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Showing papers on "Discrete sine transform published in 1990"


Book
15 Aug 1990
TL;DR: This paper presents two Dimensional DCT Algorithms and their relations to the Karhunen-Loeve Transform, and some applications of the DCT, which demonstrate the ability of these algorithms to solve the discrete cosine transform problem.
Abstract: Discrete Cosine Transform. Definitions and General Properties. DCT and Its Relations to the Karhunen-Loeve Transform. Fast Algorithms for DCT-II. Two Dimensional DCT Algorithms. Performance of the DCT. Applications of the DCT. Appendices. References. Index.

2,039 citations


Journal ArticleDOI
TL;DR: Two lapped transforms for subband/transform coding of signals are introduced: a version of the lapped orthogonal transform (LOT), which can be efficiently computed for any transform length; and the modulated lapped transform (MLT), which is based on a modulated quadrature mirror (QMF) bank.
Abstract: Two lapped transforms for subband/transform coding of signals are introduced: a version of the lapped orthogonal transform (LOT), which can be efficiently computed for any transform length; and the modulated lapped transform (MLT), which is based on a modulated quadrature mirror (QMF) bank. The MLT can also be efficiently computed by means of a type-IV discrete sine transform (DST-IV). The LOT and the MLT are both asymptotically optimal lapped transforms for coding an AR(1) signal with a high intersample correlation coefficient. The coding gains of the LOT and MLT of length M are higher than that of the discrete cosine transform (DCT) of the same length; they are actually close to the coding gains obtained with a DCT of length 2M. An MLT-based adaptive transform coder (ACT) for speech signals is simulated; the code is essentially free from frame rate noise and has a better spectral resolution that DCT-based ATC systems. >

513 citations


PatentDOI
TL;DR: In this article, a low bit-rate (192 kBits per second) transform encoder/decoder system (44.1 kHz or 48 kHz sampling rate) for high quality music applications employs short time-domain sample blocks (128 samples/block) so that the system signal propagation delay is short enough for real-time aural feedback to a human operator.
Abstract: A low bit-rate (192 kBits per second) transform encoder/decoder system (44.1 kHz or 48 kHz sampling rate) for high-quality music applications employs short time-domain sample blocks (128 samples/block) so that the system signal propagation delay is short enough for real-time aural feedback to a human operator. Carefully designed pairs of analysis/synthesis windows are used to achieve sufficient transform frequency selectivity despite the use of short sample blocks. A synthesis window in the decoder has characteristics such that the product of its response and that of an analysis window in the encoder produces a composite response which sums to unity for two adjacent overlapped sample blocks. Adjacent time-domain signal samples blocks are overlapped and added to cancel the effects of the analysis and synthesis windows. A technique is provided for deriving suitable analysis/synthesis window pairs. In the encoder, a discrete transform having a function equivalent to the alternate application of a modified Discrete Cosine Transform and a modified Discrete Sine Transform according to the Time Domain Aliasing Cancellation technique or, alternatively, a Discrete Fourier Transform is used to generate frequency-domain transform coefficients. The transform coefficients are nonuniformly quantized by assigning a fixed number of bits and a variable number of bits determined adaptively based on psychoacoustic masking. A technique is described for assigning the fixed bit and adaptive bit allocations. The transmission of side information regarding adaptively allocated bits is not required. Error codes and protected data may be scattered throughout formatted frame outputs from the encoder in order to reduce sensitivity to noise bursts.

341 citations


Journal ArticleDOI
TL;DR: Two algorithms are presented for computing the discrete cosine transform (DCT) on existing VLSI structures and a new prime factor DCT algorithm is presented for the class of DCTs of length N=N/ sub 1/*N/sub 2/, where N/sub 1/ and N/ sub 2/ are relatively prime and odd numbers.
Abstract: Two algorithms are presented for computing the discrete cosine transform (DCT) on existing VLSI structures. First, it is shown that the N-point DCT can be implemented on the existing systolic architecture for the N-point discrete Fourier transform (DFT) by introducing some modifications. Second, a new prime factor DCT algorithm is presented for the class of DCTs of length N=N/sub 1/*N/sub 2/, where N/sub 1/ and N/sub 2/ are relatively prime and odd numbers. It is shown that the proposed algorithm can be implemented on the already existing VLSI structures for prime factor DFT. The number of multipliers required is comparable to that required for the other fast DCT algorithms. It is shown that the discrete sine transform (DST) can be computed by the same structure. >

