Digital Communication Systems - Thammasat University 452 - 6 - Digital... · Transmitter Receiver...

Asst. Prof. Dr. Prapun [email protected]

Digital Modulation

1

Digital Communication SystemsECS 452

mailto:[email protected]

Elements of digital commu. sys.

2

No

ise

& In

terf

eren

ce

Information Source

Destination

Channel

Received

Signal

Transmitted

Signal

Message

Recovered Message

Source Encoder

Channel Encoder

DigitalModulator

Source Decoder

Channel Decoder

DigitalDemodulator

Transmitter

Receiver

Remove

redundancy

(compression)

Add

systematic

redundancy to

combat errors

introduced by

the channel

Map digital

sequence to

analog signal

Digital Modulation/Demodulation

3

No

ise

& In

terf

eren

ce

Information Source

Destination

Channel

Received

Signal

Transmitted

Signal

Message

Recovered Message

Source Encoder

Channel Encoder

DigitalModulator

Source Decoder

Channel Decoder

DigitalDemodulator

Transmitter

Receiver

Remove

redundancy

(compression)

Add

systematic

redundancy to

combat errors

introduced by

the channel

Map digital

sequence to

analog signal

101001

101001

𝑆 𝑡

𝑅 𝑡

0 1 2 3 4 5 6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Seconds

-8 -6 -4 -2 0 2 4 6 80

0.5

1

1.5

Frequency [Hz]

Magnitude

Simple ASK: ON-OFF Keying (OOK)

4

fc = 4 Hz

Rs = 1

M = 2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

-0.5

0

0.5

1

Seconds

-200 -150 -100 -50 0 50 100 150 2000

0.5

1

1.5

Frequency [Hz]

Magnitude

1 0 1 0 0 1 t

tfc = 100 Hz

Rs = 20 0111110010011100010110000010010010111010111010000000101110001101111001100000010111110101100011011010

0,1

cos 2𝜋𝑓c𝑡

101001Digital

Modulator?

[ASK_playTones_Demo.m]

𝑇𝑠 2𝑇𝑠 3𝑇𝑠 4𝑇𝑠 5𝑇𝑠

Simple “ASK”: “ON-OFF Keying”

5

M = 2

Smoke signal

ASK: Higher Order Modulation

6

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5

0

5

Seconds

-200 -150 -100 -50 0 50 100 150 2000

2

4

6

8

Frequency [Hz]

Mag

nitu

de

M = 6

4531145240324102302341254141102211155223215150143153135522104022505112411531551045003330455021331444

[ASK_playTones_Demo.m]

0,1,2,3,4,5

FSK

7

0 0.5 1 1.5 2 2.5 3 3.5 4-1

-0.5

0

0.5

1

Seconds

-20 -15 -10 -5 0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Frequency [Hz]

Magnitude

𝑠1 𝑡 = cos 2𝜋𝑓1𝑡

M = 4 1 2 3 4, , , 3,6,9,12 [Hz]cf f f f f

00 01 10 11

𝑠2 𝑡 = cos 2𝜋𝑓2𝑡 𝑠3 𝑡 = cos 2𝜋𝑓3𝑡 𝑠4 𝑡 = cos 2𝜋𝑓4𝑡

[FSK_playTones_Demo.m]

FSK

8

M = 4 1 2 3 4, , , 3,6,9,12 [Hz]cf f f f f

0 1 2 3 4 5 6 7 8 9 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Seconds

-20 -15 -10 -5 0 5 10 15 200

0.5

1

1.5

2

2.5

Frequency [Hz]

Magnitude

[12 12 3 12 9 3 6 9 12 12]𝑓𝑐 = Hz

[11 11 00 11 10 00 01 10 11 11]

11110011100001101111Digital

Modulator?


FSK

9

M = 4

[400 400 100 400 300 100 200 300 400 400]𝑓𝑐 = Hz

[11 11 00 11 10 00 01 10 11 11]

11110011100001101111Digital

Modulator?

