0.6 Winograd's short dft algorithms  (Page 9/11)

Here we use three methods to facilitate the construction of prime length FFT modules. First, the matrix exchange property [link] , [link] , [link] is used so that the transpose of the reduction operator can be used rather than the morecomplicated CRT reconstruction operator. This is then combined with the numerical method [link] for obtaining the multiplication coefficients rather than the direct use of the CRT. We also deviate from the Toom-Cook algorithm,because it requires too many additions for the lengths in which we are interested. Instead we use an iterated polynomial multiplication algorithm [link] . We have incorporated these three ideas into a single structural procedure that automatesthe design of prime length FFTs.

Matrix description

It is important that each step in the Winograd FFT can be described using matrices. By expressing cyclic convolution as a bilinear form, a compact form of prime length DFTs can be obtained.

If $y$ is the cyclic convolution of $h$ and $x$ , then $y$ can be expressed as

where, using the Matlab convention, $.*$ represents point by point multiplication. When $A$ , $B$ , and $C$ are allowed to be complex, $A$ and $B$ are seen to be the DFT operator and $C$ , the inverse DFT. When only real numbers are allowed, $A$ , $B$ , and $C$ will be rectangular. This form of convolution is presented with many examples in [link] . Using the matrix exchange property explained in [link] and [link] this form can be written as

where $R$ is the permutation matrix that reverses order.

When $h$ is fixed, as it is when considering prime length DFTs, the term $C^{T}Rh$ can be precomputed and a diagonal matrix $D$ formed by $D=diag\left\{C^{T}Rh\right\}$ . This is advantageous because in general, $C$ is more complicated than $B$ , so the ability to “hide" $C$ saves computation. Now $y=R{B}^{T}DAx$ or $y=R{A}^{T}DAx$ since $A$ and $B$ can be the same; they implement a polynomial reduction. The form $y=R^{T}DAxT$ can also be used for the prime length DFTs, it is only necessary to permute the entries of x and to ensure that theDC term is computed correctly. The computation of the DC term is simple, for the residue of a polynomial modulo $a-1$ is always the sum of the coefficients. After adding the $x_0$ term of the original input sequence, to the $s-l$ residue, the DC term is obtained. Now $DFT\left\{x\right\}=R{A}^{T}DAx$ . In [link] Johnson observes that by permuting the elements on the diagonal of $D$ , the output can be permuted, so that the $R$ matrix can be hidden in $D$ , and $DFT\left\{x\right\}=A^{T}DAx$ . From the knowledge of this form, once $A$ is found, $D$ can be found numerically [link] .

Programming the design procedure

Because each of the above steps can be described by matrices, the development of a prime length FFTs is made convenient with the use of a matrix oriented programminglanguage such as Matlab. After specifying the appropriate matrices that describe the desired FFT algorithm, generating code involves compiling the matrices into the desiredcode for execution.

Each matrix is a section of one stage of the flow graph that corresponds to the DFT program. The four stages are:

1. Permutation Stage: Permutes input and output sequence.
2. Reduction Stage: Reduces the cyclic convolution to smaller polynomial products.
3. Polynomial Product Stage: Performs the polynomial multiplications.
4. Multiplication Stage: Implements the point-by-point multiplication in the bilinear form.