91 citations


Journal ArticleDOI
TL;DR: A fast recursive algorithm for the discrete sine transform (DST) is developed that can be considered as a generalization of the Cooley-Tukey FFT (fast Fourier transform) algorithm.
Abstract: A fast recursive algorithm for the discrete sine transform (DST) is developed. An N-point DST can be generated from two identical N/2-point DSTs. Besides being recursive, this algorithm requires fewer multipliers and adders than other DST algorithms. It can be considered as a generalization of the Cooley-Tukey FFT (fast Fourier transform) algorithm. The structure of the algorithm is suitable for VLSI implementation. >

80 citations


Journal ArticleDOI
01 Dec 1990
TL;DR: By means of the Kronecker matrix product representation, the 1-D algorithms introduced in the paper can readily be generalised to compute transforms of higher dimensions and are more stable than and have fewer arithmetic operations than similar algorithms proposed by Yip and Rao.
Abstract: According to Wang, there are four different types of DCT (discrete cosine transform) and DST (discrete sine transform) and the computation of these sinusoidal transforms can be reduced to the computation of the type-IV DCT. As the algorithms involve different sizes of transforms at different stages they are not so regular in structure. Lee has developed a fast cosine transform (FCT) algorithm for DCT-III similar to the decimation-in-time (DIT) Cooley–Tukey fast Fourier transform (FFT) with a regular structure. A disadvantage of this algorithm is that it involves the division of the trigonometric coefficients and may be numerically unstable. Recently, Hou has developed an algorithm for DCT-II which is similar to a decimation-in-frequency (DIF) algorithm and is numerically stable. However, an index mapping is needed to transform the DCT to a phase-modulated discrete Fourier transform (DFT), which may not be performed in-place. In the paper, a variant of Hou's algorithm is presented which is both in-place and numerically stable. The method is then generalised to compute the entire class of discrete sinusoidal transforms. By making use of the DIT and DIF concepts and the orthogonal properties of the DCTs, it is shown that simple algebraic formulations of these algorithms can readily be obtained. The resulting algorithms are regular in structure and are more stable than and have fewer arithmetic operations than similar algorithms proposed by Yip and Rao. By means of the Kronecker matrix product representation, the 1-D algorithms introduced in the paper can readily be generalised to compute transforms of higher dimensions. These algorithms, which can be viewed as the vector-radix generalisation of the present algorithms, share the in-place and regular structure of their 1-D counterparts.

79 citations


Proceedings ArticleDOI
03 Apr 1990
TL;DR: Circular convolution-multiplication relationships for the discrete cosine transform (DCT) that are similar to those forThe discrete Fourier transform (DFT) are developed and can be used to filter an image in the frequency domain as an approximation of circular convolution in the spatial domain.
Abstract: Circular convolution-multiplication relationships for the discrete cosine transform (DCT) that are similar to those for the discrete Fourier transform (DFT) are developed. The relations are valid if the filter frequency response is real and even. Two fairly simple relations are developed. The multiplication of the DCT of signal sequence and the DFT of filter sequence results in circular convolution of the folded signal sequence and the filter sequence. Thus, it can be used to filter an image in the frequency domain as an approximation of circular convolution in the spatial domain. >

73 citations


Journal ArticleDOI
TL;DR: An image coding method for low bit rates based on alternate use of the discrete cosine transform and the discrete sine transform on image blocks achieves the removal of redundancies in the correlation between neighboring blocks as well as the preservation of continuity across the block boundaries.
Abstract: An image coding method for low bit rates is proposed. It is based on alternate use of the discrete cosine transform (DCT) and the discrete sine transform (DST) on image blocks. This procedure achieves the removal of redundancies in the correlation between neighboring blocks as well as the preservation of continuity across the block boundaries. An outline of the mathematical justification of the method, assuming a certain first-order Gauss-Markov model, is given. The resulting coding method is then adapted to nonstationary real images by locally adapting the model parameters and improving the block classification technique. Simulation results are shown and compared with the performance of related previous methods, namely adaptive DCT and fast Karhunen-Loeve transform (FKLT). >

73 citations


Patent
11 Aug 1990
TL;DR: In this article, the authors proposed a transform coding scheme where linear transform is performed for input signal series such as image signals, and transform coefficients are quantized from lower frequency compo-nents to higher frequency components in the transform region.
Abstract: This invention relates to a transform coding apparatus where linear transform is performed for input signal series such as image signals, and transform coefficients are quantized from lower frequency compo-nents to higher frequency components in the transform region, and then the quantized transform coefficients are coded and outputted. The quantized transform coefficients being zero are counted, and when the count value exceeds the prescribed threshold value, the quantization is terminated so that high speed in the coding is intended and variation of the code generation rate can be prevented.