1 2 3 4, , , 100, 200,300, 400 [Hz]cf f f f f

0 1 2 3 4 5 6 7 8 9 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Seconds

-800 -600 -400 -200 0 200 400 600 8000

0.5

1

1.5

2

2.5

3

Frequency [Hz]

Magnitude


Gray Code

10

Reflect-and-prefix method for 1-D Gray code construction

Gray code list for 𝑛 bits can be generated recursively from

the list for 𝑛 − 1 bits by

[http://en.wikipedia.org/wiki/Gray_code#mediaviewer/File:Binary-reflected_Gray_code_construction.svg]

reflecting the list (i.e. listing the entries in reverse order),

concatenating the original list with the reversed list,

prefixing the entries in the original list with a binary 0, and then prefixing the entries in the reflected list with a binary 1.

0

110

00

011110

0

1

00011110

00

011110

000

001011010

100101111110

n=2n=3

n=1

Asst. Prof. Dr. Prapun [email protected]

Review: Fourier Transform

11

Digital Communication SystemsECS 452

mailto:[email protected]

7 Equations

12

that changed the world

… and still rule everyday

life

Important Formulas

14

2

2

2

cos sin

2cos 1 cos 2

2sin 1 cos 2

1 1cos 2

2 2

j ft

j j

c c c

je j

x x

x

G f g t e dt

f t f f e

x

f f e

0

0

2

0

2

0

1 1cos 2

2 2

j ft

j f t

c c c

g t t e G f

e g t G f f

g t f t G f f G f f

Spectrum of ON-OFF Keying

15

fc = 100 Hz

Rs = 1

0 1 2 3 4 5 6-1

-0.5

0

0.5

1

Seconds

-200 -150 -100 -50 0 50 100 150 2000

0.5

1

1.5

Frequency [Hz]

Magnitude

1 0 1 0 0 1M = 2

Frequency-Domain Analysis

16

Modulation:

Shifting Properties: 02

0

j ftg t t e G f

02

0

j f te g t G f f

1 1

cos 22 2

c c cg t f t G f f G f f

Cos vs. Cos Pulse

17

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1

Seconds

-200 -150 -100 -50 0 50 100 150 2000

0.1

0.2

0.3

0.4

0.5

Frequency [Hz]

Magnitude

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1

Seconds

-200 -150 -100 -50 0 50 100 150 2000

0.02

0.04

0.06

0.08

0.1

Frequency [Hz]

Magnitude

𝑥 𝑡 = cos 2𝜋 100 𝑡 𝑥 𝑡 = cos 2𝜋 100 𝑡 , 0.4 ≤ 𝑡 ≤ 0.6,

0, otherwise.

0 1 2 3 4 5 6-1

-0.5

0

0.5

1

Seconds

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.5

1

1.5

2

Frequency [Hz]

Magnitude


18

fc = 5 Hz

Rs = 1

M = 2

1 0 1 0 0 1

0 1 2 3 4 5 6-1

-0.5

0

0.5

1

Seconds

-200 -150 -100 -50 0 50 100 150 2000

0.5

1

1.5

Frequency [Hz]

Magnitude


19

fc = 100 Hz

Rs = 1

M = 2

1 0 1 0 0 1

0 1 2 3 4 5 6-1

-0.5

0

0.5

1

Seconds

-200 -150 -100 -50 0 50 100 150 2000

0.5

1

1.5

Frequency [Hz]

Magnitude


20

fc = 100 Hz

Rs = 1

M = 2

1 0 1 0 0 10 1 2 3 4 5 6

-1

-0.5

0

0.5

1

Seconds

95 96 97 98 99 100 101 102 103 104 105

0.5

1

1.5

Frequency [Hz]

