Showing papers on "Non-uniform discrete Fourier transform published in 2009"

A Fourier transform method for nonparametric estimation of multivariate volatility

Abstract: We provide a nonparametric method for the computation of instantaneous multivariate volatility for continuous semi-martingales, which is based on Fourier analysis. The co-volatility is reconstructed as a stochastic function of time by establishing a connection between the Fourier transform of the prices process and the Fourier transform of the co-volatility process. A nonparametric estimator is derived given a discrete unevenly spaced and asynchronously sampled observations of the asset price processes. The asymptotic properties of the random estimator are studied: namely, consistency in probability uniformly in time and convergence in law to a mixture of Gaussian distributions.

Rotational Invariance Based on Fourier Analysis in Polar and Spherical Coordinates

Abstract: In this paper, polar and spherical Fourier analysis are defined as the decomposition of a function in terms of eigenfunctions of the Laplacian with the eigenfunctions being separable in the corresponding coordinates. The proposed transforms provide effective decompositions of an image into basic patterns with simple radial and angular structures. The theory is compactly presented with an emphasis on the analogy to the normal Fourier transform. The relation between the polar or spherical Fourier transform and the normal Fourier transform is explored. As examples of applications, rotation-invariant descriptors based on polar and spherical Fourier coefficients are tested on pattern classification problems.

A test for second order stationarity of a time series based on the Discrete Fourier Transform (Technical Report)

Abstract: We consider a zero mean discrete time series, and define its discrete Fourier transform at the canonical frequencies. It is well known that the discrete Fourier transform is asymptotically uncorrelated at the canonical frequencies if and if only the time series is second order stationary. Exploiting this important property, we construct a Portmanteau type test statistic for testing stationarity of the time series. It is shown that under the null of stationarity, the test statistic is approximately a chi square distribution. To examine the power of the test statistic, the asymptotic distribution under the locally stationary alternative is established. It is shown to be a type of noncentral chi-square, where the noncentrality parameter measures the deviation from stationarity. The test is illustrated with simulations, where is it shown to have good power. Some real examples are also included to illustrate the test.

Fractional Fourier transform of Lorentz-Gauss beams

Abstract: Lorentz-Gauss beams are introduced to describe certain laser sources that produce highly divergent beams. The fractional Fourier transform (FRFT) is applied to treat the propagation of Lorentz-Gauss beams. Based on the definition of convolution and the convolution theorem of the Fourier transform, an analytical expression for a Lorentz-Gauss beam passing through an FRFT system has been derived. By using the derived expression, the properties of a Lorentz-Gauss beam in the FRFT plane are graphically illustrated with numerical examples.

The Discrete Fourier Transform, Part 4: Spectral Leakage

Abstract: This paper is part 4 in a series of papers about the Discrete Fourier Transform (DFT) and the Inverse Discrete Fourier Transform (IDFT). The focus of this paper is on spectral leakage. Spectral leakage applies to all forms of DFT, including the FFT (Fast Fourier Transform) and the IFFT (Inverse Fast Fourier Transform). We demonstrate an approach to mitigating spectral leakage based on windowing. Windowing temporally isolates the Short-Time Fourier Transform (STFT) in order to amplitude modulate the input signal. This requires that we know the extent, of the event in the input signal and that we have enough samples to yield a sufficient spectral resolution for our application. This report is a part of project Fenestratus, from the skunk-works of DocJava, Inc. Fenestratus comes from the Latin and means "to furnish with windows".

Optical phase recovery in the dispersive Fourier transform

Abstract: The dispersive Fourier transform permits real-time acquisition of optical spectra with analog-to-digital converters. The method utilizies the property that a signal’s temporal envelope matches its spectral profile if sufficiently dispersed. Unfortunately, the dispersion demand can be substantial and signal losses in highly dispersive elements represent a significant challenge, especially outside the telecommunications band. We address this problem by experimentally demonstrating that a time-domain equivalent of the Gerchberg–Saxton algorithm removes the fundamental dispersion requirement in the dispersive Fourier transform. The algorithm recovers the phase from time-domain intensity measurements.

