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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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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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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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Analog Modulation Characteristics that be modified in sin carrier
Amplitude → Amplitude modulation Frequency Phase
→ Angle modulation
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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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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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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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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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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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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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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 t
t 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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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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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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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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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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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 Frequencymodulator
m(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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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
meAtjststs πβ=+=
)(~ ts[ ]{ } [ ]tfjtftfj
ccmc etseAts ππβπ 2)2sin(2 )(~ReRe)( == +
In-phase component Quadrature-phase component
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is periodic, expanded in Fourier series as
where
n-th order Bessel function of the first kind
Hence,
∑∞
−∞=
=n
tnfjn
mects π2)(~
( )1( ) exp sin2nJ j x nx dx
π
πβ β
π −= − ∫
)(βncn JAc =
)2sin()(~ tfjc
meAts πβ=
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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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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
[ ]
[ ]tffA
tffAtfAts
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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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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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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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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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
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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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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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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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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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Summary Spectrum of sinusoidal FM Wave
Carson rule approximation Universal curve approximation
[ ])()()(2
)( mcmcn
nc
c nfffnfffJAfS +++−−= ∑∞
−∞=
δδβ
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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) <<1 with β ≤ 0.3, we may use
Correspondingly, we may approximate s1(t) as
[ ])(2cos)( 1111 ttfAts φπ +=
∫=t
dmkt011 )(2)( ττπφ f1 = carrier frequency
k1 = frequency sensitivity
[ ][ ]
≈≈
)()(sin1)(cos
11
1
ttt
φφφ
( ) ( )( ) ( )∫−=
−=t
dmtfAktfA
ttfAtfAts
011111
111111
)(2sin22cos
)(2sin2cos)(
ττπππ
φππ
Narrow-band FW wave49
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)()()()( 1212112 tsatsatsats n
n+++=
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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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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Integrator
Messagesignal Narrow-band
phasemodulator
WidebandFM signalFrequency
multiplier
Crystal-controlledoscillator
+= ∫
t
fcc dmktfAts0
)(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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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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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 ccc
ccc
ππ
)()( 12 fHfH −=
+= ∫
t
fcc dmktfAts0
)(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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Circuit diagram and frequency response
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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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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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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
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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Suggested Reading Chapter 3 and Chapter 4.1-4.4 of Fundamentals of
Communications Systems, Pearson Prentice Hall 2005, by Proakis & Salehi
Chapter 6.1-6.3 of Fundamentals of Communications Systems, Pearson Prentice Hall 2005, by Proakis & Salehi
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