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38
Nonuniform Fast Fourier Transforms Using MinMax Interpolation
 IEEE Trans. Signal Process
, 2003
"... The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformlyspaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several pap ..."
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Cited by 90 (16 self)
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The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformlyspaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the minmax sense of minimizing the worstcase approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the minmax approach provides substantially lower approximation errors than conventional interpolation methods. The minmax criterion is also useful for optimizing the parameters of interpolation kernels such as the KaiserBessel function.
Accelerating the nonuniform Fast Fourier Transform
 SIAM REVIEW
, 2004
"... The nonequispaced Fourier transform arises in a variety of application areas, from medical imaging to radio astronomy to the numerical solution of partial differential equations. In a typical problem, one is given an irregular sampling of N data in the frequency domain and one is interested in recon ..."
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Cited by 39 (2 self)
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The nonequispaced Fourier transform arises in a variety of application areas, from medical imaging to radio astronomy to the numerical solution of partial differential equations. In a typical problem, one is given an irregular sampling of N data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain. When the sampling is uniform, the fast Fourier transform (FFT) allows this calculation to be computed in O(N log N) operations rather than O(N 2) operations. Unfortunately, when the sampling is nonuniform, the FFT does not apply. Over the last few years, a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform FFTs (NUFFTs). These rely on a mixture of interpolation and the judicious use of the FFT on an oversampled grid [A. Dutt and V. Rokhlin, SIAM J. Sci. Comput., 14 (1993), pp. 1368–1383]. In this paper, we observe that one of the standard interpolation or “gridding ” schemes, based on Gaussians, can be accelerated by a significant factor without precomputation and storage of the interpolation weights. This is of particular value in two and threedimensional settings, saving either 10dN in storage in d dimensions or a factor of about 5–10 in CPUtime (independent of dimension).
Warped DiscreteFourier Transform: Theory and Applications
 IEEE Trans. Circuits Systems I
, 2001
"... Abstract—In this paper, we advance the concept of warped discreteFourier transform (WDFT), which is the evaluation of frequency samples of thetransform of a finitelength sequence at nonuniformly spaced points on the unit circle obtained by a frequency transformation using an allpass warping funct ..."
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Cited by 9 (0 self)
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Abstract—In this paper, we advance the concept of warped discreteFourier transform (WDFT), which is the evaluation of frequency samples of thetransform of a finitelength sequence at nonuniformly spaced points on the unit circle obtained by a frequency transformation using an allpass warping function. By factorizing the WDFT matrix, we propose an exact computation scheme for finite sequences using less number of operations than a direct computation. We discuss various properties of WDFT and the structure of the factoring matrices. Examples of WDFT for first and secondorder allpass functions is also presented. Applications of WDFT included are spectral analysis, design of tunable FIR filters, and design of perfect reconstruction filterbanks with nonuniformly spaced passbands of filters in the bank. WDFT is efficient to resolve closely spaced sinusoids. Tunable FIR filters may be designed from FIR prototypes using WDFT. In yet another application, warped PR filterbanks are designed using WDFT and are applied for signal compression. Index Terms—Allpass, DFT, frequency warping, warped DFT. I.
Efficient dualtone multifrequency detection using the nonuniform discrete Fourier transform
 IEEE Signal Process. Lett
, 1998
"... recommendations for dualtone multifrequency (DTMF) signaling are not met by conventional DTMF detectors. We present an efficient DTMF detection algorithm based on the nonuniform discrete Fourier transform that meets all of the ITU recommendations. The key innovations are the use of two sliding wi ..."
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Cited by 7 (1 self)
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recommendations for dualtone multifrequency (DTMF) signaling are not met by conventional DTMF detectors. We present an efficient DTMF detection algorithm based on the nonuniform discrete Fourier transform that meets all of the ITU recommendations. The key innovations are the use of two sliding windows and development of sophisticated timing tests. Our algorithm requires no buffering of input samples. To perform DTMF detection on n telephone channels, our algorithm requires approximately n MIPS on a digital signal processor (DSP), 75 + 30n words of data memory, and 1000 words of program memory. Using the new algorithm, a single fixedpoint DSP can perform ITUcompliant DTMF detection on the 24 telephone channels of a T1 timedivision multiplexed telecommunications line. Index Terms — Modified Goertzel algorithm, multichannel DTMF detection, sliding window, spectral estimation, touchtone dialing. I.
FOURIER VOLUME RENDERING OF IRREGULAR DATA SETS
, 2002
"... Examining irregularly sampled data sets usually requires gridding that data set. However, examination of a data set at one particular resolution may not be adequate since either fine details will be lost, or coarse details will be obscured. In either case, the original data set has been lost. We p ..."
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Cited by 6 (0 self)
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Examining irregularly sampled data sets usually requires gridding that data set. However, examination of a data set at one particular resolution may not be adequate since either fine details will be lost, or coarse details will be obscured. In either case, the original data set has been lost. We present an algorithm to create a regularly sampled data set from an irregular one. This new data set is not only an approximation to the original, but allows the original points to be accurately recovered, while still remaining relatively small. This result is accompanied by an efficient ‘zooming ’ operation that allows the user to increase the resolution while gaining new details, all without regridding the data. The technique is presented in Ndimensions, but is particularly well suited to Fourier Volume Rendering, which is the fastest known method of direct volume rendering. Together, these techniques allow accurate and efficient, multiresolution exploration of volume data.
