DC Digital Communication MODULE II

8/7/2019 DC Digital Communication MODULE II

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SAMPLING AND

WAVEFORM

CODING

MODULE II


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Sampling Theorem (Low passSignals)

If a finite energy low pass signal x(t) contains nofrequencies higher than fM Hz, then it can becompletely represented by its samples, if it issampled at a minimum rate of twice its maximumfrequency, i.e., f

S

= 2 fM

.

If a finite energy low pass signal x(t) contains nofrequencies higher than fM Hz, then it can be

completely recovered from its samples, if it issampled at a minimum rate of twice its maximumfrequency, i.e., fS = 2 fM.


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Sampling theorem

MULTIPLIER

)(tx

)()()( ttxtgsT

=)(t

sT

)(tg

( ) ( ) ( )sn

g t x n t nT

=

=

( ) ( ) ( )s sn

g t x nT t nT

=

=


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Sampling theorem

Sampling of x(t) at a rate of fs Hz may be achieved by

multiplying x(t) by an impulse train Ts(t). It consists ofunit impulses repeating periodically every Ts seconds

where Ts=1/fs

The multiplication results in the sampled signal g(t)

The impulse train Ts(t) is a periodic signal of period Ts. It

may be expressed as a Fourier Series as given below.

[ ]s

1( ) 1 2cos 2cos2 2cos3 ....

TsT s s st t t t = + + + +

( ) ( ) ( ) ................(1)sT

g t x t t =


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Sampling theorem

Putting the values of Ts(t) in equation (1)

Taking the Fourier Transform of g(t) as below

[ ]s

1( ) ( ) 2 ( )cos 2 ( )cos2 2 ( )cos3 ....

Ts s sg t x t x t t x t t x t t = + + + +

}( ) ( ) FT g t G =

{2 ( )cos ( ) ( )

s s s

FT x t t X X = + +

{ }2 ( )cos2 ( 2 ) ( 2 ) s s s FT x t t X X = + +....................................................................


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Sampling theorem

This spectrum is shown in fig(2)

[ ]1

( ) ( ) ( ) ( ) ( 2 ) ( 2 ) .... s s s sG X X X X X

Ts

= + + + + + + +

1( ) ( )

n

G X nTs

=

= 1

( ) ( )n

G X nTs

=

=


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Sampling theorem

)(tx)(X

)(tg

)(G

mm

mm ss(2)Figure


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Sampling theorem

Now x(t) can be recovered from its samples by passing

the sampled signal g(t) through an ideal LPF ofbandwidth fM Hz

When we select fs=2fM the rate is called Nyquist rate. It is

the minimum sampling rate. Maximum sampling interval

is called Nyquist interval and is given by Ts=1/2fM


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Sampling theorem

For the case fs>2fM the successive cycles are not touching each other.

There is a guard band between each spectrum. In this case the originalspectrum X() can be recovered using a low pass filter easily.

fM-fM fs-fM fs+fM-fs+fM-fs-fMfs-fs 0

fs>2f

M

G()

H()


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Sampling theorem

For the case fs=2fM the successive cycles are touching each other. The

original spectrum may be recovered using an ideal LPF with sharp cut offfrequencies.

fM-fM

fs+fM-fs-fM

fs-fs0

fs=2fM

H()

G()


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Sampling theorem

For the case fs


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Reconstruction (Ideal)

A low pass filter is used to recover the original signal

from its samples. The frequency response of such an ideal lowpass filter

should be as below.

An ideal LPF is not physically realizable. The response

cannot become zero abruptly at cut off frequency. Let g(t) be the sampled signal and G() its spectrum and

x(t) be the original signal and X() be its spectrum.

fM-fM 0

Ts


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1/Ts1/Ts1/Ts

1

Ts

1

m-m

s-s m-m

s/2-s/2

m-m

Original Spectrum

Sampled Spectrum

Filter Response

Recovered Spectrum


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1( ) ( )sn

G X nTs

=

=

The goal of reconstruction is to apply some operation to

G() to convert it back to X().

Any such operation must eliminate the replicas of X() thatappear as ns.

This is accomplished by multiplying G() by H() where

2( )

02

sTH

=

( ) ( ) ( ) X G H =


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Reconstruction (Ideal) Multiplication in frequency domain is converted to

convolution in time domain.

The sampled signal g(t) can be expressed as

)()()( thtgtx = ( ) ( )( ) s sn

g t x nT t nT

=

=

( ) ( )( ) ( )s sn

x t x nT t nT h t

=

=

( ) ( )( )s sn x nT h t t nT

== ( ) ( )s

n

sh t n x nT T

=

=

Let us find out

what is

h(t-nTs)


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domainfrequency

theinpulserrectangulay thegivenis)(H

s/2-s/2

Ts

In the time domain

it is a sinc

function. That is,

the inverse FourierTransform is a

sinc function


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{ }1sin

2( ) ( )

s

s

th t F H T

t

= =

sin2

/ 2 )2 /(

s

s

s s

tT

t

= sin

2 2

s s sT c t

=

a-a

1

tat

sin

s/2-s/2

Ts sin2

s

s

tT

t


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( ) sin

2

s s

s

Th t c t

T

=

sin

2

sc t

=

( )( ) sin2

ss sh t nT c t nT

=

( )( ) sin2

ss sh t nT c t nT

=


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So in the time domain x(t) is reconstructed as a

weighted sum of sinc functions. The weights corresponds to the value of the

discrete time sequence.

