Analog Communications Archives

Receiver Characteristics

The following are the Receiver characteristics

Sensitivity.
Selectivity.
Fidelity.
Image frequency Rejection ratio.
tracking.

1.Sensitivity:- It is defined as the ability of a radio receiver to amplify weak signals. In another way that is the voltage that must be applied to the receiver input terminals to give a standard output power, measured at the output terminals.

Sensitivity is often expressed in micro volts (or) in decibels below 1 volt and measured at three points along the tuning range when a production receiver is lined up.
for example Sensitivity can be expressed as at 100 KHz a particular receiver has a sensitivity of 12.7 μV (or) -98dBV (dBV means decibels below 1V). 12.7 μV for a signal to Noise ratio of 20 dB in the output of the receiver.
For professional receivers, Sensitivity can be expressed as signal power required to produce a minimum acceptable output signal with a minimum acceptable Signal-to-noise ratio.
The curve, which is a plot of sensitivity of a good domestic receiver.
Sensitivity of a portable and other small receivers used in the broadcast band is in the presence of 150 μV.
where as Communication receivers may have better than 1 μV in the High Frequency band.

Sensitivity of a Superheterodyne receiver is determined by the following

The gain of Intermediate Frequency Amplifiers.
The gain of Radio Frequency Amplifiers.
The Noise Figure of the Receiver.

2. Selectivity:-

The Selectivity of a receiver may be defined as the ability to reject unwanted signals.
Typical Selectivity curve is shown below.
This figure shows the attenuation that the receiver offers to signals at frequencies near to the one to which it is tuned.
Selectivity is measured at the end of the Sensitivity test with the same conditions.
The curve is plotted between attenuation in dB Vs frequency.
Selectivity varies with respect to received frequency (if ordinary tuned circuits are used in Intermediate Frequency section) and this selectivity becomes worse when the received frequency increases.
Selectivity determines the adjacent channel rejection of a receiver.

3. Fidelity:-

Fidelity is the ability of a receiver to reproduce all the modulating frequencies equally.
The fidelity of the receiver basically depends on the Audio frequency Amplifier frequency response.
High fidelity is essential in order to reproduce a good quality music faithfully in audio applications that is with out introducing any distortion.
for this it is essential to have a flat-frequency response over a wide range of audio frequencies.
ideally the frequency response is flat over Audio Frequency range but practically, it decreases in the lower and higher cut off frequency sides.

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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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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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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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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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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.

Note: There is a rating embedded within this post, please visit this post to rate it. ]]> 1413 https://atomic-temporary-93025954.wpcomstaging.com/capture-effect-in-frequency-modulation/ Tue, 27 Nov 2018 10:07:21 +0000 https://atomic-temporary-93025954.wpcomstaging.com/?p=1403 Continue reading “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

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.

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.

source:-

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

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:-

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 .

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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new

single tone AM

Time-domain representation of SSB-SC signal

Let the signal produced by SSB-SC modulator is S(t), a Band pass signal

$S_{USB}(t)=S_{I}(t) cos \omega _{c}t-S_{Q}(t) sin \omega _{c}t$ , where $S_{I}(t)$ is In-Phase component of S(t) obtained by

i. Multiplying S(t) with $\cos \omega _{c}t$ .

ii. and passing the product through a LPF with suitable cut-off frequency.

$S_{I}(t) = S(t) \cos \omega _{c}t$

By finding the Fourier Transform of in-phase component

$S_{I}(f) = \frac{1}{2}(S(f-f_{c})+S(f+f_{c}))$

after restricting the signal $S_{I}(f)$ between $-B\leq f \leq B$

$S_{I}(f) =\left\{\begin{matrix} \frac{1}{2}(S(f-f_{c})+S(f+f_{c})),-B\leq f \leq B & \\0 , otherwise & \end{matrix}\right.$

Similarly $S_{Q}(t)$ is the quadrature phase component of s(t), obtained by multiplying S(t) with $\sin \omega _{c}t$ and by passing the resultant signal through a LPF .

$S_{Q}(t) = S(t) \sin \omega _{c}t$

By finding the Fourier Transform of Q-phase component

$S_{Q}(f) = \frac{1}{2j}(S(f-f_{c})-S(f+f_{c}))$

after restricting the signal $S_{Q}(f)$ between $-B\leq f \leq B$

$S_{Q}(f) =\left\{\begin{matrix} \frac{1}{2j}(S(f-f_{c})-S(f+f_{c})),-B\leq f \leq B & \\0 , otherwise & \end{matrix}\right.$

Now Let’s assume S(f) is the required frequency spectrum of SSB-SC signal when only USB has been transmitted.

i.e,

from the above figure,

one can obtain $S(f-f_{c})$ , $S(f+f_{c})$ by shifting the signal S(f) towards right by $f_{c}$ and left by $f_{c}$

Now by adding $S(f-f_{c})$ and $S(f+f_{c})$

from the above figure, $S_{I}(f)$ results to be

from the frequency spectrum of $S_{I}(f)$ , the time-domain representation turns out to be $S_{I}(t)=\frac{1}{2}A_{c}m(t)-----EQN(I)$

