Showing papers on "Fast Fourier transform published in 1999"

Option valuation using the fast Fourier transform

Abstract: This paper shows how the fast Fourier Transform may be used to value options when the characteristic function of the return is known analytically.

The Discrete Cosine Transform

Abstract: Each discrete cosine transform (DCT) uses $N$ real basis vectors whose components are cosines In the DCT-4, for example, the $j$th component of $\boldv_k$ is $\cos (j + \frac{1}{2}) (k + \frac{1}{2}) \frac{\pi}{N}$ These basis vectors are orthogonal and the transform is extremely useful in image processing If the vector $\boldx$ gives the intensities along a row of pixels, its cosine series $\sum c_k \boldv_k$ has the coefficients $c_k=(\boldx,\boldv_k)/N$ They are quickly computed from a Fast Fourier Transform But a direct proof of orthogonality, by calculating inner products, does not reveal how natural these cosine vectors are We prove orthogonality in a different way Each DCT basis contains the eigenvectors of a symmetric "second difference" matrix By varying the boundary conditions we get the established transforms DCT-1 through DCT-4 Other combinations lead to four additional cosine transforms The type of boundary condition (Dirichlet or Neumann, centered at a meshpoint or a midpoint) determines the applications that are appropriate for each transform The centering also determines the period: $N-1$ or $N$ in the established transforms, $N-\frac{1}{2}$ or $N+ \frac{1}{2}$ in the other four The key point is that all these "eigenvectors of cosines" come from simple and familiar matrices

Computing the discrete-time "analytic" signal via FFT

Abstract: Starting with a real-valued N-point discrete-time signal, frequency-domain algorithms are provided for computing (1) the complex-valued standard N-point discrete-time "analytic" signal of the same sample rate; (2) the complex-valued decimated N/2-point discrete-time "analytic" signal of half the original sample rate; and (3) the complex-valued interpolated NM-point discrete-time "analytic" signal of M times the original sample rate. Special adjustment of the transform end points are shown to be necessary in order to generate proper discrete-time "analytic" signals.

A fast Fourier transform compiler

Abstract: The FFTW library for computing the discrete Fourier transform (DFT) has gained a wide acceptance in both academia and industry, because it provides excellent performance on a variety of machines (even competitive with or faster than equivalent libraries supplied by vendors). In FFTW, most of the performance-critical code was generated automatically by a special-purpose compiler, called genfft, that outputs C code. Written in Objective Caml, genfft can produce DFT programs for any input length, and it can specialize the DFT program for the common case where the input data are real instead of complex. Unexpectedly, genfft "discovered" algorithms that were previously unknown, and it was able to reduce the arithmetic complexity of some other existing algorithms. This paper describes the internals of this special-purpose compiler in some detail, and it argues that a specialized compiler is a valuable tool.

A low-power, high-performance, 1024-point FFT processor

Abstract: This paper presents an energy-efficient, single-chip, 1024-point fast Fourier transform (FFT) processor. The 460000-transistor design has been fabricated in a standard 0.7 /spl mu/m (L/sub poly/=0.6 /spl mu/m) CMOS process and is fully functional on first-pass silicon. At a supply voltage of 1.1 V, it calculates a 1024-point complex FFT in 330 /spl mu/s while consuming 9.5 mW, resulting in an adjusted energy efficiency more than 16 times greater than the previously most efficient known FFT processor. At 3.3 V, it operates at 173 MHz-which is a clock rate 2.6 times greater than the previously fastest rate.

The Rossmann Fourier autoindexing algorithm in MOSFLM

Abstract: The fast Fourier transform (FFT) autoindexing routines written by the Rossmann group at Purdue University have been incorporated in MOSFLM, providing a rapid and reliable method of indexing oscillation images. This is a procedure which extracts direct-space information about the unit cell from the FFT. The method and its implementation in MOSFLM are discussed.

Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology

Abstract: Advantages of the lensless Fourier holography setup for the reconstruction of digitally recorded holograms in holographic interferometry are presented. This very simple setup helps to achieve a maximum lateral resolution of the object under investigation. Also, the numerical-reconstruction algorithm is very simple and fast to compute. A mathematical model based on Fourier optics is used to describe discretization effects and to determine the lateral resolution. The recording and the reconstruction processes are regarded as an optical imaging system, and the point-spread function is calculated. Results are verified by an experimental setup for a combined shape and deformation measurement.

