Showing papers on "Discrete sine transform published in 1988"
TL;DR: The circular harmonic transform (CHT) solution of the exponential Randon transform (ERT) is applied to single-photon emission computed tomography (SPECT) for uniform attenuation within a convex boundary to demonstrate that the boundary conditions are a more general property of the Radon transform and a not a property unique to rectangular coordinates.
Abstract: The circular harmonic transform (CHT) solution of the exponential Randon transform (ERT) is applied to single-photon emission computed tomography (SPECT) for uniform attenuation within a convex boundary. An important special case also considered is the linear (unattenuated) Radon transform (LRT). The solution is on the form of an orthogonal function expansion matched to projections that are in parallel-ray geometry. This property allows for efficient and accurate processing of the projections with fast Fourier transform (FFT) without interpolation or beam matching. The algorithm is optimized by the use of boundary conditions on the 2-D Fourier transform of the sinogram. These boundary conditions imply that the signal energy of the sinogram is concentrated in well-defined sectors in transform space. The angle defining the sectors depends in a direct way on the radius of the field view. These results are also obtained for fan-beam geometry and the linear Radon transform (the Fourier-Chebyshev transform of the sinogram) to demonstrate that the boundary conditions are a more general property of the Radon transform and a not a property unique to rectangular coordinates. >
107 citations
Patent•
22 Sep 1988
TL;DR: In this article, a picture encoding system which performs a prediction of picture element values within a block having a plurality of picture elements into which an original picture is divided, is presented.
Abstract: A picture encoding system which performs a prediction of picture element values within a block having a plurality of picture elements into which an original picture is divided, a discrete sine transform with respect to prediction error signals for obtaining a transform coefficient, a quantization of the transform coefficient for encoding quantized indexes, an inverse quantization of the quantized indexes for reproduction of the transform coefficient, an inverse discrete sine transform of the reproduced coefficient to reproduce the prediction error signal, and an addition thereto of the predicted picture element value for reproducing the picture element values within the block to employ them in predicting a next stage block to be encoded, whereby the block can be minimized in size while simplifying required transform operation.
78 citations
TL;DR: It is shown that the number of distinct N-point DFTs needed to calculate N*N-point two-dimensional DFT’s is equal to thenumber of linear congruences spanning the N-N grid.
Abstract: An algorithm is presented for computation of the two-dimensional discrete Fourier transform (DFT). The algorithm is based on geometric properties of the integers and exhibits symmetry and simplicity of realization. Only one-dimensional transformation of the input data is required. The transformations are independent; hence, parallel processing is feasible. It is shown that the number of distinct N-point DFTs needed to calculate N*N-point two-dimensional DFTs is equal to the number of linear congruences spanning the N*N grid. Examples for N=3, N=4, and N=10 are presented. A short APL code illustrating the algorithm is given. >
76 citations
TL;DR: An approach for realizing the N-point discrete Fourier transform (DFT) of an input sequence is presented and is combined with H.T. King's (1981) approach to construct a two-dimensional array for computing the two- dimensional DFT.
Abstract: An approach for realizing the N-point discrete Fourier transform (DFT) of an input sequence is presented. It is then combined with H.T. King's (1981) approach to construct a two-dimensional array for computing the two-dimensional DFT. This mixed model takes stream input and produces stream output. In addition, no extra I/O time delay is required before performing the row (column) transform of the two-dimensional DFT. >
55 citations
CNET1
TL;DR: Methodologies for constructing fast algorithms to compute the discrete Fourier transform of a 2-D real sequence are introduced and the resulting algorithms are shown to be in-place and butterfly-style as well as the usual 1-D FFT algorithms.
Abstract: Methodologies for constructing fast algorithms to compute the discrete Fourier transform (DFT) of a 2-D real sequence are introduced. The resulting algorithms are shown to be in-place and butterfly-style as well as the usual 1-D FFT algorithms. Above all, the computational load of these algorithms is reduced to less than one-half of their complex counterparts. Due to the in-place property, the storage requirement is exactly halved. A comparison is made on the basis of arithmetic complexity, storage, and input/output requirements. >
40 citations
TL;DR: Effective methods are proposed for calculating a multidimensional discrete Fourier transform based on a new representation of it and their application to discrete number theory is proposed.
Abstract: Effective methods are proposed for calculating a multidimensional discrete Fourier transform based on a new representation of it.
39 citations
TL;DR: In this paper, the exact expression for the discrete Fourier transform of a sum of exponentially damped sinusoids is derived, and its applicability for describing the general DFT spectrum is demonstrated.