This process is called Ideal Band limitedInterpolation and equation (1) is called interpolationformula for reconstruction of a signal.

( )( ) ( )sin (1)

2

s

s sn

x t x nT c t nT

=

=


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ORIGINAL SIGNAL x(t)


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g(t) is convolved

with sinc

function

g(t)sinc


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Original signalappears as

envelop

x(t) isreconstructed as

the weighted and

shifted sinc

functions


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RECONSTRUCTED SIGNAL x(t)


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Natural Sampling

The sampling switch s is driven by sampling function c(t)

which is a train of periodic pulses of width and frequency fs

to produce sampled signal g(t) from input x(t)

X(t)

g(t)

c(t)


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Natural Samplingx(t)

c(t)

g(t) Ts


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Natural Sampling Sampled signal g(t) is the product of c(t) and input signal x(t)

g(t) = x(t) when c(t) = A

g(t) = 0 when c(t) = 0Sampled signal is g(t)=x(t) x c(t)

The periodic pulse train c(t) may be expressed as a Fourier

series by

( )jn t

n

n

c t c e

=

=

)(sinc ss

n nfT

Ac =

0

1( ) s

T jn t

n

s

c f tWhere e dtT

=

n -

( ) sinc (n ) s jn t

s

s

Ac t f eT

=

=


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Natural Sampling

n -

( ) sinc (n ) . ( )s j nt

s

s

A g t f e x t

T

=

=

g(t)oftionrepresentadomainfrequencyget theTo

g(t)ofransformFourier tthetake

{ })()( tgFG = { }n -

sinc (n ) ( ) s j nt ss

Af FT x t e

T

=

=

n - sinc (n ) ( )s ss

A

f X nT

=

=

n -

( ) sinc (n ) ( )s ss

AG f X n

T

=

=


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Natural Sampling The spectrum X(f) of x(t) repeats periodically at

fs,2fs,etc and is weighted by the sinc function

The spectrum of impulse sampled signal isrepetitions of X() at nfs.

In the case of naturally sampled signal the spectra

repeats at nfs and is weighted by the sinc(nfs)function

In the limit as tends to zero sinc(nfs) tends to 1 andthe weighting factor disappears.

Then the naturally sampled signal spectrum becomesthe same as that of instantaneously sampled signal.


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ReconstructionX()

G()

Ho()

Xo()

-s s

-m m


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ReconstructionXo()

Hc()

X()

Aperture Effect

Distortion

Equalizing

Filter

Recovered

SignalSpectrum


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Aperture effect

Since the spectrum is multiplied by the sinc function

some distortion is introduced in to the recovered signal. The reconstructed signal contains an amount of

distortion introduced by the roll off in the sinc function.

The sinc function acts as a low pass filter andattenuates the upper portion of the message signal

spectrum.

The high frequency contents of the signal are thusattenuated.

This distortion is called aperture effect.


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Aperture effect

As the sampling pulse width increases the main lobe

width of the sinc pulse decreases. So the aperture effect becomes more prominent.

As the width of sampling pulse decreases the mainlobe width of sinc pulse increases and the apertureeffect is reduced.

Ultimately when the width approaches that of anideal impulse, aperture effect disappears.

To avoid aperture effect an equalizer whosefrequency response is opposite to that of the sincpulse is used after the signal is reconstructed.


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Aperture effect

Reconstruction

Filter

Equalizing

Filter

Message

Signal


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Flat top sampling

)(tx )(tg1G2G


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Flat top sampling

The top of the pulses are

constant and is equal to the

instantaneous value of the

baseband signal)(tg


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Flat top sampling

x(t)1

( )Ts t

)(th

)( t

)(tg

)(

t

)(th )(tg

=1

( ) ( )s sn

x nT t nT

=


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Flat top sampling

In flat top sampling the top of the samples remains constant

and is equal to the instantaneous value of the basebandsignal x(t) at the start of the sampling.

The duration of each sample is and the sampling rate is

fs=1/Ts

First the signal x(t) is multiplied by the train of impulses

to obtain

=

=n

sTs nTtt )()( )( t

= == nsTs nTttxttxt )()()()()(

=

=n

ss nTtnTx )()(


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Flat top sampling The flat top pulse is equivalent to the convolution of

instantaneous sample and a pulse h(t)

The width of the pulse in g(t) is determined by the width

of h(t) and the sampling instant is determined by the

delta function.

is not a constant amplitude delta function as

It is a varying amplitude train of impulses. When is convolved with h(t) we get a pulse whose

duration is equal to h(t) but amplitude defined by

( ) ( ) ( )g t t h t = where (t) is the instantaneous sample

)( t )(t

)( t)( t


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Flat top sampling( ) ( ) ( )g t t h t = ( ) ( ) ( )G S H =

( ) ( ) ( )sn

t x t t nT

=

= ( ) { ( )}S FT t =

( ) ( )sn

x t t FT nT

=

=

1( )s

ns

nT

=

=

1( ) ( ) ( )sns

G X n H T

=

=


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Flat top sampling

0T

FT /2sinc T/2j TATe

1

0

FT

/2

sinc /2j

e


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Flat top sampling1

( ) ( ) ( )sns

G X n H

T

=

= /2( ) sinc /2

jH e =

/21( ) sinc /2 ( )j sns

G e X nT

=

=


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Flat top sampling

H() is the FT of h(t) which is a rectangular pulse of width

and height 1. H() is a sinc function as shown in figure(2).

