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17
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 121 (22 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.
Reconstruction of nonuniformly sampled bandlimited signals by means of digital fractional delay filters
 IEEE Trans. Signal Processing
, 2002
"... Abstract – This paper deals with reconstruction of nonuniformly sampled bandlimited continuoustime signals using timevarying discretetime FIR filters. The points of departures are that the signal is slightly oversampled as to the average sampling frequency and that the sampling instances are know ..."
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Cited by 51 (8 self)
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Abstract – This paper deals with reconstruction of nonuniformly sampled bandlimited continuoustime signals using timevarying discretetime FIR filters. The points of departures are that the signal is slightly oversampled as to the average sampling frequency and that the sampling instances are known. Under these assumptions, a representation of the reconstructed sequence is derived that utilizes a timefrequency function. This representation enables a proper utilization of the oversampling and reduces the reconstruction problem to a design problem that resembles an ordinary filter design problem. Furthermore, for an important special case, corresponding to a certain type of periodic nonuniform sampling, it is shown that the reconstruction problem can be posed as a filterbank design problem, thus with requirements on a distortion transfer function and a number of aliasing transfer functions. 1.
Reconstruction of nonuniformly sampled bandlimited signals using a differentiatormultiplier cascade
 IEEE Trans. Circuits Syst. I, Reg. Papers
, 2008
"... Abstract—This paper considers the problem of reconstructing a bandlimited signal from its nonuniform samples. Based on a discretetime equivalent model for nonuniform sampling, we propose the differentiator–multiplier cascade, a multistage reconstruction system that recovers the uniform samples from ..."
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Cited by 16 (10 self)
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Abstract—This paper considers the problem of reconstructing a bandlimited signal from its nonuniform samples. Based on a discretetime equivalent model for nonuniform sampling, we propose the differentiator–multiplier cascade, a multistage reconstruction system that recovers the uniform samples from the nonuniform samples. Rather than using optimally designed reconstruction filters, the system improves the reconstruction performance by cascading stages of linearphase finite impulse response (FIR) filters and timevarying multipliers. Because the FIR filters are designed as differentiators, the system works for the general nonuniform sampling case and is not limited to periodic nonuniform sampling. To evaluate the reconstruction performance for a sinusoidal input signal, we derive the signaltonoiseratio at the output of each stage for the twoperiodic and the general nonuniform sampling case. The main advantage of the system is that once the differentiators have been designed, they are implemented with fixed multipliers, and only some general multipliers have to be adapted when the sampling pattern changes; this reduces implementation costs substantially, especially in an application like timeinterleaved analogtodigital converters (TIADCs) where the timing mismatches among the ADCs may change during operation. Index Terms—Discretetime differentiator, Farrow structure, nonuniform sampling, Taylor series expansion, timeinterleaved analogtodigital converter (TIADC), timevarying multiplier. I.
DirectFourier Reconstruction In Tomography And Synthetic Aperture Radar
 Intl. J. Imaging Sys. and Tech
, 1998
"... We investigate the use of directFourier (DF) image reconstruction in computerized tomography and synthetic aperture radar (SAR). One of our aims is to determine why the convolutionbackprojection (CBP) method is favored over DF methods in tomography, while DF methods are virtually always used in SAR ..."
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Cited by 10 (0 self)
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We investigate the use of directFourier (DF) image reconstruction in computerized tomography and synthetic aperture radar (SAR). One of our aims is to determine why the convolutionbackprojection (CBP) method is favored over DF methods in tomography, while DF methods are virtually always used in SAR. We show that the CBP algorithm is equivalent to DF reconstruction using a Jacobianweighted 2D periodic sinckernel interpolator. This interpolation is not optimal in any sense, which suggests that DF algorithms utilizing optimal interpolators may surpass CBP in image quality. We consider use of two types of DF interpolation: a windowed sinc kernel, and the leastsquares optimal Yen interpolator. Simulations show that reconstructions using the Yen interpolator do not possess the expected visual quality, because of regularization needed to preserve numerical stability. Next, we show that with a concentricsquares sampling scheme, DF interpolation can be performed accurately and efficiently...
Interpolation and denoising of nonuniformly sampled data using wavelet domain processing
 in Proc. IEEE Int. Conf. on Acoust., Speech, Signal Proc.  ICASSP '99
, 1999
"... In this paper, we link concepts from nonuniform sampling, smoothness function spaces, interpolation, and denoising to derive a suite of multiscale, maximumsmoothness interpolation algorithms. We formulate the interpolation problem as the optimization of finding the signal that matches the given sam ..."
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Cited by 9 (3 self)
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In this paper, we link concepts from nonuniform sampling, smoothness function spaces, interpolation, and denoising to derive a suite of multiscale, maximumsmoothness interpolation algorithms. We formulate the interpolation problem as the optimization of finding the signal that matches the given samples with smallest norm in a function smoothness space. For signals in the Besov space B, " (Lp), the optimization corresponds to convex programming in the wavelet domain; for signals in the Sobolev space We(&), the optimization reduces to a simple weighted leastsquares problem. An optional wavelet shrinkage regularization step makes the algorithm suitable for even noisy sample data, unlike classical approaches such as bandlimited and spline interpolation. 1.
