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Sampling and aliasing

Sampling can be modelled as a point-by-point multiplication in the time domainby a pulse train (a sequence of impulses). (Recall [link] .) While this is intuitively plausible, it is not terribly insightful.The effects of sampling become apparent when viewed in the frequency domain.When the sampling is done correctly, no information is lost. However, if the sampling is done too slowly,aliasing artifacts are inevitable. This section shows the “how” and “why” of sampling.

Suppose an analog waveform w ( t ) is to be sampled every T s seconds to yield a discrete-time sequence w [ k ] = w ( k T s ) = w ( t ) | t = k T s for all integers k . Observe the notation w ( k T s ) means w ( t ) evaluated at the time t = k T s . This is also notated w [ k ] (with the square brackets), where the samplingrate T s is implicit. This is called point sampling because it picks off the value of the function w ( t ) at the points k T s . One way to model point sampling is tocreate a continuous valued function that consists of a train of pulses that are scaled by the values w ( k T s ) . The impulse sampling function is

w s ( t ) = w ( t ) k = - δ ( t - k T s ) = k = - w ( t ) δ ( t - k T s ) = k = - w ( k T s ) δ ( t - k T s ) ,

and it is illustrated in [link] . The effect of multiplication by the pulse trainis clear in the time domain. But the relationship between w s ( t ) and w ( t ) is clearer in the frequency domain, which can be understood by writing W s ( f ) as a function of W ( f ) .

An analog signal w(t) is multiplied point-by-point by a pulse train. This effectively samples the analog signal at a rate Ts.
An analog signal w ( t ) is multiplied point-by-point by a pulse train. This effectively samples the analog signalat a rate T s .

The transform W s ( f ) is given in [link] . With f s = 1 / T s , this is

W s ( f ) = f s n = - W ( f - n f s ) .

Thus, the spectrum of the sampled signal w s ( t ) differs from the spectrum of the original w ( t ) in two ways:

  • Amplitude scaling—each term in the spectrum W s ( f ) is multiplied by the factor  f s .
  • Replicas—for each n , W s ( f ) contains a copy of W ( f ) shifted to f - n f s .

Sampling creates an infinite sequence of replicas, each separated by f s Hz. Said another way, the process of sampling in time creates a periodicity in frequency,where the period is defined by the sampling rate.Readers familiar with Fourier series will recognize this as the dual of the property thatperiodic in time is the equivalent of sampling in frequency. Indeed,  [link] shows why the relationships in [link] hold.

[link] shows these replicas in two possible cases. In (a), f s 2 B , where B is the bandwidth of w ( t ) , and the replicas do not overlap. Hence, it is possible to extract the one replica centeredat zero by using a lowpass filter. Assuming that the filtering is without error, W ( f ) is recovered from the sampled version W s ( f ) . Since[2]Be clear about this: The analog signal w ( t ) is sampled to give w s ( t ) , which is nonzero only at the sampling instants k T s . If w s ( t ) is then input into a perfect analog lowpass filter, its output is the same as the original w ( t ) . Such filtering cannot be done withany digital filter operating at the sampling rate f s . In terms of [link] , the digital filter can remove and reshape the frequenciesbetween the bumps, but can never remove the periodic bumps. the transform is invertible, this meansthat w ( t ) can be recovered from w s ( t ) . Therefore, noloss of information occurs in the sampling process. 2 This result is known as the Nyquist sampling theorem,and the minimum allowable sampling rate is called the Nyquist rate .

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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Source:  OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3
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