# 1.5 Fast convolution using the fft

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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? $y(n)=\sum_{m=0}^{N-1} x(m)h(n-m)$

There are $2N-1$ non-zero output points and each will be computed using $N$ complex mults and $N-1$ complex adds. Therefore, $\text{Total Cost}=(2N-1)(N+N-1)\approx O(N^{2})$

• Zero-pad these two sequences to length $2N-1$ , take DFTs using the FFT algorithm $x(n)\to X(k)$ $h(n)\to H(k)$ The cost is $O((2N-1)\lg (2N-1))=O(N\lg N)$
• Multiply DFTs $X(k)H(k)$ The cost is $O(2N-1)=O(N)$
• Inverse DFT using FFT $X(k)H(k)\to y(n)$ The cost is $O((2N-1)\lg (2N-1))=O(N\lg N)$

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

## Summary of dft

• $x(n)$ is an $N$ -point signal ( ).
• $X(k)=\sum_{n=0}^{N-1} x(n)e^{-(i\frac{2\pi }{N}kn)}=\sum_{n=0}^{N-1} x(n){W}_{N}^{(kn)}$ where ${W}_{N}=e^{-(i\frac{2\pi }{N})}$ is a "twiddle factor" and the first part is the basic DFT.

## What is the dft

$X(k)=X(F=\frac{k}{N})=\sum_{n=0}^{N-1} x(n)e^{-(i\times 2\pi Fn)}$ where $X(F=\frac{k}{N})$ is the DTFT of $x(n)$ and $\sum_{n=0}^{N-1} x(n)e^{-(i\times 2\pi Fn)}$ 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)

$x(n)=\frac{1}{N}\sum_{n=0}^{N-1} X(k)e^{i\frac{2\pi }{N}kn}$

• Build $x(n)$ using Simple complex sinusoidal building block signals
• Amplitude of each complex sinusoidal building block in $x(n)$ is $\frac{1}{N}X(k)$

## Dft

$↔(x(n)\mathop{\mathrm{xor}}h(n), X(k)H(k))$

## Regular convolution from circular convolution

• Zero pad $x(n)$ and $h(n)$ to $\mathrm{length}=\mathrm{length}(x)+\mathrm{length}(h)-1$
• Zero padding increases frequency resolution in DFT domain ( )

## 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: $O(N^{2})\to O(N\log_{2}N)$

## Fast convolution

• Use FFT's to compute circular convolution of zero-padded signals
• Much faster than regular convolution if signal lengths are long
• $O(N^{2})\to O(N\log_{2}N)$

See .

#### Questions & Answers

are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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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 By By Christine Zeelie By OpenStax By Brooke Delaney By David Martin By Saylor Foundation By Jonathan Long By Ryan Lowe By Katherina jennife... By Steve Gibbs