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Lecture 8Basic Modulation Techniques (IV)

Fall 2008

NCTU EE

Tzu-Hsien Sang

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Outlines

• Linear Modulation

• Angle Modulation

• Interference

• Feedback Demodulators

• Analog Pulse Modulation

• Delta Modulation and PCM

• Multiplexing

2

3

FM (Single Tone Wide-band Modulation)

0

sin

sin sin

00

Let ( ) sin .

Since it is a periodic signal, so is the FM signal.

Use Fourier series to represent the signal.

( ) cos( sin ) Re{ }

1

2

c m

m m m

m

j t j tc C c m C

T j t jn t j tmn

t t

x t A t t A e e

C e e dt e eT

( sin )

sin

1( )

2

( )

( ) Re{ ( ) } ( )cos[( ) ]

( ) ( ) ( ) ( ) ( ).

mm

m

m m

c m

jn t

j nx xn

j t jn tn

n

j t jn tc C n C n c m

n n

c n m c n m cn n

dt

e dx J

e J e

x t A e J e A J n t

X f J n J n

4

22 2 2 2 2

22 2 2 2

Power of an FM signal.

Can you guess the answer without referring to equations?

1 1( ) [ ( ) cos( ) ] ( ) .

2 2

Or,

1 1( ) ( cos( sin )) cos 2( sin ) .

2 2 2

c C n c m C n Cn n

Cc C c m C c m C

x t A J w nw t A J A

Ax t A w t w t A w t w t A

Bandwidth

When is small (narrowband FM), ( ) 0, for 2.

When is large (wideband FM), find the significant coeffients that contain

most of the signal power.

For example, a 98% power bandwidth 2(

nJ n

1) for the single tone.

1 ( )max

Peak frequency deviation 2 In general, define ,

BW of ( )

then the bandwidth 2( 1) .

m

t

f

d tdt

Dm t W

D W

5

6

• Indirect FM

• Direct FM

Voltage-Controlled Oscillator

7

Demodulation of Angle-Modulated Signals

• Ideal frequency discriminator: a device that yields an output proportional to the frequency deviation of the input.

( ) cos( ( )).

1 ( )( ) .

2

r c c

D D

x t A t t

d ty t K

dt

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• Approximation:

( ) ( )( ) ( )sin( ( )).

( )( ) output of envelope detector 2 ( ).

rC c c

D c C d

dx t d te t A t t

dt dtd t

y t A A f m tdt

To reduce the channel noise effect, an amplitude limiter and a BPF are placed before the differentiator --- a band-pass limiter.

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• Implementation examples: To approximate the transfer function of an ideal differentiator.

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Interference

• Why study interference? The short answer: we don’t have the ability to study noise yet. Let’s start with something simpler.

• Interference means unwanted signals present in the signal path.

• Recall the possible causes of interference.

• Deterministic interference can be analyzed without using stochastic tools.

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Interference in Linear ModulationAgain, we don't have the ability to deal with general-case signals.

Instead, a simple sinusoid message is assumed.

: cos

: cos( )

The received signal:

( ) cos c

mm

c

i c i

r C c m

AMessage t

A

Interference A t

x t A t A

os cos cos( )m c i c it t A t

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1

1. Coherent Detection (linear)

( ) cos cos

2. Envelope Detection (non-linear)

1 1( ) Re{ [ ]}

2 2cos cos cos cos( )

[ cos cos ] co

c m m

D m m i i

j t j t j tj tr C i m m

C c m m c i c i

C m m i i

y t A t A t

x t e A Ae A e A e

A t A t t A t

A A t A t

s sin sinc i i ct A t t

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Case (i) If (typical case)

( ) [ cos cos ] cos

Envelope of ( ) cos cos

( ) cos cos (same as the coherent detector)

Case (ii) If

( ) cos(

C i

r C m m i i c

r C m m i i

D m m i i

C i

r C c i

A A

x t A A t A t t

x t A A t A t

y t A t A t

A A

x t A

) cos cos( ) cos( )

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

cos [cos( ) cos sin( ) sin ]

[ cos cos cos

i m m c i i i c i

C c i i c i i i c i

m m c i i c i i

i C i m m i

t A t t A w t

A t t t t A t

A t t t t t

A A t A t t

] cos( )

[ sin cos sin ] sin( )

[ cos cos cos ] cos( )

Envelope of ( ) cos cos cos

( ) cos cos cos

c i

C i m m i c i

i C i m m i c i

r i C i m m i

D C i m m i

t

A t A t t t

A A t A t t t

x t A A t A t t

y t A t A t t

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• The threshold effect: when the interference is greater than a certain level, the message is totally lost. It’s a non-linear phenomenon.

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Interference in Angle Modulation

We are doing wrose here. No signal is assumed.

Assuming an interference at .

( ) cos( ( )) cos( )

cos cos cos sin sin

[ cos ]cos [ sin

c i

r C c i c i

C c i i c i i c

c i i c i

x t A t t A t

A t A t t A t t

A A t t A

2 2

1

]sin

( ) cos[ ( )]

( ) ( cos ) ( sin )where .sin

( ) tan ( )cos

i c

c

C i i i i

i i

C i i

t t

R t t t

R t A A t A t

A tt

A A t

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1

Case (i) If (typical case) and assume an ideal discriminator.

( ) cos

sin sin( ) tan ( )

For PM: ( ) sin

1For FM: ( ) ( sin ) cos

2

C i

C i i

i i i i

C C

iD D i

C

i iD D i D i

C C

A A

R t A A t

A t A tt

A A

Ay t K t

A

A Ady t K t K f

dt A A

In summary,

1AM: cos

PM: sin .

FM: cos

i

ii

C

iD i

C

iD i i

C

t

A ta A

AK t

A

AK f t

A

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i

Case (ii) If (very difficult to analyze)

Make a rough analysis based on phasor diagram.

( ) cos cos( )

Re{[ ] }

Interference phase (t)= . Resultant phase (

i c

C i

r C c i c i

j t j tC i

A A

x t A t A t

A Ae e

t

t).

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