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CHAPTER 1

Syllabus

1) Basic signal processing operation in Digital communication.

a) Channels for digital communication.

2) Sampling theorem.

3) Quadrature sampling of a band pass signal.

a) Signal distortion in sampling.

4) Practical aspects in sampling and signal recovery.

a) Natural sampling.

b) Flat top sampling.

c) Sample and hold Circuit.

Communication

The purpose of communication system is to transport an information bearing signal from

a source to user destination through communication channel.

Signal

The information bearing signal may be

1) Analog: Continuous in time and amplitude.

2) Digital: Discrete in time and amplitude.

If the information bearing signal is analog type then it is known as analog

communication, on the other hand if it is of digital type then it is known as Digital

communication.

Merits of digital over analog

1) Digital communication systems are simple and cheaper compared to analog

communication system

2) The improved reliability by the use of digital communication system.

3) The availability of wide band channels

4) Digital signals are less subjected to distortions when compared to analog signal.

5) Multiplexing of the digital signals are easier when compared to analog.

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Demerits

1) Transmission of digital data requires more system band width as compared to analog

signals

2) Synchronization is must in digital communication.

Block schematic description of a digital communication system

1) Assume the source information is digital one.

2) Source encoder:

Maps the digital signal represented at the source output into another form,

objective is to reduce redundancy.

3) Channel encoder:

Provides reliable communication over the noisy channel, this provision is

satisfied by introducing redundancy in prescribed fashion and exploiting the decoder.

4) Modulator:

Last stage of the transmitter, operates by keying shifts in amplitude, frequency or

phase.

5) Channel:

The channel is a medium through which the modulated signal passes.

6) Detector:

First stage of the receiver performs demodulation operation, inverse of

modulation process.

7) Channel decoder:

Maps the channel output into an output digital signal in such a way that the effect

of channel noise is reduced.

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8) Source decoder:

The source decoder performs the inverse mapping of the source encoder to

finally obtain the user information.

Channels for digital communication

The channel is a medium through which the modulated signal passes. Different channels

are used depending on the characteristics of the channel and the application of interest:

1) Bandwidth

2) Power

3) Amplitude response

4) Phase response

5) Linearity

6) External interferences.

Generally there are five specific channels they are

1) Telephone channel:

Voice grade communication, transmission media used are open wire lines, cables,

optical fibers, microwave radio and satellite

Sl No. Parameters Value

1 Frequency range 300 – 3.4Khz

2 SNR 30dB

3 Maximum transfer rate 16.8 kbps

2) Coaxial cables

The co-axial cables have single wire conductor centered inside an outer

conductor, insulated each other by an insulated material.

The advantages of the coaxialcables are:

1) Relatively wider bandwidth.

2) Freedom from external interferences.

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Sl No. Parameters Value

1 Frequency range 400Mhz

2 Maximum transfer rate 274kbps

3 Repeater spacing 1Km

3) Optical fibers

a) Optical fiber uses light to represents signal.

b) They work on the principle of total internal reflection.

c) They are free from external interferences.

d) They have wider bandwidth and longer repeater separation.

Sl No. Parameters Value

1 Frequency range 1014

to 1015

Hz

2 Maximum repeater spacing 2 KM

3 Effects of external noise Minimum

4) Microwave radio

a) It operates on line of sight link, consists of transmitting antenna and receiving

antenna.

b) The problem is multipath reception i.e., propagation takes places in several paths

thus receiver sees a weighted sum of delayed replicas of transmitted signal

interfering each other, hence the received signal experiences fading.

Sl No. Parameters Value

1 Frequency range 1 to 30Ghz

2 Maximum distance covered 50KM

3 Maximum transfer rate 7500Mbps

5) Satellite channels.

a) A satellite channel consists of a satellite I geostationary orbit.

b) It acts as a repeater between uplink and downlink signals.

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Sl No. Parameters Value

1 Frequency range 0.25 to 65Ghz

2 Maximum distance covered 1/8 part of the earth

3 Maximum transfer rate 2500Kbps

Sampling process

A common use of sampling theorem is for converting a continuous-time signal to an

equivalent discrete-time signal and vice versa.

