random varaibles processes and noise aug 11 - 18, 2014

8/11/2019 Random Varaibles Processes and Noise Aug 11 - 18, 2014

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Topic # 3: Random Variables & Processes & Noise

T1. B.P. Lathi, Modern Digital and Analog Communication Systems, 3rd

Edition, Oxford University Press, 1998: OR 4thEdition 2010 Chapter 8, 9 & 12

T2. Simon Haykin & Michael Moher: Communication Systems; John Wiely, 4th

Edition OR 5thEdition, 2010, 5/e. : Chapter 5

R1.DIGITAL COMMUNICATIONS Fundamentals and Applications: ERNARDSKLAR and Pabitra Kumar Ray; Pearson Education 2009, 2/e. :

( Section 5.5)

August 11- 18, 2014

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ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

What is Noise ?

Desired Signal : The one that is needed.

Undesired Signal : The one that gets added tothe desired signal when the desired signal ispassing through the medium, amplifiers, mixers,

filters and other parts of the communicationchannel between the source and the destination.

Noise : The undesired signal that adds to thedesired signal and reaches the destination.

Interference: Intentional orunintentional un desired signalsthat interfere with communicationprocess.

Effect of Noise : Since the noiseadds to the signal, it lives with it.Neither amplification nor thefiltering can alleviate the effect ofnoise on the desired signal.

The only way to keep away fromthe effects of noise is to see thatless amount of noise, relative tothe desired signal, is present atthe destination

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Noise Sources

Externally Generated

Atmospheric : Due tolightening & Thunder storms :

2MHz 10 MHz

Extra Terrestrial : Due to solar &Galactic sources

20 MHz- 1.5 GHz

Man Made Noise : Spark Plugs,engine Noise

1 MHz500 MHz

Internally Generated

Thermal noise : Random Motionof electrons due to temperaturein resistive components of thesystem

Shot Noise : Due to diffusionof carriers in semiconductorsetc.

Most of the discussion in our class willbe on Thermal Noise

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Thermal Noise

Thermal noise is an inevitable reality withwhich the received signal power has tocompete

Additive White Gaussian Noise.

Thermal Noise is AWGN

Additive :Adds to Signal

White : Its power spectral density is flat

Gaussian : The underlying probabilitydensity function is Gaussian

We talk about the probability densityfunction because, noise is random andhence to be dealt with properties of

random variables.

Gaussian or Normal probabilitydensity function

Cumulative distribution function

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Statistical Averages of Random Variable

For a Continuous RV case, the mean is

Mean of a function (y = g(x)) of a random

variable

Mean square of a random variable: use

g(x) = x2

Moments of a random variable:

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Sum of Random Variables6

If z = x + y

Then the pdf of z is

Central Limit Theorem: under certain conditions,

sum of large number of independent random

variables tends to be a Gaussian random variable,

independent of the pdfs of the random variables

involved.

Example: By adding 2 RVs, with

density function as in the figure,

the density function of the

resulting RV is

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7/26ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

Random Process

A cos (wct + ), with being a randomvariable.

Ex: Binary waveform generator, say

over 10 pulse durations

A random variable that is a function oftime is called a random process.

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Random Process

A random variable that is afunction of time is called a

random process.

Collection of all possible

waveforms is called Ensemble

A given waveform in the

Ensemble is called Sample

Function

X1, X2, .. Are the random variables generated by the amplitudes of the sample

functions at time instants t1, t2, .. respectively 8



Random Process

The n random variables X1, X2, ..aredependent, in general

The nthorder joint PDF is expressed as

If a higher order joint PDF is available,

the lower order PDF can be obtained

The mean of the random process can be

obtained from the first order PDF as 9



Auto Correlation of a Random Process

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Stationary Random Process

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Ergodic Random Process

Ensemble statistics

Time statistics

For Ergodic Process 12



Power Spectral Density of Random Process

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Transmission of a Random Process

through a Linear System.

