Showing papers on "Prime-factor FFT algorithm published in 1994"
TL;DR: Convolution theorems generalizing well known and useful results from the abelian case are used to develop a sampling theorem on the sphere, which reduces the calculation of Fourier transforms and convolutions of band-limited functions to discrete computations.
937 citations
TL;DR: The “fractional Fourier transform,” previously developed by the authors, is applied to this problem with a substantial savings in computation.
Abstract: The fast Fourier transform (FFT) is often used to compute numerical approximations to continuous Fourier and Laplace transforms. However, a straightforward application of the FFT to these problems often requires a large FFT to be performed, even though most of the input data to this FFT may be zero and only a small fraction of the output data may be of interest. In this note, the “fractional Fourier transform,” previously developed by the authors, is applied to this problem with a substantial savings in computation.
141 citations
TL;DR: A simple FFT-based algorithm for spectrum estimation using a single pass through the FFT is presented and is certainly better than the single pass FFT in separating closely spaced sinusoids.
Abstract: A simple FFT-based algorithm for spectrum estimation is presented. The major difference between this and spectrum estimation using a single pass through the FFT is that the proposed algorithm is iterative and the FFT is used many times in a systematic may to search for individual spectral lines. Using simulated data, the proposed algorithm is able to detect mulitple sinusoids in additive noise. The algorithm is certainly better than the single pass FFT in separating closely spaced sinusoids. Finally the algorithm is applied to some experimental measurements to illustrate its properties. >
123 citations
TL;DR: This modification of Temperton's (1991) self-sorting, in-place radix-p FFT algorithm reduces the required temporary working space from order of p/sup 2/ to p+1, providing a better match to the limited number of registers in a CPU.
Abstract: Presents a modification of Temperton's (1991) self-sorting, in-place radix-p FFT algorithm. This modification reduces the required temporary working space from order of p/sup 2/ to p+1, providing a better match to the limited number of registers in a CPU. >
50 citations
19 Apr 1994
TL;DR: A new fast Fourier transform algorithm, decimation-in-time-frequency (DITF) FFT algorithm, which reduces the number of real multiplications and additions, and is extended to radix-R FFT as well as the multidimensional F FT algorithm using the vector-radix FFT.
Abstract: A new fast Fourier transform algorithm is presented. The decimation-in-time (DIT) and the decimation-in-frequency (DIF) FFT algorithms are combined to introduce a new FFT algorithm, decimation-in-time-frequency (DITF) FFT algorithm, which reduces the number of real multiplications and additions. The DITF FFT algorithm reduces the arithmetic complexity while using the same computational structure as the conventional Cooley-Tukey (CT) FFT algorithm. The algorithm is extended to radix-R FFT as well as the multidimensional FFT algorithm using the vector-radix FFT. >
47 citations
IBM1
TL;DR: It is shown that the multi-dimensional formulation of the proposed FFT algorithm helps in reducing the communication costs and also improves the single node performance by effectively utilizing the memory system of the node.
Abstract: Proposes a parallel high-performance fast Fourier transform (FFT) algorithm based on a multi-dimensional formulation. We use this to solve a commonly encountered FFT based kernel on a distributed memory parallel machine, the IBM scalable parallel system, SP1. The kernel requires a forward FFT computation of an input sequence, multiplication of the transformed data by a coefficient array, and finally an inverse FFT computation of the resultant data. We show that the multi-dimensional formulation helps in reducing the communication costs and also improves the single node performance by effectively utilizing the memory system of the node. We implemented this kernel on the IBM SP1 and observed a performance of 1.25 GFLOPS on a 64-node machine. >
45 citations
TL;DR: A new prime-factor algorithm for the DCT, which deals with general (N/sub 1//spl times/ N/sub 2/)-point DCTs, is presented and the trade-off between cost and performance is illustrated.
Abstract: The prime-factor decomposition is a fast computational technique for many important digital signal processing operations, such as the convolution, the discrete Fourier transform, the discrete Hartley transform, and the discrete cosine transform (DCT). The authors present a new prime-factor algorithm for the DCT. They also design a prime-factor algorithm for the discrete sine transform based on the prime-factor DCT algorithm. Hardware implementations for the prime-factor DCT are also studied. They are especially interested in the hardware designs which are suitable for the VLSI implementations. They show three hardware designs for the prime-factor DCT, including a VLSI circuit fabricated directly according to the signal-flow graph, a linear systolic array, and a mesh-connected systolic array. These three designs show the trade-off between cost and performance. The methodology, which deals with general (N/sub 1//spl times/ N/sub 2/)-point DCTs, where N/sub 1/ and N/sub 2/ are mutually prime, is illustrated by converting a 15-point DCT problem into a (3/spl times/5)-point 2D DCT problem. >
34 citations
TL;DR: A new algorithm for fast Fourier transforms that features uniformly long vector lengths and stride one data access and is well adapted to modern vector computers like the Fujitsu VP2200 having several floating point pipelines per CPU and very fast stride oneData access.