Brief Notes in Advanced DSP: Fourier Analysis with MATLAB

Abstract: Discrete Fourier Transform Properties of the discrete Fourier transform Fourier transform splitting Fast Fourier transform Codes for the paired FFT Paired and Haar transforms Integer Fourier Transform Reversible integer Fourier transform Lifting schemes for DFT One-point integer transform DFT in vector form Roots of the unit Codes for the block DFT General elliptic Fourier transforms Cosine Transform Partitioning the DCT Paired algorithm for the N-point DCT Codes for the paired transform Reversible integer DCT Method of nonlinear equations Canonical representation of the integer DCT Hadamard Transform The Walsh and Hadamard transform Mixed Hadamard transformation Generalized bit and transformations T-decomposition of Hadamard matrices Mixed Fourier transformations Mixed transformations: continuous case Paired Transform-Based Decomposition Decomposition of 1D signals 2D paired representation Fourier Transform and Multiresolution Fourier transform Representation by frequency-time wavelets Time-frequency correlation analysis Givens-Haar transformations References Index

Random Discrete Fractional Fourier Transform

Abstract: In this letter, a new commuting matrix with random discrete Fourier transform (DFT) eigenvectors is first constructed. A random discrete fractional Fourier transform (RDFRFT) kernel matrix with random DFT eigenvectors and eigenvalues is then proposed. The RDFRFT has an important feature that the magnitude and phase of its transform output are both random. As an application example, a security-enhanced image encryption scheme based on the RDFRFT is illustrated.

Development of a non-uniform discrete Fourier transform based high speed spectral domain optical coherence tomography system

Abstract: We develop a high speed spectral domain optical coherence tomography (SD-OCT) system based on a custom-built spectrometer and non-uniform discrete Fourier transform (NDFT) to realize minimized depth dependent sensitivity fall-off. After precise spectral calibration of the spectrometer, NDFT of the acquired spectral data is adopted for image reconstruction. The spectrometer is able to measure a wavelength range of about 138nm with a spectral resolution of 0.0674nm at central wavelength of 835nm, corresponding to an axial imaging range of 2.56mm in air. Zemax simulations and sensitivity fall-off measurements under two alignment states of the spectrometer are given. Both theoretical simulations and experiments are done to study the depth dependent sensitivity of the developed system based on NDFT in contrast to those based on conventional discrete Fourier transform (DFT) with and without interpolation. In vivo imaging on human finger from volunteer is conducted at A-scan rate of 29 kHz and reconstruction is done based on different methods. The comparing results confirm that reconstruction method based on NDFT indeed improves sensitivity especially at large depth while maintaining the coherence-function-limited depth resolution.

Conjugate Symmetric Sequency-Ordered Complex Hadamard Transform

Abstract: A new transform known as conjugate symmetric sequency-ordered complex Hadamard transform (CS-SCHT) is presented in this paper. The transform matrix of this transform possesses sequency ordering and the spectrum obtained by the CS-SCHT is conjugate symmetric. Some of its important properties are discussed and analyzed. Sequency defined in the CS-SCHT is interpreted as compared to frequency in the discrete Fourier transform. The exponential form of the CS-SCHT is derived, and the proof of the dyadic shift invariant property of the CS-SCHT is also given. The fast and efficient algorithm to compute the CS-SCHT is developed using the sparse matrix factorization method and its computational load is examined as compared to that of the SCHT. The applications of the CS-SCHT in spectrum estimation and image compression are discussed. The simulation results reveal that the CS-SCHT is promising to be employed in such applications.

Hilbert Transform and Its Engineering Applications

Abstract: The Hilbert transform is useful in calculating instantaneous attributes of a time series, especially the envelope amplitude and instantaneous frequency. The instantaneous envelope is the amplitude of the complex Hilbert transform; the instantaneous frequency is the time rate of change of the instantaneous phase angle. These properties can be applied to identify dynamic characteristics of a linear as well as a nonlinear system. However, the conventional discrete Hilbert transform, which is based on fast Fourier transform and inverse Fourier transform, has shown the lack of accuracy for time-derivative calculations. In this paper, we first introduce the Hilbert transform and its applications to the nonlinear system parameter identification. Then we address the practical issues in applying the Hilbert transform to engineering applications. To increase the accuracy of the envelope detection, we propose a Hilbert transform technique based on local-maxima interpolation. Analyses and simulations are carried out to demonstrate the advantages of the proposed technique. Finally, we employ the proposed local-maxima-interpolation technique in identifying the nonlinear dynamic characteristics of industrial examples.

