Showing papers on "Split-radix FFT algorithm published in 1998"
TL;DR: Based on the (m, N, q)-regular Fourier matrix, a new algorithm is proposed for fast Fourier transform (FFT) of nonuniform (unequally spaced) data with accuracy much better than previously reported results with the same computation complexity.
Abstract: Based on the (m, N, q)-regular Fourier matrix, a new algorithm is proposed for fast Fourier transform (FFT) of nonuniform (unequally spaced) data. Numerical results show that the accuracy of this algorithm is much better than previously reported results with the same computation complexity of O(N log/sub 2/ N). Numerical examples are shown for the applications in computational electromagnetics.
251 citations
11 May 1998
TL;DR: By exploiting the spatial regularity of the new algorithm, minimal requirement for both dominant components in VLSI implementation has been achieved: only 4 complex multipliers and 1024 complex-word data memory for the pipelined 1K FFT processor.
Abstract: The design and implementation of a 1024-point pipeline FFT processor is presented. The architecture is based on a new form of FFT, the radix-2/sup 2/ algorithm. By exploiting the spatial regularity of the new algorithm, minimal requirement for both dominant components in VLSI implementation has been achieved: only 4 complex multipliers and 1024 complex-word data memory for the pipelined 1K FFT processor. The chip has been implement in 0.5 /spl mu/m CMOS technology and takes an area of 40 mm/sup 2/. With 3.3 V power supply, it can compute 2/sup n/, n=0, 1, ..., 10 complex point forward and inverse FFT in real time with up to 30 MHz sampling frequency. The SQNR is above 50 dB for white noise input.
243 citations
13 Sep 1998
TL;DR: A new VLSI-oriented fast Fourier transform (FFT) algorithm-radix-2/4/8, which can effectively minimize the number of complex multiplications and is designed for use in the DVB application in 0.3 V triple-metal CMOS process.
Abstract: In this paper, we present a new VLSI-oriented fast Fourier transform (FFT) algorithm-radix-2/4/8, which can effectively minimize the number of complex multiplications. This algorithm can be implemented efficiently using a pipelined architecture. Based on this pipelined architecture, an 8 K FFT ASIC is designed for use in the DVB (Digital Video Broadcasting) application in 0.6 /spl mu/m-3.3 V triple-metal CMOS process.
111 citations
TL;DR: The fundamentals of Fourier analysis are reviewed with emphasis on the analysis of transient signals, and the human saccade is considered to illustrate the pitfalls and advantages of various Fourier analyses.
57 citations
TL;DR: It is shown that the length- N GDFT can be computed by a split-radix algorithm of discrete Fourier transform (DFT) whose input and output sequences are rotated by twiddle factors.
41 citations
TL;DR: An algorithm called the quick Fourier transform (QFT) is developed that reduces the number of floating-point operations necessary to compute the DFT by a factor of two or four over direct methods or Goertzel's method for prime lengths.
Abstract: This paper looks at an approach that uses symmetric properties of the basis function to remove redundancies in the calculation of the discrete Fourier transform (DFT). We develop an algorithm called the quick Fourier transform (QFT) that reduces the number of floating-point operations necessary to compute the DFT by a factor of two or four over direct methods or Goertzel's method for prime lengths. By further application of the idea to the calculation of a DFT of length-2/sup M/, we construct a new O(NlogN) algorithm, with computational complexities comparable to the Cooley-Tukey algorithm. We show that the power-of-two QFT can be implemented in terms of discrete sine and cosine transforms. The algorithm can be easily modified to compute the DFT with only a subset of either input or output points and reduces by nearly half the number of operations when the data are real.
39 citations
TL;DR: This paper presents a general split-radix algorithm which can flexibly compute the discrete Fourier transforms (DFT) of length q*2/sup m/ where q is an odd integer.
