Showing papers on "Discrete sine transform published in 1989"

The Fourier transform

Abstract: To calculate a transform, just listen. The ear automatically performs the calculation, which the intellect can execute only after years of mathematical education. The ear formulates a transform by converting sound-the waves of pressure traveling through time and the atmosphere-into a spectrum, a description of the sound as a series of volumes at distinct pitches. The brain turns this information into perceived sound. Similar operations can be done by mathematical methods on sound waves or virtually any other fluctuating phenomenon, from light waves to ocean tides to solar cycles. These mathematical tools can decompose functions representing such fluctuations into a set of sinusoidal components-undulating curves that vary from a maximum to a minimum and back, much like the heights of ocean waves. The Fourier transform is a function that describes the amplitude and phase of each sinusoid, which corresponds to a specific frequency. The Fourier transform has become a powerful tool in diverse fields of science. In some cases, the Fourier transform can provide a means of solving unwieldy equations that describe dynamic response to electricity, hear or light. In other cases, it can identify the regular contributions to a fluctuating signal, thereby helping to make sense of observations inmore » astronomy, medicine and chemistry.« less

The use of orthogonal transforms for improving performance of adaptive filters

Abstract: It has been previously shown that a real-time decomposition of the incoming signal into a set of partially uncorrelated components via an orthogonal transform, and a subsequent adaptation on these individual components, leads to faster convergence rates. Here, transform domain processing is characterized by the effect of the transform on the shape of the mean-square error surface. It is shown that the effect of an ideal transform is to convert equal error contours that are initially hyperellipses in the parameter space into hyperspheres. Five specific real-valued orthogonal transforms are compared in terms of learning characteristics and computational complexity. Since the Karhunen-Loeve transform (KLT) is the ideal transform for this application, and since the KLT is defined in terms of the statistics of the input signal, it is certain that no fixed-parameter transform can deliver optimal learning characteristics for all input signals. However, the simulations suggest that transforms can be found which give much improved performance in a given situation. >

The discrete Fourier transform method of solving differential-integral equations in scattering theory

Abstract: An accurate and efficient numerical method is presented for solving many differential-integral equations arising from electromagnetic scattering theory. It uses the discrete Fourier transform technique to treat both the derivatives and the convolution integrals which often appear in these equations. As a consequence, this method is extremely simple to implement, uses less computer memory than comparable methods, and yields accurate predictions. The differential-integral equation is recast into a periodic form conducive to application of the discrete Fourier convolution theorem. The differential operators are approximated by appropriate finite-difference and discrete-convolution operators. All these quantities are computed by using the fast Fourier transform. An approximate solution is obtained by using the conjugate gradient method. Results are compared to experimental data or analytical solutions for a 3 lambda *3 lambda metal plate (where lambda is the wavelength), a homogeneous and a layered infinite circular dielectric cylinder, and a dielectric sphere. The accuracy of the method is further illustrated by comparing predictions with independent measurements by R.A. Ross (1966) on a 2 lambda *1 lambda metal plate at grazing incidence. In all cases, agreement is excellent. >

Nonlinear matched filtering

Abstract: A new class of nonlinear matched filters is discussed These filters involve the transformation of the signal spectrum and the filter transfer function through a nonlinearity before they are multiplied in the transform domain The resulting filter structures can be considered to be analogous to three-layer neural nets They have better performance in terms of signal discrimination and lack of false correlation signals and artifacts than previously known filters The matched filters are further subdivided into two major classes according to whether the filtering is based on a discrete Fourier transform (DFT) or a real discrete Fourier transform (RDFT) The DFT and the RDFT are approximations to the complex and real Fourier transforms, respectively The RDFT-based filtering gives better performance in terms of signal discrimination and lack of false correlation signals and artifacts than the DFT-based filtering

Input and output index mappings for a prime-factor-decomposed computation of discrete cosine transform

Abstract: A formal direct derivation of the prime-factor-decomposed computation algorithm is presented. The derivation is direct in the sense that it is based on the real cosine function without resort to the discrete Fourier transform expressions or the complex functions. Based on the equations obtained from the derivation, input and output index mappings are introduced in the form of tables. This tabulation enables any prime-factor-decomposable discrete cosine transform (DCT) to be implemented in a straight-forward manner. The use of the index mapping tables is demonstrated for the 12-point DCT. >

