Further results on a series representation related to the sampling theorem

During the period 1968–1971 there were several papers dealing with a class of series representations for stochastic processes on an infinite time interval. In a special case, these series reduce to the sampling theorem for bandlimited functions. The present paper continues the study of these series. Series representations of functions are studied, but most of the results should extend to stochastic processes. Sufficient conditions for convergence of the series are given and a bound on the truncation error of the series is obtained. A Kullback-Leibler measure of the interdependence of the coefficients in the series is found. Some partial results are obtained on the degree to which the coefficients depend on measurements which are localized in time. Some additional special properties of the coefficients associated with bandlimited functions are also discussed.     

Effect of sampling,quantization and noise on the performance of the second directional derivative edge detector

Berzins [2] and De Vriendt [14] studied the processes that influence the performance of the Laplacian and the second directional derivative edge detector in the continuous domain. In this paper the influence of sampling, quantization and noise is studied in the discrete domain for the second directional derivative edge detector. The results are compared with those for the Laplacian edge detector. The smoothing and derivative operations are implemented by FIR digital filters. Two sampling processes are considered: a square aperture and a Gaussian smoothing process. The influence of sampling can be limited by increasing the spread of the smoothing filter. Though, should not be chosen too large because of the influence of nearby edges. The quantization of the intensity function introduces an uncertainty in the edge location. The uncertainty is larger than the error due to sampling if the step height is small. We also prove that the second directional derivative is less sensitive to noise than the Laplacian. An increase of slightly reduces the variation of the edge location.This work was supported by the Belgian National Fund for Scientific Research (NFWO).     

Kramer's sampling theorem for multidimensional signals and its relationship with Lagrange-type interpolations

Kramer's sampling theorem, which is a generalization of the Whittaker-Shannon-Kotel'nikov (WSK) sampling theorem, enables one to reconstruct functions that are integral transforms of types other than the Fourier one from their sampled values. In this paper, we generalize Kramer's theorem toN dimensions (N 1) and show how the kernel function and the sampling points in Kramer's theorem can be generated. We then investigate the relationship between this generalization of Kramer's theorem andN-dimensional versions of both the WSK theorem and the Paley-Wiener interpolation theorem for band-limited signals. It is shown that the sampling series associated with this generalization of Kramer's theorem is nothing more than anN-dimensional Lagrange-type interpolation series.This paper was presented at the International Congress of Mathematicians in Kyoto, Japan (1990) and was supported by a CARE Grant #5915 from California Polytechnic State University, San Luis Obispo.     

A generalization of correlation functions and the Wiener-Khinchin theorem

Correlation functions and energy spectra are closely related to translation invariance properties. This can be generalized for more complex transformations, such as translations and discrete rotations or even to rotations of a sphere for functions defined on its surface (applicable to the earth's magnetic field). The concepts of correlation function and energy spectrum are extended to functions on a set M under a compact transformation group G with respect to G-invariant integrals on M and G. Two definitions of correlation are possible, which coincide in special cases. They both have Wiener-Khinchin-like relations to suitably defined energy spectra, connected with the irreducible representations of G.     

加权分数傅里叶变换在采样重构中的应用

针对频域非带限信号的重构问题,提出了一种基于加权分数傅里叶变换的采样与重构算法,并利用信号加权分数傅里叶变换在加权分数域的特性,得到了信号可完全重构的条件.理论分析和仿真结果表明,在满足给定误差的条件下,该算法能够以低于香农重构所需的采样率实现对信号的恢复,且简单易行,可利用FFT快速算法实现.