Post on 16-Mar-2020
Chapter 2 Continuous-Wave
Modulation
“Analog Modulation” is the subject concerned
in this chapter.
2.1 Introduction
Analog communication system
The most common carrier is the sinusoidal wave.
Carrier wave
(Analog)
2.1 Introduction
Modulation
A process by which some characteristic of a carrier
is varied in accordance with a modulating wave
(baseband signal).
Sinusoidal Continuous-Wave (CW) modulation
Amplitude modulation
Angle modulation
2.1 Introduction
Amplitude Modulation
Frequency Modulation
Baseband signal
Sinusoidal carrier
2.2 Double-Sideband with Carrier or simply
Amplitude Modulation
Two required conditions on amplitude sensitivity
1 + ka m(t) 0, which is ensured by |ka m(t)| ≦ 1. The case of |ka m(t)| > 1 is called overmodulation.
The value of |ka m(t)| is sometimes represented by “percentage” (because it is limited by 1), and is named (|ka m(t)|100)% modulation.
fc >> W, where W is the message bandwidth. Violation of this condition will cause nonvisualized envelope.
index modulationor y sensitivit amplitude is where
),2cos()](1[)( Signal Modulated
)( Baseband
)2cos()(Carrier
a
cac
cc
k
tftmkAts
tm
fAtc
2.2 Overmodulation
overmodulaion 1|)(| tmka
2.2 Example of Non-Visualized Envelope
2.2 Example of Visualized Envelope
2.2 Transmission Bandwidth
)()(2
)()(2
)(
)2cos()](1[)(
cc
ca
cc
c
cac
ffMffMAk
ffffA
fS
tftmkAts
2W
Transmission bandwidth BT = 2W.
2.2 Transmission Bandwidth
Transmission bandwidth of an AM wave
For positive frequencies, the highest frequency
component of the AM wave equals fc + W, and the
lowest frequency component equals fc – W.
The difference between these two frequencies
defines the transmission bandwidth BT for an AM
wave.
2.2 Transmission Bandwidth
The condition of fc > W ensures that the sidebands
do not overlap.
2.2 Negative Frequency
Operational meaning of “negative frequency”
in spectrum
If time-domain signal is real-valued, the negative
frequency spectrum is simply a mirror of the
positive frequency spectrum.
We may then define a one-sided spectrum as
Hence, if only real-valued signal is considered, it is
unnecessary to introduce “negative frequency.”
.0for )(2)(sided-one
ffSfS
2.2 Negative Frequency
So the introduction of negative frequency part is
due to the need of imaginary signal part.
Signal phase information is embedded in imaginary
signal part of the signal.
phase
mI(t)
mQ(t)
2.2 Negative Frequency
As a result, the following two spectrums contain the
same frequency components but different phases
(90 degree shift in complex plane).
fc
fc
fc
fc
)(c
ff )(c
ff
)(c
ff
)(c
ff
)2cos(2 tfc
)2sin(2 tfjc
2.2 Negative Frequency
Summary
Complex-valued baseband signal consists of
information of amplitude and phase; while real-
valued baseband signal only contains amplitude
information.
One-sided spectrum only bears amplitude
information, while two-sided spectrum (with
negative frequency part) carries also phase
information.
2.2 Virtues of Amplitude Modulation
AM receiver can be implemented in terms of simple
circuit with inexpensive electrical components.
E.g., AM receiver
)]4cos(1[)](1[2
)2(cos)](1[)()(
2
2
2222
1
tftmkA
tftmkAtstv
ca
c
cac
2.2 Virtues of Amplitude Modulation
The bandwidth of m2(t) is twice of m(t). (So to
speak, the bandwidth of m(f)*m(f) is twice of m(f).)
Lowpass filter
2.2 Virtues of Amplitude Modulation
So if 2fc > 4W,
)(2
mean zero is )( if
)](1[2
)(
)](1[2
)(
block DC
3
22
2
tmkA
tm
tmkA
tv
tmkA
tv
ac
ac
ac
By means of a squarer, the receiver can recover the information-
bearing signal without the need of a local carrier.
2.2 Limitations of Amplitude Modulation
(DSB-C)
Wasteful of power and bandwidth
Waste of power in the information-less “with-carrier” part.
)2cos()()2cos(
)2cos()](1[)(
carrier with
tftmktfA
tftmkAts
cacc
cac
2.2 Limitations of Amplitude Modulation
Wasteful of power and bandwidth
Only requires half of bandwidth after modulation
2.3 Linear Modulation
Definition
Both sI(t) and sQ(t) in
s(t) = sI(t)cos(2fct) – sQ(t) sin(2fct)
are linear function of m(t).
2.3 Linear Modulation
For a single real-valued m(t), three types of modulations can be identified according to how sQ(t) are linearly related to m(t), at the case that sI(t) is exactly m(t): (Some modulation may have mI(t) and mQ(t) that
respectively bear independent information.)
1. Double SideBand-Suppressed Carrier modulation (DSB-SC)
2. Single SideBand (SSB) modulation
3. Vestigial SideBand (VSB) modulation
2.3 DSB-SC and SSB
Type of modulation sI(t) sQ(t)
DSB-SC m(t) 0
SSB m(t) Upper side band transmission
SSB m(t) Lower side band transmission
)(ˆ tm
)(ˆ tm
DSB-SC
SSB
SSB
)( of ansformHilbert tr)(ˆ* tmtm
usb
lsb
, which is used to completely “suppress” the other sideband.
2.3 DSB-SC
Different from DSB-C, DSB-SC s(t) undergoes
a phase reversal whenever m(t) crosses zero.
)2cos()()( tftmtsc
Require a receiver that can
recognize the phase reversal!
2.3 Coherent Detection for DSB-SC
For DSB-SC, we can no longer use the “envelope
detector” (as used for DSB-C), in which no local
carrier is required at the receiver.
The coherent
detection or
synchronous
demodulation
becomes necessary.
2.3 Coherent Detection for DSB-SC
. provided
),()cos(2
1 '
Wf
tmAA
c
cc
LowPass
)()cos(2
1)()4cos(
2
1
)()2cos()2cos(
)()2cos()(
''
'
'
tmAAtmtfAA
tmtftfAA
tstfAtv
ccccc
cccc
cc
2.3 Coherent Detection for DSB-SC
Quadrature null effect of the coherent detector.
If = /2 or –/2, the output of coherent detector for DSB-
SC is nullified.
If is not equal to either /2 or –/2, the output of coherent
detector for DSB-SC is simply attenuated by a factor of
cos(), if is a constant, independent of time.
However, in practice, often varies with time; therefore, it is
necessary to have an additional mechanism to maintain the
local carrier in the receiver in perfect synchronization with
the local carrier in the transmitter.
Such an additional mechanism adds the system complexity of
the receiver.
2.3 Costas Receiver for DSB-SC
An exemplified
design of
synchronization
mechanism is the
Costas receiver,
where two
coherent
detectors are
used.
In-phase coherent detector
Quadrature-phase coherent detector
2.3 Costas Receiver for DSB-SC
Conceptually, the Costas receiver adjusts the
phase so that it is close to 0.
When drifts away from 0, the Q-channel output
will have the same polarity as the I-channel output
for one direction of phase drift, and opposite
polarity for another direction of phase drift.
The phase discriminator then adjusts through the
voltage controlled oscillator.
-pi/2 0 pi/2
)sin(
2.3 Single-Sideband Modulation
How to generate SSB signal?
1. Product modulator to generate DSB-SC signal
2. Band-pass filter to pass only one of the sideband
and suppress the other.
The above technique may not be applicable to a
DSB-SC signal like below. Why?
SSB
filter filter
2.3 Single-Sideband Modulation
For the generation of a SSB modulated signal
to be possible, the message spectrum must have
an energy gap centered at the origin.
