**"Electricity is an imaginary number which has magintude and direction."**

# Telecommunication

Power Engineering | Telecommunication | Control system Engineering | Electronics | Fields and waves | Differential equationTelecommunication comes from combination of "Tele" and "communicate" which means distant and "message" respectively. Telecommunication is a sub field of Electrical and Electronics engineering. The internal picture is quite lengthy to describe. But the important features are few and can be easily explained. Firstly communication can be both wireless and wired system. Both of the communication system can also be analog or digital. I hope that if you are reading this you already know what analog and digital system means. In short , analog system relies on quantities that vary continuously over a range. On the other hand digital system utilizes discrete quantities, which are by definition countable. All the modern systems are discrete system. They are more efficient and easier to manipulate. Analog machines are bigger in size whereas discrete system needs little space as it always deals with finite number of quantities. So it is no surprise that modern day telecommunication utilizes discrete system in them.

# Communication process and modulation

Lets try to understand the basic of communication process. There is the concept of determinism which tells a system's future state is determined at a given time when an initial condition is known. In a deterministic system the same input always creates the same output. The weird thing about the communicating system is that the information is in-deterministic . The information is in-deterministic because if otherwise, there would not need to send the them. However it is better to leave the complicity and get to know more about the fundamentals system the same input always creates the same output. The weird thing about the communicating system is that the information is in-deterministic . Th fact that information is inotherwise there would not need to send the them. However it is better to leave the complicity and get to know more about the fundamentals of the communication system.

Fig: Telecommunication

The information is converted into some form of electrical signal. Then the modulator modulates the signal to make it suitable for transmission over a channel. The channel may be a cable or free space. At the receiver
end the receiver catches the signal and demodulates it to make it to be the original form. Analog modulation is used in radio and television system. Modulation is a technique by which original signal is modified by another
signal called carrier signal. Basically it is a multiplication of two signals. In AM(Amplitude modulation) the carrier signal's amplitude is varied in accordance with message signal information.

In FM , carrier signal's
frequency is varied in accordance with message signal's information.

Phase modulation is another modulation scheme where phase or angle of carrier signal is modified by the message signal. Frequency and phase
modulation is quite similar in application. There are explanations for every part as shown in the diagram. In digital modulation , the Nyquest-Shannon sampling theorem is very important concept. The theorem says that
in order to recover the original message signal from modulated signal , the sampling rate should be at least twice the largest frequency that is present in the message signal. The sampling is the process to convert an analog signal into
discrete time signal. The process is implemented by analog-to-digital converter.

Next step is the mapping of digital bits into symbols. This is done by using code words which serves as the symbol to be transmitted. Every modulation scheme has unique number of bits to represent the symbols. In general if a digital
modulation uses 6 bits [64- QAM(Quadrature Amplitude Modulation)] to encode the message signal , there will be 64(2 ^6) unique code words in total. Now we are able to tackle the relation between baud rate and bit rate.
Baud rate is the total number of symbols that are transmitted per second whereas the bit rate is the total number of bits per second. So if we multiply the baud rate by the number
of bits per symbol we will get bit rate. The relationship is Bit rate = baud rate X N (bits per symbol).
Before transmitting the signal channel encoding of message signal becomes necessary. The purpose of the encoding is to add some redundant bits to the digitally converted signal. This transform the input signal into encoded signal which is decoded at the receiving end. Its may seem
somewhat unusual that redundant bits (1,0 ) are added to the signal sequence but the redundant bits are necessary to detect error that occurred in the signal sequence during traveling through the channel. One method of channel encoding
is related to
Parity check. Some mathematical analysis can be found
here. As I mentioned earlier that information of communication system are random in nature. Also as there are
noise and interference in ambient space, the original signal gets corrupted to some extent. As a result we need to model a telecommunication system in such a way that the it becomes as more efficient as it can be.
That forces us to incorporate random process or random variable to the communication system. Before going into details of random variables , we investigate some of the fundamental of Fourier analysis.

