<< Chapter < Page Chapter >> Page >
Efficient computation of convolution using FFTs.

Fast circular convolution

Since, m 0 N 1 x m h n m N y n is equivalent to Y k X k H k y n can be computed as y n IDFT DFT x n DFT h n

    Cost

    • Direct

    • N 2 complex multiplies.
    • N N 1 complex adds.
    • Via ffts

    • 3 FFTs + N multipies.
    • N 3 N 2 2 logbase --> N complex multiplies.
    • 3 N 2 logbase --> N complex adds.
If H k can be precomputed, cost is only 2 FFts + N multiplies.

Fast linear convolution

DFT produces cicular convolution. For linear convolution, we must zero-pad sequences so that circular wrap-around alwayswraps over zeros.

To achieve linear convolution using fast circular convolution, we must use zero-padded DFTs of length N L M 1

Choose shortest convenient N (usually smallest power-of-two greater than or equal to L M 1 ) y n IDFT N DFT N x n DFT N h n

There is some inefficiency when compared to circular convolution due to longer zero-padded DFTs . Still, O N 2 logbase --> N savings over direct computation.

Running convolution

Suppose L , as in a real time filter application, or L M . There are efficient block methods for computing fast convolution.

Overlap-save (ols) method

Note that if a length- M filter h n is circularly convulved with a length- N segment of a signal x n ,

the first M 1 samples are wrapped around and thus is incorrect . However, for M 1 n N 1 ,the convolution is linear convolution, so these samples are correct. Thus N M 1 good outputs are produced for each length- N circular convolution.

The Overlap-Save Method: Break long signal into successive blocks of N samples, each block overlapping the previous block by M 1 samples. Perform circular convolution of each block with filter h m . Discard first M 1 points in each output block, and concatenate the remaining points to create y n .

Computation cost for a length- N equals 2 n FFT per output sample is (assuming precomputed H k ) 2 FFTs and N multiplies 2 N 2 2 logbase --> N N N M 1 N 2 logbase --> N 1 N M 1 complex multiplies 2 N 2 logbase --> N N M 1 2 N 2 logbase --> N N M 1 complex adds

Compare to M mults, M 1 adds per output point for direct method. For a given M , optimal N can be determined by finding N minimizing operation counts. Usualy, optimal N is 4 M N opt 8 M .

Overlap-add (ola) method

Zero-pad length- L blocks by M 1 samples.

Add successive blocks, overlapped by M 1 samples, so that the tails sum to produce the complete linear convolution.

Computational Cost: Two length N L M 1 FFTs and M mults and M 1 adds per L output points; essentially the sames as OLS method.

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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
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?
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
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Pdf generation problem modules. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10514/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Pdf generation problem modules' conversation and receive update notifications?

Ask