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The FFT, an efficient way to compute the DFT, is introduced and derived throughout this module.

FFT
(Fast Fourier Transform) An efficient computational algorithm for computing the DFT .

The fast fourier transform fft

DFT can be expensive to compute directly k 0 k N 1 X k n 0 N 1 x n 2 k N n

For each k , we must execute:

  • N complex multiplies
  • N 1 complex adds
The total cost of direct computation of an N -point DFT is
  • N 2 complex multiplies
  • N N 1 complex adds
How many adds and mults of real numbers are required?

This " O N 2 " computation rapidly gets out of hand, as N gets large:

N 1 10 100 1000 10 6
N 2 1 100 10,000 10 6 10 12

The FFT provides us with a much more efficient way of computing the DFT. The FFT requires only " O N N " computations to compute the N -point DFT.

N 10 100 1000 10 6
N 2 100 10,000 10 6 10 12
N 10 logbase --> N 10 200 3000 6 6

How long is 10 12 sec ? More than 10 days! How long is 6 6 sec ?

The FFT and digital computers revolutionized DSP (1960 - 1980).

How does the fft work?

  • The FFT exploits the symmetries of the complex exponentials W N k n 2 N k n
  • W N k n are called " twiddle factors "

Complex conjugate symmetry

W N k N n W N k n W N k n 2 k N N n 2 k N n 2 k N n

Periodicity in n and k

W N k n W N k N n W N k N n 2 N k n 2 N k N n 2 N k N n W N 2 N

Decimation in time fft

  • Just one of many different FFT algorithms
  • The idea is to build a DFT out of smaller and smaller DFTs by decomposing x n into smaller and smaller subsequences.
  • Assume N 2 m (a power of 2)

Derivation

N is even , so we can complete X k by separating x n into two subsequences each of length N 2 . x n N 2 n even N 2 n odd k 0 k N 1 X k n 0 N 1 x n W N k n X k n 2 r x n W N k n n 2 r 1 x n W N k n where 0 r N 2 1 . So

X k r 0 N 2 1 x 2 r W N 2 k r r 0 N 2 1 x 2 r 1 W N 2 r 1 k r 0 N 2 1 x 2 r W N 2 k r W N k r 0 N 2 1 x 2 r 1 W N 2 k r
where W N 2 2 N 2 2 N 2 W N 2 . So X k r 0 N 2 1 x 2 r W N 2 k r W N k r 0 N 2 1 x 2 r 1 W N 2 k r where r 0 N 2 1 x 2 r W N 2 k r is N 2 -point DFT of even samples ( G k ) and r 0 N 2 1 x 2 r 1 W N 2 k r is N 2 -point DFT of odd samples ( H k ). k 0 k N 1 X k G k W N k H k Decomposition of an N -point DFT as a sum of 2 N 2 -point DFTs.

Why would we want to do this? Because it is more efficient!

Cost to compute an N -point DFT is approximately N 2 complex mults and adds.
But decomposition into 2 N 2 -point DFTs + combination requires only N 2 2 N 2 2 N N 2 2 N where the first part is the number of complex mults and adds for N 2 -point DFT, G k . The second part is the number of complex mults and adds for N 2 -point DFT, H k . The third part is the number of complex mults and adds for combination. And the total is N 2 2 N complex mults and adds.

Savings

For N 1000 , N 2 10 6 N 2 2 N 10 6 2 1000 Because 1000 is small compared to 500,000, N 2 2 N 10 6 2

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So why stop here?! Keep decomposing. Break each of the N 2 -point DFTs into two N 4 -point DFTs, etc. , ....

We can keep decomposing: N 2 1 N 2 N 4 N 8 N 2 m 1 N 2 m 1 where m 2 logbase --> N times

Computational cost: N -pt DFTtwo N 2 -pt DFTs. The cost is N 2 2 N 2 2 N . So replacing each N 2 -pt DFT with two N 4 -pt DFTs will reduce cost to 2 2 N 4 2 N 2 N 4 N 4 2 2 N N 2 2 2 2 N N 2 2 p p N As we keep going p 3 4 m , where m 2 logbase --> N . We get the cost N 2 2 2 logbase --> N N 2 logbase --> N N 2 N N 2 logbase --> N N N 2 logbase --> N N N 2 logbase --> N is the total number of complex adds and mults.

For large N , cost N 2 logbase --> N or " O N 2 logbase --> N ", since N 2 logbase --> N N for large N .

N 8 point FFT. Summing nodes W n k twiddle multiplication factors.

Weird order of time samples

This is called "butterflies."

Questions & Answers

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
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Fundamentals of signal processing. OpenStax CNX. Nov 26, 2012 Download for free at http://cnx.org/content/col10360/1.4
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