Showing papers on "Prime-factor FFT algorithm published in 2012"

Simple and practical algorithm for sparse Fourier transform

Abstract: We consider the sparse Fourier transform problem: given a complex vector x of length n, and a parameter k, estimate the k largest (in magnitude) coefficients of the Fourier transform of x. The problem is of key interest in several areas, including signal processing, audio/image/video compression, and learning theory.We propose a new algorithm for this problem. The algorithm leverages techniques from digital signal processing, notably Gaussian and Dolph-Chebyshev filters. Unlike the typical approach to this problem, our algorithm is not iterative. That is, instead of estimating "large" coefficients, subtracting them and recursing on the reminder, it identifies and estimates the k largest coefficients in "one shot", in a manner akin to sketching/streaming algorithms. The resulting algorithm is structurally simpler than its predecessors. As a consequence, we are able to extend considerably the range of sparsity, k, for which the algorithm is faster than FFT, both in theory and practice.

Nearly optimal sparse fourier transform

Abstract: We consider the problem of computing the k-sparse approximation to the discrete Fourier transform of an n-dimensional signal. We show: An O(k log n)-time randomized algorithm for the case where the input signal has at most k non-zero Fourier coefficients, and An O(k log n log(n/k))-time randomized algorithm for general input signals. Both algorithms achieve o(n log n) time, and thus improve over the Fast Fourier Transform, for any k=o(n). They are the first known algorithms that satisfy this property. Also, if one assumes that the Fast Fourier Transform is optimal, the algorithm for the exactly k-sparse case is optimal for any k = nΩ(1). We complement our algorithmic results by showing that any algorithm for computing the sparse Fourier transform of a general signal must use at least Ω(k log (n/k) / log log n) signal samples, even if it is allowed to perform adaptive sampling.

Superfast Fourier Transform Using QTT Approximation

Abstract: We propose Fourier transform algorithms using QTT format for data-sparse approximate representation of one- and multi-dimensional vectors (m-tensors). Although the Fourier matrix itself does not have a low-rank QTT representation, it can be efficiently applied to a vector in the QTT format exploiting the multilevel structure of the Cooley-Tukey algorithm. The m-dimensional Fourier transform of an n×⋯×n vector with n=2d has \(\mathcal{O}(m d^{2} R^{3})\) complexity, where R is the maximum QTT-rank of input, output and all intermediate vectors in the procedure. For the vectors with moderate R and large n and m the proposed algorithm outperforms the \(\mathcal{O}(n^{m} \log n)\) fast Fourier transform (FFT) algorithm and has asymptotically the same log-squared complexity as the superfast quantum Fourier transform (QFT) algorithm. By numerical experiments we demonstrate the examples of problems for which the use of QTT format relaxes the grid size constrains and allows the high-resolution computations of Fourier images and convolutions in higher dimensions without the ‘curse of dimensionality’. We compare the proposed method with Sparse Fourier transform algorithms and show that our approach is competitive for signals with small number of randomly distributed frequencies and signals with limited bandwidth.

Fast computing of discrete cosine and sine transforms of types vi and vii

Abstract: This disclosure presents techniques for implementing a fast algorithm for implementing odd-type DCTs and DSTs. The techniques include the computation of an odd-type transform on any real-valued sequence of data (e.g., residual values in a video coding process or a block of pixel values of an image coding process) by mapping the odd-type transform to a discrete Fourier transform (DFT). The techniques include a mapping between the real-valued data sequence to an intermediate sequence to be used as an input to a DFT. Using this intermediate sequence, an odd-type transform may be achieved by calculating a DFT of odd size. Fast algorithms for a DFT may be then be used, and as such, the odd-type transform may be calculated in a fast manner

Design and Simulation of 32-Point FFT Using Radix-2 Algorithm for FPGA Implementation

Abstract: The Fast Fourier Transform (FFT) is one of the rudimentary operations in field of digital signal and image processing. Some of the very vital applications of the fast fourier transform include Signal analysis, Sound filtering, Data compression, Partial differential equations, Multiplication of large integers, Image filtering etc. Fast Fourier transform (FFT) is an efficient implementation of the discrete Fourier transform (DFT). This paper concentrates on the development of the Fast Fourier Transform (FFT), based on Decimation-In-Time (DIT) domain, Radix-2 algorithm, this paper uses VHDL as a design entity, and their Synthesis by Xilinx Synthesis Tool on Vertex kit has been done. The input of Fast Fourier transform has been given by a PS2 KEYBOARD using a test bench and output has been displayed using the waveforms on the Xilinx Design Suite 12.1. The synthesis results show that the computation for calculating the 32-point Fast Fourier transform is efficient in terms of speed.