67 citations


Journal ArticleDOI
TL;DR: In this article, the authors developed the circular convolution-multiplication relationship for the discrete cosine transform (DCT) similar to that of the discrete Fourier transform (DFT) for the filter frequency response.

63 citations


Journal ArticleDOI
TL;DR: In this paper, a procedure has been developed for automated interpretation of gravity anomalies due to simple geometrical causative sources, viz., a sphere, a horizontal cylinder, and a 2-D vertical prism of large depth extent.
Abstract: Walsh functions are a set of complete and orthonormal functions of nonsinusoidal waveform. In contrast to sinusoidal waveforms whose amplitudes may assume any value between -1 to +1, Walsh functions assume only discrete amplitudes of + or -1 which form the kernel function of the Walsh transform. Because of this special nature of the kernel, computation of the Walsh transform of a given signal is simpler and faster than that of the Fourier transform. The properties of the Fourier transform in linear time are similar to those of the Walsh transform in dyadic time. The Fourier transform has been widely used in interpretation of geophysical problems. Considering various aspects of the Walsh transform, an attempt has been made to apply it to some gravity data.A procedure has been developed for automated interpretation of gravity anomalies due to simple geometrical causative sources, viz., a sphere, a horizontal cylinder, and a 2-D vertical prism of large depth extent. The technique has been applied to data from the published literature to evaluate its applicability, and the results are in good agreement with the more conventional ones.

Journal ArticleDOI
01 Jun 1990
TL;DR: In this paper, a reconstruction method for nonuniformly sampled bandlimited 1-D and 2-D discrete signals is developed for interpolating over regularly spaced points, given irregularly sampled values of the signal.
Abstract: A reconstruction method is developed for nonuniformly sampled bandlimited 1-D and 2-D discrete signals. In several applications one needs to interpolate over regularly spaced points, given irregularly sampled values of the signal. By exploiting the aliasing relationship in the DFT (sampled frequency) domain rather than in the continuous Fourier transform domain one can exactly reconstruct the discrete signal subject to the harmonically limited constraint. This approach is especially attractive in the 2-D case since a novel closed-form interpolation formula is obtained. First, a transform matrix is obtained which relates the uniformly spaced frequency samples to the specified nonuniformly spaced sample values of the signal. This is done in the 1-D as well as in the 2-D case. The required uniformly spaced interpolation is then obtained by forming the inverse DFT. Unlike in the 1-D case, in the 2-D case the transform matrix may not exist even when all the sample points are distinct. Necessary conditions for the existence of the transform matrix have been investigated for the 2-D case. An efficient algorithm to compute the transform matrix has been developed by exploiting the block structure of the matrix characterising the system of equations in the 2-D case. >

Journal ArticleDOI
TL;DR: In this article, a novel interpolation method using the type I discrete cosine transform (DCT-I) is introduced, where the original definition of the DCT is modified to suit this application.
Abstract: A novel interpolation method, using the type I discrete cosine transform (DCT-I), is introduced. The original definition of the DCT-I has been modified to suit this application. Three options for the modified DCT-I are proposed.

Journal ArticleDOI
TL;DR: A novel type of algorithms for the discrete sine transform (DST) are introduced, using a basic trigonometric identity, to realize a successive reduction of the summation size in a simple manner, and therefore cause a very simple structure.

Journal ArticleDOI
TL;DR: The fast algorithm for the (real) Hartly transform is discussed in relation to the established fast algorithmFor the (complex) Fourier transform, compared by timing comparably written programs on a given machine, and the discipline of timing is discussed as an adjunct to complexity analysis.
Abstract: The fast algorithm for the (real) Hartly transform is discussed in relation to the established fast algorithm for the (complex) Fourier transform. The two transforms are compared by timing comparably written programs on a given machine, and the discipline of timing is discussed as an adjunct to complexity analysis. With real data, one Hartley transform program can economically replace such packages as a complex-valued unilateral Fourier transform combined with a real-valued unilateral inverse Fourier transform. The Hartley transform is favorable for fast convolution of real data sets. The utility of spectral analysis into Fourier series throughout physics suggested that the Hartley transform might have less physical significance, but the construction of Hartley diffraction planes in the microwave and optical laboratories, where electromagnetic phase is encoded as real-valued field amplitudes, has revealed interesting complementarity. >