Sparse Fourier Transform via Butterfly Algorithm

Abstract: This paper introduces a fast algorithm for computing sparse Fourier transforms with spatial and Fourier data supported on curves or surfaces. This problem appears naturally in several important applications of wave scattering, digital signal processing, and reflection seismology. The main idea of the algorithm is that the interaction between a frequency region and a spatial region is approximately low rank if the product of their widths are bounded by the maximum frequency. Based on this property, we can approximate the interaction between these two boxes accurately and compactly using a small number of equivalent sources. The overall structure of the algorithm follows the butterfly algorithm. The computation is further accelerated by exploiting the tensor-product property of the Fourier kernel in two and three dimensions. The proposed algorithm is accurate and has the optimal complexity. We present numerical results in both two and three dimensions.

Optimized Least-Square Nonuniform Fast Fourier Transform

Abstract: The main focus of this paper is to derive a memory efficient approximation to the nonuniform Fourier transform of a support limited sequence. We show that the standard nonuniform fast Fourier transform (NUFFT) scheme is a shift invariant approximation of the exact Fourier transform. Based on the theory of shift-invariant representations, we derive an exact expression for the worst-case mean square approximation error. Using this metric, we evaluate the optimal scale-factors and the interpolator that provides the least approximation error. We also derive the upper-bound for the error component due to the lookup tablebased evaluation of the interpolator; we use this metric to ensure that this component is not the dominant one. Theoretical and experimental comparisons with standard NUFFT schemes clearly demonstrate the significant improvement in accuracy over conventional schemes, especially when the size of the uniform fast Fourier transform (FFT) is small. Since the memory requirement of the algorithm is dependent on the size of the uniform FFT, the proposed developments can lead to iterative signal reconstruction algorithms with significantly lower memory demands.

Nonuniform sampling, image recovery from sparse data and the discrete sampling theorem

Abstract: In many applications, sampled data are collected in irregular fashion or are partly lost or unavailable. In these cases, it is necessary to convert irregularly sampled signals to regularly sampled ones or to restore missing data. We address this problem in the framework of a discrete sampling theorem for band-limited discrete signals that have a limited number of nonzero transform coefficients in a certain transform domain. Conditions for the image unique recovery, from sparse samples, are formulated and then analyzed for various transforms. Applications are demonstrated on examples of image superresolution and image reconstruction from sparse projections.

Robust color texture features based on ranklets and discrete Fourier transform

Abstract: We present a set of multiscale, multidirectional, rotation-invariant features for color texture characterization. The proposed model is based on the ranklet transform, a technique relying on the calculation of the relative rank of the intensity level of neighboring pixels. Color and texture are merged into a compact descriptor by computing the ranklet transform of each color channel separately and of couples of color channels jointly. Robustness against rotation is based on the use of circularly symmetric neighborhoods together with the discrete Fourier transform. Experimental results demonstrate that the approach shows good robustness and accuracy.

Denoising and Signal-to-Noise Ratio Enhancement: Wavelet Transform and Fourier Transform

Abstract: This chapter introduces the methods of Fourier and wavelet analysis for enhancing the signal-to-noise ratio in typical chemometric and other measured data. Fourier analysis has been popular for many decades but is best suited for enhancing signals where most features are localized in frequency. In contrast, wavelet analysis is appropriate for signals that contain features localized in both time and frequency. It also retains the benefits of Fourier analysis such as orthono, mality and computational efficiency. Practical algorithms for off-line and on-line denoising are described and compared via simple examples. These algorithms can be used for off-line or on-line data and can remove Gaussian as well as non-Gaussian noise.

Generalized Discrete Cosine Transform

Abstract: The discrete cosine transform (DCT), introduced by Ahmed, Natarajan and Rao, has been used in many applications of digital signal processing, data compression and information hiding. There are four types of the discrete cosine transform. In simulating the discrete cosine transform, we propose a generalized discrete cosine transform with three parameters, and prove its orthogonality for some new cases. Finally, a new type of discrete cosine transform is proposed and its orthogonality is proved.

