March 2023 - ece4uplp

Probability of error of Matched Filter

Probability of error of Matched Filter:-

we know that $P_{e}$ of an optimum filter is $P_{e} = \frac{1}{2}\ erfc(\frac{x_{o1}(t)-x_{o2}(t)}{2\sqrt{2}\sigma })$

Let us consider

now chose the ratio $\rho _{max}^{2} =(\frac{x_{o1}(t)-x_{o2}(t)}{\sigma })^{2}$

from the Matched filter concept $(\frac{x_{o1}(t)-x_{o2}(t)}{\sigma })^{2}=\frac{2}{N_{o}}\int_{-\infty }^{\infty }\left | X(f) \right |^{2}df-----EQN(1)$

from Parsevel’s relation $\int_{-\infty }^{\infty }\left | X(f) \right |^{2}df =\int_{-\infty }^{\infty }\left | x(t) \right |^{2}dt----EQN(2)$

from equations (1) and (2)

$\rho _{max}^{2}=\int_{-\infty }^{\infty }\left | x(t) \right |^{2}dt$

Here x(t) is the signal $x(t) =x_{1}(t)-x_{2}(t)$ .

$x(t) =\sqrt{2P_{s}}\cos 2\pi f_{c}t$ .

over a duration of $[0, T_{b}]$ the symbols are transmitted

$\int_{0}^{T_{b}}\left | x(t) \right |^{2}dt=2P_{s}\int_{0}^{T_{b}}\cos^{2} 2\pi f_{c}t \ dt$ .

$=P_{s}T_{b}$ .

from equation(1) $\rho _{max}^{2} = \frac{2}{N_{o}} P_{s}T_{b}$ .

$\rho _{max} = \sqrt{\frac{2P_{s}T_{b}}{N_{o}}}$ .

$\therefore P_{e} = \frac{1}{2} \ erfc(\sqrt{\frac{2P_{s}T_{b}}{N_{o}}}).\frac{1}{2\sqrt{2}})$

$\therefore P_{e} = \frac{1}{2} \ erfc(\sqrt{\frac{P_{s}T_{b}}{4N_{o}}})$ .

we know that carrier signal power is $P_{s} = \frac{A^{2}}{2}$ .

$A=\sqrt{2P_{s}}$ .

$\therefore P_{e} = \frac{1}{2} \ erfc(\sqrt{\frac{A^{2}T_{b}}{8N_{o}}})$ .

$\therefore P_{e}= \frac{1}{2} \ erfc(\sqrt{\frac{E_{b}}{4N_{o}}})$ . since $A^{2}T_{b} = P_{s}T_{b} =E_{b}$ .

probability of error of coherent ASK is $P_{e}= \frac{1}{2} \ erfc(\sqrt{\frac{E_{b}}{4N_{o}}})$ (or) in terms of Q function as $P_{e}= Q(\sqrt{\frac{E_{b}}{2N_{o}}})$ .

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Optimum filter

The function of a receiver in a binary Communication system is to distinguish between two transmitted signals $x_{1}(t)\ and \ x_{2}(t)$ (or) ( $s_{1}(t)\ and \ s_{2}(t)$ ) in the presence of noise.

The performance of Receiver is usually measured in terms of the probability of error P_e an the receiver is said to be optimum if it yields the minimum probability of error.

i.e, optimum receiver is the one with minimum probability of error P_e .

optimum receiver takes the form of Matched filter when the noise at the receiver input is white noise.

optimum receiver (or) optimum filter:-

The block diagram of optimum receiver is as shown in the figure below

the decision boundary is set to $\frac{x_{o1}(T)+x_{o2}(T)}{2}$ .

Probability of error of optimum filter:-

The probability of error can be obtained as similar to Integrate and dump receiver. Here we will consider noise as Gaussian Noise.

The output of optimum filter is $y(t) = x_{o1}(t)+n_{o}(t)$ .

The output of sampler is $y(T) = \left\{\begin{matrix} x_{o1}(T)+n_{o}(T) \ for \ binary \ i/p \ '1'\\ x_{o2}(T)+n_{o}(T) \ for \ binary \ i/p \ '0' \end{matrix}\right.$

suppose if Binary ‘1’ is transmitted then the input is $x(t) = x_{1}(t)$ , to find the probability of error this transmitted ‘1’ should be received as ‘0’.

this is possible when the condition $\left | y(T) \right | <\frac{x_{o1}(T)+x_{o2}(T)}{2}$ is true.

1 will be received as 0 $\Rightarrow x_{o1}(T)+n_{o}(T) <\frac{x_{o1}(T)+x_{o2}(T)}{2}$ .

$n_{o}(T) <\frac{x_{o2}(T)-x_{o1}(T)}{2}$ .

similarly a Binary ‘0’ will be received as ‘1’ if and only if

$\Rightarrow x_{o2}(T)+n_{o}(T) >\frac{x_{o1}(T)+x_{o2}(T)}{2}$ .

$n_{o}(T) >\frac{x_{o1}(T)-x_{o2}(T)}{2}$ .

the conditions are summarized in the table

Noe the Probability Distribution Function of Gaussian noise with zero mean and standard deviation $\sigma$ is given by

$f(n_{o}(T)) = \frac{1}{\sigma \sqrt{2\pi }} e^{-\frac{n_{o}^{2}(T)}{2}}$ .

Probability of error= probability ‘1’ will be received as ‘0’ =probability ‘0’ will be received as ‘1’.

$\therefore P_{e} =$ area under the curve $n_{o}(T) >\frac{x_{o1}(T)-x_{o2}(T)}{2}$ (or) area under the curve $n_{o}(T) <\frac{x_{o2}(T)-x_{o1}(T)}{2}$ .

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ATM Reference model

ATM has its own reference model. It consists of 3 layers physical, ATM, ATM adaptation layer

physical layer:-

Physical layer deals with the physical medium like voltages, bit timing etc.
ATM has been designed to be independent of transmission medium.

ATM layer:-

ATM layer deals with cells and cell transport and defines the layout of the cell.
It deals with establishment and release of virtual circuits.
congestion control is done here.

ATM Adaptation Layer:-

It is defined to allow users to send packets larger than cell. It segments the packets into cells and reassembles them at the other end.

ATM model is 3D so it has planes.

user plane:- It deals with data transport, flow control error correction and for other issues.

control plane:- It is for connection management.

layer management:- It relates to resource management.

plane management:- It is for inter layer co-ordination.

The Physical layer has been divided into two sublayers.

Physical Medium Dependent:- It interfaces with actual cable. It moves bits and handles bit timing.

Transmission Convergence:- It converts from ATM layer into bit streams and at the other side , it has difficult job converting streams into cells.

ATM Adaptation layer has been further subdivided into two layers

Convergence layer:- It breaks coming from upper layers into cells and given ATM layer.

Segmentation And Reassembly:- at the other end the cells reassembled into segments.

Upper layer:- ATM offers different kinds of services to different Applications through this upper layers.

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Basic block diagram of analog communication system

Introduction

Communications refers to sending, receiving and processing of information by electrical means, that is it means exchanging information between transmitter and receiver.

In early 1840’s the type of communication used was Wire telegraphy later on the forms are as telephony, Radio communication (possible with the invention of triode tube, Satellite communications and fibre optics(with the invention of transistors and IC’s and semi-conductor devices), that means communications become more advanced with increasing emphasis on computer and other data communications.

A modern communication system is concerned with

before transmission

sorting:- sorting for the right message.
Processing:- processing is to make that message more suitable for transmission.
storing:- storing that message before transmission.

then the actual transmission of that message takes place (processing and filtering noise)

at the receiver

decoding:-decoding the original message.
storage:-storing a copy of that message.
interpretation:-and analyzing for the correctness of that message.

the different forms of modern communication systems includes Mobile communications,Computer communications, Radio telemetry etc.

to become familiar with communication systems one needs to know about amplifiers and oscillators that means fundamentals of electronic circuits must be known, with these concepts as a background the every day communication concepts like noise, modulation and information theory as well as various types of systems may be studied.

The most general form of Communication system ( one or two blocks may differ) is shown in the figure basic terminology used in Communication systems is message signal /information/data,channel,noise,modulation, encoding and decoding. Communication system is meant for communicating messages between Transmitter and Receiver (or) source & destination.

source

source or information source is the primary block in communication system which generates original message / actual message.

i.e, selecting one message (actual message) from a group of messages itself is called as sorting data (or) information. Source generates message which may be in any form like words, code , symbols, sound signal, images, videos etc.among these the desired message has been selected and conveyed.

A transducer is one which converts one form of energy into electrical energy because the message from information source may not be always in electrical form, a transducer is used in between source and transmitter as a separate block sometimes (or) may be a part of Tx ^r.

Transmitter

Tx^r is meant for the following tasks

restriction of range of audio frequencies (i.e, limiting the bandwidth of the message signal).
Amplification.
Modulation.

In general modulation is said to be the main function of the transmitter.

Channel

The medium that exists between transmitter and receiver is called as channel. The function of channel is to provide connection between transmitter and receiver, two types of channels are there wired/point to point and wireless/broadcasting channels.

Point to point channels are generally wired channels(i.e, a physical medium exists) like Microwave links, optical fibre links etc.

Microwave links:- these links are used in telephone transmission.In these type of links guided EM waves are used to transmit from Tx^r to Rx^r.

optical fibre links:- used in low-loss high speed data transmission and uses optical fibers as the medium .

Broadcast channels:- the medium or channel is wireless here, in broadcasting a single transmitter can send information to many receivers simultaneously, satellite broadcasting system is one such system.

during the process of transmission and reception, the signal gets distorted due to noise in the channel, noise may interfere with the signal at any point but noise in the channel has greatest effect on the signal.

Receiver

The main function of the receiver is to reproduce the message signal in electrical form from the distorted received signal. This reproduction process is called demodulation (or) detection , in general this demodulation may be assumed as the reverse process of modulation carried out in transmission.

there are a great variety of receivers in communication systems, the type of receiver chosen depends on type of modulation, operating frequency ,its range and type of destination required. Most common receiver is superheterodyne receiver .

crystal receiver with head phones

Radio receiver

so many types of receivers are available from a very simple crystal receiver with headphones to radar receiver etc.

Destination

It is the final stage of any communication system. it would be a loud speaker / a display device/simply a load etc depending up on the requirement of the system.

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Amplitude Modulation (AM)

Amplitude Modulation (AM):-

Modulation:- It is defined as the process in which one of the characeteristic of carrier signal is varied in accordance with the instantaneous values of message signal (Amplitude of the message signal).

The fundamental goal of modulation is to produce an information bearing modulated signal with efficient utilization of the channel.

Amplitude modulation:- It is defined as the process in whch the amplitude of the carrier signal is varied in accordance with the intantaneous values of message signal.

To generate a modulated signal we are in need of two signals called as message signal & carrier signal.

m(t) – message signal/modulating signal/original signal/actual signal.
C(t)- Carrier signal/unmodulated carrier signal.
S_AM(t)- Amplitude modulated signal/ Amplitude modulated Carrier.

Now these two signals are being given as inputs to an Amplitude Modulator , which in turn generates an Amplitude modulated signal S_AM(t).

Here C(t) represents carrier signal $C(t)=A _{c}cos 2\pi f_{c}t$ , the amplitude of the un modulated carrier is $A_{c}$ , when this unmodulated carrier is amplitude modulated , the new amplitude will become $A_{c}(1+k_{a}m(t))$ and the modulated carrier wave is S_AM(t)

$S_{AM}(t)=A_{c}(1+k_{a}m(t))cos 2\pi f_{c}t$ .

k_a is known as amplitude sensitivity.

In AM the frequency of the carrier signal f_c is assumed to be much larger than the highest frequency present in the base band signal and in the AM swave $\left | k_{a}m(t)\right |$ is assumed to be less than 1

i.e, $\left | k_{a}m(t)\right |< 1$ for all t

if $\left | k_{a}m(t)\right |> 1$ in any case with large value of amplitude sensitivity k_a then the envelope of the resultant signal doesn’t represent base band signal, this causes over modulation which causes a phase reversal of the carrier wave at zero-crossings.

