Fourier Transform and Communication Systems Asst. Prof. Dr ... and Commu v7 - cut - 1x2.pdf · 1...

1

ICT Elementary for Embedded SystemsSignal/Electronic Fundamental

Fourier Transform and Communication Systems

Asst. Prof. Dr. Prapun [email protected]

General Information

3

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

Lectures: July 20, 20179:00-10:20; 10:40-12:0013:30-14:50; 15:10-16:30

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

More references

6

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

7

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)

8

(Fourier transform) (Correlation)

(Spectral density) (amplitude modulation) (angle

modulation) (random process) (noise) (sampling theory)

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

[http://www.chulabook.com/description.asp?barcode=9789740327707]

. . , , 2,

, 2554. ISBN: 978-974-03-2770-7

9


From time domain to frequency domain

LED Audio Spectrum Analyzer

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

Fourier transform ( )

16

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

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

Note frequency

Tone Dialing

21

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

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leph

one.

htm

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ttp:

//bl

og.m

oder

nmec

hani

x.co

m/p

ushb

utto

ns-r

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ce-d

ials

-on-

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Dial Tone

22

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

23

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

24

Fourier transform: Example

25

f

t

t

Practice Problems

26

Sir Isaac Newton

28

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]

Electromagnetic Spectrum

32

[Gosling , 1999, Fig 1.1 and 1.2]

3 MHz 3 GHz

100 m 10 cm

c fWavelength

Frequency

Continuous Spectrum vs. Line Spectra

33

Continuous spectrum of an incandescent lamp

Discrete spectrum lines of a fluorescent lamp

(Discrete)

Line spectra

34

Remember those flame experiments from your high school chemistry class?

Line spectra

36

CD Tracks as Diffraction Gratings

37

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]

Spectral Power Distribution

42

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

Spectral Power Distribution

43

44


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

Euler’s Formula on the Complex Plane

47

1

1

unit circle

Re

Im

(real axis)

(imaginary axis)

Euler’s Formula on the Complex Plane

48

unit circle

Re

Im

(real axis)

(imaginary axis)

Rotating Vector in a Complex Plane

49

je jt ft

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

Euler’s Formula

51

je j

jA jA jA

jA jA jA

A e e e

A e e ej

Complex exponential

Euler’s Formula

52

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)

(Continuous-Time) Fourier Transform

53

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.


54


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

55

that changed the world

… and still rule everyday life

56

7 Equations


57


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)

58

(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)

59

(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)

60



j tf fe f

t ff0-f0

tf f ff f

ff0

Practice Problems (A Revisit)

61

Practice Problems (More)

62

t t

t

t t

Fourier Transform of Symbolic Expression in MATLAB

63

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


64



xxx

-0.2172

0.1284

-0.0913

sinc function

65

2--2

1

sinc function

66

-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

67

Main lobe(null to null)

sinc function

68

0 2-

1

-0.2172

0.1284

-0.0913

Amplitude of decreases

continuously as .

Fourier Transform of Symbolic Rectangular Function in MATLAB

69

>> 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

70

t

1

1-1

t

1

2-2

f

f

71

Normalized sinc function

Normalized sinc function

72

1 2-1

1

-2

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


73



-0.2172

0.1284

-0.0913


74



More realistic signal…

75

plotspect.m

plotspect.m

76

% 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

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

VoLTE Audio: AMR-WB

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

iPhone 5 supports HD Voice

81

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

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

83


Introductory concepts in communications…

Modulator: a crucial part of any communication

systems

Modulation

84

The term baseband is used to designate the band of frequencies of the signal delivered by the source.

Modulation is a process that causes a shift in the range of frequencies in a signal.

MotivationFrequency-Division Multiplexing (FDM) and Frequency-Division Multiple Access(FDMA)Reasonable antenna size for effective radiation of power over a radio linkCommunication channel matching (avoiding frequencies that suffer from large attenuation/distortion)

An Important Property of

85

Important Properties of

86

x y X Y

x y X Y

Convolution Properties:

Modulation:

x y t x y t d x t y d

Shifting Properties: tj fg t e Gt f

j tfe g t G f f

c c cf f fg t t G f G f

Note that the magnitude of this is simply

Practice Problems (Another Revisit)

87

t

t

t t

Simple Modulation: Freq. Domain

88

×

cf t

Modulator

Message(modulating signal)


c c cx t m t f t M f f M f f

c cM f f M f f

Simple Modulation: Time Domain

89

Simple Modulation

90

×

cf t

Modulator


91


So, … which frequencies do we actually use?

