Communication Systems, 5ebazuinb/ECE4600/Ch03_1.pdf · • Method1: Apply an impulse and measure...
Transcript of Communication Systems, 5ebazuinb/ECE4600/Ch03_1.pdf · • Method1: Apply an impulse and measure...
Communication Systems, 5e
Chapter 3: Signal Transmission and Filtering
A. Bruce CarlsonPaul B. Crilly
© 2010 The McGraw-Hill Companies
Chapter 3: Signal Transmission and Filtering
• Response of LTI systems (ECE 3710)• Signal distortion• Transmission Loss and decibels• Filters and filtering• Quadrature filters and Hilbert transform• Correlation and spectral density
© 2010 The McGraw-Hill Companies
Linear Time Invariant (LTI) Systems
• Principle of Superposition– Summed input create transformed summed outputs
• Time-invariant– invariant result if shifted in time
• Homogeneity– Constant multiplication of input results in identical constant
multiplication of the output
• Transfer Functions – input to output relationship• Expected for “lumped-parameter” elements (R, L, C)
– At frequencies where component sizes and frequency wavelengths are in the same order, it doesn’t work anymore. (Distributed-parameter systems)
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Transfer functions and frequency response
• System transfer function– Impulse response– Form convolution to generate output
• For a sinusoidal input:The output is a magnitude and phase shifted sinusoid
© 2010 The McGraw-Hill Companies
dttfjththfH 2exp
tftx 02cos 002cos0
fHtfAty fH
00fHA fH 000 arg fHfHfH
Example: Time Response of a First Order System
• RC Filter with 1Hz and 10Hz cutoff frequency
y(t) v(t)
RC1sRC
1
sC1R
sC1
sHsYsV
0t,RCtexp
RC1th
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Step Responses: 1 Hz and 10 Hz RC• Matlab Code Fig3_1_4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Step Response
Time (sec)
Ampl
itude
1 Hz10 Hz
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10-1 100 101 102 103 104-120
-100
-80
-60
-40
-20
0Butterworth Filters
Frequency (normalized)
Atte
nuat
ion
(dB
)
1 Hz10 Hz
10-1 100 101 102 103 104-100
-80
-60
-40
-20
0
Butterworth Filters
Phase (degrees)
Atte
nuat
ion
(dB
)
1 Hz10 Hz
Impulse and Square Wave Responses: 1 Hz and 10 Hz RC
• Matlab Code Fig3_1_4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
10
20
30
40
50
60
70
Impulse Response
Time (sec)
Ampl
itude
1 Hz10 Hz
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.5
1
Linear Simulation Results
Time (sec)
Ampl
itude
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.5
1
Linear Simulation Results
Time (sec)
Ampl
itude
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Transfer Function Methods
• For continuous-time systems, the Laplace transform is used to define the filter transfer function
– Frequency Response
– or
8
11
11
011
1
ssssss
sXsYsH n
nn
n
nn
mm
1222
222
11
1
011
1
fjfjfjfjfjfj
fXfYfH n
nn
n
nn
mm
fXj
fYj
efXefY
fXfYfH arg
arg
Determining Transfer Functions
• Method1: Apply an impulse and measure the time response.– “Impulse generator” and a time samples
• Method 2: Apply known frequencies at known amplitudes and phases and measure the frequency response.– Sine wave generator and a dual trace oscilloscope– A one instrument equivalent is a Network Analyzer
coscos fffHfY
thdthty
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Block Diagram Methods
• Extensively used to describe how signals are connected in and flow through a system– A flow chart similar to a software flow diagram– Allows engineers to handle complexity and check that
things are connected right.
• Multiple levels of block diagrams– Show the system hierarchy– Every Block can be broken into another lower-level
block diagram until you get to a single transfer function, equation, decision or operations.
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Block Diagram Analysis
• Block diagrams consist of unidirectional, operational blocks that represent the transfer function of the variables of interest
sRGsRGsY
sRGsRGsY
2221212
2121111
Notes and figures are based on or taken from materials in the ECE 371 course textbook: R.C. Dorf and R.H. Bishop, Modern Control Systems, 9th ed., Prentice Hall, 2001. ISBN 0-13-030660-6.
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Block Diagram Transformations
Notes and figures are based on or taken from materials in the ECE 3710 course textbook: R.C. Dorf and R.H. Bishop, Modern Control Systems, 9th ed., Prentice Hall, 2001. ISBN 0-13-030660-6.
