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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

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
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what is a peer
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What is meant by 'nano scale'?
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What is STMs full form?
LITNING
scanning tunneling microscope
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how nano science is used for hydrophobicity
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
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if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
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analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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?
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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why?
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what school?
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biomolecules are e building blocks of every organics and inorganic materials.
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anyone know any internet site where one can find nanotechnology papers?
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research.net
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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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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