Approximation of L\"owdin Orthogonalization to a Spectrally Efficient Orthogonal Overlapping PPM Design for UWB Impulse Radio

TL;DR

This paper proposes an approximate L"owdin orthogonalization method for designing spectrally efficient, time-limited orthogonal pulses in UWB systems, enabling practical implementation and convergence to Nyquist pulses as the number of pulses increases.

Contribution

It introduces a circulant approximation-based ALO method for L"owdin orthogonalization, providing a practical, analog-friendly approach for UWB pulse design with theoretical convergence guarantees.

Findings

ALO pulses converge to Nyquist pulses as N increases

The method enables practical, analog implementation of orthogonal UWB pulses

Connections established between orthogonalization, wavelet, and frame theories

Abstract

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 L\"owdin. Our objective is to obtain a set of N orthogonal (L\"owdin) 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 L\"owdin orthogonalization (ALO) by using circulant approximations for the Gram matrix to obtain a practical filter implementation. We show that the centered ALO and L\"owdin 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 L\"owdin 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.