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Meixia Tao @ SJTU
Principles of Communications
Meixia Tao
Dept. of Electronic EngineeringShanghai Jiao Tong University
Chapter 3: Analog ModulationSelected from Ch 3, Ch 4.1-4.4, Ch
6.1-6.2 of of Fundamentals of Communications Systems, Pearson
Prentice Hall 2005, by
Proakis & Salehi
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Meixia Tao @ SJTU
Topics to be Covered
AM/FM radio FM radio
Source Modulator Channel Demodulator Output
Amplitude modulation Angle modulation (phase/frequency) Effect
of noise on amplitude modulation Effect of noise on frequency
modulation
TV broadcast
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Meixia Tao @ SJTU
Modulation What is modulation?
Transform a message into another signal to facilitate
transmission over a communication channel
Generate a carrier signal at the transmitter Modify some
characteristics of the carrier with the information
to be transmitted Detect the modifications at the receiver
Why modulation? Frequency translation Frequency-division
multiplexing Noise performance improvement
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Meixia Tao @ SJTU
Analog Modulation Characteristics that be modified in sin
carrier
Amplitude → Amplitude modulation Frequency Phase
→ Angle modulation
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Meixia Tao @ SJTU
Baseband signal (modulating wave):
Carrier wave
Modulated wave
( )m t
( )0( ) ( ) ( ) ( ) cosc cs t c t m t A m t tω θ= = +
Amplitude ModulationDouble-sideband suppressed-carrier AM
(DSB-SC)
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Spectrum of DSB-SC Signals
[ ])()(21)( ccc ffMffMAfS ++−=
0
S(f)
ffc+Wfcfc-W-fc+W-fc-W -fc
(1/2)AcM(0)Spectrum of DSB-SC
USBUSB LSBLSB
W-W 0
M(f)
f
M(0)Spectrum of message
Translation of the original message spectrum to
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Meixia Tao @ SJTU
Bandwidth and Power Efficiency
Required channel bandwidth Required transmit power
0
S(f)
ffc+Wfcfc-W-fc+W-fc-W -fc
(1/2)AcM(0)
2cB W=
[ ]
/2 /22 2 2 20/2 /2
2 2/2 20/2
1 1lim ( ) lim ( )cos ( )
1lim ( ) 1 cos(2 2 )2 2
T T
s c cT TT T
Tc cc mTT
P s t dt A m t t dtT T
A Am t t dt PT
ω θ
ω θ
− −→∞ →∞
−→∞
= = +
= + + =
∫ ∫
∫
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Meixia Tao @ SJTU
Demodulation of DSB-SC Signals Phase-coherent demodulation
If there is a phase error φ, then
)2cos( tfcπ
Product modulator
Local oscillator
Low-pass filter
s(t) v(t) vo(t)
Scaled version of message signal Unwanted
PLL (phase-locked loop)
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Demodulation of DSB-SC Signals Pilot-tone assisted
demodulation
Add a pilot-tone into the transmitted signal
Filter out the pilot using a narrowband filter
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Conventional AM Carrier wave: Baseband signal (normalized):
Modulation index: Modulated wave
a
Modulating wave
Modulated wave 1a ≤
1a >
overmodulated10
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Meixia Tao @ SJTU
Spectrum of Conventional AM
( ) ( ) ( ) ( )( )2 2
c cc c c c
A A aS f f f f f M f f M f fδ δ= − + + + − + +
(Ac/2)δ(f-fc)
0
S(f)
ffc+Wfcfc-W-fc+W-fc-W -fc
(1/2)aAcM(0)(Ac/2)δ(f+fc)
W-W 0
M(f)
f
M(0)Spectrum of message signal
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Meixia Tao @ SJTU
Bandwidth and Power Efficiency Required channel bandwidth
Required transmit power
Modulation efficiency
message powercarrier power
22
2
2 2 22
2=1
2 2
c mm
c mc m
a A P a PEA a a PA P
= =++
power in sideband
total power
2cB W=
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Meixia Tao @ SJTU
Example Message signal: Carrier: Modulation index: a=0.85
Determine the power in the carrier component and in the
sideband components of the modulated signal
( ) 3cos(200 ) sin(600 )m t t tπ π= +5( ) cos(2 10 )c t t−=
×
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Demodulation of AM signalsEnvelop Detector
The simplicity of envelop detector has made Conventional AM a
practical choice for
AM-radio broadcasting
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Meixia Tao @ SJTU
Single Sideband (SSB) AM
Common problem in AM and DSBSC: Bandwidth wastage
SSB is very bandwidth efficient
ω
H(ω)
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Expression of SSB signals The baseband signal can be written as
the sum of finite
sinusoid signals
Then its USB component is