# Fourier Analysis

Sine wave is considered to be most fundamental signal in the Fourier analysis. In fact it is the most fundamental signal in the universe in the same way the Maxwell's equation is considered the most fundamental laws of physics.
Every sine wave has unique frequency which can not be decomposed into other frequencies. Fourier analysis is basically based on this concept of sine and cosine wave's fundamental nature. Joseph Fourier came up with the idea of Fourier analysis when he saw that the trigonometric sine and cosine
functions greatly simplify the analysis of heat equation. Heat is a kind of transport phenomena, which can be modeled with partial differential equation. He also noticed that any continuous periodic function can be represented as summation
of infinite number of cosine and sine functions. We can , for example, think of a sound of guitar's chord. This cord is composed of a number of unique frequencies or tones. If we can somehow plot the graph of
the sound of the cord in time domain , we can decompose the sound into the separate frequency components by applying Fourier transform. Fourier transform convert a time
dmoain signal into frequency domain signal.

The sine or cosine waves have the frequencies that are multiple of the original signal's(function) frequency. These frequencies are called harmonics. The applications of the
Fourier analysis in modern science is unimaginable. Slightly modified form of Fourier series is Fourier transform which is an important method for analyzing telecommunication process. Briefly, Fourier transform converts a time
domain signal into frequency domain signal. Fourier transformed function gives us a continuous graph when it is plotted as function of frequency. This helps designing an efficient telecommunication system. Fourier transform method proves or vindicates
famous Heisenberg's uncertainty principle. The time limited signal
tends to spread out in frequency domain and localized frequency domain signal tends to spread out in time domain. There is a trade-off between these pair of signals. According to Heigenberg uncertainty principle
position and momentum have similar characteristics that was proven by Heisenberg. We can never measure position and momentum of a particle to an arbitrary precision simultaneously. It is an absolute limitation imposed by
the nature of quantum world. It has nothing to do with the imprecision of measuring instruments, which some use to falsify uncertainty principle.

Although Fourier series were initially thought to be applicable to all continuous periodic functions , they are application to functions that are continuous all along except at finite number of discontinuities. An example
of such function is square wave. Fourier series method can even be extended to a-periodic signal as well. We can let the time period go to infinity to make any a-periodic signal to be a periodic one.

In the first figure basic equation of Fourier series is given. The function f(x) on the left hand side is equal to the right hand side of infinite summation of sine and cosine functions , each of which has
a coefficient. The coefficients can be determined by the given function and its period (L). Thus we can decompose any function into the original form as given in the first figure. In the
second figure sawtooth wave is approximated from individual sine waves. In the third figure a triangular wave is approximated and in the fourth a square wave is approximated by
Fourier series. There are two common phenomena involved in every approximation. One is that at the point of discontinuity where the function suddenly changes its value to larger
extent , the series converges to the the average of the two values of the functions when the variable falls in two different domain . For example the square wave has a discontinuity at point 1.0 where the average
of two values(1, -1) is zero. So the Fourier series converges to zero at this point. The other phenomena is the overshoot in the value around the discontinuity. Near the point of discontinuity
the Fourier series never converges. It always exceeds the value of the original function( square wave) even when infinite number of terms are added. When we plot the amplitude of
various terms of cosine wave in the y-axis taking frequency component in the x-axis we get a discontinuous graph. The continuous version of this graph is Fourier transform which is
applicable to a-periodic signal.

Fourier series can be interpreted in another way. Fourier series forms an infinite dimensional function space with the sine or cosine functions as basis set of that function space.
The functions in this interpretation can be thought as points in the infinite dimensional space. Space, speaking mathematically, is a set with additional structure defined on it. Some examples
of space are vector space, metric space, topological space, hilbert space and many others. Manifold is also a kind of space which is locally euclidean. The function in function space
must not be treated as usual vector of physics. This type of interpretation of functions as points in function space is
useful in quantum mechanics when superposition of quantum states is defined. This is much weirder concept as much as quantum mechanics is. Famous physicist Richard Feynman
quoted "I can safely say that nobody understand quantum mechanics" although he invented the path integral formulation of quantum mechanics.

