ecen4523 commo theory lecture #12 14 september 2015 dr. george scheets n read chapter 4.1 – 4.2 n...
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ECEN4523 Commo TheoryECEN4523 Commo TheoryLecture #12 14 September 2015Lecture #12 14 September 2015Dr. George ScheetsDr. George Scheetswww.okstate.edu/elec-engr/scheets/ecen4533www.okstate.edu/elec-engr/scheets/ecen4533
ECEN4523 Commo TheoryECEN4523 Commo TheoryLecture #12 14 September 2015Lecture #12 14 September 2015Dr. George ScheetsDr. George Scheetswww.okstate.edu/elec-engr/scheets/ecen4533www.okstate.edu/elec-engr/scheets/ecen4533
Read Chapter 4.1 – 4.2Read Chapter 4.1 – 4.2 Problems: 3.8-3 & 4Problems: 3.8-3 & 4 Quiz #3, 18 SeptemberQuiz #3, 18 September
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ECEN4523 Commo TheoryECEN4523 Commo TheoryLecture #13 16 September 2015Lecture #13 16 September 2015Dr. George ScheetsDr. George Scheetswww.okstate.edu/elec-engr/scheets/ecen4533www.okstate.edu/elec-engr/scheets/ecen4533
ECEN4523 Commo TheoryECEN4523 Commo TheoryLecture #13 16 September 2015Lecture #13 16 September 2015Dr. George ScheetsDr. George Scheetswww.okstate.edu/elec-engr/scheets/ecen4533www.okstate.edu/elec-engr/scheets/ecen4533
Read Chapter 4.3Read Chapter 4.3 Problems: 3.8-5, 4.2-2Problems: 3.8-5, 4.2-2 Quiz #3, 18 SeptemberQuiz #3, 18 September
Chapter 3Chapter 3
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OSU IEEE September General MeetingOSU IEEE September General Meeting
American Airlines (Tulsa Maintenance)American Airlines (Tulsa Maintenance) WednesdayWednesday
23 September23 September5:30 pm5:30 pmES 201bES 201b
All are invitedAll are invited Dinner will be servedDinner will be served 3 pts extra credit3 pts extra credit
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CorrelationCorrelation
x(t)y(t) dt
T Returns a numberReturns a number How similar x(t) & y(t) areHow similar x(t) & y(t) are
limT → ∞
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Laplace Transform of x(t)Laplace Transform of x(t)
X(2) = x(t) e-2t dt
0-
∞
X(s) = x(t)e-st dt; s = σ + jω
0-
∞
Evaluated at s = 2, the Laplace Transform returns a number that is a function of how alike e-2t is with the function x(t).
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Fourier Transform of x(t)Fourier Transform of x(t)
X(2) = x(t) e-j2π2t dt
0-
∞
X(f) = x(t)e-j2πft dt
-∞
∞
Evaluated at f = 2, the Fourier Transform returns a number that is a function of how alike a 2 Hz cosine & sine is with x(t).
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AutocorrelationAutocorrelation
x(t)x(t+τ) dt ≡ RX(τ)
T Returns a numberReturns a number How similar x(t) is with a time shifted How similar x(t) is with a time shifted
version, x(t+version, x(t+ττ), of itself), of itself
lim 1T → ∞ T
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Fourier Transform of x(t)Fourier Transform of x(t)
x(t) = X(f) ej2πft dt
-∞
∞
X(f) = x(t)e-j2πft dt
-∞
∞
ej2πft = cos(2πft) + jsin(2πft)
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RecapRecap x(t) volts ↔ X(f) Volts/Hzx(t) volts ↔ X(f) Volts/Hz X(f) can be complexX(f) can be complex
Angle at f1 = 0 or 180 degrees → cosine @ f1Angle at f1 = 0 or 180 degrees → cosine @ f1 Angle at f2 = 90 or 270 degrees → sine @ f2Angle at f2 = 90 or 270 degrees → sine @ f2 Otherwise → Need sine and cosineOtherwise → Need sine and cosine
x(t) ↔ X(f) is a 1 to 1 mappingx(t) ↔ X(f) is a 1 to 1 mapping LTI systemLTI system