Radio-frequency spectrum

Electromagnetic Spectrum

92

[Gosling , 1999, Fig 1.1 and 1.2]

3 MHz 3 GHz

100 m 10 cm

c fWavelength

Frequency

Radio-frequency spectrum

93

Commercially exploited bands

c fWavelength

Frequency

[http://www.britannica.com/EBchecked/topic-art/585825/3697/Commercially-exploited-bands-of-the-radio-frequency-spectrum]

Note that the freq. bands are given in decades; the VHF band has 10 times as much frequency space as the HF band.

Analog (Old) terrestrial TV in BKK

94

Channel.

Bandwidth.Picture Carrier.

Audio Carrier.

2 47 - 54 48.25 53.753 54 - 61 55.25 60.754 61 - 68 62.25 67.75

Channel.


Audio Carrier.

5 174 - 181 175.25 180.756 181 - 188 182.25 187.757 188 - 195 189.25 194.758 195 - 202 196.25 201.759 202 - 209 203.25 208.75

10 209 - 216 210.25 215.7511 216 - 223 217.25 222.7512 223 - 230 224.25 229.75

Channel.


Audio Carrier.

26 510 - 518 511.25 516.7527 518 - 526 519.25 524.7528 526 - 534 527.25 532.7529 534 - 542 535.25 540.7530 542 - 550 543.25 548.7531 550 - 558 551.25 556.7532 558 - 566 559.25 564.7533 566 - 574 567.25 562.7534 574 - 582 575.25 580.75

VHF.(Low Band)

VHF.(Hight Band)

UHF.(Band 4)

Channel.


Audio Carrier.

35 582 - 590 583.25 588.7536 590 - 598 591.25 596.7537 598 - 606 599.25 604.7538 606 - 614 607.25 612.7539 614 - 622 615.25 620.7540 622 - 630 623.25 628.7541 630 - 638 631.25 636.7542 638 - 646 639.25 644.7543 646 - 654 647.25 652.7544 654 - 662 655.25 660.7545 662 - 670 663.25 668.7546 670 - 678 671.25 676.7547 678 - 686 679.25 684.7548 686 - 694 687.25 692.7549 694 - 702 695.25 700.7550 702 - 710 703.25 708.7551 710 - 718 711.25 716.7552 718 - 726 719.25 724.7553 726 - 734 727.25 732.7554 734 - 742 735.25 740.7555 742 - 750 743.25 748.7556 750 - 758 751.25 756.7557 758 - 766 759.25 764.7558 766 - 774 767.25 772.7559 774 - 782 775.25 780.7560 782 - 790 783.25 788.75

UHF.(Band 5)

( )

Terrestrial TV in BKK

95

Channel.


Audio Carrier.

2 47 - 54 48.25 53.753 54 - 61 55.25 60.754 61 - 68 62.25 67.75

Channel.


Audio Carrier.

5 174 - 181 175.25 180.756 181 - 188 182.25 187.757 188 - 195 189.25 194.758 195 - 202 196.25 201.759 202 - 209 203.25 208.75

10 209 - 216 210.25 215.7511 216 - 223 217.25 222.7512 223 - 230 224.25 229.75

Channel.


Audio Carrier.

26 510 - 518 511.25 516.7527 518 - 526 519.25 524.7528 526 - 534 527.25 532.7529 534 - 542 535.25 540.7530 542 - 550 543.25 548.7531 550 - 558 551.25 556.7532 558 - 566 559.25 564.7533 566 - 574 567.25 562.7534 574 - 582 575.25 580.75

VHF.(Low Band)

VHF.(Hight Band)

UHF.(Band 4)

Channel.


Audio Carrier.