When the spectrum X(-ns) is multiplied by H() which is

a sinc function the resultant spectrum of H() X(-ns) is

shown in figure (4).

The reconstructed signal contains an amount of distortion

introduced by the roll off in the sinc function.

This distortion is similar to that produced in the case ofnaturally sampled signal.


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Spectrum of flat top sampled signal

X()

H()

-s s

-m m

)(G

sT/1

)(G

)( snX

(2)

(4)

2

2

(4 )Figure


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Zero order hold sampling It is difficult to generate and transmit narrow large

amplitude pulses which approximate impulses.

Zero order hold sampling eliminates this problem.

Zero order hold system samples a continuous time

signal at a given instant and holds its value until the

next instant at which another sample is taken.

ZERO ORDER HOLDx(t)x0(t)


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Zero order hold sampling

x(t)

x0(t)


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Zero order hold sampling

The output x0(t) of the zero order hold may be

considered to be generated by an impulse trainsampling followed by a continuous time LTI system with

rectangular impulse response

x(t)x0(t)

t

h0(t)1

Ts

( )Ts t

( )g t

Reconstruction of Zero order


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hold sampled signal To reconstruct a continuous time signal from the output of

zero order hold system we want a system with impulse

response hr(t).

Its frequency response may be denoted by Hr().

x(t)x0(t)

t

h0(t)1

Ts

hr(t)

Hr() xr(t)

We want the output of the reconstruction filter xr(t) to be

the same as x(t)

( )Ts t

( )t



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hold sampled signal

It is possible only when the cascade combination of

hr(t) and ho(t) is an ideal low pass filter. The impulse response ho(t) is defined as

Frequency response Ho() may be obtained by

taking the Fourier Transform of ho(t)

1 0( )

0 otherwise

s

o

t Th t

=

( ) ( ) j to oh t e dt

= ( )/2sinc / 2sj Ts sT e T

=



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hold sampled signal

sT

ss

)(oH



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hold sampled signal( ) ( ) ( )o o X G H =

1( ) ( )n

G X nTs

=

=

( )/21

( ) ( ) sinc / 2sj T

o s s s

n

X X n T e T Ts

=

=

( )/2( ) sinc / 2sj To s s H T e T

=

( )/2( ) sinc / 2 ( )sj T

o s s

n

X e T X n

==



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hold sampled signal

)(

)()(

o

rH

HH =

)()()( ro HHH =

( )

2/sinc

)()(

2/

s

Tj

s

rTeT

HH

s=

( )/2

s

( ) ( )T sinc / 2

sj T

r

s

eH HT

=

Reconstruction of zero order sampled signal


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Reconstruction of zero order sampled signalX()

G()

Ho()

Xo()

-s s

-m m

Xo()


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ReconstructionXo()

Hr()

X()

Aperture Effect

Distortion

Equalizing

Filter

Recovered

SignalSpectrum


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Sampling Theorem (Band passSignals)

A band pass signal x(t) whose maximum bandwidth

is 2fM can be completely represented and recoveredfrom its samples if it is sampled at a minimum rate of

twice its bandwidth.

Consider the spectrum of a bandpass signal which iscentered around +fc and fc The lowest and highest

frequencies present in the signal are fc-fM and fc+fMwith 2fM band width.

This signal can be represented using samples takenat minimum frequency 2 x BW = 2 x 2fM = 4fMsamples per second and recovered successfully.


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QUANTIZATION


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QUANTIZATION

QUANTIZATION


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QUANTIZATION

)(txq

QUANTIZATION


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QUANTIZATION

SOURCE

STATION

REGENERATOR

(Filter and Amplifier)

DESTINATION

STATION

REGENERATOR


NOISE

NOISE

NOISE REGENERATOR



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QUANTIZATION


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QUANTIZATION

QUANTIZATION


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QUANTIZATION

12

3

4

56

7

8

2/

2/3

2/5

2/7

2/2/3

2/5

2/7

QUANTIZATION


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QUANTIZATION

12

3

4

56

7

8

2/

2/3

2/5

2/7

2/2/3

2/5

2/7

QUANTIZATION


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Q N N

12

3

4

56

7

8

2/

2/3

2/5

2/7

2/2/3

2/5

2/7

QUANTIZATION


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Q

QUANTIZATION


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Q

LARGE NOISE

PULSE ADDED TO

TRANSMITTED

SIGNAL

QUANTIZATION


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Q

12

34

56

7

8

2/

2/3

2/5

2/7

2/2/3

2/5

2/7

QUANTIZATION


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Q

12

34

56

7

8

2/

2/3

2/5

2/7

2/2/3

2/5

2/7

QUANTIZATION


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Q

12

34

56

7

8

2/

2/3

2/5

2/7

2/2/3

2/5

2/7

QUANTIZATION


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ERROR

DUE TO

NOISE

QUANTIZATION maxx


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4

3

2

2

34

2/

2/3

2/5

2/7

2/2/3

2/5

2/7

maxx

)(tx

)(txq

0

QUANTIZATION


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INPUT

SIGNAL

OUTPUT

SIGNAL

2 3 4 2 3 4

2/

2/32/5

2/7

2/

2/3

2/5

2/7

0

QUANTIZATION ERROROUTPUT


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INPUT

SIGNAL

OUTPUT

SIGNAL

2 3 4 2 3 4

2/

2/3

2/5

2/7

2/

2/3

2/5

2/7

0

Quantization error

Input

2/

2/

2 3 40 2 3 4


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QUANTIZATION ERROR


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PDF of the quantization error is a uniformly distributed random