$\therefore \left | k_{a}m(t)\right | =1$ is the limiting (or) maximum value of AM.

this $\left | k_{a}m(t)\right |$ is called modulation index.

Note:- AM is also known as conventional AM.

Frequency spectrum of AM:-

AM signal is given as $S_{AM}(t)=A_{c}(1+k_{a}m(t))cos 2\pi f_{c}t$ to obtain the frequency spectrum of AM signal one must represent the signal in frequency domain

i.e by taking the fourier transform of S_AM(t) we will obtain S_AM(f) . Let us assume M(f) is in the figure shown below and has a bandwidth ‘B’ Hz.

$S_{AM}(t)=A_{c}(1+k_{a}m(t))cos 2\pi f_{c}t$

$S_{AM}(t)=A_{c}cos 2\pi f_{c}t+ A_{c}k_{a}m(t)cos 2\pi f_{c}t$

by taking fourier transform of s_AM(t)

$S_{AM}(f)=\frac{A_{c}}{2}(\delta (f-f_{c})\delta (f+f_{c}))+ \frac{A_{c}k_{a}}{2}(M(f-f_{c})+M(f+f_{c}))$

the frequency spectrum consists of two impulse functions at $f=\pm f_{c}$ and the frequency band $(f_{c} to(f_{c}+f_{m})) and -(f_{c} to(f_{c}+f_{m}))$ are called as Upper side band frequencies $(f_{c} to(f_{c}-f_{m})) and -(f_{c} to(f_{c}-f_{m}))$ are Lower side band frequencies.

Note:- Information or message is available in two sidebands LSB and USB.

BW os AM signal = 2 X BW of message signal.

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single tone AM

single tone AM:-

The expression for conventional AM is $S_{AM}(t)=A_{c}(1+k_{a}m(t))cos 2\pi f_{c}t$

now if the message signal is a single-tone $i.e, m(t) = A_{m}cos 2\pi f_{m}t$

$S_{Single-tone AM}(t)=A_{c}(1+k_{a}A_{m}cos2\pi f_{m}t)cos 2\pi f_{c}t$

where $\mu =k_{a}A_{m}$ is called as modulation index

$S_{Single-tone AM}(t)=A_{c}cos 2\pi f_{c}t+\mu A_{c}cos2\pi f_{m}tcos 2\pi f_{c}t$

this equation can be further simplified as follows $S_{Single-tone AM}(t)=A_{c}cos 2\pi f_{c}t+\frac{\mu A_{c}}{2}cos2\pi (f_{c}+f_{m})t+\frac{\mu A_{c}}{2}cos2\pi (f_{c}-f_{m})t$

that is by taking the fourier transform

$\dpi{150} S_{Single-tone AM}(f)=\frac{A_{c}}{2}\left \{ \delta (f-f_{c})+\delta (f+f_{c}) \right \}+\frac{\mu A_{c}}{4}\left \{ \delta (f-(f_{c}+f_{m}))+\delta (f+(f_{c}+f_{m})) \right \}+\frac{\mu A_{c}}{4}\left \{ \delta (f-(f_{c}-f_{m}))+\delta (f+(f_{c}-f_{m})) \right \}$

from the above expression the amplitude spectrum can be drawn as follows

from the spectrum single tone AM consists of 6 impulse functions located at frequencies $\pm f_{c}$ , $\pm (f_{c} + f_{m})$ and $\pm (f_{c} - f_{m})$ respectively.

Power content in AM/ Conventional AM:-

$S_{AM}(t)=A_{c}(1+k_{a}m(t))cos 2\pi f_{c}t$ represents the AM signal , here m(t) is some arbitrary signal , then the power of this signal can be calculated from its Mean Square value $\overline{}{m^{2}(t)}$

i.e, message signal power = $\overline{}{m^{2}(t)}$ Watts.

Carrier signal is $C(t)=A_{c}cos 2\pi f_{c}t$ and it’s power is $\frac{A_{c}^{2}}{2}$ Watts.

Now the total power available in the signal $S_{AM}(t)=A_{c}(1+k_{a}m(t))cos 2\pi f_{c}t$ will be $P_{TOTAL}$ .

$S_{AM}(t)=A_{c}cos 2\pi f_{c}t +A_{c}k_{a}cos 2\pi f_{c}t . m(t)$

$P_{TOTAL} =\frac{A_{c}^{2}}{2}+\frac{A_{c}^{2}k_{a}^{2}}{2} X message signal power$

$P_{TOTAL} =\frac{A_{c}^{2}}{2}+\frac{A_{c}^{2}k_{a}^{2}}{2} X\overline{m(t)^{2}}$ Watts.

Total Side Band power can be calculated from the term $A_{c}k_{a}cos 2\pi f_{c}t . m(t)$ can be denoted as $P_{SB}$ that would be $\frac{A_{c}^{2}k_{a}^{2}}{2} X\overline{m(t)^{2}}$ Watts.

from these power calculations transmission efficiency of AM can be obtained as $\eta = \frac{P_{SB}}{P_{Total}} X100$ %

$\eta = \frac{\frac{A_{c}^{2}k_{a}^{2}}{2} .\overline{m(t)^{2}}}{\frac{A_{c}^{2}}{2}+\frac{A_{c}^{2}k_{a}^{2}}{2} .\overline{m(t)^{2}}}$ X 100%.

$\eta = \frac{k_{a}^{2}.\overline{m(t)^{2}}}{1+k_{a}^{2}.\overline{m(t)^{2}}}$ X 100%.

Power content in Single-tone AM:-

In single tone AM message signal is $i.e, m(t) = A_{m}cos 2\pi f_{m}t$ , then power of the message signal is $\frac{A_{m}^{2}}{2}$ watts

carrier signal is $C(t) = A_{c}cos 2\pi f_{c}t$ implies the carrier power is $\frac{A_{c}^{2}}{2}$ watts.

$S_{Single-tone AM}(t)=A_{c}cos 2\pi f_{c}t+\frac{\mu A_{c}}{2}cos2\pi (f_{c}+f_{m})t+\frac{\mu A_{c}}{2}cos2\pi (f_{c}-f_{m})t$

then the total power of the single-tone AM signal is from the above equation given as

$P_{Total}=\frac{A_{c}^{2}}{2}+\frac{A_{c}^{2}\mu ^{2}}{8}+\frac{A_{c}^{2}\mu ^{2}}{8}$

P_Total = P_c +P_USB+P_LSB

$P_{Total}=\frac{A_{c}^{2}}{2}+\frac{A_{c}^{2}\mu ^{2}}{4}$

$P_{Total}=\frac{A_{c}^{2}}{2}(1+\frac{\mu ^{2}}{2})$ Watts.

USB and LSB has same power $P_{USB}=P_{LSB}=\frac{A_{c}^{2}\mu ^{2}}{8}$ watts.

Now total side band power is $P_{SB}=P_{USB}+P_{LSB}=\frac{A_{c}^{2}\mu ^{2}}{4}$

from these power calculations transmission efficiency of AM can be obtained as $\eta = \frac{P_{SB}}{P_{Total}} X100$ %

$\eta = \frac{\frac{A_{c}^{2}\mu ^{2}}{4}}{\frac{A_{c}^{2}}{2}(1+\frac{\mu ^{2}}{2})} X 100$ %

$\eta = \frac{\mu ^{2}}{(\mu ^{2}+2)} X100$ %.

Note:- Effeciency (or) Transmission efficiency of AM is only 33.3% only i.e, $\eta$ value calculated when $\mu$ =1.

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Effective Modulation index of a Multi-tone AM signal

In a single-tone AM, message signal is a single-tone $i.e, m(t) = A_{m}cos 2\pi f_{m}t$ being modulated by a carrier signal and generates a single-tone modulated signal, where as in Multi-tone environment message signal is a composite signal formed by number of frequencies f₁,f₂,f₃ …..f_n … being modulated by a carrier signal to generate an Amplitude Modulated signal.

i.e, Multi-tone message signal is

$\therefore m(t) = A_{1}cos 2\pi f_{1}t +A_{2}cos 2\pi f_{2}t+A_{3}cos 2\pi f_{3}t+....+A_{n}cos 2\pi f_{n}t+....$

Now from the equation of General AM signal $S_{AM}(t)=A_{c}(1+k_{a}m(t))cos 2\pi f_{c}t$

the Multi-tone modulated signal can be obtained as

$S_{AM}(t)=A_{c}(1+k_{a}(A_{1}cos 2\pi f_{1}t +A_{2}cos 2\pi f_{2}t+A_{3}cos 2\pi f_{3}t+....+A_{n}cos 2\pi f_{n}t+....))cos 2\pi f_{c}t$

$S_{AM}(t)=A_{c}(1+k_{a}A_{1}cos 2\pi f_{1}t +k_{a}A_{2}cos 2\pi f_{2}t+k_{a}A_{3}cos 2\pi f_{3}t+....+k_{a}A_{n}cos 2\pi f_{n}t+....)cos 2\pi f_{c}t$

$S_{AM}(t)=A_{c}cos 2\pi f_{c}t+k_{a}A_{1}cos 2\pi f_{1}t cos 2\pi f_{c}t +k_{a}A_{2}cos 2\pi f_{2}t cos 2\pi f_{c}t+.......$

$S_{AM}(t)=A_{c}cos 2\pi f_{c}t+A_{c}\mu _{1}cos 2\pi f_{1}t cos 2\pi f_{c}t +A_{c}\mu _{2}cos 2\pi f_{2}t cos 2\pi f_{c}t+.......$

$S_{AM}(t)=A_{c}cos 2\pi f_{c}t+\frac{A_{c}\mu _{1}}{2}cos 2\pi (f_{c}+f_{1})t+ \frac{A_{c}\mu _{1}}{2}cos 2\pi (f_{c}-f_{1})t + \frac{A_{c}\mu _{2}}{2}cos 2\pi (f_{c}+f_{2})t + \frac{A_{c}\mu _{2}}{2}cos 2\pi (f_{c}-f_{2})t + ..........$

from the above signal the total power can be obtained as

$P_{Total}=\frac{A_{c}^{2}}{2}+\frac{A_{c}^{2}\mu_{1} ^{2}}{8}+\frac{A_{c}^{2}\mu_{1} ^{2}}{8}+\frac{A_{c}^{2}\mu_{2} ^{2}}{8}+\frac{A_{c}^{2}\mu_{2} ^{2}}{8}+......$

$P_{Total}=\frac{A_{c}^{2}}{2}+\frac{A_{c}^{2}\mu_{1} ^{2}}{4}+\frac{A_{c}^{2}\mu_{2} ^{2}}{4}+......$

$P_{Total}=\frac{A_{c}^{2}}{2}(1+\frac{\mu_{1} ^{2}}{2}+\frac{\mu_{2} ^{2}}{2}+......)$

This expression can further represented in terms of effective modulation index $\mu _{eff}$ as $P_{Total}=\frac{A_{c}^{2}}{2}(1+\frac{\mu_{eff} ^{2}}{2})$ where $\mu _{eff} = \sqrt{\mu _{1}^{2}+\mu _{2}^{2}+\mu _{3}^{2}+...}$

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Indirect method of generation of FM

Indirect method of generation of FM signal is also known as Armstrong method .Here a crystal oscillator generates carrier signal , which provides very high stability compared to Direct method. this method generates a WBFM signal, i.e a phase modulator generates a NBFM signal in the first step , then in the second step NBFM will be converted to WBFM signal using a frequency multiplier.

In NBFM modulation index is small and the distortion is very low in NBFM ,here we prefer phase modulator to generate NBFM as it’s generation is easy, the frequency multiplier multiplies incoming frequency along with frequency deviation $\Delta f$ . Hence NBFM will be converted into WBFM with large frequency deviation as well.

Frequency multiplier:-

The frequency multiplier consists of a non-linear device followed by a Band Pass Filter, the non-linear device is a memory less device.

If the input to a non-linear device is an FM wave with frequency $f_{c}$ and deviation $\Delta f$ then output consists of DC component and ‘n’ frequency modulated waves with carrier frequencies $f_{c},2f_{c},3f_{c},......nf_{c}$ and frequency deviations $\Delta f,2\Delta f,3\Delta f,4\Delta f.....n\Delta f$ . The BPF designing is in such a way that it passes the FM wave centered at the frequency $nf_{c}$ with frequency deviation $n\Delta f$ and to suppress all other FM components. Thus a frequency multiplier generates a WBFM wave from a NBFM wave.

