27 Dec 2018

Cumulative Distribution Function (CDF) - Properties of CDF - CDF Definition, Basics - Continuous and Discrete CDF

Let's first understand, what is Cumulative Distribution Function (CDF) and it's Definition-

What is Cumulative Distribution Function (CDF)?

Definition of CDF-

The Cumulative Distribution Function (CDF) of a random variable 'X' may be defined as the probability that the random variable 'X' takes a value 'Less than or equal to x'.
Mathematically it can be represented as-
CDF Formula, Cumulative distribution function formula
CDF Formula

 Cumulative Distribution Function (CDF) may be defined for-
#Continuous random variables and
#Discrete random variables

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Probability Density Function (PDF) - Definition, Basics and Properties of Probability Density Function (PDF) with Derivation and Proof

Watch the Complete Video Here-

Cumulative Distribution Function (CDF) other Names

#Probability distribution function of the random variable
#Distribution function of the random variable
#Cumulative probability distribution function

Properties of Cumulative Distribution Function (CDF)

Following image discusses the 3 Properties of Cumulative Distribution Function (CDF)
Properties of CDF (Cumulative Distribution Function Properties)

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Joint Probability Density Function (Joint PDF) - Properties of Joint PDF with Derivation- Relation Between Probability and Joint PDF

Cumulative Distribution Function (CDF) for discrete random variables

If 'X' is a discrete random variable, then it takes on values at discrete points.
Therefore CDF can be defined for this case as shown in the image below-
CDF for Discrete Random Variable Explanation 

So the CDF for a discrete random variable for the complete range of x can be defined as shown in the image below-
CDF for Discrete Random Variable, Cumulative distribution function
CDF for Discrete Random Variable

 => Cumulative Distribution Function (CDF) of a discrete variable at any certain event is equal to the summation of the probabilities of random variable upto that certain event. 
As x varies from -∞ to ∞ the graph of CDF i.e. Fx(x) resembles a staircase with upward steps having height P(X=xj) at each x=xj.
But note one thing that the graph of Fx(x) CDF remains constant between the two steps or events.

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Random Variables (Discrete and Continuous Random Variables), Sample space and Random Variables Examples

 Probability Density Function (PDF) - Definition, Basics and Properties of Probability Density Function (PDF) with Derivation and Proof



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26 Dec 2018

Probability Density Function (PDF) - Definition, Basics and Properties of Probability Density Function (PDF) with Derivation and Proof

What is Probability Density Function (PDF)?

The derivative of Cumulative Distribution Function (CDF) w.r.t. some dummy variable is called as probability density function (PDF).
Probability density function can be defined mathematically as-
Relation between PDF and CDF, Formula of PDF
Relation between PDF and CDF (Formula of PDF)

 Now we will discuss the properties of probability density function. The derivation of properties of PDF is also provided here.

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Watch the Complete Video Here-

Properties of Probability Density Function (PDF) with Derivation

Property 1- Probability density function is always non-zero for all values of x.
PDF Property with Proof, Probability density function property
PDF Property 1 with Proof

 Property 2- The area under the PDF curve is always equal to Unity i.e. one.
PDF Property 2 With Proof, probability density function properties
PDF Property 2 With Proof

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 Property 3- It is possible to get Cumulative Distribution Function (CDF) by integrating PDF.
PDF Property 3 With Proof, Probability density function properties
PDF Property 3 With Proof

 Property 4- Probability of the event {x1< X<= x2} is given by the area under the PDF curve in 
{x1< X<= x2} range.
PDF Property 4 With Proof, Probability Density Function Properties
PDF Property 4 With Proof

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Random Variables (Discrete and Continuous Random Variables), Sample space and Random Variables Examples

 Probability Density Function (PDF) - Definition, Basics and Properties of Probability Density Function (PDF) with Derivation and Proof



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Random Variables (Discrete and Continuous Random Variables), Sample space and Random Variables Examples

Before understanding the concept of random variables, you need to know what is a sample space in probability.

What is Sample Space in probability (S)

The range of all possible outcomes of an experiment is known as the 'Sample Space' (s).

Now let's understand the definition of Random Variables-

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Random Variable (Random Variable Definition)

A random variable is a function which can take on any value from the sample space and having range of some set of real numbers, is known as the random variable of the experiment.

