fourier transform and communication systems and commu - part... · 2019. 7. 19. · electrical...

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ICT Elementary for Embedded SystemsSignal/Electronic Fundamental

Fourier Transform and Communication Systems

The slides to be used today can be downloaded fromhttp://www2.siit.tu.ac.th/prapun/ICTES/index.html

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ICT Elementary for Embedded SystemsSignal/Electronic Fundamental


Asst. Prof. Dr. Prapun [email protected]

Asst.Prof.Dr.Prapun Suksompong

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Chairperson of Electrical Engineering Program (and Chairperson of Electronics and Communication Engineering Curriculum and Electrical Engineering Curriculum) at Sirindhorn International Institute of Technology(SIIT)Ph.D. from Cornell University, USA

In Electrical and Computer EngineeringMinor: Mathematics (Probability Theory)Research: Neuro-Information Theory

(Communications in Human Brain)Current Research: Wireless Communications,

Localization, Game Theory

2009, 2013, and 2017 SIIT Best Teaching Awards2011 SIIT Research Award2013 TU Outstanding Young Researcher Award2017 SIIT Distinguished Teacher Award2018 TU Outstanding Teacher in Science and Technology prapun.com

General Information

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Course Website:http://www2.siit.tu.ac.th/prapun/ICTES/index.html

Lectures:July 18, 2019: 9:00-10:20, 10:40-12:00July 18, 2019: 13:00-14:20, 14:40-16:00

Textbook: Modern Digital and Analog Communication Systems

By B.P. Lathi and Zhi Ding4nd Edition

ISBN 978-0-471-27214-4

Library Call No. TK5101 L333 2009

i

Website

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prapun.com

Current version

Earlier version

Website

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More references

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Principles of CommunicationsBy Rodger E. Ziemerand William H. Tranter6th International student editionISBN 978-0-470-39878-4 Library Call No. TK5105 Z54 2010Student Companion Site: http://bit.ly/mN18kQ

Communication Systems: An Introduction to Signals and Noise in Electrical Communication

By A. Bruce Carlson and Paul B. Crilly5th International editionCall No. TK5102.5 C3 2010ISBN: 978-007-126332-0

More references

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J. G. Proakis and M. Salehi, Communication Systems Engineering, 2nd Edition, Prentice Hall, 2002. ISBN: 0-13-095007-6

S.S. Haykin, Communication Systems, 4th Edition, John Wiley & Sons, 2001. Call Number: TK5101 H38 2001.

Another Reference (in Thai)

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(Fourier transform) (Correlation)

(Spectral density) (amplitude modulation) (angle modulation)

(random process) (noise) (sampling theory) (pulse

modulation) (basenand pulse transmission) (digital passband transmission) (information)

[http://www.chulabook.com/description.asp?barcode=9789740333890]

3, 2558

ISBN: 9789740333890

10


From time domain to frequency domain

Signal (Waveform)

11[https://www.youtube.com/watch?v=cq7hXp7xUn8]

Signal in the time domain (audio)

12 [http://www.bespokenart.com/modern_art_prints/print8big.jpg]

Sound as Vibration

13 Season 3 Episode 1

Sound as Vibration

14 Season 3 Episode 1

Sound is just vibration. The speed of the vibration is called the frequency and measured in Hz. One vibration per second equals one Hertz.

Microphone

15

Microphones are a type of transducer.

They convert acoustical energy (sound waves) into electrical energy (the audio signal).

Dynamic microphones

[http://www.totalvenue.com.au/articles/microphones/microphones.html]

Dynamic Microphone

16 https://www.youtube.com/watch?v=2edewYkE_f0

Dynamic Microphone

17

A dynamic microphone gets its name from the fact that sound wave causes movement of a thin metallic diaphragm and an attached coil of wire that dynamically moves inside a permanent magnet to change acoustic energy into electronic energy.