Applications of the windowed FFT to electric power quality assessment

Abstract: This paper discusses the application of the windowed fast Fourier transform to electric power quality assessment. The windowed FFT is a time windowed version of the discrete time Fourier transform. The window width may be adjusted and shifted to scan through large volumes of power quality data. Narrow window widths are used for detailed analyses, and wide window widths are used to move rapidly across archived power quality data measurements. The mathematics of the method are discussed and applications are illustrated.

Application of Morlet wavelets to supervise power system disturbances

Abstract: A wavelet transform approach using a Morlet basis function is proposed to supervise power system disturbances in this paper. With the time-frequency localization characteristics embedded in wavelets, the time and frequency information of a waveform can be presented as a visualized scheme. Different from the fast Fourier transform, the wavelet transform approach is more efficient in monitoring various disturbances as time varies. The method has been tested on the detection of various simulated disturbances including voltage sag, voltage swell, momentary interruption and oscillatory transients and on the harmonic analysis of the arc furnace from the field test data. Testing results demonstrated the practicality and advantages of the proposed method for the applications.

Reconstruction of band‐limited signals, irregularly sampled along one spatial direction

Abstract: Seismic signals are often irregularly sampled along spatial coordinates, leading to suboptimal processing and imaging results. Least squares estimation of Fourier components is used for the reconstruction of band-limited seismic signals that are irregularly sampled along one spatial coordinate. A simple and efficient diagonal weighting scheme, based on the distance between the samples, takes the properties of the noise (signal outside the bandwidth) into account in an approximate sense. Diagonal stabilization based on the energies of the signal and the noise ensures robust estimation. Reconstruction for each temporal frequency component allows the specification of a varying spatial bandwidth dependent on the minimum apparent velocity. This parameterization improves the reconstruction capability for the lower temporal frequencies. In practical circumstances, the maximum size of the gaps in which the signal can be reconstructed is three times the (temporal frequency dependent) Nyquist interval. Reconstruction in the wavenumber domain allows a very efficient implementation of the algorithm, and takes a total number of operations a few times that of a 2-D fast Fourier transform corresponding to the size of the output data set. Quality control indicators of the reconstruction procedure can be computed which may also serve as decision criteria on in-fill shooting during acquisition. The method can be applied to any subset of seismic data with one varying spatial coordinate. Applied along the cross-line direction, it can be used to compute a 3-D stack with improved anti-alias protection and less distortion of the signal within the bandwidth.

Do's and don'ts in Fourier analysis of steady-state potentials

Abstract: Fourier analysis is a powerful tool in signal analysis that can be very fruitfully applied to steady-state evoked potentials (flicker ERG, pattern ERG, VEP, etc) However, there are some inherent assumptions in the underlying discrete Fourier transform (DFT) that are not necessarily fulfilled in typical electrophysiological recording and analysis conditions Furthermore, engineering software-packages may be ill-suited and/or may not fully exploit the information of steady-state recordings Specifically: * In the case of steady-state stimulation we know more about the stimulus than in standard textbook situations (exact frequency, phase stability), so 'windowing' and calculation of the 'periodogram' are not necessary * It is mandatory to choose an integer relationship between sampling rate and frame rate when employing a raster-based CRT stimulator * The analysis interval must comprise an exact integer number (eg, 10) of stimulus periods * The choice of the number of stimulus periods per analysis interval needs a wise compromise: A high number increases the frequency resolution, but makes artifact removal difficult; a low number 'spills' noise into the response frequency * There is no need to feel tied to a power-of-two number of data points as required by standard FFT, 'resampling' is an easy and efficient alternative * Proper estimates of noise-corrected Fourier magnitude and statistical significance can be calculated that take into account the non-linear superposition of signal and noise These aspects are developed in an intuitive approach with examples using both simulations and recordings Proper use of Fourier analysis of our electrophysiological records will reduce recording time and/or increase the reliability of physiologic or pathologic interpretations

Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm

Abstract: Recently, a pseudospectral time-domain (PSTD) algorithm was developed to simulate electromagnetic wave propagation. This technique uses the fast Fourier transform (FFT) algorithm for the spatial derivatives and uses the perfectly matched layer (PML) to eliminate the wraparound effect due to the spatial periodicity introduced by FFT. In this work, the author further analyzes this new method and compares it with the finite-difference time-domain (FDTD) and multiresolution time-domain (MRTD) methods for accuracy and efficiency. The PSTD algorithm is then applied to simulate large-scale problems for subsurface electromagnetic and acoustic measurements. For many problems encountered, since the spatial derivatives are obtained by the PSTD algorithm for continuous field components, this algorithm has a high order of accuracy in the spatial derivatives, and thus requires much fewer unknowns than the FDTD and MRTD methods. Numerical results confirm the efficacy of the PSTD method.