variable and is defined as

Noise power of the quantization noise is expressed as

Where is the mean square value of noise voltage.

Mean square value of a random variable X is expressed as

>

=2/0

2//2-/1

2/0

)(

f

RVPowerNoise n

2

=

2

nV

[ ]

== dxxfxXEX X )(222

QUANTIZATION ERROR


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Normalized value of noise power is

= dfE )()(22

=

2/

2/

22 1)( dE

2/

2/

3

31

=

( ) ( )

+

= 3

2/

3

2/133

12

2

=

( )12

V2

22

n

== E

121

22

2

==

nn V

V

QUANTIZATION ERROR


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Normalized value of noise power is the same as the mean

square value of noise voltage and is called the Quantization

error.

12

2=ErroronQuantizati

Need for non linear quantization


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In uniform quantization the step size is fixed whatever be the

amplitudes of the input signal.

The quantization error depends on step size as given by

and for fixed step size it is a constant For large amplitude signals which swings through several

quantization levels the signal to quantization noise ratio is

relatively large.

For very small amplitude signals that occupy relatively smallquantization levels the signal to quantization noise ratio

decreases to unacceptable levels.

Human speech and other voice signals are characterized by

the statistical property that large amplitude signals are rare

when compared to small amplitude signals

12

2

Need for non linear quantization


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At 50 percentage of the time the voltage characterizing the

speech energy is less than one fourth of the total range.

If we use uniform quantization higher quantization levels will

be used only rarely.

Weak signals will be confined to a limited number of

quantization levels which make their S/N ratio very small.

The solution to this problem is to use a non-uniform

quantization. We can achieve non-uniform quantization in two ways

1. Use a non-uniform quantizer characteristic.

2. Distort the original signal with a non-uniform

compressor and then use a uniform quantizer

The second method is more commonly used

Non-uniform Quantization

Small amplitude


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Signals areseverely distorted

If uniform

quantization is

used


Smaller


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quantization levelsfor weak signals

ensures high S/N

ratio



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INPUT

OUTPUT

Input-Output

relationship of non-uniform quantizer



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The second method for non-linear quantization is compandingfollowed by linear quantization.

The signal is passed through a network which has an input-

output characteristic as shown below.

Input

Output

MAXx

MINx

No compression

Compressor

Expander


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COMPRESSOR

UNIFORM

QUANTIZER

EXPANDER

Input

signal

Output

signal

TRANSMITTER

RECEIVER

Non-uniform Quantization T t l d li d


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Two most commonly used non-linear companders are:1.-Law compander

2.A-Law compander

In the -Law compander the compressor characteristics iscontinuous and is described by the relation:

-Law is neither strictly linear nor strictly logarithmic.

It is approximately linear at low input levels corresponding to

It is approximately logarithmic at high input levelscorresponding to

( )

( )

+

+=

1log

1log 12

vv

voltagOutputv 2

voltagInputv 1

11>v

Characteristics of -Law compander


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1v

2v

=100

=5

=0

Uniform

quantizationcorresponding to

=0


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Characteristics of A-Law compander


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1v

2v

A=100

A=2

A=1

Uniform

quantizationcorresponding to

A=1

CODING

3

4

3 5111 7


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0

1

2

3

1

2

3

4

3.5

2.5

1.5

0.5

-0.5

-1.5

-2.5

-3.5

111

110

101

100

011

010

001

000

7

6

5

4

3

2

1

0

Sample Values 0.5 3.1 3.2 0.6 -2.2 -2.4 -0.4 2.4

NQL 0.5 3.5 3.5 0.5 -2.5 -2.5 -0.5 2.5

Code 4 7 7 4 1 1 3 6

Binary 100 111 111 100 001 001 011 110

Code

CODING


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The analogue signal is sampled at discrete time intervals to

get sample values of the signal.

The sample values are then quantized to restrict its amplitude

to discrete values. We can transmit these sample values directly.

We can also represent the quantized sample values by code

numbers and transmit these codes.

Another alternative is to convert the code numbers to its

binary equivalent and transmit it.

Such a system is called Binary PCM.

If there are q quantization levels we require v bits to representthese levels such that vq 2=

CODING

If th ti ti l l 16 i 4 bit t t


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If the quantization levels are 16 we require 4 bits to represent

these levels.

The quality of reproduced signal increases as the number of

levels increases. 64 levels gives poor quality audio and video.

If 256 levels are used we get good quality audio and video.