Abstract: We propose a new algorithm for fast Fourier transforms. This
algorithm features uniformly long vector lengths and stride one
data access. Thus it is well adapted to modern vector
computers like the Fujitsu VP2200 having several floating point
pipelines per CPU and very fast stride one data access. It also
has favorable properties for distributed memory computers as
all communication is gathered together in one step. The
algorithm has been implemented on the Fujitsu VP2200 using the
basic subroutines for fast Fourier transforms discussed
elsewhere.
We develop the theory of index digit permutations to some
extent. With this theory we can derive the splitting formulas
for almost all mixed-radix FFT algorithms known so far. This
framework enables us to prove these algorithms but also to
derive our new algorithm. The development and systematic use of
this framework is new and allows us to simplify the
proofs which are now reduced to the application of matrix
recursions.
29 citations
TL;DR: The authors present an algorithm that calculates the minimal polynomial of s, assuming that a period of s is known, and generalises both the discrete Fourier transform and the Games-Chan algorithm.
Abstract: Let s be a periodic sequence whose elements lie in a finite field. The authors present an algorithm that calculates the minimal polynomial of s, assuming that a period of s is known. The algorithm generalises both the discrete Fourier transform and the Games-Chan algorithm. >
29 citations
19 Apr 1994
TL;DR: An algorithm is developed, called the quick Fourier transform (QFT), that will reduce 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 will look at an approach that uses symmetric properties of the basis function to remove redundancies in the calculation of discrete Fourier transform (DFT). We will develop an algorithm, called the quick Fourier transform (QFT), that will reduce 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. Further by applying the idea to the calculation of a DFT of length-2/sup M/, we construct a new O(N log N) algorithm. The algorithm can be easily modified to compute the DFT with only a subset of input points, and it will significantly reduce the number of operations when the data are real. The simple structure of the algorithm and the fact that it is well suited for DFTs on real data should lead to efficient implementations and to a wide range of applications. >
26 citations
TL;DR: A new set of input and output index mappings for DST is presented based on the Lee algorithm for the discrete cosine transform and from the relations obtained between DST and DCT.
Abstract: Presents a fast algorithm for computing the discrete sine transform (DST) when the transform size N is decomposable into mutually prime factors. Based on the Lee algorithm for the discrete cosine transform (DCT) and from the relations obtained between DST and DCT, a new set of input and output index mappings for DST is presented. >
TL;DR: This paper proposes a bit-reversal algorithm that reduces the computational effort to an extent that it becomes negligible compared with the data swapping operation for which the bit- reversal is required.
Abstract: The necessity for an efficient bit-reversal routine in the implementation of fast discrete Fourier transform algorithms is well known. In this paper, we propose a bit-reversal algorithm that reduces the computational effort to an extent that it becomes negligible compared with the data swapping operation for which the bit-reversal is required. >
01 Mar 1994
TL;DR: It is shown that these basefield transforms can be viewed as "projections" of the discrete Fourier transform (DFT) and that many of the existing real Hartley algorithms are projections of well-known FFT algorithms.
Abstract: We present a general framework for constructing transforms in the field of the input which have a convolution-like property. The construction is carried out over the reals, but is shown to be valid over more general fields. We show that these basefield transforms can be viewed as "projections" of the discrete Fourier transform (DFT). Furthermore, by imposing an additional condition on the projections, one may obtain self-inverse versions of the basefield transforms. Applying the theory to the real and complex fields, we show that the projection of the complex DFT results in the discrete combinational Fourier transform (DCFT) and that the imposition of the self-inverse condition on the DCFT yields the discrete Hartley transform (DHT). Additionally, we show that the method of projection may be used to derive efficient basefield transform algorithms by projecting standard FFT algorithms from the extension field to the basefield. Using such an approach, we show that many of the existing real Hartley algorithms are projections of well-known FFT algorithms. >
30 May 1994
TL;DR: In this paper the sliding implementation of the other useful transforms, that can also be implemented with the order of N complexity, are worked out in detail.