2.3 Single-Sideband Modulation
Example of signal with 300 Hz ~ 300 Hz energy gap
Voice : A band of 300 Hz to 3100 Hz gives good articulation
Also required for SSB modulation is a highly selective filter
2.3 Single-Sideband Modulation
Phase synchronization is also an important
issue for SSB demodulation. This can be
achieved by:
Either a separate low-power pilot carrier
Or a highly stable local oscillator (for voice
transmission)
Phase distortion that gives rise to a Donald Duck voice
effect is relatively insensitive to human ear.
2.3 Vestigial Sideband Modulation
Instead of transmitting only one sideband as
SSB, VSB modulation transmits a partially
suppressed sideband and a vestige of the other
sideband.
SSB
VSB
Still, no information loss in principle.
2.3 Requirements for VSB filter
1. The sum of values of the magnitude response |H(f)| at any
two frequencies equally displaced above and below fc is
unity. I.e., |H(fc f)| + |H(fc f)| = 1 for fv < f < fv.
2. H(f fc) + H(f + fc) = 1 for W < f < W.
fc fc
So the transmission band of VSB filter is BT = W + fv.
f f+fc ffc
2.3 Generation of VSB Signal
Analysis of VSB
Give a real baseband signal m(t) of bandwidth W.
Then,
Let where
.||for 0)( and )()( * WffMfMfM
,2/]/)(1)[()( jfHfMfMQVSB
).()( and ,
0,0
0),1,0(
,1
)(1 * fHfH
f
ff
ff
fHj
QQv
v
Q
<<
The filter is denoted by HQ because it is used to generate sQ(t) (cf. Slide 2-23)
2.3 Generation of VSB Signal
1.0
W
)(1
)( fHj
fLQQ
2/)(1
12/)](1[
fH
jfL
).(
)()(1
)(1
)(1
)(real. is )(1
)( *
*
*
fL
fLfHj
fHj
fHj
fLfHj
fL
Q
QQQQQQQ
2.3 How to recover from VSB signal?
).()( because ),(
)]()(2)[(2
1
)()( and real, is )](1[ since
,)](1)[()](1)[(2
1
)](1)[()](1)[(2
1
)()(
*
*
fLfLfM
fLfLfM
fMfMfL
fLfMfLfM
fLfMfLfM
fMfM
Q
VSBVSB
2.3 VSB upper sideband transmission
)(tmVSB
)(tsVSB
)()(22
1)(
}{)](1[}{)](1[2
1)(
}{2
)](1[}{
2
)](1[)(
}{2
)](1[}{
2
)](1[
)()(2
1
}{2
)](1[}{
2
)](1[
}{)(}{)(2
1
*
*cont.
cQcQDSB
ccQccQDSB
c
cQ
c
cQ
DSB
c
cQ
c
cQ
cc
c
cQ
c
cQ
cccccc
ffLffLfs
WffffLWffffLfs
WffffL
WffffL
fs
WffffL
WffffL
ffMffM
WffffL
WffffL
WffWfffMWffWfffM
11
11
11
11
11
(See next slide.)
.0)()()()( if
)]()()][()([)()()()(
fFfMfFfM
fFfFfMfMfFfMfFfM
LRRL
RLRLRRLL
}{)](1[ WffffL ccQ 1
}{)](1[ WffffL ccQ 12
2
2
}{)](1[
}{)](1[
WffffL
WffffL
ccQ
ccQ
1
1
)( cQ ffL
)( cQ ffL
1
1
2 2)()( cQcQ ffLffL
)()()(
2)()(2
1)()(
fHfsfs
ffLffLfsfs
DSBVSB
cQcQDSBVSB
2)()(2
1)(
cQcQffLffLfH
Consequently,
1 )( fH
1 )(
cffH
1
)(c
ffH
1
)()(cc
ffHffH
1
)()(cc
ffHffH
WfffHffHjfH
WfffHffHfL
ccQ
ccQ
||for )]()([)(
||for )()()(
2.3 Mathematical Representation of VSB
signal
)('2
1)(
2
1
)()(2
1)(
2
1
2/)](1)[()(
fjMfM
fHfjMfM
fjHfMfM
Q
QVSB
).()()()()(' where fLfjMfHfMfM QQ
.)]('[)]()([)()()(
)()()()()('
*****
*
fMfLfjMfLfMj
fLfjMfLfjMfM
Transform. Hilbert of extension an is This real. is )(' Notably, tm
2.3 Application of VSB Modulation
Television Signals
1. The video signal exhibits a large bandwidth and
significant low-frequency content.
Hence, no energy gap exists (SSB becomes
impractical).
VSB modulation is adopted to save bandwidth.
Notably, since a rigid control of the transmission VSB
filter at the very high-power transmitter is expensive, a
“not-quite” VSB modulation is used instead (a little
waste of bandwidth to save cost).
2.3 Application of VSB Modulation
VSB Filter for Television Signal Transmissions
55.25 MHz 59.75 MHz
2.3 Application of VSB Modulation
As the transmission signal is not quite VSB modulated,
the receiver needs to “re-shape” the received signal
before feeding it to a VSB demodulator.
2.3 Application of VSB Modulation
2. In order to save the cost of the receiver (i.e., in
order to use envelope detector at the receiver), an
additional carrier is added.
Notably, additional carrier does not increase
bandwidth, but just add transmission power.
)2sin()('2
1)2cos()(
2
11
)2sin()(')2cos()(2
1)2cos()(
tftmAktftmkA
tftmtftmkAtfAts
ccacac
ccaccc
2.3 Application of VSB Modulation
Distortion of envelope detector
Distortion
22
2
2LowP ass
2
22222
2
22
))('(4
1)(
2
11
2
1
)2cos()2sin()(2
11)('
2
1
)2(sin))('(4
1)2(cos)(
2
11)(
tmktmkA
tftftmktmAk
tftmAktftmkAts
aac
ccaca
ccacac
The distortion can be compensated by reducing the amplitude
sensitivity ka or increasing the width of the vestigial sideband. Both
methods are used in the design of Television broadcasting system.
2.3 Extension Usage of DSB-SC
Quadrature-Carrier Multiplexing or Quadrature
Amplitude Modulation (QAM)
Synchronization is critical in QAM modulation, which is often achieved by a
separate low-power pilot tone outside the passband of the modulated signal.
2.4 Frequency Translation
The basic operation of SSB modulation is simply a special case of frequency translation.
So SSB modulation is sometimes referred to as frequency changing, mixing, or heterodyning.
The mixer is a device that consists of a product modulator followed by a band-pass filter, which is exactly what SSB modulation does.
2.4 Frequency Translation
The process is named upconversion, if f1 + f is the wanted signal, and f1 – f is the unwanted image signal.
The process is named downconversion, if f1 f is the wanted signal, and f1 + f is the unwanted image signal.
2.5 Frequency-Division Multiplexing
Multiplexing is a technique to combine a
number of independent signals into a composite
signal suitable for transmission.
Two conventional multiplexing techniques
Frequency-Division Multiplexing (FDM)
Time-Division Multiplexing (TDM)
Will be discussed in Chapter 3.
2.5 Frequency-Division Multiplexing
Example 2.1
2.6 Angle Modulation
Angle modulation
The angle of the carrier is varied in accordance with the baseband signal.
Angle modulation provides us with a practical means of exchanging channel bandwidth for improved noise performance.
So to speak, angle modulation can provide better discrimination against noise and interference than the amplitude modulation, at the expense of increased transmission bandwidth.
2.6 Angle Modulation
Commonly used angle modulation
Phase modulation (PM)
Frequency modulation (FM)
y.sensitivit phase is where)],(2cos[)(ppcc
ktmktfAts
y.sensitivitfrequency is re whe
,)(22cos
))((2cos)(
0
0
f
t
fcc
t
fcc
k
dmktfA
dmkfAts
2.6 Angle Modulation
Main differences between Amplitude
Modulation and Angle Modulation
1. Zero crossing spacing of angle modulation no
longer has a perfect regularity as amplitude
modulation does.
2. Angle modulated signal has constant envelope; yet,
the envelope of amplitude modulated signal is
dependent on the message signal.