Efficient computation of the discrete pseudo-Wigner distribution

Abstract: A description is given of a novel algorithm, the fast Fourier transform in part (FFTP), for the computation of the discrete pseudo-Wigner distribution (DPWD). The FFTP computes the cosine and sine parts of the discrete Fourier transform (DFT) separately by employing real inverse sinusoidal twiddle factors. Unlike the conventional methods which directly utilize the complex DFT, the FFTP yields real output since the DPWD is always real. In addition, the new method reduces the computational cost by making full use of symmetries and removing redundancies in the FFTP computation. The authors also describe a simple algorithm for computing the discrete Hilbert transform (DHT) to produce the nonaliased DPWD. A pipeline structure for real-time and a bulk processing technique for offline implementations of the method are presented. >

Computation of discrete Hilbert transform through fast Hartley transform

Abstract: A fast algorithm is proposed to compute the discrete Hilbert transform via the fast Hartley transform (FHT). Instead of the conventional fast Fourier transform (FFT) approach, the processing is carried out entirely in the real domain. Also, since many efficient FHT algorithms exist, the computation complexity can be greatly reduced from two complex FFTs into two real FHTs. >

Trade-off's in the computation of mono- and multi-dimensional DCT's

Abstract: An overview of some alternative algorithms for one- and two-dimensional DCTs (discrete cosine transforms) is given. Operation counts are derived for typical examples useful in image processing. It is shown that it is possible to generalize the 2-D schemes to 3-D DCTs as well. The result is that a 3-D DCT can be obtained from a 3-D DFT (discrete Fourier transform) of the same size on reals at the cost of permutations and O(3/2N/sup 3/) multiplications. The scheme involves rotations on eight output points at a time. Improvements through scaling are discussed, and implementation issues (both in hardware and software) are addressed. >

A Fast Recursive Two Dimensional Cosine Transform

Abstract: This paper presents a recursive, radix two by two, fast algorithm for computing the two dimensional discrete cosine transform (2D-DCT). The algorithm allows the generation of the next higher order 2D-DCT from four identical lower order 2D-DCT's with the structure being similar to the two dimensional fast Fourier transform (2D-FFT). As a result, the method for implementing this recursive 2D-DCT requires fewer multipliers and adders than other 2D-DCT algorithms.

A new look at DCT-type transforms

Abstract: A computationally efficient DCT- (discrete-cosine-transform) type orthogonal transform obtained by using a construction method developed by W. Kou and H. Zu (1986) is proposed. A recursive relation exists between a higher order and a lower order form of the kernel matrix of the transform and includes the DCT matrices as submatrices. The characteristics of the proposed transform and several fast transform algorithms are discussed. The proposed transform offers a higher computational efficiency than the traditional even discrete cosine transform and yields a mean-squared error close to that of the DCT. Since the HDCT has window spectral structures, it can be used in signal filtering and speech and image processing. >

Efficient modeling of dominant transform components representing time-varying signals

Abstract: The mixed transform representation of time-varying signals uses partial sets of basis functions from the discrete Fourier transform (DFT) and the Walsh-Hadamard transform. The location, magnitude, and phase of the transform components have to be specified for proper signal reconstruction. A least-squares IIR (infinite impulse response) algorithm, in the transformed domains, which fits each of the retained subsets of the complex transform components accurately, is presented. The IIR function, while characterized by real coefficients about twice the number of the retained complex transform components, carries enough location, magnitude, and phase information for accurate signal reconstruction. To illustrate the technique's accuracy and efficiency, its application to model the DFT part of a voice speech segment is given. >

The Fourier transform and the discrete Fourier transform

Abstract: The authors establish sharp error bounds for the error committed in computing N values of the Fourier transform of a square summable function by means of the discrete Fourier transform. No assumptions are made as to the bandwidth or desired frequency resolution. Instead their error bound depends explicitly on five parameters: the number N of sampling points, the interval T where the samples are taken, the interval Omega where the Fourier transform is being approximated, and two extra parameters that account for local averaging in the time and frequency domains respectively. The resulting error bound is independent of the function in question and they find that, for an appropriate model of sampling, the error bound is minimised as a function of T, Omega by the Nyquist-Shannon choice 4T Omega =2N+1. They finally consider how these bounds are affected by some a priori knowledge about the class of functions under discussion. This is done in detail for band-limited functions.

