The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformly-spaced 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 min-max sense of minimizing the worst-case approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the min-max approach provides substantially lower approximation errors than conventional interpolation methods. The min-max criterion is also useful for optimizing the parameters of interpolation kernels such as the Kaiser-Bessel function.

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).

Examining irregularly sampled data sets usually requires gridding that data set. How-ever, 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 remain-ing 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 re-gridding the data. The technique is presented in N-dimensions, but is particu-larly well suited to Fourier Volume Rendering, which is the fastest known method of direct volume rendering. Together, these techniques allow accurate and efficient, multi-resolution exploration of volume data.

The International Telecommunication Union (ITU) recommendations for dual-tone multi-frequency (DTMF) signaling are not met by conventional DTMF detectors. We present an efficient DTMF detection algorithm based on the non-uniform 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), 120 + 30 n words of data memory, and 1000 words of program memory. Using the new algorithm, a single fixed-point DSP can perform ITU-compliant DTMF detection on the 24 telephone channels of a T1 time-division multiplexed telecommunications line. The authors are with the Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 78712-1084 (e-mail: felder@ece.utexas.edu, ma...

We introduce the design of polar-separable 2-D FIR filters by radial slice approximations (RSA). It is a two step procedure. First, 1-D 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 2-D FIR filters for signal and image processing is an important and difficult problem. Several image processing techniques [1, 2] require FIR filters that are polar-separable 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 1-D filters for the radial and angular components are de...

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...

The goal of the schemes we present in this paper is to obtain an ultra low delay audio coder with a good performance even at low bit rates (around 64 kb/s). The problem to be solved is to gain sufficient frequency resolution at low frequencies for precise low frequency psycho-acoustics and quantization noise shaping, because the ear has a higher frequency resolution at lower frequencies. Our approach is to use a warped linear noise shaping pre- and postfilter, and a short DFT for the psycho-acoustic model (length 256), but with frequency warping. We compare four different psychoacoustic versions: DFT with no warping, DFT without warping using warped pre- and post-filters, warping with the so-called NDFT (WDFT), and a DFT with an all-pass delay chain pre-processing. Listening tests show that the best performance is obtained using the WDFT. 1.

Performance of dual tone multi-frequency signal decoding algorithm using the sub-band non-uniform discrete Fourier transform on the ADSP-2192 processor

This document summarizes the author's research on adaptive image and signal representations using warped polynomial approximations and their applications in (mainly) coding and filtering. Some of the research results described in this document are explained in much greater detail in the author's Ph.D. thesis. However, this document additionally presents many recent new results concerning, e.g., adaptive filtering, and image and contour coding. It also discusses a few possible extensions and new applications of the theory. Acknowledgments

this paper the aliasing is studied by comparing the PRT with the Nonuniform Discrete Fourier Transform (NDFT). Although we find the same main principals, there are some differences. Moreover, the new points of view can provide a better understanding of the results found. The similarity between the PRT and the NDFT can also be used for the utilization of fast algorithms.