The definition of an “electronic communication” specifically excludes a visual or oral communication

1

Electronic Communication:



Any transfer of signs, signals, writing, images, sounds, data or

intelligence of any nature transmitted in whole or in part by a wire, radio, EM waves, photo electronic or photo optical system



The definition of an “electronic communication” specifically excludes a visual or oral communication

UNIT-I: CONTINUOUS WAVE (CW) MODULATION



In other words, the electronic communication may be broadly defined as the transfer of information from one point (source) to another (destination) through a succession of processes:



The generation of a thought pattern or image in the mind of an originator



The description of that image by a set of aural or visual symbols



The encoding of these symbols in a form that is suitable for transmission over a physical medium of interest



The transmission of the encoded symbols to the desired destination



The decoding & reproduction of the original symbols



The creation of original thought pattern or image

Communication Systems

 When the information is to be conveyed over any distance, a particular system is required that is known as Communication System

 Within a Communication system, the information transfer is frequently

achieved by superimposing or modulating the information on to an electrical signal or electromagnetic wave, which acts as a carrier for the information signal

 This modulated carrier is then transmitted to the required destination where it is received and the original information signal is obtained by demodulation

 The purpose of a communication system is to transmit information bearing signals from a source to a user destination

 When the message produced by the source is not electrical in nature, an input transducer is used to convert it into a time varying electrical signal (called the message signal)

 By using another transducer connected to the output end of the system, a distorted version of the message is recreated in its original form, so that it is suitable for delivery to the user destination

 The communication system consists of three basic components:

 Transmitter

 Communication Channel

 Receiver

Transmitter:

 The transmitter has the function of processing the message signal into a form suitable for transmission over the channel; such an operation is called modulation

 It involves varying some parameter like amp, frequency, or phase of a carrier wave in accordance with the message signal

Channel:

 The function of the channel is to provide a physical connection between the transmitter output & the receiver input

 It may be a wire-pair, a coaxial cable, an optical fiber, or simply free space

s the transmitted signal propagates along the channel, it is distorted due to channel imperfections



Moreover, noise and interfering signals (originating from other sources) are added to the channel output, with the result that the received signal is a corrupted version of the transmitted signal Receiver:



The receiver has the function of operating on the received signal so as to deliver it to the user destination



This operation of estimating the original message signal is called detection or demodulation



The signal processing role of the receiver is thus the reverse of that

of the transmitter

What is a Signal?



In the present context or in the field of communication, a signal is defined as a single-valued function of time that conveys information



This value may be a real number, in which case we have a real-valued signal, or it may be a complex number, in which case we have a

complex-valued signal



The signals may describe a wide variety of physical phenomena, e.g., voice, picture, pressure, temperature, etc.

Types of Signals:



Depending on the feature of interest, we may identify four different

methods of dividing into two classes:

(1) Periodic & Non-Periodic Signals:



A periodic signal g(t) is a function of time (t) that satisfies the condition g(t)= g (t+T

) for all t, where T

is a constant



The smallest value of T

that satisfies this condition is called the

period of g(t) and this period T

defines the duration of one complete cycle of g(t)



And any signal for which there is no value of T

to satisfy the above condition is called a non-periodic or aperiodic signal

(2) Deterministic Signals & Random Signals:



A deterministic signal is a signal about which there is no uncertainty w.r.t. its value at any time or the deterministic signals may be

modeled as completely specified functions of time



On the other hand, a random signal may be viewed as belonging to a set of signals, where each signal in the set having a different

waveform

(3) Analog Signals & Digital Signals:



An analog signal is a signal with an amplitude that varies continuously for all time, i.e., both amplitude & time are continuous over their

respective intervals



On the other hand, a discrete-time/digital signal is defined only at discrete instants of time



Thus, in this case, the independent variable (time) takes on only discrete values, which are usually uniformly spaced



When different samples of discrete-time signal are quantized & coded, the resulting signal is referred to as a digital signal.

