Approximating band- and energy-limited signals in the presence of jitter. (English) Zbl 0917.65119
This paper deals with the recovering of band- and energy-limited functions \(f\) from noisy samples taken at slightly wrong sampling points \(t_k\) \((k= 1,\dots, n)\). With other words, the information about \(f\) consists of finitely many values \(f(t_k+ \gamma_k)+ \delta_k\) \((k= 1,\dots, n)\), where \(|\gamma_k|\leq\gamma\) and \(\|(\delta_k)^n_{k= 1}\|\leq\delta\). This inaccuracy in reading the sampling points is called jitter.
The aim of this paper is to analyze how the minimal worst-case recovery error depends on \(\gamma\) and \(\delta\). This is accomplished by proving tight upper and lower bounds for the diameter of information. The main result implies that the jitter causes an error of order \(\gamma\Omega^{3/2}+ \delta\), where \(\Omega\) denotes the bandwidth of \(f\).
The aim of this paper is to analyze how the minimal worst-case recovery error depends on \(\gamma\) and \(\delta\). This is accomplished by proving tight upper and lower bounds for the diameter of information. The main result implies that the jitter causes an error of order \(\gamma\Omega^{3/2}+ \delta\), where \(\Omega\) denotes the bandwidth of \(f\).
Reviewer: M.Tasche (Rostock)
MSC:
65T40 | Numerical methods for trigonometric approximation and interpolation |
94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |
42A10 | Trigonometric approximation |
Keywords:
band-limited function; recovering; jitter; diameter of information; approximation of band-limited signal; energy-limited function; minimal worst-case recovery errorReferences:
[1] | Kacewicz, B. Z.; Kowalski, M. A., Approximating linear functionals on unitary spaces in the presence of bounded data errors with applications to signal recovery, J. Adaptive Control Signal Process., 9, 19-31 (1995) · Zbl 0836.94004 |
[2] | Kacewicz, B. Z.; Kowalski, M. A., Recovering linear operators from inaccurate data, J. Complexity, 11, 227-239 (1995) · Zbl 0827.65058 |
[3] | Kacewicz, B. Z.; Kowalski, M. A., Recovering signals from inaccurate data, Curves and Surfaces in Computer Vision and Graphics II. Curves and Surfaces in Computer Vision and Graphics II, Proc. SPIE, 1610 (1992), Int. Soc. Opt. Eng: Int. Soc. Opt. Eng Bellingham, p. 68-74 |
[4] | Kowalski, M. A., Optimal complexity recovery of band- and energy-limited signals, J. Complexity, 2, 239-254 (1989) · Zbl 0626.94005 |
[5] | Kowalski, M. A., On approximation of band-limited signals, J. Complexity, 5, 283-302 (1989) · Zbl 0705.94001 |
[6] | Kowalski, M. A.; Stenger, F., Optimal complexity recovery of band- and energy-limited signals II, J. Complexity, 5, 45-59 (1989) · Zbl 0672.94001 |
[7] | Micchelli, C. A.; Rivlin, T. J., A survey of optimal recovery, (Micchelli, C. A.; Rivlin, T. J., Optimal Estimation in Approximation Theory (1977), Plenum: Plenum New York) · Zbl 0386.93045 |
[8] | Oppenheim, A. V., Applications of Digital Signal Processin (1978), Prentice-Hall: Prentice-Hall Englewood Cliffs |
[9] | Plaskota, L., Noisy Information and Computational Complexity (1996), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0925.68158 |
[10] | Rudin, W., Real and Complex Analysis (1974), McGraw-Hill: McGraw-Hill New York · Zbl 0278.26001 |
[11] | Stenger, F., Numerical Methods Based on Sinc and Analytic Functions (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0803.65141 |
[12] | Traub, J. F.; Wasilkowski, G. W.; Woźniakowski, H., Information Based Complexity (1988), Academic Press: Academic Press New York · Zbl 0674.68039 |
[13] | Traub, J. F.; Woźniakowski, H., A General Theory of Optimal Algorithms (1980), Academic Press: Academic Press New York · Zbl 0441.68046 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.