Showing papers on "Non-uniform discrete Fourier transform published in 1993"

Image rotation, Wigner rotation, and the fractional Fourier transform

Abstract: In this study the degree p = 1 is assigned to the ordinary Fourier transform The fractional Fourier transform, for example with degree P = 1/2, performs an ordinary Fourier transform if applied twice in a row Ozaktas and Mendlovic [ “ Fourier transforms of fractional order and their optical implementation,” Opt Commun (to be published)] introduced the fractional Fourier transform into optics on the basis of the fact that a piece of graded-index (GRIN) fiber of proper length will perform a Fourier transform Cutting that piece of GRIN fiber into shorter pieces corresponds to splitting the ordinary Fourier transform into fractional transforms I approach the subject of fractional Fourier transforms in two other ways First, I point out the algorithmic isomorphism among image rotation, rotation of the Wigner distribution function, and fractional Fourier transforming Second, I propose two optical setups that are able to perform a fractional Fourier transform

Fractional Fourier transforms and their optical implementation. II

Abstract: The linear transform kernel for fractional Fourier transforms is derived. The spatial resolution and the space–bandwidth product for propagation in graded-index media are discussed in direct relation to fractional Fourier transforms, and numerical examples are presented. It is shown how fractional Fourier transforms can be made the basis of generalized spatial filtering systems: Several filters are interleaved between several fractional transform stages, thereby increasing the number of degrees of freedom available in filter synthesis.

What's that deal with the DCT?

Abstract: Discrete cosine transforms (DCTs) and discrete Fourier transforms (DFTs) are reviewed in order to determine why DCTs are more popular for image compression than the easier-to-compute DFTs. DCT-based image compression takes advantage of the fact that most images do not have much energy in the high-frequency coefficients. It is suggested that DCTs are more popular because fewer DCT coefficients than DFT coefficients are needed to get a good approximation to a typical signal, since the higher-frequency coefficients are small in magnitude and can be more crudely quantized than the low-frequency coefficients. >

Irregular sampling of wavelet and short-time Fourier transforms

Abstract: We obtain irregular sampling theorems for the wavelet transform and the short-time Fourier transform. These sampling theorems yield irregular weighted frames for wavelets and Gabor functions with explicit estimates for the frame bounds.

All optical nonlinear joint fourier transform correlator

Abstract: A fully parallel, self-aligning, nonlinear and all optical joint transform correlator utilizes a photorefractive four wave mixer positioned to record the joint transform spectrum interfering with a reference beam together with a phase conjugator which directs a readout beam at the four wave mixer. The signal readout of the mixer is Fourier transformed and can be recorded upon an image sensor to detect the correlation spots indicating the degree of similarity of the two input images.

Alternatives to the discrete cosine transform for irreversible tomographic image compression

Abstract: Full-frame irreversible compression of medical images is currently being performed using the discrete cosine transform (DCT). Although the DCT is the optimum fast transform for video compression applications, the authors show here that it is outperformed by the discrete Fourier transform (DFT) and discrete Hartley transform (DHT) for images obtained using positron emission tomography (PET) and magnetic resonance imaging (MRI), and possibly for certain types of digitized radiographs. This difference occurs because PET and MRI images are characterized by a roughly circular region D of non-zero intensity bounded by a region R in which the image intensity is essentially zero. Clipping R to its minimum extent can reduce the number of low-intensity pixels but the practical requirement that images be stored on a rectangular grid means that a significant region of zero intensity must remain an integral part of the image to be compressed. With this constraint imposed, the DCT loses its advantage over the DFT because neither transform introduces significant artificial discontinuities. The DFT and DHT have the further important advantage of requiring less computation time than the DCT. >

An efficient recursive algorithm for time-varying Fourier transform

Abstract: An efficient recursive algorithm for computing the time-varying Fourier transform (TVFT) or short-time Fourier transform (STFT) of a time sequence is presented. In this approach, instead of excluding the old samples, their importance is diminished by using all-pole moving windows. This recursive algorithm requires about one half of the computation and storage of the Amin's algorithm. The resulting TVFT does not possess any sidelobes. The performance of the algorithm is illustrated by two numerical examples. >

Hexagonal fast Fourier transform with rectangular output

Abstract: Hexagonal sampling is the most efficient sampling pattern for a two-dimensional circularly bandlimited function. A separable fast discrete Fourier transform (DFT) algorithm for hexagonally sampled data that directly computes output points on a rectangular lattice is reported. No interpolation is required. The algorithm has computational complexity comparable to that of standard two-dimensional fast Fourier transforms. >

Coherent signal power detector using higher-order statistics

Abstract: A signal detection system divides a data sampling run into blocks and perms a fast Fourier transform on each block, sorting results by frequency. Combinations of results of the fast Fourier transform corresponding to each frequency are processed to derive a test statistic which is unbiased by Gaussian noise while including such combinations of results of the fast Fourier transform which would be redundant over other combinations. Information concerning the frequency behavior of the signal derived in the course of detection, is accomplished with increased sensitivity.