Discrete cosine transform based regularized high-resolution image reconstruction algorithm

Abstract: While high-resolution images are required for various applica- tions, aliased low-resolution images are only available due to the physi- cal limitations of sensors. We propose an algorithm to reconstruct a high- resolution image from multiple aliased low-resolution images, which is based on the generalized deconvolution technique. The conventional approaches are based on the discrete Fourier transform (DFT) since the aliasing effect is easily analyzed in the frequency domain. However, the useful solution may not be available in many cases, i.e., the underdeter- mined cases or the insufficient subpixel information cases. To compen- sate for such ill-posedness, the generalized regularization is adopted in the spatial domain. Furthermore, the usage of the discrete cosine trans- form (DCT) instead of the DFT leads to a computationally efficient recon- struction algorithm. The validity of the proposed algorithm is both theo- retically and experimentally demonstrated. It is also shown that the artifact caused by inaccurate motion information is reduced by regular- ization. © 1999 Society of Photo-Optical Instrumentation Engineers. (S0091-3286(99)00508-5)

Nonuniform fast Fourier transform

Abstract: The nonuniform discrete Fourier transform (NDFT) can be computed with a fast algorithm, referred to as the nonuniform fast Fourier transform (NFFT) In L dimensions, the NFFT requires O(N(-lne)L+(∏𝓁=1LM𝓁)∑𝓁=1LlogM𝓁) operations, where M𝓁 is the number of Fourier components along dimension 𝓁, N is the number of irregularly spaced samples, and e is the required accuracy This is a dramatic improvement over the O(N∏𝓁=1LM𝓁) operations required for the direct evaluation (NDFT) The performance of the NFFT depends on the lowpass filter used in the algorithm A truncated Gauss pulse, proposed in the literature, is optimized A newly proposed filter, a Gauss pulse tapered with a Hanning window, performs better than the truncated Gauss pulse and the B-spline, also proposed in the literature For small filter length, a numerically optimized filter shows the best results Numerical experiments for 1-D and 2-D implementations confirm the theoretically predicted accuracy and efficiency properties of the algorithm

The Regular Fourier Matrices and Nonuniform Fast Fourier Transforms

Abstract: For any triple of positive integers (m,N,q), the matrix F(m,N,q), called the (m,N,q)-regular Fourier matrix, is defined. The regular Fourier matrices F(m,N,q) are then applied to set up new algorithms for nonuniform fast Fourier transforms. Numerical results show that the accuracies obtained by our algorithms are much better than previously reported results with the same computation complexity. The algorithms require $O(N\cdot\log_2N)$ arithmetic operations, where N is the number of data points.

Fast algorithm for electromagnetic scattering by buried 3-D dielectric objects of large size

Abstract: A fast algorithm for electromagnetic scattering by buried three dimensional (3-D) dielectric objects of large size is presented by using the conjugate gradient (CG) method and fast Fourier transform (FFT). In this algorithm, the Galerkin method is utilized to discretize the electric field integral equations, where rooftop functions are chosen as both basis and testing functions. Different from the 3-D objects in homogeneous space, the resulting matrix equation for the buried objects contains both cyclic convolution and correlation terms, either of which can be solved rapidly by the CG-FFT method. The near-scattered field on the observation plane in the upper space has been expressed by two-dimensional (2-D) discrete Fourier transforms (DFTs), which also can be rapidly computed. Because of the use of FFTs to handle the Toeplitz matrix, the Sommerfeld integrals' evaluation which is time consuming yet essential for the buried object problem, has been reduced to a minimum. The memory required in this algorithm is of order N (the number of unknowns), and the computational complexity is of order N/sub iter/N log N, in which N/sub iter/ is the iteration number, and N/sub iter//spl Lt/N is usually true for a large problem.