New fast algorithm to compute two-dimensional discrete Hartley transform

Abstract: A new fast algorithm for computing the two-dimensional discrete Hartley transform is presented. This algorithm requires the lowest number of multiplications compared with other related algorithms.

Fast cepstrum analysis using the Hartley transform

Abstract: The use of the Hartley transform (HT) in cepstrum analysis, as a substitute for the more commonly used Fourier transform (FT), is examined. With this substitution, the input to the cepstrum must be in the real domain only. The benefits of using the HT are approximately 50% less data memory required and approximately 40% faster program execution, at no loss in accuracy. >

Switched-capacitor DFT and IDFT circuit

Abstract: The switched-capacitor realization of the discrete Fourier transform (DFT) is treated in this paper as well as the inverse discrete Fourier transform (1DFT). The output of the DFT has a sinusoidal waveform including the amplitude and phase information of the required spectra. These spectra are given simultaneously and almost in real time. The output of the 1DFT is given merely by adding DFT outputs. Furthermore, the circuit configuration of this system-from input to DFT, from DFT to 1DFT, and from 1DFT to output-is a very simple configuration constructed by a non-recursive filter circuit.

2D systolic solution to discrete Fourier transform

Abstract: A number of systolic architectures have appeared over the past few years for performing the discrete Fourier transform (DFT) and fast Fourier transform (FFT) algorithms, using both linear and orthogonal processing networks. The paper shows how a rectangular array of N CORDIC (co-ordinate digital computer) processing elements can be used to carry out an efficient two-dimensional systolic implementation of the N-point DFT, offering highly attractive throughput rates in relation to other N-processor solutions, such as the conventional linear systolic array. >

Exact Interpolation of Apodized, Magnitude-Mode Fourier Transform Spectra

Abstract: A procedure is developed for the exact interpolation of apodized, magnitude-mode Fourier transform (FT) spectra. The procedure gives the true center frequency, i.e., the location of the continuous peak, from just the largest three discrete intensities in the discrete magnitude spectrum. The procedure is applicable for the peaks in the apodized magnitude spectrum of time signal of the form f(t) = cos(ωt) exp(–t/τ). There are no restrictions on the value of the damping ratio T/τ. The procedure is demonstrated for the sine-bell and Hanning windows and is gener-alizable to other windows which consist of a sum of constants and sine/cosine terms. This includes the majority of commonly used windows.

Data compression properties of the Hartley transform

Abstract: The data compression performance of the Hartley transform on a Markov-1 signal is theoretically compared to that of the Fourier transform. Covariance distribution and residue correlation measurements have been computed for the Hartley, Fourier, and cosine transforms. The Hartley transform achieves better coding performance than the Fourier transform, but is inferior to the cosine transform. >

Bidiagonal factorization of Fourier matrices and systolic algorithms for computing discrete Fourier transforms

Abstract: An algorithm is presented for factoring Fourier matrices into products of bidiagonal matrices. These factorizations have the same structure for every n and make possible discrete Fourier transform (DFT) computation via a sequence of local, regular computations. A parallel pipeline technique for computing sequences of k-point DFTs, for every k >

Inversion of the k-plane transform by orthogonal function series expansions

Abstract: The k-plane transform encompasses both the X-ray and Radon transforms. A series inversion which operates in the unified setting of the k-plane transform is presented. The author shows that with respect to either the Jacobi or the associated Laguerre polynomial bases for square integrable point functions, the k-plane transform assumes a block-diagonal-like form. Additionally, estimates are given for the minimum number of discretely sampled direction sets at which the k-plane transform must be known in order to recover a point function up to a given degree.