Abstract: Implementation of the transform domain adaptive filters is addressed. Recent results have shown that if the input data to a radix-2 fast Fourier transform (FFT) structure is sliding one sample at a time, only N-1 butterflies need to be calculated for updating the FFT structure, after the arrival of every new data sample. This is opposed to most of the previous reports that, assume order of N log N complexity, for such implementation. In this paper the sliding implementation of the other useful transforms, that can also be implemented with the order of N complexity, are worked out in detail. >
TL;DR: It is shown that the discrete short-time Fourier transform with temporal decimation (DSTFT-TD) can be evaluated using a variety of pruned FFT structures and further computational savings can be achieved by combining overlap pruning with classical frequency pruning.
Abstract: We show that the discrete short-time Fourier transform with temporal decimation (DSTFT-TD) can be evaluated using a variety of pruned FFT structures. A pruning method we refer to as overlap pruning can be used to eliminate computational overlap between consecutive FFT's for computing slices of the DSTFT-TD. When only a limited frequency range of the DSTFT-TD is of interest, further computational savings can be achieved by combining overlap pruning with classical frequency pruning. We evaluate the complexity of the overlap and frequency pruned FFT's for the DSTFT-TD in terms of the number of complex multiplications and additions required for the computation of each DSTFT-TD slice. >
09 Nov 1994
TL;DR: Vector-Radix-algorithms which decimate and transform a 2D data set simultaneous for both index directions and therefore seem suitable for parallelization are concentrated on.
Abstract: Dynamic development in digital signal processing is inseparably bound to the disclosure of the fast Fourier transform (FFT). Implications from the application of these efficient algorithms for calculating the discrete (inverse) Fourier transform are significant in many ways. Applicability of FFT algorithms ranges far into almost every aspect of physics and performs a central role in analysis, design and implementation of DSP algorithms and digital systems. Consumed computer time almost ceases to be a problem when using FFT compared with straightforward discrete Fourier transform (DFT). The cutdown on consumed computer time by usage of FFT algorithms even holds greater promise for multidimensional applications with in general more complex tasks and heavier data loads to cope with. Without multidimensional FFT algorithms for high speed convolution or spectral analysis the successes for example in SAR, tomography, data compression or picture processing could not have been achieved. Since the introduction of the Cooley-Tukey-algorithm in 1965 methods to calculate the two- or N dimensional Fourier transform of a set of data are based essentially on the separability of the 2D FFT. With a 1D FFT algorithm the data set is `combed' row- and columnwise to form the 2D transform of the calculated 1D transforms. After some basics and recalling some different conventional approaches to 1D and 2D Fourier transform the paper concentrates on Vector-Radix-algorithms which decimate and transform a 2D data set simultaneous for both index directions and therefore seem suitable for parallelization. Vector-Radix-approaches are derived for general radices and for the 2D case also for nonquadratic data sets.© (1994) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
TL;DR: The Fast Transversal Filter (ftf) algorithm as mentioned in this paper is based on the Fast Fourier Transform (FFT) and is used for recursive least squares (rls) adaptive filtering.
Abstract: We present a new fast algorithm for Recursive Least-Squares(rls) adaptive filtering that uses displacement structure and subsampled updating. Thefsu ftf algorithm is based on the Fast Transversal Filter(ftf) algorithm, which exploits the shift invariance that is present in therls adaptation of afir filter. Theftf algorithm is in essence the application of a rotation matrix to a set of filters and in that respect resembles the Levinson algorithm. In the subsampled updating approach, we accumulate the rotation matrices over some time interval before applying them to the filters. It turns out that the successive rotation matrices themselves can be obtained from a Schur type algorithm which, once properly initialized, does not require inner products. The various convolutions that thus appear in the algorithm are done using the Fast Fourier Transform(fft). For relatively long filters, the computational complexity of the new algorithm is smaller than the one of the well-known lms algorithm, rendering it especially suitable for applications such as acoustic echo cancellation.
31 Oct 1994
TL;DR: Fast Fourier transform (FFT) arrays with built-in error correction are proposed, and a time shared TMR scheme is used to achieve the error correcting capability.
Abstract: Fast Fourier transform (FFT) arrays with built-in error correction are proposed. A time shared TMR scheme is used to achieve the error correcting capability. A quarter of the original FFT array is triplicated and voted in each stage. Therefore the hardware complexity of the error correcting FFT array is a little more than 75% of the original FFT array. This is significant since the error correcting design is smaller than the original. The price for this hardware reduction is that the delay time increases by a factor of 4. However, the throughput penalty can be minimized by pipelining. A technology-independent gate-level analysis of hardware complexity and delay time is included. >
Journal Article•
TL;DR: In this paper, a new fast Fourier transform (FFT)-based algorithm was developed, suitable for spectral analysis of sparsely sampled data with a relatively small number of oscillatory components buried in background noise.
