Communication Systems, 5e - Homepages at WMUbazuinb/ECE4600/Ch03_3.pdf · Communication Systems, 5e...
Transcript of Communication Systems, 5e - Homepages at WMUbazuinb/ECE4600/Ch03_3.pdf · Communication Systems, 5e...
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• Signal distortion• Transmission Loss and decibels• Filters and filtering• Quadrature filters and Hilbert transform• Correlation and spectral density
© 2010 The McGraw-Hill Companies
3
Free-Space Loss
• As an RF signal propagates, there is path loss.
tPtG rG
f
RrP
22
cRf4R4L
• As shown above
22
2
2
44 fRcGGP
RGGP
LGGPP rt
trt
trt
tr
fc
Note
4
1st Order RF Range Estimate
• Friis Transmission Formula– Direct, line-of-sight range-power equation– No real-world effects taken into account
where: rP is the received (or transmitted)
tG is the effective transmitter (or receiver) antenna gain R is the distance between the transmitter and receiver, and is the wavelength f is the frequency
22
rtt2
2rt
tr Rf4cGGP
R4GGPP
RfdBcdBdBGdBGdBPdBP rttr
2
42
5
System Range
• Maximum Range (Pr is the receiver sensitivity)
dBmPt
dBmGt
dBmGr
dBmPr
m0 mR1
tPtG rG
f
RrP
rtr
t GGPP
f4cR
dBPdBGRfdBcdBdBGdBP rrtt
2
42
6
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
6 GHz4 GHz
dB13.1997e8.9048e3
6e369e64c
Rf4L 222
u
dB60.1957e18.6038e3
6e369e44c
Rf4L 222
d
Satellite relay system Ex. 3.3-1 (1 of 2)
Path Losses
7
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
dBW1.89dB20dB1.199dB55dBW35P rcv_sat
dBWdBdBdBdBWPout 6.110516.1951618
22
rtt
rttr R4
GGPL
GGPP
dB1.107dBW1.89dBW18gamp
Satellite relay system (2 of 2)
Pt=35 dBW
Pt=18 dBW
Error in 4th ed.Power
Received
Satellite Gain
RF Interference/Jamming
• What happens if interference is stronger than the signal of interest?
• Jamming …
– Cellular telephone: TX 824-849 and RX 869-894 MHz– Pico-cell transmitter: power +10 dBm, Gt=+3 dB, Rt=100 ft.– Jammer: 1 mW→0 dBm, Gj=0 dB
8
22
2
2
44 t
rtt
j
rjj Rf
cGGPRfcGG
P
jtt
jjt R
GPGP
R
3.22210110100 2
3 Jam cell phone if less than 22.3 ft. away ….
Example Commercial JammerManufacturer Specifications• Affected Frequency Ranges:
– CDMA/GSM: 850 to 960MHz– DCS/PCS:1805 to 1990MHz– 3G:2,110 to 2,170MHz– 4G LTE:725-770MHz– 4G Wimax:2345-2400MHz or 2620-2690MHz – WiFi:2400-2500MHz
• Total output power:3W• Jamming range: up to 20m, the jamming radius still
depends on the strength signal in given area• Power supply:50 to 60Hz, 100 to 240V AC• With AC adapter (AC100-240V-
DC12V),4000mA/H battery• Dimension:126 x 76 x 35mm not including antenna
(roughly 5” x 3” x 1.5”)• Full set weight:0.65kg
9
These are not legal in the USFCC Regulations.
~$256 from China
Filters and filtering
• Ideal filters• Bandlimiting and timelimiting• Real filters• Pulse response and risetime
© 2010 The McGraw-Hill Companies
11
The Ideal Filter
• To receive a signal without distortion, only changes in the magnitude and/or a time delay are allowed. 0ttxKty
02exp tffXKfY
• The transfer function is 0tf2expKfH
• A constant gain with a linear phase KfH 0tf2f
12
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(b) Bandpass Filter
Ideal filters
(a) Lowpass Filter
13
Ideal LPF Filter
• For no distortion, the ideal filter should have the following properties:
fjexpfHfH
u
u
fffor,0
fffor,KfH
u
u0
fffor,arbitrary
fffor,tf2f
• The impulse response for an ideal LPF is
u
u
u
u
f
f0
f
f0
dfttf2jexpKth
dftf2jexptf2jexpKth
14
Ideal Filter (2)
0uu
0
0u
0
0u
0
0u
f
f0
0
f
f0
ttf2sincKf2thtt2
ttf2sinK2th
tt2jttf2jexpK
tt2jttf2jexpKth
tt2jttf2jexpKth
dfttf2jexpKth
u
u
u
u
• Continuing
• The sinc function– A non-causal filter
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.