Generation of WBFM by Armstrong’s method:-

This Armstrong’s method is indirect method used to generate WBFM signal.It is used to generate FM signal having both the desired frequency deviation and carrier frequency.

The block diagram consists of two stage multiplier and an intermediate stage of frequency translator .

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Super heterodyne Receiver

This is the most commonly used Receiver and it uses “hetero dyning” principle which is used almost in all types of receivers like TR Receiver and Radar Receiver etc. The word hetero(≈different) dyne(≈mixing) means mixing different frequencies using a Mixer. Hence the name given as super hetero dyne Receiver.

The block diagram consists of a receiving antenna followed by an RF stage as the primary block , the receiving signal has been fed to RF stage through the antenna.

In a Super hetero dyne Receiver the incoming RF signal frequency ( $f_{s}$ ) is combined with local oscillator frequency( $f_{o}$ ) through a mixer and converts a signal of a lower fixed frequency (IF) this lower fixed frequency is called as Intermediate Frequency ( $f_{i}$ or $f_{IF}$ ). A constant frequency difference is maintained between the Local Oscillator and incoming RF signal. This is provided through Capacitance tuning that is all capacitors are ganged together and operated by a common control knob.

$\therefore$ incoming RF is down translated to IF using a mixer now this IF is given as input to the secondary stage of the block diagram that is IF amplifier. IF amplifier consists of number of transformers each consisting of a pair of mutually tuned circuits thus with a large number of double tuned circuits, operating at a specially chosen frequency the IF amplifier provides most of the gain.

Thus IF stage full fills most of the gain (sensitivity) and Band width(selectivity) requirements of the Receiver. For a Super hetero dyne receiver Sensitivity and selectivity are quite uniform throughout it’s tuning range this is one of the advantage over TRF Receiver.

The amplified IF signal is given as an input to the Detector. The Detector or the demodulator demodulates the signal and down translates the IF signal to AF(Audio Frequency) signal.

The AF signal is amplified by Audio amplifier and further by power amplifier. The last stage of the receiver is a Loud speaker , which receives AF signal. Loud speaker is in general a transducer which converts electrical signal into a voice (or) Audio.

The advantages of Super hetero dyne receiver makes it most suitable for majority of Radio Receiver applications like AM, FM, Communications, SSB, TV and even Radar Receiver.

Advantages of super hetero dyne Receiver:-

It provides high gain through IF amplifier that is more sensitivity is being provided by it.
Improved selectivity over TRF receiver.
Improved adjacent channel rejection.
BW remains constant over the entire operating range.
Selectivity and Sensitivity are uniform throughout it’s tuning range.

Note: There is a rating embedded within this post, please visit this post to rate it. ]]> 1163 https://atomic-temporary-93025954.wpcomstaging.com/tuned-radio-frequency-receiver/ Tue, 25 Sep 2018 17:34:13 +0000 https://atomic-temporary-93025954.wpcomstaging.com/?p=1155 Continue reading “Tuned Radio Frequency Receiver”]]> Tuned Radio Frequency Receiver is the primary Radio Receiver and is the most simplest form of Radio receiver.

The block diagram consists of a receiving antenna followed by an RF stage as the primary block , the receiving signal has been fed to RF stage through the antenna. This RF stage consists of two (or) three RF Amplifiers, these amplifiers are tuned RF Amplifiers.i.e they have variable tuned circuits at input and output sides.

The received signal has been amplified by the RF amplifiers and the amplified signal is being given as an input to the Detector. The Detector or the demodulator demodulates the signal and down converts the RF signal to AF(Audio Frequency) signal.

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Drawbacks of TRF Receiver:-

Selectivity of TRF Receiver is poor. This is because achieving sufficient selectivity at high frequencies is difficult due to enforced use of single-tuned Circuits.
Instability:-(RF Stage) The TRF Receiver suffers from a tendency to oscillate at a higher frequencies (i.e, instability), this is because multi-stage RF amplifiers has to provide high gain at high frequencies. RF amplifiers provides high gain which results in positive feed back leads to oscillations and then causes instability of the circuit. This positive feedback (caused by the leakage of output of RF stage back to it’s input) could result from power supply coupling through any other element common to input and output stages.
Variation of band width over tuning range:- One more draw back in TRF receiver is the BW variation over the tuning range i.e the BW of TRF receiver varies with the incoming frequency.

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Noise

Introduction:-

Noise is probably the only topic in electronics and tele communications with which everyone must be familiar, electrical disturbances that interfere with signals produces noise and this noise ever present and limits the performance of the most of the systems. Measuring noise is very controversial almost everybody has a different method of quantifying noise and its effects.

definition:- noise is unwanted energy that interfere with the required signal. In receivers :- noise is disturbance in electric nature.

Radio receivers—> noise appears as “hiss”.
TV receivers —–> it appears a snow (or) colored snow pictures.
In Pulse communication systems —->noise produces unwanted pulses.

In receivers noise effects sensitivity and band width and it decreases sensitivity as well as band width.

Basically noise can be classified as Internal and External noise .

External Noise	Internal Noise
when noise sources are external to the receiver . i.e, noise source is located outside of the receiver. It is difficult to treat quantitatively external noise.	Noise is created with in the receiver itself.i.e, noise source is internal to the receiver. internal noise can be treated quantitatively and reduction is also possible by appropriate receiver design.

External Noise:-

Atmospheric noise:-

If we try to listen to short waves on a receiver which is not well equipped to receive them, an astonishing variety of strange sounds will be heard, all tending to interfere with the program. most of these sounds are the result of spurious sources of disturbance, which represents atmospheric noise generally called as “static”.

Atmospheric noise is caused by lightning discharge in thunderstorms and other natural electric disturbances occurring in the atmosphere.
It originates in the form of amplitude modulated impulses , and are spread over most of the RF spectrum normally used for broadcasting.
i.e, It consists of spurious radio signal with components distributed over a wide range of frequencies. Atmospheric noise propagates over the earth in the same way as ordinary Radio waves of the same frequencies.
Static is more severe in the case of Radio than that of Tele-vision and it becomes less severe at frequencies above 30 MHz. Since higher frequencies are limited to line of sight propagation.
This noise is created in VHF range and above.

Extraterrestrial noise:-

This noise is generated in the earth’s outer space (atmosphere)

Extraterrestrial noise is divided into

Solar noise .
Cosmic noise.

Solar noise:-

The sun radiates so many things our way noise is noticeable among them.
Under “quiet” conditions , there is a constant noise radiation from the sun simply because its a large body at high temperature ≈ 6000^o C.
∴ The radiation consists of the frequencies which we use for communications and interferes with them.
However, the disturbances in the sun is variable and undergoes cycles at the peak of which electrical disturbances erupt. These additional disturbances are several orders of magnitude greater than the noise generated during periods of the quiet sun. The solar cycle repeats these period of great electrical disturbances approximately every 11 years, further these 11 year cycle peaks reach even a higher maximum peak every 100 years.
Thus the noise generated by sun changes periodically with the solar disturbances.

Cosmic noise:-

stars are also suns and have high temperatures, they radiate RF noise in the same manner as our sun, this refers to noise coming from distant stars other than sun.
The noise received from such stars is also called “black-body noise” and is distributed fairly uniformly over the entire sky.
Space noise is observable in the range from about 8 MHz to about 1.43 GHz this is the strongest component of noise in the range(20-120) MHz.

Industrial (or) Man-made noise:-

This noise is strongest in Industrial areas and the frequency of Man made noise spans between 1 to 600 MHz.
Man made noise is found in urban, sub-urban and industrial areas. The intensity of the noise made by human easily outstrips that created by any other source, internal or external to the receiver.
under this, sources such as Automobile, Aircraft ignition, electric motors and switching equipment leakage from high voltage lines and a multitude of other heavy electric machines are all included.
Fluorescent lights are another powerful source of such noise and therefore should not be used where sensitive receiver is installed.

Internal Noise:-

This noise is created by any of the active (or) passive devices found in receivers. It is created by various components used in processing the received signal and is completely internal to the system. The effect of this noise is significant at the front end of the receiver.This appears as thermal and shot noise caused by resistors, inductors and capacitors.

Thermal Noise:-

This noise is also known as agitation noise, Jhonson noise / white noise. thermal noise is random in nature, this mainly occurs due to random(or) rapid motion of molecules, atoms and electrons of which resistor is made up of.

from the theory of dynamics the noise generated by a resistor is proportional to it’s absolute temperature and BW over which the noise is to be measured.

$P_{n}\propto T\Delta f$ where B= BW= $\Delta f$

$P_{n}= kT\Delta f$

where k – boltzmann’s constant =1.38X10^-23 J/K.
T- Absolute temperature in Kelvin, K = 273+ ^oC.
$\Delta f$ = BW of interest.
$P_{n}$ is the maximum noise power output of a resistor.

In an ordinary resistor at the standard temp of 17^oC is not connected to any voltage source and if we are measuring voltage using a DC volt meter to measure voltage across it shows a zero. Actually a resistor itself is a noise generator, if we use a very sensitive electronic volt meter it shows a very large voltage across R.

This noise voltage is caused by the random movement of electrons with in the resistor, which constitutes a current. The rate of arrival of electrons at either end of the resistor therefore varies randomly, and so does the potential difference exists between the two ends.

from the circuit diagram,

$P_{n}=\frac{V_{n}^{2}}{4R_{L}}$ —- equation(1).

The maximum power is delivered to load when R =R_L.

$V_{n}= i(R + R_{L})$

$V_{n} = i 2R$

$i = \frac{V_{n}}{2R}$

load voltage $V = i R_{L}$

$V = \frac{V_{n}}{2 R}$

$V = \frac{V_{n}}{2}$ volts

V_n -source noise voltage.

V- ouput voltage measured across R_L.

from equation (1) $V_{n}^{2}=P_{n}4R_{L}$

$V_{n}^{2}=4k T\Delta fR_{L}$

$V_{n}=\sqrt{4k T\Delta fR_{L}}$

V_n is known as RMS noise voltage asross a resistor.

Shot Noise:-

This occurs due to shot effect, it occurs in all active and amplifying devices (diodes/transistors).

It is caused by random variations in the arrival of electrons (or holes) at the output electrode of an amplifying device and appears as a randomly varying noise current super imposed on the output.It sounds like a shower of a lead shot were falling on a metal plate. Hence named it as shot noise.

In electronic tubes shot noise is caused because of random emission of electrons from cathode.
In semi-conductors shot noise occurs due to random diffusion of minority carriers .
$i(t) = I_{o} + i_{n}(t)$

Total current = Mean DC constant current + shot noise current.

Shot noise is given by $i_{n} = \sqrt{2qi_{p}\Delta f}$

i_n -RMS shot-noise current.
q-charge of an electron 1.6 X10^-19 C.
i_p– direct diode current.
$\Delta f$ – BW of the system.

This formula for shot noise is valid for vaccum tube diode under so called temp-limited conditions.

In all other cases we use the concept of equivalent noise resistance intead of shot noise formula.

Transit-time noise (or) High-frequency noise:-

It is generally observed in semi conductor devices when the transmit time of charge carriers crossing a junction is comparable with the time-period of the signal, some charge carriers diffuse back to the source (or) emitter.

this gives rise to input admittance Y

conductance G = 1/Y , this G increases with frequency which causes noise . This is also called as high frequency noise.

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Laplace’s and Poisson’s equations(Magnetostatics)

Laplace’s equation(Magneto statics):-

From the equation $\oint_{s}\overrightarrow{B}.\overrightarrow{ds}=0\Rightarrow \overrightarrow{\bigtriangledown } .\overrightarrow{B}=0$ .

since $\overrightarrow{B}=\mu _{O}\overrightarrow{H}$

$\overrightarrow{\bigtriangledown }.(\mu _{O}\overrightarrow{H}) =0$ .