Sampling theorem

Statement

1) If a signal x(t) does not contain any frequency component beyond W Hz, then the signal

is completely described by its instantaneous uniform samples with sampling interval (or

period ) of Ts < 1/(2W) sec.

2) The signal x(t) can be accurately reconstructed (recovered) from the set of uniform

instantaneous samples by passing the samples sequentially through an ideal (brick-wall)

low pass filter with bandwidth B, where W ≤ B < fs – W and fs = 1/(Ts).

`

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( ) ∑ ( ) ( ) --------1

( ) is a dirac delta function located at time t = nTS

From the definition of delta function

( ) ( ) = ( ) ( )

Hence equation 1 may be written as

( ) ( )∑ ( ) -------------2

= ( ) (t)

( ) = ∑ ( )

Hence equation 2 may be written as

( ) ( ) ∑ ( )

Interchanging the order of summation and convolution yields

( ) ∑ ( ) ( )

From the properties of delta function

( ) ∑ ( ) --------------3

Thus ( ) represents a periodic extension of the original spectrum G(f).

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The equation 1 may also be written as

( ) ∑ ( ) ( ) ----------4

The above relations are applied to any continuous time signal g(t) of finite energy and infinite

duration. The signal is strictly band limited, with no frequency greater than ‘W’ hertz.

Suppose we choose the sampling period as “TS = 1/2W” in equation 4 we obtain

( ) ∑ (

) ( )

------------------------------5

Putting fs = 2W in equation 3 we obtain

( )

( )

Hence from equation 5 we have

( ) ∑ (

) (

) –

Hence the sequence g(n/2W) contains all the information of g(t).

Reconstruction of signal g(t)

( ) ⟨∫ ( ) ( )

⟩

∫

∑ (

) (

)

( )

Interchanging the order of summation and integration

( ) ∑ (

)

∫ [( (

)]

The integral tem in the equation is evaluated to obtain,

( ) ∑ (

)

( )

( )-------------------6

We can simplify the above expression using sinc function, defined as

The sinc function exhibits an interpolatory property, defined as

( ) {

Hence we may write equation 6 as

( ) ∑ (

)

( )--------------------------7

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Equation 7 provides an interpolation formula for reconstructing the original signal g(t) from the

sequence of sample values {g(n/2W)}. Each sample is multiplied by a delayed version of the

interpolation function, and all the resulting waveforms are added to obtain g (t).

The g(t) can be recovered from the samples by passing it through a low pass filter of bandwidth

W as shown in the block

Signal distortion in sampling

In deriving the sampling theorem, we assumed that the signal g(t) is band limited with

no frequency greater than ‘W’ hertz however the signal cannot be finite in both time and

frequency, when such signal is sampled, an error in the reconstruction occurs.

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The discrete spectrum Gδ(f) resulting from the idealized sampling, the replicas of G(f)

are shifted in frequency by multiples of sampling rate fs, two replicas can be seen are shown in

the figure above at fs and –fs , we find that portions of the frequency shifted replicas are folded

over seen by the shaded areas of the spectra. The phenomenon of a high frequency component

taking over the low frequency components is known as aliasing or foldover, when this effect

occurs, information is lost in the sampling process and hence by passing this information

through low pass reconstruction filter no longer yields an undistorted original signal g(t).

Quadrature sampling of band pass signals

In Quadrature sampling of a band pass signal, the band pass signal is represented in

terms of its in – phase and Quadrature components, each of which may then be sampled

separately.

Let g(t) is a band pass signal having highest frequency component fc + W and lowest

frequency component fc – W, centered at fc .

Any band pass signal can be represented canonically in terms of in phase and Quadrature phase

components. We may express g(t) as

( ) ( ) ( ) ( ) ( )

The in phase component ( ) and Quadrature phase component ( ) may be obtained

by multiplying the band pass signal g(t) by ( ) and ( ) respectively.we find

that ( ) and ( ) are both low pass signals limited to – W < f < W, as illustrated in the above

figure. Accordingly, each component may be sampled at the rate of 2W samples per second.

This form of sampling is called Quadrature sampling, as shown in the figure below.

