Showing papers on "Prime-factor FFT algorithm published in 2011"
TL;DR: This work shows that the fast Fourier transform, so called hyperbolic cross FFT, suffers from an increase of its condition number for both increasing refinement and increasing spatial dimension.
Abstract: A straightforward discretisation of problems in high dimensions often leads to an exponential growth in the number of degrees of freedom. Sparse grid approximations allow for a severe decrease in the number of used Fourier coefficients to represent functions with bounded mixed derivatives and the fast Fourier transform (FFT) has been adapted to this thin discretisation. We show that this so called hyperbolic cross FFT suffers from an increase of its condition number for both increasing refinement and increasing spatial dimension.
73 citations
TL;DR: Fast Fourier transform (FFT) algorithm can be introduced into the calculation of convolution format of gyrator transform in the discrete case by using convolution operation.
65 citations
TL;DR: The interpolated FFT algorithm based on the minimized sidelobe window is considered, and its calculate procedure and formulas are given, which is free of solving high-order equations, making the method a good choice for real-time applications.
Abstract: This paper focuses on the low-computation harmonic-analysis procedure with sufficient suppression of spectral leakage and picket-fence effect. The interpolated FFT algorithm based on the minimized sidelobe window is considered, and its calculate procedure and formulas are given, which is free of solving high-order equations. The implementation of the proposed algorithm in the digital-signal-processor (DSP) based three-phase harmonic ammeter is also introduced. The proposed algorithm has the major advantages that the calculate formulas for harmonic parameters can be easily implemented by hardware multipliers, making the method a good choice for real-time applications. The simulation and application results validate the accuracy and efficiency of the proposed algorithm.
62 citations
TL;DR: A novel decoupling of the least-squares problem is demonstrated which results in two systems of equations, one of which may be solved quickly by means of fast Fourier transforms (FFTs) and another that is demonstrated to be well approximated by a low-rank system.
Abstract: A new algorithm is presented which provides a fast method for the computation of recently developed Fourier continuations (a particular type of Fourier extension method) that yield superalgebraically convergent Fourier series approximations of nonperiodic functions. Previously, the coefficients of an approximating Fourier series have been obtained by means of a regularized singular value decomposition (SVD)-based least-squares solution to an overdetermined linear system of equations. These SVD methods are effective when the size of the system does not become too large, but they quickly become unwieldy as the number of unknowns in the system grows. We demonstrate a novel decoupling of the least-squares problem which results in two systems of equations, one of which may be solved quickly by means of fast Fourier transforms (FFTs) and another that is demonstrated to be well approximated by a low-rank system. Utilizing randomized algorithms, the low-rank system is reduced to a significantly smaller system of equations. This new system is then efficiently solved with drastically reduced computational cost and memory requirements while still benefiting from the advantages of using a regularized SVD. The computational cost of the new algorithm in on the order of the cost of a single FFT multiplied by a slowly increasing factor that grows only logarithmically with the size of the system.
46 citations
TL;DR: 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.
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.
40 citations
TL;DR: The computer simulation results show that the proposed image encryption algorithm is feasible, secure and robust to noise attack and occlusion.
32 citations
TL;DR: A single-path delay commutator processing element (SDC PE) has been proposed for the first time and can lead to 100% hardware utilization and 50% reduction in the overall number of adders required in the conventional pipelined FFT designs.
Abstract: We present a novel pipelined fast Fourier transform (FFT) architecture which is capable of producing the output sequence in normal order A single-path delay commutator processing element (SDC PE) has been proposed for the first time It saves a complex adder compared with the typical radix-2 butterfly unit The new pipelined architecture can be built using the proposed processing element The proposed architecture can lead to 100% hardware utilization and 50% reduction in the overall number of adders required in the conventional pipelined FFT designs In order to produce the output sequence in normal order, we also present a bit reverser, which can achieve a 50% reduction in memory usage
30 citations
13 Oct 2011
TL;DR: The number of possible algorithms for 2n-point FFTs with radix-2 butterfly operation is determined and a simple method to determine the twiddle factor indices for each algorithm based on the binary tree representation is proposed.
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.
27 citations
Proceedings Article•
01 Jan 201123 citations
TL;DR: This paper proposes composite cyclotomic Fourier transforms (CCFTs), which achieve lower overall complexities for moderate to long lengths and the improvement significantly increases as the length grows.
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.
23 citations
TL;DR: “DSP Tips and Tricks” introduces practical design and implementation signal processing algorithms that you may wish to incorporate into your designs.