$\mu _{O}(\overrightarrow{\bigtriangledown }.\overrightarrow{H}) =0$ .

we know that for zero current density $\overrightarrow{J}=0$ , the Magnetic scalar potential is $\overrightarrow{H}=-\overrightarrow{\bigtriangledown }V_{m}$

by replacing $\overrightarrow{H}$ in the equation $\mu _{O}(\overrightarrow{\bigtriangledown }.\overrightarrow{H}) =0$ ,

$\mu _{O}(\overrightarrow{\bigtriangledown }.(-\overrightarrow{\bigtriangledown }V_{m})) =0$ .

$\mu _{O}(\bigtriangledown^{2}V_{m}) =0$ .

$\bigtriangledown^{2}V_{m} =0$ . is known as Laplce’s equation in Magneto statics.

Poisson’s equation(Magneto statics):-

In vector algebra a vector can be fully defined if it’s curl and divergence are defined.

$\overrightarrow{\bigtriangledown } X\overrightarrow{H}=\overrightarrow{J}$ -point form of Ampere’s law

$\overrightarrow{\bigtriangledown } X\overrightarrow{B}=\mu _{o}\overrightarrow{J}$ because $\overrightarrow{B}=\mu _{O}\overrightarrow{H}$ .

from vector Magnetic potential $\overrightarrow{B}=\overrightarrow{\bigtriangledown } X \overrightarrow{A}$ .

$\overrightarrow{\bigtriangledown } X(\overrightarrow{\bigtriangledown } X \overrightarrow{A})=\mu _{o}\overrightarrow{J}$ .

from the vector identity $\bigtriangledown ^{2}\overrightarrow{A}=\overrightarrow{\bigtriangledown }(\overrightarrow{\bigtriangledown }.\overrightarrow{A})-\overrightarrow{\bigtriangledown } X\overrightarrow{\bigtriangledown } X \overrightarrow{A}$ .

$\overrightarrow{\bigtriangledown }(\overrightarrow{\bigtriangledown }.\overrightarrow{A})-\bigtriangledown ^{2}\overrightarrow{A}=\mu _{o}\overrightarrow{J}$ .

if $\overrightarrow{\bigtriangledown }.\overrightarrow{A}=0$ .

$\bigtriangledown ^{2}\overrightarrow{A}=-\mu _{o}\overrightarrow{J}$ – this equation is known as Poisson’s equation for Magneto statics.

$\overrightarrow{A}=A_{x}\overrightarrow{a_{x}}+A_{y}\overrightarrow{a_{y}}+A_{z}\overrightarrow{a_{z}}$ and $\overrightarrow{J}=J_{x}\overrightarrow{a_{x}}+J_{y}\overrightarrow{a_{y}}+J_{z}\overrightarrow{a_{z}}$

$\bigtriangledown ^{2}(A_{x}\overrightarrow{a_{x}}+A_{y}\overrightarrow{a_{y}}+A_{z}\overrightarrow{a_{z}})=-\mu _{o}(J_{x}\overrightarrow{a_{x}}+J_{y}\overrightarrow{a_{y}}+J_{z}\overrightarrow{a_{z}})$ .

by equating the respective components on each side

$\bigtriangledown ^{2}A_{x}=-\mu _{o}J_{x}$ , $\bigtriangledown ^{2}A_{y}=-\mu _{o}J_{y}$ and $\bigtriangledown ^{2}A_{z}=-\mu _{o}J_{z}$ are the scalar Poisson’s equations of Magneto statics.

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Relation between E and V

The potential difference between two points A and B is $V_{AB}= V_{A}-V_{B}$ and $V_{BA}= V_{B}-V_{A}$ .

i.e, $V_{AB}=-V_{BA}$

$V_{AB}+V_{BA}=0$ .

This equation implies that the total work done in moving a charge from A to B $(V_{AB})$ and then from B to A $(V_{BA})$ is zero.

i.e, $\oint_{l}\overrightarrow{E}.\overrightarrow{dl}=0$

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Single stub matching

Stub matching in Transmission lines:-

A section of Transmission line can be inserted between Load and source besides it is also possible to connect sections of open (or) short circuited lines as stub (or) tuning stubs in shunt with the main Transmission lines at a certain point to effect the matching.

Matching with the help of tuning stub (or) stub is called as stub matching and has the following advantages

Length $l,\ Z_{o}$ are unchanged.
It is possible to add adjustable susceptance in shunt with the line.

Stub matching can be possible by

Single stub matching.
Double stub matching.

Single stub matching:-

A type of transmission line frequently used in single stub matching is a short circuited section of transmission line , which is connected in parallel to the main transmission line at a particular distance from the load. By using such stub anti resonance is achieved providing impedance at resonance equal to $R_{o}$ .

The conductance at that point is equal to $Y_{o}$ and the stub length is adjusted to provide a susceptance which is equal in value but opposite in sign to the input susceptance of the main line at that point so that the total susceptance at that point is zero.

The combination of stub and line will thus represent a conductance which is equal to $Y_{o}$ of the line.

We are connecting the stub in parallel to the main Tx line , we will work out with admittance’s instead of impedance’s.

$Y_{s}=G_{o}\pm jB$

This is the admittance at a point A before stub was connected. The point A is now connected to a stub which provides a susceptance of $\mp jB$ .

$Y_{s}=G_{o}\pm jB\mp jB$ .

$Y_{s}=G_{o}$ .

$Y_{s}=\frac{1}{R_{o}}$ .

$Z_{s}=R_{o}$ .

Thus the input impedance of the line is $R_{o}$ itself up to the point A, after A to load the reflection and hence standing waves occurs but by making this distance less than the wave length the losses can be minimized.

For the single stub, it is important to know where the stub is to be connected exactly and the length of the stub for these two measurements must be made on the line it is easy to measure S and $V_{min}$ nearest to load.

The measurement is accurate in case of $V_{min}$ rather than $V_{max}$ .

From the expression of $Z_{s}$ in terms of reflection coefficient

$Z_{s}=Z_{o}(\frac{e^{\gamma l}+ke^{-\gamma l}}{e^{\gamma l}-ke^{-\gamma l}})$ .

$Z_{s}=Z_{o}(\frac{1+ke^{-2\gamma l}}{1-ke^{-2\gamma l}})$ .

Thus at point from load, input impedance is resistive and its value is minimum equal to this is point from the load

At distance

Amplitude Shift Keying(ASK)

Amplitude Shift Keying (ASK) (or) On Off Keying (OOK) is the simplest Digital Modulation technique.

In this method, carrier amplitude is switched between two voltages ON and OFF levels depending up on the input binary sequence.

The carrier signal is a continuous wave (or) sinusoidal wave form

$S(t)=A \cos 2\pi f_{c}t$ .

The normalized power is $P=\frac{A^{2}}{2}$

$A=\sqrt{2P_{s}}$ .

The carrier signal can be expresses in terms of power as $S(t)=\sqrt{2P_{s}} \cos 2\pi f_{c}t$ .

if energy per bit is $E_{b}$ and the bit interval as $T_{b}$ then the carrier signal is $S(t)=\sqrt{\frac{2E_{b}}{T_{b}}} \cos 2\pi f_{c}t$ .

Now according to ASK Binary ‘1’ is represented with carrier voltage and Binary ‘0’ is represented with zero voltage.

$\left\{\begin{matrix} S_{ASK}(t)=\sqrt{2P_{s}} \cos 2\pi f_{c}t\rightarrow \ Binary\ '1' \\ =0 \ \rightarrow \ Binary\ '0' \end{matrix}\right.$

in terms of Energy and bit duration ASK signal can be written as

$\left\{\begin{matrix} S_{ASK}(t)=\sqrt{\frac{2E_{b}}{T_{b}}} \cos 2\pi f_{c}t\rightarrow \ Binary\ '1' \\ =0 \ \rightarrow \ Binary\ '0' \end{matrix}\right.$ .

ASK Transmitter:-

The figure shows the ASK generator (or) ASK Transmitter

It is a simple product Modulator, which modulates the incoming binary sequence (in the form of a signal) with the carrier signal S(t)

i.e, $S_{ASK}(t)=b(t).S(t)$

b(t) represents the binary sequence in the form of a signal.

when the input bit (or) symbol is Binary ‘1’ product Modulator passes the carrier signal and for Binary’0′, A zero output is given which blocks the carrier signal.

$\left\{\begin{matrix} S_{ASK}(t)=\sqrt{2P_{s}} \cos 2\pi f_{c}t\rightarrow \ Binary\ '1' \\ =0 \ \rightarrow \ Binary\ '0' \end{matrix}\right.$ .

Coherent ASK Detector:-

The figure shows the Block Diagram of coherent ASK/BASK Detector. The ASK signal $S_{ASK}(t)$ is applied to the correlator ( The Block product Modulator followed up by the Integrator).

$S_{ASK}(t)$ is multiplied by local carrier C(t) this carrier C(t) is phase locked with that of the carrier used in the Transmitter. As this is coherent reception.

The product $S_{ASK}(t).C(t)$ is applied to the Integrator. The Integrator integrates the input over one bit interval $T_{b}$ and the output is given to a threshold device. If the threshold voltage is set to 0 V.

the output of threshold device v(t) (or) v is either ‘1’ (or) ‘0’ based on the following condition.

$v\leq 0\rightarrow \ a \ symbol \ '1' \ is \ detected.$

Note:- The input to demodulator is not $S_{ASK}(t)$ always most of the times it is interfered with noise n(t) in the channel.

in coherent detection input to the demodulator is simply $S_{ASK}(t)$ signal where as in Non-coherent detection the input is noisy ASK signal.

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Indirect method of generation of FM signal

Frequency multiplier:-

The frequency multiplier consists of a non-linear device followed by a Band Pass Filter, the non-linear device is a memory less device.

Generation of WBFM by Armstrong’s method:-

This Armstrong’s method is indirect method used to generate WBFM signal.It is used to generate FM signal having both the desired frequency deviation and carrier frequency.

The block diagram consists of two stage multiplier and an intermediate stage of frequency translator .

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Sampling Theorem

Sampling of signals is the fundamental operation in signal processing, a Continuous Time (CT) signal can be converted into a Discrete Time (DT) signal using Sampling process. Sampling is required since the advancement in both signals and systems which are digitized i.e, Digital systems operates only on digital signals only.

Sampling Theorem:-

A CT signal is first converted into DT signal by Sampling process. The sufficient number of samples must be taken so that the original signal is represented in it’s samples completely, and also the signal is represented from it’s samples, these two conditions representation and reconstruction depends on the sampling process ‘f_s‘ Hz.

Sampling theorem can be given into two parts

i. A band limited signal of finite energy, which has no frequency component higher than ‘f_m‘ Hz, is completely described by it’s sample values at uniform intervals less than (or) equal to 1/2f_m seconds apart.

i.e, $T_{s}\leq \frac{1}{2f_{m}}$ Seconds.

ii. A Band limited signal of finite energy, which has no frequency component higher than f_m Hz may be completely recovered from the knowledge of it’s samples if samples are taken at the rate of 2f_m samples/second.

i.e, $f_{s}\geq 2f_{m}$ Hz.

Statement:- A Continuous Time signal can be completely represented in it’s samples and recovered from it’s samples if the sampling frequency $f_{s}\geq 2f_{m}Hz.$

where $f_{s}$ is the sampling frequency.

$f_{m}$ is the highest frequency present in the original signal / Band width of the signal.

proof of Sampling theorem:-

Let us consider a CT signal x(t), which is a band limited to $f_{m}$ Hz as shown

To prove Sampling theorem, it should be shown a signal whose spectrum is band limited to f_m Hz can be reconstructed exactly without any error from it’s samples taken uniformly at a rate of $f_{s}> 2f_{m}$ Hz.

The circuit shows the sampler

Now sampling of x(t) at a rate of f_s may be achieved by multiplying x(t) with a train of impulses $\delta T_{s}(t)$ with a period ‘T_s‘ seconds.

The sampling signal is an ideal (or) instantaneous signal. This is also known as ideal (or) instantaneous sampling.

