Skip to content Skip to navigation

Connexions

You are here: Home » Content » Digital Modulation Basics

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
 

Digital Modulation Basics

Module by: Behnaam Aazhang. E-mail the author

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| X⁢ω≤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=∫−∞bf X xdx F X b x b f X x
(2)
and f X x f X x is the probability density function (pdr) (e.g., F X ⁢x F X x is differentiable and f X x=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
p X x k =PrX= x k = F X ⁢ x k −limit   x (x→ 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| (X⁢ω≤a)∧(Y⁢ω≤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∫−∞af X Y xyd x d y F X Y a b y b x a f X Y x y
(5)
(e.g., f X Y xy=∂2 F X Y ⁢xy∂ x ∂ y f X Y x y x y F X Y x y )

joint pmf if they are jointly discrete

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

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks