EE140 Introduction to Communication Systems Lecture...

3/4/2018

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EE140 Introduction to

Communication Systems

Lecture 9

Instructor: Prof. Xiliang Luo

ShanghaiTech University, Spring 20181

Architecture of a (Digital) Communication System

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Source A/Dconverter

Sourceencoder

Channelencoder Modulator

Channel

DetectorChanneldecoder

Sourcedecoder

D/Aconverter

User

Transmitter

Receiver

Absent ifsource isdigital

Noise

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Contents

• Analog Modulation

– Amplitude modulation

– Pulse modulation

– Angle modulation (phase/frequency)

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AM and PM• Amplitude modulation (AM) is linear

–

• Angle modulation (PM and FM) is nonlinear

4

m(t))t(dm

)t(ds of tindependen is

tfcos)t(mA)t(s cπ2

23

22

22

3

1

2

11

2

32

322

2

tfsin!

)t(tfcos

!

)t(tfsin)t(tfcosA

)t(!

j)t(!

)t(j A eRe

eA eRe)t(tfcosA)t(s

cccc

tfj

)t(jtfjc

c

c

πθ

πθ

πθπ

θθθ

θπ

π

θπ

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Narrowband FM (NBFM)• Narrowband FM (NBFM) – sinusoidal modulating

signal

• Given

– where is the peak frequency deviation, and is the modulation index

5

]d)(mKtfcos[A)t(st

fc 0

0

2 θττπ 0 and 2 0 θtfcosa)t(m mπ

tfsintfsinf

ftfsin

f

aKd)(mK)t( mm

mm

m

ft

f πβπππ

ττθ 22220

Δ

fΔ β

NBFM (Cont’d)• If (i.e. ), the narrowband FM

signal is given by

• Compared with DSB-LC (AM)

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1)t(θ10 β

tfjtfjtfj

cmc

mmc eeAeRe

tfsintfsinAtfcosA)t(s

πππ ββ

ππβπ

222

221

2 2 2

tfjtfjtfj

cmc

mmc em

em

AeRe

tftfmAtfcosA)t(s

πππ

πππ

222

221

2cos 2cos 2

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Wideband FM• If modulation index is NOT small, the spectral

density of a general angle-modulated signal cannot be obtained by Fourier transform.

• Wideband FM

– Modulation index

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]d)(mKtfcos[A)t(st

fc 0

0

2 θττπ

tfsinjtfj

mcmm

fc

mc eAeRe

tfsintfcosAtfsinf

KtfcosA)t(s

πβπ

πβπππ

π

22

2222

2

m

f

m f

K

f

f

πβ

2

Δ

Wideband FM• is a periodic function of time with a

fundamental frequency of . Its Fourier series representation is

– Bessel functions of the first kind

8

tfsinj me πβ 2

2

2

2222 1 where

T

T

tnfjtfsinjn

n

tnjn

tfsinj dteeT

FeFe mmmm ππβωππβ ，

)(JF nn β

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Spectra of FM Signal• Total average power in an FM signal is a constant.

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constant mf constant fΔ

Generation of Wideband FM Signals (Cont’d)

• Indirect method: Armstrong indirect FM transmitter

– As a result of the multiplication and heterodyne operations, it is difficult in this system to maintain the correct magnitude of carrier to sidebands, and thus it could not be used for an information signal with DC content.

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Demodulation of Wideband FM Signals• Demodulation: to provide an output signal whose

amplitude is linearly proportional to the instantaneous frequency of the input FM signal.

• Direct method: use frequency discriminator – Frequency discriminator is the system that has a linear

frequency-to-voltage transfer characteristic. – Ideal differentiator has a linear amplitude versus frequency

characteristic and therefore is a frequency discriminator.Input:

Output:

– If , the modulating signal can then be detected by an envelope detector.

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]d)(mktfcos[A)t(st

fc 00 2 θττπ

00

2 ]2[ θττ d)(mKtπfsin)t(mKπfA)t(sdt

d t

fcfc

cf f)t(mK π2

Demodulation of Wideband FM Signals (Cont’d)

• Examples of frequency discriminator

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• Indirect method: use phase-locked loop (PLL)

– Phase comparator detects the timing difference between the two periodic signals (with the same fundamental frequency) and produces an output voltage that is proportional to this difference.

