The Sampling Process

7/30/2019 The Sampling Process

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SamplingProcess:

The message signal is usually analog

in nature, as in a speech signal or

video signal

It has to beconvertedinto digital

form before it can be transmitted bydigital means.


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Sampling Process:

Thesampling processing is the firstprocess preformed inanalog-to-digital

conversion. In the sampling process, acontinuous-time

signal is converted into adiscrete-time

signal by measuring the signal at periodicinstants of time.


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Sampling Process:

For the sampling process to be of

practical utility, it is necessary that we

choose the sampling rate properly

So thediscrete-time resulting from

the process uniquely defines theoriginalcontinuous-time signal.


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Sampling Theorem:

Let the signal )(tx be band limited with

bandwidth W i.e., let 0)( fX .Wf for

Let )(tx be sampled at multiples of some

basic sampling interval ST, where WTS

2

1

to yield the sequence nSnTx Then it is

possible to reconstruct the original


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Sampling Theorem:

signal )(tx from the sampled values by the

reconstruction formula:

]2[sin2)( Sn

SS nTtWcnTxTWtx

WWhere ( ) is any arbitrary number thatthat satisfies .

1W

TWW

S


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Sampling Theorem:

special case whereW

TS2

1 the

reconstruction relation simplifies to:

nSn

SW

ntWc

W

nxn

T

tcnTxtx

22sin

2sin)()(

Let )(tx denote the result of the sampling

original signal by impulses atSnT time instants.


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Sampling Theorem:

Then:

n

SSnTtnTxtx )()()(

We can write )(tx as:

n

SnTttxtx )()()(

n

SS nTtnTxtx )()()(

n

SnTttxtx )()()(


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Sampling Theorem:

fW-W-fc fc0

. . . .. . . .

Wfc

ocxf

)( fx

f-W W0

ox)( fx Figure (1):

Signal spectra for low pass sampling.(a) Assumed spectrum for x(t).

(b) Spectrum of sampled signal.


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Sampling Theorem:

Now if we find the Fourier transform of

both sides of the above relation and apply

the dual of the convolution theorem to theright-hand side, we obtain:

( ) ( ) ( ) .....(4)Sn

X f X f F t nT


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Sampling Theorem:

By using Fourier Transform we obtain:

n

SnTtF )(1

....(5)

nS S

nf

T T

By substituting equation (5) into equation (4),

we obtain:

n SS T

nf

TfXfX

1)()(

n SS T

nfX

T

1


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Sampling Theorem:

Where in the last step we have employed the

convolution property of the impulse signal.

This relation shows that )( fX , the Fouriertransform of the impulse-sampled signal is a

replication of the Fourier transform of the

original signal at aS

T1 rate.

Figure (1) shows this situation.


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Sampling Theorem:

Now ifW

TS2

1 then the replicated spectrum of

)(tx overlaps, and reconstruction of the original

signal is not possible. This type of distortion

that results from under-sampling is known as

aliasing error oraliasing distortion.


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Sampling Theorem:

However, ifW

TS2

1 no overlap occurs, and by

employing an appropriate filter we can

reconstruct the original signal back. To obtainthe original signal back, it is sufficient to filter

the sampled signal by a low pass filter with

frequency response characteristic


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Sampling Theorem:

STfH )( Wf

0)( fH WT

fS

1

1. for.

2. for

For WT

fWS

1 , the filter can have any

characteristics that make its implementation easy.

Of course, one obvious (though not practical)choice is an ideal low pass filter with bandwidth

W W WT

WWS

1

where satisfies , i.e.


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Sampling Theorem:

W

fTfH S

2)(

With this choice we have:

W

fTfXfX S

2)()(

Taking inverse Fourier transform of both sides,

we obtain: tWcTWtxtx S 2sin2)()(


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Sampling Theorem:

tWcTWnTtnTx Sn

SS

2sin2

n

SSS nTtWcnTxTW 2sin2

This relation shows that if we use sine functions

for interpolation of the sampled values, we canreconstruct the original signal perfectly.


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Sampling Theorem:

The sampling rateW

fS2

1 is the minimum

sampling rate at which no aliasing occurs.

Thissampling rate is known as theNyquistsamplingrate.

If sampling is done at theNyquist rate,

then the only choice for the reconstructionfilteris an ideal low pass filter and .

2

1

STWW


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Sampling Theorem:

Then:

n

nWtcW

nxtx 2sin

2)(

n S

Sn

TtcnTxtx sin)(

In practical systems, sampling is done at a rate

higher than the Nyquist rate. This allows forthe reconstruction filter to be realizable and

easier to build.


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Sampling Theorem:

In such cases the distance between two adjacent

replicated spectra in the frequency domain; i.e.

WfWWT SS 2

1

, is known asthe guard band.Note that there exists a strong similarity

between our development of the sampling

theorem and our previous development of theFourier transform for periodic signals

(or Fourier series).


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Sampling Theorem:

In the Fourier transform for periodic signals,

we started with a time periodic signal and

showed that its Fourier transform consists ofa sequence of impulses.