Similarly,

The resultant signals $S(f-f_{c})-S(f+f_{c})$ and $S_{Q}(f)$

from the frequency spectrum of $S_{Q}(f)$ turns out to be

$S_{Q}(f) =\left\{\begin{matrix} \frac{1}{2j}A_{c}M(f),-B\leq f \leq 0 & \\ -\frac{1}{2j}A_{c}M(f),0\leq f \leq B & \end{matrix}\right.$

since Signum function is

$Sign(f) =\left\{\begin{matrix} +1,f>0 & \\ -1 f<0 & \end{matrix}\right.$

$S_{Q}(f)$ when expressed in terms of Signum function $s_{Q}(f) = \frac{j}{2}A_{c}M(f)(-sign(f))$

$s_{Q}(f) = (-jsign(f)M(f))\frac{A_{c}}{2}$

By using Hilbert transform of m(t) , the time-domain representation turns out to be $S_{Q}(t)=\frac{1}{2}A_{c}\widehat{m(t)}-----EQN(II)$

From EQN’s (I) and (II) , the time-domain representation of SSB-SC signal results

$S_{USB}(t) = \frac{A_{c}}{2}m(t)\cos \omega _{c}t-\frac{A_{c}}{2}\widehat{m(t)}\sin \omega _{c}t$ .

similarly, SSB signal when only LSB has been transmitted

$S_{LSB}(t) = \frac{A_{c}}{2}m(t)\cos \omega _{c}t+\frac{A_{c}}{2}\widehat{m(t)}\sin \omega _{c}t$

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Frequency domain representation of a Wide Band Frequency Modulation

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.

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

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

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

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

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

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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Need for modulation

Need for modulation:-

Modulation is the fundamental need for communications, the following are the basic needs for modulation

Distance:-

As low frequency signals can not travel longer distances, low frequencies can be translated into higher frequencies by using modulation schemes.

Improved Signal to Noise Ratio:-

Signal to Noise Ratio has been improved because of modulation at the Receiver.

Practicability of antennas:-

If the communication medium is free space , then messages are transmitted and received with the help of antennas.

∴ The height of the antenna is of the order of the wavelength ( λ ) of the signal being transmitted, when the signals from transmitter are transmitted without modulation then the height required for the antenna is very high, for example to transmit a message signal of frequency f = 5 KHz

height of the antenna required would be

$h \simeq \frac{\lambda }{4} \simeq \frac{c}{4f}$

$h \simeq \frac{3 X 10^{8}}{4 X 5X10^{3}}$

$h\simeq 15000 meters$

Designing an antenna with 15 Km length is almost impractical. To reduce the height of the antenna , instead of sending low frequency (5 KHz) signal as it is modulation is preferred , modulation technique reduces the height of the antenna and makes the antenna to be more practical both at Tx^r and Rx^r.

Narrow Banding:-

There is a problem caused by direct transmission of base band signal which can be explained as follows , suppose a base band signal spectrum ranges from (50 Hz-10 KHz) then the height of the antenna must be 1.5 X 10⁶ meters to receive 50 Hz signal at the receiving end , where as for 10 KHz , the height would be 7500 meters that means height of the antenna is not same for all frequencies .

∴ A wide band antenna which can operate for band edge ratio of 200 is required which is impractical, so modulation is required to use same antenna at the receiver to receive certain range of frequencies.

∴ A wide band message signal from 50 Hz to 10 KHz gets converted to a narrow band signal by a carrier frequency of 1 MHz.

This Narrow banding of Base band signal is possible with modulation which in turn eliminates the complexity of antenna height at the receiver .

Multiplexing:-

Simultaneous transmission of multiple messages over a channel is known as multiplexing. suppose number of messages from different transmitters are transmitted without modulation then there is a possibility of interference (one with other) since the base band spectrum is identical for all the messages. Hence the transmitted messages will not be received properly at the receiver.

one technique to eliminate interference is by the use of modulation and the other technique is multiplexing.

Frequency division multiplexing (FDM) which uses analog modulation techniques.
Time division multiplexing (TDM) uses pulse modulation techniques.

Multiplexing reduces number of channels needed and reduce the cost of installation and maintenance.

Radio Frequency spectrum from ELF to EHF:-

frequency Range	Description of frequency band
upto 300 Hz	Extreme Low frequency(ELF)
300 Hz-3 KHz	Voice frequency (VF)
3 KHz-30 KHz	Very Low frequency (VLF)
30 KHz-300 KHz	Low Frequency (LF)
300 KHz-3 MHz	Medium frequency (MF)
3 MHz-30 MHz	High frequency (HF)
30 MHz-300 MHz	Very High frequency (VHF)
300 MHz-3 GHz	Ultra High frequency (UHF)
3 GHz-30 GHz	Super High frequency (SHF)
30 GHz-300 GHz	Extreme High frequency (EHF)

*Audio Frequency range: 20 Hz-20 KHz.

*UHF,SHF and EHF are Micro wave frequencies.

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Analog Communication and Digital Communication Lab Manual

The following Attachment Consists of Experiments in both Analog Communication & Digital Communication using Communication kits and the same experiments were generated using Matlab Programming.

Analog Communication & Digital Communication Lab Manual