Abstract: Temporal gaps in discrete sampling sequences produce spurious Fourier components at the intermodulation frequencies of an oscillatory signal and the temporal gaps, thus signiflcantly complicating spectral analysis of such sparsely sampled data. A new fast Fourier transform (FFT)-based algorithm has been developed, suitable for spectral analysis of sparsely sampled data with a relatively small number of oscillatory components buried in background noise. The algorithm’s principal idea has its origin in the so-called \clean" algorithm used to sharpen images of scenes corrupted by atmospheric and sensor aperture efiects. It identifles as the signal’s \true" frequency that oscillatory component which, when passed through the same sampling sequence as the original data, produces a Fourier image that is the best match to the original Fourier map. Unlike the clean algorithm, it performs the search in the Fourier space. The algorithm has generally met with success on trials with simulated data with a low signal-to-noise ratio, including those of a type similar to hourly residuals for Earth orientation parameters extracted from VLBI data. For eight oscillatory components in the diurnal and semidiurnal bands, all components with an amplitude-noise ratio greater than 0.2 were successfully extracted for all sequences and duty cycles (greater than 0.1) tested; the amplitude-noise ratios of the extracted signals were as low as 0.05 for high duty cycles and long sampling sequences. When, in addition to these high frequencies, strong low-frequency components are present in the data, the low-frequency components are generally eliminated flrst, by employing a version of the algorithm that searches for noninteger multiples of the discrete FFT minimum frequency.
TL;DR: This algorithm applies a 2-D matrix factorization technique in a2-D space and offers a way to do 1-D FFT in both dimensions simultaneously and can be extended toM-D cases forM>2.
Abstract: A new 2-D FFT algorithm is described. This algorithm applies a 2-D matrix factorization technique in a 2-D space and offers a way to do 2-D FFT in both dimensions simultaneously. The computation is greatly reduced compared to traditional algorithms. This will improve the realization of a 2-D FFT on any kind of computer. However its good parallelism will especially benefit an implementation on a computer with hypercube architecture. A good arrangement of parallel processors will save a great deal of running time. Furthermore this algorithm can be extended toM-D cases forM>2.
08 Aug 1994
TL;DR: A new algorithm based on a non standard Fourier transform which allows the exact implementation of the Stolt change of variables to be implemented in synthetic aperture radar data focusing.
Abstract: Synthetic aperture radar (SAR) data focusing can be carried out by filtering the raw signal in the two-dimensional Fourier domain and by applying a change of variables often referred to as Stolt interpolation. The basic idea of the paper is to present a new algorithm based on a non standard Fourier transform which allows the exact implementation of the Stolt change of variables. Efficiency of the presented procedure is due to the possibility of applying fast Fourier transform (FFT) codes. Moreover the algorithm doesn't require any modification if the data are already range compressed and also if the SAR system transmits a signal different from a linear FM chirp. >
TL;DR: This paper evaluates the implementation of a PFFT on a network of T800 series transputers connected in the form of a linear pipeline and a binary cube and analyzes the speedup obtained, taking into account both computation load and communication overhead.
Abstract: Fast Fourier Transforms (FFTs) are frequently employed in various applications such as image processing and speech recognition. Though FFT calculations can be speeded up considerably, real time processing requirements are well above that of modern day uniprocessor systems. Computing power can be substantially increased through the exploitation of the inherent parallelism available in FFT calculations. However, experimental performance analysis of the Parallel FFT (PFFT) algorithm has not been sufficiently investigated in a loosely coupled multiprocessor environment. In this paper, we evaluate the implementation of a PFFT on a network of T800 series transputers connected in the form of a linear pipeline and a binary cube. We analyse the speedup obtained, taking into account both computation load and communication overhead. A new load balancing algorithm has been incorporated so that load balancing takes into account both computation and communication loads. Realistic performance figures obtained through ac...
20 Mar 1994
TL;DR: Using the properties of the serial operators, the authors propose a way to approach the first without going too far from the second and propose an architecture to implement it in VLSI.
Abstract: The Fourier transform is very used for its properties and because the fast fourier Transform (FFT) algorithm has allowed speeding up computations. Each step of it introduces an error caused first by the quantization of the sine and cosine coefficients, second by the necessarily limited size-increase of the results. The roundoff-phenomenon has much more important effects than the coefficient-imprecision. The arithmetic unit can treat numbers either sufficiently large to maximize the accuracy or smaller to minimize the area, in the case of parallel operators, or the computation time, in the case of serial operators. Using the properties of the serial operators, the authors propose a way to approach the first without going too far from the second and propose an architecture to implement it in VLSI. >
06 Sep 1994
TL;DR: The algorithm considered is the radix r self-sorting algorithm which does not require additional data reordering stages (digit-reversal) as this process is inherently carried out during the execution of the algorithm.