Nonuniform interpolation of noisy signals using support vector machines
 IEEE Transactions on Signal Processing
, 2007
"... Abstract—The problem of signal interpolation has been intensively studied in the Information Theory literature, in conditions such as unlimited band, nonuniform sampling, and presence of noise. During the last decade, support vector machines (SVM) have been widely used for approximation problems, i ..."
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Cited by 7 (2 self)
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Abstract—The problem of signal interpolation has been intensively studied in the Information Theory literature, in conditions such as unlimited band, nonuniform sampling, and presence of noise. During the last decade, support vector machines (SVM) have been widely used for approximation problems, including function and signal interpolation. However, the signal structure has not always been taken into account in SVM interpolation. We propose the statement of two novel SVM algorithms for signal interpolation, specifically, the primal and the dual signal model based algorithms. Shiftinvariant Mercer’s kernels are used as building blocks, according to the requirement of bandlimited signal. The sinc kernel, which has received little attention in the SVM literature, is used for bandlimited reconstruction. Wellknown properties of general SVM algorithms (sparseness of the solution, robustness, and regularization) are explored with simulation examples, yielding improved results with respect to standard algorithms, and revealing good characteristics in nonuniform interpolation of noisy signals. Index Terms—Dual signal model, interpolation, Mercer’s kernel, nonuniform sampling, primal signal model, signal, sinc
Reconstruction of periodically nonuniformly sampled bandlimited signals using timevarying FIR filters
 In Proc. Fourth Int. Workshop Spectral Methods Multirate Signal Processing
, 2004
"... This paper deals with reconstruction of nonuniformly sampled bandlimited continuoustime signals using timevarying discretetime FIR filters. The point of departure is a representation of the reconstructed sequence that utilizes a timefrequency function. This representation enables a proper utiliz ..."
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Cited by 3 (1 self)
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This paper deals with reconstruction of nonuniformly sampled bandlimited continuoustime signals using timevarying discretetime FIR filters. The point of departure is a representation of the reconstructed sequence that utilizes a timefrequency function. This representation enables a proper utilization of the slight oversampling that is assumed and reduces the reconstruction problem to a design problem that resembles an ordinary filter design problem. Of particular interest in this paper is an important special case of nonuniform sampling that corresponds to a certain type of periodic nonuniform sampling. It is shown how the reconstruction problem in this case can be posed as a multirate filterbank design problem, thus with requirements on a distortion transfer function and a number of aliasing transfer functions. (a) (c) x a(t)
A NEW INTERPOLATION TECHNIQUE FOR THE RECONSTRUCTION OF UNIFORMLY SPACED SAMPLES FROM NONUNIFORMLY SPACED ONES IN PLANERECTANGULAR NEARFIELD ANTENNA MEASUREMENTS
"... Abstract—A novel fast and accurate interpolation technique for recovering the uniformly distributed samples from the irregularly spaced samples, collected nonuniformly due to the probe position error in planar nearfield antenna measurements, is presented. The technique employs Yen’s interpolator a ..."
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Cited by 2 (0 self)
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Abstract—A novel fast and accurate interpolation technique for recovering the uniformly distributed samples from the irregularly spaced samples, collected nonuniformly due to the probe position error in planar nearfield antenna measurements, is presented. The technique employs Yen’s interpolator and tries to make it as practical as possible for the use in nearfield antenna measurements. A comprehensive simulation capability is developed and through out the simulations the speed and precision of this accurate and timely efficient interpolation technique is compared with some other techniques which are also based on Yen’s interpolators. The results well demonstrate the advantages of our technique we termed “The CrossRail Technique”. 48 Dehghanian, Okhovvat, and Hakkak 1.
Signal Processing Issues In Synthetic Aperture Radar And Computer Tomography
, 1998
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THE FAST SINC TRANSFORM AND IMAGE RECONSTRUCTION FROM NONUNIFORM SAMPLES IN kSPACE
"... A number of problems in image reconstruction and image processing can be addressed, in principle, using the sinc kernel. Since the sinc kernel decays slowly, however, it is generally avoided in favor of some more local but less precise choice. In this paper, we describe the fast sinc transform, an a ..."
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A number of problems in image reconstruction and image processing can be addressed, in principle, using the sinc kernel. Since the sinc kernel decays slowly, however, it is generally avoided in favor of some more local but less precise choice. In this paper, we describe the fast sinc transform, an algorithm which computes the convolution of arbitrarily spaced data with the sinc kernel in O.N log N / operations, where N denotes the number of data points. We briefly discuss its application to the construction of optimal density compensation weights for Fourier reconstruction and to the iterative approximation of the pseudoinverse of the signal equation in MRI. 1.