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Test 1 Review
Professor Deepa Kundur
University of Toronto
Professor Deepa Kundur (University of Toronto) Test 1 Review 1 / 87
Test 1 Review
Reference:
Sections:2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.73.1, 3.2, 3.3, 3.4, 3.5, 3.64.1, 4.2, 4.3, 4.4, 4.6, 4.7, 4.8 of
S. Haykin and M. Moher, Introduction to Analog & Digital Communications, 2nded., John Wiley & Sons, Inc., 2007. ISBN-13 978-0-471-43222-7.
Professor Deepa Kundur (University of Toronto) Test 1 Review 2 / 87
Chapter 2: Fourier Representation ofSignals and Systems
Professor Deepa Kundur (University of Toronto) Test 1 Review 3 / 87
Communication Systems: Foundational Theories
I Modulation Theory: piggy-back information-bearing signal on acarrier signal
I Detection Theory: estimating or detecting theinformation-bearing signal in a reliable manner
I Probability and Random Processes: model channel noise anduncertainty at receiver
I Fourier Analysis: view signal and system in another domain togain new insights
informationconsumption
informationsource transmitter receiver
channel
Professor Deepa Kundur (University of Toronto) Test 1 Review 4 / 87
The Fourier Transform (FT)
G (f ) =
g(t)ej2pift
g(t) =
G (f )e+j2pift
Notation:
g(t) G (f )G (f ) = F [g(t)]
g(t) = F1 [G (f )]
Professor Deepa Kundur (University of Toronto) Test 1 Review 5 / 87
Energy Signals
I The energy of a signal g(t) is given by: |g(t)|2dt
I If g(t) represents a voltage or a current, then we say that this isthe energy of the signal across a 1 ohm resistor.
I Why? Because a current i(t) or voltage v(t) exhibits thefollowing energy over a R ohm resistor.
E =
i2(t)Rdt =
v 2(t)
Rdt
Professor Deepa Kundur (University of Toronto) Test 1 Review 6 / 87
Energy Signals and the Fourier Transform
Practical physically realizable signals (e.g., energy signals) that obey: |g(t)|2dt
FT Synthesis Equation
g(t) =
G (f )e j2piftdt
I g(t) is the sum of scaled complex sinusoids
I e j2pift = cos(2pift) + jsin(2pift) complex sinusoid
Professor Deepa Kundur (University of Toronto) Test 1 Review 9 / 87
e j2pift = cos(2pift) + j sin(2pift)
cos(2pift)
0
t
sin(2pift)
0
t
Professor Deepa Kundur (University of Toronto) Test 1 Review 10 / 87
FT Analysis Equation
G (f ) =
g(t)ej2piftdt
I The analysis equation represents the inner product between g(t)
and e j2pift .
I The analysis equation states that G (f ) is a measure of similarity
between g(t) and e j2pift , the complex sinusoid at frequency f Hz.
Professor Deepa Kundur (University of Toronto) Test 1 Review 11 / 87
|G (f )| and G (f )
g(t) =
G (f )e j2pif tdf
=
|G (f )|e j(2pif t+G(f ))df
I |G (f )| dictates the relative presence of the sinusoid of frequencyf in g(t).
I G (f ) dictates the relative alignment of the sinusoid offrequency f in g(t).
Professor Deepa Kundur (University of Toronto) Test 1 Review 12 / 87
Low, Mid and High Frequency Signals
Q: Which of the following signals appears higher in frequency?
1. cos(4 106pit + pi/3)2. sin(2pit + 10pi) + 17 cos2(10pit)
A: cos(4 106pit + pi/3).
Professor Deepa Kundur (University of Toronto) Test 1 Review 13 / 87
Importance of FT Theorems and Properties
I The Fourier transform converts a signal or system representationto the frequency-domain, which provides another way tovisualize a signal or system convenient for analysis and design.
I The properties of the Fourier transform provide valuable insightinto how signal operations in the time-domain are described inthe frequency-domain.