Design Of PolarSeparable Fir Filters By Radial Slice Approximations
 in IEEE International Conference on Acoustics, Speech, and Signal Processing
, 1997
"... We introduce the design of polarseparable 2D FIR filters by radial slice approximations (RSA). It is a two step procedure. First, 1D filters for the radial and the angular components are designed. Then the desired filter response is approximated on many radial slices in a weighted mean square sen ..."
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We introduce the design of polarseparable 2D FIR filters by radial slice approximations (RSA). It is a two step procedure. First, 1D filters for the radial and the angular components are designed. Then the desired filter response is approximated on many radial slices in a weighted mean square sense. In the case of circular filters, RSA outperforms other design procedures in terms of ripple size and circularity of the passband. Examples of filters with nonconstant angular functions prove the flexibility of the new method. 1. INTRODUCTION The design of 2D FIR filters for signal and image processing is an important and difficult problem. Several image processing techniques [1, 2] require FIR filters that are polarseparable in the ideal case. We present here a general two step procedure called Radial Slice Approximations (RSA) to design FIR filters with arbitrary angular and radial specifications in the frequency domain. First 1D filters for the radial and angular components are de...
Mixed FourierRadon reconstruction of irregularly and sparsely sampled seismic data
, 1997
"... Seismic data irregularly sampled in two dimensions is transformed to the Fourier/Radon domain using a least squares formulation where the inverse transform, from the Fourier/Radon to the spatial domain is used as a forward model. By a proper choice of the region of support, the total number of para ..."
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Seismic data irregularly sampled in two dimensions is transformed to the Fourier/Radon domain using a least squares formulation where the inverse transform, from the Fourier/Radon to the spatial domain is used as a forward model. By a proper choice of the region of support, the total number of parameters is limited (yet such that the actual data is optimally contained), leading to a stable inversion. Subsequently the data can be transformed to any desired grid in the spatial domain. Also, using suitable transforms, signal and noise map to different parts of the transform domain and can be separated. The method is applied to synthetic and marine data. I. Introduction In exploration seismology structural information of the subsurface is obtained by recording the wavefield generated by a source (e.g. dynamite), using many receivers. In 2D seismics, these receivers are positioned along a line at the surface with sampling interval \Deltax r and starting at a certain 'offset' x 0 from the...
Spectral Warping Revisited
"... Spectral warping is a time domain to time domain transformation on a signal that effectively warps the frequency content of the original signal. Here we present a matrix formulation of the spectral warping transformation. The transform matrix is decomposed into three steps. The first is a DFT to con ..."
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Spectral warping is a time domain to time domain transformation on a signal that effectively warps the frequency content of the original signal. Here we present a matrix formulation of the spectral warping transformation. The transform matrix is decomposed into three steps. The first is a DFT to convert the time signal into the frequency domain. Step two is an interpolation matrix to calculate the signal content at the desired new frequency samples. This effectively provides the frequency warping. The final step is an inverse DFT to transform the signal back into the time domain. A direct consequence of this matrix representation is a direct FIR implementation of spectral warping, rather than the more commonly used IIR technique. We demonstrate that spectral warping is a generalisation of linear filtering, and show how the conventional allpass spectral warping transformation can be generalised by using either arbitrary frequency mapping functions or different interpolation schemes. Finally, the conditions for the invertibility of the spectral warping transformation are derived. 1
Connexions module: m16338 1 Algorithms for Data with Restrictions ∗
"... Many applications involve processing real data. It is ine cient to simply use a complex FFT on real data because arithmetic would be performed on the zero imaginary parts of the input, and, because of symmetries, output values would be calculated that are redundant. There are several approaches to d ..."
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Many applications involve processing real data. It is ine cient to simply use a complex FFT on real data because arithmetic would be performed on the zero imaginary parts of the input, and, because of symmetries, output values would be calculated that are redundant. There are several approaches to developing special algorithms or to modifying complex algorithms for real data. There are two methods which use a complex FFT in a special way to increase e ciency [4], [14]. The rst method uses a lengthN complex FFT to compute two lengthN real FFTs by putting the two real data sequences into the real and the imaginary parts of the input to a complex FFT. Because transforms of real data have even real parts and odd imaginary parts, it is possible to separate the transforms of the two inputs with 2N4 extra additions. This method requires, however, that two inputs be available at the same time. The second method [14] uses the fact that the last stage of a decimationintime radix2 FFT combines two independent transforms of length N/2 to compute a lengthN transform. If the data are real, the two half length transforms are calculated by the method described above and the last stage is carried out to calculate the total lengthN FFT of the real data. It should be noted that the halflength FFT does not have to be calculated by a radix2 FFT. In fact, it should be calculated by the most e cient complexdata algorithm possible, such as the SRFFT or the PFA. The separation of the two halflength transforms and
Improving Gaussian Processes Classification by Spectral Data Reorganizing
"... We improve Gaussian processes (GP) classification by reorganizing the (nonstationary and anisotropic) data to better fit to the isotropic GP kernel. First, the data is partitioned into two parts: along the feature with the highest frequency bandwidth. Secondly, for each part of the data, only the s ..."
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We improve Gaussian processes (GP) classification by reorganizing the (nonstationary and anisotropic) data to better fit to the isotropic GP kernel. First, the data is partitioned into two parts: along the feature with the highest frequency bandwidth. Secondly, for each part of the data, only the spectrally homogeneous features are chosen and used (the rest discarded) for GP classification. In this way, anisotropy of the data is lessened from the frequency point of view. Tests on synthetic data as well as real datasets show that our approach is effective and outperforms Automatic Relevance Determination (ARD). 1.