05 random signal

7/31/2019 05 Random Signal

1/40

Random Signal


2/40

Probability Theory

The probability theory is used in the analysis

of non-deterministic or random signals and

systems.

An experiment is called as random experiment

if the outcome of the experiment cannot be

predicted precisely.


3/40

Examples of random experiment are Tossing a coin

Rolling a die

Drawing a card from a deck.

A random experiment can have many different

"outcomes".

For example, a tossed coin has two possible outcomes

(H or T) or

A rolling die has six possible outcomes (1, 2, 3, .... 6).


4/40

sample space

The sample space S is defined as a" collectionof all the possible, separately identifiableoutcomes of a random experiment.

For example, the sample space for tossing a

coin will be,S = (H,T)

Similarly sample space for an experiment of

rolling of a die will be,S = { 1, 2, 3, 4, 5, 6 }


5/40

Probability

Let us assume that a specific desired event is A. Now repeat the experiment N times and record

the number of times the event A has occurredi.e. nA.

Relative frequency of occurrence = nA/N

As N then the ratio (nA / N) can be definedas the probability of occurrence of event A.


6/40

A random variable X is a process by which a(real) number x(s) is assigned to each possible

outcome of a statistical experiment

A random variable is neither random nor avariable.

Random variable


7/40

RV are of two typesDiscrete RVsand

Continuous RVs

Random Variable


8/40

Discrete Random Variables If s represents the outcome of the experiment then the RV is

represented by X(s) or simply X

RV X(s) is a function that maps the sample points into real

numbers x1, x2, x3.

If S contains a countable number of sample points, then X

will be a discrete RV having a countable number of distinctvalues.


9/40

Continuous Random Variables

A continuous RV may take on any valuewithin a certain range of the real line.

Continuous RV has an uncountablenumber

of possible values


10/40

Cumulative distribution function

The CDF of a RV is defined as the probabilitythat the RV X takes values less than or equal

to x.

Where {X x} denotes an event and x is areal number.


11/40

Properties of cdf

Since CDF represents probability so it must be

bounded between 0 and 1

With extreme values

It is an null event & probability is 0

It includes all possible outcomes or event so probability

is 100%


12/40

The CDF Fx(x) is a non-decreasing function of

x, i.e., if x1 < x2

Fx(x1) Fx(x2)

The complementary events X x and X >xencompass the entire real line, so

P(X> x) = 1 - Fx(x)


13/40

Probability density function

PDF is more convenient way of describing a

continuous RV.

Probability density function PDF is defined by


14/40

Properties of PDF

CDF can be derived from the PDF. It is non negative function for all values of x

The area under the PDF curve is always unity


15/40

Problem

A three digit message is transmitted overnoisy channel having a probability of error

P(e) = 2/5 per digit. Find out corresponding

CDF and plot it.


16/40

Write sample space Define RV

Calculate CDF

Obtain probabilities

Plot CDF


17/40

Sample space = {ccc, cce, cec, ecc, cee, ece, eec, eee}

{ccc, cce, cec, ecc, cee, ece, eec, eee}

RV X = { no error, one error, two error, three error}

x0 x1x2 x3

RV X = Number of errors


18/40

Fx(x)= P (x = x0) + P (x =x1) + P(x = x2) + P (x =x3) for x0 x x3

Fx(x0)= P (x x0) = P (x < x0) + P (x = x0)= 0+ 27/125 = 27/125

Fx(x1)= P (x x1) = P (x < x0) + P (x = x0) + P (x =x1)= 0+ 27/125 + 54/125 = 81/125

Fx(x2)= P (x x2) = P (x < x0) + P (x = x0) + P (x =x1) + P(x = x2)= 0+ 27/125 + 54/125 + 36/125 = 117/125

Fx(x3)= P (x x3) = P (x < x0)+ P (x = x0)+ P(x =x1)+ P(x = x2)+ P(x =x3)= 0+ 27/125 + 54/125 + 36/125 + 8/125 = 125/125 =1


19/40

0 1 2 3

P(X= xi )

x

FX(x))

x0 1 2 3

27/125

54/125

8/125

36/125

Probability and CDF

27/125

81/125

117/125

1


20/40

Statistical Averages of Random Variables

The Probability Density Function (PDF) providesmore information about the random variable.

But the interpretation of this information is littlecomplex.

There are other numbers which provide moreconvenient and useful information about therandom variable quickly.

These characteristic numbers are combinely called

statistical averages.


21/40

Mean/Average or Expected Value

The mean of the random variable is given bysummation of the values of X weighted by

their probabilities.

Mean value is denoted by mx and is alsocalled expected value of X.

mx = E [X]


22/40

Consider a discrete random variable X whichhas a possible values of x1, x2, with the

probabilities P(x1), P (x2) ,

Mean value of a discrete random variable


23/40

As the number of trials N approaches to , the

above equation can be written as


24/40

Mean value of continuous random variable


25/40

Moments and Variance

Thenthmoment of a random variable X is defined

as the mean value ofXn.

g (x) =Xn, then

Thus first moment of random variable X is same as

its mean value.


26/40

When n = 2

is called mean square value of random

variable X.

The central moments are the moments of the

difference between random variables X and its

mean mx.

Thus the nth central moment is defined as


27/40

Variance of random variable

The second central moment ; i.e. n = 2 is calledvariance of random variable X.

Thus variance gives an indication about randomness

of the random variable.

Variance is also denoted by 2


28/40


29/40

Standard deviation

The square root of variance is called standarddeviation of random variable X.

Standard deviation provides the measure of

spread observed over the values of X relative

to mean value

Standard deviation = (variance)


30/40

Probability Models

We know that probability density functionprovides very useful information about the

occurrence of random variable X.

It is just impossible to study all the type ofprobability distribution functions


31/40

Common PDFs.

Following are the commonly used PDFs. Binomial Distribution

Poisson Distribution.

Uniform Distribution Gaussian Distribution

Rayleighs Distribution


32/40

Uniform Distribution

If the continuous random variable X isequally likely to be observed in a finite range

and is likely to have a zero value outside this

finite range then the random variable is saidto have a uniform distribution.


33/40

The PDF for a uniform distribution is given

as


34/40

The value of PDF is same for all possible

value of a random variable. Therefore this distribution is called Uniform

Distribution.

The uniform distribution is useful in describingthe quantization noise.


35/40

Gaussian Distribution

Gaussian Distribution is also calledNormalDistribution.

It is defined for continuous random

variables. The PDF for a Gaussian random variable is

given as,


36/40

Gaussian PDF.

This function defines the bell-shaped curve


37/40

Properties of Gaussian PDF

The peak value occurs atx =m (i.e. mean

value).

The plot of Gaussian PDF has even

symmetry around mean value


38/40

Error function

The error function are some integral which can not besolved directly, that can be solved by numerical methods.

The error function is defined as

0 erf(x) 1

As x approaches to then erf (x) tends to unity i.e.


39/40

Complementary error function

The complementary error function is defined as

The value of erf (x) at some fixed values of x areavailable in the form of table.

This table is called as error function table.


40/40

The function Q (x)

The function Q (x) is closely related to errorand complementary error function

05 random signal

Documents