$g(t)=x(t)\delta T_{s}(t)$

As $\delta T_{s}(t)$ is a periodic impulse train it can be expressed in it’s Fourier Series expansion as follows

Exponential Fourier Series is

$\delta T_{s}(t) = \sum_{n=-\infty }^{\infty }F_{n}e^{jnw_{s}t}$

$F_{n}= \frac{1}{T_{s}}\int_{\frac{-T_{s}]}{2}}^{\frac{T_{s}}{2}}\delta T_{s}(t)e^{-jn\omega _{s}t}dt$

$F_{n}=\frac{1}{T_{s}}$

$F_{n}=f_{s}$

∴ Exponential Fourier Series is $\delta T_{s}(t)=\sum_{n=-\infty }^{\infty }f_{s}e^{jn\omega _{s}t}$

now the sampled signal $g(t) = x(t).\delta T_{s}(t)$

$g(t)=x(t)\sum_{n= -\infty }^{\infty }f_{s}e^{jn\omega _{s}t}$

$g(t)=\sum_{n= -\infty }^{\infty }f_{s}x(t)e^{jn\omega _{s}t}$

By finding Fourier Transform of g(t) is G(f)

$G(f)=\sum_{n= -\infty }^{\infty }f_{s}X(f-nf_{s})$

Now the frequency spectrum of the sampled signal G(f) is of the form

From G(f) spectrum the original spectrum of X(f) has been shifted to different center frequencies

i.e, when n=0 center frequency is 0.

n=1 center frequency is f_s

n=-1 center frequency is -f_s etc

Some important conclusions from frequency spectrum of sampled signal:-

The spectrum of sampled signal G(f)/G(w) will repeat periodically if $f_{s}> 2f_{m}$ without any overlapping.
G(f) is extending up to infinity and the Band width is infinity as well, out of G(f) , X(f) need to be recovered , which is band limited to f_m Hz.
X(f) is centered at f=0 and has f_m as the highest frequency, X(f) may be recovered by passing it through a Loe Pass filter with cutoff frequency approximately equals to f_m Hz.
to reconstruct x(t) from g(t) the condition that must be satisfied is $f_{s}\geq 2f_{m}$ .

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Tuned Radio Frequency Receiver

Tuned Radio Frequency Receiver is the primary Radio Receiver and is the most simplest form of Radio receiver.

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Drawbacks of TRF Receiver:-

Selectivity of TRF Receiver is poor. This is because achieving sufficient selectivity at high frequencies is difficult due to enforced use of single-tuned Circuits.
Instability:-(RF Stage) The TRF Receiver suffers from a tendency to oscillate at a higher frequencies (i.e, instability), this is because multi-stage RF amplifiers has to provide high gain at high frequencies. RF amplifiers provides high gain which results in positive feed back leads to oscillations and then causes instability of the circuit. This positive feedback (caused by the leakage of output of RF stage back to it’s input) could result from power supply coupling through any other element common to input and output stages.
Variation of band width over tuning range:- One more draw back in TRF receiver is the BW variation over the tuning range i.e the BW of TRF receiver varies with the incoming frequency.

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Super heterodyne Receiver

The block diagram consists of a receiving antenna followed by an RF stage as the primary block , the receiving signal has been fed to RF stage through the antenna.

The amplified IF signal is given as an input to the Detector. The Detector or the demodulator demodulates the signal and down translates the IF signal to AF(Audio Frequency) signal.

The advantages of Super hetero dyne receiver makes it most suitable for majority of Radio Receiver applications like AM, FM, Communications, SSB, TV and even Radar Receiver.

Advantages of super hetero dyne Receiver:-

It provides high gain through IF amplifier that is more sensitivity is being provided by it.
Improved selectivity over TRF receiver.
Improved adjacent channel rejection.
BW remains constant over the entire operating range.
Selectivity and Sensitivity are uniform throughout it’s tuning range.

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Pulse Position Modulation(PPM)

Pulse Position modulation is another type of Pulse Time modulation technique that is in PPM the position of the pulse carrier is varied in accordance with the instantaneous values of the message signal, where as the amplitude and width of the pulse remains constant. here message lies in the position(OFF periods) of the PPM signal.

PPM demodulator:-

The PPM Demodulator consists of a Transistor T₁ which acts as a switch followed by a second order Low pass filter circuit( using OP-AMP).

As the input to the demodulator is a PPM signal, the gaps between pulses contains the information in PPM signal. Let us consider a PPM signal with OFF and ON periods marked from A to F.

Here Transistor T₁ acts as a switch as follows

input to the base of T₁ is low —–> Transistor T₁ is in cut-off region.
input to the base of T₁ is high —–> Transistor T₁ is in Saturation region.

during the time inerval AB, the input to the base of T₁ is low and transistor T₁ moves into cut-off region in this condition capacitor C charges to a vlotage proportional to length of time duaration AB that is the height of the ramp is equals to duration AB.

During the time interval BC, the input to T₁ is high and T₁ moves into Saturation region in this case Capacitor ‘C’ discharges through T₁ , this discharge is rapid and the collector voltage remains low over the duartion BC.

This process continues and results a saw-tooth wave form at the output of transistor T₁ , by applying this signal to a second order LPFn Demodulated signal has been obtained as the final output.

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Band Pass Sampling

However when the given signal is a Band Pass signal then a different criterion must be used to sample the signal , the Band Pass signal x(t) whose maximum BW is ‘ $2f_{m}/W$ ‘ Hz can be completely represented and recovered from it’s samples if it is sampled at the minimum rate of greater than or equals to twice that of the BW.

then sampling rate $f_{s}\geq 2 X BW$

i.e, $f_{s}\geq 4f_{m} or f_{s}\geq 2W$

Any band pass signal in time-domain can be represented in it’s in-phase $x_{I}(t)$ and quadrature phase $x_{Q}(t)$ components as

$x(t) = x_{I}(t)cos 2\pi f_{c}t \pm x_{Q}(t)sin 2\pi f_{c}t$

after sampling the band pass signal, the signal after reconstruction is

$x(t) = \sum_{n=-\infty }^{\infty }sinc(2f_{m}t-\frac{n}{2})cos(2\pi f_{c}(t-\frac{n}{4f_{m}}))$

$T_{s} = \frac{1}{4f_{m}}$ , where BW of band pass signal is $2f_{m}$ Hz

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Generation of PWM and PPM signals using Wave forms

PWM Generator:-

The circuit that generates PWM wave is as follows, Here in this circuit Op-Amp works in comparator mode.It compares two voltages, modulating voltage with Saw-tooth Voltage. Saw-tooth voltage is taken as reference voltage.

condition	Output voltage V_o(t)
$m(t)> V_{r}(t)$	Low
$V_{r}(t)>m(t)$	High

from the graphs whenever modulating voltage dominates saw-tooth voltage corresponding output is low.

Similarly, when saw-tooth voltage dominates modulating voltage corresponding output is High.

Then the resultant output voltage is a PWM signal.

PPM Generator:-

Now, a PPM signal has been generated by passing the PWM signal through a Mono-stable Multi vibrator . Here the resultant signal is a PPM signal with the pulse starting with respect to trailing edge of PWM signal.

The width and Amplitude of Pulse remains constant only the position of the pulse changes with respect to m(t) .

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Choice Of Intermediate Frequency (or) IF Amplifier in a Radio Receiver

Choice of Intermediate Frequency of a receiving system is usually a compromise , since there are reasons why it is neither low nor high, nor in a certain range between the two.

The following are the major factors influencing the choice of the Intermediate Frequency in any particular system.

If the IF is too high poor selectivity and poor adjacent channel rejection results unless sharp cut-off filters(crystal/mechanical filters) are used in the IF stage.
A high value of Intermediate Frequency(IF) increases tracking difficulties.
If we chose IF as low frequency, image frequency rejection becomes poorer. i.e, if $\frac{f_{si}}{f_{s}}$ is more IFRR(image Frequency Rejection Ratio) has been improved, which requires a high Intermediate Frequency( $f_{si}$ ). Similarly when $f_{s}$ is more IFRR becomes worst.
Average Intermediate Frequency(IF) can make the selectivity too sharp cutting of the side bands.This problem arises because the Q must be low when the IF is low, unless crystal or mechanical filters are used and hence gain per stage is low. Thus a designer is more likely to raise Q rather than increasing the number of IF amplifiers.
If IF is very low , the frequency stability of local oscillator must be made correspondingly high.
IF must not fall in the tuning range of the receiver or else instability occurs and hetero dyne whistles (noise) will be heard.

Frequencies used:-

Standard AM broadcast receivers tuned to (540 KHz-1650 KHz) or(6 MHz-18 MHz) and European long wave band (150 KHZ- 350 KHz) uses IF in the range (438 KHz- 465 KHz). 455 KHz is the most popular value used.
FM receivers using the standard (88 MHz -108 MHz) band have an IF which is almost always 10.7 MHz.
TV Receivers in the VHF band (54 MHz-223 MHz),UHF band (470 MHz-940 MHz) uses IF between (26 MHz-46 MHz) and the popular values are 36 MHz and 46 MHz.
AM-SSB Receviers employed for short-wave reception in the short wave band / VHF band uses IF in the range (1.6 MHz to 2.3 MHz).

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Automatic Gain Control (AGC)

Why AGC is required in a Radio Receiver?

Let us discuss about the facts why we need AGC in a Radio Receiver , as we all know that the voltage gain available at the Receiver from antenna to demodulator in several stages of amplification is very high, so that it can amplify a very weak signal But what if the signal is much stronger at the front end of the receiver ?

If same gain (gain maintained for an incoming weak signal) is maintained by different stages of the Receiver for a stonger incoming signal, the signal is further amplified by these stages and the received signal strength is far beyond the expectations which can be avoided. so we need to have a mechanism which will measure the stength of the input signal and accordingly adjust the gain. AGC does precisely this job and improves the dynamic range of the antenna to (60-100)dB by adjusting the gain of the Intermediate Frequency and sometimes the Radio Frequency stages.

It is generally observed that as a result of fading, the amplitude of the IF carrier signal at the detecor input may vary as much as 30 (or) 40 dB this results in the corresponding variation in general level of reproduced signal at the receiver output.

At IF carrier minimum loud speaker output becomes inaudible and mixed up with noise.

At IF carrier maximum loud speaker output becomes intolerably large.

Therefore a properly designed AGC reduces the amplitude variation due to fading from a high value of (30-40)dB to (3-4)dB.

Basic need of AGC or AVC:-

AGC is a sub system by means of which the overall gain of a receiver is varied automatically with the variations in the stregth of the received signal to keep the output substantially constant.

i.e, the overall requirement of an AGC circuit in a receiver is to maintain a constant output level.

Some of the factors that explain why AGC is needed:-

When a Receiver without AGC/AVC is tuned to a strong station, the received signal may overload the subsequent IF and AF stages this overloading causes carrier distortion in the incoming signal this can be prevented by using manual gain control on first RF stage but now a days AGC circuits are used for this purpose.
When the Receiver is tuned from one station to another, difference in signal strengths of the two stations causes an unpleasant loud output if signal is moving from a weak station to a strong station unless we initially keep the volume control very low before changing the tuning from one station to another . Changing the volume control every time before attempting to re-tunethe receiver is howeve cumbersome. Therefore AGC/AVC enables the user to listen to a station without constantly monitoring the volume control.
AGC is particularly important for mobile Receivers.
AGC helps to smooth out the rapid fading which may occur with long distance short-wave reception.

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Capture effect in Frequency Modulation

The Amplitude Modulation schemes like AM,DSB-SC and SSB-SC systems can not handle inherent Non-linearities in a really good manner where as FM can handle it very well.

