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ELEC 350Communications Theory and
Systems: I
Review
ELEC 350 Fall 2007 1
Final Examination
• Saturday, December 15 - 3 hours
• Two pages of notes allowed
• Calculator
• Tables provided
– Fourier transforms – Table 2.1
– Bandpass to lowpass translation relations – Table 2.2
– Bessel function values – Table 3.1
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Syllabus
• Overview of Communication Systems
• Review of Signal Analysis (ELEC 260,310)
• Analog Modulation
– Linear Modulation (DSB,AM,SSB,VSB)
– Angle Modulation (FM,PM)
• Random Processes
• System Performance in Noise
3
Course Content• Text sections in Proakis and Salehi (2nd Edition)
– Chapter 1– Chapter 2– Chapter 3 (except 3.4.3 and 3.5)– Chapter 4 (except 4.5)– Chapter 5 (except 5.5.4)– Chapter 7 (only 7.7.2 – link budget analysis)
• All slides are on the website– www.ece.uvic.ca/~agullive/materials350.html
• Link Budgets– http://www.ece.uvic.ca/~agullive/350link3.pdf
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Signals and Systems
• Signal used to transmit the information over the channel
– Information Bearing Signal
• The communication channel is modelled as a Linear System
• Main analysis tool: Fourier Transform
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Power Spectral Density
o( ) cos(2 )cx t A f t
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2
0( ) cos 22
cX
AR f
2
(0)2
cX
AR
0 0
2
0
2 222
2 2
0 0
( ) {cos 2 }2
2 2
( ) ( )4 4
cX
j f j fj fc
c c
AS f F f
A e ee d
A Af f f f
Objectives of Modulation
• Convert a lowpass signal to bandpass
• Accommodate the simultaneous transmission of signals from several sources
• Expand the signal bandwidth to increase noise immunity
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Amplitude Modulation
• Double-sideband suppressed carrier (DSB-SC)
• Conventional AM
• Single-sideband (SSB) AM
• Vestigal-sideband (VSB) AM
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Conventional AM
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• Suppose that the nonlinear device is approximated by a second order polynomial
)()(2
1
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AM Modulation Summary
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ModulationPower
EfficiencySpectral
Efficiency (xW)
Modulation Complexity Demodulation
Conventional AM
low 2 low simple
DSB-SC high 2 low complex
SSB high 1 high complex
VSB high 1-2 medium complex
14
Angle Modulation
• Angle modulation
– Frequency modulation (FM): Frequency is changed by the message m(t).
– Phase modulation (PM): Phase is changed by the message m(t).
• Angle modulated signals have a high degree of noise immunity, but require larger bandwidth than AM signals.
• They are widely used in high-fidelity music broadcasting.
• They have a constant envelope, which is beneficial when using nonlinear amplifiers.
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• The message signal is used with either FM or PM for the carrier . Find the modulated signal in each case.
Solution:
we have
• Modulation index for a general m(t)
)2cos()(
)2cos( tfA cc
)2cos()()( tfaktmkt mpp )2sin()(2)(
PMPM FMFM
))2sin(2cos(
))2cos(2cos()(
/
Modulation indexModulation index
//)(max
)(max
max
max
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Varactor Diode Angle Modulator
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Phase-Locked Loop FM Demodulator
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Bandpass Processes
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• Def 4.6.1: X(t) is a bandpass or narrowband process if for , where
• Let X(t) be a bandpass process. Then is a deterministic bandpass signal whose Fourier transform is nonzero in the neighborhood of .