(4) Energy & Power Signals:



In communication system, a signal may represent a voltage or a current



Consider a voltage, v(t) is developed across a resistor R, producing a current i(t)



The instantaneous power dissipated in this resistor is defined by





Thus the total energy of a signal g(t) will be



And its average power will be,



Now, we can say that the signal, g(t), is an energy signal if the total energy of the signal satisfies the condition 0<E<∞



And the signal, g(t), is a power signal if and only if the average power of the signal satisfies the condition 0<P<∞









T T

g t dt

P T | ( ) |

2

lim 1









 





 g t dt g t dt E

T

T T

2

2

| ( ) |

| ) (

|

lim

Note:

 The energy and power classifications of signals is mutually exclusive

 In particular, an energy signal has zero average

power, whereas a power signal has infinite energy

 Also, it is of interest to note that, usually, periodic signals & random signals are power signals,

whereas signals that are both deterministic & non-

periodic are energy signals



There are many possible methods for representation of signals in frequency domain



Fourier Analysis, involving the resolution of signals into sinusoidal components, overshadows all other methods in usefulness



There are several methods of Fourier analysis available for the representation of signals



The particular version that is used in practice depends on the type of

signal being considered

(1)

If the signal is periodic (power signal), then the logical choice is to use the Fourier Series (FS) to represent the signal in terms of a set of harmonically related sinusoidal signals

(2) If the signal is an non-periodic (energy signal) then it is customary to use the Fourier Transform (FT) to represent the signal

Note:



The waveform of a signal (the time representation) & its spectrum

(frequency content) are two natural vehicles to understand the

signal



Let g

(t) denotes a periodic signal with period T

and it can be

represented by an infinite sum of sine & cosine terms using Fourier Series expansion by



g

(t)= a

+ 2 [a

cos(2πnf

t) + b

sin(2πnf

t)] … (1)

where (fundamental frequency f

=1/T

)



Each of the terms cos(2πnf

t) & sin(2πnf

t) is called a basis function which form an orthogonal set over the interval T

and satisfy the relations:

cos(2πmf

t) cos(2πnf

t) dt =

 





 n m

n m

T , 0

, 2

0

/



1 n





2 / 0

2 / 0 T

T

cos(2πmf

t) sin(2πnf

t) dt =0, m & n

sin(2πmf

t) sin(2πnf

t) dt =

 





 n m

n m

T , 0

, 2

0

/





2 / 0

2 / 0 T

T







2 / 0

2 / 0 T

T









2 /

2 0 /

0

0

0

0

( ) a 1

T

T

T

t dt

T g





















..., 1,2,3, n

; t)dt nf

sin(2 (t)

T g b 1

and

..., 1,2,3, n

; t)dt nf

cos(2 (t)

T g a 1

Similarly,

/2 T

/2 T

0 T

0 n

/2 T

/2 T

0 T

0 n

0

0 0 0

0 0





• The equation (2) is referred to as the Complex Exponential Fourier Series

& the coefficients c_n are called Complex Fourier Coefficients























 

 

















/2 T

/2 T

nf j2 T

0 n

n n

0

n n

n n

nf j2 n T

0

0

0 0

0 0

...

2, 1,

0, n

; dt e

(t) T g

c 1 And,

0 n

);

jb (a

0 n

; a

0 n

);

jb (a

c where, ...(2)

e c (t)

g

t t







Let g(t) denote a non-periodic deterministic signal, expressed as some function of time (t). By definition, the FT of the signal g(t) is given by the integral

G(f) = 

; where, f is a frequency variable





 j ft dt t

g ( ) exp( 2  )



Given the FT, G(f), the original signal g(t) is recovered exactly using the formula for the inverse FT (IFT):

; where, t is a time variable



The functions g(t) & G(f) are said to constitute a FT pair









-

df ft) (j2

exp (f)

G )

( t 

g



For the FT of a signal, g(t), to exist, it is sufficient, but not

necessary, that g(t) satisfies three conditions collectively known as Dirichlet’s conditions:

1. The function g(t) is a single-valued with a finite number of maxima

& minima in any finite time interval

2. The function g(t) has a finite number of discontinuities in any finite time interval

3. The function g(t) is absolutely integrable, i.e., < ∞ Note:



Physical realization of a signal is a sufficient condition for the existence of a FT



Indeed all energy signals, i.e., signals g(t) for which |g(t)|

dt< ∞ are Fourier transformable









dt t

g ( ) |

|





Note:



G(f)= F[g(t)] and g(t)= F

[G(f)]

where, F[ ] and F

[ ] play the role of linear operators



Another convenient shorthand notation for FT pair, represented by g(t) & G(f), is



In general, the FT, G(f) is a complex function of frequency f,



Therefore, we can express it in the form, G(f)=|G(f)| e

where |G(f)| is called the continuous amplitude spectrum of g(t), and θ(f) is called the continuous phase spectrum of g(t)

) ( )

(t G f g

FT

IFT



For the special case of a real-valued function of time g(t), we have

G(-f)=G*(f) and |G(-f)|=|G(f)|, θ(-f)= -θ(f)



Hence,

(1)

The amplitude spectrum of the signal is an even function of the frequency, i.e. symmetric about y-axis

(2) The phase spectrum of the signal is an odd function of the frequency, i.e. asymmetric about the y-axis



These two statements are summed up by saying that the

spectrum of a real-valued signal exhibits conjugate symmetry

 





 



















) arg(

) arg(

|;

|

|

|

; ,

*

*

*

z z

z z

y j x

z y

j x

z that

know

we

Sinc Function



 ) ^sin(  ⁾

sinc( 

sgn(t)





 















0

; 1

0

; 0

0

; 1 )

sgn(

t

t

t

t



 



 

 a

G f at a

g | |

) 1 ( then

G(f), g(t)

Let

NOTE:

• Function g(at) represents g(t) compressed in time by a factor

„a‟, whereas the function G (f/a) represents G(f) expanded in

frequency by the same factor „a‟

• Thus, the scaling property

states that the compression of a function g(t) in the time domain is equivalent to the expansion of its FT G(f) in the frequency domain by the same factor, or vice-versa.

Fig. (a) g(t) = sin(2πft)

Fig. (b) g(t) = sin(4πft)

Fig. (a) & (b) show the difference between two

Example-5: Sinc Pulse

 Consider a signal g(t) in the form of a sinc function

g(t)=A sinc(2Wt)

calculate its FT by applying Duality Property

  ^f

g t

G  

 G(f), then ( )

g(t) Let



 



 W rect f

W Ans A

2 : 2

NOTE:



The magnitude of FT of g(t-t

) is unaffected, i.e., the magnitude spectrum of the FTs of g(t) and g(t-t

) will be same, but phase of FT of g(t-t

) is changed by the linear factor

–2πft

ft0

-j2 0

) G(f) e t

- (t g then G(f),

g(t)

If  

NOTE:



Multiplication of a given function g(t) by the factor exp(j2πf

t) is equivalent to shifting its FT G(f) in the positive direction by the amount f



This property is called modulation theorem, because a shift of the range of frequencies in a signal is accomplished by using

modulation

constant real

a is f where );

f - G(f

(t) g e

then G(f),

g(t)

If  ^j2fct  _{c c}

 That is, the value of function g(t) is equal to its Fourier Transform G(f) at f=0

) 0 ( )

( ),

( )

( t G f then g t dt G

g

If 

 







That is, the value of function g(t) at t=0 is equal to the area under its Fourier Transform G(f)









 G f then g G f df t

g

If ( ) ( ), ( 0 ) ( )



That is, differentiation of a time function g(t) has the effect of multiplying its FT G(f) by the factor j2πf



Assuming that the FT of higher-order derivatives of g(t) exists, we can generalize the above result as

πf G(f) j

dt g(t) Then d

ble.

transforma Fourier

is g(t) of

derivative first

the that assume and

G(f), g(t)

Let

 2



πf G(f) j

dt g(t)

d

n n

) 2

 (



That is, integration of a time function g(t) has the effect of dividing its FT G(f) by the factor j2πf, assuming that G (0)=0

f G(f) d j

g

have we

G that provided

Then f

G t

g Let

t

 

 2

) 1 (

, 0 )

0 ( ).

