05 Random Signal
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Transcript of 05 Random Signal
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Random Signal
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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.
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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).
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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 }
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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.
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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
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RV are of two typesDiscrete RVsand
Continuous RVs
Random Variable
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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.
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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
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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.
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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%
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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)
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Probability density function
PDF is more convenient way of describing a
continuous RV.
Probability density function PDF is defined by
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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
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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.
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Write sample space Define RV
Calculate CDF
Obtain probabilities
Plot CDF
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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
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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
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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
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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.
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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]
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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
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As the number of trials N approaches to , the
above equation can be written as
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Mean value of continuous random variable
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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.
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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
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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
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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)
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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
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Common PDFs.
Following are the commonly used PDFs. Binomial Distribution
Poisson Distribution.
Uniform Distribution Gaussian Distribution
Rayleighs Distribution
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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.
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The PDF for a uniform distribution is given
as
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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.
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Gaussian Distribution
Gaussian Distribution is also calledNormalDistribution.
It is defined for continuous random
variables. The PDF for a Gaussian random variable is
given as,
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Gaussian PDF.
This function defines the bell-shaped curve
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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
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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.
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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.
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The function Q (x)
The function Q (x) is closely related to errorand complementary error function