Showing papers on "Fractional Fourier transform published in 1992"

Linear and quadratic time-frequency signal representations

Abstract: A tutorial review of both linear and quadratic representations is given. The linear representations discussed are the short-time Fourier transform and the wavelet transform. The discussion of quadratic representations concentrates on the Wigner distribution, the ambiguity function, smoothed versions of the Wigner distribution, and various classes of quadratic time-frequency representations. Examples of the application of these representations to typical problems encountered in time-varying signal processing are provided. >

An efficient algorithm for the calculation of a constant Q transform

Abstract: An efficient method of transforming a discrete Fourier transform (DFT) into a constant Q transform, where Q is the ratio of center frequency to bandwidth, has been devised. This method involves the calculation of kernels that are then applied to each subsequent DFT. Only a few multiples are involved in the calculation of each component of the constant Q transform, so this transformation adds a small amount to the computation. In effect, this method makes it possible to take full advantage of the computational efficiency of the fast Fourier transform (FFT). Graphical examples of the application of this calculation to musical signals are given for sounds produced by a clarinet and a violin.

A resolution comparison of several time-frequency representations

Abstract: Two signal components are considered resolved in a time-frequency representation when two distinct peaks can be observed. The time-frequency resolution limit of two Gaussian components, alike except for their time and frequency centers, is determined for the Wigner distribution, the pseudo-Wigner distribution, the smoother Wigner distribution, the squared magnitude of the short-time Fourier transform, and the Choi-Williams distribution. The relative performance of the various distributions depends on the signal. The pseudo-Wigner distribution is best for signals of this class with only one frequency component at any one time, the Choi-Williams distribution is most attractive for signals in which all components have constant frequency content, and the matched filter short-time Fourier transform is best for signal components with significant frequency modulation. A relationship between the short-time Fourier transform and the cross-Wigner distribution is used to argue that, with a properly chosen window, the short-time Fourier transform of the cross-Wigner distribution must provide better signal component separation that the Wigner distribution. >

Wavelet transform as a bank of the matched filters.

Abstract: The wavelet transform is a powerful tool for the analysis of short transient signals. We detail the advantages of the wavelet transform over the Fourier transform and the windowed Fourier transform and consider the wavelet as a bank of the VanderLugt matched filters. This methodology is particularly useful in those cases in which the shape of the mother wavelet is approximately known a priori. A two-dimensional optical correlator with a bank of the wavelet filters is implemented to yield the time-frequency joint representation of the wavelet transform of one-dimensional signals.

A generalized wavelet transform for Fourier analysis: the multiresolution Fourier transform and its application to image and audio signal analysis

Abstract: A wavelet transform specifically designed for Fourier analysis at multiple scales is described and shown to be capable of providing a local representation which is particularly well suited to segmentation problems. It is shown that, by an appropriate choice of analysis window and sampling intervals, it is possible to obtain a Fourier representation which can be computed efficiently and overcomes the limitations of using a fixed scale of window, yet by virtue of its symmetry properties allows simple estimation of such fundamental signal parameters as instantaneous frequency and onset time/position. The transform is applied to the segmentation of both image and audio signals, demonstrating its power to deal with signal events which are localized in either time/space or frequency. Feature extraction and segmentation are performed through the introduction of a class of multiresolution Markov models, whose parameters represent the signal events underlying the segmentation. >

Optical wavelet transform

Abstract: The wavelet transform is implemented using an optical multichannel correlator with a bank of wavelet transform filters. This approach provides a shift-invariant wavelet transform with continuous translation and discrete dilation parameters. The wavelet transform filters can be in many cases simply optical transmittance masks. Experimental results show detection of the frequency transition of the input signal by the optical wavelet transform.

Transform image enhancement

Abstract: Blockwise transform image enhancement techniques are discussed. Previously, transform image enhancement has usually been based on the discrete Fourier transform (DFT) applied to the whole image. Two major drawbacks with the DFT are high complexity of implementation involving complex multiplications and additions, with intermediate results being complex numbers, and the creation of severe block effects if image enhancement is done blockwise. In addition, the quality of enhancement is not very satisfactory. It is shown that the best transforms for transform image coding, namely, the scrambled real discrete Fourier transform, the discrete cosine transform, and the discrete cosine-III transform, are also the best for image enhancement. Three techniques of enhancement discussed in detail are alpha-rooting, modified unsharp masking, and filtering motivated by the human visual system response (HVS). With proper modifications, it is observed that unsharp masking and HVS-motivated filtering without nonlinearities are basically equivalent. Block effects are completely removed by using an overlap-save technique in addition to the best transform.

