Test 1 Review - University of Torontodkundur/course_info/316/KundurTest1Review... · Test 1 Review Professor Deepa Kundur University of Toronto Professor Deepa Kundur (University

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.


Chapter 2: Fourier Representation ofSignals and Systems


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


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 )]


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


Energy Signals and the Fourier Transform

Practical physically realizable signals (e.g., energy signals) that obey:∫ ∞−∞|g(t)|2dt <∞

have Fourier transforms.


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.


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


e j2πft = cos(2πft) + j sin(2πft)

cos(2πft)

0

t

sin(2πft)

0

t


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.


|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).


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).


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.


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


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


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.


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.


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



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



I Sifting Property: ∫ ∞−∞

g(t)δ(t − t0)dt = g(t0)

I Convolution with δ(t):

g(t) ? δ(t − t0) = g(t − t0)


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


Fourier Transforms of Periodic Signals

g(t) =∞∑

n=−∞

cnej2πnf0t G (f ) =

∞∑n=−∞

cnδ(f − nf0)


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


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 )


Transmission of Signals Through Linear SystemsLTI System

impulse response

LTI System

frequency response

Y (f ) = H(f )︸︷︷︸freq selective system

·X (f )


Ideal Low-Pass Filters

HLP(f ) =

{e−j2πf t0 |f | ≤ B0 |f | > B

f

f

B

B

-B

-B

STOPBAND PASSBAND STOPBAND



hLP(t) = 2Bsinc(2B(t − t0)) HLP(f ) =

{e−j2πf t0 |f | ≤ B0 |f | > B



hLP(t) = 2Bsinc(2B(t − t0))

2B

1/B

t0

t


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.


Chapter 3: Amplitude Modulation


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)


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



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)


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.



AMwave

Output+

-+

-





Double Sideband-Suppressed Carrier


= 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.



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



sDSB(t) = Acm(t) cos(2πfct)


carrier

message

amplitudemodulation

DSB-SC


carrier

message

amplitudemodulation

DSB-SC


carrier

message

amplitudemodulation

DSB-SC



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)


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


Costas Receiver


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


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.


Quadrature Amplitude Modulation

s(t) = Acm1(t) cos(2πfct) + Acm2(t) sin(2πfct)

ProductModulator

Low-passFilter

ProductModulator

Low-pass�lter

MultiplexedSignal





I Suppose m1(t) and m2(t) are two message signals both ofbandwidth W .

I QAM allows two messages to be communicated withinbandwidth 2W .




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


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


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)


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


Single Sideband

Coherent Demodulation:

ProductModulator

Low-pass�lter

Local Oscillaor

DemodulatedSignal

v (t)0s(t)

Note: Costas receiver will work for SSB demodulation.


Comparisons of Amplitude Modulation TechniquesAM:


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


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


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


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


Chapter 4: Angle Modulation


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.


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

]


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


carrier

message

amplitudemodulation

phasemodulation

frequencymodulation


carrier

message

amplitudemodulation

phasemodulation

frequencymodulation


carrier

message

amplitudemodulation

phasemodulation

frequencymodulation


Angle Modulation

Integrator PhaseModulator

m(t) s (t)FM

Di�erentiator FrequencyModulator

m(t) s (t)PM


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


Constancy of Transmitted Power: PM


Constancy of Transmitted Power: FM


Constancy of Transmitted Power: AM


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.


Irregularity of Zero-Crossings

I Zero-crossing: instants of time at which waveform changesamplitude from positive to negative or vice versa.


Zero-Crossings: PM


Zero-Crossings: FM


Zero-Crossings: AM


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).


Visualization: PM


Visualization: FM


Visualization: AM


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.


carrier

message

amplitudemodulation

phasemodulation

frequencymodulation


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.


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.


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

+-


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


Generation of FM Waves

Narrow bandModulator

FrequencyMultiplier

m(t) s(t) s’(t)Integrator

CrystalControlledOscillator

frequency isvery stable

Narrowband FM modulator

widebandFM wave


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

�


Documents

Test 1 Review - University of Torontodkundur/course_info/316/KundurTest1Review... · Test 1 Review Professor Deepa Kundur University of Toronto Professor Deepa Kundur (University