Some Structures of Parallel VLSI-Oriented Processing Units for Implementation of Small Size Discrete Fractional Fourier Transforms

Abstract

Discrete orthogonal transforms such as the discrete Fourier transform, discrete cosine transform, discrete Hartley transform, etc., are important tools in numerical analysis, signal processing, and statistical methods. The successful application of transform techniques relies on the existence of efficient fast algorithms for their implementation. A special place in the list of transformations is occupied by the discrete fractional Fourier transform (DFrFT). In this paper, some parallel algorithms and processing unit structures for fast DFrFT implementation are proposed. The approach is based on the resourceful factorization of DFrFT matrices. Some parallel algorithms and processing unit structures for small size DFrFTs such as N = 2, 3, 4, 5, 6, and 7 are presented. In each case, we describe only the most important part of the structures of the processing units, neglecting the description of the auxiliary units and the control circuits.

Discrete Pseudo-Fractional Fourier Transform and Its Fast Algorithm

“…In addition, the algorithm has a regular modular structure. Typical cores of the algorithm are standard small-size DFrFTs, which allows them to be used as unified building blocks in software or hardware implementation [13]. This will simplify and unify the process of developing programs and digital signal processors that implement the considered algorithm.…”

“…. , K have a special structure; therefore, the numbers of arithmetic operations when calculating the matrixvector product of each of them by the vector can be further reduced by about a half [12,13].…”

“…Among these transforms, the DFrFT has been applied in most practical contexts. To date, several effective algorithms have been developed to implement this transform [5,12,13]. In [14], the DFrFT was generalized, so that it depends on many fractional parameters.…”

“…As a result of this approach, a new orthogonal base was obtained and a fast algorithm for discrete transform in this base was synthesized. This transform can be implemented using small-size discrete fractional Fourier transforms, for which algorithms were described in [13]. The algorithm of the proposed transform has a parallel, simple and modular structure, which makes it convenient for hardware implementation.…”

Small-Size Algorithms for the Type-I Discrete Cosine Transform with Reduced Complexity

“…With rare exceptions, most publications known to the authors mainly deal with fast algorithms for other types of DCT. It should be noted that, as in the case of other discrete orthogonal transformations [28][29][30], the DCT-I algorithms for short sequences are also of particular interest. In the case of hardware or software implementation of digital signal processing methods, small-sized DCT-I implementation cores can serve as building blocks for the synthesis of larger-size algorithms [4,7,[31][32][33].…”