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.
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Chapter 2: Fourier Representation ofSignals and Systems
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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
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The Fourier Transform (FT)
G (f ) =
∫ ∞−∞
g(t)e−j2πft
g(t) =
∫ ∞−∞
G (f )e+j2πft
Notation:
g(t) G (f )
G (f ) = F [g(t)]
g(t) = F−1 [G (f )]
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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
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Energy Signals and the Fourier Transform
Practical physically realizable signals (e.g., energy signals) that obey:∫ ∞−∞|g(t)|2dt <∞
have Fourier transforms.
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FT Synthesis and Analysis Equations
G (f ) =
∫ ∞−∞
g(t)e−j2πft
g(t) =
∫ ∞−∞
G (f )e+j2πft
g(t) G (f )
I Since the FT is invertible both g(t) and G (f ) contain thesame information, but describe it in a different way.
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FT Synthesis Equation
g(t) =
∫ ∞−∞
G (f )e j2πftdt
I g(t) is the sum of scaled complex sinusoids
I e j2πft = cos(2πft) + jsin(2πft) ≡ complex sinusoid
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e j2πft = cos(2πft) + j sin(2πft)
cos(2πft)
0
t
sin(2πft)
0
t
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FT Analysis Equation
G (f ) =
∫ ∞−∞
g(t)e−j2πftdt
I The analysis equation represents the inner product between g(t)
and e j2πft .
I The analysis equation states that G (f ) is a measure of similarity
between g(t) and e j2πft , the complex sinusoid at frequency f Hz.
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|G (f )| and ∠G (f )
g(t) =
∫ ∞∞
G (f )e j2πf tdf
=
∫ ∞∞|G (f )|e j(2πf 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).
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Low, Mid and High Frequency Signals
Q: Which of the following signals appears higher in frequency?
1. cos(4× 106πt + π/3)
2. sin(2πt + 10π) + 17 cos2(10πt)
A: cos(4× 106πt + π/3).
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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.
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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 )e−j2πft0
Frequency Shifting: e j2πfc 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) j2πfG(f )
Time Integration :∫ t−∞ g(τ)dτ 1
j2πfG(f )
Modulation Theorem: g1(t)g2(t) ∫∞−∞ G1(λ)G2(f − λ)dλ
Convolution Theorem:∫∞−∞ g1(τ)g2(t − τ) G1(f )G2(f )
Correlation Theorem:∫∞−∞ g1(t)g∗2 (t − τ)dt G1(f )G∗2 (f )
Rayleigh’s Energy Theorem:∫∞∞ |g(t)|2dt =
∫∞∞ |G(f )|2df
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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
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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
4π
with equality achieved for a Gaussian pulse.
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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.
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Dirac Delta Function
Definition:
1. δ(t) = 0, t 6= 0
2. The area under δ(t) isunity:∫ ∞
−∞δ(t)dt = 1
Note: δ(0) = undefined
t t
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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
T1
T2
T3
T1
T2
T3
1 1
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Dirac Delta Function
I Sifting Property: ∫ ∞−∞
g(t)δ(t − t0)dt = g(t0)
I Convolution with δ(t):
g(t) ? δ(t − t0) = g(t − t0)
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The Fourier Transform and the Dirac Deltaδ(t) 1
1 δ(f )
e j2πf0t δ(f − f0)
cos(2πf1t) =e j2πf1t
2+
e−j2πf1t
2
1
2δ(f − f1) +
1
2δ(f + f1)
sin(2πf1t) =e j2πf1t
2j− e−j2πf1t
2j
1
2jδ(f − f1)− 1
2jδ(f + f1)
0
t
0
t
f
1/2 1/2
f
-0.5j
0.5jsine
-f1
-f1f1
f11
f11
cosine
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Fourier Transforms of Periodic Signals
g(t) =∞∑
n=−∞
cnej2πnf0t G (f ) =
∞∑n=−∞
cnδ(f − nf0)
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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
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Transmission of Signals Through Linear Systems
LTI System
impulse response
LTI System
frequency response
Time domain:
y(t) = x(t) ? h(t) =
∫ ∞−∞
x(τ)h(t − τ)dτ
Causality: h(t) = 0 for t < 0Stability:
∫∞−∞ |h(t)|dt <∞
Freqeuncy domain:
x(t) X (f )
y(t) Y (f )
h(t) H(f )
Y (f ) = H(f ) · X (f )
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Transmission of Signals Through Linear SystemsLTI System
impulse response
LTI System
frequency response
Y (f ) = H(f )︸︷︷︸freq selective system
·X (f )
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Ideal Low-Pass Filters
HLP(f ) =
{e−j2πf t0 |f | ≤ B0 |f | > 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 ) =
{e−j2πf t0 |f | ≤ B0 |f | > B
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Ideal Low-Pass Filters
hLP(t) = 2Bsinc(2B(t − t0))
2B
1/B
t0
t
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LTI Systems, Sinusoids and Ideal Lowpass FilteringQ: Suppose the following signals are passed through an ideal lowpass filter withcutoff frequency W such that f1 <W < f2 � fc . What are the correspondingoutputs:
1. m(t) = sin(2πf1t + π4 )
2. m(t) = sin(2πf1t + π4 ) + cos(2πf2t − π
5 )
3. m(t) = sin(2πf1t + π4 ) + cos(2πfct)
4. s(t) = sin(2πf1t + π4 ) · cos(2πfct)
5. s(t) = sin(2πf1t + π4 ) · sin(2πfct)
6. s(t) = sin(2πf1t + π4 ) + cos2(2πfct) Note: cos2 A = 1
2 + 12 cos(2A)
7. s(t) = sin(2πf1t + π4 ) · cos2(2πfct)
8. s(t) = sin(2πf1t + π4 ) · cos(2πfct) · sin(2πfct)
Note: sinA cosB = 12 sin(A + B)− 1
2 sin(B − A)
A: 1. sin(2πf1t + π4
), 2. sin(2πf1t + π4
), 3. sin(2πf1t + π4
), 4. 0, 5. 0,
6. sin(2πf1t + π4
) + 12
, 7. 12
sin(2πf1t + π4
), 8. 0.