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Block diagram types
© 2010 The McGraw-Hill Companies
(a) Parallel
(b) Cascade
(c) Feedback
Note on Zero Order Hold BD• Typical ADCs use a “sample and hold” prior to
the ADC• Sampling is typically an integration of the signal
for a fixed sampling period• Hold is to insure the ADC has a stable signal for a
defined period of time
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Frequency Selective Filtering
• Filters are intended to remove noise and interference– Maximize the amount that is removed
• Filters will modified the spectrum of the signal of interest in some way– Minimize any signal distortion that may be caused by the filter.
• The above restrictions can usually not both be performed!– Compromise between the two!!
See Figure 3.1-7 on p. 101.
See Matlab Example 15
Example: Figure 3.1-73 Filters Power Spectrum- 10, 100 & 1000 Hz, 4 pole
Signal Power Spectrum- 100 Hz, 1 pole roll-off
3 OutputPower Spectrum 16
100 101 102 103 104-45
-40
-35
-30
-25
-20
-15
-10
-5
0Signal
Freq
Am
plitu
de (d
B)
100 101 102 103 104-45
-40
-35
-30
-25
-20
-15
-10
-5
0Filters
Freq
Am
plitu
de (d
B)
FH1FH2FH3
100 101 102 103 104-45
-40
-35
-30
-25
-20
-15
-10
-5
0Output
Freq
Am
plitu
de (d
B)
O1O2O3
LTI System Energy
dffXfHdffYE22
y
dffXdffHdffXfHE
dffXfXfHfHE
dffXfHfXfHE
dffXfHfXfHE
y
y
y
y
2222
**
**
*
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Product of the Signal and Filter Power Spectrum(See the previous example)
Spectral Modifications
• Matched Filter (ECE 3800)
• “Wiener Filter” (ECE 3800)
2XXYY wHwSwS
sXsHsY
utsKh
sSsSsFsF NNXXCC
PlaneHalfLeftC
XX
C sFsS
sFsH
1
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Signal Filtering in the Real World (1)
Notes and figures are based on or taken from materials in the course textbook: Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.19
Signal Filtering in the Real World (2)
Notes and figures are based on or taken from materials in the course textbook: Bernard Sklar, Digital Communications, Fundamentals and Applications,
Prentice Hall PTR, Second Edition, 2001.20
Filter Terminology• Passband
– Frequencies where signal is meant to pass
• Stopband– Frequencies where some defined
level of attenuation is desired
• Transition-band– The transitions frequencies
between the passband and the stopband
• Filter Shape Factor– The ratio of the stopband
bandwidth to the passband bandwidth
PB
SB
BWBWSF
PBBW
SBBW
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Signal Distortion
• The RF channel (and any filtering) may distort the signal between what you think you transmitted and what you receive.
1. Amplitude Distortion
2. Delay Distortion
3. Nonlinear Distortion• Not an LTI operation
KfH
mtf2fHfHarg d
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Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Test signal x(t) = cos 0t - 1/3 cos 30t + 1/5 cos 50t
Test Signal Example
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Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) low frequency attenuated by ½ (by a HPF)
(b) high frequency attenuated by ½ (by a LPF)
Test signal with filtered, linear amplitude distortion
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Square Wave Filtering
• MATLAB Fig3_2_4.m
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10-1 100 101 102 103 104-120
-100
-80
-60
-40
-20
0
Butterworth Filters
Frequency (normalized)
Atte
nuat
ion
(dB
)
1 kHz LPF10 Hz HPF
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-2
-1
0
1
21000 Hz Low Pass Filter
Time (sec)
Ampl
itude
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-2
-1
0
1
210 Hz High Pass Filter
Time (sec)
Ampl
itude
Phase Delay and Delay Distortion
• Constant time delay results in linear phase delay and is desirable for distortionless transmissions.– See the time delay property of the Fourier Transform
• A constant phase shift is not linear in phase and results in significant signal distortion.– See the following constant phase delay of 90 deg.– Use an “equalizer” to fix the phase problems
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Test signal with constant phase distortion
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Test signal x(t) = cos(0t -90)- 1/3 cos(30t -90)+ 1/5 cos(50t-90)
shift all inputs by = -90
Not a square wave anymore
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Equalization to over come linear distortion
© 2010 The McGraw-Hill Companies
Consider a channel with linear distortion
2To overcome linear distortion equalization
( ) ( ) 0( )
overall system function is ( ) ( ) ( )
dj ft
eqC
eq C
KeH f H fH f
H f H f H f
Time Delay Distortion
• The effective time delay due to phases
• The equalized time delay can be approximated
• For distortionless transmission,
ftfffH dCC 2arg
gCEQCg tfjfHfHfHtth 2exp
gCEQC tfffHfH 2arg
gdEQC tftffHfH 2arg
gd tft 0 0arg fHfH EQC
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Phase or Delay Distortion
• If the phase shift is not linear, the phase delay time is
– For a pure time delay, the phase change is linear and the phase delay is a constant.