Fixed-Point Analysis and Parameter Selections of MSR-CORDIC With Applications to FFT Designs

Abstract: Mixed-scaling-rotation (MSR) coordinate rotation digital computer (CORDIC) is an attractive approach to synthesizing complex rotators. This paper presents the fixed-point error analysis and parameter selections of MSR-CORDIC with applications to the fast Fourier transform (FFT). First, the fixed-point mean squared error of the MSR-CORDIC is analyzed by considering both the angle approximation error and signal round-off error incurred in the finite precision arithmetic. The signal to quantization noise ratio (SQNR) of the output of the FFT synthesized using MSR-CORDIC is thereafter estimated. Based on these analyses, two different parameter selection algorithms of MSR-CORDIC are proposed for general and dedicated MSR-CORDIC structures. The proposed algorithms minimize the number of adders and word-length when the SQNR of the FFT output is constrained. Design examples show that the FFT designed by the proposed method exhibits a lower hardware complexity than existing methods.

Simple and practical algorithm for sparse fourier transform

Abstract: We consider the sparse Fourier transform problem: given a complex vector x of length n, and a parameter k, estimate the k largest (in magnitude) coefficients of the Fourier transform of x. The problem is of key interest in several areas, including signal processing, audio/image/video compression, and learning theory.We propose a new algorithm for this problem. The algorithm leverages techniques from digital signal processing, notably Gaussian and Dolph-Chebyshev filters. Unlike the typical approach to this problem, our algorithm is not iterative. That is, instead of estimating "large" coefficients, subtracting them and recursing on the reminder, it identifies and estimates the k largest coefficients in "one shot", in a manner akin to sketching/streaming algorithms. The resulting algorithm is structurally simpler than its predecessors. As a consequence, we are able to extend considerably the range of sparsity, k, for which the algorithm is faster than FFT, both in theory and practice.

Generic Mixed-Radix FFT Pruning

Abstract: Compared with traditional Fast Fourier Transform (FFT) algorithms, FFT pruning is more computationally efficient in those cases where some of the input values are zero and/or some of the output components are not needed. In this letter, a novel pruning scheme is developed for mixed-radix and high-radix FFT pruning. The proposed approach is applicable over a wide range of FFT lengths and input/output pruning patterns. In addition, it can effectively employ the benefits of high-radix FFT algorithms that have lower computational complexity.

An Algorithm for Electrical Harmonic Analysis Based on Triple-spectrum-line Interpolation FFT

Abstract: The fast Fourier transform(FFT) will have a greater error in the cases of non-synchronous sampling and in a truncated non-integral period,thus accurate harmonic parameter values can not be obtained.For this reason,an improved interpolation windowed FFT algorithm was presented in this paper to analysis electrical harmonics.The algorithm used the three neighboring spectrum lines to locate the accuracy position of the harmonic spectrum line through analyzing the discrete-time Fourier transform of the windowed signal,and then the accurate results of the frequency,phase and amplitude could be obtained.By the triple-spectrum-line Interpolation FFT,the accuracy of the harmonic analysis results could be improved.Based on the algorithm,the practical rectification formulas for harmonic analysis were obtained by using the polynomial approximation method.The simulation results have verified that the algorithm has a better precision than the double-spectrum-line interpolation algorithm using the same window.Thus the effectiveness and practicability of the algorithm have been verified.

Computation of Isotopic Peak Center-Mass Distribution by Fourier Transform

Abstract: We derive a new efficient algorithm for the computation of the isotopic peak center-mass distribution of a molecule. With the use of Fourier transform techniques, the algorithm accurately computes the total abundance and average mass of all the isotopic species with the same number of nucleons. We evaluate the performance of the method with 10 benchmark proteins and other molecules; results are compared with BRAIN, a recently reported polynomial method. The new algorithm is comparable to BRAIN in accuracy and superior in terms of speed and memory, particularly for large molecules. An implementation of the algorithm is available for download.