– Loop filter controls the dynamic response of the PLL. We have

– Voltage-controlled oscillator (VCO) generates a constant-amplitude periodic waveform whose instantaneous frequency is proportional to the input voltage, i.e., .

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Phasecomparator)(tsi

Loopfilter

Voltage-controlledoscillator (VCO)

)(tso

)(tsc

)(ts f

)t(h)t(s)t(s co

)t(sKf)t(s ofcf π2


• Indirect method: output of PLL

– Assume and the phase comparator output is then

– If is small and we have .14

)]t(tfcos[A)t(s ici νπ 21 )],t(tfsin[A)t(s fcf νπ 22

)]t()t(sin[AA

)]t(tfsin[A)]t(tfcos[A)t(s

if

LPfcicc

νν

νπνπ

21

21

2

1

2 2

)t()t( if νν )]t()t([k)t(s ifcc νν


Loopfilter


)(tso

)(tsc

)(ts f

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• Indirect method: output of PLL

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)t(h)]t()t(sin[kk)t(h)t(sk)t(sk)t(dt

difcfcfoff ννν


Loopfilter


)(tso

)(tsc

)(ts f

0 )t(kk)t(kk)t(dt

dicffcff ννν

1

)t(h

x)xsin(No PLL loop


• Indirect method: output of PLL (with loop)

– Output voltage is proportional to the instantaneous frequency (referred to the carrier) of the input wideband FM signal.

– The PLL demodulates the input wideband FM signal!

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ττνν d)(sk)t()t(t

offi 0

)t(dt

d

k)t(s i

fo ν

1

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Signal-to-noise Ratio (SNR) in FM Reception

• Input signal and input noise of the FM detector

• Output signal of the FM detector

• For convenience, assume• Output noise of the FM detector

– Unmodulated carrier with additive bandpass noise

where and

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2 2 200

/AS]d)(fktfcos[A)t(s i

t

fc θττπ

)t(fk)t(dt

d)t()t(s fcco ωθωω

)t(fkS)t(fk)t(s fofo22

)]t(tcos[)t(r

tsin)t(ntcos)t(ntcosA)t(ntcosA

c

cscccc

γω

ωωωω

)t(n)t(n)t(nNtfsin)t(ntfcos)t(n)t(n scicscc222 22 ππ

22 )]t(n[)]t(nA[)t(r sc .)t(nA

)t(ntan)t(

c

s 1

γ

SNR in FM Reception (cont’d)• Output noise of the FM detector

– Assume the noise is small, i.e. , we have

– Output noise: .

18

A)t(n&)t(n sc

A

)t(n

A

)t(ntan

)t(nA

)t(ntan)t( ss

c

s

11γ

)t(ndt

d

A)t(

dt

d)t(n so

1 γ

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Power Spectral Density / Mean Power of Noise

• Input noise of the FM detector

Bandwidth:

• Quadrature noise component

Bandwidth:

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Hzwatts)(Sn 2ηω

π

η

πω

2

2 2 2 WW

)(S)t(nN ni mnW ω 2

0 2W2W

)(snS

)t(ns

ηωωω LPcncnn )ω(S)ω(S)(Ss

π

η

πω

2

2

2 22 WW

)(S)t(nsns

mnW ω 2

Power Spectral Density / Mean Power of Noise

• Output noise of the FM detector

– where is the frequency response of the time differentiator (recall

• The discriminator output is limited by a low-pass filter (LPF) with a cutoff frequency of , the bandwidth of output noise is .

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)t(no

2

2

2

22

2

1

A)(S

A)(H)(S

A)(S

sso nnn

ηωω

ωωωω

ωω j)(H

)(Fj)t(fdt

dFT ωω

mω

mω

2

3

0

22

2

3

2

1

Ad

Ad)(S)t(nN m

noo

mm

mo π

ωηωω

π

ηωω

π

ωω

ω

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SNR in FM Reception (cont’d)• Signal-to-noise ratios (SNR) in FM reception

– For wideband FM, the output signal-to-noise ratio increases as the square of the bandwidth, which is proportional to .

