Isometric and Inner Operators

Patrick Dewilde and Alle-Jan van der Veen
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Patrick Dewilde: Delft University of Technology, DIMES
Alle-Jan van der Veen: Delft University of Technology, DIMES

Chapter 6 in Time-Varying Systems and Computations, 1998, pp 121-143 from Springer

Abstract: Abstract Lossless systems play an important role in the class of linear systems. They are causal systems which “conserve energy”. If energy is measured as the square of a quadratic norm ‖•‖, a lossless system transforms an input signal u with bounded energy ‖u‖ to an output signal y = uT which contains the same total energy: ‖u‖ = ‖y‖ In filter theory, scalar lossless systems are also known as allpass filters, with a flat amplitude spectrum but a variable phase. They have many interesting properties. One is that any passive rational filter may be realized as the partial response of a lossless filter. Another property is that lossless systems may be implemented in a locally lossless way as well, by using a state space realization in which every section is itself lossless. Such realizations do not amplify noise introduced at any point in the system, and they can be made robust with respect to parameter deviations as well.

Keywords: Transfer Operator; Hankel Operator; Lyapunov Equation; Unitary Realization; Isometric Operator (search for similar items in EconPapers)
Date: 1998
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DOI: 10.1007/978-1-4757-2817-0_6

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