Journal article
Sampling Schemes for Fourier Transform Reconstruction
SIAM journal on matrix analysis and applications, v 2(2), pp 176-191
01 Jun 1981
Abstract
We study a problem that arises in radio astronomy and other fields. Astronomers measure a function in order to recover its Fourier transform. Restrictions on the locations of telescopes limit the number of measurements made. It is generally assumed that if the measurements obtained are in some sense "equispaced" then recovery of the transform will be satisfactory. We develop more objective criteria for evaluating the appropriateness of different sampling schemes. Applying these criteria to a simplified model we show that equispaced samples are not always optimal. The problem can be abstracted as follows. Let $f$ be a function and $\hat f$ its $N$-point finite Fourier transform. If we measure $f$ at $N$ appropriate points we can recover $\hat f$ using standard techniques. Suppose instead that $f$ cannot be measured at $N$ values. Assume one has some knowledge of $\hat f$, e.g., that $f$ is band-limited so $\hat f = 0$ outside a band. Given $L$ values of $\hat f$ we need only $p = N - L$ values of $f$ to recover $\hat f$ in full. Now we ask: Which $p$ values of $f$ is it "best" to sample? Our criterion for a sampling scheme to be good is that the computation of $\hat f$ be fairly insensitive to any sampling errors in $f$. We show that this is equivalent to studying the effect of perturbations on a matrix whose entries depend upon the values where $f$ is sampled and the values where $\hat f$ is unknown. Different quantities associated with this matrix are studied to determine the effectiveness of the sampling scheme.
Metrics
26 Record Views
Details
- Title
- Sampling Schemes for Fourier Transform Reconstruction
- Creators
- Marci Perlstadt
- Publication Details
- SIAM journal on matrix analysis and applications, v 2(2), pp 176-191
- Publisher
- Society for Industrial and Applied Mathematics
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- [Retired Faculty]
- Web of Science ID
- WOS:A1981LT35800012
- Other Identifier
- 991021880193304721
InCites Highlights
Data related to this publication, from InCites Benchmarking & Analytics tool:
- Web of Science research areas
- Mathematics
- Mathematics, Applied