After manipulation
1( ) cos(2 ) ,
n
i i i i ci
m t x f t f fπ θ=
= + ≤∑
[ ]1
( ) cos 2 ( ) )2
nc
c i c i ii
Am t x f f tπ θ=
= + +∑
1 1( ) cos(2 ) cos 2 sin(2 ) sin 2
2
( )cos 2 ( )sin 22 2
n nc
c i i i c i i i ci i
c cc c
Am t x f t f t x f t f t
A Am t f t m t f t
π θ π π θ π
π π
= =
= + − +
= −
∑ ∑
Hilbert transform of m(t)
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About Hilbert Transform
1 ( ) 1( ) ( )xx t d x tt tτ τ
π τ π∞
−∞= = ∗
−∫( ) ( )x t x t⇔
1
ω
)(ωH
ω
相移
90°
-90°
, 0( ) , 0
0, 0
j fH f j f
f
− >=
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Meixia Tao @ SJTU
Generation of SSB-AM Signal
• The spectral efficiency of SSB makes it suitable for voice
communication over telephone lines (0.3~3.4 kHz)
• Not suitable for signals with significant low frequency
components
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Vestigial Sideband: VSB VSB is a compromise between SSB and
DSBSC
W-W 0
M(f)
f
0
S(f)
ffc
fv W
-fc
fvW
VSB signal bandwidth is B = W+fv
VSB is used in TV broadcasting and similar signals where low
frequency components are significant
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Meixia Tao @ SJTU
Comparison of AM Techniques DSB-SC:
more power efficient. Seldom used Conventional AM:
simple envelop detector. AM radio broadcast
SSB: requires minimum transmitter power and bandwidth. Suitable
for point-to-point and over long distances
VSB:bandwidth requirement between SSB and DSBSC. TV
transmission
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Signal Multiplexing Multiplexing is a technique where a number
of
independent signals are combined and transmitted in a common
channel
These signal are de-multiplexed at the receiver Two common
methods for signal multiplexing
TDM (time-division multiplexing) FDM (frequency-division
multiplexing)
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FDM
LPF: ensure signal bandwidth limited to W
MOD (modulator): shift message frequency range to mutually
exclusive high frequency bands
BPF: restrict the band of each modulated wave to its prescribed
range
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FDM application in telephone comm.
Voice signal: 300~3400Hz Message is SSB modulated. In 1st-level
FDM, 12 signal are stacked in frequency, with a freq.
separation of 4 kHz between adjacent carriers A composite 48 kHz
channel, called a group channel, transmits 12
voice-band signals simultaneously In the next level of FDM, a
number of group channel (typically 5
or 6) are stacked to form a supergroup channel Higher-order FDM
is obtained by combining several supergroup
channels => FDM hierarchy in telephone comm. systems
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Meixia Tao @ SJTU
Quadrature-Carrier Multiplexing Transmit two messages on the
same carrier as
cos() and sin() are two quadrature carriers Each message signal
is modulated by DSB-SC Bandwidth-efficiency comparable to
SSB-AM
Synchronous demodulation of m1(t):
( ) ( )1 2( ) ( ) cos 2 ( )sin 2c c c cs t A m t f t A m t f tπ
π= +
( ) ( ) ( ) ( )
( ) ( )
21 2
1 1 2
( ) cos 2 ( )cos 2 ( )sin 2 cos 2
( ) ( ) cos 4 ( )sin 42 2 2
c c c c c c
c c cc c
s t f t A m t f t A m t f t f tA A Am t m t f t m t f t
π π π π
π π
= +
= + +
LPF24
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Meixia Tao @ SJTU
Application: AM Radio Broadcasting Commercial AM radio uses
conventional AM Superheterodyne receiver:
from variable carrier freq of the incoming RF to fixed IF
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Topics to be Covered
Source Modulator Channel Demodulator Output
Amplitude modulation Angle modulation (phase/frequency) Effect
of noise on amplitude modulation Effect of noise on frequency
modulation
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Meixia Tao @ SJTU
Angle Modulation Either phase or frequency of the carrier is
varied according
to the message signal
The general form:
θ(t): the time-varying phase
instantaneous frequency of s(t):
1 ( )( )2i c
d tf t fdtθ
π= +
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Representation of FM and PM signals Phase modulation (PM)
Frequency modulation (FM)
The phase of FM is0
( ) 2 ( )t
ft k m dθ π τ τ= ∫
where kf = frequency deviation constant/frequency
sensitivity
1( ) ( ) ( )2i c f
df t f k m t tdtθ
π− = =
( ) ( )pt k m tθ = where kp = phase deviation constant
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Distinguishing Features of PM and FM No perfect regularity in
