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Random Variables

In: Computers and Technology

Submitted By maneel1995
Words 1408
Pages 6
Topic # 3: Random Variables & Processes & Noise

T1. B.P. Lathi, Modern Digital and Analog Communication Systems, 3rd Edition, Oxford University Press, 1998: OR 4th Edition 2010 Chapter 8, 9 & 12

T2. Simon Haykin & Michael Moher: Communication Systems; John Wiely, 4th Edition OR 5th Edition, 2010, 5/e. : Chapter 5
R1.DIGITAL COMMUNICATIONS Fundamentals and Applications: ERNARD SKLAR and Pabitra Kumar Ray; Pearson Education 2009, 2/e. : ( Section 5.5) August 11- 18, 2014

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What is Noise ?
Desired Signal : The one that is needed. Effect of Noise : Since the noise adds to the signal, it lives with it. Neither amplification nor the filtering can alleviate the effect of noise on the desired signal.

Undesired Signal : The one that gets added to the desired signal when the desired signal is passing through the medium, amplifiers, mixers, filters and other parts of the communication channel between the source and the destination. Noise : The undesired signal that adds to the desired signal and reaches the destination.

The only way to keep away from the effects of noise is to see that less amount of noise, relative to the desired signal, is present at the destination

Interference: Intentional or unintentional un desired signals that interfere with communication process.
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Noise Sources
Externally Generated Internally Generated  Thermal noise : Random Motion of electrons due to temperature in resistive components of the system  Shot Noise : Due to diffusion of carriers in semiconductors etc.

 Atmospheric : Due to lightening & Thunder storms :
2MHz – 10 MHz

 Extra Terrestrial Galactic sources

: Due to solar &

20 MHz- 1.5 GHz  Man Made Noise : Spark Plugs, engine Noise 1 MHz– 500 MHz Most of the discussion in our class will be on Thermal Noise

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Thermal Noise
Thermal noise is an inevitable reality with which the received signal power has to compete Gaussian or Normal probability density function

Thermal Noise is AWGN
Additive White Gaussian Noise.

Additive : Adds to Signal
White : Its power spectral density is flat Gaussian : The underlying probability density function is Gaussian

Cumulative distribution function We talk about the probability density function because, noise is random and hence to be dealt with properties of random variables. ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

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Statistical Averages of Random Variable
For a Continuous RV case, the mean is Moments of a random variable:

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

Mean square of a random variable: use g(x) = x2
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Sum of Random Variables z=x+y 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.

If

Then the pdf of z is

Example: By adding 2 RVs, with density function as in the figure, the density function of the resulting RV is

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Random Process
A random variable that is a function of time is called a random process. Ex: Binary waveform generator, say over 10 pulse durations A cos (wct + Φ), with Φ being a random variable.

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Random Process
A random variable that is a function 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 ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

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Random Process
The n random variables X1, X2, ..are dependent, in general The nth order 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 ELECTRICAL ELECTRONICS COMMUNICATION

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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
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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 (4th Edition)

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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 T0 W (or B) Watts T0 = Temperature in Kelvin degrees

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

= - 228.6 dBW / K-Hz
W or B = Bandwidth in Hz

Noise Power Spectral density
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N0 = (N / W ) = k T0
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Watts /Hz
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Noise Figure
All passive & active devices generate noise

Amplifiers in the system are made of active & passive devices, hence contribute to over all noise in the system
Noise Figure of Amplifier

For a lossy network, Loss is given by L = ������������������������������������ ������������������������������ Noise Figure COMMUNICATION F = L. 18 INSTRUMENTATION
������������������������������ ������������������������������

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Noise Temperature

TR0= Effective Noise Temperature of Network or Receiver ELECTRICAL

To0= Reference Temperature of the noise source, chosen to be 2900 K COMMUNICATION INSTRUMENTATION

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Composite Noise figure
Noise at the output of Network2 (Nout)2 = G2 (Nout)1 + (F2-1) G2 k 290 W (Nout)2 = G2 { G1 N1 + (F1-1) G1 k 290 W } + (F2-1) G2 k 290 W (Nout)2 = G1G2 N1 + G1G2(F1-1) k 290 W + (F2-1) G2 k 290 W

Assume the over all gain of the network is G = G1 G2 and over all noise figure is F comp The total noise power at the output of the cascaded network is given by

comp

Comparing

Let the noise at the input of Network1 be N1 (Fcomp-1) G1 G2 k 290 W Noise at the output of Network1 (Nout)1 = G1 N1+ (F1-1) G1 k 290 W ELECTRICAL ELECTRONICS = G1G2(F1-1) k 290 W + (F2-1) G2 k 290 W

Fcomp = F1 + (F2-1)/ G1
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Composite Noise figure : Feed line & Amplifier
For an N-Stage Network..