Professor Deepa Kundur (University of Toronto) Test 1 Review 14 / 87
FT Theorems and PropertiesProperty/Theorem Time Domain Frequency DomainNotation: g(t) G(f )
g1(t) G1(f )g2(t) G2(f )
Linearity: c1g1(t) + c2g2(t) c1G1(f ) + c2G2(f )Dilation: g(at) 1|aG
(fa
)Conjugation: g(t) G(f )Duality: G(t) g(f )Time Shifting: g(t t0) G(f )ej2pift0Frequency Shifting: e j2pifc tg(t) G(f fc )Area Under G(f ): g(0) =
G(f )df
Area Under g(t): g(t)dt = G(0)
Time Differentiation: ddtg(t) j2pifG(f )
Time Integration : t g()d
1j2pif
G(f )
Modulation Theorem: g1(t)g2(t)
G1()G2(f )d
Convolution Theorem: g1()g2(t ) G1(f )G2(f )
Correlation Theorem: g1(t)g
2 (t )dt G1(f )G2 (f )
Rayleighs Energy Theorem: |g(t)|2dt =
|G(f )|2df
Professor Deepa Kundur (University of Toronto) Test 1 Review 15 / 87
Time-Bandwidth Product
time-duration of a signal frequency bandwidth = constant
0 1/T
2/T
3/T
4/T
AT
-1/T
-2/T-3/T
-4/T
AT sinc(fT)
T /2-T/2
A
Arect(t/T)
t f
T larger
durationnull-to-nullbandwidth
Professor Deepa Kundur (University of Toronto) Test 1 Review 16 / 87
Time-Bandwidth Product
time-duration of a signal frequency bandwidth = constant
I the constant depends on the definitions of duration andbandwidth and can change with the shape of signals beingconsidered
I It can be shown that:time-duration of a signal frequency bandwidth 1
4pi
with equality achieved for a Gaussian pulse.
Professor Deepa Kundur (University of Toronto) Test 1 Review 17 / 87
LTI Systems and FilteringLTI System
impulse response
LTI System
frequency response
I For systems that are linear time-invariant (LTI), the Fourier transformprovides a decoupled description of the system operation on the input signalmuch like when we diagonalize a matrix.
I This provides a filtering perspective to how a linear time-invariant systemoperates on an input signal.
I The LTI system scales the sinusoidal component corresponding to frequencyf by H(f ) providing frequency selectivity.
Professor Deepa Kundur (University of Toronto) Test 1 Review 18 / 87
Dirac Delta Function
Definition:
1. (t) = 0, t 6= 02. The area under (t) is
unity:
(t)dt = 1
Note: (0) = undefined
t t
Professor Deepa Kundur (University of Toronto) Test 1 Review 19 / 87
Dirac Delta Function
I can be interpreted as the limiting case of a family of functions ofunit area but that become narrower and higher
t t
all functions haveunit area
T1T2T3
T1T2T3
1 1
Professor Deepa Kundur (University of Toronto) Test 1 Review 20 / 87
Dirac Delta Function
I Sifting Property:
g(t)(t t0)dt = g(t0)
I Convolution with (t):
g(t) ? (t t0) = g(t t0)
Professor Deepa Kundur (University of Toronto) Test 1 Review 21 / 87
The Fourier Transform and the Dirac Delta(t) 1
1 (f )e j2pif0t (f f0)
cos(2pif1t) =e j2pif1t
2+
ej2pif1t
2
1
2(f f1) + 1
2(f + f1)
sin(2pif1t) =e j2pif1t
2j e
j2pif1t
2j
1
2j(f f1) 1
2j(f + f1)
0
t
0
t
f
1/2 1/2
f
-0.5j
0.5jsine
-f1
-f1f1
f11
f11
cosine
Professor Deepa Kundur (University of Toronto) Test 1 Review 22 / 87
Fourier Transforms of Periodic Signals
g(t) =
n=cne
j2pinf0t G (f ) =
n=cn(f nf0)
Professor Deepa Kundur (University of Toronto) Test 1 Review 23 / 87
t
g(t)
A
sinc
k
kc
10 2-1-2
-3 3 4 5-4-5
sinc
0 2-1-2
-3 3 4 5-4-5
Professor Deepa Kundur (University of Toronto) Test 1 Review 24 / 87
Transmission of Signals Through Linear Systems
LTI System
impulse response
LTI System
frequency response
Time domain:
y(t) = x(t) ? h(t) =
x()h(t )d
Causality: h(t) = 0 for t < 0Stability:
|h(t)|dt B
f
f
B
B
-B
-B
STOPBAND PASSBAND STOPBAND
Professor Deepa Kundur (University of Toronto) Test 1 Review 27 / 87
Ideal Low-Pass Filters
hLP(t) = 2Bsinc(2B(t t0)) HLP(f ) ={
ej2pif t0 |f | B0 |f | > B
Professor Deepa Kundur (University of Toronto) Test 1 Review 28 / 87
Ideal Low-Pass Filters
hLP(t) = 2Bsinc(2B(t t0))
2B
1/B
t0t
Professor Deepa Kundur (University of Toronto) Test 1 Review 29 / 87
LTI Systems, Sinusoids and Ideal Lowpass FilteringQ: Suppose the following signals are passed through an ideal lowpass filter withcutoff frequency W such that f1
Amplitude Modulation
I In modulation need two things:
1. a modulated signal: carrier signal: c(t)2. a modulating signal: message signal: m(t)
I carrier:I c(t) = Ac cos(2pifct); phase c = 0 is assumed.
