Showing papers on "Discrete Fourier transform published in 1996"

Fourier transforms of colour images using quaternion or hypercomplex, numbers

Abstract: The 2D quaternion, or hypercomplex, Fourier transform is introduced. This transform makes possible the handling of colour images in the frequency domain in a holistic manner, without separate handling of the colour components, and it thus makes possible very wide generalisation of monochrome frequency domain techniques to colour images.

Identifying Fabric Structures with Fast Fourier Transform Techniques

Abstract: The Fast Fourier Transform (FFT) plays a very important role in image processing and pattern recognition. Since a woven fabric consists of regular repeating units, the FFT is particularly useful fo...

Selective discrete Fourier transform algorithm for time-frequency analysis: method and application on simulated and cardiovascular signals

Abstract: The Selective Discrete Fourier transform (DFT) Algorithm [SDA] method for the calculation and display of time-frequency distribution has been developed and validated. For each time and frequency, the algorithm selects the shortest required trace length and calculates the corresponding spectral component by means of DFT. This approach can be extended to any cardiovascular related signal and provides time-dependent power spectra which are intuitively easy to consider, due to their close relation to the classical spectral analysis approach. The optimal parameters of the SDA for cardiovascular-like signals were chosen. The SDA perform standard spectral analysis on stationary simulated signals as well as reliably detect abrupt changes in the frequency content of nonstationary signals. The SDA applied during a stimulated respiration experiment, accurately; detected the changes in the frequency location and amplitude of the respiratory peak in the heart rate (HR) spectrum. It also detected and quantified the expected increase in vagal tone during vagal stimuli. Furthermore, the HR time-dependent power spectrum displayed the increase in sympathetic activity and the vagal withdrawal on standing. Such transient changes in HR control would have been smeared out by standard heart rate variability (HRV), which requires consideration of long trace lengths. The SDA provides a reliable tool for the evaluation and quantification of the control exerted by the Central Nervous System, during clinical and experimental procedures resulting in nonstationary signals.

Generalized FFTS - A Survey of Some Recent Results

Abstract: In this paper we survey some recent work directed towards generalizing the fast Fourier transform (FFT). We work primarily from the point of view of group representation theory. In this setting the classical FFT can be viewed as a family of efficient algorithms for computing the Fourier transform of either a function defined on a finite abelian group, or a bandlimited function on a compact abelian group. We discuss generalizations of the FFT to arbitrary finite groups and compact Lie groups.

Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm

Abstract: A method for the calculation of the fractional Fourier transform (FRT) by means of the fast Fourier transform (FFT) algorithm is presented. The process involves mainly two FFT’s in cascade; thus the process has the same complexity as this algorithm. The method is valid for fractional orders varying from −1 to 1. Scaling factors for the FRT and Fresnel diffraction when calculated through the FFT are discussed.

Blind equalization of OFDM systems based on the minimization of a quadratic criterion

Abstract: Classical multicarrier systems based on the discrete Fourier transform (DFT) make use of a "guard interval" (GI) in order to enable a low complexity equalization scheme. This "guard interval" consists of a redundant prefix cyclically appended to each bloc of modulated symbols so as to exploit the cyclic convolution property of the DFT. Therefore, besides decreasing the useful transmitted symbol rate, this technique is very specific to DFT-based OFDM systems. In order to implement a digital modulator, an oversampled version of the continuous signal that would be produced by the all-analog ideal modulator is often computed. This amounts to appending null symbols to the block of symbols to be modulated. This work shows that forcing the presence of these null symbols at the appropriate places on the receiver side is sufficient to equalize the channel. Here, a linear equalizer is adapted by minimizing a quadratic criterion based on the energy of the subband signals that should be zero. Since no knowledge upon the "useful data" is required, this method performs blind equalization. Moreover, it requires neither a guard interval nor any reference symbol. As a result, for a given channel bit-rate budget, the data rate is increased.

Accuracy of the Discrete Fourier Transform and the Fast Fourier Transform

Abstract: Fast Fourier transform (FFT)-based computations can be far more accurate than the slow transforms suggest. Discrete Fourier transforms computed through the FFT are far more accurate than slow transforms, and convolutions computed via FFT are far more accurate than the direct results. However, these results depend critically on the accuracy of the FFT software employed, which should generally be considered suspect. Popular recursions for fast computation of the sine/cosine table (or twiddle factors) are inaccurate due to inherent instability. Some analyses of these recursions that have appeared heretofore in print, suggesting stability, are incorrect. Even in higher dimensions, the FFT is remarkably stable.

Calculation of Demagnetizing Field Distribution Based on Fast Fourier Transform of Convolution

Abstract: It is confirmed that the calculation of the demagnetizing field in micromagnetic simulations can be accelerated significantly by using the discrete convolution theorem and the fast Fourier transform (FFT). When the magnetization distribution is periodic, application of the theorem to the demagnetizing field calculation is straightforward. Unlike the previously reported FFT method which is based on the continuous Fourier transform of the demagnetizing field, the method can also be used in the case of non-periodic magnetization structures. It is also confirmed that the result obtained using the new FFT method coincides with that of the conventional direct method, as expected. The principle of calculation and the results of one- and two-dimensional calculations which show the validity and effectiveness of the developed method are presented.

