Showing papers on "Non-uniform discrete Fourier transform published in 1992"
TL;DR: In this article, 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.
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.
279 citations
TL;DR: A relationship between the short-time Fourier transform and the cross-Wigner distribution is used to argue that, with a properly chosen window, the long-time Fawrier transform of thecross-WIGNer distribution must provide better signal component separation that the Wigner distributions.
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. >
176 citations
TL;DR: The wavelet transform is described, which is particularly useful in those cases in which the shape of the mother wavelet is approximately known a priori and the bank of the VanderLugt matched filters is considered.
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.
145 citations
TL;DR: 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.
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. >
145 citations
TL;DR: In this paper, the wavelet transform is implemented using an optical multichannel correlator with a bank of wavelet filter filters, which provide a shift-invariant wavelet transformation with continuous translation and discrete dilation.
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.
143 citations
TL;DR: Close-form expressions are derived for the worst-case detection performance for all possible mismatch signals of a given energy, which make it possible to evaluate and compare the performance of various transient detection algorithms, for both single-channel and multichannel data.
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. >
76 citations
Book•
01 Jan 1992
TL;DR: In this article, the Discrete Fourier Transform (DFT) and Fast Fourier transform (FFT) are used to estimate the probability of a given signal in a continuous linear filter.
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.
70 citations
TL;DR: It is shown that the transformed sequence may be processed using a ROOT MUSIC based approach to estimate the elevations and azimuths of the observed sources.
Abstract: The problem of estimating the directions of arrival of narrowband plane waves impinging on a uniform circular array with M identical sensors uniformly distributed around a circle is considered. Specifically, the authors study a transformed data sequence that is equal to a reordering of the inverse discrete Fourier transform sequence corresponding to the outputs of a circular array with M elements uniformly distributed around the array circumference. It is shown that the transformed sequence may be processed using a ROOT MUSIC based approach to estimate the elevations and azimuths of the observed sources. Any root estimated via ROOT MUSIC which is on or close to the unit circle indicates the presence of a source at the elevation under consideration and an azimuth equal to the phase of the root. Experimental results are provided to demonstrate the advantages of processing the transformed data. >
67 citations
TL;DR: The sliding fast Fourier transform is reviewed and is shown to have the computational complexity of N complex multiplications per sample, as opposed to the well-cited assumption of (N/2) log/sub 2/ N complex multiplication per sample.
Abstract: The sliding fast Fourier transform (FFT) is reviewed and is shown to have the computational complexity of N complex multiplications per sample, as opposed to the well-cited assumption of (N/2) log/sub 2/ N complex multiplication per sample reported in a book by L.R. Rabiner and B. Gold (1975). >
67 citations
26 May 1992
67 citations
24 Mar 1992
TL;DR: This formulation of the inverse discrete cosine transform has several advantages over previous approaches, including the elimination of multiplies from the central loop of the algorithm and its adaptability to incremental evaluation.
Abstract: The paper presents a new realization of the inverse discrete cosine transform (IDCT). It exploits both the decorrelation properties of the discrete cosine transform (DCT) and the quantization process that is frequently applied to the DCT's resultant coefficients. This formulation has several advantages over previous approaches, including the elimination of multiplies from the central loop of the algorithm and its adaptability to incremental evaluation. The technique provides a significant reduction in computational requirements of the IDCT, enabling a software-based implementation to perform at rates which were previously achievable only through dedicated hardware. >
TL;DR: In this article, the authors proved the uncertainty relation in the general case of a complex Fourier transform and with somewhat extended definitions of durations, and showed that the uncertainty principle holds true by appropriate definitions of the durations even if discrete signals are considered.
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. >
TL;DR: It is shown that the imaging information obtained by the inversion of phased array scan data is equivalent to the image reconstructed from its synthesized array counterpart.
Abstract: The author presents a system model and inversion for the beam-steered data obtained by linearly varying the relative phase among the elements of an array, also known as phased array scan data. The system model and inversion incorporate the radiation pattern of the array's elements. The inversion method utilizes the time samples of the echoed signals for each scan angle instead of range focusing. It is shown that the temporal Fourier transform of the phased array scan data provides the distribution of the spatial Fourier transform of the reflectivity function for the medium to be imaged. The extent of this coverage is related to the array's length and the temporal frequency bandwidth of the transmitted pulsed signal. Sampling constraints and reconstruction procedure for the imaging system are discussed. It is shown that the imaging information obtained by the inversion of phased array scan data is equivalent to the image reconstructed from its synthesized array counterpart. >
01 Dec 1992
TL;DR: In this paper, the authors describe how windows modify the magnitude of a discrete Fourier transform and the level of a power spectral density computed by Welch's method, and show that the signal-to-noise ratio in a single discrete transform is related to the normal time-domain definition of the signal to noise ratio.
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. >
TL;DR: An efficient Fourier transform-based method that avoids eigenvector computation is proposed for approximating the signal subspace and yields better results than exact MUSIC and is more robust than MUSIC against overestimating the number of sinusoids.
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. >
TL;DR: In this paper, a smoothing method consisting of estimating by least-squares the coefficients of a two-dimensional discrete Fourier series with arbitrary period and arbitrary frequency resolution is presented, which can be used with non-equally-spaced and non-rectangular-domain data.
TL;DR: Preliminary studies have shown that the architecture can easily be implemented in VLSI form, and, in conjunction with a high-speed digital signal processor, it can be used for real-time transform domain LMS adaptive filtering of 8 kHz sample rate speech signals.