spacing of zero crossing
Constant envelop, i.e. amplitude of s(t) is constant
Relationship between PM and FM
Will discuss the properties of FM only
[ ]∫+ dttmktfA pcC )(2cos πintegrator
m(t)
)2cos( tfA cC π
Phasemodulator
∫ dttm )( FM wave
differentiator Frequencymodulatorm(t)
)2cos( tfA cC π
)(tmdtd
[ ])(22cos tmktfA fcC ππ +PM wave
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Example: Sinusoidal ModulationSinusoid modulating wave m(t)
FM wave
PM wave
)(tmdtd
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Example: Square ModulationSquare modulating wave m(t)
FM wave
PM wave
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FM by a Sinusoidal Signal Message
Instantaneous frequency of resulting FM wave
Frequency deviation: Carrier phase
Modulation index:
( )0
( ) 2 ( ) sin(2 )
sin(2 )
t
i c mm
m
ft f f d f tf
f t
θ π τ τ π
β π
∆= − =
=
∫
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Meixia Tao @ SJTU
Example
Problem: a sinusoidal modulating wave of amplitude 5V and
frequency 1kHz is applied to a frequency modulator. The frequency
sensitivity is 40Hz/V. The carrier frequency is 100kHz.
Calculate (a) the frequency deviation(b) the modulation
index
Solution: Frequency deviation
Modulation index
HzAkf mf 200540 =×==∆
2.01000200
==∆
=mffβ
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Spectrum Analysis of Sinusoidal FM Wave
Rewrite the FM wave as
Define the complex envelop of FM wave
retains complete information about s(t)
)2sin()()()(~ tfjcQI meAtjststsπβ=+=
)(~ ts[ ]{ } [ ]tfjtftfjc cmc etseAts ππβπ 2)2sin(2 )(~ReRe)( ==
+
In-phase component Quadrature-phase component
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Meixia Tao @ SJTU
is periodic, expanded in Fourier series as
where
n-th order Bessel function of the first kind
Hence,
∑∞
−∞=
=n
tnfjn
mects π2)(~
( )1( ) exp sin2n
J j x nx dxπ
πβ β
π −= − ∫
)(βncn JAc =
)2sin()(~ tfjc meAtsπβ=
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Meixia Tao @ SJTU 36
Substituting into
FM wave in time domain
FM wave in frequency-domain
)(~ ts
( )∑∞
−∞=
=n
mnc tnfjJAts πβ 2exp)()(~
[ ])()()(2
)( mcmcn
nc
c nfffnfffJAfS +++−−= ∑∞
−∞=
δδβ
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Meixia Tao @ SJTU
Properties of Bessel Function
∞→→ ββ as0)(nJ
−
=− odd,)(even,)(
)(nJ
nJJ
n
nn β
ββ
Property 1: for small β ≤0.3 (Narrowband FM) Approximations
Substituting above into 1,0)(
2)(1)(
1
0
>≈≈≈
nJJJ
n βββ
β
[ ]
[ ]tffAtffAtfAts
mcc
mcc
cc
)(2cos2
)(2cos2
)2cos()(
−−
++≈
πβ
πβπ
Approximate bandwidth =Discuss the similarity between the
conventional AM wave and
a narrow band FM wave37
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Meixia Tao @ SJTU
General Case Goal: to investigate how and affect the spectrum
Fix and vary
and are varied
f∆2
5=β
cf
1.0
f∆2
1=β
cf
1.0
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Meixia Tao @ SJTU
Fix and vary is fixed, but is varied
f∆2
1=β
cf
1.0
f∆2
5=β
cf
1.0
General Case
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Meixia Tao @ SJTU
Effective Bandwidth of FW Waves
For large B is only slightly greater than
For small The spectrum is limited to
Carson’s Rule:
2 2 2(1 )m mB f f fβ≈ ∆ + = +
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99% bandwidth approximation The separation between the two
frequencies beyond which
none of the side-frequencies is greater than 1% of the
unmodulated carrier amplitude
i.e where is the max that satisfies
01.0)( >βnJ
β 0.1 0.3 0.5 1.0 2.0 5.0 10 20 30
2nmax 2 4 4 6 8 16 28 50 70
Effective Bandwidth of FW Waves
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Meixia Tao @ SJTU
A universal curve for evaluating the 99% bandwidth As increases,
the bandwidth occupied by the significant side-
frequencies drops toward that over which the carrier frequency
actually deviates, i.e. B become less affected by
20
2
0.2 2
Effective Bandwidth of FW Waves
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FM by an Arbitrary Message Consider an arbitrary with highest
freq. component W
Frequency deviation:
Modulation index:
Carson’s rule applies as
Carson’s rule underestimates the FM bandwidth requirement
Universal curve yields a conservative result
max ( )ff k m t∆ =
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Meixia Tao @ SJTU
Example
In north America, the maximum value of frequency deviation is
fixed at for commercial FM broadcasting by radio.
