Communication Systems, 5e
Chapter 3: Signal Transmission and
Filtering
A. Bruce Carlson
Paul B. Crilly
© 2010 The McGraw-Hill Companies
Chapter 3: Signal Transmission and
Filtering
•
•
•
•
•
•
Response of LTI systems
Signal distortion
Transmission Loss and decibels
Filters and filtering
Quadrature filters and Hilbert transform
Correlation and spectral density
© 2010 The McGraw-Hill Companies
Filter types
• Low pass:
• High pass:
• Band pass:
rejects high frequencies
rejects low frequencies
rejects frequencies above and
below some limits
• Notch:
rejects one frequency
• Band reject: rejects frequencies between two limits
© 2010 The McGraw-Hill Companies
Filter Design Notes
•
Butterworth Filter Definition
– Poles on the unit circle
– Frequency Scaling
•
Active Audio Filter Implementations
– One Pole Op Amp design
– Sallen-Key LPF Active Filter
• 2-pole filter implementation per Op Amp (non-inverting)
– Multiple Feedback (MFB) Circuit Lowpass Filter
• Alternate 2-pole design (inverting)
– Cascading stages for higher order filters
• Texas Instruments, Active Low-Pass Filter Design, Application Report,
SLOA049B
• Passive LC filter
– T and Pi Filters
– Buy it from Coilcraft
4
Butterworth Low Pass Filter
H jw   H jw  
1
1   w 
 w0 
2n
Butterworth Filter Family
0
-20
2