I message:I m(t) (information-bearing signal)I assume bandwidth/max freq of m(t) is W
Professor Deepa Kundur (University of Toronto) Test 1 Review 33 / 87
Amplitude Modulation
Three types studied:
1. Amplitude Modulation (AM)(yes, it has the same name as the class of modulation techniques)
2. Double Sideband-Suppressed Carrier (DSB-SC)
3. Single Sideband (SSB)
Professor Deepa Kundur (University of Toronto) Test 1 Review 34 / 87
Amplitude Modulation (the specific technique)
sAM(t) = Ac [1 + kam(t)] cos(2pifct)
% Modulation = 100max(kam(t))
Suppose
I |kam(t)| < 1 (% Modulation < 100%)I [1 + kam(t)] > 0, so the envelope of sAM(t) is always positive;
no phase reversal
I fc WI the movement of the message is much slower than the sinusoid
Then, m(t) can be recovered with an envelope detector.
Professor Deepa Kundur (University of Toronto) Test 1 Review 35 / 87
Amplitude Modulation
AMwave
Output+
-+
-
Professor Deepa Kundur (University of Toronto) Test 1 Review 36 / 87
Amplitude Modulation
sAM(t) = Ac [1 + kam(t)] cos(2pifct)
Professor Deepa Kundur (University of Toronto) Test 1 Review 37 / 87
Double Sideband-Suppressed Carrier
sAM(t) = Ac [1 + kam(t)] cos(2pifct)
= Ac cos(2pifct) excess energy
+ka Acm(t) cos(2pifct) message-bearing signal
sDSB(t) = Acm(t) cos(2pifct)
I Transmitting only the message-bearing component of the AMsignal, requires more a complex (coherent) receiver system.
Professor Deepa Kundur (University of Toronto) Test 1 Review 38 / 87
Double Sideband-Suppressed Carrier
upper SSB
lower SSB
f
S (f )
f
S (f )
f
S (f )
f
S (f )
2W 2W
2W 2W
W W
W W
AM
USSB
LSSB
DSB
Professor Deepa Kundur (University of Toronto) Test 1 Review 39 / 87
Double Sideband-Suppressed Carrier
sDSB(t) = Acm(t) cos(2pifct)
Professor Deepa Kundur (University of Toronto) Test 1 Review 40 / 87
carrier
message
amplitudemodulation
DSB-SC
Professor Deepa Kundur (University of Toronto) Test 1 Review 41 / 87
carrier
message
amplitudemodulation
DSB-SC
Professor Deepa Kundur (University of Toronto) Test 1 Review 42 / 87
carrier
message
amplitudemodulation
DSB-SC
Professor Deepa Kundur (University of Toronto) Test 1 Review 43 / 87
Double Sideband-Suppressed Carrier
I An envelope detector will not be able to recover m(t); it willinstead recover |m(t)|.
I Coherent demodulation is required.
ProductModulator
Low-passlter
Local Oscillaor
DemodulatedSignal
v (t)0s(t)
Professor Deepa Kundur (University of Toronto) Test 1 Review 44 / 87
Costas Receiver
-90 degreePhase Shifter
Voltage-controlledOscillator
ProductModulator
Low-passFilter
PhaseDiscriminator
ProductModulator
Low-passlter
DSB-SC wave
DemodulatedSignal
Coherent Demodulation
Circuit for Phase Locking
v (t)0local oscillator output
Professor Deepa Kundur (University of Toronto) Test 1 Review 45 / 87
Costas Receiver
-90 degreePhase Shifter
Voltage-controlledOscillator
ProductModulator
Low-passFilter
PhaseDiscriminator
ProductModulator
Low-passlter
DSB-SC wave
DemodulatedSignal
Q-Channel (quadrature-phase coherent detector)
I-Channel (in-phase coherent detector)
v (t)I
v (t)Q
Professor Deepa Kundur (University of Toronto) Test 1 Review 46 / 87
Multiplexing and QAM
Multiplexing: to send multiple message simultaneously
Quadrature Amplitude Multiplexing (QAM): (a.k.a quadrature-carriermultiplexing) amplitude modulation scheme that enables twoDSB-SC waves with independent message signals to occupy the samechannel bandwidth (i.e., same frequency channel) yet still beseparated at the receiver.