Abstract: This paper presents a general split-radix algorithm which can flexibly compute the discrete Fourier transforms (DFT) of length q*2/sup m/ where q is an odd integer Comparisons with previously reported algorithms show that substantial savings on arithmetic operations can be made Furthermore, a wider range of choices on different sequence lengths is naturally provided
37 citations
TL;DR: A nonuniform inverse fast Fourier transform (NU-IFFT) for non ununiformly sampled data is realised by combining the conjugate-gradient fast Fouriers transform (CG-FFT) method with the newly developed NUFFT algorithms.
Abstract: A nonuniform inverse fast Fourier transform (NU-IFFT) for nonuniformly sampled data is realised by combining the conjugate-gradient fast Fourier transform (CG-FFT) method with the newly developed nonuniform fast Fourier transform (NUFFT) algorithms. An example application of the algorithm in computational electromagnetics is presented.
33 citations
Patent•
21 Jan 1998TL;DR: In this paper, a fast Fourier transform (FFT) processor is constructed using discrete-fracture transform (DFT) butterfly modules having sizes greater than 4 butterflies, and low power, fixed coefficient multipliers are employed to perform nontrivial twiddle factor multiplications in each butterfly module.
Abstract: A fast Fourier transform (FFT) processor is constructed using discrete Fourier transform (DFT) butterfly modules having, in preferred example embodiments, sizes greater than 4. In a first example embodiment, the FFT processor employs size-8 butterflies. In a second example embodiment, the FFT processor employs size-16 butterflies. In addition, low power, fixed coefficient multipliers are employed to perform nontrivial twiddle factor multiplications in each butterfly module. The number of different, nontrivial twiddle factor multipliers is reduced by separating trivial and nontrivial twiddle factors and by taking advantage of twiddle factor symmetries in the complex plane and/or twiddle factor decomposition. In accordance with these and other factors, the present invention permits construction of an FFT processor with minimal power and IC chip surface area consumption.
29 citations
06 Jul 1998
TL;DR: In this paper, the authors proposed an accurate algorithm for the non-uniform forward FFT (NUFFT) based on a new class of matrices, the regular Fourier matrices.
Abstract: Regular fast Fourier transform (FFT) algorithms require uniformly sampled data. In many practical situations, however, the input data is nonuniform, and hence the regular FFT does not apply. To overcome this difficulty the authors have proposed an accurate algorithm for the nonuniform forward FFT (NUFFT) based on a new class of matrices, the regular Fourier matrices. For the nonuniform inverse FFT (NU-IFFT) algorithm, the conjugate-gradient method and the regular FFT algorithm are combined to speed up a matrix inversion. Numerical results show that these algorithms are more than one order of magnitude more accurate than existing algorithms.
24 citations
TL;DR: This paper transforms DFT into a form expressed in discrete moments via a modular mapping and truncating Taylor series expansion and extends the use of the systolic array for fast computation of moments without any multiplications to one that computes DFT with only a few multiplications and without any evaluations of exponential functions.
Patent•
02 Jul 1998TL;DR: In this paper, the authors proposed to reduce the number of complex computations that must be performed in computing the discrete Fourier transform (DFT) and inverse DFT (IDFT) operations using the same computing device.
Abstract: The present invention significantly reduces the number of complex computations that must be performed in computing the discrete Fourier transform (DFT) and inverse DFT (IDFT) operations. In particular, the DFT and IDFT operations are computed using the same computing device. The computation operations are substantially identical for both operations with the exception that for the IDFT operation, the data are complex conjugated before and after processing. Using the same computing device/operations, both DFT and IDFT computations are optimized for maximum efficiency. A common transform process is selectively connected to first and second data processing paths. A DFT operation is performed on an N-point sequence on the first data processing path, and an IDFT operation is performed on an N-point sequence on the second data processing path using the same N-point fast Fourier transform (FFT).
TL;DR: This paper combines the classical prism representation and the Fast Fourier Transform technique for the computation of terrain corrections for a densely sampled 15 × 20 km elevation set in the Alps for which the original FFT series is divergent.