Abstract: In this work we present a study of the vectorization of the fast Fourier transform. The algorithm we have considered is the radix r self-sorting algorithm which does not require additional data reordering stages (digit-reversal) as this process is inherently carried out during the execution of the algorithm. For obtainig the vectorized version of the algorithm we employ a formulation of the FFT in terms of an operator string. Each of the operators represents an operation over the data flow of the algorithm and will have a direct implementation on the vectorial processor. The algorithm thus obtained has been implemented on the Fujitsu VP-2400/10 vector computer, resulting in reduced execution times.
TL;DR: A variable order method for the fast and accurate computation of the Fourier transform is presented and the increase in accuracy is achieved by applying corrections to the trapezoidal sum approximations obtained by the FFT method.
Abstract: In this paper, a variable order method for the fast and accurate computation of the Fourier transform is presented. The increase in accuracy is achieved by applying corrections to the trapezoidal sum approximations obtained by the FFT method. It is shown that the additional computational work involved is of orderK(2m+2), wherem is a small integer andK≤n. Analytical expressions for the associated error is also given.
TL;DR: This work uses visual analysis to describe the EEG contents in terms of frequencies, amplitudes, phases, and unique waveforms as they evolve over time to explore the relationship between the visual analysis of the EEG and the FFT.
Abstract: .There are several methods available to neurophysiologic technologists for the analysis of EEGs. Visual analysis is a technique that is comfortable and familiar to us. We use it to describe the EEG contents in terms of frequencies, amplitudes, phases, and unique waveforms as they evolve over time. The compressed spectral array (CSA) and the fast Fourier transform (FFT) are alternative analysis techniques that are becoming more common for monitoring in intensive care units and in the operating room. Additionally, with the advent of digital EEG, FFTs are available on some newer instruments. My own problems in trying to comprehend the relationship between the visual analysis of the EEG and the FFT has been my inability to get my hands on the FFT in a practical sense. I was comfortable with the general concept of transforms, but I really wanted to make small models for myself to explore the FFT. In the process of searching for methods of making small models, I began to read about the discrete Fourier ...
25 Oct 1994
TL;DR: The Battle-Lemarie scaling function is used in an algorithm for fast computation of the Fourier transform of a piecewise smooth function f and an application of this algorithm to image processing is considered.
Abstract: We use the Battle-Lemarie scaling function in an algorithm for fast computation of the Fourier transform of a piecewise smooth function f. Namely, we compute for -N
02 Dec 1994
TL;DR: In this article, the authors propose an exact relation between the Discrete Fourier Transform and the periodic sums associated with a function and its Fourier transform in a similar way as in the Poisson summation formula.
Abstract: The algorithm is based on an exact relation, due to Cooley, Lewis and Welch, between the Discrete Fourier Transform and the periodic sums, associated with a function and its Fourier Transform in a similar way as in the Poisson summation formula. It makes use of several equidistant grids, with the same number of points covering m different symmetric intervals of length L, 2L, 4L, 8L,…, where it applies FFT and spline interpolation to the midpoints of the grid.
01 Sep 1994
TL;DR: The algorithm presented is intended for use in the solution of partial differential equations, or in any situation in which a large number of forward and backward transforms must be performed and in which the Fourier Coefficients need not be ordered.
Abstract: This report deals with parallel algorithms for computing discrete Fourier transforms of real sequences of length N not equal to a power of two. The method described is an extension of existing power of two transforms to sequences with N a product of small primes. In particular, this implementation requires N = 2{sup p}3{sup q}5{sup r}. The communication required is the same as for a transform of length N = 2{sup p}. The algorithm presented is intended for use in the solution of partial differential equations, or in any situation in which a large number of forward and backward transforms must be performed and in which the Fourier Coefficients need not be ordered. This implementation is a one dimensional FFT but the techniques are applicable to multidimensional transforms as well. The algorithm has been implemented on a 128 node Intel Ipsc/860.
20 Mar 1994
TL;DR: This paper investigates the convergence properties of an identification algorithm based on recursive rational interpolation that utilizes the moving discrete Fourier transform to efficiently monitor certain points in the spectra of the system input and output signals.
Abstract: This paper investigates the convergence properties of an identification algorithm based on recursive rational interpolation. The algorithm utilizes the moving discrete Fourier transform (MDFT), which is a recursive form of the DFT, to efficiently monitor certain points in the spectra of the system input and output signals. Convergence of parameter estimates to their true values is established for the algorithm, and persistent excitation conditions are also given. >