Showing papers on "Discrete sine transform published in 1978"
TL;DR: New algorithms for computing the Discrete Fourier Transform of n points are described, which use substantially fewer multiplications than the best algorithm previously known, and about the same number of additions.
Abstract: New algorithms for computing the Discrete Fourier Transform of n points are described. For n in the range of a few tens to a few thousands these algorithms use substantially fewer multiplications than the best algorithm previously known, and about the same number of additions.
707 citations
TL;DR: An incoherent optical data-processing method is described, which has the potential for performing discrete Fourier transforms of short length at rates far exceeding those afforded by both special-purpose digital hardware and representative coherent optical processors.
Abstract: An incoherent optical data-processing method is described, which has the potential for performing discrete Fourier transforms of short length at rates far exceeding those afforded by both special-purpose digital hardware and representative coherent optical processors.
375 citations
TL;DR: An N -point discrete Fourier transform (DFT) algorithm can be used to evaluate a discrete cosine transform by a simple rearrangement of the input data.
Abstract: An N -point discrete Fourier transform (DFT) algorithm can be used to evaluate a discrete cosine transform by a simple rearrangement of the input data. This method is about two times faster compared to the conventional method which uses a 2N -point DFT.
327 citations
TL;DR: General source coding theorems are proved in order to justify using the optimal test channel transition probability distribution for allocating the information rate among the DFT coefficients and for calculating arbitrary performance measures on actual optimal codes.
Abstract: Distortion-rate theory is used to derive absolute performance bounds and encoding guidelines for direct fixed-rate minimum mean-square error data compression of the discrete Fourier transform (DFT) of a stationary real or circularly complex sequence. Both real-part-imaginary-part and magnitude-phase-angle encoding are treated. General source coding theorems are proved in order to justify using the optimal test channel transition probability distribution for allocating the information rate among the DFT coefficients and for calculating arbitrary performance measures on actual optimal codes. This technique has yielded a theoretical measure of the relative importance of phase angle over the magnitude in magnitude-phase-angle data compression. The result is that the phase angle must be encoded with 0.954 nats, or 1.37 bits, more rate than the magnitude for rates exceeding 3.0 nats per complex element. This result and the optimal error bounds are compared to empirical results for efficient quantization schemes.
85 citations
TL;DR: The performance of each of the unitary transforms in the trigonometric family is studied in terms of quantitative performance measures such as variance distribution, rate distortion, Wiener filtering and how well a transform decorrelates the data efficiently for possible bandwidth compression of a signal represented by a firstorder Markov process model.
Abstract: The feasibility of discrete sine transform (DST) and discrete sine cosine transform (DSCT) for digital image processing problems are investigated. Discrete sine transform and discrete cosine transform can be computed by using two FFT’s of original data sequence of length N. Discrete sine cosine coefficients are computed by FFT of data sequence of length N while the inverse is obtained by computing two FFT's. The performance of each of the unitary transforms in the trigonometric family is studied in terms of quantitative performance measures such as variance distribution, rate distortion, Wiener filtering and how well a transform decorrelates the data efficiently for possible bandwidth compression of a signal represented by a firstorder Markov process model. Computer simulation results on a monochrome image are presented.
74 citations
TL;DR: Efficient algorithms for computing one-dimensional convolutions and Discrete Fourier Transforms are derived from polynomial transforms, which are shown to have the convolution property and can be computed in ordinary arithmetic, without multiplications.
Abstract: Discrete transforms are introduced and are defined in a ring of polynomials. These polynomial transforms are shown to have the convolution property and can be computed in ordinary arithmetic, without multiplications. Polynomial transforms are particularly well suited for computing discrete two-dimensional convolutions with a minimum number of operations. Efficient algorithms for computing one-dimensional convolutions and Discrete Fourier Transforms are then derived from polynomial transforms.
62 citations
TL;DR: A real-time image processor which is capable of video compression using either the sequency-ordered Walsh-Hadamard transform (WHT)W, or the discrete cosine transform (DCT), is considered, which results in substantial savings in the number of multiplications and additions required to obtain the DCT, relative to its direct computation.
Abstract: A real-time image processor which is capable of video compression using either the sequency-ordered Walsh-Hadamard transform (WHT)W, or the discrete cosine transform (DCT), is considered. The processing is done on an intraframe basis in (8 X 8) data blocks. The (WHT)W coefficients are computed directly, and then used to obtain the DCT coefficients. This is achieved via an (8 X 8) transformation matrix which is orthonormal, and has a block-diagonal structure. As such, it results in substantial savings in the number of multiplications and additions required to obtain the DCT, relative to its direct computation. Some aspects of a hardware implementation of the processor are also included.
