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

(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)


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.


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

Got questions? Get instant answers now!

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

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
I'm not good at math so would you help me
what is the problem that i will help you to self with?
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris 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?
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
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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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