimina the throughpassingafter Thus,

1

The above assumption is true for any deterministic (constant) (t)!

It has been shown that Assumption 1 is justifiable for high carrier-

to-noise ratio (or equivalently, under Assumption 2).

We then obtain the desired “additive” form.

)()( 8. :6.2 Table ffGjtgdt

d )(tn

Q

cA

fjfH

2

2)(

)(tnd

)()()(

)()()()(2

2

fSA

ffS

A

fj

A

jffSfHfHfS

QQQd N

c

N

cc

NN

otherwise,0

2||,

)( 02

2

T

cN

BfN

A

f

fSd

Bandwidth W < BT/2

that is just enough

to pass m(t).

)()()( tntmktvofo

otherwise,0

||,)( 02

2

WfNA

f

fScNo

<

otherwise ,0

2/||for

),()(

)(T

cNcN

NBf

ffSffS

fSQ

2

3

02

2

02

3

2)]([

c

W

Wc

oA

WNdff

A

NtnE

Observation from the above formula:

. thenamed is This power. noise Decreasing

2/power carrier increasing system, FMan In 2

effect quieting noise

cA

).( provided ,2

3

3

2

)]([3

0

22

2

3

0

22

FM,trA

WN

PkA

A

WN

tmEkSNR

c

fc

c

f

O

),()()( As tntmktvofo

.)(2)( where)],(2cos[)(0t

fccdmktttfAts

We next turn to SNRC,FM.

.2/ is )( signal modulated theinpower average 2

cAts

.2

is bandwidth message theinpower noise Average0

0 WNdfNW

W

.2

2/

0

2

0

2

FM,WN

A

WN

ASNR cc

C

)1(21

121

12 .2

. Hence,

. ratio Deviation1.

. increasing increasing , fixed For :

Carson,

2

FM,

FM,

2/1

FM,

FM,

DWD

DWD

fB

DSNR

SNR

W

Pk

W

fD

SNR

SNRBW

T

C

O

f

C

O

TRemarks

.3

2

2

3

2

2

0

2

3

0

22

FM,

FM,

W

Pk

WN

A

WN

PkA

SNR

SNR f

c

fc

C

O

|)(|max tmkff

2.13 Summary

Specifically,

for high carrier-to-noise ratio r (equivalently to the

assumption made in Assumption 1), an increase in

transmission bandwidth BT provides a corresponding

quadratic increase in figure of merit of a FM system.

So, there is a tradeoff between BT and figure of merit.

Notably, figure of merit for an AM system is

nothing to do with BT.

Example 2.5 Single-Tone Modulation

m(t) = Am cos(2fmt)

Then we can represent the figure of merit in terms of

modulation index (or deviation ratio) as (cf. Slide 2-63):

In order to make the figure of metric for an FM system to

be superior to that for an AM system with 100%

modulation, it requires:

.2

3

2

3)2/(332

2

2

2

22

2

2

FM,

FM,

W

f

W

Ak

W

Pk

SNR

SNR mff

C

O

471.03

2

3

1

2

3 2

2.13 Capture Effect

Recall that in Assumption 1, we assume Ac >> r(t).

This somehow hints that the noise suppression of an FM

modulation works well when the noise (or other unwanted

modulated signal that cannot be filtered out by the bandpass or

lowpass filters) is weaker than the desired FM signal.

What if the unwanted FM signal is stronger than the desired FM

signal.

The FM receiver will capture the unwanted FM signal!

What if the unwanted FM signal has nearly equal strength as the

desired FM signal.

The FM receiver will fluctuate back and forth between them!

2.13 FM Threshold Effect

Recall that in Assumption 1, we assume Ac >> r(t)

(equivalently, a high carrier-to-noise ratio) to simplify (t)

so that the next formula holds.

However, a further decrease of carrier-to-noise ratio will

break the FM receiver (from a clicking sound down to a

crackling sound).

As the same as the AM modulation, this is also named the

threshold effect.

