On the Calculation of the Hilbert Transform From Interpolated Data

This correspondence studies the calculation of the Hilbert transform of continuous functions f with continuous conjugate f from a finite set of sampling points. It shows that there exists no linear operator which approximates f arbitrary well in the uniform norm from a finite number of sampling poin...

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Bibliographic Details
Published in:IEEE transactions on information theory 2008-05, Vol.54 (5), p.2358-2366
Main Authors: Boche, H., Pohl, V.
Format: Article
Language:English
Subjects:
Algorithms
Applied sciences
Approximation
Conjugates
Control theory
Error bounds
Errors
Exact sciences and technology
Hilbert transform
Information theory
Information, signal and communications theory
Interpolation
Linear approximation
Mathematical analysis
Mathematical functions
Norms
Physics
Robustness
Sampling
Sampling methods
Sampling, quantization
Signal and communications theory
Signal processing
Signal processing algorithms
Studies
Sufficient conditions
Telecommunications and information theory
Transforms
Wiener filter
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Summary:This correspondence studies the calculation of the Hilbert transform of continuous functions f with continuous conjugate f from a finite set of sampling points. It shows that there exists no linear operator which approximates f arbitrary well in the uniform norm from a finite number of sampling points for all possible continuous function f with continuous conjugate f. However for smooth functions such linear approximation operators exist and sufficient conditions on the smoothness of the functions are presented. The correspondence also examines the robustness of the calculation of the Hilbert transform from interpolated data and it gives explicit error bounds. It is shown that for a large class of algorithms the error grows at least proportional to the logarithm of the number of sampling points.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2008.920219