In this module we illustrate the processes involved in sampling and reconstruction.
To see how all these processes work together as a whole, take a look at the
system view . In
Sampling and reconstruction with Matlab we provide a Matlab script
for download. The matlab script shows the process of sampling and reconstruction
live .
Basic examples
To sample an analog signal with 3000 Hz as the
highest frequency component requires samplingat 6000 Hz or above.
The sampling theorem can also be applied in two dimensions, i.e. for image analysis.
A 2D sampling theorem has a simple physical interpretation in image analysis:Choose the sampling interval such that it is less than or equal to half of the
smallest interesting detail in the image.
The process of sampling
We start off with an analog signal. This can for example be the sound
coming from your stereo at home or your friend talking.
The signal is then sampled uniformly. Uniform sampling implies that we sample every
${T}_{s}$ seconds. In
we see an analog signal. The analog
signal has been sampled at times
$t=n{T}_{s}$ .
In signal processing it is often more convenient and easier to workin the frequency domain. So let's look at at the signal in
frequency domain,
. For illustration purposes
we take the frequency content of the signal as a triangle.(If you Fourier transform the signal in
you will not
get such a nice triangle.)
Notice that the signal in
is bandlimited.
We can see that the signal is bandlimited because
$X(i\Omega )$ is zero outside the interval
$\left[-{\Omega}_{g} , {\Omega}_{g}\right]$ . Equivalentely we can state that the signal has no angular frequencies above
${\Omega}_{g}$ , corresponding
to no frequencies above
${F}_{g}=\frac{{\Omega}_{g}}{2\pi}$ .
Now let's take a look at the sampled signal in the frequency domain.
While
proving the sampling theorem we found the the spectrum of the sampled
signal consists of a sum of shifted versions of the analog spectrum. Mathematically this isdescribed by the following equation:
In
we show the result of sampling
$x(t)$ according to
the sampling theorem .
This means that when sampling the signal in
/
we use
${F}_{s}\ge 2{F}_{g}$ .
Observe in
that we have the same spectrum as in
for
$\Omega \in \left[{\mathrm{-\Omega}}_{g} , {\Omega}_{g}\right]$ , except for the scaling factor
$\frac{1}{{T}_{s}}$ .
This is a consequence of the sampling frequency. As mentioned in the
proof the spectrum of the sampled signal
is periodic with period
$2\pi {F}_{s}=\frac{2\pi}{{T}_{s}}$ .
So now we are, according to
the sample theorem ,
able to reconstruct the original signal
exactly . How we can do this
will be explored further down under
reconstruction . But first we
will take a look at what happens when we sample too slowly.
Sampling too slowly
If we sample
$x(t)$ too slowly,
that is
${F}_{s}< 2{F}_{g}$ , we will get overlap between the repeated spectra,
see
.
According to
the resulting spectra is the sum of these. This overlap
gives rise to the concept of aliasing.
If the sampling frequency is less than twice the highest frequency component,
then frequencies in the original signal that are above half the sampling rate will be "aliased"and will appear in the resulting signal as lower frequencies.
The consequence of aliasing is that we cannot recover the original signal,so aliasing has to be avoided.
Sampling too slowly will produce a sequence
${x}_{s}(n)$ that could have orginated from a number of signals. So there is
no chance
of recovering the original signal.To learn more about aliasing, take a look at this
module .
(Includes an applet for demonstration!)
To avoid aliasing we have to sample fast enough. But if we can't sample fast enough
(possibly due to costs) we can include an Anti-Aliasing filter. This willnot able us to get an exact reconstruction but can still be a good solution.
Typically a low-pass filter that is applied before sampling to ensure that no
components with frequencies greater than halfthe sample frequency remain.
The stagecoach effect
In older western movies you can observe aliasing on a stagecoach
when it starts to roll. At first the spokes appear toturn forward, but as the stagecoach increase its speed the spokes
appear to turn backward. This comes from the fact that the sampling rate,here the number of frames per second, is too low. We can view
each frame as a sample of an image that is changing continuouslyin time. (
Applet illustrating the stagecoach effect )
Reconstruction
Given the signal in
we want to recover the original signal, but
the question is how?
When there is no overlapping in the spectrum, the spectral
component given by
$k=0$ (see
),is equal to the spectrum of the analog signal. This offers an
oppurtunity to use a simple reconstruction process. Remember what you have learned about filtering.What we want is to change signal in
into that of
.
To achieve this we have to remove all the extra components generated in the sampling process.To remove the extra components we apply an ideal analog low-pass filter as shown in
As we see the ideal filter is rectangular in the frequency domain. A rectangle in the frequency
domain corresponds to a
sinc function in time domain (and vice versa).
Then we have reconstructed the original spectrum, and as we know
if two signals are identical
in the frequency domain, they are also identical in the time domain . End of reconstruction.
Conclusions
The Shannon sampling theorem requires that the input signal prior to sampling
is band-limited to at most half the sampling frequency. Under this conditionthe samples give an exact signal representation. It is truly remarkable
that such a broad and useful class signals can be represented that easily!
We also looked into the problem of reconstructing the signals form its samples.
Again the simplicity of the
principle is striking:
linear filtering by an ideal low-pass filter will do the job. However,the ideal filter is impossible to create, but that is another story...
Go to?
Introduction
Proof
Illustrations
Matlab Example
Aliasing applet
Hold operation
System view
Exercises
Questions & Answers
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
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.
Harper
Do you know which machine is used to that process?
In this morden time nanotechnology used in many field .
1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc
2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc
3- Atomobile -MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it