1

1   s

 j w 0 
1
2n
1   j   s 
 w0 
1
1   1  s 
 w0 
n
•
2n
Attenuation (dB)
Hs  
2n
2n
-40
-60
-80
-100
-120
-1
10
1st order
2nd order
3rd order
4th order
5th order
10
0
1
10
Frequency (normalized)
10
2
10
Maximally Flat, Smooth Roll-off, Constant 3dB point for all orders
M.E. Van Valkenburg, Analog Filter Design, Oxford Univ. Press, 1982. SBN: 0-19-510734-9
5
3
Butterworth Filter PSD
Butterworth Filter Family
0
Attenuation (dB)
-20
-40
-60
-80
-100
-120
-1
10
1st order
2nd order
3rd order
4th order
5th order
10
0
1
10
Frequency (normalized)
10
2
10
3
6
Butterworth Filter PSD (2)
Butterworth Filter Family
1
0
-1
Attenuation (dB)
-2
-3
-4
-5
-6
-7
-8
-9
-1
10
1st order
2nd order
3rd order
4th order
5th order
10
Frequency (normalized)
0
7
Matlab Script: ButterPlot.m
%
% Butterworth filter plots
%
freqrange = logspace(-1,3,1024)';
wrange=2*pi*freqrange;
[B1,A1]=butter(1,2*pi,'s');
[H1] = freqs(B1,A1,wrange);
[B2,A2]=butter(2,2*pi,'s');
[H2] = freqs(B2,A2,wrange);
[B3,A3]=butter(3,2*pi,'s');
[H3] = freqs(B3,A3,wrange);
[B4,A4]=butter(4,2*pi,'s');
[H4] = freqs(B4,A4,wrange);
figure(1)
semilogx(freqrange,dB(psdg(Hmatrix)));
grid
title('Butterworth Filter Family');
xlabel('Frequency (normalized)');
ylabel('Attenuation (dB)');
legend('1st order','2nd order','3rd order','4th
order','5th order','Location','SouthWest');
axis([10^-1 10^3 -120 3]);
figure(2)
semilogx(freqrange,dB(psdg(Hmatrix)));
grid
title('Butterworth Filter Family');
xlabel('Frequency (normalized)');
ylabel('Attenuation (dB)');
legend('1st order','2nd order','3rd order','4th
order','5th order','Location','SouthWest');
axis([10^-1 3 -9 1]);
[B5,A5]=butter(5,2*pi,'s');
[H5] = freqs(B5,A5,wrange);
Hmatrix=[H1 H2 H3 H4 H5];
8
Chebyshev Type I
Filter PSD (Cheby1Plot.m)
Chebyshev Type I Filter Family
0
Attenuation (dB)
-20
-40
-60
-80
-100
-120
-1
10
1st order
2nd order
3rd order
4th order
5th order
10
0
1
10
Frequency (normalized)
10
2
10
3
9
Chebyshev Type I
Filter PSD (2)
Chebyshev Type I Filter Family
1
0
-1
Attenuation (dB)
-2
-3
-4
-5
-6
-7
-8
-9
-1
10
1st order
2nd order
3rd order
4th order
5th order
10
Frequency (normalized)
0
10
Available MATLAB Filters
(Signal Proc. TB)
• Analog or Digital
–
–
–
–
–
Butterworth
Chebyshev Type I
Chebyshev Type II
Elliptic or Cauer
Bessel
• Digital
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
barthannwin
bartlett
blackman
blackmanharris
bohmanwin
chebwin
flattopwin
gausswin
hamming
hann
kaiser
nuttallwin
parzenwin
rectwin
triang
tukeywin
11
Analog Lowpass Filter Design
Filter Comparison: Magnitude
20
• Butterworth
– Monotonic Decreasing
Magnitude
– All poles
• Chebyshev (Cheby Type 1)
– Passband Ripple
– All poles
• Inverse Chebyshev
(Cheby Type2)
– Stopband Ripple
• Elliptical or Cauer Filter
– Passband Ripple
– Stopband Ripple
• Bessel Filter
– Linear Phase Maximized
Butter
Bessel
Cheby1
Cheby2
Ellip
Spec
0
-20
-40
-60
-80
-100
-120
-140
-160
1
10
10
2
10
3
Butterworth Order Predication
Filter Order = 4
3dB BW = 1778.28 Hz
Bessel Order Predication
Filter Order = 4
3dB BW = 1778.28 Hz
Chebyshev Type I Order
Predication
Filter Order = 3
3dB BW = 1000 Hz
10
4
10
5
10
6
10
Chebyshev Type II Order
Predication
Filter Order = 3
3dB BW = 8972.85 Hz
Elliptical or Cauer Order
Predication
Filter Order = 3
