DC Chapter2a

8/12/2019 DC Chapter2a

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Digital Communications

Instructor: Hammad A. Khan


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Chapter 2: Formatting and

Baseband Modulation2.1 Baseband Systems

2.2 Formatting Textual Data (Character Coding)2.3 Messages, Characters and Symbols

2.4 Formatting of Analog Information

2.6 Pulse Code Modulation IPCM)

2.5 Sources of Corruption

2.7 Uniform and Nonuniform Quantization

2.8 Baseband Modulation

2.9 Correlative Coding: Not covered


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2.1 Baseband Systems


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Formatting


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Character Coding (Textual Information)

A textual information is a sequence of alphanumeric characters

Alphanumeric and symbolic information are encoded into digital bitsusing one of several standard formats, e.g, ASCII, EBCDIC

2.2 Formatting Textual DataCharacter Coding)


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Example: Message: USA

In ASCII alphabets, numbers, and symbols are encoded using a 7-bit code

A total of 27= 128 different characters can be represented using

a 7-bit unique ASCII code (see ASCII Table, Fig. 2.3)

2.3 Messages, Characters andSymbols


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2.4 Formatting Analog Information Structure of Digital Communication Transmitter

Analog to Digital Conversion


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Analog-to-digital conversion is (basically) a 2 step process: Sampling

Convert from continuous-time analog signalxa(t) to discrete-

time continuous value signalx(n)

Is obtained by taking the samples ofxa(t) at discrete-time

intervals, Ts

Quantization Convert from discrete-time continuous valued signal to discrete

time discrete valued signal


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Sampling Sampling is the processes of converting continuous-time analog

signal,xa(t), into a discrete-time signal by taking the samples atdiscrete-time intervals

Sampling analog signals makes them discrete in time but still

continuous valued If done properly (Nyquist theorem is satisfied), sampling does not

introduce distortion

Sampled values:

The value of the function at the sampling points Sampling interval:

The time that separates sampling points (interval b/w samples), Ts If the signal is slowly varying, then fewer samples per second will

be required than if the waveform is rapidly varying

So, the optimum sampling rate depends on the maximumfrequency component present in the signal


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Sampling Sampling Rate (or sampling frequency fs):

The rate at which the signal is sampled, expressed as thenumber of samples per second (reciprocal of the sampling

interval), 1/Ts = fs

Nyquist Sampling Theorem (or Nyquist Criterion):

If the sampling is performed at a proper rate, no info is lost about

the original signal and it can be properly reconstructed later on Statement:

If a signal is sampled at a rate at least, but not exactly equal totwice the max frequency component of the waveform, then the

waveform can be exactly reconstructed from the sampleswithout any distortion

max2sf f


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Sampling If fs < 2B, aliasing(overlapping of the spectra) results

If signal is not strictly bandlimited, then it must be passed through

Low Pass Filter(LPF) before sampling

Fundamental Rule of Sampling (Nyquist Criterion) The value of the sampling frequency fs must be greater than twice

the highest signal frequency fmax of the signal

Types of sampling

Ideal Sampling

Natural Sampling

Flat-Top Sampling


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Ideal Sampling ( or Impulse Sampling) Is accomplished by the multiplication of the signalx(t) by the uniform

train of impulses (comb function)

Consider the instantaneous sampling of the analog signalx(t)

Train of impulse functions select sample values at regular intervals

( ) ( ) ( )

( ) ( ) (2.5)

s s

n

s s

n

x t x t t nT

x nT t nT

=

=

=

=


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Ideal Sampling ( or Impulse Sampling) Take Fourier Transform (frequency convolution)

1( ) ( )* ( )s s

ns

X f X f t nTT

=

=

1

( ) ( )* ( ), 2

s

s s snsX f X f f nf fT

=

= =

1 1

( ) ( ) ( )s sn ns s s

nX f X f nf X f

T T T

= =

= =

This results in:


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Ideal Sampling ( or Impulse Sampling)This shows that the Fourier Transform of the sampled signal is theFourier Transform of the original signal at rate of 1/Ts


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Ideal Sampling ( or Impulse Sampling)This means that the output is simply the replication of the originalsignal at discrete intervals, e.g


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Ideal Sampling ( or Impulse Sampling) As long as fs> 2fm,no overlap of repeated replicasX(f - n/Ts) will

occur inXs(f)

Minimum Sampling Condition:

Sampling Theorem: A finite energy functionx(t) can be completelyreconstructed from its sampled valuex(nTs) with

provided that =>

2s m m s mf f f f f > >

2 ( )

sin 2( ) ( )

( )

s

s

s s

n s

f t nT

Tx t T x nT

t nT

=

=

( ) sin (2 ( ))s s s sn

T x nT c f t nT

=

= 1 12

s

s m

Tf f

=


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Ts is called the Nyquist interval: It is the longest time interval that can

be used for sampling a bandlimited signal and still allow

reconstruction of the signal at the receiver without distortion


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Practical Sampling In practice we cannot perform ideal sampling

It is not practically possible to create a train of impulses Thus a non-ideal approach to sampling must be used

We can approximate a train of impulses using a train of very thin rectangular

pulses:

Note:

Fourier Transform of impulse train is another impulse train

Convolution with an impulse train is a shifting operation

( ) spn

t nTx t

=

=


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Natural SamplingIf we multiplyx(t) by a train

of rectangular pulsesxp(t),

we obtain a gated waveform

that approximates the ideal

sampled waveform, known

as natural sampling or

gating (see Figure 2.8)

( ) ( ) ( )s px t x t x t=

2( ) s

j nf t

n

n

x t c e

=

= ( ) [ ( ) ( )]s pX f x t x t=

2[ ( ) ]s

n f t

n

n

c x t e

=

=

[ ]n sn

c X f n f

=

=


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Each pulse inxp(t) has width Ts and amplitude 1/Ts The top of each pulse follows the variation of the signal being

sampled

Xs (f) is the replication ofX(f) periodically every fs Hz

Xs (f) is weighted by Cn Fourier Series Coeffiecient

The problem with a natural sampled waveform is that the tops of the

sample pulses are not flat

It is not compatible with a digital system since the amplitude of eachsample has infinite number of possible values

Another technique known as flat top sampling is used to alleviate

this problem


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Flat-Top Sampling Here, the pulse is held to a constant height for the whole

sample period

Flat top sampling is obtained by the convolution of the signalobtained after ideal sampling with a unity amplituderectangular pulse,p(t)

This technique is used to realize Sample-and-Hold (S/H)

operation In S/H, input signal is continuously sampled and then the

value is held for as long as it takes to for the A/D to acquire

its value


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Flat top sampling (Time Domain)

'( ) ( ) ( )x t x t t=

( ) '( )* ( )sx t x t p t=

( )* ( ) ( ) ( )* ( ) ( )sn

t x t t p t x t t nT

=

= =


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Taking the Fourier Transform will result to

where P(f) is a sincfunction

( ) [ ( )]s sX f x t=

( ) ( ) ( )sn

P f x t t nT

=

= 1

( ) ( ) * ( )snsP f X f f nfT

=

= 1

( ) ( )sns

P f X f nf

T

=

=


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Flat top sampling (Frequency Domain)

Flat top sampling becomes identical to ideal sampling as thewidth of the pulses become shorter


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Recovering the Analog Signal One way of recovering the original signal from sampled signalXs(f)is to pass it through a Low Pass Filter (LPF) as shown below

If fs > 2B then we recoverx(t) exactly

Else we run into some problems and signalis not fully recovered


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Undersampling and Aliasing

If the waveform is undersampled(i.e. fs < 2B) then there will bespectral overlap in the sampled signal

The signal at the output of the filter will be

different from the original signal spectrum

This is the outcome of aliasing!

This implies that whenever the sampling condition is not met, anirreversible overlap of the spectral replicas is produced


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This could be due to:

1. x(t) containing higher frequency than were expected2. An error in calculating the sampling rate


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Plot of Aliased Cosine (with 90 degrees phase

shift) and Its Reconstruction


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close all

clear all

fs=8000;f0=1;

fst=4;

t=0:1/fs:2-1/fs;

x=sin(2*pi*f0*t);x2=sin(2*pi*(f0+fst)*t);

N=length(x);

plot(t,x);

hold onplot(t,x2,'g');

xs=x(1:fs/fst:N);

tn=t(1:fs/fst:N);

stem(tn,xs,'r');pause

xsn=x2(1:fs/fst:N);

stem(tn,xsn,'c');


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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1


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Solution 1: Anti-Aliasing Analog Filter

All physically realizable signals are not completely bandlimited

If there is a significant amount of energy in frequencies above

half the sampling frequency (fs/2), aliasing will occur

Aliasing can be prevented by first passing the analog signalthrough an anti-aliasing filter (also called a prefilter) before

sampling is performed

The anti-aliasing filter is simply a LPF with cutoff frequency

equal to half the sample rate


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Aliasing is prevented by forcing the bandwidth of the sampled

signal to satisfy the requirement of the Sampling Theorem


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Solution 2: Over Sampling and Filtering in the Digital

Domain

The signal is passed through a low performance (less costly)

analog low-pass filter to limit the bandwidth.

Sample the resulting signal at a high sampling frequency. The digital samples are then processed by a high

performance digital filter and down sample the resulting

signal.


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Summary Of Sampling

Ideal Sampling(or Impulse Sampling)

Natural Sampling

(or Gating)

Flat-Top Sampling

For all sampling techniques

If fs > 2B then we can recover x(t) exactly

If fs < 2B) spectral overlappingknown as aliasing will occur

( ) ( ) ( ) ( ) ( )

( ) ( )

s sn

s s

n

x t x t x t x t t nT

x nT t nT

=

=

= =

=

2( ) ( ) ( ) ( ) s

j nf t

s p n

n

x t x t x t x t c e

=

= =

( ) '( )* ( ) ( ) ( ) * ( )s sn

x t x t p t x t t nT p t

=

= =


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Example : Consider the analog signalx(t) given by

What is the Nyquist rate for this signal?

Example 2: Consider the analog signalxa(t) given by

What is the Nyquist rate for this signal?

What is the discrete time signal obtained after sampling, if

fs=5000 samples/s. What is the analog signalx(t) that can be reconstructed from the

sampled values?

( ) 3cos(50 ) 100sin(300 ) cos(100 )x t t t t = +

( ) 3cos 2000 5sin 6000 cos12000ax t t t t = + +

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