55 citations
TL;DR: It is shown that R supports a length m discrete Fourier transform if and only if m is a divisor of O(R) and this result is applied to the homomorphic images of rings-of algebraic integers.
Abstract: Necessary and sufficient conditions for a direct sum of local rings to support a generalized discrete Fourier transform are derived. In particular, these conditions can be applied to any finite ring. The function O(N) defined by Agarwal and Burrus for transforms over ZN is extended to any finite ring R as O(R) and it is shown that R supports a length m discrete Fourier transform if and only if m is a divisor of O(R) This result is applied to the homomorphic images of rings-of algebraic integers.
46 citations
TL;DR: In an algorithm proposed here, DFT coefficients are computed via the Walsh transform (WT), which is superior to the fast Fourier transform (FFT) approach in applications where L is relatively small compared with N.
Abstract: This paper presents a new computational algorithm for the discrete Fourier transform (DFT). In an algorithm proposed here, DFT coefficients are computed via the Walsh transform (WT). The number of multiplications required by the new algorithm is approximately NL/6, where N is the number of data points and L is the number of Fourier coefficients desired. As such, it is superior to the fast Fourier transform (FFT) approach in applications where L is relatively small compared with N. It is also useful in cases where the Walsh and Fourier coefficients are both desired.
44 citations
TL;DR: The concept of energy packing efficiency of the Walsh-Hadamard transform is extended to other discrete orthogonal transforms and it is shown that the EPE is invariant for the generalized transforms as long as the ordering is the same.
Abstract: The concept of energy packing efficiency (EPE) of the Walsh-Hadamard transform (WHT) first proposed by Kitajima [1], is extended to other discrete orthogonal transforms. The concept as a criterion for evaluating the transforms is discussed. It is shown that the EPE is invariant for the generalized transforms as long as the ordering is the same.
36 citations
01 Jan 1978
TL;DR: In this article, a conjugate symmetry property which generalizes the well known property of the complex DFT for real data is presented for this situation, which is used to obtain a technique for computing the DFT of μ sequences with values in a ring S using a single DFT in an extension ring R of degree μ over S.
Abstract: Often, signals which lie in a ring S are convolved using a generalized discrete Fourier transform (DFT) over an extension ring R in order to allow longer sequence lengths. In this paper, a conjugate symmetry property which generalizes the well known property of the complex DFT for real data is presented for this situation. This property is used to obtain a technique for computing the DFT of μ sequences with values in a ring S using a single DFT in an extension ring R of degree μ over S. From this result, a method to compute the convolution of length μn S-sequences using a length n DFT in R is derived. Example of the application to the complex DFT and to a number theoretic transform are presented to illustrate the theory.
01 Jan 1978
TL;DR: A general class of even/odd transforms, including the Karhunen-Loeve transform, the discrete cosine transform, and the Walsh-Hadamard transform, is presented in this paper.
Abstract: A general class of even/odd transforms is presented that includes the Karhunen-Loeve transform, the discrete cosine transform, the Walsh-Hadamard transform, and other familiar transforms. The more complex even/odd transforms can be computed by combining a simpler even/odd transform with a sparse matrix multiplication. A theoretical performance measure is computed for some even/odd transforms, and two image compression experiments are reported.
TL;DR: The results demonstrate that the performances of the discrete cosine and Karhunen-Loeve transforms compare closely, as is the case with a separable covariance model, while the corresponding performance of the Walsh-Hadamard transform is inferior.
Abstract: Intraframe transform coding of pictures for the case of a nonseparable covariance model is considered. Performances of the Walsh-Hadamard, discrete cosine and Karhunen-Loeve transforms are compared based on the compaction of signal energy in the transform components, and, the degree of decorrelation of the data. The results demonstrate that the performances of the discrete cosine and Karhunen-Loeve transforms compare closely, as is the case with a separable covariance model. The corresponding performance of the Walsh-Hadamard transform is inferior.
TL;DR: In this article, an exact representation of the scattered sound pressure field is obtained as a combination of the radiation of the image source and layer potentials, where the point spherical source is located on the interface.
IBM1
TL;DR: It is shown that, by combining the polynomial transform approach with a split nesting technique, multidimensional convolutions can be computed very efficiently in general purpose computers.