39 citations
07 Jun 1988
TL;DR: An efficient method for computing the discrete Fourier transform when only a few output points are needed is described, based on a novel factorization of the DFT, where one part is computed using standard power-of-two FFTs and the other uses a technique similar to the Goertzel algorithm.
Abstract: The authors describe an efficient method for computing the discrete Fourier transform (DFT) when only a few output points are needed. The method is shown to be more efficient than either Goertzel's method or pruning, and it allows any band in the output to be computed. It is based on a novel factorization of the DFT, where one part is computed using standard power-of-two FFTs (fast Fourier transforms) and the other uses a technique similar to the Goertzel algorithm. >
37 citations
CNET1
TL;DR: The authors indicate an apparently novel method for computing an inverse discrete Fourier transform (IDFT) through the use of a forward DFT program, and point out that, in many cases, this is obtained without any additional cost, either in terms of program length or in Terms of computational time.
Abstract: The authors indicate an apparently novel method for computing an inverse discrete Fourier transform (IDFT) through the use of a forward DFT program. They point out that, in many cases, this is obtained without any additional cost, either in terms of program length or in terms of computational time. >
36 citations
TL;DR: The authors propose an orthogonal, unitary transformation called the modified Hermite transformation (MHT) and its extension, which is called the modular modified Hermites transformation (MMHT), which is efficient computationally and comparable to the DCT for AR(1) source models with positive correlation coefficients.
Abstract: The authors propose an orthogonal, unitary transformation called the modified Hermite transformation (MHT) and its extension, which is called the modular modified Hermite transformation (MMHT). The MHT algorithm, which is an efficient algorithm, is explained and explored. The MHT is compared to the discrete cosine transform (DCT) for various AR(1) input signal source models using the performance criterion of gain over PCM, denoted by /sup N/G/sub TC/. The MHT algorithm requires only 2N real multiplications or divisions for a transformation of a signal block of N samples. It is also used for the inverse transformation, IMHT, and makes this new transform attractive. It is efficient computationally and comparable to the DCT for AR(1) source models with positive correlation coefficients, it is somewhat better than the DCT for negative correlation coefficients. >
33 citations
TL;DR: It was found that the decimation-in-frequency approach provides a viable alternative to other fast algorithms for the discrete sine and cosine transforms.
Abstract: In this paper we present results in the development of decimation-in-frequency algorithms for a family of discrete sine and cosine transforms. They are closely related to the decimation-in-time algorithms developed by Yip and Rao [1]. The complexity of the algorithms was examined through the number of multiplications and additions as well as the number of different constants required in the transforms. It was found that the decimation-in-frequency approach provides a viable alternative to other fast algorithms for the discrete sine and cosine transforms. In particular, the recursive and modular structure of the algorithms lends itself readily to possible hardware realization.
01 Dec 1988
TL;DR: It is shown that by using an index mapping scheme, the multidimensional discrete Hartley transform can be changed into convolutions that can be calculated very efficiently via the Fermat number transform.
Abstract: It is shown that by using an index mapping scheme, the multidimensional discrete Hartley transform can be changed into convolutions that can be calculated very efficiently via the Fermat number transform. Compared with existing algorithms, the number of multiplications is reduced by a factor of 8 to 20, at the expense of a slight increase in the number of shift and add operations, that are assumed to be simpler than multiplications.
TL;DR: Even-odd transforms such as the discrete cosine (DCT), the discrete sine, the slant (ST), and the discrete Legendre (DLT) transforms are developed from the Walsh-Hadamard transform (WHT).
01 Jan 1988
TL;DR: In this article, the multiplicative complexity of the discrete Fourier transform (DFT) was analyzed and the complexity of DFT for any positive integer was shown. But the complexity was not shown for any integer.
Abstract: In this chapter the multiplicative complexity of the discrete Fourier transform (DFT) is analyzed. The next several sections define the DFT and then show how the complexity of the DFT is determined when the number of inputs is prime, a power of an odd prime, a power of two, and finally for any positive integer.
TL;DR: The coding method used for the comparison utilizes a marginal bit-allocation scheme and Lloyd-Max quantizer as discussed by the authors, and the discrete cosine transform outperforms the other three transforms based on the mean-square quantization error.
Abstract: The coding method used for the comparison utilizes a marginal bit-allocation scheme and Lloyd-Max quantizer. Gamma, Laplacian, and Gaussian models are compared for the distribution of the transform coefficients. The discrete cosine transform outperforms the other three transforms based on the mean-square quantization error. >
TL;DR: It is shown that for an analysis filter length that does not exceed a given value, the optimal synthesis scheme is independent of the duration of the given MDSTT and is an extension of the weighted overlap add (WOLA) synthesis method.