Fast computation of real discrete Fourier transform for any number of data points

Abstract: In many applications, it is desirable to have a fast algorithm (FRFT) for the computation of the real discrete Fourier transform (RDFT) for any number of data points. To achieve this, the two-factor Cooley-Tukey FRFT algorithm is developed and expressed in terms of matrix factorization using Kronecker products. This is generalized to any number of factors. Each factor M involves the computation of size M RDFTs, which is carried out by the best size M FRFT algorithm available. >

New 2D systolic array algorithm for DCT/DST

Abstract: We present an algorithm and the architecture of a 2D systolic array processor for the DCT (discrete cosine transform) and the DST (discrete sine transform). It is based on the IDFT (inverse discrete Fourier transform) version of the Goertzel algorithm via Homer's rule. This 2D systolic array for the DCT/DST can be met by achieving a systematic technique for transforming algorithms to specific forms for mapping onto 2D systolic arrays.

The weighted redundancy transform

Abstract: An algorithm is introduced for computing the multidimensional finite Fourier transform. The algorithm can be applied to data samples of any size. In most cases, it offers a substantial reduction in the computational complexity. >

New fast algorithm for two-dimensional discrete Fourier transform DFT (2/sup n/; 2)

Abstract: We present a new fast algorithm for computing the two-dimensional discrete Fourier transform DFT(2n; 2) using the fast discrete cosine transform algorithm. The algorithm has a lower number of multiplications and additions compared with other published algorithms for computing the two-dimensional DFT. Because it uses only real multiplications, the algorithm is more suitable for real input data.

Computation of discrete fourier transform using recursive techniques

Abstract: The discrete Fourier transform is continuously calculated at input signal sample rate using recursive filtering, rather than transversal filtering. This reduces the number of complex digital multiplications per computational cycle to N, the number of spectral components in the discrete Fourier transform, where rectangular truncation window or a new exponential window is used. Where a triangular truncation window is used the number of complex digital multiplications per computational cycle is reduced to 2N.

Analysis of errors in the computation of Fourier coefficients using the arithmetic Fourier transform (AFT) and summation by parts (SBP)

Abstract: The computational complexity and the effects of quantization and sampling instant errors in the arithmetic Fourier transform (AFT) and the summation by parts discrete Fourier transform (SBP-DFT) algorithms are examined. The relative efficiency of the AFT and SBP-DFT algorithms is demonstrated by comparing the number of multiplications, additions, memory storage locations, and input signal samples as well as the latency time and level of parallelism of these two methods with that of more conventional single-output DFT and multiple-output fast Fourier transform (FFT) routines. The error response of the kth Fourier bin of these algorithms is analyzed as a function of increasing levels of input signal sampling errors in the AFT and coefficient quantization errors in the SBP-DFT. It is demonstrated that invalid assumptions on the bandwidth of the input signal will cause aliasing errors to occur in the AFT spectrum that are different from the aliasing errors that occur in the DFT. >

Approximate Fourier transform using square waves

Abstract: The technique of spectral analysis, by truncated approximations to the sine and cosine functions, is evaluated using a three-level approximation (-1, 0, +1), where the -1 and +1 sections are of equal length, and the length of the zero level is variable. An optimum ratio is found where the zero level together cover 120 degrees of the cycle length, to give the least amount of ringing, or leakage. The transform that converts the square-wave spectral estimates to the Fourier coefficients is evaluated. In applications where spectral estimates are required only for limited frequency bands, the technique is flexible and efficient. This analysis is also well suited to irregularly spaced samples, and for direct application to analogue signals. >

A method for computing the DFT of vector quantized data

Abstract: A method is presented for computing the discrete Fourier transform (DFT) of data compressed using vector quantization (VQ). The VQ compressed data are not reconstructed before use; instead, a codebook that has been processed with the DFT (discrete Fourier transform) algorithm is used for VQ reconstruction. An overlap-and-add technique is used to combine the processed reconstruction codebook vectors to give the DFT directly. The technique is suitable for both one-dimensional and multidimensional DFTs or, in general, any linear process. The technique is called the computation compression technique (CCT). The CCT implementation yields exactly the same result as if the compressed data had been reconstructed and the DFT performed on the data directly. CCT convolution on a 68020/6881-based computer is described. Speedups of two orders of magnitude are obtained. >