Optimal signal reconstruction

This report will use, in order to measure the performance of a (mathematical) system, the L norm for systems. For a BIBO-stable (Bounded Input Bounded Output) and Linear Continuous Time Invariant (LCTI) system usually a transfer function is defined. Using this transfer function it is possible to calculate the L norm of the system. In the process of sampling and reconstruction of a signal two systems are used: a sampler and a hold. Most of the time these systems are not LCTI but only linear and h-shift invariant or equivalently Linear Discrete Time Invariant (LDTI). For this class of systems a way of calculating the L system norm is presented. This calculation is based on the Frequency Power Response (FPR) of a system which is introduced in this report as well. This FPR is for an LDTI system what the frequency response, e.g. |G(iω)|2 is for an LCTI system. It has already been shown that the optimal combination of sampler and hold for a given sampling period h is always LCTI. This means that the L norm of the system can be calculated in a classical way. This report shows how to calculate the L norm of the optimal combination of sampler and hold. Also a graphical interpretation is given for this optimal combination. Because of the FPR, the L norm can now be calculated not only for LCTI systems but for LDTI systems as well. And it is shown how to determine the optimal hold for a given sampler and sampling period h. Additionally the L norm of the system can be calculated and graphically represented: how good (or how bad) is a certain hold in combination with the given sampler. Preface This report is part of my graduation project for the Master of Science program in Systems & Control at the University of Twente. The set up for the project came from my supervisor Dr. Ir. G. Meinsma who has done a lot of research in this area. I have had a great time working with him and I would like to thank him for his good advice, the way he tutored me and all the work and effort he put in reading parts of my report and answering my countless questions. Furthermore I would like to thank all my colleagues with whom I attended lectures and spend numerous hours of studying and playing cards. Last but certainly not least I would like to thank my friends and family for supporting me all these years. Almar Snippe September 2011