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

Random signals and Processes
ref: F. G. Stremler, Introduction to Communication Systems 3/e
NA
• Probability
P( A)  lim
N  N
• All possible outcomes (A1 to AN) are included
N
 P( A )  1
i 1
i
N AB
• Joint probability
P( AB)  lim
N  N
P( AB)  P( B | A) P( A)  P( A | B ) P( B )
• Conditional probability
N AB N AB N P( AB)
P( B | A) 


NA
NA / N
P( A)
Ya Bao
N AB N AB N P( AB)
P( A | B ) 


NB
NB / N
P( B )
Fundamentals of Communications theory
1
Examples
• Bayes’ theorem
P( B ) P( A | B )
P( B | A) 
P( A)
• Random 2/52 playing cards. After looking at the
first card, P(2nd is heart)=? if 1st 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
nx  a
F ( a )  P( x  a )  lim (
)
n 
n
• Properties of F(a)
• Nondecreasing,
• 0<=F(a)<=1,
F ( )  0
F ( )  1
Ya Bao
Fundamentals of Communications theory
3
Probability density function (PDF)
dF ( a )
f ( x) 
|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
M
PDF f ( x )  P( x ) ( x  x )

i 1
i
i
M is the number of discretely events
L
CDF
F ( a )   P( xi )
i 1
L is the largest integer such that xL  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
M
y  [h( x )]   h( xi ) P( xi )
i 1
Normalized average power
P=

y2i p(yi)
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