Abstract: Discrete transforms, defined in rings of polynomials, have been introduced recently. These polynomial transforms have the convolution property and can be computed in ordinary arithmetic, without multiplications. We show that, by combining the polynomial transform approach with a split nesting technique, multidimensional convolutions can be computed very efficiently in general purpose computers. This computation method can also be used for the evaluation of one-dimensional convolutions and discrete Fourier transforms (DFT's).
Patent•
18 Sep 1978TL;DR: In this paper, an error-limited continuous discrete Fourier transform of the input signal through the use of duplicate channels which are periodically reinitialized in time staggered relationship and alternately switched on-line and off-line in time synchronism with the reinitialization process is presented.
Abstract: To limit the accumulation of recursive computation errors while recursively calculating the discrete Fourier transform of an input signal in response to moving window sample sets of that signal, the dependent variables for the recursive calculations are periodically refreshed. One embodiment generates an error limited continuous discrete Fourier transform of the input signal through the use of duplicate channels which are periodically reinitialized in time staggered relationship and alternately switched on-line and off-line in time synchronism with the reinitialization process. Each of the channels of that embodiment is equipped to recursively calculate the discrete Fourier transform of the input signal so that while one channel is on line feeding recursively calculated transform coefficients to the outputs, the other channel or channels are off-line being reinitialized and then computing fresh sets of transform coefficients for subsequent on-line recursive calculations. Another embodiment performs essentially the same function through the use of an on-line main processing channel which recursively calculates the discrete Fourier transform coefficients of the input signal on the basis of successive sets of dependent variables which are computed in and supplied by a periodically reinitialized off-line auxiliary processing channel.
TL;DR: In this paper, a discrete Hankel transform with kernel Jo, the Bessel function of the first kind of order zero, is defined as a finite sum over the zeros of Bessel functions.
Abstract: A discrete transform with a Bessel function kernel is defined, as a finite sum, over the zeros of the Bessel function. The approximate inverse of this transform is derived as another finite sum. This development is in parallel to that of the discrete Fourier transform (DFT) which lead to the fast Fourier transform (FFT) algorithm. The discrete Hankel transform with kernel Jo, the Bessel function of the first kind of order zero, will be used as an illustration for deriving the discrete Hankel transform, its inverse and a number of its basic properties. This includes the convolution product which is necessary for solving boundary problems. Other applications include evaluating Hankel transforms, Bessel series and replacing higher dimension Fourier transforms, with circular symmetry, by a single Hankel transform
TL;DR: It is shown that substantial errors in interpolation by means of existing discrete Fourier transform techniques are generally present for finite sequences, even if the function sampled is band-limited, and a modified approach is proposed which provides significantly more accurate information.
Abstract: It is shown that substantial errors in interpolation by means of existing discrete Fourier transform (DFT) techniques are generally present for finite sequences, even if the function sampled is band-limited. A modified approach is proposed which provides significantly more accurate information.
TL;DR: In this paper, the applicability of the discrete Fourier transform (DFT) for the characterization of random and ordered solids mixtures has been tested and it has been found that the DFT power spectrum possesses more significant physical meaning than the variance itself.
TL;DR: In this paper, a comparative study of frame rate limitations, windowing, time and frequency resolution, spectral correlation, complexity, and inverse structures for the conventional chirp-z-transform (CZT) and the sliding CZT realization of the discrete sliding Fourier transform is presented.
Abstract: Two sequential short-time spectral analysis techniques, amenable to nonrecursive filter implementation, are the conventional chirp-z-transform (CZT) realization of the discrete Fourier transform and the sliding CZT realization of the discrete sliding Fourier transform. This paper presents a comparative study of frame rate limitations, windowing, time and frequency resolution, spectral correlation, complexity, and inverse structures for these methods, with particular emphasis on the recently proposed sliding transform. The sliding transform and its CZT realization are viewed as skewed output samples of a filter bank, an approach which aids in understanding the relationship between the conventional and sliding schemes. Numerous forward and inverse CZT formulations are presented to improve resolution, frame rates, and compactness.
TL;DR: Experimental results showing operation with sine and cosine inputs demonstrate processor performance in computing real and imaginary Fourier-transform components to an accuracy equivalent to 6 bits.
Abstract: A fast-Fourier-transform (f.f.t.) processor based on the chirp transform algorithm implemented with surface-acoustic-wave (s.a.w.) chirp filters is reported. Using commercially available s.a.w. devices the processor computes a 128-point (nominal) f.f.t. in 25 ?s, permitting real-time operation over 4 MHz bandwidth. Experimental results showing operation with sine and cosine inputs demonstrate processor performance in computing real and imaginary Fourier-transform components to an accuracy equivalent to 6 bits.
