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Digital Modulation Basics

Module by: Behnaam Aazhang

Summary: blargh

The outcome of a random experiment is denoted by ωω. The sample space ΩΩ is the set of all possible outcomes of a random experiment.

Random Variables

Random variable is the assignment of a real number to each outcome of a random experiment.

Figure 1
Figure 1 (Figure2-1.png)

Example 1

Roll a dice. Outcomes ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 ω 1 ω 2 ω 3 ω 4 ω 5 ω 6

ω i =i ω i i dots on the face of the dice.

X ω i =i X ω i i

PrX=i=Prω∈Ω|X ω i =i X i X ω i i ω Ω

Distributions

Probability assignments on intervals Pra<X≤b a X b

Definition 1: Cumulative distribution
The cumulative distribution function of a random variable X is a function F X F X such that ℝ→ℝ → ℝ ℝ such that
F X b=PrX≤b=Prω∈Ω|Xω≤b F X b X b X ω b ω Ω (1)
Figure 2
Figure 2 (Figure2-2.png)
Definition 2: Continuous Random Variable
A random variable XX is continuous if
F X b=∫-∞bfXxdx F X b x b f X x (2)
and fXx f X x is the probability density function (pdr) (e.g., F X x F X x is differentiable and fXx=ddx F X x f X x x F X x )
Definition 3: Discrete Random Variable
A random variable XX is discrete if it only takes at most countably many points (i.e., F X F X is piecewise constant). The probability mass function (pmf) is defined as
pX x k =PrX= x k = F X x k -limx→ x k ∧x< x k F X x p X x k X x k F X x k x x x k x x k F X x (3)

Two random variables defined on an experiment have joint distribution

F X Y ab=PrX≤aY≤b=Prω∈Ω|Xω≤a∧Yω≤b F X Y a b X a Y b X ω a Y ω b ω Ω (4)

Figure 3
Figure 3 (Figure2-4.png)

Joint pdf can be obtained if they are jointly continuous.

F X Y ab=∫-∞b∫-∞afXYxydxdy F X Y a b y b x a f X Y x y (5)
(e.g., fXYxy=∂2∂x∂y F X Y xy f X Y x y x y F X Y x y )

joint pmf if they are jointly discrete

pXY x k y l =PrX= x k Y= y l p X Y x k y l X x k Y y l (6)

Conditional density function

f Y | X y | x =fXYxyfXx f Y | X y | x f X Y x y f X x (7)
for all xx and with fXx>0 f X x 0

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