A comparison of the existence of 'cross terms' in the Wigner distribution and the squared magnitude of the wavelet transform and the short-time Fourier transform

Abstract: It is shown that cross terms comparable to those found in the Wigner distribution (WD) exist for the energy distributions of the wavelet transform (WT) and the short-time Fourier transform (STFT). The geometry of the cross terms is described by deriving mathematical expressions for the energy distributions of the STFT and the WT of a multicomponent signal. From those mathematical expressions it is inferred that the STFT and the WT cross terms: (1) occur at the intersection of the respective transforms of the two signals under consideration, whereas the WD cross terms occur at mid-time-frequency of the two signals; (2) are oscillatory in nature, as are the WD cross terms, and are modulated by a cosine whose argument is a function of the difference in center times and center frequencies of the signals under consideration; and (3) can have a maximum amplitude as large as twice the product of the magnitude of the transforms of the two signals in question, like WD cross terms. It is shown that the presence of these cross terms could lead to problems in analyzing a multicomponent signal. The consequences of this effect with respect to speech applications are discussed. >

Wavelets and their Applications

Abstract: Contributions discuss signal analysis discrete-time signal processing, wavelets for Quincunx pyramid, transform maxima and multiscale edges, among other topics; numerical analysis; other applications the optical wave transform, continuous wavelet transform, quantum mechanics; and theoretical develop

A formal definition of the Hough transform: Properties and relationships

Abstract: Shape, in both 2D and 3D, provides a primary cue for object recognition and the Hough transform (P.V.C. Hough, U.S. Patent 3,069,654, 1962) is a heuristic procedure that has received considerable attention as a shape-analysis technique. The literature that covers application of the Hough transform is vast; however, there have been few analyses of its behavior. We believe that one of the reasons for this is the lack of a formal mathematical definition. This paper presents a formal definition of the Hough transform that encompasses a wide variety of algorithms that have been suggested in the literature. It is shown that the Hough transform can be represented as the integral of a function that represents the data points with respect to a kernel function that is definedimplicitly through the selection of a shape parameterization and a parameter-space quantization. The kernel function has dual interpretations as a template in the feature space and as a point-spread function in the parameter space. A novel and powerful result that defines the relationship between parameterspace quantization and template shapes is provided. A number of interesting implications of the formal definition are discussed. It is shown that the Radon transform is an incomplete formalism for the Hough transform. We also illustrate that the Hough transform has the general form of a generalized maximum-likelihood estimator, although the kernel functions used in estimators tend to be smoother. These observations suggest novel ways of implementing Hough-like algorithms, and the formal definition forms the basis of work for optimizing aspects of Hough transform performance (see J. Princen et. al.,Proc. IEEE 3rd Internat. Conf. Comput. Vis., 1990, pp. 427–435).

Performance analysis of transient detectors based on a class of linear data transforms

Abstract: The problem of detecting short-duration nonstationary signals, which are commonly referred to as transients, is addressed. Transients are characterized by a signal model containing some unknown parameters, and by a 'model mismatch' representing the difference between the model and the actual signal. Both linear and nonlinear signal models are considered. The transients are assumed to undergo a noninvertible linear transformation prior to the application of the detection algorithm. Examples of such transforms include the short-time Fourier transform, the Gabor transform, and the wavelet transform. Closed-form expressions are derived for the worst-case detection performance for all possible mismatch signals of a given energy. These expressions make it possible to evaluate and compare the performance of various transient detection algorithms, for both single-channel and multichannel data. Numerical examples comparing the performance of detectors based on the wavelet transform and the short-time Fourier transform are presented. >

Digital Filtering: An Introduction

Abstract: Signals and Systems. Brief Review of Continuous Linear Filters. Sampling and the z -Transform. Recursive-Filter Design. Finite Impulse Response (FIR) and Nonrecursive Filters. The Discrete Fourier Transform and the Fast Fourier Transform. Basic Concepts of Probability Theory and Random Processes. Quantization Effects in Digital Filters. Digital Filtering Applied to Estimation: The Discrete Kalman Filter. Index.