2.6 Angle Modulation
Similarity between PM and FM
PM is simply an FM with in place of m(t).
Hence, the text only discusses FM in this chapter.
t
dm0
)(
t
fccFMdmktfAts
0)(22cos)(
)](2cos[)( tmktfAtspccPM
2.7 Frequency Modulation
s(t) of FM modulation is a non-linear function of m(t).
So its general analysis is hard.
To simplify the analysis, we may assume a single-tone
transmission, where
)2cos()( tfAtmmm
t
fcc
t
fcc
t
ic
dmktfA
dmkfAdfAts
0
00
)(22cos
))((2cos)(2cos)(
From the formula in the previous slide,
deviation.frequency theis where
)2cos(
)2cos(
)()(
mf
mc
mmfc
fci
Akf
tfff
tfAkf
tmkftf
)2sin(2cos
)]2cos([2cos
)(2cos)(
0
0
tff
ftfA
dfffA
dfAts
m
m
cc
t
mcc
t
ic
signal. FMof
index modulation thecalled often is / wherem
ff
Modulation index is the largest deviation from 2fct in
FM system.
As a result,
1. A small corresponds to a narrowband FM.
2. A large corresponds to a wideband FM.
)2sin(2cos)( tftfAtsmcc
mccmcicmcfffftffftfffff )2cos()(
2.7 Narrowband Frequency Modulation
)2sin()2sin()2cos(
)]2sin(sin[)2sin()]2sin(cos[)2cos(
)2sin(2cos)(
tftfAtfA
tftfAtftfA
tftfAts
mcccc
mccmcc
mcc
(Often, < 0.3.)
2.7 Narrowband Frequency Modulation
Comparison between approximate narrowband
FM modulation and AM (DSB-C) modulation
))(2cos(2
))(2cos(2
)2cos(
)2cos()]2cos(1[
)2cos()](1[)(
tffAk
tffAk
tfA
tftfAkA
tftmkAts
m
ma
m
ma
cc
cmmac
cacAM
))(2cos(2
))(2cos(2
)2cos(
)2sin()2sin()2cos()(
tffA
tffA
tfA
tftfAtfAts
mc
c
mc
c
cc
mccccFM
2.7 Narrowband Frequency Modulation
Represent them in terms of their low-pass
isomorphism.
)]2sin()2[cos(2
)]2sin()2[cos(2
)0()(~
tfjtfAk
tfjtfAk
jAts
mm
ma
mm
ma
cAM
)]2sin()2[cos(2
)]2sin()2[cos(2
)0()(~
tfjtfA
tfjtfA
jAts
mm
c
mm
c
cFM
2.7 Narrowband Frequency Modulation
Phaser diagram
)]2sin()2[cos(2
tfjtfA
mm
c
)0( jAc
))2sin(
)2(cos(2
tfj
tfA
m
m
c
)(~ tsFM
)]2sin()2[cos(2
tfjtfA
mm
c
)(~ tsAM
.Let mac
AkA
2.7 Spectrum of Single-Tone FM
Modulation
)2exp()(~Re
)2sin(2expRe
)2sin(2cos)(
tfjts
tftfjA
tftfAts
c
mcc
mcc
)2sin(exp)(~ tfjAtsmc
n
ntfj
ncmeJAts
2)()(~ )sin()( jx
n
jn
n eexJ
kind.first theof function elorder Bess nth theis )( where nJ
(See Slide 1-247)
2.7 Spectrum of Single-Tone FM
Modulation
n
mnc
n
tnffj
nc
ftj
n
ntfj
nc
ftj
nffJA
dteJA
dteeJA
dtetsfS
m
m
)()(
)(
)(
)(~)(~
)(2
22
2
2.7 Spectrum of Single-Tone FM
Modulation
n
mcmcnc
n
mcmcnc
cc
nfffnfffJA
nfffnfffJA
ffSffSfS
)()()(2
)()()(2
)(~
)(~
2
1)( *
Consequently,
2.7 Spectrum of Single-Tone FM
Modulation
The power of s(t)
By definition, the time-average autocorrelation function
is given by:
Hence, the power of s(t) is equal to:
T
TT
T
TTs
dttstsT
dttstsET
R )()(2
1lim)]()([
2
1lim)(
22
)2sin(24cos1
2
1lim
)2sin(2cos2
1lim)(
2
1lim)0(
2
2
222
cT
T
mc
cT
T
Tmcc
T
T
TTs
Adt
tftfA
T
dttftfAT
dttsT
R
2.7 Spectrum of Single-Tone FM
Modulation
The time-average power spectral density of a
deterministic signal s(t) is given by (cf. Slide 1-117)
From
we obtain:
)()(2
1lim)( *
2fSfS
TfPSD
TT
.||)( of ransform Fourier t theis )( where2
TttsfST
1
n
mcmcncTnfffTnfffTJTAfS ))(sinc(2))(2(sinc)()(
2
n
mcncTTttnffJAts ||))(2cos()()(
21
n
mmcmmcn
k
mcmck
c
p
TT
fnfffpfnfffpJ
kfffkfffJA
fSfST
fPSD
)/)(sinc()/)((sinc)(
)()()(4
lim
)()(2
1lim
)(
2
*
2
For simplicity, assume that 2T increases along the multiple of
1/fm., i.e., 2T = p/fm, where p is an integer. Also assume that fc is
a multiple of fm, i.e., fc = qfm., where q is an integer. Then
mcnff
mcfnf )1(
mcfnf )1(
p
fnff m
mc
)]()([)(4
)()()()()(
)()()()()(4
)()()/)(sinc()(
)()()/)(sinc()(
)()()/)((sinc)(
)()()/)((sinc)(lim4
)(
2
2
2
2
2
2
2
2
mcmc
n
n
c
mc
n
nmcqn
n
n
mcqn
n
nmc
n
n
c
k
mckmmc
n
n
k
mckmmc
n
n
k
mckmmc
n
n
k
mckmmc
n
np
c
nfffnfffJA
nfffJnfffJJ
nfffJJnfffJA
kfffJfnfffpJ
kfffJfnfffpJ
kfffJfnfffpJ
kfffJfnfffpJA
fPSD
2.7 Average Power of Single-Tone FM
Signal
Hence, the power of a single-tone FM signal is given by:
Question: Can we use 2f to be the bandwidth of a single-tone FM signal?
2
)()()1(12
)()()(2
)(
2
2
2
2
2
2
c
n
qnn
nc
n
qnn
n
n
c
A
JJA
JJJA
dffPSD
Example 2.2
Fix fm and kf,
but vary = f/fm =
kf Am/fm.
Example 2.2
Fix Am and kf,
but vary = f/fm
= kf Am/fm.
2.7 Spectrum of Narrowband Single-Tone
FM Modulation
.
2for 0)(
2)(
1)(
small, is When1
0
nJ
J
J
n
2.7 Spectrum of Narrowband Single-Tone
FM Modulation
This results in an approximate spectrum for narrowband single-tone FM signal spectrum as
)()()(4
)()()(4
)()()(4
)()()(4
)(
2
1
2
2
0
2
2
1
2
2
2
mcmc
c
cc
c
mcmc
c
n
mcmcn
c
ffffffJA
ffffJA
ffffffJA
nfffnfffJA
fPSD
)()(16
)()(4
)()(16
)()()(4
)()()(4
)()()(4
)(
22
2
22
2
1
2
2
0
2
2
1
2
mcmcc
ccc
mcmcc
mcmcc
ccc
mcmcc
ffffffA
ffffA
ffffffA
ffffffJA
ffffJA
ffffffJA
fPSD
)()()()1()( 22 nnn
n
nJJJJ
4
2
cA
16
22
cA
4
2
cA
16
22
cA
cf
cf
mf
16
22
cA
16
22
cA
2.7 Transmission Bandwidth of FM signals
Carson’s rule – An empirical bandwidth
An empirical rule for Transmission Bandwidth of
FM signals
For large , the bandwidth is essentially 2f.
For small , the bandwidth is effectively 2fm.