Abstract: “DSP Tips and Tricks” introduces practical design and implementation signal processing algorithms that you may wish to incorporate into your designs. We welcome readers to submit their contributions.
07 Nov 2011
TL;DR: Theoretical analysis and simulation result showed that while keep the advantages of PMF-FFT algorithm the optimized algorithm had effectively overcame the defects and will serve well in the design of receiver for high-dynamic GPS signal.
Abstract: (Partial Matching Filter)PMF-(Fast Fourier Transformation)FFT algorithm has advantages in acquisition speed and hardware complexity. However in Doppler frequency acquisition there is a contradiction between acquisition accuracy and hardware complexity. In addition the range of frequency acquisition is limited. What's worse in this limited and small range, the peak of output seriously decline with the increasing of Doppler frequency. To solve problems above, first of all this paper analyzed the source of defects, accordingly FFT was replaced by (Discrete Fourier Transformation) DFT to eliminate the contradiction between acquisition accuracy and hardware complexity. Then this paper used the known Doppler frequency got before losing track to switch the acquisition target from Doppler frequency itself to its variation during a shot time through which way the negative effects from limited acquisition range is avoided. Last this paper used a digital divider to dynamically eliminate the factor for attenuation of output peak. Theoretical analysis and simulation result showed that while keep the advantages of PMF-FFT algorithm the optimized algorithm had effectively overcame the defects sothat it will serve well in the design of receiver for high-dynamic GPS signal.
15 May 2011
TL;DR: This paper presents a novel eight-parallel 128/256-point mixed-radix multi-path delay commutator (MRMDC) FFT processor for orthogonal frequency-division multiplexing (OFDM) systems.
Abstract: This paper presents a novel eight-parallel 128/256-point mixed-radix multi-path delay commutator (MRMDC) FFT processor for orthogonal frequency-division multiplexing (OFDM) systems. The proposed FFT architecture can provide a higher throughput rate and low hardware complexity by using an eight-parallel data-path scheme, a multi-path delay commutator structure and an efficient scheduling scheme of complex multiplications. Using the modified radix-4 butterfly unit which can perform one radix-4 butterfly or two radix-2 butterflies, the proposed FFT processor can provide 128 and 256-point FFT computations. The proposed FFT processor has been designed and implemented with the 90nm CMOS technology. The proposed eight-parallel FFT processor can provide a throughput rate of up to 27.5Gsample/s at 430MHz.
23 Sep 2011
TL;DR: In this article, the authors proposed to optimize existing FFT algorithms for low-cost FPGA implementations by using short length structures to obtain higher length transforms, which can help the designer of electronic circuits to develop and to optimize their own IP in a reasonable time.
Abstract: In this paper, our objective is to detail know-how and techniques that can help the designer of electronic circuits to develop and to optimize their own IP in a reasonable time. For this reason, we propose to optimize existing FFT algorithms for low-cost FPGA implementations. For that, we have used short length structures to obtain higher length transforms. Indeed, we can obtain a VLSI structure by using log 4 (N) 4-point FFTs to construct N-point FFT rather than (N/8) log 8 (N) 8-point FFTs. Furthermore, two techniques are used to yield with VLSI architecture. Firstly, the radix-4 FFT is modified to process one sample per clock cycle. Secondly, the memory is shared and divided into 4 parts to reduce the consumed resources and to improve the overall latency. Comparisons with commercial IP cores show that the low area architecture presents the best compromise in terms of speed/area.
01 Oct 2011
TL;DR: The preliminary experimental results show that DTT has the potential to be a simpler and faster transformation for speech recognition, and an approach based on discrete orthonormal Tchebichef polynomials to analyze a vowel and a consonant in spectral frequency is proposed.
Abstract: Speech recognition is still a growing field It carries strong potential in the near future as computing power grows Spectrum analysis is an elementary operation in speech recognition Fast Fourier Transform (FFT) is the traditional technique to analyze frequency spectrum of the signal in speech recognition Speech recognition operation requires heavy computation due to large samples per window In addition, FFT consists of complex field computing This paper proposes an approach based on discrete orthonormal Tchebichef polynomials to analyze a vowel and a consonant in spectral frequency for speech recognition The Discrete Tchebichef Transform (DTT) is used instead of popular FFT The preliminary experimental results show that DTT has the potential to be a simpler and faster transformation for speech recognition
17 Oct 2011
TL;DR: Results show that multigrid-based methods begin to outperform FFT-based ones for N∼103 and the constant in front of their asymptotic complexity estimate is larger and their accuracy-efficiency tradeoffs are different.