Journal ArticleDOI
TL;DR: In this article, a 2-D systolic array algorithm for the discrete cosine transform (DCT) is presented, which is based on the inverse discrete Fourier transform (DFT) version of the Goertzel algorithm via Horner's rule.
Abstract: A 2-D systolic array algorithm for the discrete cosine transform (DCT) is presented. It is based on the inverse discrete Fourier transform (DFT) version of the Goertzel algorithm via Horner's rule. This array requires N cells and multipliers, takes square root N+2 clock cycles to produce a complete N-point DCT, and is able to process a continuous stream of data sequences. >

Journal ArticleDOI
TL;DR: The Hartley transform (HT) as discussed by the authors is an integral transform similar to the Fourier transform (FT), and it has most of the characteristics of the FT. However, the HT is a real transform and for this reason, since one complex multiplication requires four real multiplications, the discrete HT (DHT) is computationally faster than the discrete FT (DFT).
Abstract: The Hartley transform (HT) is an integral transform similar to the Fourier transform (FT). It has most of the characteristics of the FT. Several authors have shown that fast algorithms can be constructed for the fast Hartley transform (FHT) using the same structures as for the fast Fourier transform. However, the HT is a real transform and for this reason, since one complex multiplication requires four real multiplications, the discrete HT (DHT) is computationally faster than the discrete FT (DFT). Consequently, any process requiring the DFT (such as amplitude and phase spectra) can be performed faster by using the DHT. The general properties of the DHT are reviewed first, and then an attempt is made to use the FHT in some seismic data processing techniques such as one‐dimensional filtering, forward seismic modeling, and migration. The experiments show that the Hartley transform is two times faster than the Fourier transform.

Journal ArticleDOI
TL;DR: A technique is proposed for filtering multidimensional (MD) discrete signals that combines discrete Fourier transform (DFT) and linear difference equation (LDE) methods.
Abstract: A technique is proposed for filtering multidimensional (MD) discrete signals that combines discrete Fourier transform (DFT) and linear difference equation (LDE) methods. A partial P-dimensional DFT (P >

Proceedings ArticleDOI
03 Apr 1990
TL;DR: The fast approximated discrete transform is proposed as a method for reducing the time necessary to compute the discrete transform of a finite-length sequence by discarding the computations in bands that have little or no energy contribution.
Abstract: The fast approximated discrete transform is proposed as a method for reducing the time necessary to compute the discrete transform of a finite-length sequence. It is based on a subband decomposition and can be viewed as a link between the fast transform methods (like the fast Fourier transform), which compute all points in the transform domain, and the variety of methods to evaluate the discrete transforms at a given set of points. The method uses knowledge about the input signal to obtain an approximation to its transform by discarding the computations in bands that have little or no energy contribution. In a number of practical cases the proposed fast approximation is reasonably accurate, and in all cases the method can be iterated to yield the exact transform, if necessary. >

Journal ArticleDOI
TL;DR: In this paper, a method for discrete representation of signals consisting of a cascade of Chebyshev nonuniform sampling (CNS) followed by the discrete cosine transform (DCT) is presented.
Abstract: A method for discrete representation of signals consisting of a cascade of Chebyshev nonuniform sampling (CNS) followed by the discrete cosine transform (DCT) is presented. It is proven that the considered signal samples and the coefficients of the corresponding Chebyshev polynomial finite series are essentially a discrete cosine transform pair. A method for fast computation of the coefficients of the optimum interpolation formula (which minimizes the maximum instantaneous error) is provided. If the signal g(t) is band-limited and has a finite energy, the condition of convergence for interpolation can be deduced. >

Journal ArticleDOI
N.E. Wu1, G. Gu
TL;DR: In this paper, it was shown that uniform rational approximation of nonrational transfer functions can always be obtained by means of the discrete Fourier transform (DFT) as long as such approximants exist.
Abstract: It is shown that uniform rational approximation of nonrational transfer functions can always be obtained by means of the discrete Fourier transform (DFT) as long as such approximants exist. Based on this fact, it is permissible to apply the fast Fourier transform (FFT) algorithm in carrying out rational approximations without being apprehensive of convergence. The DFT is used to obtain traditional approximations for transfer functions of infinite-dimensional systems. Justification is provided for using the DFT in such approximations. It is established that whenever a stable transfer function can be approximated uniformly on the right half-plane by a rational function, its approximants can always be recognized by means of a DFT. >

Patent
19 Apr 1990
TL;DR: In this paper, a discrete Fourier transform operation is performed by a CHIRP-Z transform or a Goertzel's second order Z-transform which can accommodate any number of data lines or values.
Abstract: Magnetic resonance imaging data lines or views are generated and stored in a magnetic resonance data memory (56). The number of views or phase encode gradient steps N along each of one or more phase encode gradient directions is selected (70) to match the dimensions of the region of interest. A discrete Fourier transform algorithm (94) operates on the data in the magnetic resonance data memory to generate an image representation for storage in an image memory (96). Unlike a fast Fourier transform algorithm which requires a N views or data lines, where a and N are integers, the discrete Fourier transform has a flexible number of data lines and data values which can be accommodated. More specifically to the preferred embodiment, the discrete Fourier transform operation is performed by a CHIRP-Z transform or a Goertzel's second order Z-transform which can accommodate any number of data lines or values.

