Random signals and Processes ref: F. G. Stremler, Introduction to

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Random signals and Processes

ref: F. G. Stremler, Introduction to Communication Systems 3/e

• Probability

• All possible outcomes (A

1

N 

P ( A i

)

1 i

1

• Joint probability

P ( A )

 lim

N

 

P ( AB

P ( AB )

P ( B |

N

A

N to A

N

) are included

)

 lim

N

 

A ) P ( A )

N

N

P (

AB

A | B ) P ( B )

• Conditional probability

P ( B | A )

N

AB

N

A

N

AB

N

N

A

/ N

P ( AB )

P ( A )

P ( A | B )

N

AB

N

B

N

AB

N

N

B

/ N

P ( AB )

P ( B )

Ya Bao Fundamentals of Communications theory 1

Examples

• Bayes’ theorem

P ( B | A )

P ( B ) P ( A | B )

P ( A )

• Random 2/52 playing cards. After looking at the first card, P(2 nd is heart)=? if 1 st is or isn’t heart

• Probability of two mutually exclusive events

P(A+B)=P(A)+P(B)

• If the events are not mutually exclusive

P(A+B)=P(A)+P(B)-P(AB)

Ya Bao Fundamentals of Communications theory 2

Random variables

• A real valued random variable is a real-value function defined on the events of the probability system.

• Cumulative distribution function (CDF) of x is

F ( a )

P (

• Properties of F(a) x

 a )

 n lim

 

( n x n

 a

)

• Nondecreasing,

• 0<=F(a)<=1,

F (



)

0

F (



)

1

Ya Bao Fundamentals of Communications theory 3

Probability density function (PDF)

f ( x )

 dF ( a )

| a

 x da

Properties of PDF f ( x )

0 .

 

  f ( x ) dx

F (

)

1

Ya Bao Fundamentals of Communications theory 4

Discrete and continuous distributions

• Discrete: random variable has M discrete values

CDF or F(a) was discontinuous as a increase

Digital communications

PDF f ( x )

 i

M 

1

P ( x i

)

( x

 x i

)

M is the number of discretely events

CDF F ( a )

 i

L 

1

P ( x i

)

L is the largest integer such that x

L

 a , L

M

Ya Bao Fundamentals of Communications theory 5

• Continuous distributions: if a random variable is allowed to take on any value in some interval.

CDF and PDF would be continuous functions.

Analogue communications, noise.

• Expected value of a discretely distributed random variable y

[ h ( x )]

 i

M 

1 h ( x i

) P ( x i

)

Normalized average power

P =

 i

y i

2

p(y i

)

Ya Bao Fundamentals of Communications theory 6

Important distributions

• Binomial

• Poisson

• Uniform

• Gaussian

• Sinusoidal

Ya Bao Fundamentals of Communications theory 7

Ya Bao Fundamentals of Communications theory 8

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