Abstract: Two algorithms are given for the computation of the updated discrete cosine transform-II (DCT-II), discrete sine transform-II (DST-II), discrete cosine transform-IV (DCT-IV), and discrete sine transform-IV (DST-IV). It is pointed out that the algorithm used for running DCT-IV can also be used for computation for running DST-IV without additional computational overhead. An architecture which is common and suitable for VLSI implementation of the derived algorithms is also presented. Preliminary studies have shown that the architecture can easily be implemented in VLSI form, and, in conjunction with a high-speed digital signal processor (for example ADSP 2100A), it can be used for real-time transform domain LMS adaptive filtering (128 taps) of 8 kHz sample rate speech signals. >
01 Mar 1992
TL;DR: 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.
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.
TL;DR: A new fast radix-p-algorithm for the discrete cosine transform (DCT) and its inverse is presented, based on the divide-and-conquer method and on the arithmetic with Chebyshev polynomials.
Abstract: A new fast radix-p-algorithm (p ≧ 2) for the discrete cosine transform (DCT) and its inverse is presented. It is based on the divide-and-conquer method and on the arithmetic with Chebyshev polynomials. The algorithm can be applied for the efficient calculation of DCT's of arbitrary transform lengths and for the implementation of other discrete Vandermonde transforms withO(N logN) arithmetical operations.
27 May 1992
TL;DR: A Fourier-like transform suitable for application to digital functions is considered, and the difference operators for these functions are discussed, and a polynomial expansion fordigital functions is suggested.
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. >
TL;DR: A fast backprojection method through the use of interpolated fast Fourier transform (FFT) is presented, which allows the arbitrary control of the frequency characteristics.
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. >
26 May 1992
TL;DR: In this article, a distribution-free technique for estimating the first moment of a power density spectrum was proposed, which is based on 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.
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.
TL;DR: A fast discrete Fourier transform (DFT) computing algorithm used in situations where part of the data is zero and only the first transform elements are to be calculated is proposed.
Abstract: A fast discrete Fourier transform (DFT) computing algorithm used in situations where part of the data is zero and only the first transform elements are to be calculated is proposed. The method is based on the pruning of a split-radix decimation-time (DIT) fast Fourier transform (FFT) diagram. It has the advantage of providing gains as a result of pruning computation and the use of a split radix. >
TL;DR: In this paper, the power spectrum of a complex physiological system is estimated by smoothing the periodogram with the help of the fast Fourier transform algorithm, which can use either the whole record of the data or a number of disjoint records.
Abstract: . In this paper we consider techniques of spectral analysis for stationary point processes in order to study the behaviour of a complex physiological system. The estimates of the power spectrum are obtained by smoothing the periodogram which is computed very rapidly with the help of the fast Fourier transform algorithm. In the computation of the estimates we can use either the whole record of the data or a number of disjoint records.
TL;DR: A novel transform for spectral decomposition that uses regular square waves as the basis functions is presented and can act as a generalized frequency filter that only depends on the periodicity of the data.
Abstract: A novel transform for spectral decomposition that uses regular square waves as the basis functions is presented. The digital transform requires order N operations. The transform possesses unusually symmetry properties which may prove useful in many applications. In particular, it can act as a generalized frequency filter that only depends on the periodicity of the data. >
TL;DR: An algorithm is proposed for rapid and accurate reconstruction from data collected in Fourier space at points arranged on a grid of concentric cubes which ensures that no interpolations are needed, in contrast to methods involving backprojection with their unavoidable interpolations.
Abstract: An algorithm is proposed for rapid and accurate reconstruction from data collected in Fourier space at points arranged on a grid of concentric cubes. The Fourier transform of the object to be reconstructed is decomposed into the sum of three functions by subdividing its domain into three non-overlapping mutually orthogonal double pyramids. Each of the three functions is zero-valued outside one of the double pyramids and has values inside that double pyramid which are the same as those of Fourier transform of the object to be reconstructed at the same points. Inverse Fourier transforms of these individual functions can be calculated using the chirp z-transform. The outputs of these inverse transforms for the three functions are estimates of their values at points of the same rectangular grid. The function to be reconstructed is estimated for this grid by adding together the three inverse transforms. The whole process has computational complexity of the same order as required for the 3D fast Fourier transform and so (for medically relevant sizes of the data set) it is faster than backprojection into the same size rectangular grid. The design of the algorithm ensures that no interpolations are needed, in contrast to methods involving backprojection with their unavoidable interpolations. As an application, a 3D data collection method for MRI has been designed which directly samples the Fourier transform of the object to be reconstructed on concentric cubes as needed for the algorithm.
TL;DR: In this article, a non-linear transformation of the log-spaced data to give a new "frequency" variable enables the use of the convenient Fourier Transform (FT) approach.
TL;DR: A method for computing the Fourier coefficients which allows uniform sampling at arbitrarily chosen sampling rates is developed, although the technique still requires few multiplications, albeit at the expense of a limited amount of linear interpolation of the sample values.
Abstract: The arithmetic Fourier transform (AFT), a method for computing the Fourier coefficients of a complex-valued periodic function, is based on a formula which has the advantage of eliminating many of the multiplications usually associated with computing discrete Fourier coefficients, but has the disadvantage of requiring samples of the signal at nonuniformly spaced time values. A method for computing the Fourier coefficients which allows uniform sampling at arbitrarily chosen sampling rates is developed. The technique still requires few multiplications, albeit at the expense of a limited amount of linear interpolation of the sample values. Efficient hardware implementations of this algorithm are presented. >
Journal Article•
TL;DR: A nonlinear joint transform correlator is described where the joint power spectrum, transformed by various degrees of nonlinearity, is represented in a binary format using a multiple level threshold function.
Abstract: A joint Fourier transform optical correlator is disclosed which can have varying degrees of nonlinearity and yet employ a readily available binary spatial light modulator for producing the correlation output light signal in conjunction with a Fourier transform lens. The nonlinearly transformed joint power spectrum is binarized utilizing a multiple level threshold function which can vary from one pixel to the next.