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This module describes FFT, convolution, filtering, LTI systems, digital filters and circular convolution.

Important application of the fft

How many complex multiplies and adds are required to convolve two N -pt sequences? y n m 0 N 1 x m h n m

There are 2 N 1 non-zero output points and each will be computed using N complex mults and N 1 complex adds. Therefore, Total Cost 2 N 1 N N 1 O N 2

Got questions? Get instant answers now!
  • Zero-pad these two sequences to length 2 N 1 , take DFTs using the FFT algorithm x n X k h n H k The cost is O 2 N 1 2 N 1 O N N
  • Multiply DFTs X k H k The cost is O 2 N 1 O N
  • Inverse DFT using FFT X k H k y n The cost is O 2 N 1 2 N 1 O N N

So the total cost for direct convolution of two N -point sequences is O N 2 . Total cost for convolution using FFT algorithm is O N N . That is a huge savings ( ).

Summary of dft

  • x n is an N -point signal ( ).
  • X k n 0 N 1 x n 2 N k n n 0 N 1 x n W N k n where W N 2 N is a "twiddle factor" and the first part is the basic DFT.

What is the dft

X k X F k N n 0 N 1 x n 2 F n where X F k N is the DTFT of x n and n 0 N 1 x n 2 F n is the DTFT of x n at digital frequency F . This is a sample of the DTFT. We can do frequency domain analysis on a computer!

Inverse dft (idft)

x n 1 N n 0 N 1 X k 2 N k n

  • Build x n using Simple complex sinusoidal building block signals
  • Amplitude of each complex sinusoidal building block in x n is 1 N X k

Circular convolution


x n h n X k H k

Regular convolution from circular convolution

  • Zero pad x n and h n to length length x length h 1
  • Zero padding increases frequency resolution in DFT domain ( )

8-pt DFT of 8-pt signal
16-pt DFT of same signal padded with 8 additional zeros

The fast fourier transform (fft)

  • Efficient computational algorithm for calculating the DFT
  • "Divide and conquer"
  • Break signal into even and odd samples keep taking shorter and shorter DFTs, then build N -pt DFT by cleverly combining shorter DFTs
  • N -pt DFT: O N 2 O N 2 logbase --> N

Fast convolution

  • Use FFT's to compute circular convolution of zero-padded signals
  • Much faster than regular convolution if signal lengths are long
  • O N 2 O N 2 logbase --> N

See .

Questions & Answers

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
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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?
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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