Take , typically the maximum audio frequency of interest in FM
transmission, the modulation index is
Using Carson’s rule,
Using universal curve,
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Meixia Tao @ SJTU
Exercise
Assuming that , determine the transmission bandwidth of an FM
modulated signal with
Solution By Carson’s rule:
( )4( ) 10sinc 10m t t=
4000fk =
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Meixia Tao @ SJTU
Application: FM Radio broadcasting As with standard AM radio,
most FM radio receivers are
of super-heterodyne type
Limiter
discriminator
Audio amplifier with de-emphasis
Baseband low-pass filter
loudspeaker
Typical freq parameters– RF carrier range = 88~108 MHz– Midband
of IF = 10.7MHz– IF bandwidth = 200kHz– Peak freq. deviation =
75KHz
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Meixia Tao @ SJTU
Summary Spectrum of sinusoidal FM Wave
Carson rule approximation Universal curve approximation
[ ])()()(2
)( mcmcn
nc
c nfffnfffJAfS +++−−= ∑∞
−∞=
δδβ
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Meixia Tao @ SJTU
Generation of FM waves Direct approach
Design an oscillator whose frequency changes with the input
voltage => voltage-controlled oscillator (VCO)
Indirect approach First generate a narrowband FM signal and
then
change it to a wideband signal Due to the similarity of
conventional AM signals, the
generation of a narrowband FM signal is straightforward.
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Generation of Narrow-band FM
Consider a narrow band FM wave
where
Given φ1(t)
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Meixia Tao @ SJTU
)()()()( 1212112 tsatsatsats
nn+++=
integratorm(t)
)2sin( 11 tfA π
ProductModulator
-900 phaseshifter )2cos( 11 tfA π
Carrier wave
+-
+Narrow-bandFM wave s1(t)
Narrow-band frequency modulator
Next, pass s1(t) through a frequency multiplier
– The input-output relationship of the non-linear device is:
– The BPF is used to Pass the FM wave centred at nf1 and with
deviation n∆f1 and suppress all other FM spectra
Memorylessnonlinear device
Narrow-bandFM wave
Band-passfilter
Wideband FMWave
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Meixia Tao @ SJTU
Example: frequency multiplier with n = 2
Problem: Consider a square-law device based frequency
multiplier
with
Specify the midband freq. and bandwidth of BPF used in the freq.
multiplier for the resulting freq. deviation to be twice that at
the input of the nonlinear device
Solution:
)()()( 212112 tsatsats +=
+= ∫
tdmktfAts
01111)(22cos)( ττππ
+++
+=
++
+=
∫∫
∫∫tt
tt
dmktfAaAadmktfAa
dmktfAadmktfAats
011
212
212
01111
01122
12011112
)(44cos22
)(22cos
)(22cos)(22cos)(
ττππττππ
ττππττππ
Removed by BPF withfc=2f1BW > 2∆f = 4∆f1
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Meixia Tao @ SJTU
Integrator
Messagesignal Narrow-band
phasemodulator
WidebandFM signalFrequency
multiplier
Crystal-controlledoscillator
+= ∫
t
fcc dmktfAts 0 )(22cos)( ττππ
1
1
1
fnfnkknff
f
c
∆=∆
==
+= ∫
tdmktfAts
01111)(22cos)( ττππ
1cos(2 )cA f tπ
Generation of Wideband FM Signal
Output
)2cos( tflπ
BPF
Mixer: perform up/down conversion to shift the signal to the
desired center freq.
may not be the desired carrier frequency
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Meixia Tao @ SJTU
Exercise: A typical FM transmitter
Problem: Given the simplified block diagram of a typical FM
transmitter used to transmit audio signals containing frequencies
in the range 100Hz to 15kHz.
