On numerical realizations of Shannon's sampling theorem
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912330015571968 |
|---|---|
| author | Kircheis, Melanie Potts, Daniel Tasche, Manfred |
| author_facet | Kircheis, Melanie Potts, Daniel Tasche, Manfred |
| contents | In this paper, we discuss some numerical realizations of Shannon's sampling theorem. First we show the poor convergence of classical Shannon sampling sums by presenting sharp upper and lower bounds of the norm of the Shannon sampling operator. In addition, it is known that in the presence of noise in the samples of a bandlimited function, the convergence of Shannon sampling series may even break down completely. To overcome these drawbacks, one can use oversampling and regularization with a convenient window function. Such a window function can be chosen either in frequency domain or in time domain. We especially put emphasis on the comparison of these two approaches in terms of error decay rates. It turns out that the best numerical results are obtained by oversampling and regularization in time domain using a sinh-type window function or a continuous Kaiser-Bessel window function, which results in an interpolating approximation with localized sampling. Several numerical experiments illustrate the theoretical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_17594 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On numerical realizations of Shannon's sampling theorem Kircheis, Melanie Potts, Daniel Tasche, Manfred Numerical Analysis 94A20, 65T50 In this paper, we discuss some numerical realizations of Shannon's sampling theorem. First we show the poor convergence of classical Shannon sampling sums by presenting sharp upper and lower bounds of the norm of the Shannon sampling operator. In addition, it is known that in the presence of noise in the samples of a bandlimited function, the convergence of Shannon sampling series may even break down completely. To overcome these drawbacks, one can use oversampling and regularization with a convenient window function. Such a window function can be chosen either in frequency domain or in time domain. We especially put emphasis on the comparison of these two approaches in terms of error decay rates. It turns out that the best numerical results are obtained by oversampling and regularization in time domain using a sinh-type window function or a continuous Kaiser-Bessel window function, which results in an interpolating approximation with localized sampling. Several numerical experiments illustrate the theoretical results. |
| title | On numerical realizations of Shannon's sampling theorem |
| topic | Numerical Analysis 94A20, 65T50 |
| url | https://arxiv.org/abs/2306.17594 |