Showing papers on "Cyclotomic fast Fourier transform published in 2011"

Fast Walsh–Hadamard–Fourier Transform Algorithm

Abstract: An efficient fast Walsh-Hadamard-Fourier transform algorithm which combines the calculation of the Walsh-Hadamard transform (WHT) and the discrete Fourier transform (DFT) is introduced. This can be used in Walsh-Hadamard precoded orthogonal frequency division multiplexing systems (WHT-OFDM) to increase speed and reduce the implementation cost. The algorithm is developed through the sparse matrices factorization method using the Kronecker product technique, and implemented in an integrated butterfly structure. The proposed algorithm has significantly lower arithmetic complexity, shorter delays and simpler indexing schemes than existing algorithms based on the concatenation of the WHT and FFT, and saves about 70%-36% in computer run-time for transform lengths of 16-4096.

Generation of all radix-2 fast Fourier transform algorithms using binary trees

Abstract: In this work a systematic method to generate all possible fast Fourier transform (FFT) algorithms is proposed based on the relation to binary trees. The binary tree is used to represent the decomposition of a discrete Fourier transform (DFT) into sub-DFTs. The radix is adaptively changed according to compute sub-DFTs in proposed decomposition. In this work we determine the number of possible algorithms for 2n-point FFTs with radix-2 butterfly operation and propose a simple method to determine the twiddle factor indices for each algorithm based on the binary tree representation.

Efficient Dealiased Convolutions without Padding

Abstract: Algorithms are developed for calculating dealiased linear convolution sums without the expense of conventional zero-padding or phase-shift techniques. For one-dimensional in-place convolutions, the memory requirements are identical with the zero-padding technique, with the important distinction that the additional work memory need not be contiguous with the input data. This decoupling of data and work arrays dramatically reduces the memory and computation time required to evaluate higher-dimensional in-place convolutions. The technique also allows one to dealias the higher-order convolutions that arise from Fourier transforming cubic and higher powers. Implicitly dealiased convolutions can be built on top of state-of-the-art fast Fourier transform libraries: vectorized multidimensional implementations for the complex and centered Hermitian (pseudospectral) cases have been implemented in the open-source software FFTW++.

Composite Cyclotomic Fourier Transforms With Reduced Complexities

Abstract: Discrete Fourier transforms (DFTs) over finite fields have widespread applications in digital communication and storage systems. Hence, reducing the computational complexities of DFTs is of great significance. Recently proposed cyclotomic fast Fourier transforms (CFFTs) are promising due to their low multiplicative complexities. Unfortunately, there are two issues with CFFTs: (1) they rely on efficient short cyclic convolution algorithms, which have not been sufficiently investigated in the literature and (2) they have very high additive complexities when directly implemented. To address both issues, we make three main contributions in this paper. First, for any odd prime p, we reformulate a p -point cyclic convolution as the product of a (p-1) × (p-1) Toeplitz matrix and a vector, which has well-known efficient algorithms, leading to efficient bilinear algorithms for p-point cyclic convolutions. Second, to address the high additive complexities of CFFTs, we propose composite cyclotomic Fourier transforms (CCFTs). In comparison to previously proposed fast Fourier transforms, our CCFTs achieve lower overall complexities for moderate to long lengths and the improvement significantly increases as the length grows. Third, our efficient algorithms for p-point cyclic convolutions and CCFTs allow us to obtain longer DFTs over larger fields, e.g., the 2047-point DFT over GF(211) and 4095-point DFT over GF(212) , which are first efficient DFTs of such lengths to the best of our knowledge. Finally, our CCFTs are also advantageous for hardware implementations due to their modular structure.

On the Structure of Cyclotomic Fourier Transforms and Their Applications to Reed-Solomon Codes

Abstract: This paper is focused on cyclotomic Fourier transforms in GF(2m), and on their applications to algebraic decoding of Reed-Solomon codes, like the evaluation of syndromes and of error locator (or evaluator) polynomials. Cyclotomic transforms are much more efficient than straightforward evaluation. In particular, the number of multiplications is quite small. In this paper it is shown that also the number of additions can be considerably reduced with respect to previous analyses. A simple interpretation of the cyclotomic Fourier transform best suited for the evaluation of syndromes allows to assemble the required matrix easily and quickly, even in large fields. Fast construction of such matrices is important to obtain the best results, since as many matrices as possible must be generated and compared. It is shown that both the structure of the matrix and of bilinear convolutions need to be exploited, to reduce the complexity of the costly part of cyclotomic Fourier transforms, which is a matrix-vector product. Heuristic algorithms for matrix-vector product are to be run as many times as possible to obtain the best transform. It is shown with several examples that very good results can be obtained even with very simple algorithms.