3dB BW = 1000 Hz
12
7
Matlab Filter Generation (1)
•
•
•
•
Passband
Stopband
Passband Ripple (dB)
Stopband Ripple (dB)
•
•
•
•
•
fpass=4000;
fstop=16000;
AlphaPass=0.5;
AlphaStop=60;
w#### = 2 x pi x f####
Filter Order and other design parameters
[Nbutter, Wnbutter] = buttord(wpass, wstop, AlphaPass, AlphaStop,'s');
[Ncheby1, Wncheby1] = cheb1ord(wpass, wstop, AlphaPass, AlphaStop,'s');
[Ncheby2, Wncheby2] = cheb2ord(wpass, wstop, AlphaPass, AlphaStop,'s');
[Nellip, Wnellip] = ellipord(wpass, wstop, AlphaPass, AlphaStop,'s');
13
Matlab Filter Generation (2)
Filter Transfer Function Generation
[numbutter,denbutter] = butter(Nbutter,Wnbutter,'low','s')
[numbesself,denbesself] = besself(Nbutter,Wnbutter)
[numcheby1,dencheby1] = cheby1(Ncheby1,AlphaPass, Wncheby1,'low','s')
[numcheby2,dencheby2] = cheby2(Ncheby2,AlphaStop, Wncheby2,'low','s')
[numellip,denellip] = ellip(Nellip,AlphaPass,AlphaStop, Wnellip,'low','s');
Spectral Response from Transfer Function
[Specbutter]=freqs(numbutter,denbutter,wspace);
[Specbesself]=freqs(numbesself,denbesself,wspace);
[Speccheby1]=freqs(numcheby1,dencheby1,wspace);
[Speccheby2]=freqs(numcheby2,dencheby2,wspace);
[Specellip]=freqs(numellip,denellip,wspace);
14
Matlab Filter Generation (3)
figure(11)
semilogx((fspace),dB(psdg([Specbutter Specbesself Speccheby1 Speccheby2 Specellip])), ...
specfreq1,specmag1,'k-.',specfreq2,specmag2,'k-.',specfreq3,specmag3,'k-.');
title('Filter Comparison: Magnitude')
legend('Butter','Bessel','Cheby1','Cheby2','Ellip','Spec')
Filter Comparison: Magnitude
20
Butter
Bessel
Cheby1
Cheby2
Ellip
Spec
0
-20
-40
-60
-80
-100
-120
-140
-160
1
10
2
10
3
10
4
10
5
10
6
10
7
10
15
Using the results
Run using AnalogFilterCompare
Butterworth Filter Bode Plot
50
Magnitude (dB)
%%
% Analog Filter Comparisons
%
clear
close all
clc
0
-50
-100
-150
fprintf('\nFilter Comparison\n');
fftsize=4096;
fpass= 4000;
fstop= 16000;
AlphaPass=0.5;
AlphaStop=60;
Phase (deg)
-200
0
-180
-360
-540
10 3
10 4
10 5
10 6
Frequency (rad/s)
Butterworth Order Predication
Filter Order = 6
3dB BW = 5059.64 Hz
hbutter =
1.0323e+27
-------------------------------------------------------------------------------------(s^2 + 6.141e04s + 1.011e09) (s^2 + 4.496e04s + 1.011e09) (s^2 + 1.646e04s + 1.011e09)
16
Using the Results
Butterworth Order Predication
Filter Order = 6
3dB BW = 5059.64 Hz
Bessel Order Predication
Filter Order = 6
3dB BW = 5059.64 Hz
Chebyshev Type I Order Predication
Filter Order = 5
3dB BW = 4000 Hz
Chebyshev Type II Order Predication
Filter Order = 5
3dB BW = 11641.8 Hz
Elliptical or Cauer Order Predication
Filter Order = 4
3dB BW = 4000 Hz
Build a 6th Order Sallen-Key
Butterworth Low Pass Filter
• 3 dB cutoff 5.05964 kHz
[numbutter,denbutter] = butter(Nbutter,Wnbutter,'low','s')
roots(denbutter) % Get the roots of the denomenator
ans =
1.0e+04 *
-0.8228 + 3.0707i
-0.8228 - 3.0707i
-2.2479 + 2.2479i
-2.2479 - 2.2479i
-3.0707 + 0.8228i
-3.0707 - 0.8228i
17
Determine Poles
• 6 poles: three complex pole pairs
– Determine the roots on the unit circle.
– Multiply by the cutoff frequency
– Determine 2nd order coefficients w and 
LPFPoles =
1.0e+004 *
-0.8228 + 3.0707i
-0.8228 - 3.0707i
-2.2479 + 2.2479i
-2.2479 - 2.2479i
-3.0707 + 0.8228i
-3.0707 - 0.8228i
Carbitrary
Vout s 
K  w2