Spatial resolution analysis for discrete Fourier transform-based Brillouin optical time domain reflectometry

Abstract: Discrete Fourier transform (DFT) requires many sampled points for a spectrum. We find that the spatial resolution of the DFT-based Brillouin optical time domain reflectometry (BOTDR) is determined by the pulse width of the probe light and the time length of the sampling data used to perform the DFT. The best spatial resolution is limited by the pulse width. At a certain sampling rate, the spatial resolution increases linearly with the number of points in DFT. The frequency uncertainty improves with the increased number. Window function restrains the spectral leakage significantly and can improve the spatial resolution. But when the influence of the spectral leakage can be neglected, the frequency uncertainty without a window function is better than that with a window function for the same spatial resolution.

Method for interpolating seismic data by anti-alias, anti-leakage fourier transform

Abstract: An estimated frequency-wavenumber spectrum is generated by applying a first Anti-leakage Fourier transform method to aliased frequency components in temporal-transformed seismic data and applying a second Anti-leakage Fourier transform method to unaliased frequency components in the temporal-transformed seismic data. The second Anti-leakage Fourier transform method applies an absolute frequency-wavenumber spectrum extrapolated from unaliased frequencies to aliased frequencies to weight frequency-wavenumber components of the aliased frequencies. An inverse temporal and spatial Fourier transform is applied to the estimated frequency-wavenumber spectrum, generating trace interpolation of the seismic data.

Using nonequispaced fast Fourier transformation to process optical coherence tomography signals

Abstract: In OCT imaging the spectra that are used for Fourier transformation are in general not acquired linearly in k-space. Therefore one needs to apply an algorithm to re-sample the data and finally do the Fourier Transformation to gain depth information. We compare three algorithms (Non-Equispaced DFT, interpolated FFT and Non-Equispaced FFT) for this purpose in terms of speed and accuracy. The optimal algorithm depends on the OCT device (speed, SNR) and the object.

Digital phase contrast with the fractional Fourier transform

Abstract: A new method of digital phase contrast based on fractional-order Fourier reconstruction is proposed. We show that the diffraction patterns produced by pure phase objects exhibit linear chirp functions that can be advantageously processed using the fractional Fourier transform. The optimal fractional orders lead to the longitudinal location of the phase object, while the analysis of the reconstructed pattern leads to its diameter and to the value of the phase shift. Simulations and experimental results are given. The configuration tested in this paper is a very general Gaussian illumination.

Laguerre-Gaussian radial Hilbert transform for edge-enhancement Fourier transform x-ray microscopy

Abstract: An efficient technique to achieve isotropic edge enhancement in optics involves applying a radial Hilbert transform on the object spectrum. Here we demonstrate a simple setup for isotropic edge-enhancement in soft-x-ray microscopy, using a single diffractive Laguerre-Gaussian zone plate (LGZP) for radial Hilbert transform. Since the LGZP acts as a beam-splitter, diffraction efficiency problems usually associated with x-ray microscopy optics are not present in this system. As numerically demonstrated, the setup can detect optical path differences as small as λ/50 with high contrast.

Fourier transform based spatial outlier mining

Abstract: Outlier detection is an important problem in spatial analysis which involves finding a region of spatial locations with features significantly different from the rest of the population. In this paper, we used fast fourier transform to highlight the areas with high frequency change. The spatial points identified by the fourier transform are then reconfirmed with Z-value test and outlier regions are identified. We performed several experiments to highlight the accuracy and efficiency of the approach and compared it with some other existing approaches.

Sinusoidal Polynomial Parameter Estimation Using the Distribution Derivative

Abstract: In this paper, we present a method to estimate the parameters of a generalized sinusoidal model. A generalized sinusoid x is defined as a polynomial in the log domain, with complex coefficients alphai : x(t)=exp(Sigmai alphai t i), where i=0...Q. The method is based on the distribution derivative of the signal and operates in the transform domain. The method is very general and can use any linear transform such as the Fourier transform or the wavelet transform, or even combinations of linear transforms. Examples with the Fourier transform are given. The Fourier-based estimation methods are evaluated using synthetic signals and have performance very close to the theoretical bound.