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Chapter 3: Amplitude Modulation
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Modulation
I Modulation: to adjust or adapt to a certain proportionI Used to superimpose one signal onto another.
I In modulation need two things:
1. a modulated signal that is changed:carrier signal: c(t)
2. a modulating signal that dictates how to change:message signal: m(t)
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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(2πfct); phase φc = 0 is assumed.
I message:I m(t) (information-bearing signal)I assume bandwidth/max freq of m(t) is W
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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)
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Amplitude Modulation (the specific technique)
sAM(t) = Ac [1 + kam(t)] cos(2πfct)
% Modulation = 100×max(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.
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Amplitude Modulation
AMwave
Output+
-+
-
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Amplitude Modulation
sAM(t) = Ac [1 + kam(t)] cos(2πfct)
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Double Sideband-Suppressed Carrier
sAM(t) = Ac [1 + kam(t)] cos(2πfct)
= Ac cos(2πfct)︸ ︷︷ ︸excess energy
+ka Acm(t) cos(2πfct)︸ ︷︷ ︸message-bearing signal
sDSB(t) = Acm(t) cos(2πfct)
I Transmitting only the message-bearing component of the AMsignal, requires more a complex (coherent) receiver system.
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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
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Double Sideband-Suppressed Carrier
sDSB(t) = Acm(t) cos(2πfct)
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carrier
message
amplitudemodulation
DSB-SC
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carrier
message
amplitudemodulation
DSB-SC
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carrier
message
amplitudemodulation
DSB-SC
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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-pass�lter
Local Oscillaor
DemodulatedSignal
v (t)0s(t)
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Costas Receiver
-90 degreePhase Shifter
Voltage-controlledOscillator
ProductModulator
Low-passFilter
PhaseDiscriminator
ProductModulator
Low-pass�lter
DSB-SC wave
DemodulatedSignal
Coherent Demodulation
Circuit for Phase Locking
v (t)0local oscillator output
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Costas Receiver
-90 degreePhase Shifter
Voltage-controlledOscillator
ProductModulator
Low-passFilter
PhaseDiscriminator
ProductModulator
Low-pass�lter
DSB-SC wave
DemodulatedSignal
Q-Channel (quadrature-phase coherent detector)
I-Channel (in-phase coherent detector)
v (t)I
v (t)Q
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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.
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Quadrature Amplitude Modulation
s(t) = Acm1(t) cos(2πfct) + Acm2(t) sin(2πfct)
ProductModulator
Low-passFilter
ProductModulator
Low-pass�lter
MultiplexedSignal
-90 degreePhase Shifter
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Quadrature Amplitude Modulation
s(t) = Acm1(t) cos(2πfct) + Acm2(t) sin(2πfct)
I Suppose m1(t) and m2(t) are two message signals both ofbandwidth W .
I QAM allows two messages to be communicated withinbandwidth 2W .
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Quadrature Amplitude Modulation
s(t) = Acm1(t) cos(2πfct) + Acm2(t) sin(2πfct)
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
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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
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Single Sideband
Modulation:
sSSB(t) =Ac
2m(t) cos(2πfct)∓Ac
2m̂(t) sin(2πfct)
where
I the negative (positive) applies to upper SSB (lower SSB)
I m̂(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)
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Single Sideband
Modulation:
sSSB(t) =Ac
2m(t) cos(2πfct)∓ Ac
2m̂(t) sin(2πfct)
ProductModulator
Band-pass�lter
m(t) s(t)
f
H (f )BP
fc-fc
W W W Wupper SSBupper SSB
lower SSB lower SSB
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Single Sideband
Coherent Demodulation:
ProductModulator
Low-pass�lter
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(2πfct)
SAM(f ) =Ac
2[δ(f − fc) + δ(f + fc)] +
kaAc
2[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
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Comparisons of Amplitude Modulation TechniquesDSB-SC:
sDSB(t) = Ac cos(2πfct)m(t)
SDSB(f ) =Ac
2[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
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Comparisons of Amplitude Modulation Techniques
SSB:
sUSSB(t) =Ac
2m(t) cos(2πfct)− Ac
2m̂(t) sin(2πfct)
SUSSB(f ) =
{Ac
2 [M(f − fc) + M(f + fc)] |f | ≥ fc0 |f | < fc
sLSSB(t) =Ac
2m(t) cos(2πfct) +
Ac
2m̂(t) sin(2πfct)
SLSSB(f ) =
{0 |f | > fcAc
2 [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
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Chapter 4: Angle Modulation
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Angle Modulation
I Consider a sinusoidal carrier:
c(t) = Ac cos(2πfct + φc︸ ︷︷ ︸angle
) = Ac cos(θi(t))
θi(t) = 2πfct + φc = 2πfct for φc = 0
fi(t) =1
2π
dθi(t)
dt= fc
I Angle modulation: the message signal m(t) is piggy-backed onθi(t) in some way.