The nonuniform discrete Fourier transform and its applications in filter design. II. 2-D

Abstract: For part I see ibid., vol. 43, no. 6, p. 422-33 (1996). The concept of the nonuniform discrete Fourier transform (NDFT) is extended to two dimensions to provide a basic framework for nonuniform sampling of 2-D sequences in the frequency domain. The 2-D NDFT of a sequence of size N/sub 1//spl times/N/sub 2/ is defined as samples of its 2-D z-transform evaluated at N/sub 1/N/sub 2/ distinct points located in the 4-D (z/sub 1/, z/sub 2/) space. These points are chosen appropriately so that the inverse transform exists. We discuss two special cases in which the choice of the sampling points is constrained so that the 2-D NDFT matrix is guaranteed to be nonsingular, and the number of operations required for computing its inverse is reduced, The 2-D NDFT is applied to nonuniform frequency sampling design of 2-D finite-impulse-response (FIR) filters. Nonseparable filters with good passband shapes and low peak ripples are obtained. This is illustrated by design examples, in which 2-D filters with various shapes are designed and compared with those obtained by other existing methods.

The discrete periodic Radon transform

Abstract: In this correspondence, a discrete periodic Radon transform and its inversion are developed. The new discrete periodic Radon transform possesses many properties similar to the continuous Radon transform such as the Fourier slice theorem and the convolution property, etc. With the convolution property, a 2-D circular convolution can be decomposed into 1-D circular convolutions, hence improving the computational efficiency. Based on the proposed discrete periodic Radon transform, we further develop the inversion formula using the discrete Fourier slice theorem. It is interesting to note that the inverse transform is multiplication free. This important characteristic not only enables fast inversion but also eliminates the finite-word-length error that may be generated in performing the multiplications.

Effect of fractional Fourier transformation on time-frequency distributions belonging to the Cohen class

Abstract: We consider the Cohen (1989) class of time-frequency distributions, which can be obtained from the Wigner distribution by convolving it with a kernel characterizing that distribution. We show that the time-frequency distribution of the fractional Fourier transform of a function is a rotated version of the distribution of the original function, if the kernel is rotationally symmetric. Thus, the fractional Fourier transform corresponds to rotation of a relatively large class of time-frequency representations (phase-space representations), confirming the important role this transform plays in the study of such representations.

Formant analysis using mixtures of Gaussians

Abstract: The paper describes a new formant analysis technique whereby the formant parameters are represented in the form of Gaussian mixture distributions. These are estimated from the discrete Fourier transform (DFT) magnitude spectrum of the speech signal. The parameters obtained are the means, variances and the masses of the density functions, which are used to calculate centre frequencies, bandwidths and amplitudes of formants within the spectrum. In order to better fit the mixture distributions various modifications to the DFT magnitude spectrum, based on simple models of perception, were investigated. These include reduction of dynamic range, cepstral smoothing, use of the Mel scale and pre-emphasis of speech. Results are presented for these as well as formant tracks from analysing speech using the final formant analysis system.

Computing partial DFT for comb spectrum evaluation

Abstract: A comb spectrum evaluation problem arises in the (de)modulation for orthogonal frequency division multiplexing-based (OFDM-based) multichannel communication system. Efficient algorithms for this special type of partial discrete Fourier transform (DFT) computation are studied. For an M-component comb spectrum evaluation with transform length N, it is shown that only O(N+MlogM) multiplications are needed, compared with O(NlogM) multiplications necessary for a narrowband spectrum evaluation. Pruning radix-2 decimation-in-time fast Fourier transform (FFT) requires only (N/4+M/2log/sub 2/M-M) nontrivial complex multiplications. The frequency shift technique has also been applied to allow a modularized mixed-radix structure for the computation of comb spectrum with an initial component not starting from zero frequency point.

Analysis of the numerically implemented angular spectrum approach based on the evaluation of two‐dimensional acoustic fields. Part I. Errors due to the discrete Fourier transform and discretization

Abstract: The numerically implemented angular spectrum approach (ASA) is investigated based on the evaluation of the 2‐D fields radiated by striplike planar sources and 1‐D focusing linear‐phased arrays with different aperture sizes and on the comparison with those obtained by using the analytical solution derived with the point spread function method. Since the discrete Fourier transform (DFT) of a finite‐size source produces an angular spectrum with spatial frequency aliasing and discretization of the source in even‐numbered samples causes a half‐sample length phase shift error in the angular spectrum, the effects of the aliasing and the phase shift on calculation accuracy are dealt with. The results show that the frequency aliasing causes an overestimation of the near field and the phase shift distorts the results. A numerical algorithm for eliminating the aliasing is proposed and correction of the phase shift is confirmed to be necessary. The algorithm proposed can completely remove the errors and obtain exact ...