Signal to quantization noise ratio for

Linear Quantization


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Linear Quantization Signal to Noise ratio

Normalized noise power

If there are q quantization levels and v bits are used to

represent these levels

rNoise PoweNormalized

PowerSignalNormalized

N

S

=

12

2=

12/

2

= PowerSignalNormalizedN

S

vq 2=


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Signal to quantization noise ratio for

Linear Quantization


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Linear Quantization If the input is normalized

If the signal power is normalized, P1, i.e.

For normalized values of amplitude of input x(t) and power P,

1=MAXx

v

PN

S 2

23=

v

N

S 223

( )v

dBN

S 210 23log10

( )dBvNS

dB68.4 +

( )dBvNS

dB

68.4 +

Block Diagram of a PCM systemInpu


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Bandpass

Filter

Sample and

HoldADC

Parallel to

Serial Conv.

Regenerative

Repeater

Regenerative

Repeater

Serial toParallel Conv.

DACHold

CircuitLowpass

Filter

Sample pulses Conversion clock Line speed clock

Line speed clock OutputSignal

utSignal

Block Diagram of a PCM system with Analog

compandingIn


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compandingBand

pass

Filter

Sample

and

Hold

ADCParallel to

Serial Conv.

Regenerative

Repeater

Regenerative

Repeater

Serial to

Parallel Conv.DAC

Hold

Circuit

Lowpass

Filter

Sample pulses Conversion clock Line speed

clock

Line speed clock Outp

utSignal

nputSign

al

Analog

Compre-

ssor

Analog

Expander

Block Diagram of a PCM system with Digital

compandingIn


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compandingBand

pass

Filter

Sample

and

HoldADC

Parallel to

Serial Converter

Regenerative

Repeater

Regenerative

Repeater

Serial to

Parallel Conv.DAC

Hold

Circuit

Lowpass

Filter

Sample pulses Conversion clock Line speed

clock

Line speed clock Outp

utSignal

nputSign

al

Digital

Expander

Digital

Compressor


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Basic DPCM scheme with analog input

The main functional block in the transmitter as well as in the


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receiver is an accumulator which adds up the quantized

differences and produces an approximation of the original

signal.

At each sampling time the transmitter difference amplifier

compares and and produces an error and the

quantizer generates the signal both for transmission to

the receiver and to provide input to the transmitter

accumulator.

At the transmitter we need to know whether is larger or

smaller than and by how much.

We may then determine whether the next differenceneeds to be positive or negative and of what amplitude in

order to bring as close as possible to .

)( kx

)(tx )( kx )(te)(keq

)( kx)(tx

)(keq

)( kx )(tx


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DPCM scheme with prediction

)(tx


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Difference

Amp

Sample and

HoldQuantizer

AccumulatorPredictor

)(

)( kx

)(te )(ke )(keq

)(tso

Predictor Accumulator Filter )(tso

)( kx )( tx


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DPCM scheme with sampled input

QUANTIZER ENCODER

Sampledsignal

)( T

)( snTe )( sq nTe+)1.........()()()( sss nTxnTxnTe =


111/172

PREDICTION

FILTER

)( snTx

)( snTx

)( sq nTx

++

_

)2.........()()()( sssq nTqnTenTe +=

)3.........()()()( sqssq nTenTxnTx +=

)2()( from eqnTengSubstituti sq

)4.........()()()()( ssssq nTqnTenTxnTx ++=

(4))1( eqineqngSubstituti

)5.........()()()()()( sssssq nTqnTxnTxnTxnTx ++=

)6.........()()()( sssq nTqnTxnTx +=

to itise addedization nowith quantnTxthe signal

n ofzed versiothe quantiis indeednTthat xindicateseq

s

sq

)(

)()6(


QUANTIZER ENCODER

Sampled

signal

)(nTx

)( snTe )( sq nTe+


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PREDICTION

FILTER

)( snTx

)( snTx

)( sq nTx

++

_

The comparator finds out the difference between the actual

sample value and predicted sample value The difference is known as prediction error and is denoted by

The quantizer output signal and the previous prediction is

added and is given as input to the prediction filter. This signalis called

)( snTx )( snTx

)( snTe

)( sq nTx


QUANTIZER ENCODER

Sampled

signal

)( snTx

)( snTe )( sq nTe+_


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This makes the prediction more and more close to the actualvalue.

When is very near to the predicted value, error issmall and further, it is added or subtracted from the predictionto make it more near to

When increases suddenly so that there is much

difference between predicted value and large isproduced and it is added or subtracted from predicted value tomake it closer to

PREDICTIONFILTER

)( s)( snTx

)( sq nTx

++

_

)( snTx

)( sq nTe)( snTx

)( snTx)(

sqnTe)(

snTx

)( snTx


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Reconstruction of DPCM signal

+

)(keq)(kxq


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The decoder decodes the incoming binary DPCM signal toreproduce quantized error signal

The prediction filter generates a prediction of thepresent output based on past outputs.

The quantized error signal is added to the prediction togenerate the actual output signal

Based on this the next prediction is updated.

DecoderW

Predictor

+

+

DPCM

Input

Output

)( kxq

)(q

)(keq

)(kxq

)( kxq

Prediction

We can express a signal x(t+Ts) as a Taylor series expansion asbelow.

