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 Chapter 3 Signal Transmission and Filtering
 Outline
 3.1 Response of LTI System
Coherent AM reception and LPF
 3.2 Signal Distortion in Transmission
Multipath propagation
 3.3 Transmission Loss and Decibels
Doppler frequency shift and beating
 3.4 Filters and Filtering
Quadrature modulator and demodulator, heterodyne receiver
 3.5 Quadrature Filters and Hilbert Transform
 3.6 Correlation and Spectral Density
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 3.1 RESPONSE OF LTI SYSTEMS
 Coherent AM reception and LPF









a system
linear time-invariant system
impulse response and convolution integral
step response
LCCDE and LTI system
transfer function and frequency response
steady-state phasor response
undistorted transmission vs. distorted transmission
block diagram analysis: parallel, serial/cascade, feedback
connection
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 Example 3.3-2 Doppler Shift
beating
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 3.2 SIGNAL DISTORTION IN TRANSMISSION
 Chapter 3 is all about the channel.
 3.1 Heterodyne quadrature modulator and demodulator have LTI
filters.
 There are 4 types of channels for wireless communication using
EM wave in the RF band .
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 If interference and noise are ignored;
1. The propagation channel is modeled by a linear channel.
 Each path has the following four characters:
» Gain, Delay
» Doppler
» Angle/Direction of Departure (AOD/DOD)
» Angle/Direction of Arrival (AOA/DOA)
2. The radio channel maps the propagation channel to a CT
SISO/MISO/SIMO/MIMO linear system depending on;
 antenna pattern (directivity) and
 configurations (spacing).
» Directional antenna. Ex. Horn antenna,
» Omni-directional antenna
» uniform linear array (ULA)
» uniform circular array (UCA)
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3. The modulation channel may introduce nonlinear
distortion incurred by amplifiers.
4. The digital channel is modeled by a DT system.
 Precisely speaking, the channel becomes nonlinear with
finite precision.
 Often modeled by a linear DT system corrupted by
additive quantization noise.
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 Distortionless Transmission
A channel is distortionless iff it is an LTI system with
impulse response
h (t)  K  (t  td )  H (f )  Ke
Frequency-flat channel
 Over the desired band
 phase
 Frequency-selective channel
 Distortions
Nonlinear distortions
Linear distortions
 Amplitude distortion
 Phase distortion
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 j2 ftd
 Example: linear distortions
Test signal x(t) = cos 0t + 1/5 cos 50t
Figure 3.2-3
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 Amplitude distortion
Test signal with amplitude distortion (a) low
frequency attenuated; (b) high frequency attenuated
Figure 3.2-4
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 Phase distortion
Test signal with constant phase shift  = -90
Figure 3.2-5
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 Equalization
Multipath distortion
Intersymbol interference (ISI) in digital signal transmission
Linear equalization
 Linear zero-forcing equalization (LZF): channel inversion
 Linear minimum-mean square error equalization (LMMSE)
Nonlinear equalization
 Maximum-likelihood sequence estimator (MLSE)
 Decision-feedback equalization (DFE)
» Feedforward (FF) filter and feedback (FB) filter
» ZF-DFE
» MMSE-DFE
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CT equalizer vs. DT equalizer vs. block equalizer
 Transversal filter, tapped-delay-line equalizer
 Frequency-domain equalizer (FDE)
» One-tap equalizer for OFDM
Adaptive equalizer
 Nonlinear distortion and companding
Transfer characteristic
 Memoryless distortion
 Distortion with memory
Polynomial approximation of memoryless distortion
 Second-harmonic distortion
 Intermodulation distortion
Companding
 Compressing + expanding
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 3.3 TRANSMISSION LOSS and DECIBELS
 Power gain
g = P_out/P_in
decibels
 g_dB = 10 log_10 g
 3 dB = 1/2
 G = 10^(g_dB/10)
 Serial interconnection of amplifiers and attenuators ->
addition, subtraction in dB
 If g = 10^m, then g_dB = m*10 dB
dBm
 0 dBm = 1 mW
 10 dBm = 10 mW
 20 dBm = 100 mW = 0.1 W
 30 dBm = 1 W = 0 dBW
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 Transmission loss and repeaters
Loss L = 1/g
Path loss
Passive transmission medium
 Transmission lines
» coaxial cable: Coaxial lines confine virtually all of
the electromagnetic wave to the area inside the cable.
» Twisted(-wire) pair cable:
EMI is cancelled. Invented
by A. G. Bell.
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» Fiber-optic cables
» Waveguides
 Loss, attenuation
 Attenuation coefficient in dB per unit length
» Table 3.3-1
» Frequency bands are different.
» Fiber optic cable: 0.2-2.5 dB/km loss
» Twisted pair: 2-6 dB/km loss
» Coaxial cable: 1-6 dB/km loss
» Waveguide: 1.5-5 dB/km loss
»…
 Repeater amplifier
» Amplification of distortion, interference, and noise
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 Optical fiber cable
Total reflection, refraction index
Light propagation
down a single-mode
step-index fiber
Light propagation down a
multimode step-index fiber
Figure 3.3-3a
Figure 3.3-3b
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Light propagation down a multimode graded-index fiber
Figure 3.3-3c
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Large bandwidth and low loss
 Carrier frequencies in the range of 200 THz
» Max bandwidth 20 THz
 0.2-2 dB/km loss
» Lower than most twisted-pair and coaxial cable
systems
» Absorption
» Scattering
Less interference
 No RF interference
No noise
Low maintenance cost
Secure
Hybrid of electrical and optical components
 LED or laser
 Envelope detector
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 Correction and Announcement
 Propagation channel: Each path has gain, …
 A channel is distortionless iff it is an LTI system with impulse
response
h (t)  K  (t  td )  H (f )  Ke
 j2 ftd
 Nonlinear memoryless distortion has input output relation given
by
N
y (t)  a n x n (t)
n 0
which increases bandwidth of the output because multiplication in
TD corresponds to convolution in the FD.
 Exam on next Tuesday @LG104, 11:00-12:15
Ch. 1-3
Open book (but you will not have time to read on the site.)
T/F, filling blanks, Essay, Math
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 Radio Transmission
Line-of-sight propagation
 Free-space path loss (FSPL)
» The loss between two isotropic radiators in free
space.
 Formula  4l 2  4fl  2
L
 

