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The ideas of using the DFT to filter a signal and recover a signal from a noisy transmission are addressed based on the ideas of the DFT and convolution.


y n x n h n k x k h n k
Assume that H is specified.

How can we implement X H in a computer?

Discretize (sample) X and H . In order to do this, we should take the DFTs of x n and h n to get X k and X k . Then we will compute y ~ n IDFT X k H k Does y ~ n y n ?

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Recall that the DFT treats N -point sequences as if they are periodically extended ( ):

Compute idft of y[k]

y ~ n 1 N k N 1 0 Y k 2 k N n 1 N k N 1 0 X k H k 2 k N n 1 N k N 1 0 m N 1 0 x m 2 k N m H k 2 k N n m N 1 0 x m 1 N k N 1 0 H k 2 k N n m m N 1 0 x m h (( n m )) N
And the IDFT periodically extends h n : h ~ n m h (( n m )) N This computes as shown in :

y ~ n m N 1 0 x m h (( n m )) N
is called circular convolution and is denoted by .

The above symbol for the circular convolution is for an N -periodic extension.

Dft pair

Note that in general:

Regular vs. circular convolution

To begin with, we are given the following two length-3 signals: x n 1 2 3 h n 1 0 2 We can zero-pad these signals so that we have the following discrete sequences: x n 0 1 2 3 0 h n 0 1 0 2 0 where x 0 1 and h 0 1 .

  • Regular Convolution:
    y n m 2 0 x m h n m
    Using the above convolution formula (refer to the link if you need a review of convolution ), we can calculate the resulting value for y 0 to y 4 . Recall that because we have two length-3 signals, our convolved signal will be length-5.
    • n 0 0 0 0 1 2 3 0 0 2 0 1 0 0 0
      y 0 1 1 2 0 3 0 1
    • n 1 0 0 1 2 3 0 0 2 0 1 0 0
      y 1 1 0 2 1 3 0 2
    • n 2 0 1 2 3 0 0 2 0 1 0
      y 2 1 2 2 0 3 1 5
    • n 3
      y 3 4
    • n 4
      y 4 6

Regular convolution result

Result is finite duration, not periodic!

  • Circular Convolution:
    y ~ n m 2 0 x m h (( n m )) N
    And now with circular convolution our h n changes and becomes a periodically extended signal:
    h (( n )) N 1 0 2 1 0 2 1 0 2
    • n 0 0 0 0 1 2 3 0 1 2 0 1 2 0 1
      y ~ 0 1 1 2 2 3 0 5
    • n 1 0 0 0 1 2 3 0 0 1 2 0 1 2 0
      y ~ 1 1 1 2 1 3 2 8
    • n 2
      y ~ 2 5
    • n 3
      y ~ 3 5
    • n 4
      y ~ 4 8

Circular convolution result

Result is 3-periodic.

illustrates the relationship between circular convolution and regularconvolution using the previous two figures:

Circular convolution from regular

The left plot (the circular convolution results) has a "wrap-around" effect due to periodic extension.
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Regular convolution from periodic convolution

  • "Zero-pad" x n and h n to avoid the overlap (wrap-around) effect. We will zero-pad the two signals to a length-5 signal (5being the duration of the regular convolution result): x n 1 2 3 0 0 h n 1 0 2 0 0
  • Now take the DFTs of the zero-padded signals:
    y ~ n 1 N k 4 0 X k H k 2 k 5 n m 4 0 x m h (( n m )) 5
Now we can plot this result ( ):

The sequence from 0 to 4 (the underlined part of the sequence) is the regular convolution result. From thisillustration we can see that it is 5-periodic!

We can compute the regular convolution result of a convolution of an M -point signal x n with an N -point signal h n by padding each signal with zeros to obtain two M N 1 length sequences and computing the circular convolution (or equivalently computing the IDFT of H k X k , the product of the DFTs of the zero-padded signals) ( ).

Note that the lower two images are simply the top images that have been zero-padded.

Dsp system

The system has a length N impulse response, h n

  • Sample finite duration continuous-time input x t to get x n where n 0 M 1 .
  • Zero-pad x n and h n to length M N 1 .
  • Compute DFTs X k and H k
  • Compute IDFTs of X k H k y n y ~ n where n 0 M N 1 .
  • Reconstruct y t

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
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Akash Reply
it is a goid question and i want to know the answer as well
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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
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Intro to digital signal processing. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10203/1.4
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