Magnitude


21

fc = 100 Hz

Rs = 20

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

-0.5

0

0.5

1

Seconds

-200 -150 -100 -50 0 50 100 150 2000

0.5

1

1.5

Frequency [Hz]

Magnitude

M = 2

Five Frequencies

22

0 0.05 0.1 0.15 0.2 0.25-1

-0.5

0

0.5

1

Seconds

-6000 -4000 -2000 0 2000 4000 60000

0.01

0.02

0.03

Frequency [Hz]

Magnitude

cos 2𝜋𝑓1𝑡 cos 2𝜋𝑓2𝑡 cos 2𝜋𝑓5𝑡cos 2𝜋𝑓4𝑡cos 2𝜋𝑓3𝑡

Rate = Rs frequency-changes per second

Each tone lasts

1/Rs sec.

Spectrum of Five Frequencies (1/5)

23


300 Hz100 Hz 200 Hz 500 Hz400 Hz

0 1 2 3 4 5 6 7 8 9 10-1

-0.5

0

0.5

1

Seconds

-1000 -800 -600 -400 -200 0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

Frequency [Hz]

Mag

nitu

de

Rs = 0.5


24


300 Hz100 Hz 200 Hz 500 Hz400 Hz

Rs = 50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1

-0.5

0

0.5

1

Seconds

-1000 -800 -600 -400 -200 0 200 400 600 800 10000

0.02

0.04

0.06

0.08

0.1

Frequency [Hz]

Mag

nitu

de


25


300 Hz100 Hz 200 Hz 500 Hz400 Hz

Rs = 200 0.05 0.1 0.15 0.2 0.25

-1

-0.5

0

0.5

1

Seconds

-1000 -800 -600 -400 -200 0 200 400 600 800 10000

0.01

0.02

0.03

Frequency [Hz]

Mag

nitu

de


26

Rs = 500 0.02 0.04 0.06 0.08 0.1 0.12

-1

-0.5

0

0.5

1

Seconds

-1000 -800 -600 -400 -200 0 200 400 600 800 10000

0.005

0.01

0.015

Frequency [Hz]

Mag

nitu

de

300 Hz100 Hz 200 Hz 500 Hz400 Hz


27


300 Hz100 Hz 200 Hz 500 Hz400 Hz

0 1 2 3 4 5 6 7 8 9 10-1

-0.5

0

0.5

1

Seconds

-1000 -800 -600 -400 -200 0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

Frequency [Hz]

Mag

nitu

de

Rs = 0.5

0 1 2 3 4 5 6 7 8 9 10-1

-0.5

0

0.5

1

Seconds

85 90 95 100 105 110 115

0

0.2

0.4

0.6

0.8

Frequency [Hz]M

agnitude

Spectrum of FSK (1/2)

28

Freq. = [400 300 400 400 100 300 200 100 100 100] Hz

0 1 2 3 4 5 6 7 8 9 10-1

-0.5

0

0.5

1

Seconds

-800 -600 -400 -200 0 200 400 600 8000

0.5

1

1.5

2

Frequency [Hz]

Mag

nitu

de

Rs = 1

M = 4

Spectrum of FSK (2/2)

29

Freq. = [100 400 500 500 100 300 300 400 400 400 400 200 300 100 100 500 200 300 500 100 100 200 500 200 200 100 200 200 100 300 100 400 100 300 400 200 300 300 100 300 400 200 500 500 500 300 200 400 200 500] Hz

Rs = 50 1 2 3 4 5 6 7 8 9 10

-1

-0.5

0

0.5

1

Seconds

-1000 -800 -600 -400 -200 0 200 400 600 800 10000

0.5

1

1.5

Frequency [Hz]

Mag

nitu

de

M = 5

Frequency-Domain Analysis

30

Modulation:

Shifting Properties: 02

0

j ftg t t e G f

02

0

j f te g t G f f

1 1

cos 22 2


DSB-SC

31

𝑚 𝑡

𝑀 𝑓

𝑣 𝑡

𝑉 𝑓

𝑥 𝑡

𝑋 𝑓

𝑚 𝑡

𝑀 𝑓

0 5 10 15 20 25-1

-0.5

0

0.5

1

Seconds

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

x 104

0

0.05

0.1

0.15

0.2

Frequency [Hz]

Magnitude

0 5 10 15 20 25-2

-1

0

1

2

Seconds

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

x 104

0

0.05

0.1

0.15

0.2

Frequency [Hz]

Magnitude

0 5 10 15 20 25-2

-1

0

1

2

Seconds

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

x 104

0

0.05

0.1

0.15

0.2

Frequency [Hz]

Magnitude

0 5 10 15 20 25-2

-1

0

1

2

Seconds

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

x 104

0

0.05

0.1

0.15

0.2

Frequency [Hz]

Magnitude

𝐴

𝐴/ 2

𝐴

𝐴/2

𝐴

1 1

2 cos 22 2


DSB-SC (Zoomed in time)

32

1 1.0005 1.001 1.0015 1.002 1.0025 1.003 1.0035 1.004 1.0045 1.005-1

-0.5

0

0.5

1

Seconds

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

x 104

0

0.05

0.1

0.15

0.2

Frequency [Hz]

Magnitude

1 1.0005 1.001 1.0015 1.002 1.0025 1.003 1.0035 1.004 1.0045 1.005-1

-0.5

0

0.5

1

Seconds

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

x 104

0

0.05

0.1

0.15

0.2

Frequency [Hz]

Magnitude

1 1.0005 1.001 1.0015 1.002 1.0025 1.003 1.0035 1.004 1.0045 1.005-2

-1

0

1

Seconds

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

x 104

0

0.05

0.1

0.15

0.2

Frequency [Hz]

Magnitude

1 1.0005 1.001 1.0015 1.002 1.0025 1.003 1.0035 1.004 1.0045 1.005-1

-0.5

0

0.5

1

Seconds

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

x 104

0

0.05

0.1

0.15

0.2

Frequency [Hz]

Magnitude

𝑚 𝑡

𝑀 𝑓

𝑣 𝑡

𝑉 𝑓

𝑥 𝑡

𝑋 𝑓

𝑚 𝑡

𝑀 𝑓

Note how the high-frequency

content is riding on top of the

original baseband signal.

Note how the baseband signal

𝑚 𝑡 becomes the envelope of

the modulated signal x 𝑡 .

Binary PAM and Its Spectrum (1/4)

33

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

2

4

6

8

10

f [Hz]

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

0.2

0.4

0.6

0.8

1

f [Hz]

1

𝑇𝑠=1t

t

1

-1

𝐴= [-1,-1,1,-1,-1,1,1,-1,-1,-1,1,-1,-1,1,-1,1,1,-1,-1,-1,-1,1,-1,-1,-1,-1,-1,1,-1,1]

s t

Can you sketch the spectrum of s(t)?

1 0, sp t t T

P f


34

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

2

4

6

8

10

f [Hz]

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

0.2

0.4

0.6

0.8

1

f [Hz]

1

𝑇𝑠=1t

t

1

-1

𝐴= [-1,-1,1,-1,-1,1,1,-1,-1,-1,1,-1,-1,1,-1,1,1,-1,-1,-1,-1,1,-1,-1,-1,-1,-1,1,-1,1]

s t

1 0, sp t t T

P f

1

0

n

k s

k

s t A p t kT

This is also the spectrum of

for any .sp t kT k


35

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

2

4

6

8

10

f [Hz]

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

0.2

0.4

0.6

0.8

1

f [Hz]

1

𝑇𝑠=1t

t

1

-1

𝐴= [-1,-1,1,-1,-1,1,1,-1,-1,-1,1,-1,-1,1,-1,1,1,-1,-1,-1,-1,1,-1,-1,-1,-1,-1,1,-1,1]

s t

1 0, sp t t T

P f

1

0

n

k s

k

s t A p t kT


for any .sp t kT k

1

2

0

s

nj fkT

k

k

S f P f A e

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

2

4

6

8

10

f [Hz]

