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

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

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

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 → ∞

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).

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).

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

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)

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

Fourier Transform of Rx(τ)Fourier Transform of Rx(τ)

Rx(τ) = SX(f) ej2πfτ dτ

-∞

∞

SX(f) = Rx(τ)e-j2πfτ dτ

-∞

∞

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))

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

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

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

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

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)|

Time Domain (3 Tap Equalizer)Time Domain (3 Tap Equalizer)System Input

System OutputMultipath

Equalizer Output

Tapped Delay Line Equalizer8 Taps

Tapped Delay Line Equalizer8 Taps

1.111

0.003806

0.4938Delay

0.13 sec

Delay0.91 sec

ΣInput Output

Time Domain (8 Tap Equalizer)Time Domain (8 Tap Equalizer)System Input

System Output

Equalizer Output

Adaptive Delay Line Equalizer8 Taps

Adaptive Delay Line Equalizer8 Taps

Delay0.13 sec

Delay0.91 sec

ΣInput Output

DSB-SCDSB-SC

source: http://cnx.org/contents/b5be5e4c-4ab8-4765-a3d3-534ee2ee2ff3@1/THE-PHASE-REVERSAL-IN-DSB-SC