On the Eigenstructure of DFT Matrices [DSP Education]

Abstract: The discrete Fourier transform (DFT) not only enables fast implementation of the discrete convolution operation, which is critical for the efficient processing of analog signals through digital means, but it also represents a rich and beautiful analytical structure that is interesting on its own. A typical senior-level digital signal processing (DSP) course involves a fairly detailed treatment of DFT and a list of related topics, such as circular shift, correlation, convolution operations, and the connection of circular operations with the linear operations. Despite having detailed expositions on DFT, most DSP textbooks (including advanced ones) lack discussions on the eigenstructure of the DFT matrix. Here, we present a self-contained exposition on such.

Fast-responding measurements of power system harmonics using discrete and fast fourier transforms with low spectral leakage

Abstract: Conventional wisdom dictates that a Fast Fourier Transform (FFT) will be a more computationally effective method for measuring multiple harmonics than a Discrete Fourier Transform (DFT) approach. However, in this paper it is shown that carefully coded discrete transforms which distribute their computational load over many frames can be made to produce results in shorter execution times than the FFT approach, even for large number of harmonic measurement frequencies. This is because the execution time of the presented DFT actually rises with N and not the classical N2 value, while the execution time of the FFT rises with Nlog2N. (6 pages)

The discrete Fourier transform over a finite field with reduced multiplicative complexity

Abstract: A novel method for computation of the discrete Fourier transform over a finite field with reduced multiplicative complexity is described. The theorem about the multiplicative complexity coincidence of the Goertzel and cyclotomic algorithms is proved.

Recursive Fourier Transform hardware

Abstract: The Discrete Fourier Transform has played a fundamental role in signal analysis for radar systems, usually in its Fast Fourier Transform form. Traditionally, samples from a region of interest have been acquired, and processed in a block-processing fashion. The result is a delayed output equal to the time taken to acquire the data and then perform the Fourier Transform. To overcome this limitation, this work focuses on the development of a parallel framework for the execution of the Discrete Fourier Transform using a recursive algorithm, using many simple processing elements. The Fourier coefficients are updated every time a new sample is presented to the system, with a small latency. The system described in this work shows a 64 point Discrete Fourier Transform computation in parallel, which achieves a throughput of 817MSPS(29.42Gbit/s). The framework proposed promotes adjustable parallelism using the many simple processing elements. We show the engine is capable of dynamically correcting accumulated errors due to finite arithmetic, and allows simple adoption on existing systems.

Efficient Fractal Image Coding using Fast Fourier Transform

Abstract: The fractal coding is a novel technique for image compression. Though the technique has many attractive features, the large encoding time makes it unsuitable for real time applications. In this paper, an efficient algorithm for fractal encoding which operates on entire domain image instead of overlapping domain blocks is presented.The algorithm drastically reduces the encoding time as compared to classical full search method. The reduction in encoding time is mainly due to use of modified crosscorrelation based similarity measure. The implemented algorithm employs exhaustive search of domain blocks and their isometry transformations to investigate their similarity with every range block. The application of Fast Fourier Transform in similarity measure calculation speeds up the encoding process. The proposed eight isometry transformations of a domain block exploit the properties of Discrete Fourier Transform to minimize the number of Fast Fourier Transform calculations. The experimental studies on the proposed algorithm demonstrate that the encoding time is reduced drastically with average speedup factor of 538 with respect to the classical full search method with comparable values of Peak Signal To Noise Ratio.

Application of Fast Fourier Transform for Accuracy Evaluation of Thermal-Hydraulic Code Calculations