Professor Deepa Kundur (University of Toronto) Test 1 Review 47 / 87
Quadrature Amplitude Modulation
s(t) = Acm1(t) cos(2pifct) + Acm2(t) sin(2pifct)
ProductModulator
Low-passFilter
ProductModulator
Low-passlter
MultiplexedSignal
-90 degreePhase Shifter
Professor Deepa Kundur (University of Toronto) Test 1 Review 48 / 87
Quadrature Amplitude Modulation
s(t) = Acm1(t) cos(2pifct) + Acm2(t) sin(2pifct)
I Suppose m1(t) and m2(t) are two message signals both ofbandwidth W .
I QAM allows two messages to be communicated withinbandwidth 2W .
Professor Deepa Kundur (University of Toronto) Test 1 Review 49 / 87
Quadrature Amplitude Modulation
s(t) = Acm1(t) cos(2pifct) + Acm2(t) sin(2pifct)
upper SSB
lower SSB
f
S (f )DSB
f
S (f )
f
S (f )
f
S (f )
2W 2W
2W 2W
W W
W W
USSB
LSSB
QAM
Professor Deepa Kundur (University of Toronto) Test 1 Review 50 / 87
Is there another way to gain this bandwidth efficiency?
upper SSB
upper SSBlower SSB
lower SSB
f
S (f )
f
S (f )
f
S (f )
2W 2W
W W
W W
USSB
LSSB
QAM
Professor Deepa Kundur (University of Toronto) Test 1 Review 51 / 87
Single Sideband
Modulation:
sSSB(t) =Ac2m(t) cos(2pifct)Ac
2m(t) sin(2pifct)
where
I the negative (positive) applies to upper SSB (lower SSB)
I m(t) is the Hilbert transform of m(t)
H(f ) = -j sgn(f )M(f ) M(f )
f
H(f )j
-j
h(t) = 1/( t)m(t) m(t)
Professor Deepa Kundur (University of Toronto) Test 1 Review 52 / 87
Single Sideband
Modulation:
sSSB(t) =Ac2m(t) cos(2pifct) Ac
2m(t) sin(2pifct)
ProductModulator
Band-passlter
m(t) s(t)
f
H (f )BP
fc-fc
W W W Wupper SSBupper SSB
lower SSB lower SSB
Professor Deepa Kundur (University of Toronto) Test 1 Review 53 / 87
Single Sideband
Coherent Demodulation:
ProductModulator
Low-passlter
Local Oscillaor
DemodulatedSignal
v (t)0s(t)
Note: Costas receiver will work for SSB demodulation.