Watch the Complete Video Here-

Types of Random Variables 

Random variables can be classified as-
#Discrete Random Variables and
#Continuous Random Variables

 Now we will understand the Discrete Random Variables with the help of an example-

Discrete Random Variables

These are the random variable which can take on only finite number of values in a finite observation interval. So we can say that to discrete random variable has distinct values that can be counted.
We Will understand this with the help of an example-

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Example of Discrete Random Variable

Let's take an example (experiment) of tossing 3 coins at the same time (simultaneously).
Now for this experiment the sample space is
S= {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
Here let's suppose that the number of tails is the random variable X
So X= {0    1    1   1   2   2   2  3}
       = {x1  x2  x3 x4 x5 x6 x7 x8}

 Now let's discuss other type of random variables i.e.  Continuous random variables.

Continuous Random Variables

A random variable that takes on an infinite number of values is known as a continuous random variable. Many physical systems (experiments) can produce infinite number of outputs in a finite time of observation. In such cases we use continuous random variables to define outputs of such systems.

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Examples of Continuous Random Variables

Example 1- A random variable that measures the time taken in completing a job, is continuous random variable, since there are infinite number of times (different times) to finish that job.

 Example 2 - Noise voltage that is generated by an electronic amplifier has a continuous amplitude. Therefore sample space (S) and random variable (X) both are continuous.
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 Random Variables (Discrete and Continuous Random Variables), Sample space and Random Variables Examples
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24 Dec 2018

Dispersion in Optical Fiber - Intramodal Dispersion (Chromatic Dispersion) and Intermodal Dispersion

Dispersion in optical fibres

Broadening of the transmitted light pulses take place as the light rays move along the optical fibre. This broadening of light pulses is known as dispersion.

Intersymbol Interference (ISI) in Optical Fibers

Let's understand the dispersion with help of diagram given below-
Dispersion in Optical Fiber, Intersymbol Interference in optical fiber, ISI in optical fiber
Dispersion in Optical Fiber (Intersymbol Interference ISI)

 You can see in this diagram that the light pulses that are sharp before transmission, gets broadened after travelling through the optical fibre. This Increase in width of the pulses makes it very difficult to distinguish them at the receiving end. Because of this light pulse broadening, these pulses overlap with its neighboring light pulses and it becomes hard to identify them as separate pulses at the receiving side. This effect is known as the Intersymbol Interference (ISI).
Now observe the same diagram carefully. Due to this dispersion effect (broadening of light pulses) the digital bit pattern 1011 at the input side is not indistinguishable at the output side as the same bit pattern. Because of this effect, '0' level is missing at the output side.

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Optical fiber communication (Complete)

Types of Dispersion in optical fibers

Dispersion in optical fibers can be categorized into two parts -
#Intramodal Dispersion (Chromatic dispersion) and
#Intermodal Dispersion (Modal or Mode dispersion)

Intramodal Dispersion (Chromatic Dispersion)

Intramodal dispersion may occur in all types of optical fibers. As we know that optical sources emit a band of frequencies so do not emit just a single frequency. Therefore different spectral components present in the optical source take different propagation delay while travelling through the optical fiber. This phenomena results in the broadening of each transmitted mode and is responsible for the intramodal dispersion. Intramodal dispersion is also popular by another name 'chromatic dispersion'.
Intramodal dispersion (chromatic dispersion) is found more in LED sources in comparison to LASER sources.
This delay difference may be caused by the dispersive properties of the material of the waveguide (material dispersion) and also guidance effects within the fibre structure (waveguide dispersion). 

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Material Dispersion

Pulse broadening because of material dispersion is caused due to different group velocities of various spectral components that are launched into the fibre from the optical source.
It occurs when the phase velocity of a plane wave that is propagating in the dielectric medium varies non-linearly with wavelength.

Waveguide Dispersion

Intramodal dispersion may also be caused due to waveguiding of the optical fibre. As the group velocity varies with change in wavelength for a particular mode, the waveguide dispersion takes place.
When the angle between the ray and the fibre axis varies with wavelength then it results in different transmission times for the rays which is responsible for dispersion.
Now we will discuss another kind of dispersion known as intermodal dispersion.

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Intermodal Dispersion (Modal or Mode Dispersion)

Intermodal dispersion is found in multimode optical fibres. Multimode fiber are the fibres that allow various modes to propagate through it. Therefore it is not observed in single mode fibers as only a single mode is allowed to propagate through the single mode fibre. But single mode fibres suffer from the intramodal dispersion (chromatic dispersion). 
The intermodal dispersion results due to propagation delay difference between various modes propagating through the optical fibre.
Intermodal Dispersion in Step Index Fibers, Intermodal Dispersion, Dispersion in Step Index Fibers
Intermodal Dispersion in Step Index Fibers  

Intermodal dispersion is found more in case of multimode step index fibres in comparison to graded index fibres. As in case of multimode step index fibres, the refractive index of the core is uniform. Because of this same refractive index throughout the core of the multimode step index fibre, different modes propagating through the core travel with same speed. Because of this same speed, different light rays launched into the optical fibre at different angles at the transmitting end takes different times to reach to the other end of the optical fibre as their propagation path (path length) changes with change in angle while launching.

