Two-dimensional discrete Fourier transform with small multiplicative complexity using number theoretic transforms (original) (raw)

Very fast discrete Fourier transform using number theoretic transform

IEE Proceedings G (Electronic Circuits and Systems)

It is shown that number theoretic transforms (NTT) can be used to compute discrete Fourier transform (DFT) very efficiently. By noting some simple properties of number theory and the DFT, the total number of real multiplications for a length-P DFT is reduced to (P -1). This requires less than one real multiplication per point. For a proper choice of transform length and NTT, the number of shift adds per point is approximately the same as the number of additions required for FFT algorithms.

FPGA Architecture for 2D Discrete Fourier Transform Based on 2D Decomposition for Large-sized Data

Journal of Signal Processing Systems, 2011

Applications based on Discrete Fourier Transforms (DFT) are extensively used in several areas of signal and digital image processing. Of particular interest is the two-dimensional (2D) DFT which is more computation- and bandwidth-intensive than the one-dimensional (1D) DFT. Traditionally, a 2D DFT is computed using Row-Column (RC) decomposition, where 1D DFTs are computed along the rows followed by 1D DFTs

Algorithms and pipeline architectures for 2-D FFT and FFT-like transforms

Digital Signal Processing, 2010

In this paper, efficient pipeline architectures that implement the 2-D FFT are presented. Based on the Vector Radix approach, the new structures alleviate the use of memory banks and the transposition of data of the row-column technique. Architectures for Vector Radix 2 × 2 algorithm and for a modified Vector Radix 4 × 4, called Vector Radix 2 2 × 2 2 algorithm, which has been devised and constructed from Vector Radix 2 × 2, are presented.

Fast algorithms for the multidimensional discrete Fourier transform

IEEE Transactions on Acoustics, Speech, and Signal Processing, 1986

In this paper, the prime factor algorithm for the evaluation of a one-dimensional discrete Fourier transform is generalized to the evaluation of multidimensional discrete Fourier transforms defined on arbitrary periodic sampling lattices. It is shown that such an algorithm is equivalent in computational complexity to the evaluation of a rectangular discrete Fourier transform. As a sidelight to the derivation of the algorithm, a Chinese remainder theorem is derived for integer lattices. I. INTRODUCTION T HERE is a need for the efficient calculation of multidimensional discrete Fourier transforms (MDFT's) in applications ranging from image processing to antenna design, geophysics, and optics. In the past decade, bounds on the computational complexity of one-dimensional (1-D) DFT calculations have been established. In this paper, these one-dimensional results are extended to arbitrary multidimensional transform calculations and efficient algorithms are presented. Historically, the first approach to the evaluation of multidimensional discrete Fourier transforms was the rowcolumn algorithm [l]. This algorithm evaluates rectangular samples of the Fourier transform of a multidimensional sequence by evaluating 1-D DFT's on the rows and columns of a multidimensional array. The vector-radix algorithm [2]-[6] represents a slightly more efficient algorithm which attacks the DFT computation with a divide-and-conquer strategy similar to the 1-D Cooley-Tukey FFT algorithm. Nussbaumer [7]-[9] has used polynomial theory to extend the 1-D Winograd algorithm to the multidimensional case. All of these algorithms, however, are applicable only to the evaluation of rectangular DFT's; that is, they compute samples of a multidimensional Fourier transform taken on a hypercubic lattice from samples of a band-limted multidimensional function taken on a similar'lattice. Mersereau and Speake [lo] demonstrated that discrete Fourier transforms could be defined for signals defined on Manuscript

Rapid Computation of the Discrete Fourier Transform

SIAM Journal on Scientific Computing, 1996

Algorithms for the rapid computation of the forward and inverse discrete Fourier transform for points which are nonequispaced or whose number is unrestricted are presented. The computational procedure is based on approximation using a local Taylor series expansion and the fast Fourier transform (FFT). The forward transform for nonequispaced points is computed as the solution of a linear system involving the inverse Fourier transform. This latter system is solved using the iterative method GMRES with preconditioning. Numerical results are given to confirm the efficiency of the algorithms.