Dynamic Microphone

18

This construction gives the dynamic mic its robustness but because the diaphragm is relatively heavy, it means that it can't response to sound wave quickly which means that its high frequency response beyond 10 kHz is usually limited.

LED Audio Spectrum Analyzer

19 [http://www.instructables.com/id/100-LED-10-band-Audio-Spectrum-atmega32-MSGEQ7-wit/]


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low-pitched sound


21

high-pitched sound


22

amount

Fourier transform ( )

23

The Fourier transform is a frequency domain representation of the original signal.

The term Fourier transform refers to both the frequency domain representation and the corresponding mathematical operation ( ).

f t f f f f

t ff0-f0

The (Fundamental) Frequencies of Musical Instruments

24 [http://www.psbspeakers.com/articles/The-Frequencies-of-Music]

Note frequency

A440 on Different Instruments

25 [https://www.youtube.com/watch?v=9iGjo2cd69s][http://www.philvarner.com/2015/01/27/why-does-a-tuning-fork-sound-different-than-a-piano-even-if-theyre-playing-the-same-note/]

[GarageBand]

“Same” timbre of a tuning fork (“pure” tone)

Any physical instrument is not only going to play the fundamental but also harmonics. These harmonics are frequencies in the sound that are integer multiples of the fundamental tone.

Ex.1: A440 on a Cathedral Organ

26[http://www.philvarner.com/2015/01/27/why-does-a-tuning-fork-sound-different-than-a-piano-even-if-theyre-playing-the-same-note/]

Left track

Right track

t

f

Ex.2: A440 on a Grand Piano

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Left track

Right track

t

f

[http://www.philvarner.com/2015/01/27/why-does-a-tuning-fork-sound-different-than-a-piano-even-if-theyre-playing-the-same-note/]

Tone Dialing

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Most modern telephones use a dialing system known as Touch-Tone.

Dual-tone multifrequency (DTMF) system.

Use pairs of audio (voice-frequency) tones to create signals representing the numbers to be dialed.First developed in the Bell System in the United States, and became known under the trademark Touch-Tone for use in push-button telephones starting in 1963.

Replace the use of rotary dial.Standardized by ITU-T Recommendation Q.23.

Also known in the UK as MF4.[Apr, 1964]

[htt

p://

sam

anth

amye

rs.t

ypep

ad.c

om/p

hoto

s/te

leph

ones

/b19

64_t

ouch

_ton

e_te

leph

one.

htm

l][h

ttp:

//bl

og.m

oder

nmec

hani

x.co

m/p

ushb

utto

ns-r

epla

ce-d

ials-

on-t

elep

hone

/]

Dial Tone

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North American and UK: A continuous mix of 350 Hz and 440 Hz

These two frequencies correspond to the standard concert pitch of A440, and approximately an “F”.@ -12dBm

Most of Europe: constant single tone (425 Hz)

Encoding

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Each number corresponds to a mix of two audio frequencies associated with each row and column of the corresponding pushbutton.

Most telephones use a standard keypad with 12 buttons or switches for the numbers 0 through 9 and the special symbols * and #.

Four additional keys for special applications.

[Fre

nzel

, 201

6, F

igur

e 18

-5, p

. 702

]

The “1” tone

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Fourier transform: Example

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f

t

t

Practice Problems

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f

f

34

Sir Isaac Newton

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Our modern understanding of light and color begins with Isaac Newton (1642-1726) and a series of experiments that he publishes in 1672.

He refracts white light with a prism, resolving it into its component colors.

[https://www.britannica.com/biography/Isaac-Newton/images-videos/Sir-Isaac-Newton-dispersing-sunlight-through-a-prism-for-a/153369][http://www.webexhibits.org/colorart/bh.html]

Sir Isaac Newton

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[http://sirisaacne.weebly.com/accomplishments.html]

A triangular prism, dispersing light

37 [http://www.astromia.com/astronomia/newtonluz.htm]

A triangular prism, dispersing light

38[https://en.wikipedia.org/wiki/Prism#/media/File:Light_dispersion_conceptual_waves.gif]

Waves shown to illustrate the differing wavelengths of light.