Abstract: Various computational techniques for the numerical implementation of the Newtonian attraction integral of the terrain exist. Two representative methods are the classical prism representation, accurate but tedious, and the Fast Fourier Transform technique which is efficient, but for which certain convergence criteria have to be met. We combine both methods for the computation of terrain corrections for a densely sampled 15 × 20 km elevation set in the Alps for which the original FFT series is divergent. The computation area is divided into two zones. An inner zone around the computational point, where the vertical component of the Newtonian attraction is computed by summing the individual effect of right rectangular prisms. The outer area is treated with the FFT approach, after an appropriate modification of the kernel function. We show that this modified approach has two positive effects. On the one hand, the convergence of the FFT series is regained, which is not always the case when slopes greater than 45° exist in the terrain. Furthermore, it approximates the terrain correction of the prism summation method better than its linear approximation computed by FFT.
TL;DR: The FFT butterfly processor reported here consists of one parallel-parallel multiplier and two adders, capable of computing one butterfly computation every 100 ns thus it can compute a 256-point complex FFT in 102.4 /spl mu/s excluding data input and output processes.
Abstract: This paper describes in detail the design of a CMOS custom fast Fourier transform (FFT) processor for computing a 256-point complex FFT. The FFT is well-suited for real-time spectrum analysis in instrumentation and measurement applications. The FFT butterfly processor reported here consists of one parallel-parallel multiplier and two adders. It is capable of computing one butterfly computation every 100 ns thus it can compute a 256-point complex FFT in 102.4 /spl mu/s excluding data input and output processes.
Patent•
02 Jan 1998
TL;DR: In this paper, a method and apparatus for calculating fast Fourier transforms FFTs are described, and a tensor product based implementation of the FFT of a given size is presented.
Abstract: A method and apparatus for calculating fast Fourier transforms FFTs are disclosed. An FFT of a given size is formatted using tensor product principles for implementation in apparatus or by software such that the same reconfigurable hardware or software can calculate FFTs of any dimension for the selected FFT size. The FFT is factored and presented to a first permutation block (10), then to first computation blocks (12, 14, 16, 18) for computing tensor products of dimensionless Fourier transforms of a relatively small base size and twiddle factors, then to a second permutation block (20), then to second computation blocks (22, 24, 26, 28), and finally to a third permutation block (30). The basic building blocks (10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30) of the circuitry can be reconfigurable for maximizing use-flexibility of the hardware or software.
15 Apr 1998
TL;DR: The goal is to improve size-performance requirements of an FFT core function using modular and hierarchical VHDL description combined with IP-core library elements from Xilinx.
Abstract: Design and implementation of parallel pipelined Fast Fourier Transform (FFT), using Decimation in Frequency (DIF) algorithm on FPGAs is presented. The FFT core for 1024 complex data point is implemented on the X-CIM which is a Re-configurable Acceleration Subsystem (RAS) with a TMS320C4x DSP-processor and two XC4013 FPGA as its processing units. The proposed FFT machine is an alternative to the bit serial-parallel FFT algorithm using Distributed Arithmetic Look Up Table (DALUT) method. The advantage of proposed design is mainly in its cost effective and hardware-efficient parallel implementations of the N-point DFT, offering highly attractive throughput rates in relation to the conventional DSP processors. Moreover, the processor's data-path structure is independent of sampled data-paints, and it has a self-sorting property where the output is in properly ordered form. Our goal is to improve size-performance requirements of an FFT core function using modular and hierarchical VHDL description combined with IP-core library elements from Xilinx.
TL;DR: A logic design, based on RNS units, to perform the N point FFT on a continuous data stream is proposed, and its performance is evaluated in terms of asymptotic VLSI complexity.
TL;DR: In this paper, the authors introduce the continuous-time Fourier transform (CFT), discretetime Fourier Transform (DFT), and Discrete Fourier Transformer (DFT) and present an example to illustrate the relation between CFT and DFT.
Abstract: Fast Fourier transform (FFT) is an efficient algorithm for computing the discrete Fourier transform. The discovery of the FFT algorithm paved the way for widespread use of digital methods of spectrum estimation which influenced the research in almost every field of engineering and science. In this article, we will first introduce the continuous-time Fourier transform (CFT), discretetime Fourier transform and discrete Fourier transform (DFT) and then present an example to illustrate the relation between CFT and DFT. In particular, we bring out the fact that the DFT is a tool to estimate the samples of the CFT at uniformly spaced frequencies. Next, we introduce the FFT algorithm giving certain key steps in its development.