Infrared intensities of liquids. IX. The Kramers–Kronig transform, and its approximation by the finite Hilbert transform via fast Fourier transforms

Abstract: It is well known that the infinite Kramers–Kronig transform is equivalent to the infinite Hilbert transform, which is equivalent to the allied Fourier integrals. The Hilbert transform can thus be i...

Image reconstruction from localized phase

Abstract: The authors present a novel approach to image representation using partial information defined by the localized phase. The scheme is implemented using the short-time (short-distance) Fourier transform. This is a generalization of the Gabor scheme which is well-established with regard to biological representation of visual information at the level of the visual cortex. Similar to processing in vision, the DC component is first extracted from the signal and treated separately. Computational results and theoretical analysis indicate that image reconstruction from the localized phase representation requires fewer computer operations and yields an improved rate of convergence compared to reconstruction from the global phase representation. It is also implementable with fast algorithms using highly parallel architecture. >

The Hausdorff mean of a Fourier-Stieltjes transform

Abstract: It is shown that the integral Hausdorff mean TA of the FourierStieltjes transform of a measure on the real line is the Fourier transform of an LI function if and only if TP vanishes at infinity or the kernel of T has mean value zero. Also a sufficient condition on the kernel of T and a necessary and sufficient condition on the measure is established in order for -i sign(x) TAi(x) to be the Fourier transform of an LI-function. These results yield an improvement of Fejer's and Wiener's formulas for the inversion of Fourier-Stieltjes transforms, the uniqueness property of certain generalized Fourier transforms, and a generalization of the mean ergodic theorem for unitary operators. Let M(R) be the space of complex bounded regular Borel measures on the real line R and A(x) be the Fourier-Stieltjes transform of a measure tu in M(R), A(X) = je-ixt dt(t), x E R. We consider the integral Hausdorff mean T/u generated by a Borel measurable kernel V in L1(R), which is defined for x E R by Tf (x) = J (XS) s (s) ds x JR ( (y)dy, for x $ 0, and Tg(O) = gi(O) fR yi(s) ds. When the kernel V is the characteristic function of [0, 1], Tfu reduces to the integral arithmetic average of A over [0, x]. The summability properties of T are well known [8, pp. 275-278]. The continuity properties and the spectrum of T as a bounded operator on LP(R) have been studied by Schur [14], Hardy, Littlewood, and Polya [9], Rhoades [12], Fabes, Jodeit, and Lewis [3], and Leibowitz [11]. Goldberg in [5, 6] studied the properties of the transformation T on the Fourier transform for the Lebesgue class LP(R) for 1

A two-dimensional time-domain near-zone to far-zone transformation

Abstract: A 2D version of a time-domain transformation useful for extrapolating 3D near-zone finite-difference time-domain (FDTD) results to the far zone is outlined. While the 3D transformation produced a physically observable far-zone time-domain field, this is not convenient for the 2D case. However, a representative, 2D far-zone time-domain result can be obtained directly. This result can then be transformed to the frequency domain using a fast Fourier transform, corrected with a simple multiplicative factor, and used, for example, to calculate the complex wideband scattering width of a target. If an actual time-domain far zone result is required, it can be obtained by inverse Fourier transform of the final frequency-domain result. >

On the uncertainty principle in discrete signals

Abstract: It has recently been shown that the uncertainty principle holds true by appropriate definitions of the durations even if discrete signals are considered. A basic inequality was derived in the particular case where the Fourier transform is real. As an extension to this work, the authors prove the uncertainty relation in the general case of a complex Fourier transform and with somewhat extended definitions of durations. >

The effects of windowing and quantization error on the amplitude of frequency-domain functions

Abstract: The author describes how windows modify the magnitude of a discrete Fourier transform and the level of a power spectral density computed by Welch's method. For white noise, the magnitude of the discrete Fourier transform at a fixed frequency has a Rayleigh probability distribution. For sine waves with an integer number of cycles and quantization noise, the theoretical values of the amplitude of the discrete Fourier transform and power spectral density are calculated. The authors show the signal-to-noise ratio in a single discrete Fourier transform or power spectral density frequency bin is related to the normal time-domain definition of the signal-to-noise ratio. The answer depends on the discrete Fourier transform length, the window type, and the function averaged. >