Abstract: The discrete short-time transform (DSTT) is a generalization of the discrete short-time Fourier transform (DSTFT). The necessary and sufficient conditions on the analysis filter, under which perfect reconstruction of the input signal is possible (when the DSTT is not modified), are presented. The class of linear modifications for which the original input can be reconstructed when the modification is applied is characterized. The synthesis of an optimal (in the minimum-mean-square-error sense) signal from a modified DSTT (MDSTT) of finite duration is presented. It is shown that for an analysis filter length that does not exceed a given value, the optimal synthesis scheme is independent of the duration of the given MDSTT and is an extension of the weighted overlap add (WOLA) synthesis method. For longer analysis filters, the optimal synthesis scheme becomes quite cumbersome, and therefore, a steady-state solution (as the duration of the MDSTT approaches infinity) is presented for this case. It is shown that this solution can be approximated with arbitrarily small reconstruction error. >
TL;DR: Based on a multichannel incoherent optical correlator, a new simple scheme was proposed for performing a complex discrete Fourier transform as mentioned in this paper, where a complex value is represented by using three nonnegative reals, and every real is encoded with the area of a rectangular aperture.
TL;DR: In this article, the effects of a discrete Fourier transform (DFT) filter on a time series of ocean currents and sea levels were investigated and it was shown that ringing in the retransformed time series can be minimized by careful choice of filter bandwidth and the amount of tapering of the sides of the filter.
Abstract: Many different filters exist to remove unwanted signals from time series of ocean currents and sea levels, and in this paper we examine two: a tidal harmonic “filter” and a discrete Fourier transform (DFT) filter. DFT filters are particularly easy to use if working in the frequency domain, but as with all filters it is necessary to understand the effects of a DFT filter before it can be used with complete confidence. A number of sample time series are used, some artificial and some real, to test DFT filters. It is shown that “ringing” in the retransformed time series can be minimized by careful choice of the filter bandwidth and the amount of tapering of the sides of the filter.
11 Apr 1988
TL;DR: The fast real Fourier transform (FRFT) algorithms discussed are the radix-2 decimation-in-time (DIT), theRadix-4 DIT, the split-radix DIF, the prime factor, and the Winograd FRFT algorithm.
Abstract: Fast algorithms for the computation of the real discrete Fourier transform (RDFT) are discussed. Implementations based on the RDFT are always efficient, whereas the implementations based on the DFT are efficient only when signals to be processed are complex. The fast real Fourier transform (FRFT) algorithms discussed are the radix-2 decimation-in-time (DIT), the radix-4 DIT, the split-radix DIT, the split-radix DIF, the prime factor, and the Winograd FRFT algorithm. >
01 Jan 1988
20 Mar 1988
TL;DR: In this paper, the authors examined the effect of these dangers when the FFT algorithm is supplied to power system load variation and recommended the use of a DFT algorithm to evaluate the frequency spectrum of power system loads.
Abstract: The discrete Fourier transform (DFT) and the fast Fourier transform (FFT) are based on certain assumptions that must be understood and satisfied, or misleading results will be obtained. The authors examine these assumptions and qualitatively analyze the effect of these dangers when the FFT algorithm is supplied to power system load variation. They recommend the use of a DFT algorithm to evaluate the frequency spectrum of power system load variation. >
TL;DR: The Hartley Transform not only decreases the computer time of the WDF but also simplifies the convolution of two WDFs, which is used here to simulate a blurred image and its restoration.
TL;DR: A computationally efficient parallel algorithm is derived by using the fast discrete sine transform to solve the restoration problem for blurred images based on the two-dimensional semi-causal model and the Kalman filtering technique.
Patent•
03 Jun 1988TL;DR: In this article, an electronic digital system for performing discrete Fourier transforms in real time is presented, where read only memory (ROM) modules are used as look-up tables for providing inputs for multiplier stages corresponding to sequences of sample signals as well as for logic and other input converting elements of the system.
Abstract: An electronic digital system for performing discrete Fourier transforms in real time. Read only memory (ROM) modules are used as look-up tables for providing inputs for multiplier stages corresponding to sequences of sample signals as well as for logic and other input converting elements of the system. Two such systems are coupled together so that their real and imaginary output signal components may be additively and subtractively combined. The entire transform output is available one cycle period after the last sample input.
TL;DR: The present authors provide additional information on the structure of G/sub w/ and generalize some results by C.J. Zarowski and M. Yunik (see ibid., vol.ASSP-33, p.1246-52, Oct. 1985).