Abstract: To study the behaviour of nuclear power plants, sophisticated and complex computer codes are needed. Before the computer codes are used for safety evaluations they have first to be validated. The assessment process of system codes involves the comparison of code results against experimental data and measured plant data. The accuracy of the code is the capability of the code to correctly predict the physical behaviour. Therefore the evaluation of accuracy coincides with code validation. In the past a few methods to quantify the code accuracy of thermal-hydraulic system codes have been proposed. Among the proposed methods, the approach using the fast Fourier transform has been proposed as one of the most effective approaches in 1990s (Ambrosini et al., 1990; D’Auria et al., 1994). The fast Fourier transform based method (FFTBM) shows the measurement-prediction discrepancies, i.e. the accuracy quantification, in the frequency domain. From the amplitudes of the component frequencies, the average amplitude (AA) is calculated. AA sums the difference between experimental and calculated signal discrete Fourier transform amplitudes at each frequency. To get a dimensionless accuracy measure, the sum of the amplitudes of the experimental signal is used for normalization. The closer the non-dimensional AA value is to zero, the better the agreement between the calculated results and the experimental measurements is judged. However, some problems involved in FFTBM, such as proper selection of time windows, weighting factors, number of discrete data used, consistency of the method in all cases and time dependent accuracy, still remain open and partly limit its application, especially those requiring a consistent accuracy judgement. For example, in early applications of FFTBM problems in evaluating signals, where experimental or error signal has the shape similar to triangle (i.e. first increases and then decreases), the accuracy value regularly overshoots at triangle peak, stabilising at lower value when discrepancy decease (Mavko et al. 1997). Not being aware about the reasons of such or some other strange behaviour, the FFTBM method has been also criticized. In general it is required that at any time into the transient the accuracy measure should remember the previous history. But on the other hand, the original FFTBM method has been effectively applied in obtaining information on code accuracy by several researchers in the literature. More recently, an automated code assessment program (ACAP) has been developed to provide a quantitative comparison between nuclear reactor system code results and

Dealiased convolutions for pseudospectral simulations

Abstract: Summary Efficient algorithms have recently been developed for calculating dealiased linear convolution sums without the expense of conventional zero-padding or phase-shift techniques. For one-dimensional in-place convolutions, the memory requirements are identical with the zero-padding technique, with the important distinction that the additional work memory need not be contiguous with the input data. This decoupling of data and work arrays dramatically reduces the memory and computation time required to evaluate higher-dimensional in-place convolutions. The memory savings is achieved by computing the in-place Fourier transform of the data in blocks, rather than all at once. The technique also allows one to dealias the n-ary convolutions that arise on Fourier transforming cubic and higher powers. Implicitly dealiased convolutions can be built on top of state-of-the-art adaptive fast Fourier transform libraries like FFTW. Vectorized multidimensional implementations for the complex and centered Hermitian (pseudospectral) cases have already been implemented in the open-source software FFTW++. With the advent of this library, writing a high-performance dealiased pseudospectral code for solving nonlinear partial differential equations has now become a relatively straightforward exercise. New theoretical estimates of computational complexity and memory use are provided, including corrected timing results for 3D pruned convolutions and further consideration of higher-order convolutions.

A New Overlap Save Algorithm for Fast Block Convolution and Its Implementation Using FFT

Abstract: Convolution of data with a long-tap filter is often implemented by overlap save algorithm (OSA) using fast Fourier transform (FFT). But there are some redundant computations in the traditional OSA because the FFT is applied to the overlapped data (concatenation of previous block and the current block) while the DFT computations are recursive. In this paper, we first analyze the redundancy by decomposing the OSA into two processes related to the previous and current block. Then we eliminate the redundant computations by introducing a new transform which is applied only to the current data, not to the overall overlapped data. Hence the size of transform is reduced by half compared to the traditional OSA. The new transform is in the form of DFT and it can be implemented by defining a new butterfly structure. However we implement it by a cascade of twiddle factor and conventional FFT in this paper, in order to use the FFT libraries in PC and DSP. The computational complexity in this case is analyzed and compared with the existing methods. In the experiment, the proposed method is applied to several block convolutions and partitioned-block convolutions. The CPU time is reduced more than expected from the arithmetic analysis, which implies that the reduced transform size gives additional advantage in data manipulation.

Addition Chains Algorithm Based on Fast Fourier Transform

Abstract: Some algorithms for generating addition chains are discussed, Fast Fourier Transform is considered thoroughly, and then a new algorithm based on FFT is proposed to quickly generate addition chains. The new addition chains given by the FFT based algorithm consist lots of multiply 2 operations, which can generate the addition chains at the fastest speed and can make the addition chains shortest. The new addition chain is close to the shortest addition chain. Besides, the FFT based algorithm for addition chains can be parallel implemented to improve its efficiency. More important, this algorithm gives a new direction for the research and application of FFT algorithm.

A fast algorithm for binary image restoration

Abstract: In this paper, we consider the problem to recover a binary digital image u ∈ Rm×n when the observed image g is corrupted with noise such as Gaussian or salt and pepper. By explicitly using the a priori knowledge for binary image, we propose a novel minimization model. Based on this new model, we present a novel fast iterative projection algorithm for binary image restoration. The main computation at each iteration is one fast Fourier transform (FFT) and one inverse fast Fourier transform (IFFT). Experimental results show that the proposed method is feasible and effective for binary image restoration. All of the results can be obtained within one second.

