# 0.4 The dft as convolution or filtering

 Page 1 / 4

A major application of the FFT is fast convolution or fast filtering where the DFT of the signal is multiplied term-by-term by the DFT ofthe impulse (helps to be doing finite impulse response (FIR) filtering) and the time-domain output is obtained by taking the inverse DFT ofthat product. What is less well-known is the DFT can be calculated by convolution. There are several different approaches to this,each with different application.

## Rader's conversion of the dft into convolution

In this section a method quite different from the index mapping or polynomial evaluation is developed. Rather than dealingwith the DFT directly, it is converted into a cyclic convolution which must then be carried out by some efficient means. Those meanswill be covered later, but here the conversion will be explained. This method requires use of some number theory, which can befound in an accessible form in [link] or [link] and is easy enough to verify on one's own. A good general reference on numbertheory is [link] .

The DFT and cyclic convolution are defined by

$C\left(k\right)=\sum _{n=0}^{N-1}x\left(n\right)\phantom{\rule{4pt}{0ex}}{W}^{nk}$
$y\left(k\right)=\sum _{n=0}^{N-1}x\left(n\right)\phantom{\rule{4pt}{0ex}}h\left(k-n\right)$

For both, the indices are evaluated modulo $N$ . In order to convert the DFT in [link] into the cyclic convolution of [link] , the $nk$ product must be changed to the $k-n$ difference. With real numbers, this can be done with logarithms, but it is more complicated when working in a finite set of integersmodulo $N$ . From number theory [link] , [link] , [link] , [link] , it can be shown that if the modulus is a prime number, a base (called aprimitive root) exists such that a form of integer logarithm can be defined. This is stated in the following way. If $N$ is a prime number, a number $r$ called a primitive roots exists such that the integer equation

$n={\left(\left({r}^{m}\right)\right)}_{N}$

creates a unique, one-to-one map of the $N-1$ member set $m=\left\{0,...,N-2\right\}$ and the $N-1$ member set $n=\left\{1,...,N-1\right\}$ . This is because the multiplicative group of integers modulo a prime, $p$ , is isomorphic to the additive group of integers modulo $\left(p-1\right)$ and is illustrated for $N=5$ below.

 r m= 0 1 2 3 4 5 6 7 1 1 1 1 1 1 1 1 1 2 1 2 4 3 1 2 4 3 3 1 3 4 2 1 3 4 2 4 1 4 1 4 1 4 1 4 5 * 0 0 0 * 0 0 0 6 1 1 1 1 1 1 1 1

[link] is an array of values of ${r}^{m}$ modulo $N$ and it is easy to see that there are two primitiveroots, 2 and 3, and [link] defines a permutation of the integers $n$ from the integers $m$ (except for zero). [link] and a primitive root (usually chosen to be the smallest of those that exist) can be used to convert the DFT in [link] to the convolution in [link] . Since [link] cannot give a zero, a new length-(N-1) data sequence is defined from $x\left(n\right)$ by removing the term with index zero. Let

$n={r}^{-m}$

and

$k={r}^{s}$

where the term with the negative exponent (the inverse) is defined as the integer that satisfies

${\left(\left({r}^{-m}{r}^{m}\right)\right)}_{N}=1$

If $N$ is a prime number, ${r}^{-m}$ always exists. For example, ${\left(\left({2}^{-1}\right)\right)}_{5}=3$ . [link] now becomes

$C\left({r}^{s}\right)=\sum _{m=0}^{N-2}x\left({r}^{-m}\right)\phantom{\rule{4pt}{0ex}}{W}^{{r}^{-m}{r}^{s}}+\phantom{\rule{4pt}{0ex}}x\left(0\right),$

for $s=0,1,..,N-2$ , and

$C\left(0\right)=\sum _{n=0}^{N-1}x\left(n\right)$

New functions are defined, which are simply a permutation in the order of the original functions, as

${x}^{\text{'}}\left(m\right)=x\left({r}^{-m}\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{C}^{\text{'}}\left(s\right)=C\left({r}^{s}\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{W}^{\text{'}}\left(n\right)={W}^{{r}^{n}}$

[link] then becomes

${C}^{\text{'}}\left(s\right)=\sum _{m=0}^{N-2}{x}^{\text{'}}\left(m\right)\phantom{\rule{4pt}{0ex}}{W}^{\text{'}}\left(s-m\right)\phantom{\rule{4pt}{0ex}}+\phantom{\rule{4pt}{0ex}}x\left(0\right)$

which is cyclic convolution of length N-1 (plus $x\left(0\right)$ ) and is denoted as

${C}^{\text{'}}\left(k\right)={x}^{\text{'}}\left(k\right)*{W}^{\text{'}}\left(k\right)+x\left(0\right)$

Applying this change of variables (use of logarithms) to the DFT can best be illustrated from the matrix formulation of the DFT. [link] is written for a length-5 DFT as

#### Questions & Answers

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
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 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

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Fast fourier transforms. OpenStax CNX. Nov 18, 2012 Download for free at http://cnx.org/content/col10550/1.22
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fast fourier transforms' conversation and receive update notifications?

 By OpenStax By Briana Knowlton By Marriyam Rana By OpenStax By OpenStax By Kevin Amaratunga By Stephanie Redfern By Stephanie Redfern By OpenStax By OpenStax