pr a tsks01 digital communication lecture 2 · 2011-09-12 · 2010-09-07 tsks01 digital...
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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
2010-09-07 TSKS01 Digital Communication - Lecture 2 3
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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