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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


    • 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

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
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
for teaching engĺish at school how nano technology help us
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
Smarajit Reply
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