15
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) Transfer function (b) Impulse response
For a causal approximation, eliminate negative time from h(t).
t
dttB2sincKB2th
Ideal lowpass filter
16
Band-limiting and Time-limiting
• Band-limiting and Time-limiting are mutually exclusive!!– Easy to show with rect <==>sinc transform pair
• The engineering solution– Negligibly small can be ignored– Values less than a defined value are ignored– The non-ideal design is used and,
if it isn’t good enough, a smaller threshold to ignore value is set (repeating until the desired result achieved)
Filter types
• Low pass: rejects high frequencies• High pass: rejects low frequencies• Band pass: rejects frequencies above and
below some limits• Notch: rejects one frequency• Band reject: rejects frequencies between two limits
© 2010 The McGraw-Hill Companies
18
Real Filters: 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
19
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Typical amplitude ratio of a real bandpass filterFigure 3.4-3
Real Bandpass Filter
The -3 dB or half-power bandwidth is shown
20
Bandwidths that are Used
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.
Note: Sinc freq. domain is appropriate for digital symbols
21
Bandwidth Definitions
(a) Half-power bandwidth. This is the interval between frequencies at which Gx(f ) has dropped to half-power, or 3 dB below the peak value.
(b) Equivalent rectangular or noise equivalent bandwidth. The noise equivalent bandwidth was originally conceived to permit rapid computation of output noise power from an amplifier with a wideband noise input; the concept can similarly be applied to a signal bandwidth. The noise equivalent bandwidth WN of a signal is defined by the relationship WN = Px/Gx(fc), where Px is the total signal power over all frequencies and Gx(fc) is the value of Gx(f ) at the band center (assumed to be the maximum value over all frequencies).
(c) Null-to-null bandwidth. The most popular measure of bandwidth for digital communications is the width of the main spectral lobe, where most of the signal power is contained. This criterion lacks complete generality since some modulation formats lack well-defined lobes.
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.
22
Bandwidth Definitions (2)
(d) Fractional power containment bandwidth. This bandwidth criterion has been adopted by the Federal Communications Commission (FCC Rules and Regulations Section 2.202) and states that the occupied bandwidth is the band that leaves exactly 0.5% of the signal power above the upper band limit and exactly 0.5% of the signal power below the lower band limit. Thus 99% of the signal power is inside the occupied band.
(e) Bounded power spectral density. A popular method of specifying bandwidth is to state that everywhere outside the specified band, Gx(f ) must have fallen at least to a certain stated level below that found at the band center. Typical attenuation levels might be 35 or 50 dB.
(f) Absolute bandwidth. This is the interval between frequencies, outside of which the spectrum is zero. This is a useful abstraction. However, for all realizable waveforms, the absolute bandwidth is infinite.
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.
23
Selecting RF/IF Filter Types Based on Shape Factors
Vectron International, General technical information, http://www.vectron.com/products/saw/pdf_mqf/TECHINFO.pdf
Ban
dwid
th (k
Hz)
Center Freq. (MHz)
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
24
25
Butterworth Low Pass Filter
• Maximally Flat, Smooth Roll-off, Constant 3dB point for all orders
n2
0ww1
1jwHjwH
n2
0
n
n2
0
n2
n2
0
2
ws11
1w
sj1
1wj
s1
1sH
M.E. Van Valkenburg, Analog Filter Design, Oxford Univ. Press, 1982. SBN: 0-19-510734-9
10-1 100 101 102 103-120
-100
-80
-60
-40
-20
0
Butterworth Filter Family
Frequency (normalized)
Atte
nuat
ion
(dB
)
1st order2nd order3rd order4th order5th order
26
Butterworth Filter PSD
10-1 100 101 102 103-120
-100
-80
-60
-40
-20
0
Butterworth Filter Family
Frequency (normalized)
Atte
nuat
ion
(dB
)
1st order2nd order3rd order4th order5th order
27
Butterworth Filter PSD (2)
10-1 100-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1Butterworth Filter Family
Frequency (normalized)
Atte
nuat
ion
(dB
)
1st order2nd order3rd order4th order5th order
28
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);
[B5,A5]=butter(5,2*pi,'s');[H5] = freqs(B5,A5,wrange);
Hmatrix=[H1 H2 H3 H4 H5];
figure(1)semilogx(freqrange,dB(psdg(Hmatrix)));gridtitle('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)));gridtitle('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]);
29
Chebyshev Type IFilter PSD (Cheby1Plot.m)
10-1 100 101 102 103-120
-100
-80
-60
-40
-20
0
Chebyshev Type I Filter Family
Frequency (normalized)
Atte
nuat
ion
(dB
)
1st order2nd order3rd order4th order5th order
30
Chebyshev Type IFilter PSD (2)
10-1 100-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1Chebyshev Type I Filter Family
Frequency (normalized)
Atte
nuat
ion
(dB
)
1st order2nd order3rd order4th order5th order
31
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
32
Analog Lowpass Filter Design • 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
101 102 103 104 105 106 107-160
-140
-120
-100
-80
-60
-40
-20
0
20Filter Comparison: Magnitude
ButterBesselCheby1Cheby2EllipSpec
Butterworth Order PredicationFilter Order = 4 3dB BW = 1778.28 Hz
Bessel Order PredicationFilter Order = 4 3dB BW = 1778.28 Hz
Chebyshev Type I Order PredicationFilter Order = 3 3dB BW = 1000 Hz
Chebyshev Type II Order PredicationFilter Order = 3 3dB BW = 8972.85 Hz
Elliptical or Cauer Order PredicationFilter Order = 3 3dB BW = 1000 Hz
33
Matlab Filter Generation (1)
• Passband • Stopband• Passband Ripple (dB)• Stopband Ripple (dB)
• fpass=1000;• fstop=10000;• AlphaPass=0.5;• AlphaStop=60;• w#### = 2 x pi x f####
[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');
Filter Order and other design parameters
34
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);
35
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')
101 102 103 104 105 106 107-150
-100
-50
0
Filter Comparison: Magnitude
ButterBesselCheby1Cheby2EllipSpec
36
Matlab Code
• AnalogFilterCompare.m
• 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
37
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
38
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.