 2
Vins  s  a  j  b   s  a  j  b 
s  2    w  s  w2
2   w  2  a
w2  a 2  b 2
18
The Pole Pairs
LPFPoles =
K1  w 2
 2
H 1 s   2
s + 6.141e04  s + 1.011e09
s  2   1  w  s  w2
Carbitrary
1.0e+004 *
-0.8228 + 3.0707i
-0.8228 - 3.0707i
-2.2479 + 2.2479i
-2.2479 - 2.2479i
-3.0707 + 0.8228i
-3.0707 - 0.8228i
K 2  w2
 2
H 2 s   2
s + 4.496e04  s + 1.011e09
s  2   2  w  s  w2
Carbitrary
K 3  w2
H 3 s   2
 2
s + 1.646e04  s + 1.011e09
s  2   3  w  s  w2
Carbitrary
Solve for the damping factors … ζ1, ζ2, and ζ3
The value of w2 is given.
19
Determine Rs and Cs
C2
Let R=10 kohm
Find C and R4
+Vdc
R1
R2
V1
+
-
C1
C1  C 2  C
Vout
MaxGain  K 
-Vdc
w
R1  R2  R3  R
R3  R 4
K
R3
OPAmp
R4
R3
 
R3  R 4
R3
1
CR
3 K
2
1
 R3  R 4 



Vout s 
K  w2
R3  C1  C 2  R1  R 2

 2

R4
V 1s 
s  2    w  s  w2
 1

1
1
R3  
s 2  s  


 C 2  R1 C 2  R 2 C1  R 2  C1  C 2  R1  R 2


20
Collect Results
Sallen-Key LPF Design Summary
Filter Gain = 4.20476, Filter Gain (dB) = 12.4748
Stage R
C
R3
R4
Gain
1 10.0 kohm 3145.6 pF 10.0 kohm 0.6815 kohm 1.07
2 10.0 kohm 3145.6 pF 10.0 kohm 5.8579 kohm 1.59
3 10.0 kohm 3145.6 pF 10.0 kohm 14.8236 kohm 2.48
21
The Easier Way
• see BW_SKLPF_Gen.m
%%
% BW Filter generation demonstration
%
close all
clear all
clc
BW Goal
Final
0
Magnitude (dB)
BWn=6;
% Butterworth Filter Order
PB_Hz=5059.64 ;
% 3dB BW Derived from AnalogLPFCompare
PBfreq=2*pi*PB_Hz;
…
After execution you get:
Sallen-Key LPF Design Summary
Butterworth SK LPF with Fc = 5.05964 kHz and Gain = 12.47 dB
50
-50
-100
Filter Gain = 4.20476, Filter Gain (dB) = 12.4748
-150
Stage R
C
R3
R4
Gain
1 10.0 kohm 3145.6 pF 10.0 kohm 0.6815 kohm 1.07
2 10.0 kohm 3145.6 pF 10.0 kohm 5.8579 kohm 1.59
3 10.0 kohm 3145.6 pF 10.0 kohm 14.8236 kohm 2.48
-200
10 0
10 1
10 2
10 3
10 4
10 5
Freq (Hz)
22
10 6
Check the Spice Design
23
The Spectral Plot
24
Taking it to the next level
What resistors are actually available?
What Capacitors are actually available?
• Standard “E-Series” Values for resistors and capacitors:
– selected to be evenly spaced logarithmically across one decade
– typically related to value tolerance (+/- % accuracy) where a value
is available within the tolerance
– Example” E3 has three values available (10, 22, 47)
• Normally think in 10%, 5%, or 1% tolerance
– E12 – for 10%, E24 for 5%, and E96for 1%
http://www.vishay.com/docs/28372/e-series.pdf
25
Matlab Code
• AnalogFilterCompare.m
• BW_SKLPG_Gen.m
– subroutines
• Additional Resources
– Dr. Bazuin’s Filter Notes – on web site
– Dr. Bazuin’s Draft Filter Manual – for ECE 4810 folks
(or pre 4810) see me
26
Pulse Response and Risetime
• Low Pass Filters cause sharp signal edges to be
smoothed.
• The amount of smoothing is based on the
bandwidth of the filter
– More smoothing  smaller bandwidth
• Fourier relationship:
– a narrow rect function in time results in a broad (wide
bandwidth) sinc function in frequency
– a wide rect function in time results in a narrow (small
bandwidth) sinc function in frequency
27
Filter Step Response
• 1 Hz and 10 Hz 4th order Butterworth LPF Filters
• The step response can be used to help define the
bandwidth required for pulse signals.
Butterworth Filters
Step Response
0
1.4
1.2
1
-40
Amplitude
Attenuation (dB)
-20
-60
0.8
0.6
-80
0.4
-100
0.2
1 Hz
10 Hz
-120
-1
10
10
0
1 Hz
10 Hz
1
2
10
10
Frequency (normalized)
10
3
10
4
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Time (sec)
28
1.8
Filter Bandwidth for Pulses
• Pulse of length T
t
rect   T  sin cf  T 
T
2
1.5
1
• Null-to-null BW of
2
null  to  null 
T
• Single Sided BW desired
B
0.5
0
-3
-2
-1
0
1
2
3
1
T
• B/2 may be acceptable in some cases
– See textbook discussion
29
Pulse Filtering
Butterworth Filters
20
2.5
5.0
10.
20.
0
-20
Hz
Hz
Hz
Hz
•
•
Attenuation (dB)
-40
Four one-sided BW filters
0.1 sec pulse responses
-60
-80
Butterworth Filters
-100
1.2
Test Signal
2.5 Hz
5.0 Hz
10. Hz
20. Hz
-120
1
-140
0
10
20
30
40
50
60
70
Frequency (fs = 100 Hz)
80
90
Amplitude (dB)
-160
0.8
100
0.6
0.4
0.2
PulseTest1.m
(digital filters)
0
-0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (fs=100Hz)
0.7
0.8
0.9
1
30
Text Comparison Chart
(2.5, 5.0 and 20 Hz Plots)
Pulse response of an ideal LPF
Figure 3.4-10
See PulseTest2.m or PulseTest3.m
(digital filters)
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
31
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Pulse resolution of an ideal LPF.
B = 1/2
See Fig03_4_11.m
(Butterworth filters)
The inherent time delay has been removed from the output
32
Pulse Resolution: Matlab
• Using a 1st order and 6th order Butterworth Filter
– The text uses an “ideal filter”
– Filters have group and phase delay!
Linear Simulation Results
Amplitude
1
0.5
0
0
0.5
1
1.5
2
2.5
3
2
2.5
3
Time (sec)
Linear Simulation Results
Amplitude
1
0.5
0
0
0.5
1
1.5
Time (sec)
33
The Hilbert Transform
• It is a useful mathematical tool to describe the
complex envelope of a real-valued carrier
modulated signal.
– Make a real signal complex complex Positive frequency
• The precise definition is as follows:
1
1 x  
 d
 
