# 0.4 The dft as convolution or filtering

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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

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

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

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

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
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da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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Rafiq
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Damian
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LITNING
scanning tunneling microscope
Sahil
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Santosh
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Rafiq
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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
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Anam
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Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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