Fast Fourier optimization

Abstract: Many interesting and fundamentally practical optimization problems, ranging from optics, to signal processing, to radar and acoustics, involve constraints on the Fourier transform of a function. It is well-known that the fast Fourier transform (fft) is a recursive algorithm that can dramatically improve the efficiency for computing the discrete Fourier transform. However, because it is recursive, it is difficult to embed into a linear optimization problem. In this paper, we explain the main idea behind the fast Fourier transform and show how to adapt it in such a manner as to make it encodable as constraints in an optimization problem. We demonstrate a real-world problem from the field of high-contrast imaging. On this problem, dramatic improvements are translated to an ability to solve problems with a much finer grid of discretized points. As we shall show, in general, the “fast Fourier” version of the optimization constraints produces a larger but sparser constraint matrix and therefore one can think of the fast Fourier transform as a method of sparsifying the constraints in an optimization problem, which is usually a good thing.

A framework for low-communication 1-D FFT

Abstract: In high-performance computing on distributed-memory systems, communication often represents a significant part of the overall execution time. The relative cost of communication will certainly continue to rise as compute-density growth follows the current technology and industry trends. Design of lower-communication alternatives to fundamental computational algorithms has become an important field of research. For distributed 1-D FFT, communication cost has hitherto remained high as all industry-standard implementations perform three all-to-all internode data exchanges (also called global transposes). These communication steps indeed dominate execution time. In this paper, we present a mathematical framework from which many single-all-to-all and easy-to-implement 1-D FFT algorithms can be derived. For large-scale problems, our implementation can be twice as fast as leading FFT libraries on state-of-the-art computer clusters. Moreover, our framework allows tradeoff between accuracy and performance, further boosting performance if reduced accuracy is acceptable.

Multiplier-less based architecture for variable-length FFT hardware implementation

Abstract: This paper presents a new variable-length, multiplierless Fast Fourier Transform (FFT) hardware implementation. The proposed implementation handles with 16, 64 and 128 point FFT frames. The architecture is designed based on Single-path Delay Feedback (SDF) scheme which implements the mixed radi×2, 22 FFT algorithms. In order to improve the performance, the complex multipliers are replaced by simpler and faster units which use only shift and addition/subtraction operations. As consequence, we obtain twofold increase in the speed.

An iterative FFT based flat-top footprint pattern synthesis method with planar array

Abstract: A pattern synthesis technique for the flat-top pattern with an arbitrary footprint is proposed in this paper. This technique is based on an iterative fast Fourier transform (FFT) method and the Elliott–Stern method. Due to the Fourier transformation-pair relationship between the array excitations and the array factor, the iterative FFT method is used to adjust the array factor iteratively to obtain the desired pattern. To accelerate this method and to make this method more robust, the initial element excitations are obtained by applying Fourier transform to a pattern designed by the Elliott–Stern method. Several numerical simulations are applied to validate the effectiveness of this technique.

On the additive complexity of the cyclotomic FFT algorithm

Abstract: The problem of efficient evaluation of the discrete Fourier transform over finite fields is considered. The techniques for additive complexity reduction of the cyclotomic FFT algorithm are proposed. The first one is based on the classical simultaneous reduction algorithm. The second one is based on a factorization of the presummation matrix into a sparse and block-diagonal ones. The proposed methods provide smaller asymptotic complexity, although for small-sized problems the required number of operations appears to be higher than the complexity of computer-optimized algorithms.

Parallel pipelined FFT architectures with reduced number of delays

Abstract: This paper presents a novel approach to design four and eight parallel pipelined fast Fourier transform (FFT) architectures using folding transformation. The approach is based on use of decimation in time algorithms which reduce the number of delay elements by 33% compared to the decimation in frequency based designs. The number of delay elements required for an N-point FFT architecture is N - 4 which is comparable to that of delay feedback schemes. The number of complex adders required is only 50% of those in the delay feedback designs. The proposed approach can be extended to any radix-2n based FFT algorithms. The proposed architectures are feed-forward designs and can be pipelined by more stages to increase the throughput. Further, a novel four parallel 128-point FFT architecture is derived using the proposed approach. It is shown that a radix-24 4-parallel 128-point design requires 124 delay elements, 28 complex adders, and four full complex multipliers.

A high performance split-radix FFT with constant geometry architecture

Abstract: High performance hardware FFTs have numerous applications in instrumentation and communication systems. This paper describes a new parallel FFT architecture which combines the split-radix algorithm with a constant geometry interconnect structure. The split-radix algorithm is known to have lower multiplicative complexity than both radix-2 and radix-4 algorithms. However, it conventionally involves an "L-shaped" butterfly datapath whose irregular shape has uneven latencies and makes scheduling difficult. This work proposes a split-radix datapath that avoids the L-shape. With this, the split-radix algorithm can be mapped onto a constant geometry interconnect structure in which the wiring in each FFT stage is identical, resulting in low multiplexing overhead. Further, we exploit the lower arithmetic complexity of split-radix to lower dynamic power, by gating the multipliers during trivial multiplications. The proposed FFT achieves 46% lower power than a parallel radix-4 design at 4.5GS/s when computing a 128-point real-valued transform.