Multiple-Valued Logic Networks with Regular Structure Obtained from Fast Fourier Transforms on Finite Groups

Abstract: In this paper, we discuss the Fast Fourier transform (FFT) on finite groups as a useful method in synthesis for regularity. FFT is the algorithm for efficient calculation of the Discrete Fourier transform (DFT) and has been extended to computation of various Fourier-like transforms. The algorithm has a very regular structure that can be easily mapped to technology by replacing nodes in the corresponding flow-graphs by circuit modules performing the operations in the flow-graphs. In this way, networks with highly regular structure for implementing functions from their spectra are derived. Fourier transforms on non-Abelian groups offer additional advantages for reducing the required hardware due to matrix-valued spectral coefficients and the way how such coefficients are used in reconstructing the functions. Methods for optimization of spectral representations of functions on finite groups may be applied to improve networks with regular structure.

Correcting DFT Codes with Modified Berlekamp-Massey Algorithm and Syndrome Extension

Abstract: Real number block codes derived from the discrete Fourier transform (DFT) are corrected by coupling a very modified Berlekamp-Massey algorithm with a syndrome extension process Enhanced extension recursions based on Kalman syndrome extensions are examined

Fast transform for multi-carrier wireless communications

Abstract: The proposed fast transform is originated from the Haar wavelet. The N-point (N=2 power d) fast transform requires no multiplications in case d is even, and N real multiplications with constant in case d is odd, and it uses at least 33 percent less real additions than the Fast Fourier Transform. The proposed fast transform is developed to reduce complexity of Wavelet Packet Multiplexing (WPM). The same fast transform algorithm can be used for both multiplexing and demultiplexing of data streams. Simulations show that multi-carrier wireless communication systems can profit from use of WPM based on the proposed transform, because, in terms of complexity, WPM outperforms the most used now-a-days Orthogonal Frequency Division Multiplexing (OFDM).

A fast recursive algorithm for computing cyclotomic polynomials

Abstract: The nth cyclotomic polynomial, Φn(z), is the minimal polynomial over Q of the nth complex primitive roots of unity. We let the order of Φn(z) be the number of distinct odd prime divisors of n and denote by A(n) the height of Φn(z), that is, the magnitude of the largest coefficient of Φn(z). In [5], Erdős proved that, for any c > 0, A(n) is not bounded by nc. Maier showed in [3] that the set of n for which A(n) > nc has positive lower density. We were originally motivated by the question: what is the least n for which A(n) > n? n2? n3? and so forth. Towards that aim, we wanted to develop and implement faster algorithms for computing Φn(z). One approach to compute Φn(z) is to compute Φn(z) using

The efficient computation DHT using cyclic convolutions

Abstract: The efficient discrete Hartley transform (DHT) algorithm is introduced. The algorithm are determined the structure of discrete basis matrix. Thel cyclic convolutions is fundamental of algorithm for computation discrete Hartley transform.

Fast fourier transforms over poor fields

Abstract: We present a new algebraic algorithm for computing the discrete Fourier transform over arbitrary fields. It computes DFTs of infinitely many orders n in O(n log n) algebraic operations, while the complexity of a straightforward application of the known FFT algorithms can be Ω(n1.5) for such n. Our algorithm is a novel combination of the classical FFT algorithms, and is never slower than any of the latter.As an application we come up with an efficient way of computing DFTs of high orders in finite field extensions which can further boost polynomial multiplication algorithms. We relate the complexities of the DFTs of such orders with the complexity of polynomial multiplication.

How to use the Fast Fourier Transform in Large Finite Fields

Abstract: The article contents suggestions on how to perform the Fast Fourier Transform over Large Finite Fields. The technique is to use the fact that the multiplicative groups of specific prime fields are surprisingly composite.

A Parallel Architecture for Radix-2 Fast Hartley Transform/Real-valued Fast Fourier Transform*

Abstract: Fast Hartley transform (FHT) /Real-valued Fast Fourier trans- form (RFFT) algorithms are important Fourier-related transforms, because they lower twice the operational and memory requirements when input data is real-valued. However, these types of algorithms, have irregular computational structure, which makes their parallel implementation a difficult tas The aim of this paper is to show that these algorithms have the same potential for parallel implementation as the complex fast Fourier transform (FFT), as well as they share common natural architectures with FFT, based on the perfect shuffle permutation method. Presented is a processor array architecture based on the "indirect hypercube" concept which is suitable for realization of fast FHT/RFFT processors for real time applications.