So Carson proposes that:
11222 fffB
mT
2.7 Transmission Bandwidth of FM signals
“Universal-curve” transmission bandwidth
The transmission bandwidth of an FM wave is the
minimum separation between two frequencies
beyond which none of the side frequencies is
greater than 1% of the carrier amplitude obtained
when the modulation is removed.
n
mcmcn
c nfffnfffJA
fS )()()(2
)(
)()(2
)2cos(cc
c
ccffff
AtfA
0.1
0.3
0.5
1.0
2.0
5.0
10.0
20.0
30.0
2
4
4
6
8
16
28
50
70
max
2n
maxmax
22 n
f
fn
f
B
m
mT
.larger a causes smaller a , fixedFor T
Bf
20.0
13.3
8.0
6.0
4.0
3.2
2.8
2.5
2.3
fBT
/
2.7 Bandwidth of a General FM wave
Now suppose m(t) is no longer a single tone but a
general message signal of bandwidth W.
Hence, the “worst-case” tone is fm = W.
For nonsinusoidal modulation, the deviation ratio D = f / W is used instead of the modulation index .
The derivation ratio D plays the same role for nonsinusoidal
modulation as the modulation index for the case of
sinusoidal modulation.
We can then use Carson’s rule or “universal curve” to
determine the transmission bandwidth BT.
2.7 Bandwidth of a General FM wave
Final notes
Carson’s rule usually underestimates the
transmission bandwidth.
Universal curve is too conservative in bandwidth
estimation.
So, a choice of a transmission bandwidth in-
between is acceptable for most practical purposes.
Example 2.3
FM radio in North America requires the
maximum frequency derivation f = 75 kHz.
If some message signal has bandwidth W = 15 kHz,
then the deviation ratio D = f / W = 75/15 = 5.
Then
kHz 180 1
17521
12Carson,
DDfB
T
kHz 240755
162max
Curve Universal, f
D
nB
T
Example 2.3
In practice, a bandwidth of 200 kHz is allocated to
each FM transmitter.
So Carson’s rule underestimates BT, while
“Universal Curve” overestimates BT.
2.7 Generation of FM Signals
Direct FM
Carrier frequency is directly varied in accordance with the message signal as accomplished using a voltage-controlled oscillator.
Indirect FM
The message is first integrated and sent to a phase modulator.
So, the carrier frequency is not directly varied in accordance to the message signal.
2.7 Generation of FM Signals
dm )( (See next slide.)
2.7 Generation of FM Signals
Frequency multiplier
))cos(10)3cos(5)5(cos(16
1)(cos
))cos(3)2cos(4)4(cos(8
1)(cos
))cos(3)3(cos(4
1)(cos
)1)2(cos(2
1)(cos
5
4
3
2
xxxx
xxxx
xxx
xx
2.7 Demodulation of FM Signals
Indirect Demodulation – Phase-locked loop
Will be introduced in Section 2.14
Direct Demodulation
Balanced frequency discriminator
)(ts
)(1
ts
)(2
ts
|)(~|1
ts
|)(~|2
ts
)(~ tso
differentiation filters
elsewhere,02
||,2
2
2||,
22
)(1
T
c
T
c
T
c
T
c
Bff
Bffaj
Bff
Bffaj
fH
elsewhere,02
||,2
2
2||,
22
)(2
T
c
T
c
T
c
T
c
Bff
Bffaj
Bff
Bffaj
fH
2.7 Analysis of Direct Demodulation in
terms of Low-Pass Equivalences
otherwise0,2
||,2
4
otherwise0,2
||),(2)(
~1
1
TT
T
c
Bf
Bfaj
BfffH
fH
elsewhere,02
||),(~
22
)(~
)(~
2
1)(
~11
TTB
ffSB
fajfSfHfS
)(~)(~
)(~1
tsBjdt
tsdats
T
t
fccdmktfAts
0)(22cos)(
t
fcdmkjAts
0)(2exp)(~
t
f
T
f
cT
t
fcT
t
ffc
T
dmkjtmB
kaABj
dmkjABj
dmkjtmkjAa
tsBjdt
tsdats
0
0
0
1
)(2exp)(2
1
)(2exp
)(2exp)(2
)(~)(~)(~
t
fc
T
f
cT
t
fc
T
f
cT
c
t
f
T
f
cT
c
dmktftmB
kaAB
dmktftmB
kaAB
tfjdmkjtmB
kaABj
tfjtsts
0
0
0
11
2)(22cos)(
21
)(22sin)(2
1
)2exp()(2exp)(2
1Re
)2exp()(~Re)(
message. equivalent lowpass theof amplitude theobtain to
used be candetector envelope then, and 1)(2
If WftmB
kc
T
f<
)(
21|)(~|
1tm
B
kaABts
T
f
cT
otherwise0,2
||,2
4)(
~2
TTB
fB
fajfH
elsewhere,02
||),(~
22
)(~
)(~
2
1)(
~22
TTB
ffSB
fajfSfHfS
2 Similarly,
t
f
T
f
cT
T
dmkjtmB
kaABj
tsBjdt
tsdats
0
2
)(2exp)(2
1
)(~)(~)(~
t
fc
T
f
cT
c
dmktftmB
kaAB
tfjtsts
0
22
2)(22cos)(
21
)2exp()(~Re)(
)(
21|)(~|
2tm
B
kaABts
T
f
cT
)(4|)(~||)(~|)(~21
tmaAktststscfo
Final Note: a is a parameter of the filters, which can be used to
adjust the amplitude of the resultant output.
2.7 FM Stereo Multiplexing
How to do Stereo Transmission in FM radio?
Two requirements:
Backward compatible with monophonic radio receivers
Operate within the allocated FM broadcast channels
To fulfill these requirements, the baseband message
signal has to be re-made.
fm = 19 kHz
detectioncoherent For reception monophonicFor
)2cos()4cos()]()([)]()([)( tfKtftmtmtmtmtm mmrlrl
m
m
m
Demultiplexer in receiver of FM stereo.
2.8 Nonlinear Effects in FM Systems
The channel (including background noise, interference and circuit imperfection) may introduce non-linear effects on the transmission signals.
For example, non-linearity due to amplifiers.
<
1)(,3
2
1)(,3
2
1|)(|),(3
1)(
)(
3
tv
tv
tvtvtv
tv
i
i
iii
o
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-1.5 -1 -0.5 0 0.5 1 1.5
2.8 Nonlinear Effects in FM Systems
Suppose
t
f
cci
iiio
dmkt
ttfAtv
tvatvatvatv
0
3
3
2
21
)(2)(
)(2cos)(
)()()()(
Then
)](2[cos
)](2[cos)](2cos[
)()()()(
33
1
22
11
3
3
2
21
ttfAa
ttfAattfAa
tvatvatvatv
cc
cccc
iiio
WfB
cc
WfB
cc
WfB
cccc
ccc
cccco
TT
T
ttfAattfAa
ttfAaAaAa
ttfttfAa
ttfAattfAatv
26
3
1
24
2
2
22
3
31
2
1
3
3
2
21
Carson,Carson,
Carson,
)](36cos[4
1)](24cos[
2
1
)](2cos[4
3
2
1
)](36cos[)](2cos[34
1
)](24cos[12
1)](2cos[)(
Thus, in order to recover s(t) from vo(t) using band-pass filter
(i.e., to remove 2fc and 3fc terms from vo(t)), it requires:
2/)22(2/)24(2 WffWffcc
.23 ly,equivalentor Wffc
The filtered output is therefore:
)](2cos[4
3)( 3
31filtered,ttfAaAatv
ccco
Observations
Unlike AM modulation, FM modulation is not affected
by distortion produced by transmission through a channel
with amplitude nonlinearities.
So the FM modulation allows the usage of highly non-
linear amplifiers and power transmitters.
2.8 AM-to-PM Conversion
Although FM modulation is insensitive to amplitude nonlinearity, it is indeed very sensitive to phase nonlinearity.