# Antenna Theory

Antenna is very vital to communication system as it is to the television and radio technology. Antenna is essential for both sending and receiving electromagnetic radiation. Antenna is nothing but a straight or curved conductor through which
the electrical signal is applied. The electric signal that is applied is a type of modulated signal. So it is actually a voltage or current wave form. When it is applied to the antenna at the transmission end, the electrons in the conductor of the antenna begin to
flow. They become accelerated. We know that an accelerated electric charge emits electromagnetic wave. So the conductor acts like a emitting source of electromagnetic field, strength of which is proportional to the voltage of the signal.
At the receiving end, another antenna is exposed to the transmitted electromagnetic field. The antenna conductor now does the opposite of what the antenna at the transmission end does. The electrons in the conductor is displaced by the incoming
electromagnetic field and begin to flow through the circuit connected to the antenna. We know that any charge that is placed in a electric field experiences a force. This force acts on the electrons of the conductor to make it flow through it.
This is the basic idea of antenna theory. The mathematical analysis is somewhat complicated. There are various parameters and conditions to be taken in consideration to design the antenna.

The radiation pattern of antenna can be plotted using radiation intensity which is a function of two angle θ and φ which are spherical coordinates.In some directions the intensity of radiation is more than that in other directions. It depends on the strength of the electric and magnetic field that the anttenna creates. The electric field has special mathematical form for particular type of antenna. The polarization of the far field can be graphed using the mathematical formula. Polarization is the geometric orientation of the oscillation which forms transverse wave. Normal electromagnetic wave is linearly polarized. The direction of oscillation is usually confined to one direction in a plane for linear polarization.

There are various types of antennas like omni-directional , bi-directional and parabolic antenna. Now in telecommunication system MIMO(multiple input-multiple output) antenna arrays are used.
In MIMO system a number of antennas are grouped together to increase efficiency. To effectively transmit signal across large area sometimes repeaters are used in between the transmission and the receiving station.
The repeaters amplify the signal and re transmit it to the destination. In mobile or cellular communication antennas are very essential component. Even in satellite communication antennas are indispensable components.
In RADAR, antenna is also used to detect reflected signal. Maxwell's equations are the driving strategy behind all antenna theory.

In radio astronomy the uses of antenna is necessary. The antennas are used in radio astronomy to detect various radio signal that may be coming from deep space outside our solar system.
The existence of alien civilization is a mystery. There are billions of galaxies having countless number of stars and solar systems like ours in the universe. Scientist are optimist that there are
alien life outside our galaxy or even in our own galaxy. The odds are high. Radio telescopes are the only means to detect any signal that the aliens may be sending towards us.

# Random variable and Random process

Random variable is the value of a variable, that is determined from a random experiment. Random variable usually denoted by X, are mapping from elements of a set to real numbers (R); Suppose a dice is rolled then
the outcome of the event is a set containing 6 elements each with probability 1/6. We can say X is the random variable corresponding to each element in the set consisting of all those elements. In this particular case X(1)=X(2)=X(3)= X(4)=X(5)=X(6)=1/6 ;
The sample space is the set of all outcomes of a random event. When we associate probability to each outcome or a group of outcomes F, we get a probability space(X, F, P()) of that random event. And Random process is a realization of a
random variable at a particular instant of time. The signals generated at the transmission station are the example of a random process. Random process are function of random variable. Random processes are also known as stochastic processes. Continuous random variables have
probability density functions associated with them , which, if integrated, gives the likelihood that the value of the random variable will fall in some interval. We know that the integration finds the area under a curve. So the probability
of finding a specific value of random variable is zero as the area of a single line is zero.

Probability mass function are used in case of discrete random variables.
Probability density function is the differential or derivative of some other function which describes cumulative probability of random variable . This means we must integrate probability density function over an interval in order to find the probability that the value of a random variable falls into that.

