On bandlimited signals with minimal product of effective spatial and spectral widths
It is known that signals (which could be functions of space or time) belonging to 𝕃2-space cannot be localized simultaneously in space/time and frequency domains. Alternatively, signals have a positive lower bound on the product of their effective spatial andeffective spectral widths, f...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2005-01-01
|
| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/IJMMS.2005.1589 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850232185565478912 |
|---|---|
| author | Y. V. Venkatesh S. Kumar Raja G. Vidya Sagar |
| author_facet | Y. V. Venkatesh S. Kumar Raja G. Vidya Sagar |
| author_sort | Y. V. Venkatesh |
| collection | DOAJ |
| description | It is known that signals (which could be functions of space or time) belonging to 𝕃2-space cannot be localized simultaneously in space/time and frequency domains. Alternatively, signals have a positive lower bound on the product of their effective spatial andeffective spectral widths, for simplicity, hereafter called the effective space-bandwidthproduct (ESBP). This is the classical uncertainty inequality (UI), attributed to many, but, from a signal processing perspective, to Gabor who, in his seminal paper, established the uncertainty relation and proposed a joint time-frequency representation in which the basis functions have minimal ESBP. It is found that the Gaussian function is the only signal that has the lowest ESBP. Since the Gaussian function is not bandlimited, no bandlimited signal can have the lowest ESBP. We deal with the problem of determining finite-energy, bandlimited signals which have the lowest ESBP. The main result is as follows. By choosing the convolution product of a Gaussian signal (with σ as the variance parameter) and a bandlimited filter with a continuous spectrum, we demonstrate that there exists a finite-energy, bandlimited signal whose ESBP can be made to be arbitrarily close (dependent on the choice of σ) to the optimal value specified by the UI. |
| format | Article |
| id | doaj-art-0995c478b4134904bbe95f878a380aed |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2005-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-0995c478b4134904bbe95f878a380aed2025-08-20T02:03:16ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252005-01-012005101589159910.1155/IJMMS.2005.1589On bandlimited signals with minimal product of effective spatial and spectral widthsY. V. Venkatesh0S. Kumar Raja1G. Vidya Sagar2Department of Electrical Engineering, Indian Institute of Science, Bangalore, Karnataka 560012, IndiaDepartment of Electrical Engineering, Indian Institute of Science, Bangalore, Karnataka 560012, IndiaDepartment of Electrical Engineering, Indian Institute of Science, Bangalore, Karnataka 560012, IndiaIt is known that signals (which could be functions of space or time) belonging to 𝕃2-space cannot be localized simultaneously in space/time and frequency domains. Alternatively, signals have a positive lower bound on the product of their effective spatial andeffective spectral widths, for simplicity, hereafter called the effective space-bandwidthproduct (ESBP). This is the classical uncertainty inequality (UI), attributed to many, but, from a signal processing perspective, to Gabor who, in his seminal paper, established the uncertainty relation and proposed a joint time-frequency representation in which the basis functions have minimal ESBP. It is found that the Gaussian function is the only signal that has the lowest ESBP. Since the Gaussian function is not bandlimited, no bandlimited signal can have the lowest ESBP. We deal with the problem of determining finite-energy, bandlimited signals which have the lowest ESBP. The main result is as follows. By choosing the convolution product of a Gaussian signal (with σ as the variance parameter) and a bandlimited filter with a continuous spectrum, we demonstrate that there exists a finite-energy, bandlimited signal whose ESBP can be made to be arbitrarily close (dependent on the choice of σ) to the optimal value specified by the UI.http://dx.doi.org/10.1155/IJMMS.2005.1589 |
| spellingShingle | Y. V. Venkatesh S. Kumar Raja G. Vidya Sagar On bandlimited signals with minimal product of effective spatial and spectral widths International Journal of Mathematics and Mathematical Sciences |
| title | On bandlimited signals with minimal product of effective spatial and spectral widths |
| title_full | On bandlimited signals with minimal product of effective spatial and spectral widths |
| title_fullStr | On bandlimited signals with minimal product of effective spatial and spectral widths |
| title_full_unstemmed | On bandlimited signals with minimal product of effective spatial and spectral widths |
| title_short | On bandlimited signals with minimal product of effective spatial and spectral widths |
| title_sort | on bandlimited signals with minimal product of effective spatial and spectral widths |
| url | http://dx.doi.org/10.1155/IJMMS.2005.1589 |
| work_keys_str_mv | AT yvvenkatesh onbandlimitedsignalswithminimalproductofeffectivespatialandspectralwidths AT skumarraja onbandlimitedsignalswithminimalproductofeffectivespatialandspectralwidths AT gvidyasagar onbandlimitedsignalswithminimalproductofeffectivespatialandspectralwidths |