– Note: constant time delay is desirable, while constant phase delay is not
ffHftd
2
arg
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Envelope or Group Delay
• The group delay is a constant for all frequencies, it is approximated as
• This is commonly envisioned to apply to the envelope of the signal, while the phase delay is applied to the carrier.
df
fdtg
21
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Time Delay – Signal Distortion
• For a test signal
• The filtered output would become
tf2sintxtf2costxtx c2c1
cdcg2
cdcg1
fttf2sinttxA
fttf2costtxAty
Expect the carrier delay to be greater than the signal envelope delay!
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Equalization
• Apply a filter that corrects for amplitude and delay distortion in the receiver.
• Compensate so that the difference between the phase and group delay becomes zero
021
2
ftdf
fdf
ffttft compcompgd
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Logarithms and Decibels• RF signal power and voltage magnitudes are typically
described in:– Decibel-Watts (dBW)– Decibel-milliwatts (dBm)– Decibel-voltage (dBv)
• The Ratio of RF power or voltage is typically describe in:– Decibels (dB)
• The relative antenna power (transmitting or receiving) for a particular direction as compared to an “isotropic antenna” is describe in:– Decibles-isotropic (dBi)
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• The “bel” is defined as the log base 10 of power ratios
• The magnitude was too small, therefore the Decibel was defined
Alexander Graham Bell
in
out
PPbels log
in
out
PPdBdecibels log10
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Derived Power/Voltage Measures• Decibels for Power, Voltage, and Current
– Power
– Voltage
– For matched impedances where Rout=Rin
in
out
PPdBdecibels log10
in
out
RV
RV
dBdecibels
RVP
in
out
2
2
2
log10
in
out
VVdBdecibels log20
36
Derived Power/Current Measures• Decibels for Power, Voltage, and Current
– Power
– Current
– For matched impedances where Rout=Rin
in
out
PPdBdecibels log10
in
out
RIRI
dBdecibels
RIP
in
out
2
2
2
log10
in
out
IIdBdecibels log20
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Power Measurement• Watts (P in Watts)
• Milliwatts (P in Watts)
• Milliwatts (Pm in milliwatts)
PdBWwattsdecibels log10
310)(log10 mWPdBmmilliwattsdecibels
mWPdBmmilliwattsdecibels log10
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Common Decibel Values
Power Ratio (in
outP
P ) Decibels (dB)
100 20 dB
10 10 dB
5 6.99 dB
4 6.02 dB
2 3.01 dB
1 0 dB
0.5 -3.01 dB
0.25 -6.02 dB
0.1 -10 dB
0.01 -20 dB
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Common Decibel Power ValuesPower (Watts) Decibels-Watts (dBW) Decibels-MilliWatts (dBm)
1 megaWatt 60 dBW 90 dBm
1 kilowatt 30 dBW 60 dBm
10 watts 10 dBW 40 dBm
1 watt 0 dBW 30 dBm
0.1 watt -10 dBW 20 dBm
ohmsvolt 501 2 -16.99 dBW +13 dBm
1 milliwatt -30 dBW 0 dBm
1 microwatt -60 dBW -30 dBm
1 nanowatt -90 dBW -60 dBm
1 picowatt -120 dBW -90 dBm
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Suggested Matlab Functions
valuevaluedB log10)(
valuevaluedBv log20)(
1010)(value
valueidB
2010)(value
valueidBv
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Logarithm/Decibel Reminders baba logloglog
baba logloglog
bdBadBbadB
bdBadBbadB
cbacb
a logloglog1log
cdBbdBadBcb
adB
1
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