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W

A

N

SSNR

i

ii

2

η

π

3

222

3

m

f

o

oo

)t(fKA

N

SSNR

ωη

π

fK

SNR in PM Reception• Input signal of the PM detector

• Output signal of the PM detector

• For convenience, assume • Input noise of the PM detector

• Output noise of the PM detector

– Assume the noise is small,

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2 20 AS])t(fKtcos[A)t(s ipc θω

)t(fkt)t()t(s pco 0θωθ

)t(fkS)t(fk)t(s popo22

)t(n

π

η

πωηω

2

2 2 2 2 WW

)(S)t(nNHzwatts)(S nin

)t(no

)t(nA)t(ntan)t()t(n cso 1γ

.A

)t(n)t( sγ

222

22

22

1

AA)t(nN

A)(S

A)(S mm

oonn so π

ωη

π

ωηηωω

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SNR in PM Reception (cont’d)• Signal-to-noise ratios (SNR) in PM reception

• For sinusoidal modulating signal, i.e. , we have

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W

A

N

SSNR

i

ii

2

η

π

m

p

o

oo

)t(fkA

N

SSNR

ωη

π

222

mmm

p

o

o A)(AakA

N

S

ωη

βπ

ωη

θπ

ωη

π

2

2

2

2222222

Δ

i

i

o

o

N

Sn

N

S 2β

tωcosa)t(m m

SNR in PM Reception (cont’d)• Compared to the AM signal with 100% modulation,

we have

– Conclusion: the output SNR can be made much higher in PM than in DSB-LC (AM) by increasing the modulation index .

– Assumption: (1) ; (2) .– Expense: an increase in also increases the signal

bandwidth.

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AMo

o

AMi

c

PMo

o

N

S

N

S

N

S

22 ββ

β

A)t(n&)t(n sc 1ββ

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Comparison of CW Modulation Systems

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Contents

• Sampling

– periodic sampling

– frequency domain representation of sampling

– Reconstruction of a bandlimited signal from samples

– change the sampling rate using discrete-time processing

• Quantization

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Periodic Sampling

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• ：sample period• 1/ :sample rate• Ω 2 / :sample rate

Mathematical Model for Ideal Sampling

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[ ] ( ) | ( )c t nT cx n x t x nT

( )n

t nT

( ) ( ) ( )

( ) ( )

s cn

cn

x t x t t nT

x nT t nT

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Frequency Domain Representation• Fourier transform

• Nyquist sampling theorem: for a band-limited signal– 2 , no aliasing– o.w., aliasing

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*

/

s c

n

X f X f S f

X f n T

Aliasing

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0 0~ 2k

0 2 4 6 8 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

Am

plit

ude

cos(0.1n), continuous time


cos(0.1n), discrete time


0 2 4 6 8 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

Am

plit

ude





[ ] cosx n A n

0 0~ 2k

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Symmetry

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0 2 4 6 8 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

Am

plit

ude





0 0~ 2k

Anyreal ∗ conjugatesymmetric

[ ] cosx n A n

0 00.1 ~ 0.1

0 00.1 ~ 2 1.9

Reconstruct Bandlimited Signal From Samples

• Impulse-train modulated signal

• Pass through LPF with impulse response to reconstruct

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Ideal reconstruction

filter

Convert from sequence to

impulse train

sn

x t x n t nT

r rn

x t x n h t nT

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Ideal Reconstruction Filter• Ideal LPC with cut of frequency: / or 2/

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sin /

/r

t Th t

t T

Reconstructed Signal

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sin /

/rn

t nT Tx t x n

t nT T

j Tr rX j X e H j

Sinc function is 1 at t=0

Sinc function is 0 at nT

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Example• Reconstruct the sinusoidal signal from samples

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0 2 4 6 8 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

Am

plit

ude





[ ] cos 0.1x n n

Exercise• Sampling the signal cos 4000

– 1/6000

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Exercise• Sampling the signal cos 4000

– 1/1500

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Thanks for your kind attention!

Questions?

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EE140 Introduction to Communication Systems Lecture...

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