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 You can observe this phenomena in the diagram shown above. Because of this, intermodal dispersion is found more in multimode step index fibres.
Propagation of Light Ray inside Graded Index Fibers, Total Internal Reflection in optical fiber
Propagation of Light Ray inside Graded Index Fibers (Total Internal Reflection)

On the other hand, graded index fibres offer less intermodal dispersion as the refractive index of the core is not uniform in it. Refractive Index is maximum at the core axis and decreases as we move away from the core axis. So the refractive index is maximum at the core axis in case of graded index fibers. The refractive index of the cladding is uniform.
But how does this non-uniform refractive index of the core in graded index fibres help in reducing intermodal dispersion?
To understand it, carefully observe the diagram shown below.
Intermodal Dispersion in Graded Index Fibers, Dispersion in optical Fibers
Intermodal Dispersion in Graded Index Fibers 

As the light rays travel slower in denser mediums (high refractive index) and we also know that refractive index in case of graded index fibers is maximum at the core axis and decreases as we move radially away from the core axis towards the core-cladding interface. 

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 We can transmit light rays at different angles into the optical fibre. So the light rays that travel near the core axis move slower in comparison to the light rays that travel near the core-cladding interface. 

 You can easily understand this, that the light rays that travel near the core axis have to cover smaller distance in comparison to the rays that are close to the core-cladding interface. This creates a compensating effect in dispersion of light rays.

This phenomena is responsible for lower dispersion (broadening of light pulses) in case of graded index fibres in comparison to step index fibres because the light rays travelling at different angles in graded index fiber reach at the receiving end, almost at the same time. Because the light rays that need to travel longer distances (moving close to core-cladding interface) propagate at high speed because of lower refractive index (rarer medium) near the core-cladding interface. This reduces broadening (dispersion) of the light pulses.

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Meridional and Skew Rays (Optical Fiber Communication)

There are two types of light rays on the basis of propagation inside the optical fiber- 
#Meridional Rays and
#Skew Rays (Helical Rays)

 Here we will discuss propagation of both types of rays-

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Meridional Rays

*Meridional rays enter into the optical fiber through it's axis. 
*These rays cross the fiber axis at each reflection.
Meridional Rays, Propagation Inside Optical Fiber
Meridional Rays Propagation Inside Optical Fiber

 The two diagrams given above show the propagation of meridional rays inside the optical fiber. The first diagram provides the ray path view along the fiber axis. It is clear from the diagram that the light ray is crossing the fiber axis at each reflection. These reflections are marked as 1, 2 and 3.

Meridional and Skew Rays Video

 

 Another diagram is also of meridional ray propagation but with a different view. It is ray path view along the plane normal to the fiber axis.

 Now let's discuss another kind of light ray propagation inside optical fibers i.e. Skew rays.

Skew Rays (Helical Rays)

*Skew rays are also known as helical rays as they move on helical path inside the optical fiber. 
*Skew rays do not cross the fibre axis and propagate around the optical fiber axis on zigzag path.
*Skew rays greatly outnumber the meridional rays. *Skew rays enter the optical fiber off the fiber axis.
It should be noted here that, in case of skew rays, the point of emergence from the fiber in air depends upon the number of reflections inside the optical fibre. It does not depend upon the input conditions to the fiber.

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This concept is made clear with the help of two diagrams given below. 
Skew Rays, Helical Rays, Propagation Inside Optical Fiber
Skew Rays (Helical Rays) Propagation Inside Optical Fiber

 The first diagram of skew ray shows the ray path view along the fiber axis and the second diagram shows the ray path view along the plane normal to the fiber axis.
It is clear from the second diagram that the skew ray (helical ray) is not crossing the optical fiber axis and propagating around the axis.

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AMPLITUDE MODULATION (TIME DOMAIN EQUATIONS AND WAVEFORMS)

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PULSE MODULATION TECHNIQUES (PAM, PWM, PPM, PCM)

OPTICAL FIBER: STRUCTURE AND WORKING PRINCIPLE

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COMPARISON OF PAM, PWM, PPM MODULATION TECHNIQUES

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PULSE POSITION MODULATION (PPM)

OPTICAL FIBERS IN COMMUNICATION: COVERS ALL IMPORTANT POINTS

OPTICAL FIBER SOURCES (DESIRABLE PROPERTIES)

AMPLITUDE MODULATION Vs FREQUENCY MODULATION (ADVANTAGES AND DISADVANTAGES)

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