Abstract: The effectiveness of multigrid and fast Fourier transform (FFT) based methods are investigated for accelerating the solution of volume integral equations encountered in bioelectromagnetics (BIOEM) analysis. The typical BIOEM simulation is in the mixed-frequency regime of analysis because the field variations in the simulation domain are dictated by a combination of the free space wavelength, geometrical features, and the wavelengths/skin depths in tissues. In this case, multigrid-based methods (when appropriately truncated at high-frequency levels) can achieve O(N) complexity that is asymptotically superior to the O(NlogN) complexity of FFT-based ones. Nevertheless, the constant in front of their asymptotic complexity estimate is larger and their accuracy-efficiency tradeoffs are different. Numerical experiments are performed to compare these methods and the results show that multigrid-based methods begin to outperform FFT-based ones for N∼103.
22 May 2011
TL;DR: The development of a truly centered DFT is coupled with a method for computing the Centered DFT to provide an FFT that requires no complex multiplications and which allows a highly parallel implementation.
Abstract: This paper describes a novel method for the computation of the Discrete Fourier Transform (DFT). The development of a truly centered DFT is coupled with a method for computing the Centered DFT to provide an FFT that requires no complex multiplications and which allows a highly parallel implementation.
07 Apr 2011
TL;DR: An optimized implementation of the 8-point FFT processor with radix-2 algorithm in R2MDC architecture is presented, showing that this module significantly achieves a better performance with lower resource usage.
Abstract: The Fast Fourier Transform (FFT) and its inverse transform (IFFT) processor are key components in many communication systems. An optimized implementation of the 8-point FFT processor with radix-2 algorithm in R2MDC architecture is presented in this paper. The butterfly — Processing Element (PE) used in the 8-FFT processor reduces the multiplicative complexity by using a real constant multiplication in one method and eliminates the multiplicative complexity by using add and shift operations in other proposed method. The pipeline architecture R2MDC has been implemented with the 8-point module and simulation results show that this module significantly achieves a better performance with lower resource usage.
TL;DR: The results show that the implementation by Radix-4 FFT is simple, the effect is ideal and lower time-consuming, and the feasibility and the advantage of Fourier transform for image compression is discussed.
Abstract: Image compression is a crucial step in image processing area. Image Fourier transforms is the classical algorithm which can convert image from spatial domain to frequency domain. Because of its good concentrative property with transform energy, Fourier transform has been widely applied in image coding, image segmentation, image reconstruction. This paper adopts Radix-4 Fast Fourier transform (Radix-4 FFT) to realize the limit distortion for image coding, and to discuss the feasibility and the advantage of Fourier transform for image compression. It aims to deal with the existing complex and time-consuming of Fourier transform, according to the symmetric conjugate of the image by Fourier transform to reduce data storage and computing complexity. Using Radix-4 FFT can also reduce algorithm time-consuming, it designs three different compression requirements of non-uniform quantification tables for different demands of image quality and compression ratio. Take the standard image Lena as experimental data using the presented method, the results show that the implementation by Radix-4 FFT is simple, the effect is ideal and lower time-consuming.
25 Jul 2011
TL;DR: This paper presents an efficient analysis of MSE as well as an optimization algorithm for CORDIC-based FFT units, which is applicable to other Linear-Time-Invariant (LTI) circuits as well.
Abstract: Fixed-point Fast Fourier Transform (FFT) units are widely used in digital communication systems. The twiddle multipliers required for realizing large FFTs are typically implemented with the Coordinate Rotation Digital Computer (CORDIC) algorithm to restrict memory requirements. Recent approaches aiming to optimize the bit-widths of FFT units while satisfying a given maximum bound on Mean-Square-Error (MSE) mostly focus on the architectures with integer multipliers. They ignore the quantization error of coefficients, disabling them to analyze the exact error defined as the difference between the fixed-point circuit and the reference floating-point model. This paper presents an efficient analysis of MSE as well as an optimization algorithm for CORDIC-based FFT units, which is applicable to other Linear-Time-Invariant (LTI) circuits as well.
12 Dec 2011
TL;DR: This paper proposes a novel image encryption algorithm based on the discrete fractional Fourier transform and an improved magic cube rotation scrambling algorithm that can achieve double image encryption in the time-frequency domain.
