# 0.7 Dft and fft: an algebraic view  (Page 4/5)

 Page 4 / 5

## Matrix notation

We denote the $N×N$ identity matrix with ${I}_{N}$ , and diagonal matrices with

${diag}_{0\le k

The $N×N$ stride permutation matrix is defined for $N=KM$ by the permutation

${L}_{M}^{N}:\phantom{\rule{4pt}{0ex}}iK+j↦jM+i$

for $0\le i . This definition shows that ${L}_{M}^{N}$ transposes a $K×M$ matrix stored in row-major order. Alternatively, we can write

${L}_{M}^{N}\phantom{\rule{3pt}{0ex}}\text{:}\phantom{\rule{3pt}{0ex}}i\phantom{\rule{3pt}{0ex}}↦\phantom{\rule{3pt}{0ex}}\mathrm{iM}\phantom{\rule{3pt}{0ex}}\text{mod}\phantom{\rule{3pt}{0ex}}N-1,\phantom{\rule{5pt}{0ex}}\text{for}\phantom{\rule{3pt}{0ex}}0\le i

For example ( $·$ means 0),

${L}_{2}^{6}=\left[\begin{array}{cccccc}1& ·& ·& ·& ·& ·\\ ·& ·& 1& ·& ·& ·\\ ·& ·& ·& ·& 1& ·\\ ·& 1& ·& ·& ·& ·\\ ·& ·& ·& 1& ·& ·\\ ·& ·& ·& ·& ·& 1\end{array}\right].$

${L}_{N/2}^{N}$ is sometimes called the perfect shuffle.

Further, we use matrix operators; namely the direct sum

$A\oplus B=\left[\begin{array}{c}A\\ & B\end{array}\right]$

and the Kronecker or tensor product

$A\otimes B={\left[{a}_{k,\ell }B\right]}_{k,\ell },\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}A=\left[{a}_{k,\ell }\right].$

In particular,

${I}_{n}\otimes A=A\oplus \cdots \oplus A=\left[\begin{array}{c}A\\ & \ddots \\ & & A\end{array}\right]$

is block-diagonal.

We may also construct a larger matrix as a matrix of matrices, e.g.,

$\left[\begin{array}{cc}A& B\\ B& A\end{array}\right].$

If an algorithm for a transform is given as a product of sparse matrices built from the constructs above, then an algorithm for the transpose orinverse of the transform can be readily derived using mathematical properties including

$\begin{array}{cc}{\left(AB\right)}^{T}={B}^{T}{A}^{T},\hfill & {\left(AB\right)}^{-1}={B}^{-1}{A}^{-1},\hfill \\ {\left(A\oplus B\right)}^{T}={A}^{T}\oplus {B}^{T},\hfill & {\left(A\oplus B\right)}^{-1}={A}^{-1}\oplus {B}^{-1},\hfill \\ {\left(A\otimes B\right)}^{T}={A}^{T}\otimes {B}^{T},\hfill & {\left(A\otimes B\right)}^{-1}={A}^{-1}\otimes {B}^{-1}.\hfill \end{array}$

Permutation matrices are orthogonal, i.e., ${P}^{T}={P}^{-1}$ . The transposition or inversion of diagonal matrices is obvious.

## Radix-2 fft

The DFT decomposes $\mathcal{A}=\mathbb{C}\left[s\right]/\left({s}^{N}-1\right)$ with basis $b=\left(1,s,\cdots ,{s}^{N-1}\right)$ as shown in [link] . We assume $N=2M$ . Then

${s}^{2M}-1=\left({s}^{M}-1\right)\left({s}^{M}+1\right)$

factors and we can apply the CRT in the following steps:

$\begin{array}{ccc}& & \mathbb{C}\left[s\right]/\left({s}^{N}-1\right)\hfill \\ & \to & \mathbb{C}\left[s\right]/\left({s}^{M}-1\right)\oplus \mathbb{C}\left[s\right]/\left({s}^{M}+1\right)\hfill \end{array}$
$\begin{array}{ccc}& \to & \underset{0\le i
$\begin{array}{ccc}& \to & \underset{0\le i

As bases in the smaller algebras $\mathbb{C}\left[s\right]/\left({s}^{M}-1\right)$ and $\mathbb{C}\left[s\right]/\left({s}^{M}+1\right)$ , we choose $c=d=\left(1,s,\cdots ,{s}^{M-1}\right)$ . The derivation of an algorithm for ${DFT}_{N}$ based on [link] - [link] is now completely mechanical by reading off the matrix for each of the threedecomposition steps. The product of these matrices is equal to the ${DFT}_{N}$ .

