signals and systems lecture 14

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Signals and SystemsLecture 14

•Multiplication Property•Communication System

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Chapter 4 Fourier Transform

§4.5 The Multiplication property (modulation property)

（调制特性）

r t s t p t 1

2R j S j P j

0 F0 j ts t e S j

F0 0 0

1 1cos

2 2x t t X j X j

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Example 4.23

Determine the Fourier transform of the signal

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sin sin / 2t tx t

t

2/1 jX

2

1

2

3

2

3

2

1 0

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例某连续时间 LTI 系统的单位冲激响应为

若输入信号，

试求整个系统的输出。

sincos6

th t t

t

k

x t t k

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1h t 2h t x t

cos 4 t cos 6 t

A B C

例如图所示系统中， 1

sin 4

t

h t tt

2

sin,

d th t

dt t

若输入信号 sin 2

t

x tt，试分别 A、 B、 C各点信号

的频谱。

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§4.5.1 Frequency-Selective Filtering with Variable Center Frequency

中心频率可调的频率选择性滤波器

tytj ce

txtj ce

tf

0

1

WW

jH tr

jX

0

jY

0 c

jH

WW

0 WW

jR

W2

c

0 W2

jF

c

jG

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Chapter 8 Communication Systems

Communication Channel信道

SourceSignals

Modulator 调制器

Modulated Signals

Be best suited for transmitting

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Modulation : The general process of embedding an information

-bearing signal into a second signal.

Extracting the information-bearing signal.Demodulation :

Modulation :

Amplitude Modulation (AM)

Frequency Modulation (FM)

Phase Modulation (PM)

ModulatingSignal

Carrier Signal

ModulatedSignal

tctxty tx

tc

Amplitude Modulation

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Modulating system model:

x(t)

c(t)

y(t)=x(t)c(t)

x(t) --- modulating signal

c(t) --- Carrier signal


§8.1 Complex Exponential and Sinusoidal Amplitude Modulation

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8.1.1 Amplitude Modulation with Complex Exponential Carrier

Exponential carrier:

For convenience, let c=0, so

)()( cctjetc tj cetc )(

Output signal(modulated signal):tj cetxty )()(

)()(

)(2)(

)(*)(2

1)(

c

c

jjXjY

jC

jCjXjY

(1) Modulation Theory


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12

ttjxttx

etxty

cc

tj c

sin)(cos)(

)()(

(1) Implementation


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8.1.2 Amplitude Modulation with Sinusoidal signal

For convenience, choose c=0, so

ttxtctxty ccos)()()()(

)(2

1)(

2

1)(

)()()(

)(*)(2

1)(

cc

cc

jjXjjXjY

jC

jCjXjY


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§8.1 Complex Exponential and Sinusoidal Amplitude Modulation

1. cj tc t e

2. cos cc t t

t

tx

t

ty Envelope

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8.2 Demodulation for Sinusoidal AM

8.2.1 Synchronous demodulation

(1) Demodulation process


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)2(4

1)2(

4

1)(

2

1)( cc jjXjjXjXjW

In frequency domain:

ttxtx

ttx

tctytw

c

c

2cos)(2

1)(

2

1

cos)(

)()()(2

In time domain:

Expected signal: )()( jWortx


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(2) Synchronous problem


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Time domain:

)2cos()(2

1)cos()(

2

1

)cos()cos()(

)cos()()(

ccccc

cccc

cc

ttxtx

tttx

ttytw

The output of lowpass filter: )cos()( cctx

Ideal output: x(t) )( desirediscc

When , it is referred to as synchronous

demodulation.cc


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8.2.2 Asynchronous demodulation

Amplitude-modulated signal:

tAtxty ccos])([)(


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1. Suppose the modulator and demodulator are not synchronized in phase.

① Complex Exponential Carrier

§8.2.1 Synchronous Demodulation

c cj te

tx ty CommunicationChannel

Transmitter Modulation

ReceiverSynchronous Demodulation

LPF tx tw

dj te

2. Suppose the modulator and demodulator are not synchronized in frequency.

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② Sinusoidal Carrier

cct cos

tx ty CommunicationChannel

Transmitter Modulation

ReceiverSynchronous Demodulation

LPF tx tw

cos dt

1. Suppose the modulator and demodulator are not synchronized in phase.

2. Suppose the modulator and demodulator are not synchronized in frequency.

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§8.3 Frequency-Division Multiplexing（频分复用）


t1cos tx1

t2cos tx2

ty1BPF

1

ty2BPF

2

tncos

txn tynBPF

ty

n

jX1

M 0 M

M 0 M

jX 2

M 0 M

jX n

22 11 0

jY

nn

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Quadrature Multiplexing（正交多路复用）

tccos tx1

tcsin

tx2

tr

tcsin

tccos

LPF

LPF

ty1

ty2

M

MCommunication

Channel tr tw1

tw2

txty 11

txty 22

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§8.4 Single-Sideband Sinusoidal Amplitude Modulation

（单边带）

M 0

jX

Mtccos

tx ty

c 0

jY

cM2

M2

Uppersideband

Uppersideband

Lower sideband

Double Sideband Modulation (DSB)

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Retain the upper sideband

c 0 cM

Uppersideband

Uppersideband

M

c 0 c

Lower sideband

MM

Retain the lower sideband

Single Sideband Modulation (SSB)

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M 0

jX

M M 0

jX p

M

j

j

Figure 8.21 SSB system using a 90o phase-shift network.

0 j

0 j-

jH

tx ty

tcsin

ty2

ty1

txp

90o phase-shift

cos ct

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cpcp jjXj

jjXj

jY 2

1

2

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ttxty cp sin2

M 0

jX p

M

j

j

c 0

jY1

c

2/1

c cLower Sideband

jY

2/1

2/1 jY2

c c

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Summary


Multiplication Property and Communication Systems

AM and DSB、 SSB System

Modulation and Demodulation

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Readlist

• Signals and Systems:– 5.1, 5.2

• Question: – DTFT

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Homework:

8.1 8.3 8.22 8.28

signals and systems lecture 14

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