{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:14:38Z","timestamp":1760148878151,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,6,13]],"date-time":"2023-06-13T00:00:00Z","timestamp":1686614400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["RGPIN-2023-03392"],"award-info":[{"award-number":["RGPIN-2023-03392"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>Time and frequency concentrations of waveforms are often of interest in engineering applications. The Slepian basis of order zero is an index-limited (finite) vector that is known to be optimally concentrated in the frequency domain. This paper proposes a method of mapping the index-limited Slepian basis to a discrete-time vector, hence obtaining a time-limited, discrete-time Slepian basis that is optimally concentrated in frequency. The main result of this note is to demonstrate that the (discrete-time) Slepian basis achieves minimum time-bandwidth compactness under certain conditions. We distinguish between the characteristic (effective) time\/bandwidth of the Slepians and their defining time\/bandwidth (the time and bandwidth parameters used to generate the Slepian basis). Using two different definitions of effective time and bandwidth of a signal, we show that when the defining time-bandwidth product of the Slepian basis increases, its effective time-bandwidth product tends to a minimum value. This implies that not only are the zeroth order Slepian bases known to be optimally time-limited and band-concentrated basis vectors, but also as their defining time-bandwidth products increase, their effective time-bandwidth properties approach the known minimum compactness allowed by the uncertainty principle. Conclusions are also drawn about the smallest defining time-bandwidth parameters to reach the minimum possible compactness. These conclusions give guidance for applications where the time-bandwidth product is free to be selected and hence may be selected to achieve minimum compactness.<\/jats:p>","DOI":"10.3390\/computation11060116","type":"journal-article","created":{"date-parts":[[2023,6,14]],"date-time":"2023-06-14T01:32:15Z","timestamp":1686706335000},"page":"116","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On the Time Frequency Compactness of the Slepian Basis of Order Zero for Engineering Applications"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8218-5114","authenticated-orcid":false,"given":"Zuwen","family":"Sun","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, University of Ottawa, Ottawa, ON K1N 6N5, Canada"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7025-7501","authenticated-orcid":false,"given":"Natalie","family":"Baddour","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, University of Ottawa, Ottawa, ON K1N 6N5, Canada"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1274","DOI":"10.1109\/TAES.2020.3046086","article-title":"Adaptive Transmit Waveform Design Using Multitone Sinusoidal Frequency Modulation","volume":"57","author":"Hague","year":"2021","journal-title":"IEEE Trans. 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