First, we derive the base change matrix $B$ corresponding to [link] . To do so, we have to express the base elements ${s}^{n}\in b$ in the basis $c\cup d$ ; the coordinate vectors are the columns of $B$ . For $0\le n , ${s}^{n}$ is actually contained in $c$ and $d$ , so the first $M$ columns of $B$ are

$B=\left[\begin{array}{cc}{I}_{M}& *\\ {I}_{M}& *\end{array}\right],$

where the entries $*$ are determined next. For the base elements ${s}^{M+n}$ , $0\le n , we have

$\begin{array}{ccc}\hfill {s}^{M+n}& \equiv & {s}^{n}\phantom{\rule{4.pt}{0ex}}\text{mod}\phantom{\rule{4.pt}{0ex}}\left({s}^{M}-1\right),\hfill \\ \hfill {s}^{M+n}& \equiv & -{s}^{n}\phantom{\rule{4.pt}{0ex}}\text{mod}\phantom{\rule{4.pt}{0ex}}\left({s}^{M}+1\right),\hfill \end{array}$

which yields the final result

$B=\left[\begin{array}{cc}{I}_{M}& \phantom{-}{I}_{M}\\ {I}_{M}& -{I}_{M}\end{array}\right]={DFT}_{2}\otimes {I}_{M}.$

Next, we consider step [link] . $\mathbb{C}\left[s\right]/\left({s}^{M}-1\right)$ is decomposed by ${DFT}_{M}$ and $\mathbb{C}\left[s\right]/\left({s}^{M}+1\right)$ by ${DFT\text{-3}}_{M}$ in [link] .

Finally, the permutation in step [link] is the perfect shuffle ${L}_{M}^{N}$ , which interleaves the even and odd spectral components (even and odd exponents of ${W}_{N}$ ).

The final algorithm obtained is

${DFT}_{2M}={L}_{M}^{N}\left({DFT}_{M}\oplus {DFT\text{-3}}_{M}\right)\left({DFT}_{2}\otimes {I}_{M}\right).$

To obtain a better known form, we use ${DFT\text{-3}}_{M}={DFT}_{M}{D}_{M}$ , with ${D}_{M}={diag}_{0\le i , which is evident from [link] . It yields

$\begin{array}{ccc}\hfill {DFT}_{2M}& =& {L}_{M}^{N}\left({DFT}_{M}\oplus {DFT}_{M}{D}_{M}\right)\left({DFT}_{2}\otimes {I}_{M}\right)\hfill \\ & =& {L}_{M}^{N}\left({I}_{2}\otimes {DFT}_{M}\right)\left({I}_{M}\oplus {D}_{M}\right)\left({DFT}_{2}\otimes {I}_{M}\right).\hfill \end{array}$

The last expression is the radix-2 decimation-in-frequency Cooley-Tukey FFT. The corresponding decimation-in-time version isobtained by transposition using [link] and the symmetry of the DFT:

${DFT}_{2M}=\left({DFT}_{2}\otimes {I}_{M}\right)\left({I}_{M}\oplus {D}_{M}\right)\left({I}_{2}\otimes {DFT}_{M}\right){L}_{2}^{N}.$

The entries of the diagonal matrix ${I}_{M}\oplus {D}_{M}$ are commonly called twiddle factors .

The above method for deriving DFT algorithms is used extensively in [link] .

## General-radix fft

To algebraically derive the general-radix FFT, we use the decomposition property of ${s}^{N}-1$ . Namely, if $N=KM$ then

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
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
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
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 By Danielrosenberger