Professor Deepa Kundur (University of Toronto) Test 1 Review 54 / 87
Comparisons of Amplitude Modulation TechniquesAM:
sAM(t) = Ac [1 + kam(t)] cos(2pifct)
SAM(f ) =Ac2
[(f fc) + (f + fc)] + kaAc2
[M(f fc) + M(f + fc)]
f
S (f )
f
S (f )
f
S (f )
f
S (f )
2W 2W
2W 2W
W W
W W
AM
USSB
LSSB
DSB
I highest power
I BT = 2W
I lowest complexity
Professor Deepa Kundur (University of Toronto) Test 1 Review 55 / 87
Comparisons of Amplitude Modulation TechniquesDSB-SC:
sDSB(t) = Ac cos(2pifct)m(t)
SDSB(f ) =Ac2
[M(f fc) + M(f + fc)] f
S (f )
f
S (f )
f
S (f )
f
S (f )
2W 2W
2W 2W
W W
W W
AM
USSB
LSSB
DSB
I lower power
I BT = 2W
I higher complexity
Professor Deepa Kundur (University of Toronto) Test 1 Review 56 / 87
Comparisons of Amplitude Modulation Techniques
SSB:
sUSSB(t) =Ac2m(t) cos(2pifct) Ac
2m(t) sin(2pifct)
SUSSB(f ) =
{Ac2 [M(f fc) + M(f + fc)] |f | fc
0 |f | < fcsLSSB(t) =
Ac2m(t) cos(2pifct) +
Ac2m(t) sin(2pifct)
SLSSB(f ) =
{0 |f | > fcAc2 [M(f fc) + M(f + fc)] |f | fc
Professor Deepa Kundur (University of Toronto) Test 1 Review 57 / 87
Comparisons of Amplitude Modulation TechniquesSSB:
upper SSB
lower SSB
f
S (f )
f
S (f )
f
S (f )
f
S (f )
2W 2W
2W 2W
W W
W W
AM
USSB
LSSB
DSB
I lowest power
I BT = W
I highest complexity
Professor Deepa Kundur (University of Toronto) Test 1 Review 58 / 87
Chapter 4: Angle Modulation
Professor Deepa Kundur (University of Toronto) Test 1 Review 59 / 87
Angle Modulation
I Consider a sinusoidal carrier:
c(t) = Ac cos(2pifct + c angle
) = Ac cos(i(t))
i(t) = 2pifct + c = 2pifct for c = 0
fi(t) =1
2pi
di(t)
dt= fc
I Angle modulation: the message signal m(t) is piggy-backed oni(t) in some way.
Professor Deepa Kundur (University of Toronto) Test 1 Review 60 / 87
Angle ModulationI Phase Modulation (PM):
i (t) = 2pifct + kpm(t)
fi (t) =1
2pi
di (t)
dt= fc +
kp2pi
dm(t)
dtsPM(t) = Ac cos[2pifct + kpm(t)]
I Frequency Modulation (FM):
i (t) = 2pifct + 2pikf
t0
m()d
fi (t) =1
2pi
di (t)
dt= fc + kfm(t)
sFM(t) = Ac cos
[2pifct + 2pikf
t0
m()d
]
Professor Deepa Kundur (University of Toronto) Test 1 Review 61 / 87
PM vs. FM
sPM(t) = Ac cos[2pifct + kpm(t)]
sFM(t) = Ac cos
[2pifct + 2pikf
t0
m()d
]sPM(t) = Ac cos[2pifct + kp
dg(t)
dt]
sFM(t) = Ac cos [2pifct + 2pikf g(t)]
FMwave
Modulatingwave Integrator
PhaseModulator
PMwave
Modulatingwave
Dieren-tiator
FrequencyModulator
Professor Deepa Kundur (University of Toronto) Test 1 Review 62 / 87
carrier
message
amplitudemodulation
phasemodulation
frequencymodulation
Professor Deepa Kundur (University of Toronto) Test 1 Review 63 / 87
carrier
message
amplitudemodulation
phasemodulation
frequencymodulation
Professor Deepa Kundur (University of Toronto) Test 1 Review 64 / 87
carrier
message
amplitudemodulation
phasemodulation
frequencymodulation
Professor Deepa Kundur (University of Toronto) Test 1 Review 65 / 87
Angle Modulation
Integrator PhaseModulator
m(t) s (t)FM
Dierentiator FrequencyModulator
m(t) s (t)PM
Professor Deepa Kundur (University of Toronto) Test 1 Review 66 / 87
Properties of Angle Modulation
1. Constancy of transmitted power
2. Nonlinearity of angle modulation
3. Irregularity of zero-crossings
4. Difficulty in visualizing message
5. Bandwidth versus noise trade-off
Professor Deepa Kundur (University of Toronto) Test 1 Review 67 / 87
Constancy of Transmitted Power: PM
Professor Deepa Kundur (University of Toronto) Test 1 Review 68 / 87
Constancy of Transmitted Power: FM
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Constancy of Transmitted Power: AM
Professor Deepa Kundur (University of Toronto) Test 1 Review 70 / 87
Nonlinearity of Angle Modulation
Consider PM (proof also holds for FM).
I Suppose
s1(t) = Ac cos [2pifct + kpm1(t)]
s2(t) = Ac cos [2pifct + kpm2(t)]
I Let m3(t) = m1(t) + m2(t).
s3(t) = Ac cos [2pifct + kp(m1(t) + m2(t))]
6= s1(t) + s2(t) cos(2pifct + A + B) 6= cos(2pifct + A) + cos(2pifct + B)
Therefore, angle modulation is nonlinear.