35 582 - 590 583.25 588.7536 590 - 598 591.25 596.7537 598 - 606 599.25 604.7538 606 - 614 607.25 612.7539 614 - 622 615.25 620.7540 622 - 630 623.25 628.7541 630 - 638 631.25 636.7542 638 - 646 639.25 644.7543 646 - 654 647.25 652.7544 654 - 662 655.25 660.7545 662 - 670 663.25 668.7546 670 - 678 671.25 676.7547 678 - 686 679.25 684.7548 686 - 694 687.25 692.7549 694 - 702 695.25 700.7550 702 - 710 703.25 708.7551 710 - 718 711.25 716.7552 718 - 726 719.25 724.7553 726 - 734 727.25 732.7554 734 - 742 735.25 740.7555 742 - 750 743.25 748.7556 750 - 758 751.25 756.7557 758 - 766 759.25 764.7558 766 - 774 767.25 772.7559 774 - 782 775.25 780.7560 782 - 790 783.25 788.75

UHF.(Band 5)

( )

MUX 1

MUX 2

MUX 3

MUX 4

MUX 5

Cellular Support in iPhone 7

96

FDD-LTE (Bands 1, 2, 3, 4, 5, 7, 8, 12, 13, 17, 18, 19, 20, 25, 26, 27, 28, 29, 30)

TD-LTE (Bands 38, 39, 40, 41)

UMTS/HSPA+/DC-HSDPA (850, 900, 1700/2100, 1900, 2100 MHz)

GSM/EDGE (850, 900, 1800, 1900 MHz)

http://www.apple.com/iphone-7/specs/

Model A1778Model A1784

Announced on September 7, 2016. Released on September 16, 2016.

Additionally, only in models A1660 and A1661

TD-SCDMA 1900 (F), 2000 (A)CDMA EV-DO Rev. A (800, 1900, 2100 MHz)

FDD and TDD LTE frequency bands

97

TDD LTE frequency band allocationsFDD LTE frequency band allocations

[http://www.radio-electronics.com/info/cellulartelecomms/lte-long-term-evolution/lte-frequency-spectrum.php]

Operating bands specified for LTE in 3GPP below 1 GHz

98 [Dahlman, Parkvall, and Skold, 2016]

Operating bands specified for LTE in 3GPP above 1 GHz

99 [Dahlman, Parkvall, and Skold, 2016]

Spectrum Allocation

100

Spectral resource is limited.Most countries have government agencies responsible for allocating and controlling the use of the radio spectrum.Commercial spectral allocation is governed

globally by the International Telecommunications Union (ITU)ITU Radiocommunication Sector (ITU-R) is responsible for radio communication.

in the U.S. by the Federal Communications Commission (FCC)in Europe by the European Telecommunications Standards Institute (ETSI) in Thailand by the National Broadcasting and Telecommunications Commission (NBTC;

; .)Blocks of spectrum are now commonly assigned through spectral auctions to the highest bidder.

101[http://www.ntia.doc.gov/page/2011/united-states-frequency-allocation-chart]2011

Thailand Freq. Allocations Chart

102http://www.ntc.or.th/uploadfiles/freq_chart_thai.htm

Spectrum Allocation

103

Spectrum is a scarce resource.“Radio spectrum will be the first of our finite resources to run out, long before oil, gas or mineral deposits.”

Spectrum is allocated in “chunks” in frequency domain.“Chunks” are licensed to (cellular/wireless) operators.

Within a single cellular operator, the chunk is further divided into many channels.

Each channel has its own band of frequency.

Oct 2012: Thailand 2.1GHz Auction

104

4.5bn baht per license (freq chunk)1 license (chunk) = 5 MHz (UL) 5 MHz (DL)

450 million baht per MHz30 million baht per MHz per year

Nov 2015: Thailand 1800MHz Auction

105

40bn baht 15 MHz (UL) + 15 MHz (DL)

1.3 billion baht per MHz74 million baht per MHz per year

(18 years)(15 years)

( 2.5 )

Dec 2015: Thailand 900MHz Auction

106

76bn baht 10 MHz (UL) + 10 MHz (DL)

3.8 billion baht per MHz250 million baht per MHz per year

Dec 201515 years

Nov 201518 years

( 3 )

re-auction

(forfeit)

Digital TV License Auction in 2012

107[ http://hilight.kapook.com/view/95367 ]

Channels for variety TV in high definition (HD) and standard definition (SD)

Digital TV License Auction in 2012

108 [ http://hilight.kapook.com/view/95408 ]

News and children's/family channels

Cognitive radios: Motivation

109

Traditional Rule: Allow predetermined licensed users the right to transmit at given frequencies.

Frequency bands were sold at auction, bringing considerable revenue to the government.Unlicensed users are regarded as “harmful interference.”