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

Tcomp0 = (LF-1) 2900 K
Tcomp0 = (LF-1) 2900 K
= (LF-1 + L -L) 2900 K = (L -1 + L(F-1) ) 2900 K

Tcomp0 = TL0 + L TR0
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System Effective Temperature
Manmade noises: Radiation from Automobile ignition and electrical machinery and Radio transmissions from other users that fall into the BW.

The system effective Temperature is F

TS0 = TA0 + TL0 + TR0 / (1/L) = TA0 + (L-1)290 + L(F-1) 290 TS0 = TA0 + (LF-1) 2900 K

TA0 is the antenna noise temperature Natural Sources including : Lightening, Celestial radio sources, Atmospheric sources, Thermal radiation from The ground and other structures. ELECTRICAL ELECTRONICS

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Example Problem on Lossy Line

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Example on NF & Noise Temp

and the overall Noise Figure of the system

Nout = G k TS0 W = 108 X 1.38 X 10 -23 X 2760 X 6 X 106 = 22.8 mw

TR0 = (F-1)2900 K = 26100 K
TS0 = TA0 + TL0 + LTR0 = 150 + 2610 = 2760 K
ELECTRICAL ELECTRONICS COMMUNICATION 29.1 – 16.4 = 12.7 dB INSTRUMENTATION
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Improving SNR - Benefit of using Pre Amplifier

Fig. 5.19a

SNRout = 16.4 dB

TR20 = (F2-1)2900 K

= 26100 K

Tcomp0 = TR10 + TR20 / G1 = 290 + 2610/20 = 420.5 0K
Fig. 5.19a

TS0 = TA0 + Tcomp0

= 150 + 420.5 0K = 570.5 0K = 2+ 9/20 = 2.5 (4dB)

SNRout = 23.3 dB

Fcomp = F1 + (F2-1)/ G1

Fig. 5.19b

TR10 = (F1-1)2900 K = 2900 K

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Problem

<

75 feet Lossy Cable 3dB/100 ft Loss (a) Fcomp

Receiver F = 13 dB

Fcomp =?

Pre-amp G = 20 dB F = 3 dB (c) Fcomp = F1 + (F2-1)/ G1 + (F2-1)/ G1G2 = 2 + (1.68-1) /100

(b) Fcomp = F1 + (F2-1)/ G1 + (F3-1)/ G1G2 = 1.68 + (2-1) X 1.68

= 3 X 0.75
= 2.25 dB

= F1 + (F2-1)/ G1

= 1.68 +
(20-1) X 1.68 = 33.6 = 15.26 dB

= 1.68 =F

+ (20-1) X 1.68 /100 = 3.68 = 5.65dB

+ (20-1) X 1.68 /100 = 2.32 = 3.66dB

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...ANALYSIS OF DATA Sessions 1-5 2 Basic Concepts-I • An experiment is an occurrence whose result, or outcome, is uncertain. • The set of all possible outcomes is called the sample space for the experiment. 3 Example 4 One Die Two Dice 5 Basic Concepts-II • Given a sample space S, an event E is a subset of S. • The outcomes in E are called the favourable outcomes. • We say that E occurs in a particular experiment if the outcome of that experiment is one of the elements of E. 6 Examples • When a fair die is rolled, probability of each outcome is 1/6. • So probability of E: Even numbers is P(E) = 3/6 = 0.5. • But a die does not need to be fair, and outcomes are not always equally likely! • For example, suppose we have a loaded die with probabilities of 1, 2, 3, 4, 5 and 6 resp. 0.25, 0.15, 0.15, 0.15, 0.15 and 0.15. • Then P(E) = 0.45. 7 Basic Concepts For any event A: i) 0 ≤ P(A) ≤ 1. ii) P(Φ) = 0, where Φ is the null event (no outcomes). iii) P(S) = 1, where S is the sample space (all outcomes). iv) If A and B are two events with no common outcome, (i.e. “mutually exclusive”) then P(A  B)  P(A)  P(B) v) If A and B are two mutually exclusive events such that P(A B)  1 then A and B are called complements of each other. We denote B as A or AC. 8 General Formula P(A  B)  P(A)  P( B)  P(A B) 9 Example • Consider the fair die being rolled twice • P(sum is even) = • P(product is......

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