( )

(











f) ( G (t)

g

have we

g(t) function

time valued

complex a

for then G(f),

g(t) If

*

*

 





We conclude that the multiplication of two signals in the time domain is transformed into the convolution of their individual FTs in the frequency domain



This property is known as multiplication theorem



The shorthand notation for convolution:

G

(f) = G

(f) * G

(f)



Therefore, we can write



















 G f d G

t g t g

then f

G t

g f

G t

g Let

) (

) ( )

( )

(

), (

) (

&

) ( )

(

2 1

2 1

2 2

1 1

) (

* ) ( )

( )

(

1

t g t G f G f

g 



The convolution of two signals in the time domain is transferred into the multiplication of their individual FTs in the frequency domain



This property is known as convolution theorem



Its use permits us to exchange a convolution operation for a

transform multiplication, an operation that is ordinarily easier to manipulate



In shorthand notation, we can write:

) ( )

( )

( )

(

), (

) (

&

) ( )

(

2 1

2 1

2 2

1 1

f G

f G d

t g g

then f

G t

g f

G t

g Let





















) ( )

( )

(

* )

(

1

t g t G f G f

g 

• It is denoted by δ(t) and is defined as

having zero amplitude everywhere except t=0, where it is

infinitely large in such a way that it contains unit area under its curve, i.e.













1 )

(

0 ,

0 )

(

dt t and

t t





) 2

(t e ^t g  ^

• No function in the ordinary sense can satisfy two rules of above two equations

• However, we can imagine a sequence of functions that have progressively taller &

thinner peaks at t=0, with the area under the curve remaining equal to unity, whereas the value of the function tends to zero at every point, except at t=0 where it tends to infinity

• That is, we may view the delta function as the limiting form of a unit-area pulse as the pulse-duration approaches zero

pulse) r

rectangula as

expressed is

function delta

the if

( τ ;

rect t τ

lim 1

δ(t) τ 0 



 



 







τ exp lim 1

δ(t) ^{2 2}

τ 0  t 

 

OR

τ varying for

) / exp(- τ

g(t) 1 pulse Gaussian

(b)

Fig.   t² ²

1. The delta function is an even function of time, i.e. δ (t)= δ (-t)

2. The integral of the product of δ (t) & any time function g(t) that is continuous at t=0 is equal to g(0); thus







 ( 0 ) )

( )

( t t dt g g 

• We refer to this statement as the sifting property of the delta function because single value of g(t) is sifted out

3. The sifting property of the delta function may be generalized by writing

• Since the delta function is an even function of t, we may rewrite the above equation in a way emphasizing resemblance to the

convolution integral as follows:

• That is, the convolution of any function with the delta function leaves that function unchanged

• We refer to this statement as the replication property of the delta









 ) ( ) (

)

( t t t

dt g t

g 

) ( )

(

* ) (

) ( )

( ) (

t g t

t g or

t g d

t g





 

















4. The FT of the delta function is given by

 

1 )

(

1 ) ( )

(

0

2



















t or

e

dt e

t t

F







• This relations states that the spectrum of the delta function δ(t)

extends uniformly over the entire frequency interval from -∞ to ∞ as shown in fig. (2.11b)

1. DC Signal: If g(t)=1, then G(f) = ?

 By applying the duality property, we get







 ( ) )

2

cos( ft dt  f

• Invoking the definition of FT, we can deduce the following useful relation:









 2 ) ( ) exp( j  ft dt  f

• As the delta function is real-valued, we may simplify the above relation:

)