01 Dec 1998
TL;DR: A new method of implementing the fast Fourier transform (FFT), the "real-time FFT algorithm", which efficiently utilizes the computer time to perform the FFT computation while the data acquisition proceeds, is presented.
Abstract: In most electrophysiological signals like EEG (electroencephalograph), frequency characteristics play an important role in quantifying the signals. This paper presents a new method of implementing the fast Fourier transform (FFT), the "real-time FFT algorithm", which efficiently utilizes the computer time to perform the FFT computation while the data acquisition proceeds. The main idea is to build the local butterfly modules using the data points available. The algorithm is based on the decimation-in-time split-radix FFT (DIT sr-FFT) butterfly structure. The algorithm is superior to the conventional whole-block FFT algorithm in synchronizing with the on-line process. The time delay is about 2/r that of the whole-block algorithm considering the FFT size N=2/sup r/.
Journal Article•
16 May 1998
TL;DR: This paper presents a high performance frequency domain filter implementation for a moving window-type processing that has the advantage of being parallel in nature and can be used in various real-time frequency processing, continuous data flow, single or multiple channel applications.
Abstract: This paper presents a high performance frequency domain filter implementation for a moving window-type processing. The computational structure consists of three stages: a sliding discrete Fourier transform (SDFT) for a vectorized updating of the DFT; a frequency domain filter; and a one-point inverse discrete Fourier transform (IDFT). The total computation required for generating one filtered output point is 2/spl times/N multiplications (N is the frequency window length) and 3/spl times/N additions compared to 2/spl times/N/spl times/log/sub 2/N multiplications and additions if using FFT and IFFT. The proposed structure also has the advantage of being parallel in nature and can be used in various real-time frequency processing, continuous data flow, single or multiple channel applications.
TL;DR: A fast algorithm of realizing a method of inverting a long liner convolution based on the procedure of sectionalization combined with effective real-valued split- radix fast Fourier transformation (FFT) algorithm for solving problems of restoration digital signals (images).
Abstract: A fast algorithm of realizing a method of inverting a long liner convolution is presented. It is based on the procedure of sectionalization combined with effective real-valued split- radix fast Fourier transformation (FFT) algorithm for solving problems of restoration digital signals (images). The minimal multiplicative complexity of such algorithm is obtained.
TL;DR: An new algorithm for the fast Fourier transform is proposed that is similar to the conventional algorithm for small transforms, and about three times faster for large transforms.
21 Apr 1998
TL;DR: This work has implemented and benchmarked a 2-dimensional parallel FFT code on the APE100/Quadrics parallel computer, where–due to a rigid next-neighbour connectivity and lack of local addressing–efficient FFT implementations could not be realized so far.
Abstract: We describe a novel practical parallel FFT scheme designed for SIMD systems and/or data parallel programming A bit-exchange of elements between the processors is avoided by means of the ‘Transpose Algorithm’ Our transposition is based on the assignment of the data field onto a 1-dimensional ring of systolic cells which subsequently is mapped onto a ring of processors, realized as a subset of the system's connectivity We have implemented and benchmarked a 2-dimensional parallel FFT code on the APE100/Quadrics parallel computer, where–due to a rigid next-neighbour connectivity and lack of local addressing–efficient FFT implementations could not be realized so far
01 Dec 1998
TL;DR: A new algorithm is obtained that uses 2 n−2(3n−13)+4n−2 real multiplications and 6n+2 real additions for a real data N=2n point DFT, comparable to the number of operations in the Split-Radix method, but with slightly fewer multiply and add operations in total.
Abstract: This paper presents a new fast Discrete Fourier Transform (DFT) algorithm. By rewriting the DFT, a new algorithm is obtained that uses 2n?2(3n?13)+4n?2 real multiplications and 2n?2(7n?29)+6n+2 real additions for a real data N=2n point DFT, comparable to the number of operations in the Split-Radix method, but with slightly fewer multiply and add operations in total. Because of the organization of multiplications as plane rotations in this DFT algorithm, it is possible to apply a pipelined CORDIC algorithm in a hardware implementation of a long-point DFT, e.g., at a 100 MHz input rate, a 1024-point transform can be realized with a 200 MHz clocking of a single CORDIC pipeline.
