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![Page 1: 1 Chapter 3 Signal Transmission and Filtering Outline 3.1 Response of LTI System Coherent AM reception and LPF 3.2 Signal Distortion in Transmission.](https://reader036.fdocuments.us/reader036/viewer/2022062409/5697c0091a28abf838cc711c/html5/thumbnails/1.jpg)
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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 FilteringQuadrature 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 Shiftbeating
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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 TransmissionA channel is distortionless iff it is an LTI system with
impulse response
Frequency-flat channelOver the desired bandphaseFrequency-selective channel
DistortionsNonlinear distortionsLinear distortions
Amplitude distortionPhase distortion
dftjd KefHttKth 2)()()(
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Example: linear distortions
Test signal x(t) = cos 0t + 1/5 cos 50t
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 distortionIntersymbol interference (ISI) in digital signal transmission
Linear equalizationLinear zero-forcing equalization (LZF): channel inversionLinear minimum-mean square error equalization
(LMMSE)
Nonlinear equalizationMaximum-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 equalizerTransversal filter, tapped-delay-line equalizerFrequency-domain equalizer (FDE)
» One-tap equalizer for OFDM
Adaptive equalizer
Nonlinear distortion and compandingTransfer characteristic
Memoryless distortionDistortion with memory
Polynomial approximation of memoryless distortionSecond-harmonic distortionIntermodulation distortion
CompandingCompressing + expanding
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3.3 TRANSMISSION LOSS and DECIBELS Power gain
g = P_out/P_indecibels
g_dB = 10 log_10 g3 dB = 1/2G = 10^(g_dB/10)Serial interconnection of amplifiers and attenuators ->
addition, subtraction in dBIf g = 10^m, then g_dB = m*10 dB
dBm0 dBm = 1 mW10 dBm = 10 mW20 dBm = 100 mW = 0.1 W30 dBm = 1 W = 0 dBW
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Transmission loss and repeatersLoss L = 1/gPath lossPassive 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, attenuationAttenuation 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 cableTotal reflection, refraction index
Light propagation down a multimode step-index fiber
Figure 3.3-3b
Light propagation down a single-mode step-index fiber
Figure 3.3-3a
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Light propagation down a multimode graded-index fiber
Figure 3.3-3c
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Large bandwidth and low lossCarrier frequencies in the range of 200 THz
» Max bandwidth 20 THz0.2-2 dB/km loss
» Lower than most twisted-pair and coaxial cable systems
» Absorption » Scattering
Less interferenceNo RF interference
No noiseLow maintenance costSecure 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
Nonlinear memoryless distortion has input output relation given by
which increases bandwidth of the output because multiplication in TD corresponds to convolution in the FD.
Exam on next Tuesday @LG104, 11:00-12:15Ch. 1-3 Open book (but you will not have time to read on the site.)T/F, filling blanks, Essay, Math
dftjd KefHttKth 2)()()(
N
n
nn txaty
0
)()(
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Radio TransmissionLine-of-sight propagation
Free-space path loss (FSPL)» The loss between two isotropic radiators in free
space.Formula
» far-field» It is a function of frequency. However, it does not say
that free space is a frequency-selective channel.
2 2
dB 10 10 km
4 4
where path length, = wavelength, frequency, speed of light
92.4 20log 20logGHz
l flL
c
l fc
L f l
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Example 3.3-1
Satellite repeater system: uplink, downlink, frequency translation, geostationary, low orbit, OBP
Figure 3.3-5
ampTu Ru Td Rdout in
u d
g g g g gP P
L L
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3.4 FILTERS and FILTERING Ideal Filters
LPFBPF
Lower and upper cutoff frequenciesPassband and stopband
HPFNF
Transfer function of a ideal bandpass filter
Figure 3.4-1
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Realizability, noncausality
Bandlimiting and timelimitingIt is impossible to have perfect bandlimiting and timelimiting
at the same time.
Ideal lowpass filter (a) Transfer function (b) Impulse response
Figure 3.4-2
Ideal filters are noncausal.
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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
1 0( ) sgn ( )
0Q
j fH f j f h t
j f t
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Quadrature Filtering and Hilbert Transform
Hilbert tranform
1 1 ( )ˆ ( ) ( )
Fourier transform of Hilbert transform
ˆ ( ) ( sgn ) ( ) ( ) ( )Q
xx t x t d
t t
x t j f X f H 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
0
0 0
0 0
10
( ) cos( )
ˆ ( ) sgn ( ) ( ) ( ) sgn2
= ( ) ( )2
ˆˆ( ) ( ) sin( )
x t A t
jAX f j fX f f f f f f
Af f f f
j
x t X f A t
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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
ˆ1. ( ) and ( ) have same amplitude spectrum
2. Energy and power in a signal and its Hilbert tranform are equal
ˆ3. ( ) and ( ) are orthogonal
ˆ ( ) ( ) 0 (energy)
li
x t x t
x t x t
x t x t dt
1ˆm ( ) ( ) 0 (power)
2
T
TT
x t x t dtT
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3.6 CORRELATION AND SPECTRAL DENSITY Stochastic Process = signal with uncertainty described
probabilistically
Two ways to describe: 1) probability space and mapping to sample path ,2) Kolomgorov’ s extension theorem
Non-periodic signal
Non-energy signal
Ex)Bit Stream
Noise
Voice Signal
t
)(tv
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Ensemble Average
<At time t>
Correlation
Autocorrelation Function
dvtvfvtVEtV V
);()]([)(
)}()({),( 2121 tWtVEttRVW
21212121
2121
),;,(
)]()([),(
2,1dvdvttvvfvv
tVtVEttR
VV
VV
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Time Average vs. Ensemble Average
ensemble average
time average
Power Spectral Density Definition.
Theorem.
2
2
)(1
lim)(T
TTdttV
TtV
dvtvfvtVEtV V
);()]([)(
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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
mtVE )]([
)()]()([ 2121 ttRtVtVE VV
21 tt
)]()([)]()([)( tVtVEtVtVERVV
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Power Spectral Density of Real-Valued WSS Random Process (Wiener-Kinchine Theorem)
Property 1.
Property 2.
When X(t) and h(t) are real,
)]([)(
)()]([)(
1
2
fSFR
deRRFfS
VVVV
fjVVVVVV
dffSRPV VVVVVV )()0(2
0)( fSVV
)()( fSfS VVVV
)()()(
)()()()(2
fSfHfS
RhhR
XXYY
XXYY
YYXX RR)()( tYtX
)()( fSfS YYXX
H
th )(
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White Noise
따라서
Noise : White & Gaussian
practically non-white
온도의 함수
2)( 0NfSNN
)(22
)( 020 Ndfe
NR fj
NN
“uncorrelated”
20N
)(fSNN
)(NNR
20N
f
0N
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Noise Equivalent Bandwidth
dffHfH
B
dffHNdffHN
P
oN
oYY
2
max
2
0
20
)()(
1
)()(2
?
)(
NB
th