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Angle ModulationI Phase Modulation (PM):
θi (t) = 2πfct + kpm(t)
fi (t) =1
2π
dθi (t)
dt= fc +
kp2π
dm(t)
dtsPM(t) = Ac cos[2πfct + kpm(t)]
I Frequency Modulation (FM):
θi (t) = 2πfct + 2πkf
∫ t
0
m(τ)dτ
fi (t) =1
2π
dθi (t)
dt= fc + kfm(t)
sFM(t) = Ac cos
[2πfct + 2πkf
∫ t
0
m(τ)dτ
]
Professor Deepa Kundur (University of Toronto) Test 1 Review 61 / 87
PM vs. FM
sPM(t) = Ac cos[2πfct + kpm(t)]
sFM(t) = Ac cos
[2πfct + 2πkf
∫ t
0
m(τ)dτ
]sPM(t) = Ac cos[2πfct + kp
dg(t)
dt]
sFM(t) = Ac cos [2πfct + 2πkf g(t)]
FMwave
Modulatingwave Integrator
PhaseModulator
PMwave
Modulatingwave
Di�eren-tiator
FrequencyModulator
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carrier
message
amplitudemodulation
phasemodulation
frequencymodulation
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carrier
message
amplitudemodulation
phasemodulation
frequencymodulation
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carrier
message
amplitudemodulation
phasemodulation
frequencymodulation
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Angle Modulation
Integrator PhaseModulator
m(t) s (t)FM
Di�erentiator FrequencyModulator
m(t) s (t)PM
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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
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Constancy of Transmitted Power: PM
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Constancy of Transmitted Power: FM
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Constancy of Transmitted Power: AM
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Nonlinearity of Angle Modulation
Consider PM (proof also holds for FM).
I Suppose
s1(t) = Ac cos [2πfct + kpm1(t)]
s2(t) = Ac cos [2πfct + kpm2(t)]
I Let m3(t) = m1(t) + m2(t).
s3(t) = Ac cos [2πfct + kp(m1(t) + m2(t))]
6= s1(t) + s2(t)
∵ cos(2πfct + A + B) 6= cos(2πfct + A) + cos(2πfct + B)
Therefore, angle modulation is nonlinear.
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Irregularity of Zero-Crossings
I Zero-crossing: instants of time at which waveform changesamplitude from positive to negative or vice versa.
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Zero-Crossings: PM
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Zero-Crossings: FM
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Zero-Crossings: AM
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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).
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Visualization: PM
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Visualization: FM
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Visualization: AM
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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.
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carrier
message
amplitudemodulation
phasemodulation
frequencymodulation
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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 = 2∆f + 2fm vs. BT ,AM = 2W or W
I AM is lower complexity than FM.
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Narrow Band Frequency Modulation
I Suppose m(t) = Amcos(2πfmt).
fi (t) = fc + kf Amcos(2πfmt) = fc + ∆f cos(2πfmt)
∆f = kf Am ≡ frequency deviation
θi (t) = 2π
∫ t
0
fi (τ)dτ
= 2πfct +∆f
fmsin(2πfmt) = 2πfct + βsin(2πfmt)
β =∆f
fmsFM(t) = Ac cos [2πfct + βsin(2πfmt)]
For narrow band FM, β � 1.
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Narrowband FM
Modulation:
sFM(t) ≈ Ac cos(2πfct)︸ ︷︷ ︸carrier
−β Ac sin(2πfct)︸ ︷︷ ︸−90oshift of carrier
sin(2πfmt)︸ ︷︷ ︸2πfmAm
∫ t0 m(τ)dτ︸ ︷︷ ︸
DSB-SC signal
Modulatingwave
IntegratorNarrow-band
FM waveProduct
Modulator
-90 degreePhase Shifter carrier
+-
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Transmission Bandwidth of FM Waves
A significant component of the FM signal is within the followingbandwidth:
BT ≈ 2∆f + 2fm = 2∆f
(1 +
1
β
)I called Carson’s 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 β � 1, BT ≈ 2∆f = 2kfAm
I For β � 1, BT ≈ 2∆f 1β
= 2∆f∆f /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
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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
�
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