Desired FM wave: fc = 100MHz, ∆f = 75kHz. Set β1 = 0.2 in the
narrowband phase modulation to limit harmonic
distortion. Specify the two-stage frequency multiplier factors
n1 and n2
Integrator
Messagesignal Narrow-band
phasemodulator
FM signalFrequencymultiplier
n1
Crystal-controlledoscillator
Mixer
Crystal-controlledoscillator
Frequencymultiplier
n2
0.1MHz ?
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Meixia Tao @ SJTU
Demodulation of FMBalanced Frequency Discriminator
Given FM wave
Differentiator + envelop detector = FM demodulator Frequency
discriminator: a “freq to amplitude” transform device
FMwave
Basebandsignal
Envelopdetector
Slope circuitH1(f)
Slope circuitH2(f)
Envelopdetector
∑+
-
( )( )
+−≤≤−−−++≤≤−+−
=elsewhere ,0
2/2/,2/22/2/ ,2/2
)(1 BffBfBffajBffBfBffaj
fH cccccc
ππ
)()( 12 fHfH −=
+= ∫
t
fcc dmktfAts 0 )(22cos)( ττππ
0( ) 2 2 ( ) sin 2 2 ( )
t
c c f c fd s t A f k m t f t k m ddt
π π π π τ τ = − + + ∫Hybrid-modulated wave with AM and FM
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Meixia Tao @ SJTU
Circuit diagram and frequency response
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Meixia Tao @ SJTU
Application: FM Radio broadcasting As with standard AM radio,
most FM radio receivers are
of super-heterodyne type
Limiter
discriminator
Audio amplifier with de-emphasis
Baseband low-pass filter
loudspeaker
Typical freq parameters– RF carrier range = 88~108
MHz– Midband of IF = 10.7MHz– IF bandwidth = 200kHz– Peak freq.
deviation = 75KHz
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Meixia Tao @ SJTU10/4/2004 57
FM Radio Stereo Multiplexing Stereo multiplexing is a form of
FDM
designed to transmit two separate signals via the same
carrier.
Widely used in FM broadcasting to send two different elements of
a program (e.g. vocalist and accompanist in an orchestra) so as to
give a spatial dimension to its perception by a listener at the
receiving end
∑
∑
Frequencydoubler
K
∑
+
++
-
ml(t)
mr(t)+
+ m(t)
)2cos( tf cπ
[ ][ ]
)2cos()4cos()()(
)()()(
tfKtftmtm
tmtmtm
c
crl
rl
ππ
+−+
+=
The sum signal is left unprocessed in its baseband form
The difference signal and a 38-kHz subcarrier produce a DSBSC
wave
The 19-kHz pilot is included as a reference for coherent
detection
fc = 19kHz
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Meixia Tao @ SJTU10/4/2004 58
FM-Stereo Receiver
Baseband LPF
BPF centered at 2fc=38kHz
Narrow-band filter tuned to
fc=19kHz
Frequency doubler
m(t)
∑
∑
+
++
-
2ml(t)
2mr(t)
Baseband LPF
ml(t)+mr(t)
ml(t)-mr(t)
To two loudspeakers
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Meixia Tao @ SJTU
Think …
Compared with AM, FM requires a higher implementation complexity
and a higher bandwidth occupancy. What is the advantage of FM
then?