Abstract: A.E. Kahveci and E.L. Hall (see IEEE Trans. Comput., vol.C-23, no.9, p.976-81, Sept. 1974) introduced the concept of filtering discrete Fourier transform (DFT) spectra in the Walsh sequency domain. This is accomplished by finding a real and block-diagonal Walsh filter matrix G/sub w/ in the Walsh domain that performs the sample filtering operation as the prototype complex diagonal Fourier filter matrix G/sub f/ in the Fourier domain. The present authors provide additional information on the structure of G/sub w/ and generalize some results by C.J. Zarowski and M. Yunik (see ibid., vol.ASSP-33, p.1246-52, Oct. 1985). They consider a more general class of transforms, the T transforms, and the structure of the resulting T transform filter matrices G/sub t/. Examples of T besides T=W are considered, such as the Harr transform and fourth-order Chrestenson transform. The implementation of the presented DFT spectrum filtering techniques using linear systolic arrays is briefly considered. >
11 Apr 1988
TL;DR: A systolic-network architecture for the computation of the FFT is presented, and a one-chip VLSI design consideration for AT/sup 2/ optimal fast Fourier transform (FFT) shuffle-exchange architecture is considered.
Abstract: One-chip VLSI design consideration for AT/sup 2/ optimal fast Fourier transform (FFT) shuffle-exchange architecture is considered, and a systolic-network architecture for the computation of the FFT is presented. This architecture has the same asymptotically optimal theoretical O(N/sup 2/log/sup 2/N) AT/sup 2/ complexity as the FFT shuffle-exchange architecture, but is more suitable for one-chip VLSI design. Architectures which are feasible for a one-chip FFT design, as well as for shuffle-exchange-type fast discrete orthogonal transforms such as the generalized transform, cosine transform, and slant transform are also discussed. >
TL;DR: A coherent optical system composed of a holographic mask and two Fourier lenses is described for performing an arbitrary linear transform that is optically made in 1-D space.
Abstract: A coherent optical system composed of a holographic mask and two Fourier lenses is described for performing an arbitrary linear transform. A set of equations for determining the amplitude-phase distribution of the mask is given. As a specific transform, the Walsh-Hadamard transform for orders 32 and 64 is optically made in 1-D space.
11 Apr 1988
TL;DR: On taking the 2D Fourier transform of the output of a linear array, a ridge will appear on the frequency-wave number plane if a source is present and two methods are proposed to estimate this slope.
Abstract: On taking the 2D Fourier transform of the output of a linear array, a ridge will appear on the frequency-wave number plane if a source is present. The slope of this ridge is determined by the direction of arrival of the wavefront. Two methods are proposed to estimate this slope. The first one takes the sums of the squared magnitude of the transform along predetermined slopes to find the maximum sum. The second one requires less computation; it first locates all the maxima of the transform. A weighted-least-squares fit is then taken through these maxima to give a slope estimation. The multisource case is considered, and properties and statistics of the beamformer, together with simulation results, are given. >
TL;DR: According to this method, called the Running Fourier Transform (RFT) in this paper, running power spectra with arbitrary center frequencies and arbitrary Q values can be obtained.
Abstract: A short-time Fourier transform can be derived by low-pass filtering of the product of an input signal and exp(j2°ft). According to this method, called the Running Fourier Transform (RFT) in this paper, running power spectra with arbitrary center frequencies and arbitrary Q values can be obtained.
This paper proposes a fast algorithm of discrete RFT (FRFT). In the FRFT, a first-order lag system is adopted as the low-pass filter (LPF), and by approximating the impulse response of the LPF with a step function, the amount of multiplications is reduced. The calculation of the complex exponentials is omitted by referring to a table of sin(2°k/K) (k =0, …, K).
TL;DR: A comparison of the characteristics of the DFT and the FFT shows that, whereas the latter is clearly advantageous in terms of calculation time, it does not allow for precise localization of the spectral lines, which can be overcome by interpolating with Spline functions.
Abstract: Biological rhythms are often studied by complementary explorations in the temporal and the frequency domains. This provides a means of investigating purely frequential features such as periods and phases, as well as those related to the shape of the curve. A comparison of the characteristics of the DFT (Discrete Fourier Transform) and the FFT (Fast Fourier Transform) shows that, whereas the latter is clearly advantageous in terms of calculation time, it does not allow for precise localization of the spectral lines. This drawback can be overcome by interpolating with Spline functions. The determination of Splines associated with an FFT is detailed for microcomputer application. The algorithm proposed here is then tested on a concrete example of measurement of biological rhythms of activity on an inbred mouse.