1.3 Ffts of prime length and rader's conversion

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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

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!

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.

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.

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