Review Chapter 1.1 - 1.4Review Chapter 1.1 - 1.4Problems: 1.1a-c, 1.4, 1.5, Problems: 1.1a-c, 1.4, 1.5, 1.91.9

Review Chapter 1.5 - 1.8Problems: 1.13 - 1.16, 1.20

Quiz #1

Quiz #1

Read: 5.1 - 5.3 Problems: 5.1 - 5.3 Quiz #1

Read 5.4 & 5.5 Problems 5.7 & 5.12 Quiz #1

Local: Thursday, 28 September, Lecture 6

Off Campus DL: < 11 OctoberStrictly Review (Chapter 1)

Full Period, Open Book & Notes

In Class: 2 Quizzes, 2 Tests, 1 Final ExamOpen Book & Open NotesWARNING! Study for them like they’re closed book!

Graded Homework: 2 Design Problems Ungraded Homework:

Assigned most every classNot collectedSolutions ProvidedPayoff: Tests & Quizzes

An Analogy: Commo Theory vs. Football Reading the text = Reading a playbook Working the problems =

playing in a scrimmage Looking at the problem solutions =

watching a scrimmage

Quiz = Exhibition Game Test = Big Game

Show some self-discipline!! Important!!For every hour of class...

... put in 1-2 hours of your own effort.

PROFESSOR'S GUIDEIf you put in the timeYou should do fine.You have only three days in your life one is today do it today second is yesterday that has gone forget about it third is tomorrow but many of you do not have tomorrow so do every thing today Imam Ali

Digital Analog

Binary M-ary

Wide Band Narrow Band

Digital M-Ary System M = 8 x 8 x 4 = 256

Source:January 1994Scientific American

Source: January 1994Scientific American

Phonograph → Compact Disk Analog NTSC TV → Digital HDTV Video Cassette Recorder

→ Digital Video Disk AMPS Wireless Phone → 4G LTE Terrestrial Commercial AM

& FM Radio Last mile Wired Phones

Fourier Transforms X(f)Table 2-4 & 2-5

Power SpectrumGiven X(f)

Power SpectrumUsing Autocorrelation Use Time Average Autocorrelation

Autocorrelations deal with predictability over time. I.E. given an arbitrary point x(t1), how predictable is x(t1+tau)?

time

Volts

t1

tau

Autocorrelations deal with predictability over time. I.E. given an arbitrary waveform x(t), how alike is a shifted version x(t+τ)?

Voltsτ

time

Volts

0

Vdc = 0 v, Normalized Power = 1 watt

If true continuous time White Noise, no predictability.

• The sequence x(n)x(1) x(2) x(3) ... x(255)

• multiply it by the unshifted sequence x(n+0)x(1) x(2) x(3) ... x(255)

• to get the squared sequencex(1)2 x(2)2 x(3)2 ... x(255)2

• Then take the time average[x(1)2 +x(2)2 +x(3)2 ... +x(255)2]/255

• The sequence x(n)x(1) x(2) x(3) ... x(254) x(255)

• multiply it by the shifted sequence x(n+1)x(2) x(3) x(4) ... x(255)

• to get the sequencex(1)x(2) x(2)x(3) x(3)x(4) ... x(254)x(255)

• Then take the time average[x(1)x(2) +x(2)x(3) +... +x(254)x(255)]/254

• If the average is positive...– Then x(t) and x(t+tau) tend to be alike

Both positive or both negative• If the average is negative

– Then x(t) and x(t+tau) tend to be oppositesIf one is positive the other tends to be negative

• If the average is zero– There is no predictability

tau (samples)

Rxx

0

Time

Volts

23 points

0

tau samples

Rxx

0

23

Rx(τ)

tau seconds0

A

Gx(f)

Hertz0

A watts/Hz

Rx(τ) & Gx(f) form a Fourier Transform pair.

They provide the same infoin 2 different formats.

Rx(tau)

tau seconds0

A

Gx(f)

Hertz0

A watts/Hz

Average Power = ∞D.C. Power = 0A.C. Power = ∞

Rx(tau)

tau seconds0

A

Gx(f)

Hertz0

A watts/Hz

-WN Hz

2AWN

1/(2WN)Average Power = 2AWN wattsD.C. Power = 0A.C. Power = 2AWN watts

Time Average Autocorrelation Easier to use & understand than

Statistical Autocorrelation E[X(t)X(t+τ)] Fourier Transform yields GX(f)

Autocorrelation of a Random Binary Square Wave Triangle riding on a constant term Fourier Transform is sinc2 & delta function

Linear Time Invariant Systems If LTI, H(f) exists & GY(f) = GX(f)|H(f)|2

X

=

Cos(2πΔf)

If input is x(t) = Acos(ωt)output must be of form

y(t) = Bcos(ωt+θ)

Filterx(t) y(t)

Maximum Power Intensity Average Power Intensity

WARNING!Antenna Directivity is NOT =

Antenna Power Gain10w in? Max of 10w radiated.

Treat Antenna Power Gain = 1 Antenna Gain = Power Gain * Directivity

High Gain = Narrow Beam

Antenna Gain is what goes in RF Link Equations

In this class, unless specified otherwise, assume antennas are properly aimed. Problems specify peak antenna gain

High Gain Antenna = Narrow Beam

sou

rce:

en

.wik

iped

ia.o

rg/w

iki/

Par

abol

ic_a

nte

nn

a

EIRP = PtGt

Path Loss Ls = (4*π*d/λ)2

Final Form of Analog Free Space RF Link EquationPr = EIRP*Gr/(Ls*M*Lo) (watts)

Derived Digital Link EquationEb/No = EIRP*Gr/(R*k*T*Ls*M*Lo)

(dimensionless)

• Models for Thermal Noise: *White Noise & Band limited White Noise*Gaussian Distributed

• Noise Bandwidth– Actual filter that lets A watts of noise thru?– Ideal filter that lets A watts of noise thru?– Peak value at |H(f = center freq.)|2 same?• Noise Bandwidth = width of ideal filter (+ frequencies).

