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Exercises to TTT4110: Information and Signal Theory, Sampling theorem.

Problems related to the Sampling Theorem module.

Express the sampling theorem in words.

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Theoretically, why is the sinc-function so important for reconstruction? Sketch a sinc(t). What are the values for integer values of t?

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Argue that the sampling rate for CD should be over 40KHz.

The human ear can hear frequencies up to 20 KHz, so according to the sampling theorem we should sample at a rate equal to or exceeding 40KHz. In practice we always have to sampleat more than the double rate, partly due to finite precision.

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(by don johnson)

What is the simplest bandlimited signal? Using this signal, convince yourself that less than twosamples/period will not suffice to specify it. If the sampling rate 1 T s is not high enough, what signal would your resulting undersampled signal become? Hint: Try the aliasing applet .

The simplest bandlimited signal is the sine wave. At the Nyquist frequency, exactly two samples/period wouldoccur. Reducing the sampling rate would result in fewer samples/period, and these samples would appear to havearisen from a lower frequency sinusoid.

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Are the filter h(t) described by the sinc function the only filterwe can use as a perfect reconstruction filter? If not what are the condition that would allow us to use another filter?

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If you found that it is possible to use another filter in specify such a filter. Hint: Try using the domain which usually simplifies things...

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What are the difficulties introduced when we want to apply the results of thischapter in practice?

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If a real signal has frequency content up to F 1 . What is then the bandwith of the signal?

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If a real signal has frequency content confined in the interval F 1 F 1 . What is then the bandwith of the signal?

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What can be said in general for the spectrum of a discrete signal whichis the result of sampling an analog signal that is NOT bandlimited?

The spectrum will ALWAYS overlap,there will always be aliasing.

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Link to the aliasing applet (Right click if you want to open it in a new window).

In the following problems, as in the aliasing applet, we are studying a sinusoidal signal, x t 2 f t , which is sampled at F s 8000 .

What is the frequency limitation of an analog sinusoidalsignal if we want to avoid aliasing, given F s 8000 ?

With a sampling frequency of 8000 Hz, the maximum frequency of the analog signal is 4000 Hz, as given by the sampling theorem .

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Describe with words the type of signal we "reconstruct" from the sampleswhen the input frequency (of the sinusoidal signal) is higher than the sample rate can deal with?

The signal we "reconstruct" is a sinusoidal signal with a frequency that is lower than the original because of aliasing.

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Find an expression the signal we "reconstruct" from the sampleswhen the input frequency is 6000 Hz.

When the input frequency is 6000 Hz, a sampling frequency of 8000 Hz is to low, i.e aliasing will occur. The sampled signal willhave frequency components at +6000 Hz and -6000 Hz plus some new frequency components as a result of aliasing.

We know from the proof of the sampling theorem that the sampled signal is periodic with F s 8000 . Thus a frequency component at 6000 Hz implies frequencies at -2000 Hz, -10000 Hz, 14000 Hz and so on.Similarly a frequency component at -6000 Hz give rise to(among others) a 2000 Hz component. Looking only at the positive frequencies the "reconstructed" signal will only have a 2000Hz frequency component. The removal of the 6000 Hz and above frequencies are due to the reconstruction filter. The filter is designed based on a maximum input signal frequency of 4000 Hz.Thus the "reconstructed" signal can be written as: 2 2000 t .

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Explain the "strange" sample points when the input input frequency is 4000 Hz.

The sampled signal can be written as x s n 2 4000 n 8000 n 0 . Thus all the samples are zero-valued.

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Explain the "strange" sample points when the input input frequency is 8000 Hz.

The sampled signal can be written as x s n 2 8000 n 8000 2 n 0 . Thus all the samples are zero-valued.

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Find an expression for the signal we can reconstruct from the sampleswhen the input frequency is 4000 Hz.

As shown in problem 14, the samples are zero valued. A reconstructing filter cannot distinguish this from the all zerosignal so the reconstructed signal will be the all zero signal.

Note that a small change in the sinusoidal signals phase would produce samples that are not only zero-valued. The "reconstructed" signal willthen be a equal to the original signal. This problem illustrates that sampling twice the signals highest frequency component doesnot always guarantee perfect recontstruction. If we could increase the sampling frequency to, say, F s 8000.00001 , we could reconstruct the original signal. I.e sampling at a rate greater than twice the highest frequency component yields the desiredreconstruction.

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Find an expression for the "reconstructed" signal from the sampleswhen the input frequency is 8000 Hz.

As shown in problem 15, the samples are zero valued. A reconstructing filter cannot distinguish this from the all zerosignal so the reconstructed signal will be the all zero signal.

Note that a small change in the sinusoidal signals phase would produce samples that are not only zero-valued. The "reconstructed" signal willthen be a signal with aliased components.

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Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
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Source:  OpenStax, Information and signal theory. OpenStax CNX. Aug 03, 2006 Download for free at http://legacy.cnx.org/content/col10211/1.19
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