If either or both of them are zero

mean processes,

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Home Work

Solve & understand the following worked examples:

9.2

9.5 from Lathi (4thEdition)

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System Noise Characterization

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Thermal Noise Power

The thermal noise is AWGN in nature and its power is

N = k T0W (orB) Watts

T0= Temperature in Kelvin degrees

k = Boltzman Constant = 1.38 X 10-23J/K or W / K-Hz

= - 228.6 dBW / K-Hz

W or B = Bandwidth in Hz

Noise Power Spectral density N0

= (N / W ) = k T0 Watts /Hz17



Noise Figure

Amplifiers in the system are made ofactive & passive devices, hencecontribute to over all noise in the system

All passive & active devicesgenerate noise

Noise Figure of Amplifier

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L =

For a lossy network, Loss is given by

Noise Figure F = L.



Noise Temperature

TR0= Effective Noise

Temperature of

Network or Receiver

To0= Reference Temperature of

the noise source, chosen to be

2900

K 19



CompositeNoise figure

Noise at the output of Network1

Let the noise at the input of Network1 be N1

(Nout)1= G1N1+ (F1-1) G1k 290 W

Noise at the output of Network2

(Nout)2= G2(Nout)1+ (F2-1) G2k 290 W

(Nout)2= G2{G1N1 + (F1-1) G1k 290 W}

+ (F2-1) G2k 290 W

= G1G2N1

+ G1G2(F1-1) k 290 W

+ (F2-1) G2k 290 W

The total noise power at the output of thecascaded network is given by

Assume the over all gain of the network

is G = G1G2and over all noise figure isF comp

comp

(Fcomp-1) G1 G2k 290 W

= G1G2(F1-1) k 290 W + (F2-1) G2k 290

Comparing

(Nout)2

Fcomp

= F1

+ (F2

-1)/ G1

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Composite Noise figure : Feed line & Amplifier

Tcomp0= (L-1)290 + (F-1) 290/(1/L)

= (L-1)290 + L(F-1) 290

Tcomp0= (LF-1) 2900K

Tcomp0= (LF-1) 2900K

= (LF-1 + L -L) 2900K

= (L -1 + L(F-1) ) 2900K

Tcomp0= TL

0+ L TR0

For an N-Stage Network..

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System Effective Temperature

TA0is the antenna noise

temperature

The system effective Temperature is

Natural Sources including : Lightening,

Celestial radio sources, Atmospheric

sources, Thermal radiation from The

ground and other structures.

Manmade noises: Radiation fromAutomobile ignition and electrical

machinery and Radio transmissions from

other users that fall into the BW.

TS0= TA

0+ (LF-1) 2900K

= TA0 + (L-1)290 + L(F-1) 290

TS0= TA

0+ TL0+ TR

0/ (1/L)F

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Example on NF & Noise Temp

TR0= (F-1)2900K = 26100K

TS0= TA

0+ TL0+ LTR

0

= 150 + 2610 = 2760 K

Nout= G k TS0W

= 108X 1.38 X 10 -23X 2760 X 6 X 106

= 22.8 mw

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29.1 16.4 = 12.7 dB

and the overall Noise Figure of the system



Improving SNR - Benefit of using Pre Amplifier

TR10= (F1-1)290

0K = 2900K

TR20= (F2-1)290

0K = 26100K

Tcomp0= TR1

0+ TR20/ G1 = 290 + 2610/20 = 420.5

0K

TS0= TA

0+ Tcomp0 = 150 + 420.5 0K = 570.5 0K

Fcomp= F1+ (F2-1)/ G1 = 2+ 9/20 = 2.5 (4dB)SNRout= 23.3 dB

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Fig. 5.19a

Fig. 5.19b

Fig. 5.19a

SNRout= 16.4 dB

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Problem

75 feet Lossy

Cable

3dB/100 ft

Receiver

F = 13 dB

random varaibles processes and noise aug 11 - 18, 2014

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