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To reconstruct the original band pass signal from its Quadrature sampled version, first

reconstruct the in - phase and Quadrature – phase components from their respective samples,

multiply them respectively by ( ) and ( ), and then add the results, depicted as

shown in the figure below

Practical aspects of sampling

The sampling function (pulses) will have a finite duration rather than an impulse

1) The practical reconstruction filters are not ideal these filters need a guard band, ie.,

gap between the spectral components.

2) The signals to be sampled are not band limited, therefore there should be a proper

selection of sampling frequency fs.

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There are two types of practical sampling

1) Natural sampling

2) Flat top sampling

Natural sampling

Let an arbitrary analog signal is applied to a switching circuit controlled by a sampling

function c(t) that consists of finite succession of rectangular pulses of amplitude A, duration T,

occurring with a period Ts .The output of the switching circuit is denoted by s(t).The waveforms

for g(t), c(t) and s(t) are as shown in the figure below.

Mathematically we have

( ) ( ) ( )--------------------------------------1

c (t) may be expressed in the form of Fourier series as

( ) ∑ ( ) ( ) -------------------------2

Substituting equation 2 in 1, we get

( ) ∑ ( ) ( ) ( )

Taking Fourier transform to above equation, we obtain

( ) ∑ ( ) ( )

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The relation between S(f) and G(f) is illustrated in the below figure assuming that g(t)

contains no frequency components greater than ‘W’ hertz and sampling frequency as fs > 2W, so

that there is no aliasing. The original signal can be recovered by passing the s(t) signal through

an ideal low pass filter.

Inference: since the original signal can be recovered from s(t), it concludes that the use of

sampling process of finite duration has no important effect on sampling process. It is seen that

as the pulse duration approaches to zero s (f) approaches Gδ(f).

Flat top sampling

Consider an analog signal which is instantaneously sampled at athe rate fs = 1/Ts and

duration of the each sample is lengthened to ‘T’, as shown in the figure.

s(t) denotes the sequence of flat top samples generated, we may write

( ) ∑ ( ) ( )-------------------------------1

h(t) is a rectangular pulse of unit amplitude and duration T

we have

( ) ∑

( ) ( )

Convolving ( ) with the pulse h(t),we ge

( ) ( ) ∫

( ) ( )

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( ) ( ) ∫ ∑ ( ) ( )

( )

( ) ( ) ∑

( ) ∫ ( )

( )

From the shifting property of delta function we hav

( ) ( ) ∑ ( ) ∫

( )-----------2

Comparing the equations 1 and 2 it follows that s(t) is mathematically equivalent to the

convolution of ( ) , hence

s(t) = ( ) ( )

Taking Fourier transform on both sides we have,

S (f) = ( ) ( )

Hence substituting the value of ( ) we obtain

S (f) = ∑ ( ) ( )

If the signal g(t) is strictly band limited and sampling frequency fs > 2W. Then, passing

s(t) through a low – pass filter to obtain the original signal g(t).

We may give h(t) as

( ) ( ) ( )

This is plotted in the figure as

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Thus flat top sampling introduces amplitude distortion is introduced in reconstructed

signal g(t) from s(t), this effect is called “aperture effect”. The distortion may be corrected by

connecting the equalizers in cascade with low pass reconstruction filter; ideally the amplitude

response of the equalizer is given by

( )

( )

Swamy

Sample and hold circuit Lecturer

The circuit is as shown in the figure Dept. of E&C

The circuit consists of unity gain and low output impendence, a switch and a capacitor.

The switch is timed to close only for a small duration ‘T’ of each sampling pulse, during that

time the capacitor rapidly charges up to a voltage level equal to that of the input sample. When

the switch is open, the capacitor retains its voltage level until the next closure of the switch.

Thus the sample and hold circuit, in its ideal form, produces an output waveform that represents

a staircase interpolation of the original analog signal as shown in the figure

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The output may be defined as

( ) ∑

( ) ( )

Where h(t) is an impulse response representing the action of sample and hold circuit

( ) {

Taking Fourier transform we obtain

( ) ∑ ( ) ( )

H(f) is the transfer function given by

( ) ( ) ( )

The signal g(t) can be reconstructed by passing the output of the sample and hold circuit

through a low pass filter designed to remove components of spectrum U(f) at multiples of

sampling rate fs and an equalizer whose amplitude response equals 1/|H(f)|.