communication systems, 5e
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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
Power Gain
• Gain is defined as the power out divided by the power in
3
2
in
out 0HP
Pg
• Cascaded gain is the gain of the receiver chain
in
out10dB P
Plog10g
321 GGGG
1in
3out
3in
3out
2in
2out
1in
1out
P
P
P
P
P
P
P
PG
dB3dB2dB1dB GGGG
Noise Figure (App. A p. 828)
• The Noise Figure Described the expected output noise power reflected to the input of the device. Thus, it defines the additional “equivalent input” noise contribution. (p. 840)
4
0
device
0
device0
N
N1
N
NNF
• Cascaded noise figure is effected by the gain
21
3
1
21 GG
1F
G
1FFF
Receiver Stages Gain/NF Example
• T/R Switch
• RF Filter and Amplification
• Zero-IF Quadrature Downconversion
• Anti-Aliasing Filters
• ADC
5
LNAFilter
ADC
ADC
Quad.Hybrid
LPF10 MHz
LPF10 MHz
InPhase
QuadPhase
T/R Ctrl LO
1 3a 4 5a 6
RF Input RF Filter and LNAQuadrature
DownconversionDigitization with
Anti-Aliasing Filter
AMP
3b
Attn
Attn
5b
RFT/RSW
Xmt
2
• Isolate Transmit and Receive (TDD comm)
• Reduce Interference BW and Limit NF
• Gain Adjustment
• Downconvert
• ADC Input Signal Conditioning
Receiver Cascaded Analysis
6
LNAFilter
ADC
ADC
Quad.Hybrid
LPF10 MHz
LPF10 MHz
InPhase
QuadPhase
T/R Ctrl LO
1 3a 4 5a 6
RF Input RF Filter and LNAQuadrature
DownconversionDigitization with
Anti-Aliasing Filter
AMP
3b
Attn
Attn
5b
RFT/RSW
Xmt
2
Stage 1 2 3a 3b 4 5a 5b
Gain (dB) -0.90 -0.50 20.00 20.00 14.00 -6.00 -3.00Noise FIgure (dB) 0.90 0.50 2.90 5.30 35.00 6.00 3.00
Noise Figure (linear) 1.23 1.12 1.95 3.39 3162.28 3.98 2.00
Total Gain (dB) -0.90 -1.40 18.60 38.60 52.60 46.60 43.60Total Gain (linear) 0.81 0.72 72.44 7.2E+03 1.8E+05 4.6E+04 2.3E+04
Total NF (linear) 1.230 1.380 2.692 2.725 3.161 3.161 3.161Total NF (dB) 0.90 1.40 4.30 4.35 5.00 5.00 5.00
Receiver Sensitivity (Advanced)
• Receiver Noise Floor– kTB (p. 413)
– Cascaded NF
• Min. Detectable Signal– Detection Threshold
• ADC Considerations– Max. Signal
– ADC SNR
• Dynamic Range– Instantaneous
– Spur Free Dynamic Range
7
-174 dBm/Hz
-104.00 dBm
-33.60 dBm
-103.60 dBm
10 MHz - 70.00dB
Receiver Gain43.6 dB
AD
C S
NR
70 dB
-99.00 dBm
NF 5.00 dB
+10.0 dBm
Receiver NoiseFloor
-91.00 dBm
DT 8 dB
Min. DetectableSignal (MDS)
Maximum InputSignal Power
Thermal NoiseFloor
Maximum SignalPower to ADC
Received SignalPower
Dynamic Range
kT Noise Floor
57.4 dB
ECE 6640 8
RF Propagation & Range Equations
• The power density in a sphere from a “point source” antenna (surface area of a sphere)
• Receiving power collected by an antenna (using the effective area of the receiving antenna so that p(d) can be collected)
sphereofarea
P
r
Prp tt
__4 2
𝑃 = 𝑝 𝑟 ⋅ 𝐴 =𝑃 ⋅ 𝐴
4𝜋 ⋅ 𝑟densityfluxpowerincident
extractedpowertotalAer
• Effective Antenna Area
𝐴 =𝐺 ⋅ 𝜆
4𝜋𝑃 = 𝑝 𝑟 ⋅ 𝐴 =
𝑃 ⋅ 𝐺 ⋅ 𝜆
4𝜋 ⋅ 𝑟
9
Free-Space Path Signal Loss
• As an RF signal propagates, there is path loss.
tPtG rG
f
R
rP
22
c
Rf4R4L
• As shown above
22
2
2
44 fR
cGGP
R
GGP
L
GGPP rt
trt
trt
tr
fc
Note
10
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
11
System Range
• Maximum Range (Pr is the receiver sensitivity - MDS)
dBmPt
dBmGt
dBmGr
dBmPr
m0 mR1
tPtG rG
f
R
rP
dBPdBGRfdBc
dBdBGdBP rrtt
2
42
Note: as f increases range decreases
12
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
6 GHz4 GHz
dB13.1997e8.9048e3
6e369e64
c
Rf4L 2
22
u
dB60.1957e18.6038e3
6e369e44
c
Rf4L 2
22
d
Satellite relay system Ex. 3.3-1 (1 of 2)
Path Losses
13
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
GGP
L
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
14
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 of 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:0.65kg15
These are not legal in the USby FCC Regulations.