An Implementation of Parallel 1-D FFT on the K Computer

Abstract: In this paper, we propose an implementation of a parallel one-dimensional fast Fourier transform (FFT) on the K computer. The proposed algorithm is based on the six-step FFT algorithm, which can be altered into the recursive six-step FFT algorithm to reduce the number of cache misses. The recursive six-step FFT algorithm improves performance by utilizing the cache memory effectively. We use the recursive six-step FFT algorithm to implement the parallel one-dimensional FFT algorithm. The performance results of one-dimensional FFTs on the K computer are reported. We successfully achieved a performance of over 18 TFlops on 8192 nodes of the K computer (82944 nodes, 128 GFlops/node, 10.6 PFlops peak performance) for a 2^41-point FFT.

Fast Fourier transform harmonic analysis based on nine terms minimum side-lobe cosine-sum window

Abstract: Since it's difficult to do synchronous sampling when making the signal sampling, the spectral leakage and fence effect will occur during fast Fourier transform (FFT) processing, which leads to greater measurement errors. To solve these problems, this paper proposes an interpolation FFT algorithm based on the nine terms cosine-sum window with minimum side-lobe and deduces the correction formula of this algorithm using the dual interpolated theory. Compare the measurement result of this algorithm by simulation with the measurement results of interpolation algorithm based on the Hanning window `Blackman-Harris window' Nuttall window, which verifies that this algorithm is more accurate at measurement and has the actual application value.

Minimizing the error: A study of the implementation of an Integer Split-Radix FFT on an FPGA for medical imaging

Abstract: Fixed-point arithmetic is used to provide faster and smaller implementations in many digital signal processing applications, including medical imaging, at the expense of decreased accuracy. In particular, when a Fast Fourier Transform (FFT)-Inverse Fast Fourier Transform (IFFT) pair are required as part of the calculation, the error introduced into the calculations can be significant. For some applications, such as Fourier Domain Optical Coherence Tomography (FD-OCT), this degradation is unacceptable. Our study shows that using a conventional fixed-point FFT-IFFT pair, such as Xilinx's FFT core, can produce an average 6-bit error for a 1024-point FFT using 12-bit input data in a 32-bit arithmetic system. The majority of the error is caused by quantization effects, particularly on the phase information of input signal. For this reason, in phase sensitive applications such as FD-OCT, the error dominates the fixed-point calculation: 78% in 16-bit and 51% in 32-bit systems. This work presents a parameterized 32 to 4096-point integer FFT implementation for FPGAs that uses a Split-Radix algorithm to reduce the number of multiplies and improve latency. Integer FFTs are perfectly reconstructible, with zero reconstruction error. Here, we specifically analyze a 1024-point Integer Split-Radix FFT (Int-SRFFT) and IFFT pair that perfectly reconstructs the original 12-bit input data using 22-bit arithmetic, compared to the average 6-bit error for a 1024-point FFT using 32-bit fixed-point arithmetic. The pipelined architecture of this design has a latency of 29.06us for a 1024-point FFT, and a throughput of more than 34 thousand 1024-point FFTs/second for a 22-bit datapath at an operating frequency of 274MHz. Although our Int-SRFFT is perfectly reconstructible, compared to Xilinx's fixed-point FFT, it has ∼6% more flipflops, ∼63% more LUTs, 5.3x more BRAMs, and a ∼44% increase in latency. However, compared to a previous fixed-point SRFFT design on an FPGA, our throughput is 15.5x greater.

A transpose-free in-place SIMD optimized FFT

Abstract: A transpose-free in-place SIMD optimized algorithm for the computation of large FFTs is introduced and implemented on the Cell Broadband Engine. Six different FFT implementations of the algorithm using six different data movement methods are described. Their relative performance is compared for input sizes from 217 to 221 complex floating point samples. Large differences in performance are observed among even theoretically equivalent data movement patterns. All six implementations compare favorably with FFTW and other previous FFT implementations.