Gravitational waves from inspiraling compact binaries: Validity of the stationary-phase approximation to the Fourier transform

Abstract: We prove that the oft-used stationary-phase method gives a very accurate expression for the Fourier transform of the gravitational-wave signal produced by an inspiraling compact binary. We give three arguments. First, we analytically calculate the next-order correction to the stationary-phase approximation, and show that it is small. This calculation is essentially an application of the steepest-descent method to evaluate integrals. Second, we numerically compare the stationary-phase expression to the results obtained by fast Fourier transform. We show that the differences can be fully attributed to the windowing of the time series, and that they have nothing to do with an intrinsic failure of the stationary-phase method. And third, we show that these differences are negligible for the practical application of matched filtering. @S0556-2821~99!00414-2#

Method and apparatus for enhancing noise-corrupted speech

Abstract: A noise suppression device receives data representative of a noise-corrupted signal which contains a speech signal and a noise signal, divides the received data into data frames, and then passes the data frames through a pre-filter to remove a dc-component and the minimum phase aspect of the noise-corrupted signal. The noise suppression device appends adjacent data frames to eliminate boundary discontinuities, and applies fast Fourier transform to the appended data frames. A voice activity detector of the noise suppression device determines if the noise-corrupted signal contains the speech signal based on components in the time domain and the frequency domain. A smoothed Wiener filter of the noise suppression device filters the data frames in the frequency domain using different sizes of a window based on the existence of the speech signal. Filter coefficients used for Wiener filter are smoothed before filtering. The noise suppression device modifies magnitude of the time domain data based on the voicing information outputted from the voice activity detector.

Discrete wavelet transform for tool breakage monitoring

Abstract: Detection of tool breakage is of vital importance in automated manufacturing. Various methods have been attempted, and it is considered that the use of discrete wavelet transform (DWT), which is much more efficient and just as accurate wavelet analysis, may provide a realistic solution to the detection of tool breakage in operation. The DWT uses an analyzing wavelet function which is localized in both time and frequency to detect a small change in the input signals. In addition, it requires less computation than Fast Fourier Transformation (FFT). This paper discusses a tool breakage monitoring system based on DWT of an acoustic emission (AE) and an electric feed current signal using an effective algorithm. The experiment results show overall 98.5% reliability and the good real-time monitoring capability of the proposed methodology for detecting tool breakage during drilling.

Perceptual speech coding and enhancement using frame-synchronized fast wavelet packet transform algorithms

Abstract: This paper presents new wideband speech coding and integrated speech coding-enhancement systems based on frame-synchronized fast wavelet packet transform algorithms. It also formulates temporal and spectral psychoacoustic models of masking adapted to wavelet packet analysis. The algorithm of the proposed FFT-like overlapped block orthogonal wavelet packet transform permits us to efficiently approximate the auditory critical band decomposition in the time and frequency domains. This allows us to make use of the temporal and spectral masking properties of the human auditory system to decrease the average bit rate of the encoder while perceptually hiding the quantization error. The same wavelet packet representation is used to merge speech enhancement and coding in the context of auditory modeling. The advantage of the method presented in this paper over previous approaches is that perceptual enhancement and coding, which is usually implemented as a cascade of two separate systems, are combined. This leads to a decreased computational load. Experiments show that the proposed wideband coding procedure by itself can achieve transparent coding of speech signals sampled at 16 kHz at an average bit rate of 39.4 kbit/s. The combined speech coding-enhancement procedure achieves higher bit rate values that depend on the residual noise characteristics at the output of the enhancement process.

Comparison of various periodograms for sinusoid detection and frequency estimation

Abstract: With the advent of the fast Fourier transform (FFT) algorithm, the periodogram and its variants such as the Bartlett's procedure and Welch method, have become very popular for spectral analysis. However, there has not been a thorough comparison of the detection and estimation performances of these methods. Different forms of the periodogram are studied here for single real tone detection and frequency estimation in the presence of white Gaussian noise. The threshold effect in frequency estimation, that is, when the estimation errors become several orders of magnitude greater than the Cramer-Rao lower bound (CRLB), is also investigated. It is shown that the standard periodogram gives the optimum detection performance for a pure tone while the Welch method is the best detector when there is phase instability in the sinusoid. As expected, since the conventional periodogram is a maximum likelihood estimator of frequency, it generally provides the minimum mean square frequency estimation errors.

