random signals and processes ref: f. g. stremler, introduction to communication systems 3/e
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Random signals and Processes ref: F. G. Stremler, Introduction to Communication Systems 3/e. Probability All possible outcomes (A 1 to A N ) are included Joint probability Conditional probability. Examples. Bayes’ theorem - PowerPoint PPT PresentationTRANSCRIPT
Ya Bao Fundamentals of Communications theory 1
Random signals and Processesref: F. G. Stremler, Introduction to Communication Systems 3/e
• Probability
• All possible outcomes (A1 to AN) are included
• Joint probability
• Conditional probability
N
NAP A
N lim)(
1)(1
N
iiAP
N
NABP AB
N lim)(
)(
)(
/)|(
AP
ABP
NN
NN
N
NABP
A
AB
A
AB )(
)(
/)|(
BP
ABP
NN
NN
N
NBAP
B
AB
B
AB
)()|()()|()( BPBAPAPABPABP
Ya Bao Fundamentals of Communications theory 2
Examples• Bayes’ theorem
• 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)
)(
)|()()|(
AP
BAPBPABP
Ya Bao Fundamentals of Communications theory 3
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
• Properties of F(a)
• Nondecreasing,
• 0<=F(a)<=1,
)(lim)()(n
anaxPaF x
n
1)(
0)(
F
F
Ya Bao Fundamentals of Communications theory 4
Probability density function (PDF)
xada
adFxf |
)()(
Properties of PDF
.0)( xf
1)()(
Fdxxf
Ya Bao Fundamentals of Communications theory 5
Discrete and continuous distributions• Discrete: random variable has M discrete values
CDF or F(a) was discontinuous as a increase
Digital communications
CDF
events discretely ofnumber theisM
)()()(1
M
iii xxxPxf
MLax
xPaF
L
L
ii
, that suchinteger largest theis L
)()(1
Ya Bao Fundamentals of Communications theory 6
• 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
)()()]([1
i
M
ii xPxhxhy
Normalized average power P =
i
yi2 p(y
i)
Ya Bao Fundamentals of Communications theory 7
Important distributions• Binomial
• Poisson
• Uniform
• Gaussian
• Sinusoidal
Ya Bao Fundamentals of Communications theory 8