TL;DR: An inductive proof for the block-diagonal structure of the cyclic time e-shift matrix, under various types of generalized orthogonal discrete transforms, was provided in this paper.
Abstract: An inductive proof is provided for the block-diagonal structure of the cyclic timee-shift matrix, under various types of generalized orthogonal discrete transforms.
01 Apr 1978
TL;DR: A conventional chirp-z-transform realization of the discrete Fourier transform and an analogous CZT realization of a sliding transform better suited to CCD technology than the DFT are considered.
Abstract: In this paper we discuss real time spectral analysis using charge-coupled device (CCD) nonrecursive filters with an application to homomorphic speech analysis-synthesis. We consider the conventional chirp-z-transform (CZT) realization of the discrete Fourier transform (DFT) and an analogous CZT realization of a sliding transform, a modified DFT, better suited to CCD technology than the DFT. Novel CCD configurations based on both transforms are proposed for improved spectral resolution and frame rate; such schemes utilize parallel spectral processing and interpolation. Stationarity conditions and windowing techniques required by these CCD structures are discussed within the context of short-time spectral analysis. A number of homomorphic schemes and experimental results are presented which illustrate the trade-offs that CCD technology imposes between implementational complexity and synthetic speech quality.
IBM1
TL;DR: It is shown that for small word lengths and/or moderate to large transform lengths NTT filtering achieves a better SNR than FFT filtering with fixed-point arithmetic.
Abstract: Number Theoretic Transforms (NTT's), unlike the Discrete Fourier Transform (DFT), are defined in finite rings and fields rather than in the field of complex numbers. Some NTT's have a transform structure like the Fast Fourier Transform (FFT) and can be used for fast digital signal processing. The computational effort and the signal-to-noise ratio (SNR) performance of linear filtering in finite rings and fields are investigated. In particular, the effect of limited word lengths, i.e., b \leq 16 , and long transform lengths on the SNR is analyzed. It is shown that for small word lengths and/or moderate to large transform lengths NTT filtering achieves a better SNR than FFT filtering with fixed-point arithmetic. Some new NTT's with a single- or mixed-radix fast transform structure are presented. While these NTT's may require special modulo arithmetic they achieve optimum transform length for any given word length b in the range 8 \leq b \leq 16 .
TL;DR: In this paper, a two-stage approach to the computation of the discrete Fourier transform is described, which can offer a number of distinct advantages: fewer inherent multiplications over a given range, no complex arithmetic for real data, and flexibility of output.
Abstract: A new two-stage approach to the computation of the discrete Fourier transform is described, which, relative to the fast Fourier transform (f.f.t.), can offer a number of distinct advantages: fewer inherent multiplications over a given range, no complex arithmetic for real data, and flexibility of output. The mathematical foundations and related algorithms are discussed in detail and a guide to the advantages over both the f.f.t. and the recently reported Winograd Fourier transform are included.
01 Jan 1978
TL;DR: A general class of even/odd transforms is presented that includes the Karhunen-Loevetransform, the discrete cosine transform, the Walsh-Hadamard transform, and other familiar transforms.
Abstract: A general class of even/odd transforms is presented that includes the Karhunen-Loeve transform, the discrete cosine transform, the Walsh-Hadamard transform, and other familiar transforms. The more complex even/odd transforms can be computed by combining a simpler even/odd transform with a sparse matrix multiplication. A theoretical performance measure is computed for some even/odd transforms, and two image compression experiments are reported.
07 Dec 1978
TL;DR: The product separable model for the transform coefficient variances forms the basis of a transform coding algorithm and is described and tested on real sampled images.
Abstract: Image transform coding is a technique whereby a sampled image is divided into blocks. A two-dimensional discrete transform of each block is taken, and the resulting transform coefficients are coded. Coding of the transform coefficients requires their quantization and, consequently, a model for the transform coefficient variances that is based in turn on a correlation model for the image blocks. In the proposed correlation model each block of image data is formed by an arbitrary left and right matrix multiplication of a stationary white matrix. One consequence of this correlation model is that the transform coefficient variances are product separable in row and column indexes. The product separable model for the transform coefficient variances forms the basis of a transform coding algorithm. The algorithm is described and tested on real sampled images.© (1978) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.