Showing papers on "Discrete-time Fourier transform published in 2009"
TL;DR: A new convolution structure for the LCT is introduced that preserves the convolution theorem for the Fourier transform and is also easy to implement in the designing of filters.
Abstract: The linear canonical transform (LCT) plays an important role in many fields of optics and signal processing. Many properties for this transform are already known, however, the convolution theorems don't have the elegance and simplicity comparable to that of the Fourier transform (FT), which states that the Fourier transform of the convolution of two functions is the product of their Fourier transforms. The purpose of this letter is to introduce a new convolution structure for the LCT that preserves the convolution theorem for the Fourier transform and is also easy to implement in the designing of filters. Some of well-known results about the convolution theorem in FT domain, fractional Fourier transform (FRFT) domain are shown to be special cases of our achieved results.
120 citations
TL;DR: In this article, a nonparametric method for the computation of instantaneous multivariate volatility for continuous semi-martingales, which is based on Fourier analysis, is presented. But the method is not suitable for the analysis of the entire market.
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.
114 citations
Book•
08 Jan 2009
TL;DR: In this article, Fourier Transforms in Probability, Random Variables and Stochastic Processes are used for time-frequency representation of signal and image synthesis in the context of Fourier analysis.
Abstract: 1. Introduction 2. Fundamentals of Fourier Analysis 3. Fourier Analysis in Systems Theory 4. Fourier Transforms in Probability, Random Variables and Stochastic Processes 5. The Sampling Theory 6. Generalizations of the Sampling Theorem 7. Noise and Error Effects 8. Multidimensional Signal Analysis 9. Time-Frequency Representations 10. Signal Recovery 11. Signal and Image Synthesis: Alternating Projections Onto Convex Sets 12. Mathematical Morphology and Fourier Analysis on Time Sales 13. Applications 14. Appendices 15. Reference
111 citations
TL;DR: This paper derives the requisite polar version of the standard Fourier operations for convolution-two dimensional, circular, and radial one dimensional-and shows that standard multiplication/convolution rules do apply as long as the correct definition of convolution is applied.
Abstract: For functions that are best described in terms of polar coordinates, the two-dimensional Fourier transform can be written in terms of polar coordinates as a combination of Hankel transforms and Fourier series-even if the function does not possess circular symmetry. However, to be as useful as its Cartesian counterpart, a polar version of the Fourier operational toolset is required for the standard operations of shift, multiplication, convolution, etc. This paper derives the requisite polar version of the standard Fourier operations. In particular, convolution-two dimensional, circular, and radial one dimensional-is discussed in detail. It is shown that standard multiplication/convolution rules do apply as long as the correct definition of convolution is applied.
78 citations
Posted Content•
TL;DR: In this article, the authors considered a zero mean discrete time series and defined its discrete Fourier transform at the canonical frequencies, and constructed a Portmanteau type test statistic for testing stationarity of the time series.
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.
77 citations
TL;DR: Independence of the evolution time domain size (in the terms of both: dimensionality and evolution time reached), suggests that random sampling should be used rather to design new techniques with large time domain than to accelerate standard experiments.
77 citations
TL;DR: An analytical expression for a Lorentz-Gauss beam passing through an FRFT system has been derived based on the definition of convolution and the convolution theorem of the Fourier transform.
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.
66 citations
TL;DR: This paper demonstrates an approach to mitigating spectral leakage based on windowing and states that spectral leakage applies to all forms of DFT, including the FFT and the IFFT (Inverse Fast Fourier Transform).
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".
65 citations
TL;DR: This letter proposes a random discrete fractional Fourier transform (RDFRFT) kernel matrix with random DFT eigenvectors and eigenvalues, which is illustrated as a security-enhanced image encryption scheme based on the RDFR FT.
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.
54 citations
TL;DR: A fast computational algorithm able to evaluate the Rayleigh-Sommerfeld diffraction formula, based on a special formulation of the convolution theorem and the fast Fourier transform with a more general type of convolution with a scale parameter is described.
Abstract: We describe a fast computational algorithm able to evaluate the Rayleigh-Sommerfeld diffraction formula, based on a special formulation of the convolution theorem and the fast Fourier transform. What is new in our approach compared to other algorithms is the use of a more general type of convolution with a scale parameter, which allows for independent sampling intervals in the input and output computation windows. Comparison between the calculations made using our algorithm and direct numeric integration show a very good agreement, while the computation speed is increased by orders of magnitude.
43 citations
Book•
17 Jun 2009TL;DR: This book expands the domains of existence of the most important transforms and thus opens the door to a new world of applications using novel, powerful mathematical tools.