• Noise out of an Antenna = k*Tant*WN

Radio Static (Thermal Noise) Analog TV "snow"

2 secondsof White Noise

Probability Density Functions (PDF's), of which a Histograms is an estimate of shape, frequently (but not always!) deal with the voltage likelihoods

Time

Volts

time

Volts

0

Vdc = 0 v, Normalized Power = 1 watt

If true continuous time White Noise, No Predictability.

Volts

BinCount

Vol

ts

Bin

Cou

nt

Time

Volts

0

Volts

BinCount

00

200

When bin count range is from zero to max value, a histogram of a uniform PDF source will tend to look flatter as the number of sample points increases.

Time

Volts

0

Volts

BinCount

Time

Volts

0

Volts

BinCount

Volts

BinCount

0

400

Are all 0 mean, 1 watt, White Noise

0

0

Rx(tau)

tau seconds0

A

Gx(f)

Hertz0

A watts/Hz

The previous WhiteNoise waveforms all

have same Autocorrelation& Power Spectrum.

Autocorrelation: Time axis predictability PDF: Voltage liklihood Autocorrelation provides NO information about

the PDF (& vice-versa)... ...EXCEPT the power will be the same...

PDF second moment E[X2] = Rx(0) = area under Power Spectrum = A{x(t)2}

...AND the D.C. value will be related. PDF first moment squared E[X]2 = constant term in autocorrelation = E[X]2δ(f) = A{x(t)}2

x

WinterSun is belowsatelliteorbital plane.

x

Fall Sun → sameplane assatellite.

x

Spring Sun→ sameplane asSatellite.

x

SummerSun is abovesatelliteorbital plane.

Source: www.ses.com/4551568/sun-outage-data

x

Time

Volts

0

If AC power = 4 watts & BW = 1,000 GHz...

fx(x)

Volts0

.399/σx = .399/2 = 0.1995Time

Volts

0

Rx(tau)

tau seconds0

Gx(f)

Hertz0

2(10-12) watts/Hz

-1000 GHz

4

500(10-15)

Time

Volts

3

AC power = 4 watts

0

Gx(f)

Hertz0-1000 GHz

9

Gx(f)

Hertz0

2(10-12) watts/Hz

-1000 GHz

2(10-12) watts/Hz

No DC

3 vdc → 9 watts DC Power

Rx(tau)

tau seconds0

13

9

Rx(tau)

tau seconds0

4

500(10-15)

500(10-15)

No DC

3 vdc → 9 watts DC Power

fx(x)

Volts0

σ2x = E[X2] -E[X]2 = 4

0

fx(x)

Volts3

σ2x = E[X2] -E[X]2 = 4

Time

Volts

3

AC power = 4 wattsDC power = 9 wattsTotal Power = 13 watts

0

Sin

&Nin

GSin

&G(Nin + Nai)

G

Namp = kTampWn

+

+

G > 1

F = SNRin/SNRout WARNING! Use with caution.

If input noise changes, F will change.

F = 1 + Tamp/Tin Tin = 290o K (default)

Sin

&Nin

GSin

&G(Nin + Nai)

G

Namp = kTpassiveWn

+

+

G < 1

Tpassive = (L-1)Tphysical

Active Device (Tamp) From Spec Sheet (may have F)

Passive Device (Tcable or T passive)

(L-1)*Tphysical

Noise Striking Antenna = NoWThermal

= kTsurroundings1000*109 = k*290*1000*109

= 4.00 n watts

Much of this noise doesn't exit system.Blocked by system filters. kTantWN = ???

SystemCable + Amp

Noise exiting Antenna that will exit the System =kTant6*106 = 12.42*10-15 watts

Noise Antenna "Sees" = Noise exiting antenna = NoWAntenna

≈ kTant1000*109 = 2.07 n watts

(Tantenna = 150 Kelvin)

SystemCable + Amp

Noise Actually Exiting Antenna = Noise Antenna "Sees" ≠ Noise Exiting Antenna that will exit the System = kTantWN = 12.42*10-15 watts

AntennaPower

Gain = 1Signal Power in =Signal Power out

This is the model we use.

We don't worry aboutnoise that won't make the output.

Noise Seen by Antenna = NoWAntenna

= kTant1000*109 = 2.07 n wattsSignal Power Picked Up by Antenna = 10-11 watts

SystemCable + Amp

SNR at "input" of antenna = 10-11/(4*10-9) = 0.0025SNR at output of antenna = 10-11/(2.07*10-9) = 0.004831SNR at System Output = 43.63

Noise seen by Antenna TCRO = NoWN

= kTant6*106 = 12.42 femto wattsSignal Power Picked Up by Antenna = 10-11 watts

SystemCable + Amp

SNR at output of antenna = 805.2

SNR at System Output = 43.63

This is the noise we're

worried about.

Filtering...Removes noise power outside signal BWLets the signal power through

SystemCable + Amp

SNR at Antenna Input = 0.0025SNR at Antenna Output = 0.004831SNR at System Output = 43.67

Only considers input noise that is in the signal BW & can reach the output.Cable & electronics dump in more

noise.

SystemCable + Amp

SNR at antenna output = 805.2 SNR at System Output = 43.67