Abstract: This paper proposes a novel image encryption algorithm based on the discrete fractional Fourier transform and an improved magic cube rotation scrambling algorithm. Through fractional Fourier transform and position scrambling, the proposed algorithm can achieve double image encryption in the time-frequency domain. Compared the encrypted images and decrypted images, the proposed image encryption algorithm has better performance than only using fractional Fourier transform.
TL;DR: The spline-interpolation-based fast Fourier transform (FFT) algorithm, designated as the SFFT algorithm, is proposed in the present paper to further enhance the computational speed of simulating the multivariate stochastic processes.
Abstract: The spline-interpolation-based fast Fourier transform (FFT) algorithm, designated as the SFFT algorithm, is proposed in the present paper to further enhance the computational speed of simulating the multivariate stochastic processes. The proposed SFFT algorithm first introduces the spline interpolation technique to reduce the number of the Cholesky decomposition of a spectral density matrix and subsequently uses the FFT algorithm to further enhance the computational speed. In order to highlight the superiority of the SFFT algorithm, the simulations of the multivariate stationary longitudinal wind velocity fluctuations have been carried out, respectively, with resorting to the SFFT-based and FFT-based spectral representation SR methods, taking into consideration that the elements of cross-power spectral density matrix are the complex values. The numerical simulation results show that though introducing the spline interpolation approximation in decomposing the cross-power spectral density matrix, the SFFT algorithm can achieve the results without a loss of precision with reference to the FFT algorithm. In comparison with the FFT algorithm, the SFFT algorithm provides much higher computational efficiency. Likewise, the superiority of the SFFT algorithm is becoming more remarkable with the dividing number of frequency, the number of samples, and the time length of samples going up.
TL;DR: In this paper, the authors present a method of estimation of frequency groups with 200-Hz bandwidth in the frequency range from the 50th harmonic up to 9 kHz, which consists of the application of a fast Fourier transform (FFT) for wavelet coefficients after input signal decomposition.
TL;DR: It is shown that the paired-transform-based algorithm of the FFT is faster than the radix-2 FFT; consequently, it is useful for higher sampling rates.
Abstract: Frequency analysis plays a vital role in the applications like cryptanalysis, steganalysis, system identification, controller tuning, speech recognition, noise filters, etc. Discrete Fourier transform (DFT) is a principal mathematical method for the frequency analysis. The way of splitting the DFT gives out various fast algorithms. In this paper, we present the implementation of two fast algorithms for the DFT for evaluating their performance. One of them is the popular radix-2 Cooley-Tukey fast Fourier transform (FFT) algorithm and the other one is the Grigoryan FFT based on the splitting by the paired transform. We evaluate the performance of these algorithms by implementing them on the TMS320C5416 DSP and also on the Virtex-II FPGAs. Finally, we show that the paired-transform-based algorithm of the FFT is faster than the radix-2 FFT; consequently, it is useful for higher sampling rates. We also discuss the performances of TMS DSP and Xilinx FPGAs and tradeoffs.
06 Jul 2011
TL;DR: A new harmonic analysis algorithm based on interpolating windowed FFT and wavelet transform that can not only analyze the steady-state harmonic accurately, but also inspect non-steady harmonics effectively is proposed.
Abstract: Harmonics in power system are harmful to power network. In order to analyze power system harmonic more accurately, a new harmonic analysis algorithm based on interpolating windowed FFT and wavelet transform is proposed. This approach overcomes the shortcoming of FFT having frequency domain localization ability, but without time-domain localization ability. In this paper, the original signal is decomposed into the high-frequency component and the low-frequency component by the discrete wavelet transform. The low-frequency part is analyzed by interpolating windowed FFT algorithm to get the amplitude and phase of every steady harmonic. Simulation experiments are conducted with Matlab. The experimental results show that this method can not only analyze the steady-state harmonic accurately, but also inspect non-steady harmonics effectively.
Patent•
12 Oct 2011
TL;DR: In this paper, a processor consisting of a top layer, an intermediate layer, and a bottom layer was proposed, where the top layer resolved 3780-point by using a mixed-based number algorithm, the intermediate layer resolved 63-point and 60-point FFT using a prime factor algorithm, and the bottom layer complete 7-point, 9-point.