.2

33

0

22

FM,WN

PkASNR

fc

O


Consider a simplified case with m(t) = 0 (no message

signal).

)()](cos[)(2

)(')()](sin[)(')](cos[)(')(

)](cos[)(

)](sin[)(tan

)(')(2

22

2

1

trttrAA

ttrttrAtttrA

dt

ttrA

ttrd

ttv

cc

cc

c

To facilitate the understanding of “clicking” sound effect,

we let r(t) = l Ac, a constant ratio of Ac.


2)](cos[21

)](cos[)(')(')(2

ll

ll

t

ttttv

)('1)](cos[21

)](cos[)(')(')(2

2t

t

ttttv

l

l

ll

ll

1but 1 and ,)( say, time,at the Then ll t

Thus a sign change in ’(t) will cause a spike!

Notably, when l = 0 (no noise), the output equals m(t)

= 0 as desired.


2)]sin(cos[21

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

ll

ll

t

ttttvtt

05.1l

05.0l

5l

2.13 Numerical Experiment

ratio. noise carrier to theis 2/

and 2

,2

3

and 2

,4

3

)( provided ,2

3

0

23

2

3

0

22

3

0

22

FM,

NB

ABf

W

B

AkfA

PWN

fA

trAWN

PkASNR

T

cTT

mf

mc

c

fc

O

rr

Take BT/(2W) = 5.

The average output signal power is calculated in the absence of noise.

The average output noise power is calculated when there is no signal present.

2.13 Numerical of the

formula in Slide 2-156

Curve I: The average noise power is calculated assuming an unmodulated carrier.

Curve II: The average noise power is calculated assuming a sinusoidally modulated carrier.

As text mentioned, for r < 11 dB, the output signal-to-noise ratio deviates appreciably from the linear curve.

A true experiment found that occasional clicks are heard at r around 13 dB, only slightly larger than what theory indicates.

The deviation of curve II from curve I shows that the average noise power is indeed dependent on the modulating signal.

2.13 How to avoid “clicking” sound?

For given modulation index (or deviation ratio) and message

signal bandwidth W,

1. Determine BT by Carson’s rule or Universal curve.

2. For a specified noise level N0, select Ac to satisfy:

.202

y,equvalentlor dB 132

log100

2

0

2

10

NB

A

NB

A

T

c

T

c

2.13 Threshold Reduction

After our learning

that FM

modulation has

threshold effect,

the next question

is naturally on

“how to reduce

the threshold?”

2.13 Threshold Reduction

Threshold reduction in FM receivers may be

achieved by

1. negative feedback (commonly referred to as an

FMFB demodulator), or

2. phase-locked loop demodulator.

Why PLL can reduce threshold effect is not covered in

this course. Notably, the PLL system analysis in

Section 2.14 assumes noise-free transmission.

)(tv)(tx

.)(2)( where)],(2cos[)(0t

fccdmktttfAts

.)(2)( where)],(2cos[2)(0t

fvcovcovcovcodmktttfts

)]()1()(2cos[

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

Bandpass

ttffA

ttfttfAtsts

vcocc

vcovcoccvco

).(at centered ,)1( as wideas

passbandsmaller a havely conceptual canfilter bandpass theThus,

vcocTffB

.)1( deviationfrequency new Theoriginalnew

ff

noise. einput whit theas level noise

same the with white treatedbe canoutput Mixer at the noise The

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

ttftwtstw vcovcovco

y.probabilithiger withholds )()()( of condition the

same, theremains and smaller, is )]([)]([ Since

22

22

tntntrA

AtnEtnE

QIc

cQI

Experiments show that an FMFB receiver is capable of realizing a

threshold extension on the order of 5~7 dB.

2.13 Threshold Reduction of an FMFB

receiver

To sum up:

An FMFB demodulator is essentially a tracking

filter that can track only the slowly varying

frequency of a wideband FM signal (i.e., the

message signal part), and consequently it responds

only to a narrowband of noise centered about the

instantaneous carrier frequency.