AM vs. FM
Why AM radio is mostly for news broadcasting while FM radio is
mostly for music program
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Meixia Tao @ SJTU
Suggested Reading Chapter 3 and Chapter 4.1-4.4 of Fundamentals
of
Communications Systems, Pearson Prentice Hall 2005, by Proakis
& Salehi
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Meixia Tao @ SJTU
Topics to be Covered
Source Modulator Channel Demodulator Output
Amplitude modulation Angle modulation (phase/frequency) Effect
of noise on amplitude modulation Effect of noise on frequency
modulation
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Meixia Tao @ SJTU
A Benchmark System Baseband system:
No carrier demodulation The receiver is an ideal LPF with
bandwidth W Noise power at the output of the receiver
Baseband SNR is given by
+ LPFn(t)
)(tsm
0
002
W
n W
NP df N W−
= =∫
0
R
b
PSN N W
=
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Meixia Tao @ SJTU
Example: Find the SNR in a baseband system with a bandwidth
of 5 kHz and with W/Hz. The transmitter power is 1kW and the
channel attenuation is
Solution:
140 10N
−=1210−
12 3 910 10 10 WattsRP− −= × =
9
140
10 2010 5000
R
b
PSN N W
−
−
= = = ×
1010 log 20 13dB= =
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Effect of Noise on DSBSC
Modulated signal Input to the demodulator
+ BPF demodn(t)
( )s t
)(tni
)(0 tm
)(0 tn( ) ( ) cosc cs t A m t tω=
( ) ( ) ( )( ) cos ( ) cos ( )sin
i
c c c c s c
r t s t n tA m t w t n t w t n t w t
= += + −
Here ni(t) is a Gaussian narrow-band noise
0 / 2( )0 otherwisei
cn
N f f WS f
− ≤=
( )s t
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In the demodulator, the received signal is first multiplied by a
locally generated sinusoid signal
Assume coherent detector, we have
( ) cos( ) ( ) cos cos( )( ) cos cos( ) ( )sin cos( )
c c c c
c c c s c c
r t w t A m t w t w tn t w t w t n t w t w t
φ φφ φ
+ = +
+ + − +
[ ]
[ ]
1 1( )cos ( ) cos(2 )2 21 ( )cos ( )sin21 ( )cos(2 ) ( )sin(2
)2
c c c
c s
c c s c
A m t A m t w t
n t n t
n t w t n t w t
φ φ
φ φ
φ φ
= + +
+ +
+ + − +
0φ =65
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Meixia Tao @ SJTU
Then the signal is passed through a LPF with bandwidth W
The output SNR can thus be defined as
Since the received power of DSBSC in baseband is The output SNR
can be rewritten as
[ ]1( ) ( ) ( )2 c c
y t A m t n t= +
22
0
141 24
oc
c mo c m
o nn
A PP A PSN P WNP
= = =
( ) ( ) ( )c i in n c n c
S f S f f S f f
for f W
= − + +
≤where
2
2c m
RA PP =
0
R
oDSB b
PS SN WN N
= =
DBSSC does not provide any SNR improvement over a simple
baseband systems
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Meixia Tao @ SJTU
Effect of Noise on SSB Modulated signal: Input to the
demodulator
Output of LPF:
Therefore, the output SNR is
( ) ( ) cos ( )sinc c c cs t A m t w t A m t w t= ±
[ ] [ ]( ) ( ) cos ( ) ( ) sinc c c c s cA m t n t w t A m t n t
w t= + + ± −
1( ) ( ) ( )2 2
cc
Ay t m t n t= +
0 / 2 / 2( )0 otherwisei
cn
N f f WS f
− ≤=
( ) ( ) ( )ir t s t n t= + where
22
0
1414
oc
c mo c m
o nn
A PP A PSN P WNP
= = =
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Meixia Tao @ SJTU
But in this case,
Thus,
2R c mP A P=
0
R
oSSB b
PS SN WN N
= =
SNR in an SSB system is equivalent to that of a DSBSC system
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Effect of Noise on Conventional AM Modulated signal
Input to the demodulator
With coherent detector, after mixing and LPF:
Removing DC component
[ ]( ) 1 ( ) cosc cs t A am t w t= +
[ ]( ) ( ) ( )
( ) ( ) cos ( )sini
c c c c s c
r t s t n tA A am t n t w t n t w t
= +
= + + −
[ ]11( ) ( ) ( )2 c c c
y t A A am t n t= + +
[ ]1( ) ( ) ( )2 c c
y t A am t n t= +
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Meixia Tao @ SJTU
Exercise: please show that the output SNR is
Hint: the received signal power is
Modulation efficiency is
2 21 12R c m
P A a P = +
SNR in conventional AM is always smaller than that in
baseband.
