Advantages of PSWF-based models for UWB systems Wouter Dullaert 1 Hendrik Rogier 2 Freek Boeykens 3 Abstract—The frequency dependency of signals plays a very important role in Ultra Wideband (UWB) systems. Being able to accurately model this frequency dependency can provide a multitude of benefits, especially for UWB systems in medical applications. The model, based on prolate spheroidal wave functions (PSWFs), presented in this contribution improves greatly upon the model relying on discrete prolate spheroidal sequences that was previously presented by the authors: it offers a much more compact representation, while retaining the same robustness to noise. Furthermore, with the improved model, a cascade of signals can be evaluated directly using the model coefficients. All of these properties can be of use when analysing UWB systems. 1 I NTRODUCTION Ultra Wideband (UWB) communication systems play a very important role in the current movement to ubiquitous computing. Its low power requirements and high bandwidth makes it ideal for close range communications where battery life is of the utmost importance. Especially in medical applications it could enable the permanent monitoring of patients without the need for excessive wiring. However, the advantages of UWB systems, being high bandwidth and low power consumption, provide various new challenges for the system design. An- tenna design is tradiationally handled as a smallband problem: apart from the return loss, all the frequency dependency of all antenna parameters is ignored. In the case of UWB systems, this assumption is no longer valid, the frequency dependency of all parameters must be taken into account, which greatly increases the amount of data that needs to be handled during the design phase. Furthermore, the low spectral density of UWB systems means that great care needs to be taken to prevent noise from disturbing the data. Any form of representing UWB data should be able to handle, and preferably reduce, the noise contribution. In this paper we would like to propose a model for the frequency depedency of UWB systems, based on prolate spheroidal wave functions (PSWfs). It is an improvement of the model presented by the authors in [1], which modeled frequency dependency using 1 Department of Information Technology, Ghent University, Sint-Pietersnieuwstraat 41 B-9000 Ghent, email: wouter.dullaert@intec.ugent.be 2 Department of Information Technology, Ghent University, Sint-Pietersnieuwstraat 41 B-9000 Ghent, email: hendrik.rogier@intec.ugent.be 3 Department of Information Technology, Ghent University, Sint-Pietersnieuwstraat 41 B-9000 Ghent, email: freek.boeykens@intec.ugent.be discrete prolate spheroidal sequences (DPSSs), the discrete counterpart of PSWFs. The model has a lot of properties which help to overcome the difficulties of UWB system design and analysis. In Section 2 we will briefly present the theory behind the model. In the following section, Section 3, the various properties and advantages of the model are illustrated. Finally, in Section 4, we present our conclusions. 2 PSWF- BASED MODEL 2.1 Prolate Spheroidal Wave Functions The basic asumption of the model is that all UWB data have a very short time response s(t), i.e they are timelimited, but at the same time have a frequency spectrum S(f ) that is concentrated in a given region [−B/2,B/2]. This means that the data can be effi- ciently respresented as a series of prolate spheroidal wave functions (PSWFs), as in [2], S(f )= ∞  k=0 A k ψ k,c (f ) (1) where the coefficients A k are obtained as follows: A k = B/2  -B/2 S(f )ψ k,c (f )df (2) where ψ k,c (f ) is the PSWF of order k with a time- bandwidth product c =2t 0 B. The PSWFs were found to be the solution to the following maximisation problem over all timelimited functions S(f ) by Slepian et al. in [3]: max B/2  -B/2 |S(f )| 2 df ∞  -∞ |S(f )| 2 df (3) In other words, the zero-th order PSWF ψ 0,c is the timelimited funtion with the most energy in the fre- quency interval [−B/2,B/2]. When (3) is evaluated 978-1-4673-0405-4/12/$31.00 ©2012 IEEE 1024