Journal ArticleDOI
TL;DR: The linear complexityL2(G) of a finite groupG is the minimal number of additions, subtractions and multiplications by complex constants of absolute value ≦2 sufficient to evaluate a suitable Fourier transform of ℂG.
Abstract: The linear complexityL2(G) of a finite groupG is the minimal number of additions, subtractions and multiplications by complex constants of absolute value ≦2 sufficient to evaluate a suitable Fourier transform of ℂG. Combining and modifying several classical FFT-algorithms, we show thatL2(G)≦8|G|log2|G| for any finite metabelian groupG.

Proceedings ArticleDOI
21 Mar 1990
TL;DR: This study concentrates on discrete orthogonal transforms such as the discrete Fourier transform (DFT), the discrete cosine transform (DCT), the Walsh-Hadamard transform (WHT), and the Karhunen-Loeve transform (KLT) for low-rate seismic data compression.
Abstract: An investigation of low-rate seismic data compression using transform techniques is presented. This study concentrates on discrete orthogonal transforms such as the discrete Fourier transform (DFT), the discrete cosine transform (DCT), the Walsh-Hadamard transform (WHT), and the Karhunen-Loeve transform (KLT). Uniform and subband transform coding schemes are implemented, and comparative results are given for data rates ranging from 150 to 550 b/s. Results are also compared with existing linear prediction techniques. >

Journal ArticleDOI
TL;DR: A new algorithm is introduced such that a discrete cosine transform by correlations can be applied to any odd prime length DCT and is most suitable for VLSI implementation.
Abstract: A new algorithm is introduced such that we can realise a discrete cosine transform by correlations. This algorithm can be applied to any odd prime length DCT and is most suitable for VLSI implementation.

Journal ArticleDOI
TL;DR: The authors explain theoretically this asymmetrical performance of the DCT as well as the behavior of the modified Hermite transform (MHT) and the discrete Hadamard transform (DHT), which perform symmetrically regardless of the sign of the autocorrelation coefficient.
Abstract: The discrete cosine transform (DCT) is considered as a suboptimum transform for many practical source-coding applications. Autoregressive order 1 (AR(1)) source models are good first approximations to several natural signals. It is known that the performance of the DCT depends on the sign of the autocorrelation coefficient of the AR(1) source. The authors explain theoretically this asymmetrical performance of the DCT as well as the behavior of the modified Hermite transform (MHT) and the discrete Hadamard transform (DHT), which perform symmetrically regardless of the sign of the autocorrelation coefficient. >


Journal ArticleDOI
TL;DR: A FDRT-based algorithm is presented for computing 2-D DFTs, which has the advantages of having the lowest number of multiplications and being more suitable for parallel implementation compared with other related algorithms.
Abstract: The discrete Radon transform (DRT) has been known to convert two-dimensional discrete Fourier transforms (2-D DFTs) into 1-D DFTs. A fast discrete Radon transform (FDRT) algorithm is presented. A FDRT-based algorithm is presented for computing 2-D DFTs, which has the advantages of having the lowest number of multiplications and being more suitable for parallel implementation compared with other related algorithms.

Journal ArticleDOI
TL;DR: In this article, the EM algorithm is applied to each scalar subsystem derived from the state-space model via the discrete sine transform (DST) to obtain a scheme of estimating the AR parameters of transformed image.

Proceedings ArticleDOI
03 Apr 1990
TL;DR: The performances of five discrete orthogonal transforms in speech encryption systems are compared and a figure of merit based on all the four objective measures is formed that gives good correlation to the subjective results of residual intelligibility and recovered speech quality.
Abstract: The performances of five discrete orthogonal transforms in speech encryption systems are compared. The transforms considered are the discrete Fourier transform, discrete cosine transform, Walsh-Hadamard transform, Karhunen-Loeve transform, and discrete prolate spheroidal transform. Four objective measures are used to grade the encryption systems with respect to residual intelligibility and recovered voice quality. A figure of merit based on all the four objective measures is formed. It gives good correlation to the subjective results of residual intelligibility and recovered speech quality. >