Abstract: Signals, Systems, Transforms, and Digital Signal Processing with MATLAB has as its principal objective simplification without compromise of rigor. Graphics, called by the author, "the language of scientists and engineers", physical interpretation of subtle mathematical concepts, and a gradual transition from basic to more advanced topics are meant to be among the important contributions of this book. After illustrating the analysis of a function through a step-by-step addition of harmonics, the book deals with Fourier and Laplace transforms. It then covers discrete time signals and systems, the z transform, continuous- and discrete-time filters, active and passive filters, lattice filters, and continuous- and discrete-time state space models. The author goes on to discuss the Fourier transform of sequences, the discrete Fourier transform, and the fast Fourier transform, followed by Fourier-, Laplace, and z-related transforms, including WalshHadamard, generalized Walsh, Hilbert, discrete cosine, Hartley, Hankel, Mellin, fractional Fourier, and wavelet. He also surveys the architecture and design of digital signal processors, computer architecture, logic design of sequential circuits, and random signals. He concludes with simplifying and demystifying the vital subject of distribution theory. Drawing on much of the authors own research work, this book expands the domains of existence of the most important transforms and thus opens the door to a new world of applications using novel, powerful mathematical tools.
TL;DR: In this paper, an efficient numerical scheme is presented to compute the response of piezoelectric composite materials with arbitrary complex microstructures, making use of fast Fourier transform to solve iteratively the coupled periodic Lippmann-Schwinger equations for a heterogeneous electroelastic medium.
Abstract: An efficient numerical scheme is presented to compute the response of piezoelectric composite materials with arbitrary complex microstructures. It makes use of fast Fourier transform to solve iteratively the coupled periodic Lippmann-Schwinger equations for a heterogeneous electroelastic medium. The method is assessed in the case of a two-phase composite by comparison with analytic solutions and finite-element results taken from the literature.
TL;DR: A memory efficient approximation to the nonuniform Fourier transform of a support limited sequence is derived based on the theory of shift-invariant representations and an exact expression for the worst-case mean square approximation error is derived.
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.
Book•
01 Jan 2009
TL;DR: Fourier transform methods have emerged as one of the major methodologies for the evaluation of derivative contracts, largely due to the need to strike a balance between the extension of existing pricing models beyond the traditional Black-Scholes setting and a need to evaluate prices consistently with the market quotes as mentioned in this paper.
Abstract: In recent years, Fourier transform methods have emerged as one of the major methodologies for the evaluation of derivative contracts, largely due to the need to strike a balance between the extension of existing pricing models beyond the traditional Black-Scholes setting and a need to evaluate prices consistently with the market quotes.Fourier Transform Methods in Finance is a practical and accessible guide to pricing financial instruments using Fourier transform. Written by an experienced team of practitioners and academics, it covers Fourier pricing methods; the dynamics of asset prices; non stationary market dynamics; arbitrage free pricing; generalized functions and the Fourier transform method.Readers will learn how to:
compute the Hilbert transform of the pricing kernel under a Fast Fourier Transform (FFT) technique
characterise the price dynamics on a market in terms of the characteristic function, allowing for both diffusive processes and jumps
apply the concept of characteristic function to non-stationary processes, in particular in the presence of stochastic volatility and more generally time change techniques
perform a change of measure on the characteristic function in order to make the price process a martingale
recover a general representation of the pricing kernel of the economy in terms of Hilbert transform using the theory of generalised functions
apply the pricing formula to the most famous pricing models, with stochastic volatility and jumps.
Junior and senior practitioners alike will benefit from this quick reference guide to state of the art models and market calibration techniques. Not only will it enable them to write an algorithm for option pricing using the most advanced models, calibrate a pricing model on options data, and extract the implied probability distribution in market data, they will also understand the most advanced models and techniques and discover how these techniques have been adjusted for applications in finance.
TL;DR: An optical image watermarking scheme using fractional Fourier transform that offers additional degrees of freedom to enlarge the key size, thus enhancing the level of security is proposed.
Abstract: An optical image watermarking scheme using fractional Fourier transform is proposed. The watermark is encrypted using double random fractional order Fourier domain encoding scheme. Encrypted image is water marked into a host image. Embedding watermark sequences into fractional Fourier domain has an important advantage over embedding in spatial domain or in frequency domain. The watermark is recovered by applying corresponding correct fractional orders and random phase masks. The use of fractional Fourier transform offers additional degrees of freedom to enlarge the key size, thus enhancing the level of security. The effect of occlusion of watermarked image on the recovered watermark is also studied. The proposed idea is supported with simulation results.