Time-frequency perspectives: the 'chirplet' transform

Abstract: The authors have developed an expansion they call the chirplet transform. It has been successfully applied to a wide variety of signal processing applications, including radar and image processing. There has been a recent debate as to the relative merits of an affine-in-time (wavelet) transform and the classical short-time Fourier transform (STFT) for the analysis of nonstationary phenomena. Chirplet filters embody both the wavelet and STFT as special cases by decoupling the filter bandwidths and center frequencies. Chirplets, by their embodiment of affine geometry in the time-frequency (TF) plane, may also include shears in time and frequency (chirps) and even time-bandwidth product variation (noise bursts) if desired. The most general chirplets may be derived from one or more basic ('mother') chirplets by the transformations or perspective geometry in the TF plane. >

Sinusoidal frequency estimation by signal subspace approximation

Abstract: An efficient Fourier transform-based method that avoids eigenvector computation is proposed for approximating the signal subspace. The resulting signal subspace estimate can be used directly to define a MUSIC-type frequency estimator or as a very good initial guess in context with adaptive or iterative eigenvector computation schemes. At low signal-to-noise ratios, the approximation yields better results than exact MUSIC. It is also more robust than MUSIC against overestimating the number of sinusoids. Some variations of the basic method are briefly discussed. >

Optical N4 implementation of a two-dimensional wavelet transform

Abstract: A two-dimensional wavelet transform is implemented by a bank of wavelet transform filters in the Fourier domain. An optical N 4 multichannel correlator architecture is proposed to perform parallel optical 2-D wavelet transforms. A holographic recording scheme is proposed to implement such a wavelet transform filter array. The optical experimental results are presented using the computer-generated transmittance masks as the wavelet transform filters.

Resolvent approach for the nonstationary Schrodinger equation

Abstract: The spectral transform for the nonstationary Schrodinger equation is considered. The resolvent operator of the Schrodinger equation is introduced and the Fourier transform of its kernel (called the resolvent function) is studied. It is shown that it can be used to construct a generalized version of the theory of the spectral transform which enables one to handle also potentials approaching zero in every direction except a finite number, which corresponds to the physical situation of long waves mutually interacting in the plane.

Performance of the short-time Fourier Transform and Wavelet Transform to phonocardiogram signal analysis

Abstract: In this paper the shortages of using the Nan&ml Fourier Transform to analyze the Phonocardiogram (KG) signals is first pointed out and the need for time-varyingdigitat signal processing techniquesto conectly analyzeKG signals is discussed.‘I%vo timefrequency analysis techniques am presented in this pape~ namely, the Spectrogram and the Wavelet Transform. Furthermore, a comparisonstudy between these two techniques has shown the ~lution diffenmces between them. The Wavelet Transform is shown to be capable to detect the two components, aortic valve component A2 and pulmonary valve compment P2, of the second sound S2 of a normal PCG signal which am not detectable neitherusing the standardFourier‘fkansfonnmr the Spectrogram.In addition to thz the Wavelet ‘fYansfomtenables Physicians to obtain qualitative and quantitative measurements of time-fiwpency characteristicsof phonocadiogram (KG) signals.

New number theoretic transform

Abstract: A new number theoretic transform is introduced. This transform is defined modulo the Mersenne primes, has long transform length which is a power of two, a fast algorithm, and the inverse transform has within a factor of (1/N) the same form as the forward transform. Thus, it is well suited for the calculation of error free convolutions and correlations.

Multiresolution Estimation of 2-d Disparity Using a Frequency Domain Approach

Abstract: An efficient algorithm for the estimation of the 2-d disparity between a pair of stereo images is presented. Phase based methods are extended to the case of 2-d disparities and shown to correspond to computing local correlation fields. These are derived at multiple scales via the frequency domain and a coarse-to-fine 'focusing' strategy determines the final disparity estimate. Fast implementation is achieved by using a generalised form of wavelet transform, the multiresolution Fourier transform (MFT), which enables efficient calculation of the local correlations. Results from initial experiments on random noise stereo pairs containing both 1-d and 2-d disparities, illustrate the potential of the approach.