A common type of phase nonlinearity encountered in microwave radio transmission is the AM-to-PM conversion.
The AM-to-PM conversion is owing to that the phase characteristic of amplifiers (or repeaters) also depends on the instantaneous amplitude of the input signal.
Notably, the nonlinear amplifiers discussed previously will leave the phase of the input unchanged.
Often, it requires that the peak phase change for a 1-dB change in input envelope is less than 2%.
2.9 Superheterodyne Receiver
A commercial radio communication system
contains not only the “transmission” but also
some other functions, such as:
Carrier-frequency tuning, to select the desired
signals
Filtering, to separate the desired signal from other
unwanted signals
Amplifying, to compensate for the loss of signal
power incurred in the course of transmission
2.9 Superheterodyne Receiver
A superheterodyne receiver or superhet is designed to facilitate the fulfillment of these functions, especially the first two. It overcomes the difficulty of having to build a tunable highly
selective and variable filter (rather a fixed filter is applied on IF section).
heterodyning function
First detector
2.9 Superheterodyne Receiver
Example AM Radio FM Radio
RF carrier range 0.535-1.605 MHz 88-108 MHz
Midband frequency of IF section 0.455 MHz 10.7 MHz
IF bandwidth 10 kHz 200 kHz
Second detector
2.9 Image Interference
Fix fIF and fLO at the receiver end. What is the fRF that will survive at the IF section output?
Answer:
Example. Suppose the receiver use 1.105MHz local oscillator, and receives two RF signals respectively centered at 0.65MHz and 1.56MHz.
||IFLORF
fff
MHz455.0
IFf
MHz455.0
IFf
MHz65.0
RFf
MHz56.1
ceInterferenf
MHz105.1
LOf
MHz.4550
RFLOIFfff MHz.7551
ageIm
RFLOfff
MHz.4550ceInterferen Image
LOceInterferen
fff MHz665.2LOceInterferen
ff
2.9 Image Interference
Then at the output of the IF section, we respectively obtain:
2.9 Image Interference
A cure of image interference is to employ a highly
selective stages in the RF session in order to favor the
desired signal (at fRF) and discriminate the undesired
signal (at fRF + 2fIF or fRF – 2fIF).
2.9 Advantage of Constant Envelope for FM
modulation
Observations
For FM modulation, any variation in amplitude is
caused by noise or interference.
For FM modulation, the information is resided on
the variations of the instantaneous frequency.
So we can use an amplitude limiter to remove the
amplitude variation, but to retain the frequency
variation after the IF section.
2.9 Advantage of Constant Envelope for FM
modulation
Amplitude limiter
Clipping the modulated wave at the IF section
output (almost to the zero axis) to result in a near-
rectangular wave.
Pass the rectangular wave through a bandpass filter
centered at fIF to suppress harmonics (due to
clipping).
Then the filter output retains the frequency
variation with constant amplitude.
2.10 Noise in CW Modulation Systems
To simplify the system analysis, we assume: ideal band-pass filter (that is just wide enough to pass the
modulated signal s(t) without distortion),
ideal demodulator,
Gaussian distributed white noise process.
So the only source of imperfection is from the noise.
TB
2.10 Noise in CW Modulation Systems
),()()( tntstx
So after passing through the ideal bandpass filter, s(t) is
unchanged but w(t) becomes a narrowband noise n(t).
Hence,
).2sin()()2cos()()( where tftntftntncQcI
2.10 Noise in CW Modulation Systems
Input signal-to-noise ratio (SNRI)
The ratio of the average power of the modulated
signal s(t) to the average power of the filtered noise
n(t).
Output signal-to-noise ratio (SNRO)
The ratio of the average power of the demodulated
message signal to the average power of the noise,
measured at the receiver output.
2.10 Noise in CW Modulation Systems
It is sometimes advantageous to look at the lowpass equivalent model.
Channel signal-to-noise ratio (SNRC)
The ratio of the average power of the modulated signal s(t) to the average power of the channel noise in the message bandwidth, measured at the receiver input (as illustrated below).
2.10 Noise in CW Modulation Systems
Notes
SNRC is nothing to do with the receiver structure, but depends on the channel characteristic and modulation approach.
SNRO is however receiver-structure dependent.
Finally, define the figure of merit for the receiver as:
C
O
SNR
SNRmerit of figure
2.11 Noise in Linear Receivers Using
Coherent Detection
Recall that for AM demodulation
when the carrier is suppressed, linear coherent
detection is used. (Section 2.11)
when the carrier is additionally transmitted,
nonlinear envelope detection is used. (Section 2.12)
The noise analysis of the above two cases are
respectively addressed in Sections 2.11 and
2.12.
2.11 Noise in Linear Receivers Using
Coherent Detection
. todbandlimite )( PSDand meam zero withstationary:)( WfStmM
)2cos()()( tftmCAts cc
)2cos( tfc
2.11 Noise in Linear Receivers Using
Coherent Detection
Average signal power
2
)2(cos2
1lim
)()2(cos2
1lim
)()2(cos2
1lim)]([
2
1lim
22
222
2222
22222
PAC
dttfT
PAC
dttmEtfACT
dttmtfACET
dttsET
c
T
Tc
Tc
T
Tcc
T
T
Tcc
T
T
TT
power. message theis )()]([ where 2
W
WM dffStmEP
2.11 Noise in Linear Receivers Using
Coherent Detection
Noise power in the message bandwidth
00
2)( WNdf
NdffS
W
W
W
Ww
)2cos( tfc
2.11 Noise in Linear Receivers Using
Coherent Detection
Channel SNR for DSB-SC and coherent detection (observed at x(t))
Next, we calculate output SNR (observed at y(t)) under the condition
that the transmitter and the receiver are perfectly synchronized.
o
c
o
c
CWN
PAC
WN
PACSNR
2
2 2222
SC-DSB,
)2sin()()2cos()()2cos()(
)()()(
tftwtftwtftmCA
twtstx
cQcIcc
)(2
1)(
2
1
)2cos()2sin()()2(cos)()2(cos)(
)2cos()]2sin()()2cos()()2cos()([
)2cos()()(
LowPass
22
twtmCA
tftftwtftwtftmCA
tftftwtftwtftmCA
tftxtv
Ic
ccQcIcc
ccQcIcc
c
)(2
1)(
2
1)( twtmCAty Ic
0
22
2
22
2
222
SC-DSB,2)]([4/)(
4/)(
WN
PAC
twE
PAC
twE
tmACESNR cc
I
cO
)].)([)]([)]([ (Recall 222 twEtwEtwE QI
(See Slide 1-219.)
.1detectioncoherent and SC-for DSBmerit of Figure
Similar derivation on SSB and coherent detection yields the
same figure of merit.
Conclusions
Coherent detection for SSB performs the same as
coherent detection for DSB-SC.
There is no SNR degradation for SSB and DSB-SC
coherent receivers. The only effect of these modulation
and demodulation processes is to translate the message
signal to a different frequency band to facilitate its
transmission over a band-pass channel.
No trade-off between noise performance and bandwidth.
This may become a problem when high quality
transceiving is required.
2.12 Noise in AM Receivers Using Envelope
Detection
)2cos()](1[)( tftmkAtscac
mean.) zero )( (Assume )1(2
)2(cos2
1lim))(1()]([
2
1lim
2
2
2222
tmPkA
dttfT
tmkEAdttsET
a
c
T
Tc
Tac
T
TT
Hence, channel SNR for DSB-C is equal to:
Next, we calculate output SNR (observed at y(t)) under the condition that the transmitter and the receiver are perfectly synchronized.
0
0
2)( Also, WNdf
NdffS
W
W
W
Ww
o
ac
CWN
PkASNR
2
)1( 22
AM,
)2sin()()2cos()]())(1([
)2sin()()2cos()()2cos()](1[
)()()(
tftntftntmkA
tftntftntftmkA
tntstx
cQcIac
cQcIcac
)]()([2
1
)]())(1([2
1
)()]())(1([2
1
)()(
block DC
22
LowP ass
2
tntmkA
tntmkA
tntntmkA
txty
Iac
Iac
QIac
envelop detector
x
x
)(~)](1[ if tntmkAac
(Refer to Slide 2-136 and 2-138.)
