Principles of Communications - 4...In the time domain… 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2-1 0 1 2 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-4-2 0 2 4 I P 2 I P I P H2cos2 B Ö P L T P 2

1

Principles of CommunicationsECS 332

Asst. Prof. Dr. Prapun [email protected]

4. Amplitude Modulation

Office Hours: BKD, 4th floor of Sirindhralai building

Monday 9:30-10:30Monday 14:00-16:00Thursday 16:00-17:00

DSB-SC

2

× ×Channel

2 cos 2 cf t

y

2 cos 2 cf t

vLPF

Modulator Demodulator

Message(modulating signal)

22

2 cos 2 2 cos 2c cx t

v t

m t f t f t

LPF m tKey equation:

In the time domain…

3

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

-1

0

1

2

3

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

-2

0

2

4

2

2cos 2

2

2cos 2

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

0

5

Seconds

Note the oscillation at twice the carrier frequency

In the time domain…

4

2

2cos 2

2

2cos 2

When the sampling rate is not fast enough,…

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

0

5

Seconds

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

-2

0

2

4

Seconds

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

-1

0

1

2

3

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

-2

-1

0

1

2

3

4

5

The problem with sampling rate

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

0

5

Seconds

This is the plot of when we don’t connect the dots

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

-2

-1

0

1

2

3

4

5

The problem with sampling rate

6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5

0

5

Seconds

DSB-SC

7

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]

Mag

nitu

de

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]

Mag

nitu

de0 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]

Mag

nitu

de

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]

Mag

nitu

de

/ 2

/2

[Demo_DSBSC_Sound_ReadWAV.m]

DSB-SC (Zoomed in time)

8

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]

Mag

nitu

de

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]

Mag

nitu

de1 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]

Mag

nitu

de

Note how the baseband signal becomes the envelope of

the modulated signal x .

Note the delay caused by the LPF.

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

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]

Mag

nitu

de

Fourier Series: Ex 1

9

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

ECS332_4_Amplitude_Modulation_Fourier_Ex1.fig


10

-1-0.8

-0.6-0.4

-0.20

0.20.4

0.60.8

1 -0.8-0.6

-0.4-0.2

00.2

0.40.6

0.81

1.2

0

2

4

6

8

10

12

14

16

18

20

inde

x

t

Fourier Series: Ex 1 (interactive)

11 ECS332_4_Amplitude_Modulation_Fourier_Ex1.jar [http://www.tomasboril.cz/hobbies_programs_en.html]


12

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

t


13

0 2 4 6 8 10 12-1.5

-1

-0.5

0

0.5

1

1.5

t

ECS332_4_Amplitude_Modulation_Fourier_Ex2.fig


14

0

2

4

6

8

10

12-1.5

-1

-0.5

0

0.5

1

1.5

0

2

4

6

8

10

12

14

16

18

20

inde

x

t


15


16 [http://codepen.io/anon/pen/jPGJMK/]

Fourier Series visualization

17 [http://bl.ocks.org/jinroh/7524988]

Fourier Series: Drawing

18

The same technique, but now tracing the whole trajectory and not just the vertical displacement, can be used to draw “anything”.

[https://www.youtube.com/watch?v=QVuU2YCwHjw&t=1m]

Fourier Series: Drawing

19[http://devpost.com/software/draw-anything]

Draw Anything is an iOS app that harnesses the computational power of the Wolfram Programming Cloud to automatically create step-by-step drawing guides.


20

0 2-

1

1/1/

sincsin 1

sin

sin 2 sinc 2

-3 113 2

2 13

-2 0 0

-1 -11

1 2

2

0 0 1

1 11

1 2

2

2 0 0

3 -11

3 2

2 13

2

2 13

2 15


21

0

1

1/1/

2

2 13

2 15

2

21


22

0

1/2

1

1 13

1 15

2

Note that this is the scaled Fourier transform of the restricted (one period) version of your signal.


23

0

1/2

1

1 13

1 15

2

These “lines” are collectively referred to as the (two-sided) line spectrum of the periodic signal.

Usually, you will get complex numbers and hence the spectrum is represented by two plots: the amplitude (magnitude) and the phase.

Here, we “happen” to have all of the Fourier coeff. being real-valued. So, one plot is OK.


24

0

1/2

1

1 13

1 15

2

Simply changing them to “arrows”. Collectively, they are now the Fourier transform of your periodic signal.

Effect of Duty Cycle

25


26

Note that it is not always the case that the 2nd

harmonic (along with its muliples) is suppressed.

Duty cycle = 0.070


27

Duty cycle = 0.125

When duty cycle = 1/8, the 8th harmonic (along with its muliples) is suppressed.