(

1   f

2. Complex Exponential Function: If g(t) = exp(j2πf_ct), then G(f) = ?

 By applying the frequency shifting property, we get

 That is, the complex exponential function, exp(j2πf_ct) is transformed in the frequency domain into a delta function δ(f-f_c) occurring at f = f_c

) (

) 2

exp( j  f

t   f  f

Fig. Frequency Spectrum of exponential function

G(f)

0 f_c f

3. Sinusoidal Functions: If g(t) = cos(2πf_ct), then G(f) = ?

 First, express cos(2πf_ct) function into exponential form by using Euler’s formula and then by using the frequency shifting property, we obtain

 Similarly, we can get the FT of sin(2πf

[ (

) ( )]

2 ) 1

2

cos(  f

t   f  f

  f  f

)]

( )

( 2 [

) 1 2

sin(

f f

f f

t j

f      



Figure 2.15 (a) Sine Function. (b) Spectrum.

4. Signum Function: The signum function can be defined as:



The signum function does not satisfy the Dirichlet

conditions, and strictly

speaking, it does not have FT



However, we can find FT of signum function by viewing it as the limiting form of the

asymmetric double exponential pulse as:

 

 















0

; 1

0

; 0

0

; 1 )

sgn(

t

t

t

t

Applications of Delta Function

) , ( lim )

sgn( t

g a t

a



 

 















0

;

0

; 0

0

; )

, (

t e

t t e

t a g where

at at

• We know that



^{4 4}



) ,

( a f

f t j

a

g 





 

• Therefore,





0 0

0

4 lim 4

)]

, ( [ lim

)]

, ( lim [ )]

[sgn(

f a

f j

t a g F

t a g F

t F

a a

a







 











Applications of Delta Function

• Hence,

f t j



) 1

sgn( 

• Another useful FT pair

involving a signum function in frequency domain is obtained by applying the duality property:

) 1 sgn(

) 1 sgn(

f t j

or

t f j













Fig. Spectrum of signum function

Applications of Delta Function

5. Unit Step Function:

 

 











0 t

0,

0 t

1/2,

0 t

1, u(t)

] 1 )

[sgn(

2 ) 1

( t  t 

u

• We can represent unit step function in terms of signum function as:

• Therefore, using linearity

property, the FT is given by:

( )

2 1 2

) 1

( f

f t j

u 

 ^



6. Integration in Time Domain (Revisited):









 g  u t  d  t

y

or ( ) ( ) ( )

Applications of Delta Function







t

d g

t y

Let ( ) () 



















t t t t

u where









; 0

2; 1

; 1 )

( ,

• Now, the FT of y(t) will be





 



 

 ( )

2 1 2

) 1 ( )

( f

f f j

G f

Y 



) ( ) 0 2 (

) 1 2 (

) 1

( G f G f

f f j

Y

or 

 ^



Modulation:



The purpose of a communication system is to deliver a message signal from an information source in recognizable form to a user destination



To do this, the transmitter modifies the carrier signal according to message/modulating signal suitable for transmission over the

channel



This modification is achieved by means of a process known as Modulation, which involves varying some parameter (e.g.

amplitude, frequency, or phase) of a high frequency carrier wave according to the instantaneous value of the message signal



We may classify the modulation process into Continuous Wave (CW) Modulation & Pulse Modulation



In CW Modulation, a sinusoidal wave is used as the carrier wave



In Pulse Modulation, on the other hand, a periodic sequence of

rectangular pulses is used as the carrier wave



The CW modulation is divided into AM, FM, & PM



The pulse modulation can be of an analog or digital type



In analog pulse modulation, the amplitude, duration or position of the pulse is varied in accordance with instantaneous values of the

message signal



In such case, we have PAM (pulse amplitude modulation), PDM (pulse duration modulation), & PPM (pulse position modulation)



The standard digital pulse modulation is known as PCM (pulse coded modulation), which is basically PAM with an important modification:

the amplitude of each modulated pulse (i.e. sample of the original message signal) is quantized & then coded into a corresponding sequence of binary symbols ‘0’ & ‘1’



These binary symbols are suitably shaped for transmission over the

channel

 I

 in the field of communication, there exist various difficulties, which make it mandatory to modulate the signal before transmitting to the receiver from transmitting side. Some of these are given below:

(1) At the audio frequencies (20Hz-20KHz), for efficient radiation

(transmission) & reception the transmitting and receiving antennas should have lengths comparable to a quarter-wavelength of the frequency used.