0
22
2
22
2
222
AM,2)]([2/)(
2/)(
WN
PkA
tnE
PkA
tnE
tmkAESNR acac
I
ac
O
11)2()1(
)2(2
2
22
0
22
AM,
AM, <
Pk
Pk
WNPkA
WNPkA
SNR
SNR
a
a
oac
ac
C
O
Conclusion
Even if the noise power is small compared to the
average carrier power at the envelope detector output,
the noise performance of a full AM receiver is
inferior to that of a DSB-SC receiver due to the
wastage of transmitter power.
Example 2.4 Single-Tone Modulation
Assume
)2cos()]2cos(1[)(
)2cos()(
tftfAkAts
tfAtm
cmmac
mm
)1(24
02
1
)2(cos)2(cos
)2(cos)2cos(2)2(cos2
1lim
)2(cos)2cos(12
1lim)]([
2
1lim
2
222
2
2222
222
2222
PkAAk
A
dttftfAk
tftfAktfT
A
dttftfAkT
AdttsET
a
cma
c
cmma
T
Tcmmac
Tc
T
Tcmma
Tc
T
TT
Hence,
.2/1
2/
1
,discussion previous as proceduresimilar Following
.2
)2(cos1
lim where
22
22
2
2
AM,
AM,
2
0
22
ma
ma
a
a
C
O
mT
mmT
Ak
Ak
Pk
Pk
SNR
SNR
AdttfA
TP
So even if for 100% percent modulation (kaAm = 1), the figure of merit
= 1/3. This means that an AM system with envelope detection must
transmit three times as much average power as DSB-SC with coherent
detector to achieve the same quality of noise performance.
2.12 Threshold Effect
What if is violated (in AM modulation
with envelope detection)?
)2sin()()2cos()]())(1([
)2sin()()2cos()()2cos()](1[
)()()(
tftntftntmkA
tftntftntftmkA
tntstx
cQcIac
cQcIcac
)(~)](1[ tntmkAac
|~||~||~|2|~|2)(|~|
22)(|~|
222222
2222222
nBnBnBnBnBnnBnB
nBnBnBnnBnBnnBnB
IQI
IIIQIIQI
<<
|))(~|))(1((2
1
|))(~|))(1((2
1
|)(~|))(1(|)(~|2))(1(2
1
|)(~|))(1()(2))(1(2
1
)()]())(1([2
1
)()(
2
222
222
22
LowP ass
2
tntmkA
tntmkA
tntmkAtntmkA
tntmkAtntmkA
tntntmkA
txty
ac
ac
acac
acIac
QIac
|])(~|)([2
1block DC
tntmkAac
.)(~)](1[ Assume tntmkAac
<<
<<
|)(~|)](1[|,)(~|)(
|)(~|)](1[),()()(2
tntmkAtntmkA
tntmkAtntmkAty
acac
acIac
0
22
22
22
2
222
AM,4)]([)]([|)(~|
)(
WN
PkA
tnEtnE
PkA
tnE
tmkAESNR ac
QI
acac
O
2
1
)1(2)2()1(
)4(2
2
22
0
22
AM,
AM, <
Pk
Pk
WNPkA
WNPkA
SNR
SNR
a
a
oac
ac
C
O
2.12 Threshold Effect
Threshold effect
For AM with envelope detection, there exists a
carrier-to-noise ratio r (namely, the power ratio
between unmodulated carrier Ac cos(2fct) and the
passband noise n(t) ) below which the noise
performance of a detector deteriorates rapidly.
0
2
0
2
42
2/
WN
A
WN
Acc r
2.12 General Formula for SNRO in Envelope
Detection
For envelope detector, the noise is no longer additive; so the
original definition of SNRO (which is based on additive
noise) may not be well-applied.
A new definition should be given:
Definition. The (general) output signal-to-noise ratio
for an output y(t) due to a carrier input is defined as
)]([Var
2
ty
sSNR o
O
alone. noise of presense thein
)( toequal is )( and )],([)]([ where tytytyEtyEsooo
)()( tysty oo
2.12 General Formula for SNRO in Envelope
Detection
so is named the mean output signal.
Var[y(t)] is named the mean output noise power.
Example. y(t) = A + nI(t), where nI(t) is zero mean..
)]([)]([Var)]([Var
)]([)]([
2 tnEtnty
AtnEtnAEs
II
IIo
)]([ 2
2
tnE
ASNR
I
O
This somehow shows the backward compatibility of the new
definition.
2.12 General Formula for SNRO in Envelope
Detection
Now for an envelope detector, the output due to a carrier
input and additive Gaussian noise channel is given by:
)())(()( 22 tntnAtyQI
pdf withddistribute Rayleighis )()()( 22 tntntyQIo
pdf withddistribute Ricianis )(ty
I0( ) = modified Bessel function
of the first kind of zero order.
Appendix 4 Confluent Hypergeometric
Functions
The Kummer confluent hypergeometric function is a
solution of Kummer’s equation
For b 0, 1, 2, …., the Kummer confluent
hypergeometric function is equal to 1F1(a;b;x).
complex ,for 0)(2
2
baaydx
dyxb
dx
ydx
.1)(
)1()1()( where,
!)()()(
)()()();;(
function trichypergeome dGeneralize
00 21
21
a
kaaaa
k
x
bbb
aaaxbaF
kk
k kqkk
kpkk
qp
rr
./)0(' and 1)0( conditionsboundary with bayy
A4.2 Properties of the Confluent
Hypergeometric Function
);;();;()exp( .5
4
1;1;
22
)2/()()exp( .4
. as 2
22)1(
2exp);1;2/1( .3
.1);1;1( 2.
.0 as 1);;( .1
1111
2110
0
221
2011
11
11
uFuFu
b
mF
b
mduuIubu
xx
xxI
xIx
xxF
xxF
xxb
axbaF
m
m
2.12 General Formula for SNRO in Envelope
Detection
Hence,
r
rr
rr
rr
rr
r
;1;2
1
2
;1;2
3)exp(
2
;1;2
3
)4(2
)2/3()exp(
)2(
)(4
exp)exp()2(
)]([
11
11
112/32/3
00
2
2
2/3
F
F
F
duuIu
utyE
N
N
N
N
By Property 4
By Property 5
2.12 General Formula for SNRO in Envelope
Detection
As a result,
Similarly, we can obtain:
.12
;1;2
1
2)]([)]([
2
2
11
N
Noo
AFtyEtyEs
2
2
2
112
2
2
2
2
2
112
2
11
2
2;1;
2
1
4212
2;1;
2
1
42;1;12)](Var[
NN
N
NN
N
AF
A
AF
AFty
By Property 2
2.12 General Formula for SNRO in Envelope
Detection
This concludes to:
(Continue from the previous slide.)
2.12 General Formula for SNRO in Envelope
Detection
rPkSNR aO
2
AM,
es.approximat limiting two theand
;1;2/1)1(4
1;1;2/1 of Curve
2
11
2
11
rr
r
F
FSNR
O
rPkSNR aO
2
AM, 2
0.1
1
10
100
1000
0.1 1 10 100
2.12 General Formula for SNRO in Envelope
Detection
Remarks
For large carrier-to-noise ratio r, the envelope detector
behaves like a coherent detector in the sense that the output
SNR is proportional to r.
For small carrier-to-noise ratio r, the (newly defined) output
signal-to-noise ratio of the envelope detector degrades faster
than a linear function of r (decrease at a rate of r2).
From “threshold effect” and “general formula for SNRO,” we
can see that the envelope detector favors a strong signal. This
is sometimes called “weak signal suppression.”
2.13 Noise in FM Receivers
To simplify the system analysis, we assume: ideal band-pass filter (that is just wide enough to pass the
modulated signal s(t) without distortion),
ideal demodulator,
Gaussian distributed white Noise process.