Let us suppose un Modulated FM carrier $S(t) = A_{c}cos\omega _{c}(t)$

$S(t) = A_{c}cos(\omega _{c}(t)+\phi (t))$

By considering un modulated FM carrier in terms of frequency(by neglecting phase) i.e $S(t) = A_{c}cos (\omega _{c}t)$ has been interfered by a near by interference located at a frequency $(\omega _{c}+\omega )$ where $\omega$ is a small deviation from $\omega _{c}$ .

the nearby inerference is $I cos(\omega _{c} + \omega )t$

when the original signal got interfered by this near by interference , the received signal is $r(t)= A_{c}cos \omega _{c}t + I cos(\omega _{c}+\omega )t$

$r(t)= (A+ I cos \omega t)cos \omega _{c}t -I sin \omega t sin\omega _{c}t$ Let $A_{c}=A$

$r(t) = E_{r}(t) cos (\omega _{c}t+\Psi _{d}(t))$

now the phase of the signal is $\Psi _{d}(t) = tan^{-1} (\frac{I sin \omega t}{A+I cos \omega t })$

as $A> > I$ implies $\frac{I}{A}< < 1$

$\Psi _{d}(t) = tan^{-1} (\frac{I sin \omega t}{A})$

since $\frac{I}{A}< < 1$ , $\tan ^{-1}\theta = \theta$

$\Psi _{d}(t) \approx \frac{I sin \omega t}{A}$

As the demodulated signal is the output of a discriminator $y _{d}(t) =\frac{d}{dt} (\frac{I sin \omega t}{A})$

$y _{d}(t) =\frac{I\omega }{A} ({cos \omega t})$ , which is the detected at the output of the demodulator.

the detected output at the demodulator is $y_{d}(t)$ in the absence of message signal i.e, $m(t)=0$ .

i.e, when message signal is not being transmitted at the transmitter but detected some output $y_{d}(t)$ which is nothing but the interference.

As ‘A’ is higher the interference is less at t=0 the interference is $\frac{I\omega }{A}$ and is a linear function of $\omega$ , when $\omega$ is small interference is less. That is $\omega$ is closer to $\omega _{c}$ interference is less in FM.

Advantage of FM :- is Noise cancellation property , any interference that comes closer with the carrier signal (in the band of FM) more it will be cancelled. Not only that it overridden by the carrier strength $A_{c}$ but also exerts more power in the demodulated signal.

This is known as ‘Capture effect’ in FM which is a very good property of FM. Over years it has seen that a near by interference is 35 dB less in AM where as the near by interference in FM is 6 dB less this is a big advantage.

Two more advantages of FM over AM are:

Non-linearity in the Channel ,FM cancels it very nicely due to it’s inherent modulation and demodulation technique.
Capture effect( a near by interference) FM overrides this by $A_{c}$ .
Noise cancellation.

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Phase Locked Loop (PLL)

Demodulation of an FM signal using PLL:-

Let the input to PLL is an FM signal $S(t) = A_{c} \sin (2 \pi f_{c}t+2\pi k_{f} \int_{0}^{t}m(t)dt)$

let $\Phi _{1} (t) = 2\pi k_{f} \int_{0}^{t}m(t)dt ------Equation (I)$

Now the signal at the output of VCO is FM signal (another FM signal, which is different from input FM signal) Since Voltage Controlled Oscillator is an FM generator.

$\therefore b(t) = A_{v} \cos (2 \pi f_{c}t+2\pi k_{v} \int_{0}^{t}v(t)dt)$

the corresponding phase $\Phi _{2} (t) = 2\pi k_{v} \int_{0}^{t}v(t)dt ------Equation (II)$

It is observed that S(t) and b(t) are out of phase by $90^{o}$ . Now these signals are applied to a phase detector , which is basically a multiplier

$\therefore$ the error signal $e(t) =S(t) .b(t)$

$e(t) =A_{c} \sin (2 \pi f_{c}t+2\pi k_{f} \int_{0}^{t}m(t)dt). A_{v} \cos (2 \pi f_{c}t+2\pi k_{v} \int_{0}^{t}v(t)dt)$

$e(t) =A_{c}A_{v} \sin (2 \pi f_{c}t+\phi _{1}(t)). \cos (2 \pi f_{c}t+\phi _{2}(t))$

on further simplification , the product yields a higher frequency term (Sum) and a lower frequency term (difference)

$e(t) =A_{c}A_{v}k_{m} \sin (4 \pi f_{c}t+\phi _{1}(t)+\phi _{2}(t))- A_{c}A_{v}k_{m}\sin (\phi _{1}(t)-\phi _{2}(t))$

$e(t) =A_{c}A_{v}k_{m} \sin (2 \omega _{c}t+\phi _{1}(t)+\phi _{2}(t))- A_{c}A_{v}k_{m}\sin (\phi _{1}(t)-\phi _{2}(t))$

This product e(t) is given to a loop filter , Since the loop filter is a LPF it allows the difference and term and rejects the higher frequency term.

the over all output of a loop filter is

]]> 1652 https://atomic-temporary-93025954.wpcomstaging.com/frequency-domain-representation-of-a-wide-band-fm/ Thu, 21 Feb 2019 13:48:15 +0000 https://atomic-temporary-93025954.wpcomstaging.com/?p=1639 Continue reading “Frequency domain representation of a Wide Band FM”]]> To obtain the frequency-domain representation of Wide Band FM signal for the condition $\beta > > 1$ one must express the FM signal in complex representation (or) Phasor Notation (or) in the exponential form

i.e, Single-tone FM signal is $S_{FM}(t)=A_{c}cos(2\pi f_{c}t+\beta sin 2\pi f_{m}t).$

Now by expressing the above signal in terms of Phasor notation ( $\because \beta > > 1$ , None of the terms can be neglected)

$S_{FM}(t) \simeq Re(A_{c}e^{j(2\pi f_{c}t+\beta sin 2\pi f_{m}t)})$

$S_{FM}(t) \simeq Re(A_{c}e^{j2\pi f_{c}t}e^{j\beta sin 2\pi f_{m}t})$

$S_{FM}(t) \simeq Re(e^{j2\pi f_{c}t} A_{c}e^{j\beta sin 2\pi f_{m}t})-------Equation(I)$

Let $\widetilde{s(t)} =A_{c}e^{j\beta sin 2\pi f_{m}t}$ is the complex envelope of FM signal.

$\widetilde{s(t)}$ is a periodic function with period $\frac{1}{f_{m}}$ . This $\widetilde{s(t)}$ can be expressed in it’s Complex Fourier Series expansion.

i.e, $\widetilde{S(t)} = \sum_{n=-\infty }^{\infty }C_{n} e^{jn\omega _{m}t}$ this approximation is valid over $[-\frac{1}{2f_{m}},\frac{1}{2f_{m}}]$ . Now the Fourier Coefficient $C_{n} = \frac{1}{T} \int_{\frac{-T}{2}}^{\frac{T}{2}} \widetilde{S(t)} e^{-jn2\pi f_{m}t}dt$

$T= \frac{1}{f_{m}}$

$C_{n} = \frac{1}{\frac{1}{f_{m}}} \int_{\frac{-1}{2f_{m}}}^{\frac{1}{2f_{m}}} \widetilde{S(t)} e^{-jn2\pi f_{m}t}dt$

$C_{n} = f_{m} \int_{\frac{-1}{2f_{m}}}^{\frac{1}{2f_{m}}} A_{c}e^{j\beta sin 2\pi f_{m}t} e^{-jn2\pi f_{m}t}dt$

$C_{n} = f_{m} \int_{\frac{-1}{2f_{m}}}^{\frac{1}{2f_{m}}} A_{c}e^{{j\beta sin 2\pi f_{m}t-jn2\pi f_{m}t}}dt$

$C_{n} = f_{m} \int_{\frac{-1}{2f_{m}}}^{\frac{1}{2f_{m}}} A_{c}e^{j({\beta sin 2\pi f_{m}t-n2\pi f_{m}t})}dt$

let $x=2\pi f_{m}t$ implies $dx=2\pi f_{m}dt$

as $x\rightarrow \frac{-1}{2f_{m}} \Rightarrow t\rightarrow -\pi$ and $x\rightarrow \frac{1}{2f_{m}} \Rightarrow t\rightarrow \pi$

$C_{n} = \frac{A_{c}}{2\pi } \int_{-\pi }^{\pi } e^{j({\beta sin x-nx})}dx$

let $J_{n}(\beta ) = \frac{1}{2\pi } \int_{-\pi }^{\pi } e^{j({\beta sin x-nx})}dx$ as $n^{th}$ order Bessel Function of first kind then $C_{n} = A_{c} J_{n}(\beta )$ .

Continuous Fourier Series expansion of

$\widetilde{S(t)} = \sum_{n=-\infty }^{\infty }C_{n} e^{jn\omega _{m}t}$

$\widetilde{S(t)} = \sum_{n=-\infty }^{\infty }A_{c} J_{n} (\beta )e^{jn\omega _{m}t}$

Now substituting this in the Equation (I)

$S_{WBFM}(t) \simeq Re(e^{j2\pi f_{c}t} \sum_{n=-\infty }^{\infty }A_{c} J_{n} (\beta )e^{jn\omega _{m}t})$

$S_{WBFM}(t) \simeq A_{c} Re( \sum_{n=-\infty }^{\infty }J_{n} (\beta ) e^{j2\pi f_{c}t} e^{jn\omega _{m}t})$

$S_{WBFM}(t) \simeq A_{c} Re( \sum_{n=-\infty }^{\infty }J_{n} (\beta ) e^{j2\pi (f_{c}+nf _{m}t)})$

$\therefore S_{WBFM}(t) \simeq A_{c} \sum_{n=-\infty }^{\infty }J_{n} (\beta ) cos 2\pi (f_{c}+nf _{m}t)$

The Frequency spectrum can be obtained by taking Fourier Transform

$S_{WBFM}(f) = \frac{A_{c}}{2}\sum_{n=-\infty }^{\infty }J_{n}(\beta )$

n value	wide Band FM signal
0	$S_{WBFM}(f) = \frac{A_{c}}{2}\sum_{n=-\infty }^{\infty }J_{0}(\beta )$
1	$S_{WBFM}(f) = \frac{A_{c}}{2}\sum_{n=-\infty }^{\infty }J_{1}(\beta )$
-1	$S_{WBFM}(f) = \frac{A_{c}}{2}\sum_{n=-\infty }^{\infty }J_{-1}(\beta )$
…	….

From the above Equation it is clear that

FM signal has infinite number of side bands at frequencies $(f_{c}\pm nf_{m})$ for n values changing from $-\infty$ to $\infty$ .
The relative amplitudes of all the side bands depends on the value of $J_{n}(\beta )$ .
The number of significant side bands depends on the modulation index $\beta$ .
The average power of FM wave is $P=\frac{A_{c}^{2}}{2}$ Watts.

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Reconstruction filter(Low Pass Filter)

Reconstruction filter (Low Pass Filter) Procedure to reconstruct actual signal from sampled signal:-

Low Pass Filter is used to recover original signal from it’s samples. This is also known as interpolation filter.

An LPF is that type of filter which passes only low frequencies up to cut-off frequency and rejects all other frequencies above cut-off frequency.

For an ideal LPF, there is a sharp change in the response at cut-off frequency as shown in the figure.

i.e, Amplitude response becomes suddenly zero at cut-off frequency which is not possible practically that means an ideal LPF is not physically realizable.

i.e, in place of an ideal LPF a practical filter is used.

In case of a practical filter, the amplitude response decreases slowly to zero (this is one of the reason why we choose $f_{s}>2f_{m}$ )

This means that there exists a transition band in case of practical Low Pass Filter in the reconstruction of original signal from its samples.

Signal Reconstruction (Interpolation function):-

The process of reconstructing a Continuous Time signal x(t) from it’s samples is known as interpolation.

Interpolation gives either approximate (or) exact reconstruction (or) recovery of CT signal.

One of the simplest interpolation procedures is known as zero-order hold.

Another procedure is linear interpolation. In linear interpolation the adjacent samples (or) sample points are connected by straight lines.

We may also use higher order interpolation formula for reconstructing the CT signal from its sample values.

If we use the above process (Higher order interpolation) the sample points are connected by higher order polynomials (or) other mathematical functions.

For a Band limited signal, if the sampling instants are sufficiently large then the signal may be reconstructed exactly by using a LPF.

In this case an exact interpolation can be carried out between sample points.

Mathematical analysis:-

A Band limited signal x(t) can be reconstructed completely from its samples, which has higher frequency component f_m Hz.

If we pass the sampled signal through a LPF having cut-off frequency of f_m Hz.

From sampling theorem

$g(t) = x(t).\delta _{T_{s}}(t)$ .

$g(t)=\frac{1}{T_{s}}\left \{ 1+2\cos \omega _{s}t+2\cos 2\omega _{s}t+2\cos 3\omega _{s}t+..... \right \}$ .

g(t) has a multiplication factor $\frac{1}{T_{s}}$ . To reconstruct x(t) (or) X(f) , the sampled signal must be passed through an ideal LPF of Band Width of $f_{m}$ Hz and gain $T_{s}$ .

