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Wavelet Transforms and Time-Frequency Signal Analysis pp 235–287Cite as

Birkhäuser

Quadratic Time-Frequency Analysis of Linear Time-Varying Systems

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Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

Viewed from a time-frequency (TF) perspective, the action of a linear time-varying (LTV) system consists of TF weightings and TF displacements. In this chapter, we study TF weighting and displacement effects using known and novel quadratic TF representations of LTV systems. In particular, we define and study two novel TF representations termed “input Wigner distribution” (IWD) and “output Wigner distribution” (OWD). The interpretations, various expressions, and fundamental properties of the IWD and OWD are discussed. In the case of a normal system, the IWD and OWD coincide and reduce to a single TF representation, the “Wigner distribution (WD) of an LTV system.”

In addition to the IWD and OWD, we define displacement spread quantities that globally characterize TF displacement effects. It is shown that positive semidefinite systems feature minimum TF displacements. The IWD and OWD of an “under spread” system (i.e., a system whose TF displacements are small) are shown to be approximately equal to each other and also to the squared Weyl symbol, to be approximately nonnegative, and to satisfy an approximate composition property. The application of the IWD and OWD to random LTV systems is briefly considered. Finally, we describe a WD-based TF design method for (normal) LTV systems with prescribed TF weightings and minimum TF displacements.

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Authors
  1. Franz Hlawatsch
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  2. Gerald Matz
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Editors and Affiliations

  1. Department of Mathematics, University of Central Florida, Orlando, Fl, 32816, USA

    Lokenath Debnath

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Hlawatsch, F., Matz, G. (2001). Quadratic Time-Frequency Analysis of Linear Time-Varying Systems. In: Debnath, L. (eds) Wavelet Transforms and Time-Frequency Signal Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0137-3_9

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  • DOI: https://doi.org/10.1007/978-1-4612-0137-3_9

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6629-7

  • Online ISBN: 978-1-4612-0137-3

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