2 2 3 3( ) ( ) ( )d T d T d


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For small values of Ts,

If we know the derivatives we can predict the future value of a

signal. Putting t=kTs in eq(1)

Let the kth sample of x(t) be x(k). Otherwise x(kTs)=x(k) andx(kTsTs)=x(k1) and so on.

2 2 3 3

2 3

( ) ( ) ( )( ) ( )

2! 3!

s ss s

dx t T d x t T d x t x t T x t T

dt dt dt + = + + + +

( )( ) ( ) ( 1 )s s

dx tx t T x t T

dt+ +

( )( 1) ( ) ( 3 )s

dx kx k x k T

dk+ +

( )( ) ( ) ( 2 )s s s s s

s

dx kT x kT T x kT T

dkT+ +

Prediction

( ) ( ) ( 1)( 4 )

dx k x k x k


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Substituting (4) in (3)

Equation (6) indicates that we can obtain a crude prediction ofthe (k+1)th sample from the two previous samples.

The approximation improves as we add more and more terms.

In general we can express the prediction formula as

( )sdk T

[ ]( 1) ( ) ( ) ( 1) ( 5 )x k x k x k x k + +

( 1) 2 ( ) ( 1) ( 6 )x k x k x k +

1 2( ) ( 1) ( 2) ( ) ( 7 )Nx k a x k a x k a x k N + + +


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Linear Delta Modulation

The input signal x(t) and its quantized approximation areapplied as input to a comparator.

)( tx


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The comparator produces a high level VH when

and a low level VL when

The updown counter increments or decrements its count by 1

according to the count direction.

When count direction is VH the counter counts up and when it

is VL the counter counts down. The state of the counter direction control is the transmitted

signal.

Thus when the step is reduced 0 is transmitted and when the

step is increased 1 is transmitted. For each sample only one bit is transmitted.

)()( txtx


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Clock

LINEAR DELTA MODULATOR TRANSMISSION


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)(tx

)( tx

+ )()( txtx + + + + + + + + + + Counter

Control1 1 1 1 1 1 1 1 1 1 10 0 0 0 0

What is transmitted: 1 1 1 1 1 0 1 1 1 1 0 1 0 1 0 0

Clock

LINEAR DELTA MODULATOR RECONSTRUCTION

What is Received: 1 1 1 1 1 0 1 1 1 1 0 1 0 1 0 0


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)(tx

)( tx

ERRORS IN DELTA MODULATION SYSTEMS

When the slope of the signal is too large , the approximationcannot catch up with the original signal. As a result the error

becomes progressively larger The excessive difference


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becomes progressively larger. The excessive difference

between when slope is large is called slope

overload error. When the signal x(t) remain constant swings up and

down about x(t) as and x(t) are alternately greater and

smaller. This produces a distortion called granular noise or

hunting. At the start up there is a brief interval when is a poor

approximation of x(t) as it takes some time for to catch up

with x(t).

)()( txandtx

)( tx

)( tx

)( tx)( tx

Clock

DELTA MODULATOR SLOPE OVERLOAD ERROR


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)(tx

)( tx

The approximationcannot catch up with

original signal due to

slope overload error

Clock

GRANULAR NOISE OR HUNTING


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)(tx

)( tx

The approximation

oscillates about the

original signal due to

granular error when x(t)is constant

Clock

ERROR AT START UP


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Large error between the

approximation and

original signal at start up

LINEAR DELTA MODULATION-SAMPLED

INPUT

Sampled


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One-Bit

Quantizer

Delay Ts

W

W

+

_+

+

)( snTx

)( snTe )( snTb

)( snTx

)( snTu

])1[( sTnu

Sampled

Input

LINEAR DELTA MODULATION-SAMPLED INPUT

One-Bit

QuantizerW

W

+

_+

)( snTx

)( snTe )( snTb

)( snTx

lttT )(

+


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The error between the sampled value of x(t) and the last

approximated sample is given by

Let u(nTs) be the present sample approximation.

b(nTs) is the quantizer output which is + or - depending onwhether x(nTs) is less than or greater than . That is

b(nTs)=

Delay Ts

W_

)( snTu

])1[( sTnu

)()()( sss nTxnTxnTe =

samplepresentaterrornTe s )(

)()( txofvaluesamplednTx s

t)ion of x(approximatsampledlastnTx s )(

)(])1[( ss nTxTnu =

)( snTx

LINEAR DELTA MODULATION-SAMPLED INPUT

One-Bit

QuantizerW

W

+

_+

_

)( snTx

)( snTe )( snTb

)( snTx

+


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The summer adds the quantizer output i.e. b(nTs) with the

previous sample approximation.

The previous sample approximation u[(n-1)Ts] is obtained by

delaying the sample by one time period Ts. The sampled signal x(nTs) minus the approximated signal

gives the error signal.

Delay Ts

W_

)( snTu

])1[( sTnu

)(])1[()( += ss TnunTu)(])1[()( sss nTbTnunTu +=

)( snTx

LINEAR DELTA MODULATION-SAMPLED INPUT-

RECEIVER

LOWPASS

FILTER

W+

INPUT OUTPUT


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At the receiving end an accumulator generates the staircase

approximated signal output and is delayed by one sampling periodTs.

It is then added to the input signal.

If the input is binary 1 it adds + step to the previous output.

