1
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
Fourier Transform and Communication Systems
Asst. Prof. Dr. Prapun [email protected]
Asst.Prof.Dr.Prapun Suksompong
3
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
5
prapun.com
Current version
Earlier version
Website
6
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
8
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)
9
(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
Fourier Transform and Communication Systems
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/]
LED Audio Spectrum Analyzer
20
low-pitched sound
LED Audio Spectrum Analyzer
21
high-pitched sound
LED Audio Spectrum Analyzer
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
27
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
28
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]
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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
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. 702
]
The “1” tone
31
Fourier transform: Example
32
f
t
t
Practice Problems
33
f
f
34
Sir Isaac Newton
35
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
36
[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
43
CD Tracks as Diffraction Gratings
44
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
46
Compact Fluorescent Lamp
47
CF vs LED
48
Spectral Power Distribution
49
Plot of the relativepower emitted bythe light source ateach wavelengthover the visiblespectrum.
Spectral Power Distribution
50
51
Fourier Transform and Communication Systems
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
55
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.
(Continuous-Time) Fourier Transform
62
j ft j ftg t G f e df G f g t e dt
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
(Continuous-Time) Fourier Transform
65
j ft j ftg t G f e df G f g t e dt
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)
68
j ft j ftg t G f e df G f g t e dt
Time Domain Frequency Domain
j tf fe f
t ff0-f0
tf f ff f
ff0
Practice Problems (A Revisit)
69
f
f
Practice Problems (More)
70
t t
t
t t
f
f
f
Fourier Transform of Symbolic Expression in MATLAB
71
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
Fourier Transform Pairs (2)
72
j ft j ftg t G f e df G f g t e dt
Time Domain Frequency Domain
xxx
-0.2172
0.1284
-0.0913
sinc function
73
2--2
1
sinc function
74
-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
76
0 2-
1
-0.2172
0.1284
-0.0913
Amplitude of decreases
continuously as .
Fourier Transform of Symbolic Rectangular Function in MATLAB
77
>> 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
78
t
1
1-1
t
1
2-2
f
f
79
Normalized sinc function
Normalized sinc function
80
1 2-1
1
-2
Its zero crossings are at non-zero integer values of its argument.
Fourier Transform Pairs (2)
81
j ft j ftg t G f e df G f g t e dt
Time Domain Frequency Domain
-0.2172
0.1284
-0.0913
Fourier Transform Pairs (3)
82
j ft j ftg t G f e df G f g t e dt
Time Domain Frequency Domain
More realistic signal…
83
plotspect.m
plotspect.m
84
% 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
86
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
87[https://www.youtube.com/watch?v=CPFiufYmtAc]
VoLTE Audio: AMR-WB
88[https://www.youtube.com/watch?v=CPFiufYmtAc]
iPhone 5 supports HD Voice
89
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]