Henri J. Nussbaumer

Fast polynomial transform algorithms for digital convolution

We have recently introduced new transforms, called polynomial transforms, which are defined in rings of polynomials and give efficient algorithms for the computation of multidimensional DFTs and convolutions. In this paper we present a method for computing one-dimensional convolutions by polynomial transforms. We show that this method is computationally efficient, even for large convolutions, and can be implemented with FFT-type algorithms, while avoiding the use of trigonometric functions and complex arithmetic. We then extend this technique to complex convolutions and to multidimensional convolutions.

Fast computation of discrete Fourier transforms using polynomial transforms

Polynomial transforms, defined in rings of polynomials, have been introduced recently and have been shown to give efficient algorithms for the computation of two-dimensional convolutions. In this paper we present two methods for computing discrete Fourier transforms (DFT) by polynomial transforms. We show that these techniques are particularly well adapted to multidimensional DFTs as well as to some one-dimensional DFTs and yield algorithms that are, in many instances, more efficient than the fast Fourier transform (FFT) or the Winograd Fourier Transform (WFTA). We also describe new split nesting and split prime factor techniques for computing large DFTs from a small set of short DFTs with a minimum number of operations.

Computation of convolutions and discrete Fourier transforms by polynomial transforms

Discrete transforms are introduced and are defined in a ring of polynomials. These polynomial transforms are shown to have the convolution property and can be computed in ordinary arithmetic, without multiplications. Polynomial transforms are particularly well suited for computing discrete two-dimensional convolutions with a minimum number of operations. Efficient algorithms for computing one-dimensional convolutions and Discrete Fourier Transforms are then derived from polynomial transforms.

Digital filtering using pseudo fermat number transforms

In this paper pseudo Fermat number transforms (FNTs) are discussed. These transforms are defined in a ring of integers modulo an integer submultiple of a pseudo Fermat number, and can be computed without multiplications while allowing a great flexibility in word length selection. Complex pseudo FNTs are then introduced and are shown to relieve some of the length limitations of conventional Fermat number transforms (FNTs). These transforms, which under certain conditions can be computed via fast transform algorithms allow the implementation of digital filters with better efficiency and accuracy than the fast Fourier transform (FFT).

New polynomial transform algorithms for multidimensional DFT's and convolutions

This paper introduces a new method for the computation of multidimensional DFTs by polynomial transforms. The method, which maps mtiltidimensional DFTs into one-dimensional odd-time DFTs by use of inverse polynomial transforms, is shown to be significantly more efficient than the conventional row-column method from the standpoint of the number of arithmetic operations and quantization noise. The relationship between DFT and convolution algorithms using polynomial transforms is clarified and new convolution algorithms with reduced computational complexities are proposed.

Relative evaluation of various number theoretic transforms for digital filtering applications

The main existence conditions for pseudo Mersenne and pseudo Fermat number transforms defined in a ring submultiple of a pseudo Mersenne or pseudo Fermat number are defined. The computational complexity of various multiplication-free number theoretic transforms (NTTs) used for implementing digital filters is evaluated. It is shown that Fermat number transforms (FNTs) with root \sqrt{2} and some complex pseudo Mersenne and pseudo Fermat number transforms with root 1 + j yield optimum processing efficiency and allow significant computational savings over direct filter evaluation.

Microcoded Modem Transmitters

This paper describes various microcoded designs for modem transmitters. The digital echo modulation technique, originally introduced by J-M. Pierret, is applied to cover the case of a fully digital universal modem. The capabilities of several microcoded modem designs are presented and their limitations are discussed.

New algorithms for convolution and DFT based on polynomial transforms

Discrete transforms, defined in rings of polynomials, have been introduced recently. These polynomial transforms have the convolution property and can be computed in ordinary arithmetic, without multiplications. We show that, by combining the polynomial transform approach with a split nesting technique, multidimensional convolutions can be computed very efficiently in general purpose computers. This computation method can also be used for the evaluation of one-dimensional convolutions and discrete Fourier transforms (DFTs).

Inverse polynomial transform algorithms for DFTs and convolutions

In this paper, we introduce a new fast computation algorithm for multidimensional DFTs. This method uses inverse polynomial transforms to perform an efficient mapping of multidimensional DFTS into one-dimensional DFTs in a way similar to earlier polynomial transform techniques, but with all operations performed in reversed order. This is shown to yield fast DFT algorithms which retain the basic advantages related to the use of polynomial transforms while allowing a significant reduction in round-off noise. We then combine the direct and inverse polynomial transform methods to derive new fast algorithms for multidimensional convolutions.

New polynomial transform algorithms for fast DFT computation

Polynomial transforms defined in rings of polynomials, have been introduced recently and shown to give efficient algorithms for the computation of two-dimensional convolutions. In this paper, we present two methods for computing discrete Fourier transforms (DFT) by polynomial transforms. We show that these techniques are particularly well adapted to multidimensional DFTs and yield algorithms that are, in many instances, more efficient than the fast Fourier transform (FFT) or the Winograd Fourier Transform (WFTA).