# 1.3 Ffts of prime length and rader's conversion

 Page 1 / 1
FFTs of prime length can be computed efficiently via Rader's conversion, via the chirp z-transform, or by use of Winograd methods.

The power-of-two FFT algorithms , such as the radix-2 and radix-4 FFTs, and the common-factor and prime-factor FFTs, achieve great reductions in computational complexity of the DFT when the length, $N$ , is a composite number.DFTs of prime length are sometimes needed, however, particularly for the short-length DFTs in common-factor or prime-factor algorithms.The methods described here, along with the composite-length algorithms, allow fast computation of DFTs of any length.

There are two main ways of performing DFTs of prime length:

• Rader's conversion, which is most efficient, and the
• Chirp-z transform , which is simpler and more general.
Oddly enough, both work by turning prime-length DFTs into convolution! The resulting convolutions can then be computed efficiently by either
• fast convolution via composite-length FFTs (simpler) or by
• Winograd techniques (more efficient)

## Rader's conversion

Rader's conversion is a one-dimensional index-mapping scheme that turns a length- $N$ DFT ( $N$ prime) into a length-( $N-1$ ) convolution and a few additions. Rader's conversion works only for prime-length $N$ .

An index map simply rearranges the order of the sum operation in the DFT definition . Because addition is a commutative operation, the same mathematical result is producedfrom any order, as long as all of the same terms are added once and only once. (This is the condition that defines an index map.)Unlike the multi-dimensional index maps used in deriving common factor and prime-factor FFTs , Rader's conversion uses a one-dimensional index map in a finite group of $N$ integers: $k=r^{m}\mod N$

## Fact from number theory

If $N$ is prime, there exists an integer " $r$ " called a primitive root , such that the index map $k=r^{m}\mod N$ , $m$

0 1 2 N 2
, uniquely generates all elements $k$
1 2 3 N 1

$N=5$ , $r=2$ $2^{0}\mod 5=1$ $2^{1}\mod 5=2$ $2^{2}\mod 5=4$ $2^{3}\mod 5=3$

## Another fact from number theory

For $N$ prime, the inverse of $r$ (i.e. $r^{(-1)}r\mod N=1$ is also a primitive root (call it $r^{(-1)}$ ).

$N=5$ , $r=2$ $r^{(-1)}=3$ $2\times 3\mod 5=1$ $3^{0}\mod 5=1$ $3^{1}\mod 5=3$ $3^{2}\mod 5=4$ $3^{3}\mod 5=2$

So why do we care? Because we can use these facts to turn a DFT into a convolution!

## Rader's conversion

Let $\forall mn, (m)$

0 1 N 2
n
1 2 N 1
n r m N , $\forall pk, (p)$
0 1 N 2
k
1 2 N 1
k r p N $X(k)=\sum_{n=0}^{N-1} x(n){W}_{N}^{nk}=\begin{cases}x(0)+\sum_{n=1}^{N-1} x(n){W}_{N}^{nk} & \text{if k\neq 0}\\ \sum_{n=0}^{N-1} x(n) & \text{if k=0}\end{cases}()$ where for convenience ${W}_{N}^{nk}=e^{-(i\frac{2\pi nk}{N})}$ in the DFT equation. For $k\neq 0$
$X(r^{p}\mod N)=\sum_{m=0}^{N-2} x(r^{-m}\mod N){W}^{{r}^{p}{r}^{-m}}+x(0)=\sum_{m=0}^{N-2} x(r^{-m}\mod N){W}^{{r}^{p-m}}+x(0)=x(0)+(x(r^{-l}\mod N), {W}^{{r}^{l}})$
where $l$
0 1 N 2

$N=5$ , $r=2$ , $r^{(-1)}=3$ $\begin{pmatrix}X(0)\\ X(1)\\ X(2)\\ X(3)\\ X(4)\\ \end{pmatrix}=\begin{pmatrix}0 & 0 & 0 & 0 & 0\\ 0 & 1 & 2 & 3 & 4\\ 0 & 2 & 4 & 1 & 3\\ 0 & 3 & 1 & 4 & 2\\ 0 & 4 & 3 & 2 & 1\\ \end{pmatrix}\begin{pmatrix}x(0)\\ x(1)\\ x(2)\\ x(3)\\ x(4)\\ \end{pmatrix}$ $\begin{pmatrix}X(0)\\ X(1)\\ X(2)\\ X(4)\\ X(3)\\ \end{pmatrix}=\begin{pmatrix}0 & 0 & 0 & 0 & 0\\ 0 & 1 & 3 & 4 & 2\\ 0 & 2 & 1 & 3 & 4\\ 0 & 4 & 2 & 1 & 1\\ 0 & 3 & 4 & 2 & 3\\ \end{pmatrix}\begin{pmatrix}x(0)\\ x(1)\\ x(3)\\ x(4)\\ x(2)\\ \end{pmatrix}$ where for visibility the matrix entries represent only the power , $m$ of the corresponding DFT term ${W}_{N}^{m}$ Note that the 4-by-4 circulant matrix $\begin{pmatrix}1 & 3 & 4 & 2\\ 2 & 1 & 3 & 4\\ 4 & 2 & 1 & 1\\ 3 & 4 & 2 & 3\\ \end{pmatrix}$ corresponds to a length-4 circular convolution.

Rader's conversion turns a prime-length DFT into a few adds and a composite-length ( $N-1$ ) circular convolution, which can be computed efficiently using either

• fast convolution via FFT and IFFT
• index-mapped convolution algorithms and short Winograd convolution alogrithms. (Rather complicated, and trades fewer multipliesfor many more adds, which may not be worthwile on most modern processors.) See R.C. Agarwal and J.W. Cooley

## Winograd minimum-multiply convolution and dft algorithms

S. Winograd has proved that a length- $N$ circular or linear convolution or DFT requires less than $2N$ multiplies (for real data), or $4N$ real multiplies for complex data. (This doesn't count multiplies by rational fractions, like $3$ or $\frac{1}{N}$ or $\frac{5}{17}$ , which can be computed with additions and one overall scaling factor.) Furthermore, Winograd showed how toconstruct algorithms achieving these counts. Winograd prime-length DFTs and convolutions have the followingcharacteristics:

• Extremely efficient for small $N$ ( $N< 20$ )
• The number of adds becomes huge for large $N$ .
Thus Winograd's minimum-multiply FFT's are useful only for small $N$ . They are very important for Prime-Factor Algorithms , which generally use Winograd modules to implement the short-length DFTs. Tables giving themultiplies and adds necessary to compute Winograd FFTs for various lengths can be found in C.S. Burrus (1988) . Tables and FORTRAN and TMS32010 programs for these short-length transforms canbe found in C.S. Burrus and T.W. Parks (1985) . The theory and derivation of these algorithms is quite elegant but requires substantialbackground in number theory and abstract algebra. Fortunately for the practitioner, all of the shortalgorithms one is likely to need have already been derived and can simply be looked up without mastering thedetails of their derivation.

## Winograd fourier transform algorithm (wfta)

The Winograd Fourier Transform Algorithm (WFTA) is a technique that recombines the short Winograd modules in a prime-factor FFT into a composite- $N$ structure with fewer multiplies but more adds. While theoretically interesting,WFTAs are complicated and different for every length, and on modern processors with hardware multipliers the trade of multiplies for manymore adds is very rarely useful in practice today.

#### Questions & Answers

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
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
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, The dft, fft, and practical spectral analysis. OpenStax CNX. Feb 22, 2007 Download for free at http://cnx.org/content/col10281/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'The dft, fft, and practical spectral analysis' conversation and receive update notifications?  By By      By By