So the only source of imperfection is from the noise.
.)(2)( where)],(2cos[)(0t
fccdmktttfAts
)](2cos[)]()([sin)()]()(cos[)(
)](2sin[)]()(sin[)(
)](2cos[)]()(cos[)(
)]()()(2cos[)()](2cos[
)](2cos[)()](2cos[)(
222ttftttrtttrA
ttftttr
ttftttrA
ttttftrttfA
ttftrttfAtx
cc
c
cc
ccc
ccc
)]()(cos[)(
)]()(sin[)(tan)()( where 1
tttrA
tttrtt
c
))(sin()()(
))(cos()()(
ttrtn
ttrtn
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I
)](2cos[
)](2cos[)]()([sin)()]()(cos[)()(
Limiter
222
ttfA
ttftttrtttrAtx
c
cc
)('2)(~Output ))(2cos()(Input taAtsttfAtsoc
Recall on Slides 2-94 ~ 2-101, we have talked about the
Balanced Frequency Discriminator, whose input and output
satisfy:
Next, the signal will be passed through a Discriminator.
)(exp)(')(~)(~)(~
1tjBtaAjtsBj
dt
tsdats
TT
)(exp)(')(~)(~)(~
2tjBtaAjtsBj
dt
tsdats
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)('2|)(~||)(~|)(~21
taAtststso
Specifically, with :have we)),(exp()(~ tjAts
dt
tttrA
tttrtd
aAtaAtvc
)]()(cos[)(
)]()(sin[)(tan)(
2)('2)(
tor discrimina the throughpassingafter Thus,
1
The above assumption is true for any deterministic (constant) (t)!
It has been shown that Assumption 1 is justifiable for high carrier-
to-noise ratio (or equivalently, under Assumption 2).
We then obtain the desired “additive” form.
)()( 8. :6.2 Table ffGjtgdt
d )(tn
Q
cA
fjfH
2
2)(
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that is just enough
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Observation from the above formula:
. thenamed is This power. noise Decreasing
2/power carrier increasing system, FMan In 2
effect quieting noise
cA
).( provided ,2
3
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2
)]([3
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We next turn to SNRC,FM.
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DWD
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DSNR
SNR
W
Pk
W
fD
SNR
SNRBW
T
C
O
f
C
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TRemarks
.3
2
2
3
2
2
0
2
3
0
22
FM,
FM,
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Pk
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SNR
SNR f
c
fc
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O
|)(|max tmkff
2.13 Summary
Specifically,
for high carrier-to-noise ratio r (equivalently to the
assumption made in Assumption 1), an increase in
transmission bandwidth BT provides a corresponding
quadratic increase in figure of merit of a FM system.
So, there is a tradeoff between BT and figure of merit.
Notably, figure of merit for an AM system is
nothing to do with BT.
Example 2.5 Single-Tone Modulation
m(t) = Am cos(2fmt)
Then we can represent the figure of merit in terms of
modulation index (or deviation ratio) as (cf. Slide 2-63):
In order to make the figure of metric for an FM system to
be superior to that for an AM system with 100%
modulation, it requires:
.2
3
2
3)2/(332
2
2
2
22
2
2
FM,
FM,
W
f
W
Ak
W
Pk
SNR
SNR mff
C
O
471.03
2
3
1
2
3 2
2.13 Capture Effect
Recall that in Assumption 1, we assume Ac >> r(t).
This somehow hints that the noise suppression of an FM
modulation works well when the noise (or other unwanted
modulated signal that cannot be filtered out by the bandpass or
lowpass filters) is weaker than the desired FM signal.
What if the unwanted FM signal is stronger than the desired FM
signal.
The FM receiver will capture the unwanted FM signal!
What if the unwanted FM signal has nearly equal strength as the
desired FM signal.
The FM receiver will fluctuate back and forth between them!
2.13 FM Threshold Effect
Recall that in Assumption 1, we assume Ac >> r(t)
(equivalently, a high carrier-to-noise ratio) to simplify (t)
so that the next formula holds.
However, a further decrease of carrier-to-noise ratio will
break the FM receiver (from a clicking sound down to a
crackling sound).
As the same as the AM modulation, this is also named the
threshold effect.
.2
33
0
22
FM,WN
PkASNR
fc
O
2.13 FM Threshold Effect
Consider a simplified case with m(t) = 0 (no message
signal).
)()](cos[)(2
)(')()](sin[)(')](cos[)(')(
)](cos[)(
)](sin[)(tan
)(')(2
22
2
1
trttrAA
ttrttrAtttrA
dt
ttrA
ttrd
ttv
cc
cc
c
To facilitate the understanding of “clicking” sound effect,
we let r(t) = l Ac, a constant ratio of Ac.
2.13 FM Threshold Effect
2)](cos[21
)](cos[)(')(')(2
ll
ll
t
ttttv
)('1)](cos[21
)](cos[)(')(')(2
2t
t
ttttv
l
l
ll
ll
1but 1 and ,)( say, time,at the Then ll t
Thus a sign change in ’(t) will cause a spike!
Notably, when l = 0 (no noise), the output equals m(t)
= 0 as desired.
2.13 FM Threshold Effect
2)]sin(cos[21
)]sin(cos[)cos()(')(2)sin()(
ll
ll
t
ttttvtt
05.1l
05.0l
5l
2.13 Numerical Experiment
ratio. noise carrier to theis 2/
and 2
,2
3
and 2
,4
3
)( provided ,2
3
0
23
2
3
0
22
3
0
22
FM,
NB
ABf
W
B
AkfA
PWN
fA
trAWN
PkASNR
T
cTT
mf
mc
c
fc
O
rr
Take BT/(2W) = 5.
The average output signal power is calculated in the absence of noise.
The average output noise power is calculated when there is no signal present.
2.13 Numerical of the
formula in Slide 2-156
Curve I: The average noise power is calculated assuming an unmodulated carrier.
Curve II: The average noise power is calculated assuming a sinusoidally modulated carrier.
As text mentioned, for r < 11 dB, the output signal-to-noise ratio deviates appreciably from the linear curve.
A true experiment found that occasional clicks are heard at r around 13 dB, only slightly larger than what theory indicates.
The deviation of curve II from curve I shows that the average noise power is indeed dependent on the modulating signal.
2.13 How to avoid “clicking” sound?
For given modulation index (or deviation ratio) and message
signal bandwidth W,
1. Determine BT by Carson’s rule or Universal curve.
2. For a specified noise level N0, select Ac to satisfy:
.202
y,equvalentlor dB 132
log100
2
0
2
10
NB
A
NB
A
T
c
T
c
2.13 Threshold Reduction
After our learning
that FM
modulation has
threshold effect,
the next question
is naturally on
“how to reduce
the threshold?”
2.13 Threshold Reduction
Threshold reduction in FM receivers may be
achieved by
1. negative feedback (commonly referred to as an
FMFB demodulator), or
2. phase-locked loop demodulator.
Why PLL can reduce threshold effect is not covered in
this course. Notably, the PLL system analysis in
Section 2.14 assumes noise-free transmission.
)(tv)(tx
.)(2)( where)],(2cos[)(0t
fccdmktttfAts
.)(2)( where)],(2cos[2)(0t
fvcovcovcovcodmktttfts
)]()1()(2cos[
)](2cos[)](2cos[2)()(
Bandpass
ttffA
ttfttfAtsts
vcocc
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).(at centered ,)1( as wideas
passbandsmaller a havely conceptual canfilter bandpass theThus,
vcocTffB
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noise. einput whit theas level noise
same the with white treatedbe canoutput Mixer at the noise The
)](2cos[)(2)()(
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y.probabilithiger withholds )()()( of condition the
same, theremains and smaller, is )]([)]([ Since
22
22
tntntrA
AtnEtnE
QIc
cQI
Experiments show that an FMFB receiver is capable of realizing a
threshold extension on the order of 5~7 dB.