An OpenMP Parallelized Multilevel Green's Function Interpolation Method Accelerated by Fast Fourier Transform Technique

Abstract: A parallelized multilevel Green's function interpolation method (MLGFIM) accelerated by fast Fourier transform (FFT) technique is proposed The difficulties in applying various improved interpolation schemes to effectively reduce the number of interpolation points are overcome by using the FFT technique In order to accelerate the convergence property of the iterative solution using the proposed algorithm, a recently proposed preconditioning scheme, ie, multiplicative Calderon preconditioner is adopted to transform the first kind integral operator to the second kind, albeit an increase of computer memory storage requirement An OpenMP parallel implementation of the MLGFIM-FFT algorithm on a share-memory computer system is developed to analyze various electrically large electromagnetic scattering problems including a NASA almond, a 20-wavelength cylinder capped with two half spheres, and a 37-wavelength cylinder array Numerical results illustrate good computational performance of the proposed algorithm

Recursive Discrete Fourier Transform based SMT receivers for cognitive radio applications

Abstract: This paper presents a modified Staggered MultiTone (SMT) receiver structure tailored to cognitive radio applications. SMT schemes employing Fast Fourier Transform (FFT) in the receiver for demodulation are discussed in the first part. The second part deals with Recursive Discrete Fourier Transform (R-DFT) originating from measurement systems. The recursive structure is suggested as an alternative to FFT for demodulation and spectral sensing. The paper concludes with a discussion of the advantages of the proposed receiver architecture especially in cognitive radio applications, and the investigation of the required signal processing complexity.

Reducing power consumption in FFT architectures by using heuristic-based algorithms for the ordering of the twiddle factors

Abstract: This paper addresses the exploration of different heuristic-based algorithms for a better manipulation of coefficients in Fast Fourier Transform (FFT). Due to the characteristics of the FFT algorithms, which involve multiplications of input data with appropriate coefficients, the best ordering of these operations can contribute for the reduction of the switching activity, what leads to the minimization of power consumption in the FFTs. The heuristic-based algorithm named Bellmore and Nemhauser and a new proposed one named Anedma are used to get as near as possible to the optimal solution for the ordering of coefficients in FFTs with larger number of points. As will be shown, the appropriate ordering of coefficients, based on the guidance given by the Anedma heuristic algorithm, can contribute for the reduction of power consumption of the FFT architectures.

Coherent optical OFDM systems based on the fractional Fourier transform

Abstract: An innovative Orthogonal Frequency Division Multiplexing (OFDM) scheme based on orthogonal chirped subcarriers is proposed. We show that the fractional Fourier transform (FrFT) of the input signal can be electronically implemented with a complexity equivalent to the conventional fast Fourier transform (FFT); on the other hand, the planar device that implements the FrFT in the optical domain is similar to the passive arrayed waveguide grating (AWG) device that performs the FFT. We analyze the spectral efficiency, the peak-to-average power ratio (PAPR) and the frequency offset sensitivity of a FrFT-based optical OFDM system, and make an accurate comparison with the standard FFT-based implementation.

In-place simultaneous prime factor algorithm-based 3780-point discrete Fourier transform processing device and method

Abstract: The invention relates to an in-place simultaneous prime factor algorithm-based 3780-point discrete Fourier transform processing device, which is characterized by comprising 3780 complex share memories Cache 1, a conventional 4-point Winograd Fourier Transform Algorithm (WFTA) module, a conventional 5-point WFTA module, a conventional 7-point WFTA module and a modified 27-point two-dimensional Cooley-Tukey module, wherein the modified 27-point two-dimensional Cooley-Tukey module is used for executing 140 times of 27-point two-dimensional Cooley-Turkey Fourier transformation, sequence of addresses which are read by the modified 27-point two-dimensional Cooley-Tukey module from the data is identical to the address sequence of a written result of the Cachel, and the written address and the read address of the modified 27-point two-dimensional Cooley-Tukey module in Cachel are identical to each other.

A faster fast fourier transform

Abstract: Gilbert Strang, author of the classic textbook Linear Algebra and Its Applications, once referred to the fast Fourier transform, or FFT, as "the most important numerical algorithm in our lifetime." No wonder. The FFT is used to process data throughout today's highly networked, digital world. It allows computers to efficiently calculate the different frequency components in time-varying signals- and also to reconstruct such signals from a set of frequency components. You couldn't log on to a Wi-Fi network or make a call on your cellphone without it. So when some of Strang's MIT colleagues announced in January at the ACM-SIAM Symposium on Discrete Algorithms that they had developed ways of substantially speeding up the calculation of the FFT, lots of people took notice.