01 Sep 1998
TL;DR: This paper shows on how the real algorithms for the reduction a modulo arbitrary polynomial and fast Vandermonde transform (FVT) are realized on computer using fast Fourier transform (FFT).
Abstract: This paper shows on how the real algorithms for the reduction a modulo arbitrary polynomial and fast Vandermonde transform (FVT) are realized on computer using fast Fourier transform (FFT). This real-valued FVT algorithm on the developed fast reduction polynomial algorithm is based. The realization of FVT algorithm on computer with real multiplicative complexity O(2Nlog 2 2N) and real additive complexity O(6Nlog 2 2N) is obtained. New FVT algorithm is applied in digital signal, filtering and interpolation problems.
21 Jun 1998
TL;DR: Based on the CG-FFT method and the NUFFT (nonuniform FFT) algorithm, a new non-uniform inverse FFT (NU-IFFT) was developed for nonuniform data with a comparable complexity of O(N log/sub 2/N).
Abstract: Based on the CG-FFT method and the NUFFT (nonuniform FFT) algorithm, a new nonuniform inverse FFT (NU-IFFT) algorithm is developed for nonuniform data. With a comparable complexity of O(N log/sub 2/N), this algorithm is much more accurate than the previously reported results since it is optimal in the least squares sense.
TL;DR: A method for the computation of the Stokes formula using the Fast Hartley Transform CFHT techniques, which shows that the resulting values of geoidal undulations by FHT techniques are almost the same as by FFT techniques, and the computational speed of F HT techniques is about two times faster than that of F FT techniques.
Abstract: This paper presents a method for the computation of the Stokes formula using the Fast Hartley Transform CFHT) techniques. The algorithm is most suitable for the computation of real sequence transform, while the Fast Fourier Transform (FFT) techniques are more suitable for the computa ton of complex sequence transform. A method of spherical coordinate transformation is presented in this paper. By this method the errors, which are due to the approximate term in the convolution of Stokes formula, can be effectively eliminated. Some numerical tests are given. By a comparison with both FFT techniques and numerical integration method, the results show that the resulting values of geoidal undulations by FHT techniques are almost the same as by FFT techniques, and the computational speed of FHT techniques is about two times faster than that of FFT techniques.
TL;DR: The commenter states that the fast Fourier transform (FFT) processor of the aforementioned paper by C.C. Hui et al., contains many interesting and novel features, but it is pointed out that bit reversed input/output FFT algorithms, matrix transposers, and bit reversers have been noted in the literature.
Abstract: For the original paper see ibid., vol. 31, no. 11, p. 1751-61 (Nov. 1996). The commenter states that the fast Fourier transform (FFT) processor of the aforementioned paper by C.C. Hui et al., contains many interesting and novel features. However, it is pointed out that bit reversed input/output FFT algorithms, matrix transposers, and bit reversers have been noted in the literature. In addition, lower radix algorithms can be modified to be made computationally equivalent to higher radix algorithms. Many FFT ideas, including those of the above paper, can also be applied to other important algorithms and architectures.
TL;DR: Improved fast Fourier transform (FFT) spectra are obtained by multiplying a temporal or spatial series by twice the reciprocal length of useful, windowed data as mentioned in this paper, which provides spectral amplitudes that compare with average amplitudes in the original data.
Abstract: Improved fast Fourier transform (FFT) spectra are obtained by multiplying a temporal or spatial series by twice the reciprocal length of useful, windowed data. This scaling provides spectral amplitudes that compare with average amplitudes in the original data. The inverse‐transform scale is the reciprocal of the forward scaling divided by the FFT length. True‐amplitude processing is convenient particularly when evaluating sparse data consisting of few traces and waveform types that are to be filtered or compensated for amplitude attenuations.