2011/09/14 by Philipp Walk, Peter Jung · 7 citations
Computer Science · Engineering · Mathematics · #Algorithm #Circulant matrix #Computer science #Digital Filter Design and Implementation #Mathematical Analysis and Transform Methods #Mathematical analysis #Mathematics #Nyquist–Shannon sampling theorem #Orthogonalization #Orthonormal basis #Physics #Ultra-Wideband Communications Technology #Wavelet #cs.IT #math.IT
paper · pdf · doi:10.1016/j.sigpro.2011.09.006
published in Signal Processing 92(3), 649-666 (Elsevier BV) · 33 pages, 11 figures. Accepted for publication 9 Sep 2011
arxiv created 2011/09/14 · arxiv updated 2011/09/15 · openalex publication_date 2011/09/16 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
In this paper we consider the design of spectrally efficient time-limited pulses for ultrawideband (UWB) systems using an overlapping pulse position modulation scheme. For this we investigate an orthogonalization method, which was developed in 1950 by Per-Olov Löwdin. Our objective is to obtain a set of N orthogonal (Löwdin) pulses, which remain time-limited and spectrally efficient for UWB systems, from a set of N equidistant translates of a time-limited optimal spectral designed UWB pulse. We derive an approximate Löwdin orthogonalization (ALO) by using circulant approximations for the Gram matrix to obtain a practical filter implementation. We show that the centered ALO and Löwdin pulses converge pointwise to the same Nyquist pulse as N tends to infinity. The set of translates of the Nyquist pulse forms an orthonormal basis or the shift-invariant space generated by the initial spectral optimal pulse. The ALO transform provides a closed-form approximation of the Löwdin transform, which can be implemented in an analog fashion without the need of analog to digital conversions. Furthermore, we investigate the interplay between the optimization and the orthogonalization procedure by using methods from the theory of shift-invariant spaces. Finally we develop a connection between our results and wavelet and frame theory.