Therefore, to define the signal, it was enough

to give the weights of these Impulses(Fourier series coefficients).


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Sampling Theorem:

In the sampling theorem, we started with an

impulse-sampled signal, or a sequence of

impulses in the time domain, and showed thatthe Fourier transform is a periodic function in

the frequency domain. Here again, the values

of the samples are enough to define the signalcompletely.


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Sampling Theorem:

This similarity is a consequence of the duality

between the time and frequency domains and

the fact that both the Fourier series expansionand reconstruction from samples are orthogonal

expansions, one in terms of the exponential

signals and the other in terms of the sineFunctions.


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Analog Pulse Modulation:

In the sampling theory section we show that

continuous band limited signals can be

represented by a sequence of discrete samplesand that the continuous signal can be

reconstructed with negligible error if the

sampling rate is sufficiently high.Consideration of the sampled signals leads us

to the topic of the pulse modulation.


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Pulse modulation can be either analog, in which

some attribute of a pulse varies continuously in

one-to-one correspondence with sample value,or digital, in which some attribute of a pulse can

take on a certain value from a set of allowable

values.


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t

t

t

tAnalog

Signal

(Samples)

PAM Signal

PWM

Signal

PPM

SignalTs 2Ts 9Ts0

Figure (2): illustration ofPAM, PWM, and PPM


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As mentionedAnalog Pulse Modulation

results when some attribute of a pulse values

continuously in one-to-one correspondencewith a sample value. There are three pulse

attributes that can be readily varied:

Amplitude, Width, andPosition.


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These lead to pulse amplitude modulation

(PAM), pulse width modulation (PWM), and

pulse position modulation (PPM), as illustratedin figure (2).


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Pulse Amplitude Modulation:

A (PAM) waveform consists of a sequence of

a flat-topped pulses designating sample value.

The amplitude of each pulse corresponds to thevalue of the message signal at the leading edge

of the pulse.

The essential difference between PAM andsampling operation is that in PAM we allow the

sampling pulse to have finite width.


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The finite-width pulse can be generated from

impulse-train sampling function by passing

the impulse-train sample through a holdingnetwork as shown in figure (3). The holding

network transforms the impulse function

samples, given by:

n

sS nTtnTxtx )(


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(a)

Input PAM Output

)(th (b)

)(th

0 t

Slope= -

(d)

f

)( fH

2/1/-1/-2/

f0

(c)

)(fH

2/1/-1/-2/f

0

(c)

)(fH

Figure (3): Generation of PAM.(a) Holding network.(b) Impulse response of holding network.(c) Amplitude response of holding network.(d) Phase response of holding network.


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From figure (2) a PAM signal can be written as:

n

S

SPAM

nTt

nTxtx

2

1

)(

The waveform is generated by placing the

impulse function in (11) on the output of aholding network having the impulse response.


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2

1

)(

SnTt

th

And the transfer function is:

fjefcfH

sin)(


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Since the holding network dose not have a

constant amplitude response over the bandwidth

of)(tx , unless of course the pulse widthis sufficiently narrow, amplitude distortion

results. This amplitude distortion can be

removed by passing the samples, prior toreconstruction of )(tx


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through a filter having an amplitude response

equal to )(1 fH , over the bandwidth of ).(tx

.

Since the phase response of the holdingnetwork is linear, the effect is a time delay and

can usually be neglected.


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Pulse Width Modulation:

A (PWM) waveform, as illustrated in figure (2),

consists the sequence of pulse width each pulse

having a width proportional to the values of thea message signal at the sampling instants.

If the message is (0) at the sampling time, the

width of the (PWM) pulse is .21

ST


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Thus, pulse widths less thanST

2

1correspond

to negative sample values and the pulse widths

greater than correspond to positive sampleST21

values.

PWM is seldom used in modern communications

systems.


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PWM is used extensively for DC motor

control in which motor speed is proportional

to the width of the pulses.Since thee pulses have equal amplitude, the

energy in a given pulse is proportional to the

pulse width. Thus, the sample values can berecovered from a PWM waveform by low pass

filtering.


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Pulse Position Modulation:

A (PPM) signal consists of a sequence of

pulses in which the pulse displacement from

a specified time reference is proportional tothe sample values of the information-bearing

signal.

A (PPM) signal is illustrated in figure (2),and can be represented by the expression:


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n

nttgtx )()(

Where )(tg represents the shape of theindividual pulses, and occurrence times nt

are related to the values of the message signal

)(txSnTat the sampling instants , as discussed

previously.


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The spectrum of a PPM signal is very similar to

the spectrum of a PWM signal.

If the time axis is slotted so that a given rangeof sample values is associated with each slot,

the pulse positions are quantized and pulse is

assigned to given slot depending on the samplevalue.


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Slots are non-overlapping and are therefore

orthogonal.

If a given sample value is assigned to oneof (M) slot, the result is (M-ary) orthogonal

communications. PPM is finding new

applications in area of ultra-widebandcommunications.

The Sampling Process

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