Abstract: The invention relates to a method for realizing 3780-point fast Fourier transform/inverse fast Fourier transform (FFT/IFFT) and a processor thereof. The processor consists of a top layer, an intermediate layer and a bottom layer, wherein the top layer resolves 3780-point by using a mixed based number algorithm, the intermediate layer resolves 63-point and 60-point FFT by using a prime factor algorithm, and the bottom layer complete 7-point, 9-point, 3-point, 4-point and 5-point FFT calculation by using a winograd fourier transform algorithm (WFTA) algorithm. The method realizes 3780-point FFT by combining the mixed base number algorithm, the prime factor algorithm and the WFTA algorithm, avoids errors caused by calculating 4096-point by using an interpolation method, and reduces rotation factors and a chaotic unit in the mixed base number algorithm. Furthermore, an index structure completed by a multiplex memory in the design has a simple circuit, is easy to realize, and can save chip resources.
TL;DR: This paper extends the discontinuous fast Fourier transform (DFFT) algorithm to deal with the two dimensional function with a discontinuous boundary of arbitrary shape and discretizes the support domain of the function by triangle mesh, which reduces the stair-casing error of an orthogonal grid required by FFT.
Abstract: In computational electromagnetics and other areas of computational science, Fourier transforms of discontinuous functions are frequently encountered This paper extends the discontinuous fast Fourier transform (DFFT) algorithm which was presented previously by Fan and Liu to deal with the two dimensional (2-D) function with a discontinuous boundary of arbitrary shape First, the proposed algorithm discretizes the support domain of the function by triangle mesh, which reduces the stair-casing error of an orthogonal grid required by FFT Second, the algorithm adopts the basic idea of double interpolation used by the original 1-D DFFT algorithm in the literature, but with a significant modification that the nonuniform fast Fourier transform (NUFFT) with the least square error (LSE) interpolation other than a Lagrange interpolation is used to process nonuniformly spaced samples of the exponentials The proposed 2-D DFFT algorithms obtain much higher accuracy than the conventional 2-D FFT for the discontinuous
22 May 2011
TL;DR: A method for reducing the number of multiplications and additions is proposed, making it possible to noticeably improve the arithmetic complexity as compared with the existing FHT algorithms.
Abstract: In this paper, a new fast Hartley transform (FHT) algorithm-radix-22 suitable for pipeline implementation of the discrete Hartley transform (DHT) is presented. The proposed algorithm is developed by integrating two stages of the twiddle factor decomposition together into single butterfly, and applying the multidimensional index mapping technique. Radix-22 algorithm achieves at the same time both a simple and regular butterfly structure as a radix-2 algorithm and a reduced number of twiddle factor multiplication provided by a radix-4 algorithm and, unlike radix-4, can be applied to any transform length that is power-of-two with simple bit reversing for ordering the output sequence. The algorithm performance is analyzed and the number of multiplications and additions are calculated. Furthermore, a method for reducing the number of multiplications and additions is proposed, making it possible to noticeably improve the arithmetic complexity as compared with the existing FHT algorithms.
01 Dec 2011
TL;DR: The proposed architecture is compared to an implemented behavioral, unrestricted architecture synthesized using the CADENCE Encounter RTL Compiler for the UMC130nm technology and results show reductions up to 31% in area and 15% in power when using the proposed solution.
Abstract: This paper reports the optimization of area and power for a 32-point radix-2 hybrid FFT (Fast Fourier Transform). The strategy consists of using the Constant Matrix Multiplication (CMM) method along the stages of the 8-point FFT architecture, which is implemented with Carry Save Adders (CSA). The use of CMM at gate level enables the replacement of the multiplication operations by addition/subtractions and shifts for each stage of the real and imaginary parts of the butterflies. The 32-point FFT is obtained through the composition of the optimized 8-point FFT modules. The partial decomposition of coefficients allows the computation of all coefficients necessary for the 32-point through a control unit. We have compared our proposed architecture to an implemented behavioral, unrestricted architecture synthesized using the CADENCE Encounter RTL Compiler for the UMC130nm technology. The results show reductions up to 31% in area and 15% in power when using our proposed solution.
03 Nov 2011
TL;DR: Comparisons of the computational complexity for the proposed split-radix FFT pruning algorithm with time shift for consecutive partial inputs with other algorithms show that the proposed method is more computationally efficient.
Abstract: Eliminating computations on zeros when the number of nonzero inputs is considerably less than the length of Fast Fourier Transform (FFT) is considered as one of the methods to increase the computational efficiency of an FFT algorithm. This paper proposes a new split-radix FFT pruning algorithm with time shift for consecutive partial inputs. The shifting simplifies the flow graph in the first few stages of the pruning algorithm and makes the algorithm architecturally efficient. Comparisons of the computational complexity for the proposed split-radix FFT pruning algorithm with other algorithms show that the proposed method is more computationally efficient. Optimized hardware design based on this algorithm is also devised.