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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

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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

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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

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fjVVVVVV

dffSRPV VVVVVV )()0(2

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)()( 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