DFT interpolation kernels and error bounds

IEEE Transactions on Instrumentation and Measurement, 2000

This paper describes the discrete Fourier transform (DFT) interpolation algorithm for arbitrary windows and its application and performance for optimal noncosine Kaiser-Bessel and Dolph-Chebyshev windows. The interpolation algorithm is based on the polynomial approximation of the window's spectrum that is computed numerically. Two-and three-point (2p and 3p) interpolations are considered. Systematic errors and noise sensitivity are analyzed for the chosen Kaiser-Bessel and Dolph-Chebyshev windows and compared with Rife-Vincent class I windows. Index Terms-Discrete Fourier transform (DFT), frequencydomain measurements, frequency estimation, interpolated DFT, signal processing, windowing. I. INTRODUCTION T HE ESTIMATION of frequency, amplitude, and phase of single-frequency and multifrequency signal has applications in many fields of engineering. Estimation methods are based on Fourier analysis or parametric modeling [1], [2]. The advantage of Fourier-based methods is their computational efficiency. The limitations of discrete Fourier transform (DFT) application in measurements are due to spectral leakage and the picket-fence effect. Spectral leakage is typically reduced by selection of the proper window [3]. The picket-fence effect errors are compressed by interpolated DFT. The DFT interpolation formula for the rectangular window was introduced in [4]. It was then extended for a Hanning window in [5] and for Rife-Vincent class I windows in [6]. It was shown in [6] that the noise sensitivity for Rife-Vincent class I windows increases with the window order. It was shown in [7] that the interpolation formulas are practically independent of the signal length N for N ≥ 32. The reduction of the number of signal samples was significant, as compared with 2048 in previous works [4]-[6]. It was also noticed in [7] that the DFT bins of the window may be used in interpolation formulas instead of the DFT bins of the signal. On that basis, the polynomial approximation was proposed for cosine windows. A comparative study in [8] that included a rectangular window, a Hanning window, and a second-order Rife-Vincent class I window showed that systematic errors can be neglected in Manuscript

cs.tut.fi

IEEE Transactions on Signal Processing, 2000

Recently, the convolution of the sinc kernel with the infinite sequence of a periodic function was expressed as a finite summation. The expression obtained, however, is not numerically stable when evaluated at or near integer values of time. This correspondence presents a numerically stable formulation equivalent to the results reported in the above cited paper and which, when sampled, is also shown to be equivalent to the inverse discrete Fourier transform (IDFT). To appear in the IEEE Transactions on Signal Processing 2 , , , K for real x(n), we obtain eqn. (A1). We had previously shown how to get eqns. (2a) and (2b) from eqn. (A1).

2009

This article comments on a frequency estimator which was proposed by [6] and shows empirically that it exhibits a much larger mean squared error than a well known frequency estimator by [8]. It is demonstrated that by using a heuristical adjustment [2] the performance can be greatly improved. Furthermore, references to two modern techniques are given, which both nearly attain the Cramér-Rao bound for this estimation problem.

Electronics and Communications in Japan (Part III: Fundamental Electronic Science), 1990

2009

This article comments on a frequency estimator which was proposed by

Fourier Transform - Signal Processing, 2012

Advances in Computational Mathematics, 2010

It is often observed that interpolation based on translates of radial basis functions or non-radial kernels is numerically unstable due to exceedingly large condition of the kernel matrix. But if stability is assessed in function space without considering special bases, this paper proves that kernel-based interpolation is stable. Provided that the data are not too wildly scattered, the L 2 or L ∞ norms of interpolants can be bounded above by discrete ℓ 2 and ℓ ∞ norms of the data. Furthermore, Lagrange basis functions are uniformly bounded and Lebesgue constants grow at most like the square root of the number of data points. However, this analysis applies only to kernels of limited smoothness. Numerical examples support our bounds, but also show that the case of infinitely smooth kernels must lead to worse bounds in future work, while the observed Lebesgue constants for kernels with limited smoothness even seem to be independent of the sample size and the fill distance.

SEP-94: Stanford Exploration Project, 1997

2008

In [2] we proved the stability, in L2 and L∞ norms, of kernel–based interpolation, essentially based on standard error estimates for radial basis functions. In this note we give a different proof based on a sampling inequality (cf. [14, Th. 2.6]). A few numerical examples, supporting the results, will be presented too.