Large dft modules: 11, 13, 16, 17, 19, and 25

Introduction

This report describes three large DFT modules (17,19,25) which were developed by the first author, Howard Johnson, in June of 1981, and two previously undocumented modules (11,13) which were originally generated at Stanford in 1978 [link] .

The length 17 and 19 modules were created in the style of Winograd's convolutional DFT programs with strict adherence to three additional module development principles. First, as much code as possible was automatically generated. This included use of FORTRAN programs to generate the input and output mapping statements and the multiplication statements, and heavy use of EDIT commands to copy redundant sections of code. The code for imaginary data manipulation was copied directly from a working listing of code for the real part. All discussion below therefore centers on producing code only for the real part of the input data array. Even the EDIT commands for copying sections of code and substituting variable names were themselves listed in a command file. In this way, the programmer was prevented from introducing occasional typographical errors which are the bane of the DFT module debugger. Errors which did occur tended to be very large and obvious. Test routines were written to test particularly difficult sections of code before they were inserted into the DFT module (such as the modulo $z^8+1$ convolution subsection).

Once the reduction, or PRE-WEAVE, section was written, the reconstruction, or POST-WEAVE, section was arranged to be the transpose of the reduction equations, according to the method of 'transposing the tensor' [link] . Although the problem of minimizing the number of additions in a module is not necessarily solved by transposing the tensor, due to the inordinate difficulty of finding suitable substitutions which would abate the addition count, and the high probability of error involved in making such substitutions, it was decided to use this method. This method also provides a convenient way to check the correctness of the reconstruction procedure by computing the matrices of the reduction and reconstruction subroutines and testing to see that they are indeed a transpose pair.

Intrinsic to the method of transposing the tensor is the fact that the matrix B used to compute the algorithm's multiplication coefficients from the Nth roots of unity is generally more complicated than either the reduction matrix or its transpose, the reconstruction matrix. This result is a consequence of B having been generated from Toom-Cook polynomial reconstruction procedures and also CRT polynomial reconstructions, which are both known to be more complicated than their associated reduction procedures. The problem of finding B in order to compute a set of multipliers may be neatly circumvented by directly solving a set of linear equations to find a coefficient vector which makes the algorithm work. The details of this trick are not reported here, but may be found in [link] . Suffice to say that given working FORTRAN subroutines for the reduction and reconstruction procedures, a FORTRAN program exists which will solve for the correct coefficients.