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

0.2

0.4

0.6

0.8

1

f [Hz]


36

1

𝑇𝑠=1t

t

1

-1

𝐴= [-1,-1,1,-1,-1,1,1,-1,-1,-1,1,-1,-1,1,-1,1,1,-1,-1,-1,-1,1,-1,-1,-1,-1,-1,1,-1,1]

s t

1 0, sp t t T

P f


for any .sp t kT k

1 1

2

0 0

s

n nj fkT

k s k

k k

s t A c t kT S f P f A e

S f

GSOP

37

𝑢(1) = 𝑣(1)

𝑒(1)

GSOP

38

𝑢(1)

𝑣(2)

GSOP

39

𝑢(1)

proj𝑢(1) 𝑣(2)

𝑣(2)

𝑢(2) = 𝑣(2) − proj𝑢(1) 𝑣(2)

GSOP

40

𝑢(1)

𝑢(2)

𝑒(1)

𝑒(2)

GSOP

41

𝑢(1)

𝑢(2)

𝑒(1)

𝑒(2)

GSOP

42

𝑢(1)

𝑢(2)

𝑒(1)

𝑒(2)

GSOP

43

𝑢(1)

𝑢(2)

𝑣(3)

GSOP

44

𝑢(1)

𝑢(2)

𝑣(3)

projspan 𝑢(1),𝑢(2) 𝑣(3)

GSOP

45

𝑢(1)

𝑢(2)

𝑣(3)

projspan 𝑢(1),𝑢(2) 𝑣(3)

𝑢(3) = 𝑣(3) − projspan 𝑢(1),𝑢(2) 𝑣(3)

GSOP

46

𝑢(1)

𝑢(2)

𝑣(3)

proj𝑢(1) 𝑣(3)

proj𝑢(2) 𝑣(3) projspan 𝑢(1),𝑢(2) 𝑣(3)=proj𝑢(1) 𝑣(3) + proj𝑢(2) 𝑣(3)

GSOP

47

proj𝑢(1) 𝑣(3)𝑢(1)

𝑢(2)

𝑣(3)

proj𝑢(2) 𝑣(3) projspan 𝑢(1),𝑢(2) 𝑣(3)=proj𝑢(1) 𝑣(3) + proj𝑢(2) 𝑣(3)

𝑢(3)

𝑢(3) = 𝑣(3) − proj𝑢(1) 𝑣(3) + proj𝑢(2) 𝑣(3)

GSOP

48

𝑢(1)

𝑢(2)𝑢(3)

GSOP

49

𝑢(1)

𝑢(2)𝑢(3)

𝑒(1)

𝑒(2) 𝑒(3)

Modulator and Waveform Channel

50

Waveform Channel:

Transmission of the message 𝑊 = 𝑖 is done by inputting the

corresponding waveform 𝑠𝑖 𝑡 into the channel.

Goal: Want to transmit the message (index) W {1, 2, 3, …, M}

Prior Probabilities: ip P W i

R t S t N t

Received waveformTransmitted waveform

Additive White Noise

(Independent of S(t))

M-ary Scheme

M possible messages requires

M possibilities for S(t):

1 2, , , Ms t s t s t

Prior Probabilities: i ip P W i P S t s t

M =2: Binary

M = 3: Ternary

M = 4: QuaternaryDigital

Modulator

N t

R t S tW

Digital

DeModulator W

Energy: ,i i is tE s t 2

1

logM

i i

i

s bp EEE M

Digital Communication Systems - Thammasat University 452 - 6 - Digital... · Transmitter Receiver...

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Transcript of Digital Communication Systems - Thammasat University 452 - 6 - Digital... · Transmitter Receiver...