2.13 Pre-Emphasis and De-emphasis in FM

Recall that the noise PSD at the output shapes like a bowel.

If we can “equalize” the signal-to-noise power ratios over the entire message band, a better noise performance should result.


In order to produce an undistorted version of the original

message at the receiver output, we must have:

This relation guarantees the intactness of the message power.

Next, we need to find Hde(f) such that the noise power is

optimally suppressed.

.for 1)()( WfWfHfHdepe


Under the assumption of high carrier-to-noise ratio, the

noise PSD at the de-emphasis filter output is given by:

otherwise,0

||,|)(|)(|)(|

2

2

2

02 WffH

A

fN

fSfH de

cNde o

W

Wde

c

dffHA

fN 2

2

2

0 |)(| power noise Average


Since the message power remains the same, the

improvement factor of the output signal-to-noise ratio after

and before pre/de-emphasis is:

W

Wde

W

Wde

W

W

W

Wde

c

W

Wc

dffHf

W

dffHf

dff

dffHA

fN

dfA

fN

I22

3

22

2

2

2

2

0

2

2

0

|)(|3

2

|)(||)(|

Example 2.6

Pre-emphasis filter De-emphasis filter

).2/(1 where,)/(1

1)( and )/(1)(

0

0

0Crf

ffjfHffjfH

depe

.||for 12 and Assume WffCrrR <<<<

Example 2.6

)]/(tan)/[(3

)/(

)/(13

2

|)(|3

2

0

1

0

3

0

2

0

2

3

22

3

fWfW

fW

dfff

f

W

dffHf

WI

W

W

W

Wde

With f0 = 2.1 KHz and W = 15 KHz, we obtain I = 22 = 13 dB.

A significant noise performance improvement is therefore obtained.


Final remarks:

The previous example uses a linear pre-emphasis

and de-emphasis filter to improve the noise.

A non-linear pre-emphasis and de-emphasis filters

have been applied successfully to applications like

tape recording. These techniques, known as Dolby-

A, Dolby-B, and DBX systems, use a combination

of filtering and dynamic range compression to

reduce the effects of noise.

2.14 Computer Experiments: Phase-Locked

Loop

Phase-locked loop

.)(2)( where)],(2sin[)(

011

t

fccdmktttfAts

.)(2)( where)],(2cos[)(0

22 t

vcvdvktttfAtr

mk

)],(sin[2

)]()(sin[)]()(4sin[2

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

)()()(

Low P ass

2121

21

tAAk

tttttfAAk

ttfAttfAk

trtskte

e

vcm

c

vcm

cvccm

m

mk

Loop filter = low pass filter + filter h().

).()()( where21

ttte

.)()}({)( Also,LowP ass

dthetv

t

vedvktttt

0121

)(2)()()()(

dthkdt

td

dthekdt

td

tvkdt

td

dt

td

e

v

v

e

)()](sin[2)(

)()}({2)(

)(2)()(

0

1

LowP ass

1

1

.2/ where0 vcvm

AAkkk

duuhukt

dudssuhskdudu

ud

dudu

udt

t

e

t

e

t

te

e

001

00

0

1

0

)(*)](sin[2)(

)()](sin[2)(

)()(

The previous formula suggests an equivalent analytical model for PLL.

When e(t) = 0, the system is said to be in phase-lock.

In this case, 1(t) = 2(t) or equivalently, kvv(t) = kfm(t).

When e(t) is small (< 0.5 radians), the system is said to be in near

phase-locked.

In this case, we can approximate sin[e(t)] by e(t); hence, a linear

approximate model is resulted.

Linearization approximation model for PLL.

We can transform the above time-domain system to its

equivalent frequency domain to facilitate its analysis.

Linearization approximation model for PLL.