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Meixia Tao @ SJTU
Performance of Envelope-Detector Input to the
envelope-detector
Envelope of r(t)
If signal component is much stronger than noise
After removing DC component, we obtain
[ ]( ) ( ) ( ) cos ( )sinc c c c s cr t A A am t n t w t n t w
t= + + −
[ ]2 2( ) ( ) ( ) ( )r c c c sV t A A am t n t n t= + + +
( ) ( ) ( )r c c cV t A A am t n t≈ + +
( ) ( ) ( )c cy t A am t n t= +
At high SNR, performance of coherent detector and envelop
detector is the same
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Meixia Tao @ SJTU
Performance of Envelope-Detector If noise power is much stronger
than the signal power
( ) ( )22 2 2( ) 1 ( ) ( ) ( ) 2 ( ) 1 ( )r c c s c cV t A am t
n t n t A n t am t= + + + + +
( ) ( )2 2 2 22 ( )( ) ( ) 1 1 ( )( ) ( )
c cc s
c s
A n tn t n t am tn t n t
≈ + + + +
Ignore 1st term
( )
( )
2
( )( ) 1 1 ( )( )
( )( ) 1 ( )( )
c cn
n
c cn
n
A n tV t am tV t
A n tV t am tV t
≈ + +
= + +
1 12εε+ ≈ +
2 2( ) ( ) ( )n c sV t n t n t= +
The system is operating below the threshold, no meaningful SNR
can be defined.
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Meixia Tao @ SJTU
Exercise Consider that the message is a WSS r.p M(t) with
autocorrelation function . It is given that .
We want to transmit this message to a destination via a channel
with a 50dB attenuation and additive white noise with PSD . . We
also want to achieve an SNR at the modulator output of at least
50dB.
What is the required transmitted power and the channel bandwidth
if we employ the following modulation schemes? DSB-SC SSB AM with
modulation index = 0.8
2( ) 16sinc (10000 )MR τ τ=max
( ) 6m t =
120( ) / 2 10 W/HznS f N
−= =
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Meixia Tao @ SJTU
Topics to be Covered
Source Modulator Channel Demodulator Output
Amplitude modulation Angle modulation (phase/frequency) Effect
of noise on amplitude modulation Effect of noise on frequency
modulation
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Meixia Tao @ SJTU
Effect of Noise in FM The effect of additive noise is described
by the
changes of frequency, or the changes in the zero-crossings of
the modulated FM single.
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Meixia Tao @ SJTU
Effect of Noise on Angle Modulation
Block diagram of an angle demodulator
Input to the demodulator is
BPFBW=Bc Demod
LPFBW=W
( ) ( )ws t n t+( ) ( ) ( )r t s t n t= + ( )y t
o
SN
[ ]cos ( ) ( )cos ( )sinc c c c s cA t t n t t n t tω φ ω ω= + +
−( )r t =
( ) cos( ( ))n c nR t w t tθ= +
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Meixia Tao @ SJTU
Assume that the signal is much larger than the noise
( )cn t
( )nR t
( )sn t
( )tφ ( )y tθ
( )e tθ () ( )
nt tθ
φ−
( )nR t
( )n tθ
( )yR
t
2 2 ( )cA E n t
cA
( )( )( )
1
( ) ( ) cos ( ) ( )
( )sin ( ) ( )cos ( )
( ) cos ( ) ( )
c n n
n nc
c n n
r t A R t t t
R t t tw t t tg
A R t t t
θ φ
θ φφ
θ φ−
≈ + − ⋅ −
+ + + −
( )( )( ) ( ) sin ( ) ( )nr nc
R tt t t tA
θ φ θ φ= + −
The phase term can be further approximated as
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Meixia Tao @ SJTU
Therefore, the output of the demodulator is
The noise component is inversely proportional to the signal
amplitude Ac. (This is not the case for AM system)
( )( )( ) ( ) ( ) sin ( ) ( )2 2
( ) ( )2
nr f n
c
f n
R td dy t t k m t t tdt dt A
dk m t Y tdt
θ θ φπ π
π
= = + −
= +
Desired signal Noise
( ) [ ]( ) 1( ) sin ( ) ( ) ( ) cos ( ) ( )sin ( )nn n s cc
c
R tY t t t n t t n t tA A
θ φ φ φ= − = −
[ ]1 ( )cos ( )sins cc
n t n tA
φ φ= − (Since is slowly varying)( )tφ
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Meixia Tao @ SJTU
The power spectral density of is 1 ( )2 n
d Y tdtπ
2 22 22
2
22
022
4 cos sin( ) ( ) ( )4
2( )
0 otherwise
n s c
c
Y n nc c
Cn c
c
f S f f S f S fA A
f N f Bf S f AA
π φ φπ
= +
≤= =
xf− xf2TB− 2TB f
( )nfS f( )noS f
W-W
At the output of LPF, the noise is limited to the freq. range
[-W, W]
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Output SNR in FM Now we can determine the output SNR in FM
First, the output signal power is
The output noise power is
Then, the output SNR is
2os f m
P k P=
320 0
2 2
23o
W
n Wc c
N N WP f dfA A−
= =∫
0
2 2
20
32
o
s f c m
o n
P k A PSN P W N W
= = ( )
2
2
3
max ( )f m
b
P SNm t
β =
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Meixia Tao @ SJTU
Key Observations Rewrite
Increasing increases the output SNR, in contrast to AM
Increasing the bandwidth (by Carson rule )
increases the output SNR. Thus, FM provides a way to trade off
bandwidth for transmitted power
Having a large B means having a large noise power. Thus,
Increasing cannot continue improving the performance indefinitely
due to the threshold effect
Increasing the transmitted power increases output SNR in both FM
and AM systems, but the mechanisms are totally different.