$\left | H(\omega ) \right |=T_{s} \ for \ -\omega _{m}\leq \omega \leq \omega _{m}$ .

$h(t) = \frac{1}{2\pi } \int_{-\omega _{m}}^{\omega _{m}}T_{s}e^{j\omega t}\ d\omega$ .

$h(t) = 2f_{m}T_{s} \ sinc(2\pi f_{m}t)$ .

If sampling is done at Nyquist rate , then Nyquist interval is $T_{s} = \frac{1}{2f_{m}}$ .

therefore $h(t) = \ sinc(2\pi f_{m}t)$ .

h(t) = 0. at all Nyquist instants $t= \pm \frac{n}{2f_{m}}$ , when g(t) is applied at the input to this filter the output will be x(t) .

Each sample in g(t) results a sinc pulse having amplitude equal to the strength of sample. If we add all these sinc pulses that gives the original signal x(t) .

$g(t) = x(kT_{s})\delta (t-kT_{s})$ .

$x(t) =\sum_{k} x(kT_{s})\ h (t-kT_{s})$ .

$x(t) =\sum_{k} x(kT_{s})\ sinc(2\pi f_{m} (t-kT_{s}))$ .

$x(t) =\sum_{k} x(kT_{s})\ sinc(2\pi f_{m}t-k\pi )$ .

This is known as interpolation formula

It is assumed that the signal x(t) is strictly band limited but in general an information signal may contain a wide range of frequencies and can not be strictly band limited this means that the maximum frequency in the signal can not be predictable.

then it is not possible to select suitable sampling frequency f_s .

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Figure of merit of FM

The block diagram of FM Receiver in the presence of noise is as follows

The incoming signal at the front end of the receiver is an FM signal $S_{FM}(t) = A_{c} cos(2\pi f_{c}t+2\pi k_{f}\int m(t )dt )--------Equation(1)$ got interfered by Additive noise $n(t)$ , since the FM signal has a transmission band width $B_{T}$ ,the Band Pass filter characteristics are also considered over the band of interest i.e from $\frac{-B_{T}}{2}$ to $\frac{B_{T}}{2}$ .

The output of Band Pass Filter is $x(t) = S_{FM}(t)+n_{o}(t)------------------Equation(2)$ is passed through a Discriminator for simplicity simple slope detector (discriminator followed by envelope detector) is used, the output of discriminator is $v(t)$ this signal is considered over a band of $(-W,W)$ by using a LPF .

The input noise to the BPF is n(t), the resultant output noise is band pass noise $n_{o}(t)$

$n_{o}(t) = n_{I}(t)cos \omega _{c}t-n_{Q}(t)sin \omega _{c}t-----------Equation(3)$

phasor representation of Band pass noise is $n_{o}(t)=r(t)cos (\omega _{c}t +\Psi (t))$ where $r(t)=\sqrt{n_{I}^{2}(t)+n_{Q}^{2}(t)}$ and $\Psi (t) = \tan ^{-1}(\frac{n_{Q}(t)}{n_{I}(t)})$ .

$n_{I}(t),n_{Q}(t)$ are orthogonal, independent and are Gaussian.

$r(t)$ – follows a Rayleigh’s distribution and $\Psi (t)$ is uniformly distributed over $(0,2\pi )$ . $r(t) and \Psi (t)$ are separate random processes.

substituting Equations (1), (3) in (2)

$x(t) = S_{FM}(t)+n_{o}(t)$

$x(t)= A_{c} cos(2\pi f_{c}t+2\pi k_{f}\int m(t )dt )+n_{I}(t)cos \omega _{c}t-n_{Q}(t)sin \omega _{c}t$

$x(t)= A_{c} cos(2\pi f_{c}t+2\pi k_{f}\int m(t )dt )+r(t)cos (\omega _{c}t +\Psi (t))$

$x(t)= A_{c} cos(2\pi f_{c}t+\Phi (t) )+r(t)cos (\omega _{c}t +\Psi (t))-----Equation(4)$ where $\Phi (t)=2\pi k_{f}\int m(t )dt$ .

now the analysis is being done from it’s phasor diagram/Noise triangle as follows

$x(t)$ is the resultant of two phasors $A_{c} cos(2\pi f_{c}t+\Phi (t) )$ and $r(t)cos (\omega _{c}t +\Psi (t))$ .

$\theta (t)-\Phi (t) = \tan ^{-1}(\frac{r(t)\sin (\Psi (t)-\Phi (t))}{A_{c}+r(t)\cos (\Psi (t)-\Phi (t))})$

$\theta (t)-\Phi (t) = \tan ^{-1}(\frac{r(t)\sin (\Psi (t)-\Phi (t))}{A_{c}})$ since $r(t)< <A _{c}$

$\theta (t)=\Phi (t) +\tan ^{-1}(\frac{r(t)\sin (\Psi (t)-\Phi (t))}{A_{c}})$

$\theta (t)=\Phi (t) +\frac{r(t)\sin (\Psi (t)-\Phi (t))}{A_{c}}$ because $\frac{r(t)}{A_{c}}< < 1\Rightarrow \tan ^{-1}\theta =\theta$ .

$\theta (t)$ is the phase of the resultant signal $x(t)$ and when this signal is given to a discriminator results an output $v(t)$ .

i.e, $v(t)=\frac{1}{2\pi }\frac{d\theta (t)}{dt}$

i.e, $v(t)= \frac{1}{2\pi }\frac{d}{dt}(\Phi (t) +\frac{r(t)\sin (\Psi (t)-\Phi (t))}{A_{c}})---------Equation(5)$

As $\Phi (t) =2\pi k_{f}\int m(t)dt$

$\frac{d\Phi (t)}{dt}=2\pi k_{f}m(t)$

the second term in the Equation $n_{d}(t)=\frac{1}{2\pi }\frac{d}{dt}(\frac{r(t)\sin (\Psi (t)-\Phi (t))}{A_{c}})$ where $n_{d}(t)$ – denotes noise after demodulation.

this can be approximated to $n_{d}(t)=\frac{1}{2\pi A_{c} }\frac{d}{dt}(r(t)\sin \Psi (t))-------Equation(6)$ , which is a valid approximation. In this approximation $r(t)\sin \Psi (t)$ is Quadrature-phase noise with power spectral density $S_{NQ}(f)$ over $(\frac{-B_{T}}{2},\frac{B_{T}}{2})$

the power spectral density of $n_{d}(t)$ will be obtained from Equation (6) using Fourier transform property $\frac{d}{dt}\leftrightarrow j2\pi f$

$S_{Nd}(f)=\frac{1}{(2\pi A_{c})^{2}}(2\pi f)^{2}S_{NQ}(f)$

$S_{Nd}(f)=(\frac{f}{A_{c}})^{2}S_{NQ}(f)$ , $\left | f \right |\leq \frac{B_{T}}{2}$

$S_{Nd}(f)=0$ elsewhere.

the power spectral density functions are drawn in the following figure

$\therefore v(t) = k_{f}m(t)+n_{d}(t)-------Equation(7))$ , from Carson’s rule $\frac{B_{T}}{2}\geq W$

the band width of v(t) has been restricted by passing it through a LPF.

Now, $S_{Nd}(f)=(\frac{f}{A_{c}})^{2}S_{NQ}(f),\left | f \right |\leq W$

$S_{Nd}(f)=0 elsewhere$ .

To calculate Figure of Merit $FOM = \frac{(SNR)_{output}}{(SNR)_{input}}$

Calculation of $(SNR)_{output}$ :-

output Noise power $P_{no} = \int_{-W}^{W}(\frac{f}{A_{c}})^{2} N_{o}df$

$P_{no} = \frac{N_{o}}{A_{c}^{2}} \left ( \frac{f^{3}}{3} \right )^{W}_{-W}$

$P_{no} = \frac{N_{o}}{A_{c}^{2}} \left ( \frac{2W^{3}}{3} \right )------Equation(I)$

The output signal power is calculated from $k_{f}m(t)$ tha is $P_{so} = k_{f}^{2}P--------Equation(II)$

$(SNR)_{output} = \frac{P_{so}}{P_{no}}$

From Equations(I) and (II)

$(SNR)_{output} =\frac{\frac{N_{o}}{A_{c}^{2}} \left ( \frac{2W^{3}}{3} \right )}{k_{f}^{2}P}$

$(SNR)_{output} =\frac{3}{2}\frac{k_{f}^{2}PA_{c}^{2}}{N_{o}W^{3}}-------Equation(8)$

Calculation of $(SNR)_{input}$ :-

$(SNR)_{input} = \frac{P_{si}}{P_{ni}}$

input signal power $P_{si}= \frac{A_{c}^{2}}{2}---------------Equation(III)$

noise signal power $P_{ni}=N_{o}W--------------Equation(IV)$

from Equations (III) and (IV)

$(SNR)_{input} = \frac{A_{c}^{2}}{2WN_{o}}-------------------Equation(9)$

Now the Figure of Merit of FM is $FOM = \frac{(SNR)_{output}}{(SNR)_{input}}$

$FOM = \frac{\frac{3}{2}\frac{k_{f}^{2}PA_{c}^{2}}{N_{o}W^{3}}}{\frac{A_{c}^{2}}{2WN_{o}}}$

$FOM_{FM} = \frac{3k_{f}^{2}P}{W^{2}}--------Equation(10)$

to match this with AM tone(single-tone) modulation is used i.e, $m(t) = cos \omega _{m}t$ then the signal power $P = \frac{1}{2}$ and $W = f_{m}$

$FOM_{FM} = \frac{3k_{f}^{2}}{f_{m}^{2}}\frac{1}{2}$

$FOM_{FM} = \frac{3}{2} (\frac{k_{f}}{f_{m}})^{2}$

$FOM_{FM} = \frac{3}{2}\beta ^{2}$

since for tone(single-tone) modulation $\beta = \frac{k_{f}}{f_{m}}$ .

when you compare single-tone FM with AM $FOM_{FM (single-tone)} =FOM_{AM(single-tone)}$

$\frac{3}{2}\beta ^{2} > \frac{1}{3}$

$\beta > \frac{\sqrt{2}}{3}$

$\beta > 0.471$ .

the modulation index $\beta >0.471.$ will be beneficial in terms of noise cancellation, this is one of the reasons why we prefer WBFM over NBFM.

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Asynchronous Transfer Model(ATM)-Introduction and design goals

Introduction:-

Asynchronous Transfer Model is another important connection oriented Network.

Why we call it asynchronous is most of the transmission in telephone systems is synchronous (closed tied to a clock) but ATM is not such type.

ATM was designed in 1990’s, it was the cell ray protocol designed by the ATM forum and was adopted by ITU-T.

Design goals:-

A technology is required that provides large data rates with the high data rate transmission media available (Optical Fiber Communication) and this media requires less susceptible to noise.
The system must interface with existing systems to provide wide-area inter connectivity.
The cost for such a system should not be more.
The new system must be connection-oriented type.
The new system must be able to work with the existing tele-communication hierarchies like local loops, long distance carriers etc.

The problems associated with existing networks:-

The design goals come into picture for ATM, since there exists some problems that are associated with the existing systems.

Frame Networks:-

Before ATM we have data communications at DLL are based on frame switching and frame networks .

i.e, different protocols use frames of varying size (frame has data and header). If header size is more than that of actual data there is a burden so some protocols have enlarged the size of data unit relative to the header.

if there is no data in such cases there is a wastage , so there is to provide variable frame sizes to the users.

Mixed N/w Traffic:-

If there exists variable frame sizes

The switches Multiplexers and routers must require an elaborate Software to manage variable size frames.
Internet working among different frame N/w ‘s become slow and expensive too.

suppose we have two networks generating frames of variable sizes that is N/W 1 is connected to line 1 and the frame is X. N/W 2is connected to line 2 and of having 3 frames of equal sizes A,B,C are connected to a TDM.

