Adaptive Delta Modulation(ADM)

Adaptive Delta Modulation, a modification of Linear Delta Modulation (LDM) is a scheme that circumvents the deficiency of DM. In ADM step size Δ of the Quantizer  is not a constant but varies with time , we shall express Δ as \Delta (n)= 2 \delta (n) .

where \delta (n) increases during a steep segment of input and decreases for a slowly varying segment of input.

The adaptive step size control which forms the basis of an ADM scheme can be classified in various ways such as

  • Discrete or Continuous.
  • instantaneous (or) syllabic (fairly gradual change).
  • forward (or) backward.

we shall describe an adaptation scheme that is backward, instantaneous and discrete in practical implementation , the step size \delta (n) is constrained in between some pre-determined minimum and maximum values.

\delta _{min} \leq \delta (n)\leq \delta _{max}

The upper limit \delta _{max} controls the amount of Slope over load distortion and the lower limit \delta _{min} controls the granular noise (or) Idle noise.

The adaptive rule for  \delta (n)can be expressed in the general form \delta (n) = g(n) \delta (n-1)

where the time varying gain g(n) depends on the present binary output b(n) and M previous values b(n-1),b(n-2) ……….b(n-M). The algorithm is usually initiated with \delta _{min}.

when M=1, b(n) and b(n-1) are compared to detect probable slope over load {b(n) = b(n-1)} (or) probable granularity  {b(n) ≠ b(n-1)} then g(n) is 

  • g(n) = P if     b(n) =b(n-1).
  • g(n)=\frac{1}{P} if   b(n) \neq b(n-1).

when b(n) =b(n-1) Slope overload distortion is detected and when b(n) \neq b(n-1) Idle noise is detected.

where P\geq 1, note that P=1 represents LDM. P_{optimum}=1.5 minimizes the Quantization noise for speech signal, where 1< P< 2 is for broad class of signals.

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Maxwell’s First Equation in Electrostatics

From the Divergence theorem, we have

\overrightarrow{\bigtriangledown }.\overrightarrow{D}= div \overrightarrow{D}=( \frac{\partial D _{x}}{\partial x}+ \frac{\partial D _{y}}{\partial y}+ \frac{\partial D _{z}}{\partial z})

\lim_{dv->0}\frac{\oint_{s}\overrightarrow{D}.\overrightarrow{ds}}{dv}=\lim_{dv->0}( \frac{\partial D _{x}}{\partial x}+ \frac{\partial D _{y}}{\partial y}+ \frac{\partial D _{z}}{\partial z})

{\oint_{s}\overrightarrow{D}.\overrightarrow{ds}}=\( \frac{\partial D _{x}}{\partial x}+ \frac{\partial D _{y}}{\partial y}+ \frac{\partial D _{z}}{\partial z})dv

from Gauss’s law \oint_{s}\overrightarrow{D}.\overrightarrow{ds} =Q_{enclosed}

{\oint_{s}\overrightarrow{D}.\overrightarrow{ds}}=\( \frac{\partial D _{x}}{\partial x}+ \frac{\partial D _{y}}{\partial y}+ \frac{\partial D _{z}}{\partial z})dv = Q_{enclosed}

dividing it by dv(or) \Delta v differential volume on both sides

 \frac{{\oint_{s}\overrightarrow{D}.\overrightarrow{ds}}}{dv} = \frac{Q_{enclosed}}{dv}

by applying limit on both  sides

\lim_{dv->0}\frac{{\oint_{s}\overrightarrow{D}.\overrightarrow{ds}}}{dv} = \lim_{dv->0} \frac{Q_{enclosed}}{dv}

div\overrightarrow{D} = \rho _{v}

\overrightarrow{\bigtriangledown }.\overrightarrow{D} = \rho _{v}

This equation is known as Maxwell’s first equation and is also known as point form of Gauss’s law /Differential form of Gauss’s law.



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Tutorial problems Analog communications

Pb1. A broadcast radio transmitter radiates 5KW power when the modulation percentage is 60% , how much is the carier power?  [Ans: 4.23 KW]

pb2. A 400 Watts carrier is modulated to a depth of 75%, calculate the total power in the modulated wave by assuming the modulating wave as sinusoidal signal. [Ans: 512.5 Watts].

Pb3. The antenna current of an AM transmitter is 8 A when only carrier is being transmitted , but is increases to 8.96 A , when the carrier is modulated by a single-tone sinusoid, find the percentage of modulation? find the antenna current when the depth of modulation changes to 0.8.  [Ans: 0.713, 9.19A].

Pb4. A 300 Watts carrier is simultaneously modulated by two audio waves with modulation percentages of 50 and 60 respectively. what will be the total side band power radiated? [Ans: 91.5 Watts].

pb5. Find the power of the signal V(t)= cos\omega _{c}t + cos\omega _{c}tcos\omega _{m}t  [Ans: 3/4 Watts].

pb6. Find the power of the signal V(t)= cos\omega _{l}t + cos\omega _{c}tcos\omega _{m}t  [Ans: 3/4 Watts].



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 …..fn … 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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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}

PTotal = Pc +PUSB+PLSB

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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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.
  • SAM(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 SAM(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 SAM(t)

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

ka is known as amplitude sensitivity.

In AM the frequency of the carrier signal fc 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 ka 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 SAM(t) we will obtain SAM(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 sAM(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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Need for modulation

Need for modulation:-

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


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 Txr and Rxr.

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 .


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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Basic block diagram of analog communication system


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 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.


Txr 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.


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 Txr to Rxr.

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.


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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