An algorithm for fast computation of the multiresolution discrete Fourier transform

ArXiv, 2015

The article presents a computationally effective algorithm for calculating the multiresolution discrete Fourier transform (MrDFT). The algorithm is based on the idea of reducing the computational complexity which was introduced by Wen and Sandler [10] and utilizes the vectorization of calculating process at each stage of the considered transformation. This allows for the use of a computational process parallelization and results in a reduction of computation time. In the description of the computational procedure, which describes the algorithm, we use the matrix notation. This notation enables to represent adequately the space-time structures of the implemented computational process and directly map these structures into the constructions of a high-level programming language or into a hardware realization space.

On the computation of discrete Fourier transform using Fermat number transform

In the paper the results of a study using Fermat number transforms (FNTs) to compute discrete Fourier transforms (DFTs) are presented. Eight basic FNT modules are suggested and used as the basic sequence lengths to compute long DFTs. The number of multiplications per point is for most cases not more than one, whereas the number of shift-adds is approximately equal to the number of additions in the Winograd-Fourier-transform algorithm and the polynomial transform. Thus the present technique is very effective in computing discrete Fourier transforms.

Triple-Matrix Product-Based 2D Systolic Implementation of Discrete Fourier Transform

Circuits, Systems, and Signal Processing, 2015

Realization of N-point discrete Fourier transform (DFT) using onedimensional or two-dimensional systolic array structures has been developed for power of two DFT sizes. DFT algorithm, which can be represented as a triple-matrix product, can be realized by decomposing N into smaller lengths. Triple-matrix product form of representation enables to map the N-point DFT on a 2D systolic array. In this work, an algorithm is developed and is mapped to a two-dimensional systolic structure where DFT size can be non-power of two. The proposed work gives flexibility to choose N for an application where N is a composite number. The total time required to compute N-point DFT is 2(N 1 − 1) + N 2 + N for any N = N 1 N 2. The array can be used for matrix-matrix multiplication and also to compute the diagonal elements of triple-matrix multiplication for other applications. The proposed architecture produces in-order stream of DFT sequence at the output avoiding need for reordering buffer. Large sized DFT can be computed by repeatedly using the proposed systolic array architecture.

An Area Efficient 2D Fourier Transform Architecture for FPGA Implementation

2018

Two-dimensional Fourier transform plays a significant role in a variety of image processing problems, such as medical image processing, digital holography, correlation pattern recognition, hybrid digital optical processing, optical computing etc. 2D spatial Fourier transformation involves large number of image samples and hence it requires huge hardware resources of field programmable gate arrays (FPGA). In this paper, we present an area efficient architecture of 2D FFT processor that reuses the butterfly units multiple times. This is achieved by using a control unit that sends back the previous computed data of N/2 butterfly units to itself for {log_2(N) - 1} times. A RAM controller is used to synchronize the flow of data samples between the functional blocks.The 2D FFT processor is simulated by VHDL and the results are verified on a Virtex-6 FPGA. The proposed method outperforms the conventional NxN point 2D FFT in terms of area which is reduced by a factor of log_N(2) with neglig...

FAST MERSENNE NUMBER TRANSFORMS FOR THE COMPUTATION OF DISCRETE FOURIER TRANSFORMS

In this paper we present results on the computation of Discrete Fourier Transforms (DFT) using Mersenne Number Transforms (MNT). It is shown that in the case of Mersenne-composite Number Transforms, the number of multiplications per point for real input data is never more than one, even for sequence lengths exceeding one thousand points. The computation time per point for a length (2P+ 1)-point DFT is simply equal to the time for one MNT multiplication and 3 MNT additions if a high-speed, parallel hardware module is used to implement the MNT unit. This new approach allows a large choice of wordlengths and in addition the control of data flow is extremely simple. We also present the results obtained by using Winograd's Fourier Transform Algorithm and the nested MNT to compute efficiently the DFT's of long sequences. We also show that the number of additions can be reduced significantly if Pseudo Mersenne-Number Transforms are used for the computation of DFTs.