Some remarks on Fourier transform and differential operators for digital functions

Abstract: A Fourier-like transform suitable for application to digital functions is considered, and the difference operators for these functions are discussed. Special attention is focused to the relationship between the Fourier transform of a function and the Fourier transform of its difference relative to a particular variable. Using the transform introduced, a polynomial expansion for digital functions is suggested. >

Backprojection by upsampled Fourier series expansion and interpolated FFT

Abstract: A fast backprojection method through the use of interpolated fast Fourier transform (FFT) is presented. The computerized tomography (CT) reconstruction by the convolution backprojection (CBP) method has produced precise images. However, the backprojection part of the conventional CBP method is not very efficient. The authors propose an alternative approach to interpolating and backprojecting the convolved projections onto the image frame. First, the upsampled Fourier series expansion of the convolved projection is calculated. Then, using a Gaussian function, it is projected by the aliasing-free interpolation of FFT bins onto a rectangular grid in the frequency domain. The total amount of computation in this procedure for a 512*512 image is 1/5 of the conventional backprojection method with linear interpolation. This technique also allows the arbitrary control of the frequency characteristics. >

Obtaining Spectral Moments By Discrete Fourier Transform With Noise removal In Radar Meteorology

Abstract: We present a distribution-free technique for estimating the first moment of a power density spectrum. The method determines a noise level by the Kolmogorov-Smirnov (K-S) test applied to the periodogram obtained from a disarete Fourier Transform (DFT) obtained from the complex envelope of radar echo. The K-S test gives results that are independent of the noise distribution and depend only upon the pulse to pulse noise samples being uncorrelated, a conditional almost always satisfied in a radar. After the noise level has been determined, it is removed from the periodogram. In simulation experiments we started with 64 data points in each replication, computed eight 8-point periodograms, averaged these to get a smoothed periodogram. The K-S test was applied to the smoothed periodogram and the first moment obtained from the noise removed smoothed periodogram. Results are tabulated for, specifically, an example of an asymmetrical power density spectrum. The results show quite acceptable accuracy for most radar meteorology applications. GLOSSARY OF PRINCIPAL SYMBOLS S(f,) = periodogram evaluated at frequency f, f, = n-th doppler frequency in received complex envelope Sck) = k-th ordered value of S(f,) M = total number of frequency components NI = number of frequency components used a step i. F,(k) = i-th empirical spectral distribution function TI, = K-S two sided statistic at step i S,,,,, = estimate of noise power density spectrum wIpa = critical value of TI, = quantile of the 2-sided K-S statistic K-S = Kolmogorov-Smirnov a = significance level of K-S test INTRODUCTION Methods for estimating spectral moments include time domain methods [ 1, 21 and frequency domain methods [ 1, 3-51. Although time domain methods have been popular, there are circumstances in which a frequency domain method must be used. One such circumstance arises in a technique for suppressing range sidelobes in pulse compression radar [6]. In the frequency domain methods, it becomes important to determine the noise level. Hildebrand and Sekhon [3] have described a method of determining the spectral density level of the noise. Their method involves eliminating, by successively decreasing thresholds, those spectral components with values greater than the noise spectral level. What remains is subjected to a sequence of tests assuming that the residue is white Gaussian noise. If the tests are passed, the spectral level obtained is taken as the noise power density spectrum. In this paper we present an alternative algorithm in which a distribution-free test is applied to each step of decreasing threshold. The test (or sequence of tests) is a Kolmogorov-Smirnov (K-S) test. The algorithm involves discarding high spectral values (or retaining low spectral values) until the application of the K-S test shows that a flat power density spectrum remains with a pre-determined probability of lying within a specified interval around the correct value: The advantages over the Hildebrand-Sekhon algorithm are: 1. A Gaussian process is not assumed. 2. There are definite, quantitative, criteria for deciding when the noise level has been reached and what that noise level is. An explanation of the algorithm follows. THE ALGORITHM FOR ESTIMATING MEAN DOPPLER The steps in the algorithm are best explained in connection with Figure I. We start with a sequence of pulse to pulse complex envelope values from each of a set of range bins. For one specific range bin at a time we obtain the discrete Fourier transform (DFT), usually by means of a Fast Fourier Transform (FFT), of a sequence of M such complex envelope values. Of course, the algorithm is to be applied to a large number of range bins, but the procedure is the same for all of the range bins. Now we enumerate the steps in the algorithm.