If the signal is binary 0 then is subtracted from the delayed signal.

A lowpass filter is used to reconstruct the original signal.

DELAY

Ts

+

ACCUMULATOR

ADAPTIVE DELTA MODULATION

In delta modulation there are three type of errors:1. Start up error

2 Slope overload error


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2. Slope overload error

3. Granular noise

The reason for all these errors is that the step size in deltamodulation is fixed.

In adaptive delta modulation the step size is not fixed but changes

according to the slope of the signal.


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Sample and HoldComparator

)(tx

)( tx


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The processor has an accumulator and at each active edge of theclock waveform generates a step s which increases or decreases theaccumulator.

The step s is not fixed but is always a multiple of the basic step size.

In response to the kth

active clock edge the processor generates astep equal in magnitude to the step generated in the (k-1) th clock

edge.

Digital to AnalogConverter

Digital Processor

Clock


Sample and HoldComparator

)(tx

)( tx


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This step is added to or subtracted from the accumulator as required tomove towards

If the direction of the step at clock edge k is the same as at (k-1) then theprocessor increases the step size by an amount s(0) where s(0) is the basic

step size.

If the directions are opposite then the processor decreases the magnitude of

the step size by s(0). As the algorithm is carried out there are clock edges when the step size is

zero.

Digital to AnalogConverter

Digital Processor

Clock

)( tx )(tx

Clock


13

14

15


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0

1

2

3

4

5

67

8

9

10

11

12

13

0 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 202


edgekthebeforeyimmediateltxtxifke th)()(1)( >+=

edgekthebeforeyimmediateltxtxifke th)()(1)(


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gfyf )()()(

)1()0()()1()( += keSkekSkSstepktheofsizeStepth

01)0(1)0()0()0()1()0()1( =+=+= + SSeSeSS

)0(1)0(10)1()0()2()1()2( SSeSeSS =+=+= 01)0(1)0()2()0()3()2()3( =+=+= + SSeSeSS

)0(1)0(10)3()0()4()3()4( SSeSeSS =+=+= ++

)0(21)0(1)0()4()0()5()4()5( SSSeSeSS =+=+= ++

)0(31)0(1)0(2)5()0()6()5()6( SSSeSeSS =+=+= ++


138/172


As long as the condition persists the jumps in becomes

progressively larger.

The estimate catches up with x(t) sooner than would be the case withlinear delta modulation.

)( tx)()( txtx >

)( tx


139/172

On the other side when develops large jumps in response to a large

slope in x(t) it may require a large number of clock cycles for these jumps todecay in amplitude when these jumps are no longer required. Quantization

error is larger in this case.

Note that

The ADM system reduces slope error but it increases quantizationerror.

When x(t) remains constant the estimate oscillates about x(t) butthe oscillation frequency is half of the clock frequency.

The noise frequency components introduced by slope overload error

is in low frequency range where as the error introduced byquantization is in high frequency range.

Depending on which frequency components we wish to preserve wemay select DM or ADM.

)( tx


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PULSE WIDTH MODULATION


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142/172


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PWM DEMODULATION

Vcc

A B C

R2


144/172

+

LPF

Q1

R1A B C

A B C

A B C

C1Q2

PWM DEMODULATION

Transistor T1 acts as an inverter.

When the PWM signal is high Q1 is ON and Q2 is OFF.

Capacitor C1 is charged through R2.

The voltage built up in the capacitor depends on the pulse


145/172

The voltage built up in the capacitor depends on the pulse

width of the signal.

When PWM signal is low Q1 is OFF and Q2 is ON.

Capacitor quickly discharges through Q2.

This process repeats for other pulses also.

The amplitude of the saw tooths thus developed is directlyproportional to the width of the pulse.

The envelope of the saw tooth waveform gives the original

signal.

This signal is passed through a second order LPF to filter out

the envelop which is the original signal.

PULSE POSITION MODULATION


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PULSE POSITION MODULATION

In PWM the leading edges are fixed and the trailing edgesvary according to the instantaneous value of the wave form.

The leading edges convey no information.

In PPM we use pulses of equal width for modulation


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In PPM we use pulses of equal width for modulation.

The position of these pulses relative to the reference positionis varied according to the instantaneous value of the

modulating wave form.

Thus we get a train of constant amplitude constant widthpulses whose position at any instant depends on theinstantaneous modulating signal.

PPM can be generated by triggering a monostable with fixedtime period by the trailing edge of the PWM waveform.


148/172


149/172

PPM DEMODULATION

The gaps between the pulses of a PPM signal contain theinformation regarding the modulating signal.

During the gap the transistor is cut off and the capacitor C1

charges through R.


150/172

charges through R.

During the high period of the pulse Q1 is ON and thecapacitor is discharged through Q1.

Hence the waveform at the collector is approximately a saw

tooth waveform whose envelope is the modulating signal.

When it is passed through a second order LPF the originalsignal is recovered.

FEATURES OF PWM AND PPM

The instantaneous power of the signal varies in PWM .

Transmitted power is large in PWM.

In PPM the power is constant.

Transmitted power is small in PPM


151/172

Transmitted power is small in PPM.

Noise interference is minimum in both cases. BW depends on the rise time of the pulses.