28

When duty cycle = 1/5, the 5th harmonic (along with its muliples) is suppressed.

Duty cycle = 0.203


29

When duty cycle = 1/3, the 3rd harmonic (along with its muliples) is suppressed.

Duty cycle = 0.336


30

When duty cycle = 1/2, the 2nd harmonic (along with its muliples) is suppressed.

Duty cycle = 0.500

Square Wave

31

1

44

0

1/2

1

1 13

1 15

2

Fourier series expansion:

: the scaled Fourier

transform of the restricted (one period) version of .

12

1 13

15 ⋯

1 13

15 ⋯

period

Fundamental frequency = 1/T0

Square Wave

32

1

44Fourier series expansion:

These “lines” are collectively referred to as the (two-sided) line spectrum of the periodic signal.

0

1/2

1

1 13

1 15

2

12

1 13

15 ⋯

1 13

15 ⋯

Square Wave

33

1

44Fourier series expansion:12

1 13

15 ⋯

1 13

15 ⋯

Simply changing them to “arrows” (representing the delta functions). Collectively, they are now the Fourier transform of your periodic signal.

0

1/2

1

1 13

1 15

2

Square Wave

34

1


1 13

15 ⋯

1 13

15 ⋯

12

1 13 3

15 5 ⋯

1 1

3 315 5 ⋯

Square Wave

35

1


1 13

15 ⋯

1 13

15 ⋯

12

2cos 2

23 cos 2 3

25 cos 2 5 ⋯

Trigonometric Fourier series expansion: 2cos

Square Wave

36

1

44

Compact expression based on the cosine function:

Trigonometric Fourier series expansion:

1 cos 2 0 1, cos 2 0,0, otherwise.

12

2cos 2

23 cos 2 3

25 cos 2 5 ⋯

Switching Operation

37

44

OFF ON OFF ON OFF ON OFF ON OFF ON OFF

Switching Operation

38

1

44

OFF ON OFF ON OFF ON OFF ON OFF ON OFF

Multiplying a signal by the square-wave is equivalent to switching on (for half a period) and off periodically.

Switching Modulator

39

12

2cos 2

23 cos 2 3

25 cos 2 5 ⋯

12

2cos 2

23 cos 2 3

25 cos 2 5 ⋯

0

2

53

355 3

Set =

Switching Modulator

40

0

2

53

355 3

BPF2

cos 2

12

2cos 2

23 cos 2 3

25 cos 2 5 ⋯

12

2cos 2

23 cos 2 3

25 cos 2 5 ⋯

Switching Demodulator

41

12

2cos 2

23 cos 2 3

25 cos 2 5 ⋯

LPFcos 2

12

2cos 2

23 cos 2 3

25 cos 2 5 ⋯


42

1 2 2 2cos 2 cos 2 3 cos 2 52 3 512

2 cos 2

2 cos 2 332

cos 2

cos 2

cos 2

co cos 55

s 2 2

c c c

c

c

c c

c c

c c

c cc

y t y t y t y t y t

A m t f t

A

f t f t f t

f t

f t

m t f t

A m t f t

A m t

r

f

t

f tt

1 cos 2 2

cos 2 2 cos 2 4

c

cos 212

1

131

5os 2 4 cos 2 6

c c

c

c c c

cc

c

c

f t

f t

A m t f t

A m t

A m t

A m t

f t

f t f t

12

1 1 cos 2 2

cos 2 2 cos 2 4

co

c

s 2 4 cos 2 6

1 13 31 1

5

2

5

osc c

c c

c c

c cc

c

c c

c

f t

f t

A m t f t

A m t A m t

A m t A m t

A m

f t

f t f tt A m t

cos cos12 cos

12 cos


43

cos 2

1 2 2 2cos 2 cos 2 3 cos 2 52 3 512

2 cos 2

2 cos 2 332

cos 2

cos 2

cos 2

co cos 55

s 2 2

c c c

c

c

c c

c c

c c

c cc

y t y t y t y t y t

A m t f t

A

f t f t f t

f t

f t

m t f t

A m t f t

A m t

r

f

t

f tt

Now, recall that cos cos cos cos


44

1 cos 2 2

1 co

12

s 2 cos 2 5

1 cos 2 3 cos 2

cos

57

2

1

3

c

c c

c c

c c

c

c

c

r t

f t

f

y t A m t f t

A m t

A m t

A m t

t f t

f t f t

cos 2


45

cos 2 2

1 1cos 2 2 cos 2 4

1 1co

12

1 1

3 3

5s 2 4 cos 2 6

5

cos 2c c

c c

c c

c c

c

c c

c c

r t

f t

f

y t A m t f t

A m t A m t

A m t A mt f t

f t

t

A m t A m tt f

cos 2


46

LPFcos 2

cos 2 2

cos 2 2 cos 2 4

cos

12

1 1

1 13

2 4

31 1 cos 2 6

5 5

cos 2c c

c c

c c

c c

c

c c

c c

r t

f t

f

y t A m t f t

A m t A m t

A m t A mt f t

f t

t

A m t A m tt f

Part A

47

outL

+vin-+vin-

240 V

[Slides from basic EE lab]