** Cumulative probability function**

The blue curve is the probability density function whereas the red curve in cumulative probability function. The properties of the cumulative probability function is that we can take the difference of two values of the function at two distinct values to discard the probability of the cumulative probability of the values of random variable
up to the lower limit. Thus difference is the
likelihood that any value of random variable falls in a interval, which is given as a value of definite integral between two limits. In
this way we can model
telecommunication system as they are inherently random in nature. There are many standard types of probability distribution like Gaussian distribution , Pearson distribution
, each of which has application in modeling different phenomena. Many systems are modeled by normal distribution which is a bell shaped curved extending to infinity to both sides. Noise in telecommunication
system follows a normal distribution curve. So what is normal distribution and what is so special about it? I already said how the distribution look when plotted in graph.The best way to explain it is through by explaing
the central limit theorem. We talk about samples when dealing with statistics. If we consider a sample or collection of things, the property of something may be more frequent than other. But there is a relationship
between sample size and their mean value (i.e property). When we plot the frequency as a function of mean of the samples , we get a normal distribution curve. The exact expression is that if we continue to increase the sample size to infinity ,the mean value follows a curve known as normal distribution.
It has wide range of applicationa in both engineering and physics, especially in telecommunication engineering.

** Normal distribution**

The normal distribution , as seen in the figure, is symmetric about the mean (u). It is a function of a variable x whose value can be anything between negative infinity to positive infinity. It is also
a function of standard deviation sigma() which tells how data is distributed around the mean. There is a statistical formula for sigma() too. If the value of sigma increases the curve get squeezed vertically more.
The normal distribution is widely used in both physics and engineering disciplines to characterize random phenomena.

is said to be inside 1 standard deviation ( σ = 1 and μ = 0) around the mean (0=average).

### Shannon entropy formula

Entropy is such an idea which applies almost everywhere. A black hole has an entropy. Similarly electrical system also has entropy. In telecommunication system entropy is defined in terms of information or digital bytes.### Correlation functions

One of most important concepts in science and engineering is the correlation function which is statistical correlation between two random variables, contingent on the spatial and temporal distances between these two variables.## Deriving Fourier transform

Fourier transform is quite similar to Fourier series except that it has a continuous frequency spectrum. Before going into details , a few things to know about periodic functions is necessary. A periodic function is a function whose value repeats after some interval in the domain over which it is defined. A periodic function can be periodic of one periods or doubly periodic ( periodic of two period). F(x) = F(x+a ) = F(a + b) which has two periods a and b.A periodic function of real variable x can be defined on a circle whose circumstances represents the period of the function and x measures the distance around the circle. This special property can be more analysed using the argument of Laurent series and Fourier series.

Rather than simply going oV in a straight line, these distances now wrap around the circle, so that the periodicity is automatically taken into account. For convenience (at least for the time being), I take this circle to be the unit circle in the complex plane, whose circumference is 2p, and I take the period l to be 2p. Accordingly ω = 1, so z = e^(ix) {exponential)

We think of f (w) = F(z) as deWned on the unit circle in the z-plane, with z = e^(ix), and then the Fourier decomposition is just the Laurent series description of this function, in terms of a complex variable z.

Fourier decomposition of a periodic function f (x), of period l, as given above:

(the angular frequency o being given by ω = 2π/l). Let us take the period to be initially 2π, so ω = 1. Now we are going to try to increase the period by some large integer factor N (whence l = 2πN), so the frequency is reduced by the same factor (i.e. w = N^(-1)). A pure tone that used to be an nth harmonic would now be an (nN)th harmonic. It is r/N that keeps track of this frequency, and we need a new variable to label this. For Wnite N, I write p = r/N

In the limit as N -> &infinity; , the parameter p becomes a continuous variable and, since the â€˜coefficients arâ€™ in our sum will then depend on the continuous real-valued parameter p rather that on the discrete integer-valued parameter r, it is better to write the dependence of the coefficients ar on r by using the standard type of functional notation, say g(p), rather than just using a suffix (e.g. gp), as in ar. Effectively, we shall make the replacement α(r) -> g(p)

in our summation ∑α(r)z^r, but we must bear in mind that, as N gets larger, the number of actual terms lying within some small range of p-values gets larger (basically in proportion to N, because we are considering fractions n/N that lie in that range). Accordingly, the quantity g(p) is really a measure of density, and it must be accompanied by the differential quantity dp in the limit as the summation P becomes an integral ∫ . Finally, consider the term z^r in our sum ∑α(r)z^r. We have z = e^(ix), with x = N^(-1); so z = e^(ix/N). Thus z^r = e^(ixr/N) = e^(ixp); so putting these things together, in the limit as N tends to infinity , we get the expression

Including a scaling factor we end up with this final equation of Fourier transform:

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