    c 
where l  path length,  = wavelength, f  frequency,
c  speed of light
LdB  92.4  20log10 f GHz  20log10 lkm
» far-field
» It is a function of frequency. However, it does not say
that free space is a frequency-selective channel.
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 Example 3.3-1
Satellite repeater system:
uplink, downlink, frequency translation, geostationary, low
orbit, OBP
Figure 3.3-5
Pout 
gTu g Ru g amp gTd g Rd
Lu Ld
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Pin
 3.4 FILTERS and FILTERING
 Ideal Filters
LPF
BPF
 Lower and upper cutoff frequencies
 Passband and stopband
HPF
Transfer function of a ideal bandpass filter
NF
Figure 3.4-1
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 Realizability, noncausality
Ideal lowpass filter (a) Transfer function (b) Impulse response
Figure 3.4-2
Ideal filters are noncausal.
 Bandlimiting and timelimiting
It is impossible to have perfect bandlimiting and timelimiting
at the same time.
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 Real-World Filters
 Half-power or 3 dB bandwidth
 Passband, transition band/region, and stopband
Typical amplitude ratio of a real bandpass filter
Figure 3.4-3
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 3.5 QUADRATURE FILTERS and HILBERT TRANSFORMS
The quadrature filter is an allpass network that shifts the phase of
positive frequencies by -900 and negative frequencies by +900
H Q ( f )   j sgn f 

j
j
f 0
f 0
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 h(t ) 
1
t
 Quadrature Filtering and Hilbert Transform
Hilbert tranform