Electromagnetic Spectrum

39

[Gosling , 1999, Fig 1.1 and 1.2]

3 MHz 3 GHz

100 m 10 cm

c fWavelength

Frequency

Continuous Spectrum vs. Line Spectra

40

Continuous spectrum of an incandescent lamp

Discrete spectrum lines of a fluorescent lamp

(Discrete)

Line spectra

41

Remember those flame experiments from your high school chemistry class?

Line spectra

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Line spectra

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CD Tracks as Diffraction Gratings

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The tracks of a compact disc can act as a diffraction grating, producing a separation of the colors of white light.

[http://3.14.by/en/read/cd-dvd-microscope]

CD Tracks as Diffraction Gratings

45

Sunshine

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Compact Fluorescent Lamp

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CF vs LED

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Spectral Power Distribution

49

Plot of the relativepower emitted bythe light source ateach wavelengthover the visiblespectrum.

Spectral Power Distribution

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51


Mathematically speaking…

Euler’s Formula

j je

[https://www.youtube.com/watch?v=N321EcSzNas]

15 April 1707 – 18 September 1783

a Swiss mathematician, physicist, astronomer, logician and engineer

Made important and influential discoveries in many branches of mathematics

Complex exponential function

Sinusoids

Euler's number

The Most Beautiful Equation

53

Euler’s identity (Euler’s equation)

[http://www.scientificamerican.com/article/equations-are-art-inside-a-mathematicians-brain/]

Relate the three fundamental constants e, and i.

Fact: When mathematicians describe equations as beautiful, they are not lying. Brain scans show that their minds respond to beautiful equations in the same way other people respond to great paintings or masterful music.

Euler’s Formula on the Complex Plane

54

1

1

unit circle

Re

Im

(real axis)

(imaginary axis)

Euler’s Formula on the Complex Plane

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unit circle

Re

Im

(real axis)

(imaginary axis)

Rotating Vector in a Complex Plane

56

je jt ft

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

Euler’s Formula

58

je j

jA jA jA

jA jA jA

A e e e

A e e ej

Complex exponential

Euler’s Formula

59

je j

jA jA jA

jA jA jA

A e e e

A e e ej

Complex exponential

x x

x x

x x xd x xdx

x y x y x

x x

x x

y(product-to-sum formula)

Fourier transform: Example

60

f

t

tOne the left side, note how we decompose the signal into a linear combination of componentsat different frequencies. One the right side, note how each “arrow” corresponds to one component.

(Continuous-Time) Fourier Transform

61

j ft j ftg t G f e df G f g t e dt

Complex exponential: j fte ft j ftsinusoids

The relationship on the left is simply a decomposition of the signal into a linear combination of (potentially infinitely many) components at different frequencies.


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From the decomposition point of view, the value of at a particular frequency is simply the weight (scaling/coefficient) which tells how much component there is in .

By the orthogonality among complex exponential functions, the value of at a particular frequency can be calculated by the formula above.

This coefficient considered as a function of frequency is the Fourier transform of our signal.

7 Equations

63

that changed the world

… and still rule everyday life

64

7 Equations


65


Time Domain Frequency Domain

Capital letter is used to represent corresponding signal in the frequency domain.

direct transform

inverse transform

Signals in this form is “easy” to work with under LTI system.

g G f df G g t dt

Delta function (f)

66

(Dirac) delta function or (unit) impulse function

Usually depicted as a vertical arrow at the origin

Not a true functionUndefined at f = 0

Intuitively we may visualize (f) as an infinitely tall, infinitely narrow rectangular pulse of unit area

ff

Area = 1

smaller f

Area = 1 Area = 1

A (f)