TL;DR: The generalized Poisson sum formula is obtained based on the relationship of the FrFT and the Fourier transform and some new results associated with this novel sum formula have been derived and the potential applications of these new results in estimating the bandwidth and the fractional spectrum shape of a signal in the fractionsal Fourier domain are proposed.
Patent•
12 Mar 2009
TL;DR: In this paper, 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-LEF method to unaliased frequency component in the temporaltransformed data.
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.
TL;DR: A Fourier analytical condition linking alias-free sampling with the Fourier transform of the indicator function defined on the given frequency support is proposed, and a computational testing procedure is proposed that can efficiently search for the optimal alias- free sampling lattices for a given polygonal or polyhedral shaped frequency domain.
Abstract: We propose a Fourier analytical condition linking alias-free sampling with the Fourier transform of the indicator function defined on the given frequency support. Our discussions center around how to develop practical computation algorithms based on the proposed analytical condition. We address several issues along this line, including the derivation of simple closed-form expressions for the Fourier transforms of the indicator functions defined on arbitrary polygonal and polyhedral domains; a complete and nonredundant enumeration of all quantized sampling lattices via the Hermite normal forms of integer matrices; and a quantitative analysis of the approximation of the original infinite Fourier condition by using finite computations. Combining these results, we propose a computational testing procedure that can efficiently search for the optimal alias-free sampling lattices for a given polygonal or polyhedral shaped frequency domain. Several examples are presented to show the potential of the proposed algorithm in multidimensional filter bank design, as well as in applications involving the design of efficient sampling patterns for multidimensional band-limited signals.
TL;DR: It is demonstrated that the AFT-MC system effectively minimizes interference in time-varying multipath channels with line-of-sight component and narrow beamwidth of scattered components that often occurs in aeronautical and satellite communications.
Abstract: Multicarrier techniques based on affine Fourier transform (AFT) have been recently proposed for transmission in the wireless channels. The AFT represents a generalization of the Fourier and fractional Fourier transform. We derive the exact and approximated interference power, upper bound and measure of applicability for the AFT based multicarrier (AFT-MC) system. It is demonstrated that the AFT-MC system effectively minimizes interference in time-varying multipath channels with line-of-sight component and narrow beamwidth of scattered components that often occurs in aeronautical and satellite communications.
14 Jun 2009
TL;DR: In this paper, the authors compare three algorithms (non-equispaced DFT, interpolated FFT and non-Equispaced FFT) for OCT imaging in terms of speed and accuracy.
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.
TL;DR: In this new algorithm, the analytical formulas for the harmonic frequency are obtained by applying the Chebyshev best approximation theory and can be easily implemented by hardware multipliers, which is convenient for real-time measurement.
20 Aug 2009
TL;DR: This paper shows that hidden shifts can be extracted efficiently from Boolean functions that are quadratic forms, and shows how to identify an unknown quadRatic form on n variables using a linear number of queries, in contrast to the classical case were this takes ?
Abstract: Most quantum algorithms that give an exponential speedup over classical algorithms exploit the Fourier transform in some way. In Shor's algorithm, sampling from the quantum Fourier spectrum is used to discover periodicity of the modular exponentiation function. In a generalization of this idea, quantum Fourier sampling can be used to discover hidden subgroup structures of some functions much more efficiently than it is possible classically. Another problem for which the Fourier transform has been recruited successfully on a quantum computer is the hidden shift problem. Quantum algorithms for hidden shift problems usually have a slightly different flavor from hidden subgroup algorithms, as they use the Fourier transform to perform a correlation with a given reference function, instead of sampling from the Fourier spectrum directly. In this paper we show that hidden shifts can be extracted efficiently from Boolean functions that are quadratic forms. We also show how to identify an unknown quadratic form on n variables using a linear number of queries, in contrast to the classical case were this takes ?(n 2) many queries to a black box. What is more, we show that our quantum algorithm is robust in the sense that it can also infer the shift if the function is close to a quadratic, where we consider a Boolean function to be close to a quadratic if it has a large Gowers U 3 norm.
TL;DR: It turns out that the two-scale compactness results are easily obtained by the use of the two -scale Fourier transform, and the Floquet–Bloch eigenvalue problems for differential operators is recovered in a natural and straight forward way.
Abstract: A two-scale Fourier transform for periodic homogenization in Fourier space is introduced. The transform connects the various existing techniques for periodic homogenization, i.e., two-scale convergence, periodic unfolding and the Floquet– Bloch expansion approach to homogenization. It turns out that the two-scale compactness results are easily obtained by the use of the two-scale Fourier transform. Moreover, the Floquet–Bloch eigenvalue problems for differential operators is recovered in a natural and straight forward way by the use of this transform. The transform is generalized to the (N + 1)-scale case.