x̂ t   x t  
  t   t  

h Q t  
1
 t
xHilbert t   xt   j  xˆ t 
 j,

H Q f    j  sgn f   0,
j

f 0
f 0
0f
H Q f   H Q f   H Q f   1
2
http://en.wikipedia.org/wiki/Hilbert_transform
*
34
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Hilbert transform of a rectangular pulse
(a) Convolution; (b) Result in time domain
y t   xt   j  xˆ t 
35
Hilbert Transform of Cos
x t   A  cos2  f 0  t 
A
 f  f 0   f  f 0  H Q f 
2
A
 j    f  f 0   f  f 0 
2
X̂f  
x̂ t   A  sin 2  f 0  t 
• This is useful in generating a complex signal from
a real input signal as follows …
36
Real to Complex Conversion
 t 
x t 
x t 
yt   x t   j  x̂ t 
1
t
x̂ t 
37
Real to Complex Mixing
• Analog Devices AD8347: 0.8 GHz to 2.7 GHz
Direct Conversion Quadrature Demodulator
– RF input and LO input
– Quadrature Output
38
Hilbert Transform Real to
Complex Conversion
• Original Real
xt   X  f 
• Hilbert Transform Complex
htct   xt   j  xˆ t   X  f   j  j  sgn  f   X  f 
htct   xt   j  xˆ t   X  f   sgn  f   X  f 
2  X  f ,
htct   xt   j  xˆ t   
0,
for f  0
for f  0
The Hilbert Transform can be used to create a single sided
spectrum! The complex representation of a real signal.
39
Quadratic Filters
• We may want to process real signals using
complex filtering or translated into the complex
domain.
• Quadrature Signal Processing involves creating an
“In-Phase” and “Quadrature-Phase” signal
representation.
– Usually this is done by “quadrature mixing” which
creates two outputs from a real data stream by mixing
one by a cosine wave and the over by a sine wave.
x t   exp j  2  f  t   x t   cos2  f  t   j  sin 2  f  t 
 in  phase  j  quadrature  phase
40
Correlation and Spectral Density
• Using Probability and the 1st and 2nd moments
– Assuming an ergodic, WSS process we use the time
average
2

Pv  vt   vt   vt   0
• Properties:
zt   zt 
*
*
zt  t 0   zt 
a 1  z1 t   a 2  z 2 t   a 1  z1 t   a 2  z 2 t 
• Schwarz’s Inequality
Pv  Pw  vt   w t 

2
41
Autocorrelation and Power
• Autocorrelation Function
R vv   vt   vt    vt    vt 


• Properties
R vv 0   Pv
R vv 0   R vv 
R vv    R vv 

42
Crosscorrelation
• Crosscorrelation Function
R vw   vt   w t    vt    w t 


• Properties
R vv 0   R ww 0   R vw  
2
R vw   R wv 

43
Application
• Correlation of phasors
T
1 2
exp j  w 1  t   exp j  w 2  t   lim   exp j  w 1  w 2   t   dt
T  T
T
2
sin cw 1  w 2   T 
T 
2
exp j  w 1  t   exp j  w 2  t   lim
1,
exp j  w 1  t   exp j  w 2  t   
0,
w1  w 2
else
44
Power Spectral Density
• The Fourier Transform of the Autocorrelation
• Remember ECE 3800!
45
System analysis in τ domain
x (t )
R x ()
y  
R y  
h (t )

y (t )  x(t )  h(t ) 
 h() x(t  )d 


Ryx ()  h()  Rx () 
 h( ) R (    ) d 
x

and with    output autocorrelation is:

Ry ()  h ( )  Ryx () 
*
 h () R
*
yx
(   ) d 

 Ry ()  h* ()  h( )  Rx ( )
© 2010 The McGraw-Hill Companies
System Analysis in f domain
Gx  f  x ( t )
R x ()
y  
Gy  f 
R y  
h (t ) H  f 

Gx  f   Rx  

G y  f    R y  
Y  f   H  f  X  f 
G y  f   H  f   Gx  f 
2
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Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Interpretation of spectral density
functions
Power or Energy Spectral
Density is energy per unit
frequency
48