Professor Deepa Kundur (University of Toronto) Test 1 Review 71 / 87
Irregularity of Zero-Crossings
I Zero-crossing: instants of time at which waveform changesamplitude from positive to negative or vice versa.
Professor Deepa Kundur (University of Toronto) Test 1 Review 72 / 87
Zero-Crossings: PM
Professor Deepa Kundur (University of Toronto) Test 1 Review 73 / 87
Zero-Crossings: FM
Professor Deepa Kundur (University of Toronto) Test 1 Review 74 / 87
Zero-Crossings: AM
Professor Deepa Kundur (University of Toronto) Test 1 Review 75 / 87
Difficulty of Visualizing Message
I Visualization of a message refers to the ability to glean insightsabout the shape of m(t) from the modulated signal s(t).
Professor Deepa Kundur (University of Toronto) Test 1 Review 76 / 87
Visualization: PM
Professor Deepa Kundur (University of Toronto) Test 1 Review 77 / 87
Visualization: FM
Professor Deepa Kundur (University of Toronto) Test 1 Review 78 / 87
Visualization: AM
Professor Deepa Kundur (University of Toronto) Test 1 Review 79 / 87
Bandwidth vs. Noise Trade-Off
I Noise affects the message signal piggy-backed as amplitudemodulation more than it does when piggy-backed as anglemodulation.
I The more bandwidth that the angle modulated signal takes,typically the more robust it is to noise.
Professor Deepa Kundur (University of Toronto) Test 1 Review 80 / 87
carrier
message
amplitudemodulation
phasemodulation
frequencymodulation
Professor Deepa Kundur (University of Toronto) Test 1 Review 81 / 87
AM vs. FM
I AM is an older technology first successfully carried out in themid 1870s than FM was developed in the 1930s (by EdwinArmstrong).
I FM has better performance than AM because it is lesssusceptible to noise.
I FM takes up more transmission bandwidth than AM; Recall,
BT ,FM = 2f + 2fm vs. BT ,AM = 2W or W
I AM is lower complexity than FM.
Professor Deepa Kundur (University of Toronto) Test 1 Review 82 / 87
Narrow Band Frequency Modulation
I Suppose m(t) = Amcos(2pifmt).
fi (t) = fc + kf Amcos(2pifmt) = fc + f cos(2pifmt)
f = kf Am frequency deviationi (t) = 2pi
t0
fi ()d
= 2pifct +f
fmsin(2pifmt) = 2pifct + sin(2pifmt)
=f
fmsFM(t) = Ac cos [2pifct + sin(2pifmt)]
For narrow band FM, 1.
Professor Deepa Kundur (University of Toronto) Test 1 Review 83 / 87
Narrowband FM
Modulation:
sFM(t) Ac cos(2pifct) carrier
Ac sin(2pifct) 90oshift of carrier
sin(2pifmt) 2pifmAm
t0 m()d
DSB-SC signal
Modulatingwave
IntegratorNarrow-band
FM waveProduct
Modulator
-90 degreePhase Shifter carrier
+-
Professor Deepa Kundur (University of Toronto) Test 1 Review 84 / 87
Transmission Bandwidth of FM Waves
A significant component of the FM signal is within the followingbandwidth:
BT 2f + 2fm = 2f(
1 +1
)I called Carsons Rule
I f is the deviation of the instantaneous frequency
I fm can be considered to be the maximum frequency of themessage signal
I For 1, BT 2f = 2kfAmI For 1, BT 2f 1 = 2ff /fm = 2fm
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Generation of FM Waves
Narrow bandModulator
FrequencyMultiplier
m(t) s(t) s(t)Integrator
CrystalControlledOscillator
frequency isvery stable
Narrowband FM modulator
widebandFM wave
Professor Deepa Kundur (University of Toronto) Test 1 Review 86 / 87
Demodulation of FM Waves
Ideal EnvelopeDetector
ddt
I Frequency Discriminator: uses positive and negative slopecircuits in place of a differentiator, which is hard to implementacross a wide bandwidth
I Phase Lock Loop: tracks the angle of the in-coming FM wavewhich allows tracking of the embedded message
Professor Deepa Kundur (University of Toronto) Test 1 Review 87 / 87