This is 5 km at 15 kHz and a vertical antenna of this size is unthinkable.

Therefore, some form of frequency-band shifting must be used, which is accomplished by the process of modulation.

(2) Most of the message signals have frequency spectrum in lower frequency range, e.g., all sound signals frequency lies within the range 20 Hz – 20 kHz, so all signals from different sources would be inseparably mixed up.

Therefore, in order to separate the various signals, it is necessary to

translate them all to different portions of the electromagnetic spectrum.

(3) The unmodulated carriers of various frequencies can’t by themselves, be used to transmit intelligence (information) because they have constant

c(t)

s(t)

t

s(t) m(t)

t

t t

(a)

(b) (d)

(c)

Fig. (3.1) Illustrating the AM process (a) Baseband signal, m(t) (b) Carrier signal, c(t) (c) AM wave for |k_am(t)|<1 for all t (d) AM wave for |k_am(t)|>1 for some t

 Amplitude Modulation (AM) is defined as a process in which the

amplitude of the carrier wave is varied about a mean value, linearly with the message signal

 Consider a sinusoidal carrier wave, c(t), defined by c(t)= A_c cos(ω_ct) = A_ccos(2πf_ct)

 For simplicity, the phase angle has been ignored

 Let m(t) be the message signal (baseband signal) that specifies the information

 Thus, an amplitude-modulated wave may be described (in its most general form) as a function of time as follows:

 s(t) = A_c[1+k_a m(t)] cos(2πf_ct) …(1)

where, k_a is a constant called the amplitude sensitivity of the amplitude modulator and measured in volt^-1

 We observe that the envelope of s(t) has essentially the same phase as the message signal m(t) provided that two requirements are satisfied:

(1) |k_a m(t)|<1 for all t, which ensures that the function 1+k_am(t) is always positive

 When the amplitude sensitivity k_a of the modulator is large enough to make |ka m (t)|>1 for any t, the carrier wave becomes overmodulated, resulting in carrier phase reversals wherever the factor ‘1+k_am(t)’

crosses zero and AM wave then exhibits envelope distortion Note:

The absolute maximum value of k_am(t) is modulation factor, that is, μ

= |k_a m(t)|_max and when multiplied by 100 is referred to as the percentage modulation factor/index

(2) The carrier frequency f_c is much greater than the highest frequency component W (bandwidth) of the message signal m(t), i.e., f_c>>W



From equation (1), assuming base band signal m(t) is band limited to the interval –W ≤ f ≤ W, the spectrum of the modulated signal is given by

Note:



For positive frequencies, the highest frequency component of the wave equals f

+ W, and the lowest frequency component is f

– W



The difference between these two frequencies defines the

transmission between B

for an AM wave, which is exactly twice the message bandwidth (W), i.e.,

B

= (f

+W) - (f

-W)= 2W

)]

( )

( 2 [

)]

( )

( 2 [

)

(

k

A

M f f

M f f

f f

f A f

f

S          



In practice, modulation of a carrier by several modulating signals is a rule rather than the exception



Let V

, V

, ….V

be the simultaneous modulating voltages, then the total modulating voltage V

will be equal to the square root of the sum of squares of the individual voltages, i.e.,





And

 

   





2 2 2

1

2 / 2 1 2 2

3 2 2

2 2 2

1 2 2

/ 2 1 2

3 2

2 2

1

2 / 2 1 2

3 2

2 2

1

...

,

...

...

