Hilbert space formulation of symbolic systems for signal representation and analysis

Yicheng Wen, Asok Ray, Shashi Phoha

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


Probabilistic finite state automata (PFSA) have been widely used as an analysis tool for signal representation and modeling of physical systems. This paper presents a new method to address these issues by bringing in the notion of vector-space formulation of symbolic systems in the setting of PFSA. In this context, a link is established between the formal language theory and functional analysis by defining an inner product space over a class of stochastic regular languages, represented by PFSA models that are constructed from finite-length symbol sequences. The norm induced by the inner product is interpreted as a measure of the information contained in the respective PFSA. Numerical examples are presented to illustrate the computational steps in the proposed method and to demonstrate model order reduction via orthogonal projection from a general Hilbert space of PFSA onto a (closed) Markov subspace that belongs to a class of shifts of finite type. These concepts are validated by analyzing time series of ultrasonic signals, collected from an experimental apparatus, for fatigue damage detection in polycrystalline alloys.

Original languageEnglish (US)
Pages (from-to)2594-2611
Number of pages18
JournalSignal Processing
Issue number9
StatePublished - Sep 2013

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Software
  • Signal Processing
  • Computer Vision and Pattern Recognition
  • Electrical and Electronic Engineering


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