• X(t) and its Hilbert transform have the form
• are lowpass processes
( ) 0XS f
0| |f f W 0W f
( )XR
( )XS f
0f
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0 0
0 0
( ) ( )cos(2 ) ( )sin(2 )
ˆ ( ) ( )sin(2 ) ( )cos(2 )
c s
c s
X t X t f t X t f t
X t X t f t X t f t
( ) and ( )c sX t X t
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Narrowband Noise Properties
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1. are zero-mean, lowpass, jointly stationary and jointly Gaussian random processes
2. If the power in X(t) is
3. have common PSDs obtained by shifting the positive frequencies in to the left by and the negative frequencies in to the right by and adding the two spectra
XP
c sX X X XP P P S f df
and ( ) ( )c sX t X t
0f0f( )XS f
( )XS f
and ( ) ( )c sX t X t
Noise in Analog Systems
• Most analog continuous-wave systems are bandpass -> suffer from bandpass noise
• Design the BP filter just wide enough to pass u(t) without distortion
– Minimize the noise power input to the demodulator
• Figure of Merit – SNR at demodulator output
• Reference – baseband SNR
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AM Signal to Noise Ratio (SNR)
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2
0 0
2
0 0
2 22 2 2
2
0 0
2 2
2 2
0
2
/ 2 1
2 1
1 1
o
o
nn n
o n
n n
n n
o c m
oDSB bn
o c m
oSSB n
c mc m mo
oAM n m
m m
b bm m
R
R
R
PP A PS S
N P N W N W N
PP A PS
N P N W N W
A a PA a P a PPS
N P N W a P N W
Pa P a P S S
a P N W a P N N
Angle Modulation
• The modulated signal
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))(2cos()(
FMdmktfA
PMtmktfAt
fcc
pcc
))(22cos(
))(2cos(
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Angle Modulation Noise PSD
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0
0
0
2
20
2
0
2
3
0
2
PM
( )
FM
2PM
2FM
3
c
n
c
c
n
c
N
AS f
Nf
A
WN
AP
W N
A
Noise PSD for PM and FM
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FM Signal to Noise Ratio (SNR)
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2
2
2
2
PMN
3 FMN
12 PM
max m(t)
123 FM
max m(t)
n
n
p M
b
f M
b
M
M
b
b
o
o
SP
S
N SP
SP
N
S
N
SP
N
Observations• The output SNR is proportional to the square of the
modulation index• Angle modulation allows a tradeoff between SNR and
bandwidth• The relationship between the output SNR and the
bandwidth expansion factor is quadratic• Increasing too much results in the threshold effect
where the signal is lost in noise• Compared with AM, increasing the transmitted power
increases the output SNR – but the mechanisms differ• In FM, noise affects higher frequencies more than
lower frequencies
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Threshold Effect in FM• At threshold
• Carson’s rule
• Assume
• then
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,
20( 1)f
b th
S
N
2( 1)c fB W
2
1
(max | ( ) |) 2n
MM
PP
m t
23
2f
o b
S S
N N
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Pre-emphasis and De-emphasis
• The phase of the transmitted signal is
• At high frequencies use PM
– At the transmitter, a differentiator followed by an FM modulator
– At the receiver, an FM demodulator followed by an integrator
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FM)(2
PM)()( t
f
p
dmk
tmkt
Pre-emphasis and De-emphasis Filters
• In order to produce an undistorted version of the original message at the receiver output, we must have
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( ) ( ) 1 for W Wp dH f H f f
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Noise PSD and SNR Gain
• The noise power at the demodulator output
• The ratio of output SNRs is
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3
0
0 0
3 arctan
oPD
D
no
n
o
S W
PN f
S P W W
N f f
3
0 0
2
0 0
2( ) arctan
PD PD
W
n nW
c
N f W WP S f df
A f f
Comparison of Analog Modulation
• Linear modulation
– DSB-SC
– SSB-SC
– VSB
– Conventional AM
• Nonlinear modulation
– PM
– FM
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Comparison Criteria
• Bandwidth efficiency
• Power efficiency
– SNR at demodulator output
• Simplicity of the transmitter and receiver implementation
– Receiver complexity is most important
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Bandwidth Efficiency
• SSB-SC is best
• VSB
• DSB-SC and Conventional AM
• FM is worst – using Carson’s rule:
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cB W
cW B 2W
cB 2W
c2 B 6Wf
c5 B 12Wf
Power Efficiency
• Output SNR for a given received signal power
• Angle modulation and in particular FM provides the highest SNR gain
• Conventional AM and VSB+carrier are worst
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Implementation Complexity
• Conventional AM and VSB+C demodulators have extremely simple receiver structures
– envelope detector
• FM also has a simple structure
– discriminator + AM demodulator
• To obtain better FM performance use a PLL
• SSB-SC and DSB-SC require coherent detectors (Squaring Loop or a Costas Loop)
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Final Comments
• SSB modulation provides optimum noise performance and bandwidth efficiency with amplitude modulation
• Conventional AM provides the simplest receiver structure making it the most common wireless communication technique
• FM improves the noise performance at the expense of increased transmission bandwidth
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