TIME DIVISION MULTIPLEXING

When we transmit a sampled signal through a channel thetransmission of the message signal engages the

transmission channel for only a fraction of the sampling

interval.


152/172

Some of the time interval between adjacent samples is free touse by other independent message sources on a time shared

basis.

This technique by which a number of independent messages

are transmitted through a common communication channelwithout mutual interference on a time sharing basis is called

time division multiplexing.


Each message is first restricted to a pre-defined BW by a lowpass filter which removes the high frequencies that are non-

essential to an adequate signal representation.

The LPF outputs are then applied to a commutator that is


153/172

The LPF outputs are then applied to a commutator that is

implemented using electronic switching circuitry. The commutator takes a narrow sample of each of the N input

messages at a rate 1/Ts that is higher than 2fM.

It sequentially interleaves these N samples inside a sampling

interval Ts.

Following the commutation process the multiplexed signal is

then applied to a pulse modulator which transforms the

multiplexed signal in to a form suitable for transmission over

the channel.


At the receiving end of the system the received signal isapplied to a pulse modulator which performs the inverse

operation of the pulse modulator.

The narrow samples produced at the output are distributed to


154/172

e a o sa p es p oduced at t e output a e d st buted to

the appropriate LP Filters by means of a decommutatorwhich operates in synchronism with the commutator in the

transmitter.


155/172



156/172

Frame1 Frame2 Frame3 Frame4 Frame5 Frame6 Frame7 Frame8


If the highest frequency present in all the channels is fM themby sampling theorem fs>2fM.

Therefore the time interval between successive samples from

any one input will be Ts=1/fs where Ts 1/2fM.


157/172

s s M

Hence the time interval Ts contains one sample from eachinput, This time interval is called a frame.

If there are N input channels then in each frame there will be

one sample from each of the N channels.

In one frame of Ts seconds there are total N samples.

Pulse to pulse spacing between two samples in the frame will

be equal to Ts/N

pulsestwobetweenSpacing

1secondperpulsesofNumber =


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Transmission BW of the TDM system is equal to the BW ofsuch a lowpass filter.

sNf2

1BW =


159/172

If sampling rate is equal to Nyquist rate Ms ff 2=MM NffN == 2

2

1BW

MNf=BW


160/172

FDM RECEIVER SIDE

BPF

BPF

DEMODULATOR

DEMODULATOR

LPF

LPF

1f

)(1 tx

)(2 tx


161/172

BPF

BPF

DEMODULATOR

DEMODULATOR

LPF

LPF

CARRIER

SUPPLY

..

2f

3f

nf

)(3 tx

)(txn

FDM BAND ALLOCATION

1f 2f 3f1f2f3f


162/172

1111 MM ffff + 2222 MM ffff + 3333 MM ffff +1111 MM ffff +2222 MM ffff +3333 MM ffff +

Frequency

Band 1

Frequency

Band 2

Frequency

Band 3

Frequency

Band 4

Frequency

Band 5

Total Bandwidth

Guard band


163/172

FREQUENCY DIVISION MULTIPLEXING

7.156.127.15- 6.12-

7.116.87.11- 6.8-


164/172

7.196.167.19- 6.16-

7.116.87.11- 6.8- 7.156.127.15- 6.12- 7.196.167.19- 6.16-

7.196.86.8-7.19-


165/172


4.1113.1004.111- 3.100-

4.1513.1403.140-4.151-


166/172

4.1913.180-4.191- 3.180

4.1113.100

4.1513.140

4.1913.1804.111- 3.100-

3.140-4.151-

3.180-4.191-

4.191- 3.100- 3.100 4.191


LPF MOD BPF CHAN

Ch 1


167/172

LPF

LPF

LPF

MOD

MOD

MOD

BPF

BPF

BPF

NELCOM

BINER

Ch 2

Ch 3

Ch n

Group1


BPF MOD BPFGROUPCO

Group 1

Group 2


168/172

BPF

BPF

BPF

MOD

MOD

MOD

BPF

BPF

BPF

MBININGNETWORK

Super

Group1Group 3

Group n


BPF MOD BPF SUPER

Super

Group 1

Super

Group 2


169/172

BPF

BPF

BPF

MOD

MOD

MOD

BPF

BPF

BPF

GROUPCOM

BINING

NETWORK

Master

Group1

Super

Group 3

Super

Group n


BPF MOD BPFMASTER

Master

Group 1

Master

Group 2


170/172

BPF

BPF

BPF

MOD

MOD

MOD

BPF

BPF

BPF

GROUPCO

MBINING

NETWORK

FDM

OUTPUTMaster

Group 3

Master

Group n

Block Diagram of a Digital Baseband

Communication System

Information

Source

Formatter/

ADC

Source

Encoder

Channel

Encoder


171/172

Base band

ProcessorChannel

Base band

decoder

Channel

Decoder

Source

Decoder

Deformatter/

DAC

Destination/

Output signal

Block Diagram of a Modulated Digital

Communication System

Information

Source

Formatter/

ADC

Source

Encoder

Channel

Encoder


172/172

Bandpass

ModulatorChannel

Bandpass

Demodulator

Channel

Decoder

Source

Decoder

Deformatter/

DAC

Destination/

Output signal

Documents

DC Digital Communication MODULE II