2.13 Threshold Reduction of an FMFB
receiver
To sum up:
An FMFB demodulator is essentially a tracking
filter that can track only the slowly varying
frequency of a wideband FM signal (i.e., the
message signal part), and consequently it responds
only to a narrowband of noise centered about the
instantaneous carrier frequency.
2.13 Pre-Emphasis and De-emphasis in FM
Recall that the noise PSD at the output shapes like a bowel.
If we can “equalize” the signal-to-noise power ratios over the entire message band, a better noise performance should result.
2.13 Pre-Emphasis and De-emphasis in FM
In order to produce an undistorted version of the original
message at the receiver output, we must have:
This relation guarantees the intactness of the message power.
Next, we need to find Hde(f) such that the noise power is
optimally suppressed.
.for 1)()( WfWfHfHdepe
2.13 Pre-Emphasis and De-emphasis in FM
Under the assumption of high carrier-to-noise ratio, the
noise PSD at the de-emphasis filter output is given by:
otherwise,0
||,|)(|)(|)(|
2
2
2
02 WffH
A
fN
fSfH de
cNde o
W
Wde
c
dffHA
fN 2
2
2
0 |)(| power noise Average
2.13 Pre-Emphasis and De-emphasis in FM
Since the message power remains the same, the
improvement factor of the output signal-to-noise ratio after
and before pre/de-emphasis is:
W
Wde
W
Wde
W
W
W
Wde
c
W
Wc
dffHf
W
dffHf
dff
dffHA
fN
dfA
fN
I22
3
22
2
2
2
2
0
2
2
0
|)(|3
2
|)(||)(|
Example 2.6
Pre-emphasis filter De-emphasis filter
).2/(1 where,)/(1
1)( and )/(1)(
0
0
0Crf
ffjfHffjfH
depe
.||for 12 and Assume WffCrrR <<<<
Example 2.6
)]/(tan)/[(3
)/(
)/(13
2
|)(|3
2
0
1
0
3
0
2
0
2
3
22
3
fWfW
fW
dfff
f
W
dffHf
WI
W
W
W
Wde
With f0 = 2.1 KHz and W = 15 KHz, we obtain I = 22 = 13 dB.
A significant noise performance improvement is therefore obtained.
2.13 Pre-Emphasis and De-emphasis in FM
Final remarks:
The previous example uses a linear pre-emphasis
and de-emphasis filter to improve the noise.
A non-linear pre-emphasis and de-emphasis filters
have been applied successfully to applications like
tape recording. These techniques, known as Dolby-
A, Dolby-B, and DBX systems, use a combination
of filtering and dynamic range compression to
reduce the effects of noise.
2.14 Computer Experiments: Phase-Locked
Loop
Phase-locked loop
.)(2)( where)],(2sin[)(
011
t
fccdmktttfAts
.)(2)( where)],(2cos[)(0
22 t
vcvdvktttfAtr
mk
)],(sin[2
)]()(sin[)]()(4sin[2
)](2cos[)](2sin[
)()()(
Low P ass
2121
21
tAAk
tttttfAAk
ttfAttfAk
trtskte
e
vcm
c
vcm
cvccm
m
mk
Loop filter = low pass filter + filter h().
).()()( where21
ttte
.)()}({)( Also,LowP ass
dthetv
t
vedvktttt
0121
)(2)()()()(
dthkdt
td
dthekdt
td
tvkdt
td
dt
td
e
v
v
e
)()](sin[2)(
)()}({2)(
)(2)()(
0
1
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1
1
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AAkkk
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ud
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udt
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e
t
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e
001
00
0
1
0
)(*)](sin[2)(
)()](sin[2)(
)()(
The previous formula suggests an equivalent analytical model for PLL.
When e(t) = 0, the system is said to be in phase-lock.
In this case, 1(t) = 2(t) or equivalently, kvv(t) = kfm(t).
When e(t) is small (< 0.5 radians), the system is said to be in near
phase-locked.
In this case, we can approximate sin[e(t)] by e(t); hence, a linear
approximate model is resulted.
Linearization approximation model for PLL.
We can transform the above time-domain system to its
equivalent frequency domain to facilitate its analysis.
Linearization approximation model for PLL.
)()()(21
1
)()()(2)(
)(
)()]()([
)(
)(
)(
00
0
2211
fHkjf
jf
fGfHk
fGfHfkf
f
fff
f
f
f
ee
e
ee
)(2
f
)(1
f )( fH
)( fG
fjfG
2
1)(
)( fe
2.14 Experiment 0: First-Order PLL
.1)( fH
)/(1
)/(
)(
)(
0
0
1kfj
kfj
f
fe
A parameter k0 controls both the loop gain and bandwidth of the
filter. In other words, it is impossible to adjust the loop gain
without changing the filter bandwidth.
2.14 Experiment 1: Second-Order
Acquisition PLL
2
2
00
2
2
001
)/()/(21
)/(
)()(
)(
))/(1()()(
)(
nn
n
e
fjffjf
fjf
akjfkjf
jf
jfakjf
jf
fHkjf
jf
f
f
.)4/(factor damping and frequency natural where00
akakfn
2.14 Experiment 1: Second-Order
Acquisition PLL
Fast response but require
longer time to stablelize
Slow response but quick
stablelizatoin.
2.14 Experiment 2: Phase-Plain Portrait
model. PLLnonlinear using and )/(1)( jfafH
damping) (critical 2
1
Herz22
50
Take
n
f
Hz.22
50 where),cos(2)(Let
mmmftfAtm
dthtfk
Ak
dthdt
td
kdt
td
k
em
mf
e
e
)()](sin[)2cos(
)()](sin[)(
2
1)(
2
1
0
1
00
Phase-Plane Portrait
)(te
)(2
1
0
tdt
d
ke
Initial value
Remarks on the previous figure
Each curve corresponds to different initial frequency error
(i.e., boundary condition for the differential equation).
The phase-plane portrait is periodic in e(t) with period 2,
but the phase-plane portrait is not periodic in de(t)/dt.
There exists an initial frequency error (such as 2 in the
previous figure) at which a saddle line appears (i.e., the
solid line in the previous figure).
In certain cases, the PLL will ultimately reach an
equilibrium (stable) points at (0,0) or (2,0).
A slightest perturbation to the “saddle line” causes it to shift
to the equilibrium points.
2.15 Summary and Discussion
Four types of AM modulations are introduced (expensive) DSB-C transmitter + (inexpensive) envelope
detector, which is good for applications like radio broadcasting.
(less expensive) DSB-SC transmitter + (more complex) coherent detector, which is good for applications like limited-transmitter-power point-to-point communication.
(less bandwidth) VSB transmitter + coherent detector, which is good for applications like television signals and high speed data.
(minimum transmission power/bandwidth) SSB transmitter + coherent detector, which is perhaps only good for applications whose message signals have an energy gap on zero frequency.
2.15 Summary and Discussion
FM modulation, a representative of Angle
Modulation
A nonlinear modulation process
Carson’s rule and universal curve on transmission
bandwidth
2.15 Summary and
Discussion
Noise performance
I. Full AM (DSB-C) with
100 percent modulation
(See Slide 2-5)
II. Coherent DSB-SC &
SSB
III. FM with = 2 and 13
dB pre/de-emphasis
improvement
IV. FM with = 5 and 13
dB pre/de-emphasis
improvement
Threshold effect
2.15 Summary and Discussion
Normalized transmission bandwidth in the previous figure
DSB-C, DSB-SC SSB FM(=2) FM(=5)
Bn 2 1 8 16
f
B
fW
fB
W
BB TTT
n
/
/urveUniversalC,
42 2.35
(Refer to Slide 2-85.)
2.15 Summary and Discussion
Observations from the figure
1. SSB modulation is optimum in noise performance
and bandwidth conservation in AM family.
2. FM modulation improves the noise performance of
AM family at the expense of an excessive
transmission bandwidth.
3. Curves III an IV indicate that FM modulation
offers the tradeoff between transmission bandwidth
and noise performance.