A 64-point Fourier transform chip for video motion compensation using phase correlation

Abstract: Details of a new low power fast Fourier transform (FFT) processor for use in digital television applications are presented. This has been fabricated using a 0.6-/spl mu/m CMOS technology and can perform a 64 point complex forward or inverse FFT on real-time video at up to 18 Megasamples per second. It comprises 0.5 million transistors in a die area of 7.8/spl times/8 mm/sup 2/ and dissipates 1 W. The chip design is based on a novel VLSI architecture which has been derived from a first principles factorization of the discrete Fourier transform (DFT) matrix and tailored to a direct silicon implementation.

Pattern collation apparatus based on spatial frequency characteristics

Abstract: In a pattern processing apparatus, a registration data preparation unit performs two-dimensional discrete Fourier transform of image data of a registration pattern to prepare registration Fourier image data. A collation data preparation unit performs two-dimensional discrete Fourier transform of image data of a collation pattern to prepare collation Fourier image data. A data synthesizing unit synthesizes the registration Fourier image data with the collation Fourier image data, both of which consist of phase and amplitude information, to output first synthesized Fourier image data. An image processing unit performs Fourier transform of the first synthesized Fourier image data to output second synthesized Fourier image data representing intensities of correlation components. A pattern collation unit collates the registration pattern with the collation pattern on the basis of the intensities of the correlation components of pixels in a correlation component area set in the second synthesized Fourier image data. An amplitude suppressing processing unit performs amplitude suppressing processing of one of the registration/collation Fourier image data and the first synthesized Fourier image data to extract only the phase information.

Discrete Fourier Transform

Abstract: In chapter 6, we investigated the definition and properties of the discrete-time Fourier transform X(e jω ), with ω being a continuous frequency variable, and found it to be very useful for analyzing a wide variety of signals and systems of theoretical interest. However, much of the practice of digital signal processing is done in computers where we cannot evaluate a continuum of frequencies ω, nor can we input and store an infinite-duration sequence x(n). Hence, for actual data sequences, as opposed to theoretically defined signals, we cannot compute the Fourier transform, in general.

Optical illustration of a varied fractional Fourier-transform order and the Radon–Wigner display

Abstract: Based on an all-optical system, a display of a fractional Fourier transform with many fractional orders is proposed. Because digital image-processing terminology is used, this display is known as the Radon–Wigner transform. It enables new aspects for signal analysis that are related to time- and spatial-frequency analyses. The given approach for producing this display starts with a one-dimensional input signal although the output signal contains two dimensions. The optical setup for obtaining the fractional Fourier transform was adapted to include only fixed free-space propagation distances and variable lenses. With a set of two multifacet composite holograms, the Radon–Wigner display has been demonstrated experimentally.

Frequency spectrum analyzing apparatus and transmitter characteristics measuring apparatus using the same

Abstract: In a frequency spectrum analyzing apparatus which utilizes a DFT (discrete Fourier transform) filter bank, and can detect the peak value of the amplitude of an input signal, a time sequence signal generation circuit samples an input signal to convert it into a time sequence signal. A memory stores the time sequence signal. The DFT filter bank receives a time sequence signal read out from the memory by a control device, and performs DFT filter bank processing corresponding to a plurality of desired frequencies while shifting a window function by a predetermined number of sample point units. An absolute value calculation unit calculates absolute values in units of outputs from the DFT filter bank. A peak value detection unit detects peak values in units of outputs from the absolute value calculation unit. A display device displays the frequency spectrum on the basis of these peak values.

The discrete fractional Fourier transformation

Abstract: Based on the fractional Fourier transformation of sampled periodic functions, the discrete form of the fractional Fourier transformation is obtained. It is found that for a certain dense set of fractional orders it is possible to define a discrete transformation. Also, for its efficient computation a fast algorithm, which has the same complexity as the FFT, is given.

Constrained imaging: overcoming the limitations of the Fourier series

Abstract: Magnetic resonance imaging (MRI) is usually implemented as a Fourier transform-based technique. During data acquisition, spatially resolved information relating to spin density, relaxation rates, chemical shifts, and other parameters is phase and frequency encoded in the measured data. Image reconstruction is accomplished through the use of the Fourier series model, which can be evaluated efficiently using a fast Fourier transform (FFT) algorithm. Theoretically, the Fourier series is capable of producing perfect images if the data space (often called k-space) is sufficiently covered. In practice, several problems arise with this model due to finite sampling. Specifically, finite sampling leads to a truncation or the Fourier series, which results in image blurring and ringing. Image blurring is attributed to a loss of spatial resolution. In fact, with the Fourier series model, the resulting image resolution is limited to roughly the reciprocal of the frequency interval over which the data are sampled. The ringing artifact is due to the well-known Gibbs phenomenon, which is more pronounced for images with sharp edges. In order to overcome these limitations associated with the direct application of the Fourier series model, many alternatives have been proposed in the past decade to incorporate a priori information into the imaging process. This article discusses the constrained imaging concept. Specifically, the authors review 3 model-based imaging techniques that the authors have developed in the past few years. An essential feature of these methods is that a parametric model in the form of a generalized series is superimposed on the underlying measured data or image.