TSKS01 Digital Communication

Lecture 2

Repetition of Probability Theory & Introduction to Stochastic Processes

Mikael Olofsson

Department of EE (ISY)

Div. of Communication Systems

2010-09-07 TSKS01 Digital Communication - Lecture 2 2

A One-way Telecommunication System

Channel

Source encoder

Source decoder

Source

Destination

Channel encoder

Modulator

Channel decoder

De-modulator

Source

coding

Channel

coding

Packing

Unpacking

Error control

Error correction

Digital to analog

Analog to digital

Medium

Digital

modulation

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Probabilities and Distributions

Probability: Pr{A} [0,1]

Joint prob.: Pr{A,B}

Cond. Prob.: Pr{ A|B } =

Prob. distr.: FX(x)=Pr{ !"} [0,1]

Prob. density.: fX(x) = FX(x)

Properties: FX(x) is non-decreasing

fX(x !"!#!!!for all x

!-" fX(x) dx = 1

Pr{x1# !"2} = !x1+ fX(x) dx

Pr{A,B} Pr{B}

d dx

"

x2+

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Example Game based on tossing two coins:

2 heads +400

2 tails –100

1 tail, one head –200

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Expectations

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Example cont’d

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Gaussian Distributions, N(m, )

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Example of the Q Function

Q(1.96) ! 2.4998 ·10-2

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Example of the Q Function cont’d

Q(x) = 10-6

x ! 4.75

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Other Common Distributions

Exponential distribution:

Binary distribution:

Uniform distribution:

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Two-Dimensional Stochastic Variables

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Dependencies

Definition: X & Y are independent if FX,Y(x,y) = FX(x)FY(y) holds.

Theorem: Independent " fX,Y(x,y) = fX(x) fY(y) holds.

Definition: Covariance: Cov{X,Y} = E{(X – mX )(Y – mY )}

Theorem: Cov{X,Y} = E{XY} – mXmY

Definition: X & Y are uncorrelated if Cov{X,Y} = 0 holds.

Theorem: Independent uncorrelated.

Note: Var{X} = Cov{X,X}

Theorem: Uncorrelated " E{XY} = E{X}E{Y}

" Var{X+Y} = Var{X} + Var{Y}

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Bayes’ Rule

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Multi-Dimensional Stochastic Variables

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Jointly Gaussian Variables

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Stochastic Process

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Examples of Stochastic Processes

Ex 1: Finite number of realizations:

X(t) = sin(t+ ), ! { 0, "/2, ", 3"/2 }

Ex 2: Infinite number of realizations:

X(t) = A·sin(t), A # N(0,1)

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Examples of Stochastic Processes cont’d

$cos("t), |t|<1/2 %&0, elsewhere

Ex 2: Infinite number of realizations:

X(t) = Ak·g(t – k), g(t) =

{Ak} independent, N(0,1)

A realization:

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Distributions and Densities

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Examples of Distributions and Densities

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