Spectral analysis of cervical cells using the discrete Fourier transform.

Abstract: This paper contains results from a preliminary study of spectral analysis techniques applied to the classification of cervical cells from routinely prepared Papanicolaou cervical smears. Experiments were conducted using a subset of 110 normals and 110 dyskaryotic single cell images randomly selected from a larger cell image data base. An assessment was made of the contribution of different regions within a cell image to the frequency spectrum. Three image sets were used, the original image itself plus two derived from it. In the first derived set, only nuclear size and shape were used. In the second set nuclear morphology and texture were included. Nuclear masking was performed using an interactive segmentation procedure. The discrete Fourier transform was applied to each image in the three image sets and classification experiments were performed using 80 features derived from the frequency spectra. An optimum set of features was selected for each experiment by canonical analysis. Good classification results were obtained when features extracted solely from nuclear shape were used. The inclusion of information relating to nuclear texture improved the results. However, inclusion of the extra nuclear region degraded the classifier's ability to discriminate between cell groups.

Efficient computation of the time-domain scattering from bodies of revolution

Abstract: An approach obtaining the time-domain scattering from metal, dielectric, or dielectric-coated bodies of revolution (BORs) is presented. The axial symmetry of the BORs enables a Fourier mode decomposition in the frequency domain, thereby allowing three-dimensional problems to be treated in two dimensions. By implementing the moment method and inverse discrete Fourier transform techniques, the proposed approach enables the time-domain analysis of more complicated objects, such as multiple metal scatterers and dielectric-coated scatterers. The effects of lossy and dispersive materials can be easily included in the time-domain analysis. Several examples are presented to illustrate the capabilities of the proposed approach. >

The poorman's transform: approximating the Fourier transform without multiplication

Abstract: A time-domain to frequency-domain transformation for sampled signals which is computed with only additions and trivial complex multiplications is described. This poorman's transform is an approximation to the usual Fourier transform, obtained by quantizing the Fourier coefficients to the four values (+or-1, +or-j), and is especially useful when multiplication is expensive. For the general case of an N-point quantization, an analytic formula is given for the error in the approximation, which involves only contributions from aliased harmonics. Continuous-time signals are considered; in this case the approximation is exact for bandlimited signals. >

Two-dimensional image reconstruction from Fourier coefficients computed directly from zero crossings

Abstract: Two-dimensional image reconstruction using Fourier coefficients that are computed directly from the sampled representation of zero crossings is demonstrated. A two-dimensional image of dimensions Nx × Ny is interpreted as a set of Ny independent x-space lines (in gray-scale format) that are arranged uniquely along the y direction. Each line has Nx elements. Reconstruction is achieved first by computing the entire set of Ny one-dimensional Fourier transforms from the measured zero crossings using Newton’s formula. Each Nyth line spectra has Nx Fourier coefficients. The inverse Fourier transform is then applied to each of the line spectra to obtain a set of Ny reconstructed x-space lines. The reconstructed image is obtained by arranging the reconstructed lines properly along the y direction.

Audio signal frequency analysis method - using window functions to provide sample signal blocks subjected to Fourier analysis to obtain respective coefficients.

Abstract: The frequency analysis method involves dividing an input signal, represented by discrete sample values, into overlapping blocks using a window function and subjecting them to a Fourier transformation, to obtain a set of coefficients. Each input signal block is evaluated via a set of differing symmetrical window functions of equal length, their Fourier transformations having different bandwidths. The simultaneously generated blocks are used to provide a set of characteristic Fourier transformations, with a coefficient provided for each block, dependent on the window function Fourier transformation bandwidth. The selected coefficients have frequency bands which are only slightly overlapping, or only slightly spaced apart. USE/ADVANTAGE - Accurate simulation of human hearing characteristics. At least first five spectral coefficients processed together, e.g. in fast fourier transformation.

Bispectral analysis and reconstruction in the frequency domain of mono- and bidimensional deterministic sampled signals

Abstract: Deterministic sampled signals bispectra are periodic and hold more information than analog signal bispectra. After showing this difference, the communication presents two algorithms for reconstructing a sampled signal Fourier transform from its bispectrum: the first is a least squares reconstruction method deducing the Fourier transform logarithm from the bispectrum logarithm through a simple average; the second is an algorithm for reconstructing the Fourier transform from a restricted number of values on the bispectrum diagonal slice by a simple resolution of linear equations. The resistance of both algorithms to the measurement noise is given.

Dynamic range enhancement of Fourier transform infrared spectrum measurement using delta sigma modulation.