Part A: Half-Wave Rectifier (HWR)

48

A rectifier is an electrical device that converts alternating current (AC) to direct current (DC).

240 V


Part A: Half-Wave Rectifier (HWR)

49

=


Part A: Full-Wave Rectifier (FWR)

50

T1

220 V50 Hz

A

B

C

D

+

_Vout

D11N4001

D21N4001

RL10 k

S1+vin-+vin-

S2

240 V


Part A: Full-Wave Rectifier (FWR)

51

=


T1

240 V50 Hz +

_

Vout

D11N4001

D21N4001

A

B

C R110k

C1100 F50 V

+

_

D

Part B: Filter Capacitor

52

240 V

ripple waveform


53


Problem with the angle

54

Not as easy as it looks

Problem with the angle

55

atan: Inverse tangent (arctangent)

56

atan

[rad

ians

]

Return values in the interval [-/2,/2]. not (-,]

>> atan(1/1)*180/pians =

45>> atan((-1)/(-1))*180/pians =

45>> atan(-1/1)*180/pians =

-45>> atan(1/-1)*180/pians =

-45

Want this to be -135.

Want this to be 135.

x

y

atan2: Four-quadrant inverse tangent

57

>> atan(1/1)*180/pians =

45>> atan((-1)/(-1))*180/pians =

45>> atan(-1/1)*180/pians =

-45>> atan(1/-1)*180/pians =

-45

Want this to be -135.

Want this to be 135.

x

y

>> atan2(1,1)*180/pians =

45>> atan2(-1,-1)*180/pians =

-135>> atan2(-1,1)*180/pians =

-45>> atan2(1,-1)*180/pians =

135

atan2(y,x) returns values in the interval (-,].

atan2: Four-quadrant inverse tangent

58

atan2 ,

arctan , 0,

arctan , 0 , 0,

arctan , 0 , 0,

2 , 0 , 0,

2 , 0 , 0,

undefined, 0 , 0,

Supplementary Reference

59

Modem Theory: An Introduction to Telecommunications

By Richard E. Blahut Date Published: December 2009 ISBN: 9780521780148 http://www.cambridge.org/us/ac

ademic/subjects/engineering/communications-and-signal-processing/modem-theory-introduction-telecommunications

https://books.google.co.th/books?id=ApmsJAvnMc0C

Richard Blahut

60

Former chair of the Electrical and Computer Engineering Department at the University of Illinois at Urbana-Champaign

Best known for Blahut–Arimotoalgorithm

Claude E. Shannon Award

61

Claude E. Shannon (1972)

David S. Slepian (1974)

Robert M. Fano (1976)

Peter Elias (1977)

Mark S. Pinsker (1978)

Jacob Wolfowitz (1979)

W. Wesley Peterson (1981)

Irving S. Reed (1982)

Robert G. Gallager (1983)

Solomon W. Golomb (1985)

William L. Root (1986)

James L. Massey (1988)

Thomas M. Cover (1990)

Andrew J. Viterbi (1991)

Elwyn R. Berlekamp (1993)

Aaron D. Wyner (1994)

G. David Forney, Jr. (1995)

Imre Csiszár (1996)

Jacob Ziv (1997)

Neil J. A. Sloane (1998)

Tadao Kasami (1999)

Thomas Kailath (2000)

Jack KeilWolf (2001)

Toby Berger (2002)

Lloyd R. Welch (2003)

Robert J. McEliece (2004)

Richard Blahut (2005)

Rudolf Ahlswede (2006)

Sergio Verdu (2007)

Robert M. Gray (2008)

Jorma Rissanen (2009)

Te Sun Han (2010)

Shlomo Shamai (Shitz) (2011)

Abbas El Gamal (2012)

Katalin Marton (2013)

János Körner (2014)

Arthur Robert Calderbank (2015)

Berger plaque

62

Documents

Principles of Communications - 4...In the time domain… 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2-1 0 1 2 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-4-2 0 2 4 I P 2 I P I P H2cos2 B Ö P L T P 2