)()()(21

1

)()()(2)(

)(

)()]()([

)(

)(

)(

00

0

2211

fHkjf

jf

fGfHk

fGfHfkf

f

fff

f

f

f

ee

e

ee

)(2

f

)(1

f )( fH

)( fG

fjfG

2

1)(

)( fe

2.14 Experiment 0: First-Order PLL

.1)( fH

)/(1

)/(

)(

)(

0

0

1kfj

kfj

f

fe

A parameter k0 controls both the loop gain and bandwidth of the

filter. In other words, it is impossible to adjust the loop gain

without changing the filter bandwidth.

2.14 Experiment 1: Second-Order

Acquisition PLL

2

2

00

2

2

001

)/()/(21

)/(

)()(

)(

))/(1()()(

)(

nn

n

e

fjffjf

fjf

akjfkjf

jf

jfakjf

jf

fHkjf

jf

f

f

.)4/(factor damping and frequency natural where00

akakfn

2.14 Experiment 1: Second-Order

Acquisition PLL

Fast response but require

longer time to stablelize

Slow response but quick

stablelizatoin.

2.14 Experiment 2: Phase-Plain Portrait

model. PLLnonlinear using and )/(1)( jfafH

damping) (critical 2

1

Herz22

50

Take

n

f

Hz.22

50 where),cos(2)(Let

mmmftfAtm

dthtfk

Ak

dthdt

td

kdt

td

k

em

mf

e

e

)()](sin[)2cos(

)()](sin[)(

2

1)(

2

1

0

1

00

Phase-Plane Portrait

)(te

)(2

1

0

tdt

d

ke

Initial value

Remarks on the previous figure

Each curve corresponds to different initial frequency error

(i.e., boundary condition for the differential equation).

The phase-plane portrait is periodic in e(t) with period 2,

but the phase-plane portrait is not periodic in de(t)/dt.

There exists an initial frequency error (such as 2 in the

previous figure) at which a saddle line appears (i.e., the

solid line in the previous figure).

In certain cases, the PLL will ultimately reach an

equilibrium (stable) points at (0,0) or (2,0).

A slightest perturbation to the “saddle line” causes it to shift

to the equilibrium points.

2.15 Summary and Discussion

Four types of AM modulations are introduced (expensive) DSB-C transmitter + (inexpensive) envelope

detector, which is good for applications like radio broadcasting.

(less expensive) DSB-SC transmitter + (more complex) coherent detector, which is good for applications like limited-transmitter-power point-to-point communication.

(less bandwidth) VSB transmitter + coherent detector, which is good for applications like television signals and high speed data.

(minimum transmission power/bandwidth) SSB transmitter + coherent detector, which is perhaps only good for applications whose message signals have an energy gap on zero frequency.


FM modulation, a representative of Angle

Modulation

A nonlinear modulation process

Carson’s rule and universal curve on transmission

bandwidth

2.15 Summary and

Discussion

Noise performance

I. Full AM (DSB-C) with

100 percent modulation

(See Slide 2-5)

II. Coherent DSB-SC &

SSB

III. FM with = 2 and 13

dB pre/de-emphasis

improvement

IV. FM with = 5 and 13

dB pre/de-emphasis

improvement

Threshold effect


Normalized transmission bandwidth in the previous figure

DSB-C, DSB-SC SSB FM(=2) FM(=5)

Bn 2 1 8 16

f

B

fW

fB

W

BB TTT

n

/

/urveUniversalC,

42 2.35

(Refer to Slide 2-85.)


Observations from the figure

1. SSB modulation is optimum in noise performance

and bandwidth conservation in AM family.

2. FM modulation improves the noise performance of

AM family at the expense of an excessive

transmission bandwidth.

3. Curves III an IV indicate that FM modulation

offers the tradeoff between transmission bandwidth

and noise performance.

Analog Modulation” is the subject concerned · Quadrature null effect of the coherent detector....

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Transcript of Analog Modulation” is the subject concerned · Quadrature null effect of the coherent detector....