TL;DR: The classical technique for numerical dispersion compensation, usually used with a Michelson interferometer, cannot be efficiently used with wedge interferometers as it is limited to the cases of weak dispersion.
Abstract: In this paper we study the effect of material dispersion on the performance of a moving-optical-wedge Fourier transform spectrometer. The spectrum is thus evaluated numerically using a test spectrum source. The obtained numerical results show that the classical technique for numerical dispersion compensation, usually used with a Michelson interferometer, cannot be efficiently used with wedge interferometers as it is limited to the cases of weak dispersion. The error in this technique is thus evaluated in different cases and a new numerical technique is proposed to overcome this error. We also notice shrinkage in the interferogram spread in the spatial domain in contradiction with the normal dispersion effect in a Michelson interferometer.
TL;DR: In this article, it is shown that the coefficients of the wrapped filter can also be obtained by applying a discrete Fourier transform to a set of ordinates sampled from the frequency response function, which results in results that are identical to those obtained by a multiplication in the frequency domain.
01 Dec 2009
TL;DR: This paper shows that the Slice Theorem is also valid within the NTT and that it can be utilized as a new exact, integer-only and fast inversion scheme for the FRT, with the same computational complexity as the FFT.
Abstract: This paper presents a new fast method to map between images and their digital projections based on the Number Theoretic Transform (NTT) and the Finite Radon Transform (FRT). The FRT is a Discrete Radon Transform (DRT) defined on the same finite geometry as the Finite or Discrete Fourier Transform (DFT). Consequently, it may be inverted directly and exactly via the Fast Fourier Transform (FFT) without any interpolation or filtering [1]. As with the FFT, the FRT can be adapted to square images of arbitrary sizes such as dyadic images, prime-adic images and arbitrary-sized images. However, its simplest form is that of prime-sized images [2]. The FRT also preserves the discrete versions of both the Fourier Slice Theorem (FST) and Convolution Property of the Radon Transform (RT). The NTT is also defined on the same geometry as the DFT and preserves the Circular Convolution Property (CCP) of the DFT [3, 4]. This paper shows that the Slice Theorem is also valid within the NTT and that it can be utilized as a new exact, integer-only and fast inversion scheme for the FRT, with the same computational complexity as the FFT. Digital convolutions and exact digital filtering of projections can also be performed using this Number Theoretic FRT (NFRT).
TL;DR: In this article, an analytical expression for a Lorentz beam passing through a fractional Fourier transform (FRFT) system has been derived based on the deflnition of convolution and the convolution theorem of the Fourier Transform.
Abstract: This paper introduces Lorentz beams to describe certain laser sources that produce highly divergent flelds. The fractional Fourier transform (FRFT) is applied to treat the propagation of Lorentz beams. Based on the deflnition of convolution and the convolution theorem of the Fourier transform, an analytical expression for a Lorentz beam passing through a FRFT system has been derived. By using the derived formula, the properties of a Lorentz beam in the FRFT plane are illustrated numerically.
18 May 2009
TL;DR: This paper discusses the efficient, non-redundant evaluation of a Discrete Fourier Transform on the three dimensional Body-Centered and Face- Centered Cubic lattices by using an axis aligned window to truncate and periodize the sampled function which leads to separable transforms.
Abstract: This paper discusses the efficient, non-redundant evaluation of a Discrete Fourier Transform on the three dimensional Body-Centered and Face-Centered Cubic lattices. The key idea is to use an axis aligned window to truncate and periodize the sampled function which leads to separable transforms. We exploit the geometry of these lattices and show that by choosing a suitable non-redundant rectangular region in the frequency domain, the transforms can be efficiently evaluated using the Fast Fourier Transform.
TL;DR: In this article, it is shown that an ideal filter with gain of unity throughout the pass band and a gain of zero throughout the stop band can be realized both in the time domain and in the frequency domain.
Abstract: This paper expounds some of the results of Fourier theory that are essential to the statistical analysis of time series. It employs the algebra of circulant matrices to expose the structure of the discrete Fourier transform and to elucidate the filtering operations that may be applied to finite data sequences.An ideal filter with a gain of unity throughout the pass band and a gain of zero throughout the stop band is commonly regarded as incapable of being realised in finite samples. It is shown here that, to the contrary, such a filter can be realised both in the time domain and in the frequency domain.The algebra of circulant matrices is also helpful in revealing the nature of statistical processes that are band limited in the frequency domain. In order to apply the conventional techniques of autoregressive moving-average modelling, the data generated by such processes must be subjected to anti-aliasing filtering and sub sampling. These techniques are also described.It is argued that band-limited processes are more prevalent in statistical and econometric time series than is commonly recognised.