Abstract: A method for reconstructing a spectrum from a binary interferogram observed with an Fourier transform IR spectrometer is described. With this method an interferogram is quantized with a 1-bit analog-to-digital converter with a differentiator and an integrator. This method, called delta sigma modulation, features an oversampling of a signal at a rate much higher than the Nyquist sampling rate. We show experimental examples of IR spectra reconstructed by the method, which demonstrate the potential applications of the method to Fourier transform IR analysis. The results imply that the method may exceed the dynamic range over the one attainable with a conventional Fourier transform spectrometer with an analog-to-digital converter of a finite bit number.

Using an improved procedure of fast discrete Fourier transform to analyse Mössbauer spectra hyperfine parameters

Abstract: In this paper, the authors propose to supplement the mathematical processing of Mossbauer spectra (MS) by means of Fourier transforms using a regular algorithm with an iteration refinement method. The use of a priori information concerning the solution in the form of the condition of its nonnegativity allows to increase the resolution in the spectrum without the appearance of oscillations characteristic of the solutions obtained by the Fourier transform method alone. The MS of Fe-Si alloys of low concentration were processed according to the given computational scheme to evaluate the influence of the II and III coordination shells on the parameters of the hyperfine interaction on the Fe nucleus.

2 – Fourier analysis

Abstract: Publisher Summary This chapter discusses Fourier analysis and associated transform methods for both discrete-time and continuous-time signals and system. Fourier methods are based on using real or complex sinusoids as basic functions, and they allow signals to be represented in terms of sums of sinusoidal components. In order for a digital computer to manipulate a signal, the signal must be sampled at a chosen sampling rate, 1/TS, giving rise to a set of numbers called a sequence. Special analysis and many other applications often require discrete Fourier transforms (DFTs) to be performed on data sequences in real time and on contiguous sets of input samples. The theory of discrete-time, linear, time invariant systems forms the basis for digital signal processing, and a discrete-time system performs an operation on the input signal according to defined criteria to produce a modified output signal. The z-transform of the sum of two sequences multiplied by arbitrary constants is the sum of the z-transforms of the individual sequences, where Z represents the z transform operator. Data windows are introduced to smooth out the ripples introduced by the rectangular window.

Optical wavelet transform classifier with positive real Fourier transform wavelets

Abstract: A wavelet transform is used to transform data so as to improve discrimination between objects for classification. The wavelet transform is implemented in optics using the Fourier transform of the wavelets. Further, the wavelet chosen has a Fourier transform that is real and positive to avoid the use of holographic techniques. An optical setup is described in which an electronic analog sensor signal drives an acousto-optic cell that controls the intensity of an argon laser. A mechanical scanner writes the information as a line onto a spatial light rebroadcaster module containing an optical liquid crystal light valve. A lens system expands the line into a 2-D array. A real positive Fourier transform wavelet filter is placed in the Fourier transform plane of a 4- f correlator. An optical demonstration shows the formation of a wavelet transform and agreement with computer simulations. An approximation of the inverse wavelet transform is possible using only a real filter, and this is demonstrated in an optical experiment.

Accurate determination of frequency and angle of arrival from undersampled signals

Abstract: The problem of determining the frequency and angle of arrival of an unknown signal that is undersampled when incident on a linear array is considered. A simple technique to deal with the ambiguities arising from undersampling is coupled with classic monopulse techniques to provide excellent accuracy, even in the presence of noise. Excellent results were obtained in a computer simulation. Sampling was done at 465.45 MHz and 568.88 MHz with a 512 point discrete Fourier transform. >

Accurate interpolation via the discrete W transform

Abstract: Two transform interpolation schemes which take advantage of the periodicity and antiperiodicity of different types of discrete W transform (DWT) are proposed. Results show that these new schemes achieve much higher accuracy than the standard approach.

Electromagnetic analysis of complex microstrip structures using a CG-FFT scheme

Abstract: This work presents the extensions required in a Conjugate Gradient Fast Fourier Transform (CG-FFT) scheme to obtain scattering parameters of arbitrary geometry planar microstrip structures directly from the equivalent currents. These extensions imply the development of an appropriate method to fed and load the system. A new set of basis functions which improves the speed of convergence is also introduced. Besides, almost all computations are performed in the spectral domain where operators have simpler forms. Results are compared with measurements. The method is accurate and efficient from a computational point of view.

Determination of sampling points for nearly DFT-equivalent almost-periodic fourier transforms

Abstract: This paper presents analytical formulas for choosing sampling points required for almost-periodic Fourier transforms (APFT) of waveforms for the case of two fundamental frequencies. The generated sampling points are such that the resulting APFT is an optimal approximation of a corresponding two-dimensional discrete Fourier transform with respect to phase error, resulting in a numerically well conditioned transformation. The formulas may easily be implemented into existing APFT-based software packages.