...

n

n a a

a a

n a

m a

n m

Hence

V k V

k V

k V

k V

V V

V k V

k

V V

V V

V









     































1 2







t

P P



Amplitude Modulator is simple to design and build



Popular



Less complex



Receiver (Amplitude Demodulator) is cheap to build

1. AM is wasteful of Power:



In the standard form of AM, the carrier wave c(t) is completely independent of the message signal m(t), which means that the

transmission of the carrier wave represents wastage of power (only a fraction of total power is affected by m(t))

2. AM is wasteful of Bandwidth:



The upper & lower sidebands of AM are uniquely related to each other by virtue of their symmetry about the carrier frequency, i.e., given the amplitude and phase spectra of either sideband, we can uniquely determine the other



This means if both the carrier & one sideband are suppressed at

the transmitter, no information is lost and thus the transmission

bandwidth becomes equal to the message signal bandwidth

 To overcome the limitations of AM, we make certain changes, which results in increased system complexity of AM process

 Starting with AM as standard, we can differentiate three modified forms of AM as follow:

1. DSBSC (Double Side Band Suppressed Carrier) Modulation:

 Here, the transmitted wave consists of only the upper & lower sidebands

 Transmitted power is saved through the suppression of the carrier wave, but the channel bandwidth requirement is same as that of standard AM (i.e. twice of message bandwidth, 2W)

2. SSBSC (Single Side Band Suppressed Carrier) Modulation:

 SSBSC modulated signal consists only of the upper sideband or lower

sideband, therefore, it only translates the spectrum of the message signal to a new frequency location

 This modulation is particularly suited for the transmission of voice signals

 It is an optimum form of modulation in that it requires the minimum transmitted power and minimum channel bandwidth (i.e., equal to message bandwidth, W)

 The principal disadvantage of SSBSC modulation is increased cost &

complexity

3. VSB (Vestigial Side Band) Modulation:

 In this technique, one sideband is passed almost completely & just a trace (or vestige) of the other sideband is retained

 The required channel bandwidth is therefore in excess of the message bandwidth by an amount equal to the width of the vestigial sideband

 This form of modulation is well suited for the transmission of wideband signals such as TV signals that contain significant components at

extremely low frequencies

f

Gain (A)

f_L 0

(a) Low Pass Filter

f

Gain (A)

f_H 0

(b) High Pass Filter

f

Gain (A)

f_L 0

(c) Band Pass Filter

f_H f

Gain (A)

f_L 0

(d) Band Reject Filter f_H

Fig. Gain versus frequency characteristics of Ideal Filters

 The generation of an AM signal may be accomplished using various devices, one of them is Switching Modulator

 As shown in fig. (3.5a), the message signal m(t) & carrier signal c(t) are applied to diode which acts as an ideal switch (i.e., it presents zero

impedance in forward bias)

 The amplitude of c(t) is large enough, which keeps the diode always in forward linear region

 That is, v₂(t) varies periodically between the values v₁(t) & 0 at a rate equal to the carrier frequency f_c

) 2 ( 0 ...

) (

; 0

0 )

(

; ) ) (

(

) ( ,

) (

) 1 ( ...

) ( )

2 cos(

) (

1 2

2 1

 





 







t c

t c t

t v v

by given is

t v voltage load

resulting the

A t

m where

t m t

f A

t v

c c

c



 In this way, by assuming a message signal that is weak compared with carrier signal, we have effectively replaced the non-linear behavior of the diode by an approximately equivalent piecewise-linear time varying

operation

 We may express eqn.(2) mathematically as

where, g_To(t) is a periodic pulse train of 50% duty cycle & period T_o=1/f_c as in fig.(3.6)

 The g_To(t) can be represented by its Fourier Series as

 Thus, putting the value of g_To(t) from eqn.(4) into eqn.(3), we find that v₂(t) consists of two components:

) 3 ( ...

) ( )]

( )

2 cos(

[ )

2

( t A f t m t g t

v 





) 4 ( ...