Radio-frequency resources are not fully utilizedThere exists a large number of frequency bands that have considerable, and sometimes periodic, idle time intervals. For example, some TV stations do not work at night. Spectrum holes.

Cognitive radios

110

A revolutionary communication paradigm that can utilize the existing wireless spectrum resources more efficiently.

New Assumption: Users are intelligent and have the ability to observe, learn, and act to optimize their performance.

Game theory has been recognized as an important tool in studying, modeling, and analyzing the cognitive interaction process.

Game Theory

111

A branch of applied mathematics as well as of applied sciences.

A formal framework with a set of mathematical tools

To study the complex interactionsanalyze competition and cooperation

among interdependent rational players (having individual self-interests.)

For more than half a century, game theory has led to revolutionary changes in economics.

Three Nobel Prizes have been given in the economic sciences for work primarily in game theory.

John Nash

112

June 13, 1928 – May 23, 2015an American mathematicianShare the 1994 Nobel Memorial Prizein Economic Sciences with game theorists Reinhard Selten and John Harsanyi.In 1959, Nash began showing clear signs of mental illness, and spent several years at psychiatric hospitals being treated for paranoid schizophrenia.

His struggles with his illness and his recovery became the basis for Sylvia Nasar's biography, A Beautiful Mind, as well as a film of the same name starring Russell Crowe.

[https://en.wikipedia.org/wiki/John_Forbes_Nash_Jr.#/media/File:John_Forbes_Nash,_Jr._by_Peter_Badge.jpg][https://en.wikipedia.org/wiki/A_Beautiful_Mind_(book)]

113


Demodulation

DSB-SC

114

× ×Channel

cf t cf t

LPF

Modulator Demodulator


c c

x t

v t

m t f t f tLPF m t

Key equation:

In the time domain…

115

Note the oscillation at twice the carrier frequency

Scaling and Suppressing Frequency Components

116

Important Properties of

117

x y X Y

x y X Y

Convolution Properties:

Modulation:

x y t x y t d x t y d

Shifting Properties: tj fg t e Gt f

j tfe g t G f f


Note that the magnitude of this is simply

Filter Property of

x y X Y

x y X Y

Convolution Properties:x y t x y t d x t y d

Filterx t x h tFilterX f X f H f

Time Domain View: Frequency Domain View:

DSB-SC

119

[Demo_DSBSC_Sound_ReadWAV.m]

DSB-SC (Zoomed in time)

120

Note how the baseband signal becomes the envelope of

the modulated signal .

Note the delay caused by the LPF.

(Zoomed in time)

121

In the time domain… we expect

122

Note the oscillation at twice the carrier frequency

In the time domain…

123

When the sampling rate is not fast enough,…

The problem with sampling rate

124

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

The problem with sampling rate

125

BW Inefficiency in Our System

126

Conjugate symmetry property of Fourier transform:

Recall .

If is real-valued, then

127


Quadrature Amplitude Modulation (QAM)

m tv t

LPH f m t

m tv t

LPH f m t

cf t

cf t

h ty tx t

cf t

cf t

QAM

128

Send two messages over the same bandwidth of 2B Hz.

A more general formula:

129


cj

cj

cg t t G f e eGf fff

QAM Demodulation

130

When ,

c

c c

c c

c

c

c

c

c

c

m t f

v t x t f t

m t f t f t

m t f t f t

m t f t

m t

m t f t

m t f t

t

m t f tm t f t

c

c

x t f t m t

x t f t m t

cB f

Complex form of QAM

131

where .

: complex envelope or complex baseband signal

: in-phase component

: quadrature(-phase) component

c

c c

j f t

x t m t f t m t f t

m t e

QAM vs. DSB-SC Key Equations

132

LPF m tc c

x t

m t f t f t

c cj f t j f t

x t

m t e eLPF m t

DSB-SC Key Equation

QAM Key Equation

Derivation of the QAM Key Equation (1)

133

Recall that .

Find and simplify the Fourier transform of .

Find and simplify the Fourier transform of .

Derivation of the QAM Key Equation (2)

134

c cj f t j f t

x t

m t e eLPF m t

135


OFDM

OFDM Applications

136

802.11 Wi-Fi: a/g/n/ac versionsDVB-T (Digital Video Broadcasting —Terrestrial)

terrestrial digital TV broadcast system used in most of the world outside North America

DMT (the standard form of ADSL - Asymmetric Digital Subscriber Line)WiMAX, LTE (OFDMA)

Overview: Baseband OFDM Symbol

137

Let S = (S1, S2, …, SN) be the information vector.