xˆ (t )
1 1 x ( )
x(t )   
d
t   t  
Fourier transform of Hilbert transform
 xˆ (t )  ( j sgn f ) X ( f )  H Q ( f ) X ( f )
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 Example. Hilbert transform of a rectangular pulse
(a) Convolution; (b) Result
Figure 3.5-2
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 Example. Hilbert transform of cosine signal
x(t )  A cos(0t  )
 jA
ˆ
 X ( f )   j sgn fX ( f ) 
( f  f 0 )  ( f  f 0 ) sgn f
2
A
=  ( f  f 0 )  ( f  f 0 ) 
2j


xˆ (t )  1 Xˆ ( f )  A sin(0t  )
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
Instead of separating signals based on frequency
content we may want to separate them based on phase
content.  Hilbert transform
Hilbert transform used for describing single sideband (SSB)
signals and other bandpass signals
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 Properties of the Hilbert transform
1. x(t ) and xˆ (t ) have same amplitude spectrum
2. Energy and power in a signal and its Hilbert tranform are equal
3. x(t ) and xˆ (t ) are orthogonal 

 x(t ) xˆ(t )dt  0
(energy)

1
lim
T  2T
T
 x(t ) xˆ(t )dt  0
(power)
T
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 3.6 CORRELATION AND SPECTRAL DENSITY
 Stochastic Process = signal with uncertainty described
probabilistically
Non-periodic signal
Non-energy signal
Ex)Bit Stream
Noise
Voice Signal
Two ways to describe: 1) probability space and mapping to sample path ,2)
Kolomgorov’ s extension theorem
v(t )
t
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 Ensemble Average
<At time t>
V (t)  E [V (t)] 



v fV (v ;t)dv
 Correlation R VW (t1,t2 )  E {V (t1)W (t2 )}
 Autocorrelation Function
R VV (t1,t2 )  E [V (t1)V (t2 )]


 

 
v 1v 2fV V (v 1,v 2 ;t1,t2 )dv 1dv 2
1, 2
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 Time Average vs. Ensemble Average
ensemble average
V (t)  E [V (t)] 



v fV (v ;t)dv
time average
 V (t)  lim
T 
1
T
T
T
2

V (t)dt
2
 Power Spectral Density
 Definition.
 Theorem.
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Interpretation of spectral density functions
Figure 3.6-2
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 Real-Valued Wide-Sense Stationary Processes
 Def. A real-valued random process is called WSS if following two
properties are met.
Property 1.
Property 2.
따라서
E [V (t)]  m
E [V (t1)V (t2 )]  R VV (t1  t2 )
t1  t 2  
R VV ( )  E [V (t)V (t   )] E [V (t   )V (t)]
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 Power Spectral Density of Real-Valued WSS Random Process
(Wiener-Kinchine Theorem)
S VV (f )  F [R VV ( )] 



R VV ( )e  j2 f d 
R VV ( )  F 1[S VV (f )]
V
2
 PVV  R VV (0) 



S VV (f )df
Property 1.
S VV (f )  0
Property 2.
S VV (f )  S VV (f )
 When X(t) and h(t) are real,
X (t)
R XX
S XX (f )
h(t )
H
Y (t)
R YY
S YY (f )
R YY ( )  h ( ) h ( ) R XX ( )
2
S YY (f )  H (f ) S XX (f )
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 White Noise
S NN
(f )  N

0
S NN (f )
N0
2
2
f
따라서
R NN ( ) 

N0

2

e
j2 f
df 
N0
2
 ( )
R NN ( )
N
0
2
“uncorrelated”

 Noise : White & Gaussian

practically non-white

N0
온도의 함수
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
 Noise Equivalent Bandwidth
h(t )
BN ?

2 
1

H (f ) max
PYY 
BN
N0
2

2
H (f ) df  N 0  H (f ) df
o

o
2
H (f ) df
38