β
β
Power content of the normalized message
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Threshold Effect There exists a specific SNR at the input of
the
demodulator below which the signal is not distinguishable from
the noise
0
FM
DSB
0
0
NS
i
i
NS
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Comparison of Analog-Modulation Bandwidth efficiency
SSB is the most bandwidth efficient, but cannot effectively
transmit DC
VSB is a good compromise PM/FM are the least favorable
systems
Power efficiency (reflected in performance with noise) FM
provides high noise immunity Conventional AM is the least power
efficient
Implementation complexity (transmitter and receiver) The
simplest receiver structure is conventional AM FM receivers are
also easy to implement DSB-SC and SSB-SC requires coherent detector
and hence is
much more complicated.
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Think …
Compared with AM, FM requires a higher implementation complexity
and a higher bandwidth occupancy. What is the advantage of FM
then?
AM vs. FM
FM provides high noise immunity. This is why AM radio is mostly
for news broadcasting while FM radio is mostly for music
program.
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Meixia Tao @ SJTU
Suggested Reading Chapter 6.1-6.3 of of Fundamentals of
Communications Systems, Pearson Prentice Hall 2005, by Proakis
& Salehi
85
Principles of CommunicationsTopics to be CoveredModulationAnalog
ModulationAmplitude ModulationSpectrum of DSB-SC SignalsBandwidth
and Power EfficiencyDemodulation of DSB-SC SignalsDemodulation of
DSB-SC SignalsConventional AMSpectrum of Conventional AMBandwidth
and Power EfficiencyExampleDemodulation of AM signals�Envelop
DetectorSingle Sideband (SSB) AM Expression of SSB signalsAbout
Hilbert TransformGeneration of SSB-AM SignalVestigial Sideband:
VSBComparison of AM TechniquesSignal MultiplexingFDMFDM application
in telephone comm.Quadrature-Carrier MultiplexingApplication: AM
Radio BroadcastingTopics to be CoveredAngle
ModulationRepresentation of FM and PM signalsDistinguishing
Features of PM and FMExample: Sinusoidal ModulationExample: Square
ModulationFM by a Sinusoidal SignalExample幻灯片编号 34幻灯片编号 35幻灯片编号
36Properties of Bessel FunctionGeneral CaseGeneral CaseEffective
Bandwidth of FW WavesEffective Bandwidth of FW WavesEffective
Bandwidth of FW WavesFM by an Arbitrary
MessageExampleExerciseApplication: FM Radio
broadcastingSummaryGeneration of FM wavesGeneration of Narrow-band
FM 幻灯片编号 50幻灯片编号 51幻灯片编号 52Exercise: A typical FM
transmitterDemodulation of FM�Balanced Frequency Discriminator幻灯片编号
55Application: FM Radio broadcastingFM Radio Stereo
Multiplexing幻灯片编号 58Think …Suggested ReadingTopics to be CoveredA
Benchmark SystemExample:Effect of Noise on DSBSC 幻灯片编号 65幻灯片编号
66Effect of Noise on SSB幻灯片编号 68Effect of Noise on Conventional
AM幻灯片编号 70Performance of Envelope-DetectorPerformance of
Envelope-DetectorExercise Topics to be CoveredEffect of Noise in FM
Effect of Noise on Angle Modulation幻灯片编号 77幻灯片编号 78幻灯片编号 79Output
SNR in FMKey ObservationsThreshold EffectComparison of
Analog-ModulationThink …Suggested Reading