67

(Dirac) delta function or (unit) impulse function

Usually depicted as a vertical arrow at the origin

Not a true functionUndefined at f = 0

Intuitively we may visualize A (f) as an infinitely tall, infinitely narrow rectangular pulse of area A

ff

Area = A

smaller f

Area = A Area = A

Fourier Transform Pairs (1)

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j tf fe f

t ff0-f0

tf f ff f

ff0

Practice Problems (A Revisit)

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f

f

Practice Problems (More)

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t t

t

t t

f

f

f

Fourier Transform of Symbolic Expression in MATLAB

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function G = fourierf(g)syms fG = simplify(subs(fourier(g),'w',2*pi*f));end

>> syms t; g = exp(1j*2*pi*5*t);>> G = fourierf(g)G =dirac(f - 5)

>> syms t; g = cos(2*pi*5*t);>> G = fourierf(g)G =dirac(f - 5)/2 + dirac(f + 5)/2>> pretty(G)

dirac(f - 5) dirac(f + 5) ------------ + ------------

2 2

>> syms t f0; g = cos(2*pi*f0*t);>> G = fourierf(g)G =dirac(f + f0)/2 + dirac(f - f0)/2>> pretty(G)

dirac(f + f0) dirac(f - f0) ------------- + -------------

2 2


72



xxx

-0.2172

0.1284

-0.0913

sinc function

73

2--2

1

sinc function

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-0.2172

0.1284

-0.09132--2

Zero crossings are at all non-zero integer multiples of because = 0.

Sinusoidal oscillations of period 2

1

As , we have . Using L'Hospital's Rule, we set .

sinc function

75

Main lobe(null to null)

sinc function

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0 2-

1

-0.2172

0.1284

-0.0913

Amplitude of decreases

continuously as .

Fourier Transform of Symbolic Rectangular Function in MATLAB

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>> syms a t>> g = rectangularPulse(-a,a,t)g =rectangularPulse(-a, a, t)>> G = fourierf(g)G =sin(2*pi*a*f)/(pi*f)

t

1

f

2a

Practice Problems

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t

1

1-1

t

1

2-2

f

f

79

Normalized sinc function

Normalized sinc function

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1 2-1

1

-2

Its zero crossings are at non-zero integer values of its argument.


81



-0.2172

0.1284

-0.0913


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More realistic signal…

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plotspect.m

plotspect.m

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% plotspec(x,t) plots the spectrum of the signal x% whose values are sampled at time (in seconds) specified in t function plotspect(x,t)N=length(x); % length of the signal xTs = t(2)-t(1); % find the sampling intervalssf=((-N/2):(N/2-1))/(Ts*N); % frequency vectorfx=Ts*fft(x(1:N)); % do DFT/FFTfxs=fftshift(fx); % shift it for plottingsubplot(2,1,1);set(plot(t,x),'LineWidth',1.5); % plot the waveformxlabel('Seconds'); % label the axessubplot(2,1,2);set(plot(ssf,abs(fxs)),'LineWidth',1.5); % plot magnitude spectrumxlabel('Frequency [Hz]'); ylabel('Magnitude') % label the axes

Phone/Cellphone (Muffled) Audio

85[https://www.youtube.com/watch?v=CPFiufYmtAc]

PILOT'S ALPHABET

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International Radiotelephony Spelling Alphabet.

Pilots use it so that essential letter combinations can be easily understood by individuals transmitting and receiving voice messages.

VoLTE Audio


VoLTE Audio: AMR-WB


iPhone 5 supports HD Voice

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in the form of AMR-WB over 3G (UMTS)

In Thailand, dtac is the first mobile operator and the only one that provides HD Voice via 3G network.

[https://www.dtac.co.th/en/network/hd-voice.html]

Digital Audio Watermarking

90 [http://www.sersc.org/journals/IJSIA/vol5_no2_2011/3.pdf]

fourier transform and communication systems and commu - part... · 2019. 7. 19. · electrical...

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