)]

1 2

( 2

1 cos[

2 ) 1 ( 2

2 ) 1

(

1

1





 

 





n t

n f t

g

n

n

T_o





1. The component

which is desired AM wave with k_a=4/πA_c

 The switching modulator is therefore made more sensitive by reducing A_c; however, it must be maintained large enough to make the diode act like an ideal switch

2. Unwanted components, the spectrum of which contains delta functions at 0, ±2f_c, ±4f_c, …

 The unwanted terms are removed from v₂(t) by means of band-pass filter with mid-band frequency f_c and bandwidth 2W; provided f_c>2W, which ensures that the frequency separations between the desired AM wave &

the unwanted components are large enough for the band-pass filter to suppress the unwanted components

) 2

cos(

) 4 (

2 1 m t f t

A A

c c

c



 ^ _

 



 

 The process of demodulation is used to recover the original message signal from the incoming modulated signal, i.e., demodulation is the reverse of modulation process

 The simple & effective way of demodulating the AM signal is by using envelope detector; provided (i) AM wave is narrow-band, which requires f_c>>W (ii) the percentage modulation is less than 100%

 The envelope detector (shown in fig.(3.7a)) consists of a diode & an RC filter Working of Envelope Detector:

 On a +ve half-cycle of the input signal, the diode is forward biased & the capacitor C charges up rapidly to the peak value of the i/p signal

 When input signal falls below this value, the diode becomes reverse-biased

& capacitor C discharges slowly through load resistor R_L & the discharging process continues until the next +ve half-cycle

 When the input signal becomes greater than the voltage across the capacitor, the diode conducts again & the process is repeated

 We assume that the diode is ideal, presenting resistance r_f to current in forward-biased region & infinite resistance in the reverse-biased region

 We further assume that the AM signal applied to envelope detector is supplied by a voltage source of internal impedance R_s

 The charging time constant (r_f + R_s)C must be short compared with the carrier period 1/f_c [i.e., (r_f + R_s)C <<1/f_c], so that the capacitor C charges rapidly & thereby follows the applied voltage up to the +ve peak when the diode is conducting

 On the other hand, the discharging time constant R_LC must be long enough to ensure that the capacitor discharges slowly through the load R_L between +ve peaks of the carrier signal, but not so long that capacitor voltage will not discharge at the maximum rate of change of message signal, i.e., 1/f_c <<

R_LC << 1/W

 The result is that the capacitor voltage or detector output is nearly the same as the envelope of the AM signal

79

UNIT-II: Angle Modulation



In this process, the angle of the carrier is varied

according to the base band (message) signal and the amplitude is maintained constant



An important feature of angle modulation is that it can provide better discrimination against noise &

interference than amplitude modulation at the expense of increased transmission bandwidth



Let θ

(t) be the instantaneous angle of a modulated carrier, assumed to be a function of the message signal m(t)



We express the resulting angle-modulated wave as



s(t) = A

cos[θ

(t)] ( where A

is carrier amplitude)

Note: A complete oscillation occurs whenever θ

(t) changes by 2π radians

t=0 t=t t=t+∆t

θ(t) θ(t+∆t)

Fig. Phasor Diagram



If θ

(t) increases monotonically with time, the average frequency (in Hz) over an interval from t to t+Δt is given by



And the instantaneous frequency of the angle-modulated signal

s(t) is defined as

 Thus, according to the equation for the modulated signal, we may

interpret the angle modulated signal s(t) as a rotating phasor of length A_c

& angle θ_i(t) with angular velocity,

 In the simple case of an unmodulated carrier, θ_i(t) = 2πf_ct + Φ_c

 And the corresponding phasor rotates with a constant angular velocity equal to 2πf_c

 The constant Φ_c (initial phase angle) is the value of θ_i(t) at t = 0

Note:

 Two most commonly used methods are: Phase Modulation & Frequency Modulation in which we vary θ_i(t) according to message signal

s dt rad

t d

) /

 (