~$256 from China
Filters and filtering
• Ideal filters
• Bandlimiting and timelimiting
• Real filters
• Pulse response and risetime
© 2010 The McGraw-Hill Companies
17
The Ideal Filter
• To receive a signal without distortion, only changes in the magnitude and/or a pure time delay are allowed. 0ttxKty
02exp tffXKfY
• The transfer function is
0tf2expKfH
• A constant gain with a linear phase KfH 0tf2f
18
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(b) Bandpass Filter
Ideal filters
(a) Lowpass Filter
19
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
f
0
f
f
0
dfttf2jexpKth
dftf2jexptf2jexpKth
20
Ideal Filter (2)
0uu
0
0u
0
0u
0
0u
f
f0
0
f
f
0
ttf2sincKf2th
tt2
ttf2sinK2th
tt2j
ttf2jexpK
tt2j
ttf2jexpKth
tt2j
ttf2jexpKth
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.
21
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
22
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
24
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
BW
BWSF
PBBW
SBBW
25
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Typical amplitude ratio of a real bandpass filter
Figure 3.4-3
Real Bandpass Filter
The -3 dB or half-power bandwidth is shown
26
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
27
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.
28
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.
29
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 (
kHz)
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 Coilcraft30
31
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
1
wsj1
1
wjs1
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)
Att
enua
tion
(dB
)
1st order
2nd order
3rd order4th order
5th order
32
Butterworth Filter PSD
10-1
100
101
102
103
-120
-100
-80
-60
-40
-20
0
Butterworth Filter Family
Frequency (normalized)
Att
enua
tion
(dB
)
1st order
2nd order
3rd order4th order
5th order
33
Butterworth Filter PSD (2)
10-1
100
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1Butterworth Filter Family
Frequency (normalized)
Att
enua
tion
(dB
)
1st order
2nd order
3rd order4th order
5th order
34
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]);
35
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)
Att
enua
tion
(dB
)
1st order
2nd order
3rd order4th order
5th order
36
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)
Att
enua
tion
(dB
)
1st order
2nd order
3rd order4th order
5th order
37
Available MATLAB Filters(Signal Proc. TB)
• Analog or Digital– Butterworth
– Chebyshev Type I
– Chebyshev Type II
– Elliptic or Cauer
• Special – Linear Phase– Bessel
• Digital Windows/Filters– barthannwin
– bartlett
– blackman
– blackmanharris
– bohmanwin
– chebwin
– flattopwin
– gausswin
– hamming
– hann
– kaiser
– nuttallwin
– parzenwin
– rectwin
– triang
– tukeywin
38
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
Butter
Bessel
Cheby1Cheby2
Ellip
Spec
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
39
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
40
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);
41
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
Butter
Bessel
Cheby1Cheby2
Ellip
Spec
42
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
43
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
44
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)
Att
enua
tion
(dB
)
1 Hz
10 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)
Am
plitu
de
1 Hz
10 Hz
45
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
TfcsinTT
trect
T
2nulltonull
T
1B
-3 -2 -1 0 1 2 3
0
0.5
1
1.5
2
46
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)
Att
enua
tion
(dB
)
2.5 Hz
5.0 Hz
10. Hz
20. 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 (
dB)
Test Signal
2.5 Hz
5.0 Hz10. Hz
20. Hz
PulseTest1.m
(digital filters)
47Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Pulse response of an ideal LPF
Figure 3.4-10
Text Comparison Chart(2.5, 5.0 and 20 Hz Plots)
48
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 = 1/2t
49
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
dt
x1
t
1txtx̂
t
1thQ
f0j
0f,0
0f,j
fsgnjfHQ
1fHfHfH *QQ
2
Q
50
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) Convolution; (b) Result
Hilbert transform of a rectangular pulse
51
Hilbert Transform of Cos
• This is useful in generating a complex signal from a real input signal as follows …
tf2cosAtx 0
00
Q00
ffff2
Aj
fHffff2
AfX̂
tf2sinAtx̂ 0
52
Real to Complex Conversion
t
1
t
tx
tx
tx̂
tx̂jtxty
53
Hilbert Transform Real to Complex Conversion
• Original Real
• Hilbert Transform Complex
fXtx
fXfjjfXtxjtxthtc sgnˆ
fXffXtxjtxthtc sgnˆ
0,0
0,2ˆ
ffor
fforfXtxjtxthtc
The Hilbert Transform can be used to create a single sided spectrum! The complex representation of a real signal.
54
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
55
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
0tvtvtvP2
v
tzatzatzatza
tzttz
tztz
22112211
0
**
2
wv twtvPP
56
Autocorrelation and Power
• Autocorrelation Function
• Properties ttt tvtvtvtvR vv
tt
t
vvvv
vvvv
vvv
RR
R0R
P0R
57
Crosscorrelation
• Crosscorrelation Function
• Properties ttt twtvtwtvR vw
tt
t
wvvw
2
vwwwvv
RR
R0R0R
58
Application
• Correlation of phasors
2T
2T
21T
21 dttwwjexpT
1limtwjexptwjexp
2
Twwcsinlimtwjexptwjexp 21
T21
else,0
ww,1twjexptwjexp 21
21
59
Power Spectral Density
• The Fourier Transform of the Autocorrelation
• Remember ECE 3800!
60
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Interpretation of spectral density functions
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