One baseband OFDM modulated symbol can be expressed as

Note that:N

k kk s s

kt kts t S ST TN

s

k

N

k sk s

N

Tkk s

c t

kts t S j t TTN

ktS jTN

t

Some references may use different constant in the front

Some references may start with different time interval, e.g. [-Ts/2, +Ts/2]

Single-Carrier Digital Transmission

138

Baseband:

Passband:

N

k sk

s t s p t kT

s

sT

t Tp t t

cj f tx t s t e

Multipath Propagation

139

In a wireless mobile communication system, a transmitted signal propagating through the wireless channel often encounters multiple reflective paths until it reaches the receiver We refer to this phenomenon as multipath propagation and it causes fluctuation of the amplitude and phase of the received signal.We call this fluctuation multipath fading.

Wireless Comm. and Multipath Fading

140

v

i ii

y t x t h t n t x t n t

v

i ii

h t t

s s sh t t t T t T t T

s s sh t t t T t T t T

The signal received consists of a number of reflected rays, each characterized by a different amount of attenuation and delay.

ISI(Intersymbol Interference)

Observation and a Solution

141

Observation: Delay spread causes ISI

A general rule of thumb is that a delay spread of less than 5 or 10 times the symbol width will not be a significant factor for ISI.

Solution: The ISI can be mitigated by reducing the symbol rate and/or including sufficient guard times between symbols.

Multi-Carrier Transmission

142

Convert a serial high rate data stream on to multiple parallel low rate sub-streams.

Each sub-stream is modulated on its own sub-carrier.

Time domain perspective: Since the symbol rate on each sub-carrier is much less than the initial serial data symbol rate, the effects of delay spread, i.e. ISI, significantly decrease, reducing the complexity of the equalizer.

[Fazel and Kaiser, 2008, Fig 1-4]

Multi-Carrier Modulation

143

0 0 1 0

1 1 0 10 1 1 1

010011101011Old

New

Frequency Division Multiplexing

144

Frequency Domain Perspective: Even though the fast fading is frequency-selective across the entire OFDM signal band, it is effectively flat in the band of each low-speed signal.

[Myung and Goodman, 2008]

[The flatness assumption is the same one that you used in Riemann approximation of integral.]

Frequency Division Multiplexing (FDM)

145

To facilitate separation of the signals at the receiver, the carrier frequencies were spaced sufficiently far apart so that the signal spectra did not overlap. Empty spectral regions between the signals assured that they could be separated with readily realizable filters.

The resulting spectral efficiency was therefore quite low.

Single Carrier vs. Multi-Carrier (FDM)

146

Single Carrier Multi-Carrier (FDM)

Single higher rate serial scheme Parallel scheme. Each of the parallel subchannels can carry a low signaling rate, proportional to its bandwidth.

Multipath problem: Far more susceptible to inter-symbol interference (ISI) due to the short duration of its signal elements and the higher distortion produced by its wider frequency band

Complicated equalization

Long duration signal elements and narrow bandwidth in sub-channels.

Complexity problem: If built straightforwardly as several (N) transmitters and receivers, will be more costly to implement.

BW efficiency problem: The sum of parallel signalling rates is less than can be carried by a single serial channel of that combined bandwidth because of the unused guard space between the parallel sub-carriers.

OFDM

147

OFDM = Orthogonal frequency division multiplexingOne of multi-carrier modulation (MCM) techniques

Parallel data transmission (of many sequential streams)A broadband is divided into many narrow sub-channelsFrequency division multiplexing (FDM)

High spectral efficiencyThe sub-channels are made orthogonal to each other over the OFDM symbol duration Ts.

Spacing is carefully selected.

Allow the sub-channels to overlap in the frequency domain.Sub-carriers are spaced as close as theoretically possible.

Subcarrier Spacing

148

s

fT

Spectrum Overlap in OFDM

N separate QAM signals, at N frequencies separated by the signaling rate.

Each QAM signal carries one of the original input complex numbers.

The spectrum of each QAM signal is of the form with nulls at the center of the other sub-carriers.

s

s

N

kk s

T

TjN k f

sk

f

ks

t

S f e

kts t S j

T T f

TN

S k fN

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