Y(f) = X(f)H(f)Y(f) = X(f)H(f) y(t) = x(t) ☺ h(t)y(t) = x(t) ☺ h(t)
Negative Frequencies don't existNegative Frequencies don't exist Don't count when measuring BWDon't count when measuring BW
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Fourier Transform of Rx(τ)Fourier Transform of Rx(τ)
Rx(τ) = SX(f) ej2πfτ dτ
-∞
∞
SX(f) = Rx(τ)e-j2πfτ dτ
-∞
∞
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RecapRecap RRXX((ττ) watts ↔ S) watts ↔ SXX(f) Watts/Hz(f) Watts/Hz
SSXX(f) is a real function (Not Complex)(f) is a real function (Not Complex) RRXX((ττ) is an even function) is an even function
Only requires cosines to constructOnly requires cosines to construct
SSXX(f) (f) >> 0 0
x(t) → Rx(t) → RXX((ττ) is a many to 1 mapping) is a many to 1 mapping RRXX((ττ) ↔ S) ↔ SXX(f) is a 1 to 1 mapping(f) is a 1 to 1 mapping
LTI systemLTI system SSYY(f) = S(f) = SXX(f) |H(f)|(f) |H(f)|22
RRYY((ττ) = R) = RXX((ττ) ☺ h() ☺ h(ττ) ☺ h(-) ☺ h(-ττ))
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EqualizationEqualization Seeks to reverse effects of channel Seeks to reverse effects of channel
filtering Hfiltering Hchannel channel (f)(f)
Ideally HIdeally Hequalizerequalizer(f) = 1/H(f) = 1/Hchannelchannel(f)(f) Result will be flat spectrumResult will be flat spectrum Not always practical if parts of |HNot always practical if parts of |Hchannelchannel(f)| (f)|
have small magnitudehave small magnitude
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System with MultipathSystem with Multipath
0 20 40 60 80 1000.4
0.6
0.8
1
1.2
1.4
Hf i
i
h(t) = 0.9h(t) = 0.9δδ(t) – 0.4(t) – 0.4δδ(t - 0.13)(t - 0.13) H(f) = .9H(f) = .9 - - .4e .4e -j-jωω0.130.13
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Required Equalizer Filter|Heq(f)| = 1/|H(f)|
Required Equalizer Filter|Heq(f)| = 1/|H(f)|
0 20 40 60 80 1000.5
1
1.5
2
1
Hf i
i
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Heq(f) = 1 / (.9 - .4e -jω0.13 ) Heq3(f) = 1.111 + 0.4938e-jω0.13 + 0.2194e -jω0.26 + ...
Heq(f) = 1 / (.9 - .4e -jω0.13 ) Heq3(f) = 1.111 + 0.4938e-jω0.13 + 0.2194e -jω0.26 + ...
Impulse Response of a 3 tap Equalizing filter.h(t) = 1.111δ(t) + 0.4938δ(t – 0.13) + -.2194δ(t – 0.26)
Heq3(f)of a
3 tap filter
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Tapped Delay Line Equalizera.k.a. FIR Filter and Moving Average Filter
Tapped Delay Line Equalizera.k.a. FIR Filter and Moving Average Filter
1.111
0.2194
0.4938Delay
0.13 sec
Delay0.26 sec
ΣInput Output
Ideally |H(f)Heq(f)| = 1Was 0.5 < |H(f)| < 1.3Now 0.9 < |H(f)Heq3(f)| < 1.1
|H(f)*Heq3(f)|
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Time Domain (3 Tap Equalizer)Time Domain (3 Tap Equalizer)System Input
System OutputMultipath
Equalizer Output
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Tapped Delay Line Equalizer8 Taps
Tapped Delay Line Equalizer8 Taps
1.111
0.003806
0.4938Delay
0.13 sec
Delay0.91 sec
ΣInput Output
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Time Domain (8 Tap Equalizer)Time Domain (8 Tap Equalizer)System Input
System Output
Equalizer Output
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Adaptive Delay Line Equalizer8 Taps
Adaptive Delay Line Equalizer8 Taps
Delay0.13 sec
Delay0.91 sec
ΣInput Output
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DSB-SCDSB-SC
source: http://cnx.org/contents/b5be5e4c-4ab8-4765-a3d3-534ee2ee2ff3@1/THE-PHASE-REVERSAL-IN-DSB-SC
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