If X has arrived a bit earlier than A,B,C (having more priority than X) on the output line . The frames has to wait for a time to move on to the output line, this causes delay for line 2 N/W.

i.e, Audio and video frames are small so mixing them with conventional data traffic often creates unacceptable delays and makes shared frame links unusable for audio and video information.

but we need to send all kinds of traffic over the same links.

Cell Networks:-

so a solution to frame internet working is by adopting a concept called cell networking.

In a cell N/W we use a small data unit of fixed size called cell so all types of data are loaded into identical cells and are multiplexed with other cells and are routed through the cell N/W.

because each cell is small and of same size the problems associated with multiplexing different sized frames are avoided.

Asynchronous TDM:-

ATM uses asynchronous TDM- hence the name Asynchronous Transfer Model.

i.e, it multiplexes data coming from different channels. it also uses fixed size slots called cells.

ATM Mux’rs fill a slot with a cell from any input channel that has a cell and slot is empty if there is no cell.

ATM architecture:-

ATM was going to solve all the world’s networking and tele-communications problems by merging voice, data, cable TV, telex,telegraph…… and everything else into a single integrated system that could do everything for everyone.

i.e, ATM was much successful than OSI and is now widely used in telephone system for moving IP packets.

ATM is a cell-switched N/W the user access devices are connected through a user-to- N/W interface (UNI) to the switches inside the N/W. The switches are connected through N/W-to-N/W interface (NNI) as shown in the following figure

Virtual Connection:-

two end points is accomplished through transmission paths (TP’s), Virtual Paths (VP’s) and Virtual Circuits (VC’s)

ATM Virtual Circuits:-

Since ATM N/w’s are connection-oriented, sending data requires a connection , first sending a packet to setup the connection.

as setup packet travels though the sub net all the routers on the path make an entry in their internal tables noting for existence and reserving the resources.
connections are often called virtual circuits and most ATM N/W’s support permanent virtual circuits. i.e, for permanent connections b/w two hosts.
after establishing a connection either side can transmit data.
all information is in small, fixed size packets called cells.
cell routing is done in Hard ware at high speed.
fixed size cells makes the building of Hard ware routers easier with short, fixed length cells.
variable length IP packets have to be routed by Software which is a slower process.
ATM uses the Hardware that can setup to copy one incoming cell to multiple output lines (ex:-TV).
All cells follow the same route to the destination.
cell delivery is not guaranteed but their order is.
if cells lost along the way it is up to higher protocol levels to recover from lost cells but this also not guarantee.
ATM N/W’s are organized like traditional WAN’s with lines and switches.
the most common speeds for ATm are

155 Mbps-used for high definition TV.
155.52 Mbps-used for AT & T’s SONET transmission system
622 Mbps-4 X 155 Mbps channels can be sent over it.

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Difference between Congestion control and flow control

congestion control	flow control
Main theme of congestion control is to carry the offered traffic without any loss.	whereas flow control relates the point-to-point traffic between a fast sender and a slow receiver.
It is global issue involving Routers, hosts behavior.	It involves the behavior of hosts that is sender and receiver.
It uses the concept of store and forward.	It uses the signals acknowledgements and negative acknowledgements.
a store and forward n/w with 1 Mbps lines that uses 50 computers 25 wants to send data to the half(25) at a time causes congestion.	A N/w with a capacity of 1 Gbps, on which a superfast computer tries to transfer a file to a normal PC. we require flow control but not congestion control.

Network Hardware

Now we turn our attention from the applications and social issues (fun stuff) to technical issues involved in N/w design (the work stuff).

generally CN are designed by 2 important things

1.. Transmission technology.

2. Scale(size).

within a N/w , now we need to specify how data are transmitted from one device to other, which we will discuss later.

1. Transmission Technology:-

Broadly speaking there exists two types of Transmission technology that are widely in use

i. Braodcast links.

ii. Point to Point links.

Broadcast links(or) Networks:-

These type of N/w’s will have a single communication channel.
this channel is shared by all the machines on the network.
short messages called ‘packets’ sent by any machine are received by all the others.
for intended reception from a machine a address field with in the packet is used.
A machine the packet, if the packet is intended for it , it will process it otherwise it will simply ignores it.

ex:-i. calling a person who stands in the corridor from one end to other end.

ii. In the airport the announcement asking for the flight 644 passengers to report gate12 for immediate boarding.

Broadcasting Networks also allow to send a packet to all the users of the Network by using a special code in the address field, that is in this case a packet is received by all the machines and is processed by every machine . This is known as Broadcasting.

Multicasting:-

Broadcast Networks also supports transmission to a subset of machines known as Multicasting.

its address field contains n bits.

onebit—> reserved for indication of multicasting.

(n-1) bits—> can hold a group number.

and each machine in a multicast N/w is able to subscribe any number of groups and a packet send to a group can be delivered to all machines subscribing that group.

Point to Point links:-

Point to Point N/ws consists of many connections between individual pair of machines.
to go from source to destination a packet on this type of N/w may go from intermediate machines often multiple roots of different lengths. so finding good one is important.
smaller geographically localized networks use broadcasting while larger networks uses point to point.
A point to Point with one sender and one receiver is called Unicasting.
It provides a dedicated communication link between two devices most of the cases these uses wire (or) cable to connect two ends but Microwave (or) Satellite links may also possible.

ex:- T.V controlling with infrared remote control.

Entire channel capacity is reserved for Transmission between Source and Destination.

2. Scale(size):-

Another alternative criterion for classifying networks is their scale, that is we classify multiple processor systems by their physical size as follows PAN,LAN,WAN,MAN.

(write about PAN,LAN,WAN,MAN).

Flooding(static)

This is another type of static algorithm.

the main concept of flooding is to sent every incoming packet on a line to every other outgoing line except the line it arrived on.

flooding generates a large no.of duplicate packets, sometimes infinite unless we may take certain measures.

the measures are as follows:-

one measure is use of hop count in the header of each packet and decrement this count at each hop when count reaches to zero discard the packet.
How to take this hop count is another problem. Generally it is set to the length of path from source to destination and in worst cases the full diameter of the subnet.

another way is avoid sending a packet more than once through a router this is possible by using sequence no.
i.e, a source router (which generates packets) can put a sequence no. to each packet and each router will maintain a list of sequence nos. and if sees a packet with same sequence no in the list that packet is discarded (not flooded).

another way of flooding is of use selective flooding.

i.e, with this the router wouldn’t send every incoming packet on every line instead the router will send packets in a particular direction only.

i.e, east bound packets are sent on east side routers and similarly on west side by west side routers.

even flooding is cumbersome, it has some uses

i.e,

used in military applications.
used in distributive data base applications in which to update all data bases concurrently.
used in broadcast Routing.

flooding is used rather than any other algorithm since flooding chooses shorter path between two nodes where other algorithms may not.

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Broadcast Routing(dynamic)

In some applications hosts need to send messages to many (or) all other hosts like weather reports, stock market updates (or) live radio programs.

i.e, sending a packet to all destinations simultaneously is called Broadcasting.

Different methods of Broadcasting:-

first method is to send a packet to all destinations. This is a method wasteful of Band width and source needs to know the complete list of all destinations.

so this is least desirable one.

flooding is another way to broadcast a packet, the problem with flooding is that it generates too many packets and also consumes too much of Band width.
Third way is to use multi destination routing

In this technique each packet contains a list of destinations (or) a bit map for those destinations.

when a packet arrives at a router, the router checks all the output lines it requires. The router generates a new copy of the packet for each output line after sufficient number of hops each packet will carry only one destination.

i.e, multi destination routing is like separately addressed packets (to B,C,D,E & D) must follow the same route one of them pays full fare and rest are free.

The fourth type of method is to use sink tree (or) spanning tree.

A spanning tree is a subset of subnet that includes all the routers but contains no loops.

if each router knows which of it’s lines belong to spinning tree then it broadcasts packet to all the lines except the one it arrived on.

This is efficient method in terms of Band width usage but problem is to maintain the knowledge of all the nodes of spanning tree at a routes.

Last method is to use Reverse path forwarding to approximate behavior of spanning tree.

Consider a subnet and it’s sink tree for router I as root node and how reverse path algorithm works in figure (C)

on the first hop I sends packets to F, H, J & N. on the second hop eight packets are generated among them 5 are given to preferred paths indicated as circles (A,D,G,O,M)

of the 6 packets generated in third hop only 3 are given to preferred paths (C,E & K) the others are duplicates.

in the fourth hop to B and L after this broadcasting terminates.

advantages of reverse path forwarding:-

it is easy to implement.
it does not require routers to known about spanning trees.
it does not require any special mechanism to stop the process (as like flooding).

The principle is : if a packet arrives on a line if it is preferred one to reach source it gets forwarded.

if it arrives on a line that is not preferred one that packet is discarded as a duplicate.

ex:-

when a packet arrives at ‘L’ the preferred paths are N and P so it forward the packets to both N and P and if a packet arrives at ‘K’ , there the preferred path is M and N is not preferred so it forwards packet to M and discards to N.

This is reverse path forwarding.

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Distance Vector Routing(dynamic)

Modern Computer Networks uses dynamic algorithms rather than static algorithms.

dynamic algorithms may consider the current traffic (or) load on the Network.

Two types of dynamic routing algorithms are there

Distance Vector Routing (DVR).
Link state Routing.

Distance Vector Routing operates by the following way

each Router maintains a table (gives the information about distance to other routers) and updates these routing tables by exchanging information with it’s neighbors.

It is also known as Bellman-Ford (or) Ford Fulkerson algorithm.

DVR is used in ARPANET and also as RIP.

In DVR each Router will maintain a Routing Table regarding to each Router in the subnet and the estimate of the time (or) distance to the destination.

one can use different design metrics like no.of hops, time delay in (milli Seconds), no.of packets Queued etc.

Here time delay is used as a metric.

Therefore a Router knows a delay to each of it’s neighbors and once every T milli Seconds these delays get updated by exchanging information with it’s neighboring Routers.

Consider a subnet with Routers A,B,…..L . Now choose a Router J with immediate neighbors (directly connected to J) are A, I, H and K.

Now the estimated delay of J to A, I, H & K are 8, 10, 12 & 6 milli Seconds respectively.

Suppose J wants to calculate a new route from J to G this is possible by finding the delay from J to G using the neighbors to J.

i.e, J to G delay (through A) = J to A delay +A to G delay = 8+18=26 mSec.

J to G delay (through I) = J to I delay +I to G delay = 10+31=41 mSec.

J to G delay (through H) = J to H delay +H to G delay = 12+6=18 mSec.

J to G delay (through K) = J to K delay +K to G delay = 6+31=37 mSec.

The best among the 4 possibilities is through H with less delay 18 mSec and makes as entry in it’s Routing table.

In this way Router J computes all possible delays to each router and updates it in it’s Routing table.

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Shortest Path Routing (static)

The idea of this shortest path routing is simple, which is used to build a graph with each node as router and arc represents a communication line (or) link.

This algorithm just finds the shortest path between them on the graph.

There exists many design metrics to choose to get the shortest path are no.of hops, queue length,transmission delay etc.

for example if we choose no.of hops as metric, the paths ABC, ABE have equal no of hops means that those are equally long but ABC is much larger than ABE.

The labels on the above graph (2,2,7) are computed as a function of the distance, Band width, average traffic, cost etc.

one of the algorithm used for computing the shortest path between 2 nodes is Dijkstra’s algorithm.

it is as follows, Initially all nodes are labeled with infinite distance.

Let us consider the figure as shown below

to find the shortest path from A to D.

step 1:- choose the source node as A and mark it as permanent node.

step 2:- find the adjacent nodes to A those are B and G then choose the node with the smallest label as the permanent node.

Now this node B becomes the new working node.

step 3:- Now start at B and repeat the same procedure

by following above procedure two paths are available ABEGHD with a distance of 11 from A and ABEFHD with a distance of 10 from A.

so the second path is chosen as a shortest path.

therefore the final shortest path is ABEFHD with nodes A,B,E,F,H and D as permanent nodes.

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Distance Vector Routing(dynamic)

Modern Computer Networks uses dynamic algorithms rather than static algorithms.

dynamic algorithms may consider the current traffic (or) load on the Network.

Two types of dynamic routing algorithms are there

Distance Vector Routing (DVR).
Link state Routing.