10-1 100 101 102 103 104-120
-100
-80
-60
-40
-20
0
Butterworth Filters
Frequency (normalized)
Atte
nuat
ion
(dB
)
1 Hz10 Hz
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
0.2
0.4
0.6
0.8
1
1.2
1.4
Step Response
Time (sec)
Ampl
itude
1 Hz10 Hz
39
Filter Bandwidth for Pulses
• Pulse of length T
• Null-to-null BW of
• Single Sided BW desired
• B/2 may be acceptable in some cases– See textbook discussion
TfcsinTTtrect
T2nulltonull
T1B
-3 -2 -1 0 1 2 3
0
0.5
1
1.5
2
40
Pulse Filtering
• Four one-sided BW filters• 0.1 sec pulse responses
0 10 20 30 40 50 60 70 80 90 100-160
-140
-120
-100
-80
-60
-40
-20
0
20Butterworth Filters
Frequency (fs = 100 Hz)
Atte
nuat
ion
(dB
)
2.5 Hz5.0 Hz10. Hz20. Hz
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2
0
0.2
0.4
0.6
0.8
1
1.2Butterworth Filters
Time (fs=100Hz)
Am
plitu
de (d
B)
Test Signal2.5 Hz5.0 Hz10. Hz20. Hz
PulseTest1.m
(digital filters)
41Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Pulse response of an ideal LPFFigure 3.4-10
Text Comparison Chart(2.5, 5.0 and 20 Hz Plots)
42
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
See PulseTest2.m or PulseTest3.m
(digital filters)
Pulse resolution of an ideal LPF. B = 1/2
43
Hilbert Transform
• It is a useful mathematical tool to describe the complex envelope of a real-valued carrier modulated signal in communication theory.
• The precise definition is as follows:
http://en.wikipedia.org/wiki/Hilbert_transform
dtx1
t1txtx̂
t
1thQ
f0j0f,00f,j
fsgnjfHQ
1fHfHfH *QQ
2Q
44
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) Convolution; (b) Result
Hilbert transform of a rectangular pulse
45
Hilbert Transform of Cos
• This is useful in generating a complex signal from a real input signal as follows …
tf2cosAtx 0
00
Q00
ffff2Aj
fHffff2AfX̂
tf2sinAtx̂ 0
46
Real to Complex Conversion
t1
t
tx
tx
tx̂
tx̂jtxty
47
Hilbert Transform Real to Complex Conversion
• Original Real
• Hilbert Transform Complex fXtx
fXfjjfXtxjtxthtc sgnˆ
fXffXtxjtxthtc sgnˆ
0,00,2
ˆfforfforfX
txjtxthtc
The Hilbert Transform can be used to create a single sided spectrum! The complex representation of a real signal.
48
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 “Qudrature-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.
phasequadraturejphasein
tf2sinjtf2costxtf2jexptx
49
Correlation and Spectral Density
• Using Probability and the 1st and 2nd moments– Assuming an ergodic, WSS process we use the time
average
• Properties:
• Schwarz’s Inequality
0tvtvtvP 2v
tzatzatzatza
tzttz
tztz
22112211
0
**
2
wv twtvPP
50
Autocorrelation and Power
• Autocorrelation Function
• Properties tvtvtvtvR vv
vvvv
vvvv
vvv
RR
R0RP0R
51
Crosscorrelation
• Crosscorrelation Function
• Properties twtvtwtvR vw
wvvw
2vwwwvv
RR
R0R0R
52
Application
• Correlation of phasors
2T
2T
21T21 dttwwjexpT1limtwjexptwjexp
2Twwcsinlimtwjexptwjexp 21
T21
else,0
ww,1twjexptwjexp 21
21
53
Power Spectral Density
• The Fourier Transform of the Autocorrelation
• Remember ECE 3800!
54
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Interpretation of spectral density functions