Power system harmonics estimation using linear least squares method and SVD

Abstract: The paper examines the singular value decomposition (SVD) for estimation of harmonics in signals, in the presence of high noise. The proposed approach results in a linear least squares method. The methods developed for locating the frequencies as closely spaced sinusoidal signals are appropriate tools for the investigation of power system signals containing harmonics and interharmonics differing significantly in their multiplicity. The SVD approach is a numerical algorithm to calculate the linear least squares solution. The methods can also be applied for frequency estimation of heavy distorted periodical signals. To investigate the methods several experiments have been performed using simulated signals and the waveforms of a frequency converter current. For comparison, similar experiments have been repeated using the FFT with the same number of samples and sampling period. The comparison has proved superiority of the SVD for signals buried in the noise. However, the SVD computation is much more complex than FFT, and requires more extensive mathematical manipulations.

Efficient algorithms for burst error recovery using FFT and other transform kernels

Abstract: We show that the problem of signal reconstruction from missing samples can be handled by using reconstruction algorithms similar to the Reed-Solomon (RS) decoding techniques. Usually, the RS algorithm is used for error detection and correction of samples in finite fields. For the case of missing samples of a speech signal, we work with samples in the field of real or complex numbers, and we can use FFT or some new transforms in the reconstruction algorithm. DSP implementation and simulation results show that the proposed methods are better than the ones previously published in terms of the quality of recovered speech signal for a given complexity. The burst error recovery method using the FFT kernel is sensitive to quantization and additive noise like the other techniques. However, other proposed transform kernels are very robust in correcting bursts of errors with the presence of quantization and additive noise.

Numerical Analysis for the Elastic Contact of Real Rough Surfaces

Abstract: The elastic contact of rough surfaces and the subsurface stresses caused by the contact have been analyzed by means of a numerical model based on fast Fourier transforms (FFT) and minimization of complementary energy The elastic contact has been modeled mathematically as a linear complementarity problem and solved by a robust algorithm, Conjugate Gradient Method, while the force-displacement relation is determined through a FFT approach After solving for the pressure distribution, the subsurface stress field is obtained by calculating the stresses due to the application of a point force, and then integrating over the contact region In comparison with the matrix based method published in recent years, the numerical approach presented in this study is more efficient, more stable and requires less memory It has great potential in application to problems with general contact geometry and three-dimensional surface roughness The results show that high frequency roughness could lead to very sharp impulses a

The use of the wavelet transform to describe embolic signals.

Abstract: A number of methods to detect cerebral emboli and differentiate them from artefacts using Doppler ultrasound have been described in the literature. In most, Fourier transform-based (FT) spectral analysis has been used. The FT is not ideally suited to analysis of short-duration embolic signals due to an inherent trade-off between temporal and frequency resolution. An alternative approach that might be expected to describe embolic signals well is the wavelet transform. Wavelets are ideally suited for the analysis of sudden short-duration signal changes. Therefore, we have implemented a wavelet-based analysis and compared the results of this with a conventional FFT-based analysis. The temporal resolution, as measured by the half-width maximum, was significantly better for the continuous wavelet transform (CWT), mean (SD) 8.40 (8.82) ms, compared with the 128-point FFT, 12.92 (9.70) ms, and 64-point FFT, 10.80 (5.69) ms. Time localization of the CWT for the embolic signal was also significantly better than the FFT. The wavelet transform appears well suited to the analysis of embolic signals offering superior time resolution and time localization to the FFT.

Radar system and method of operating same

Abstract: A method of operating a radar system, including the steps of digitally sampling a received signal at a predetermined sampling rate, to periodically provide a set of selected samples, the set of selected samples including positive going ramp samples, negative going ramp samples and CW burst samples and performing a first fast Fourier transform (FFT) on the positive going ramp samples, performing a second fast Fourier transform on the negative going ramp samples and performing a third fast Fourier transform on the CW burst samples is described. Utilizing the subsequent radar operations the method further includes the steps of tracking each resulting signal from the first fast Fourier transform performing steps, tracking each resulting signal from the second fast Fourier transform performing steps and tracking each resulting signal from the third fast Fourier transform performing steps and associating any resulting signals from the tracking steps to periodically provide output signals indicative of other vehicles.

A fast Fourier transform compiler

Abstract: The FFTW library for computing the discrete Fourier transform (DFT) has gained a wide acceptance in both academia and industry, because it provides excellent performance on a variety of machines (e...