Frequency analysis and norms of distributed spatially periodic systems

Makan Fardad, Mihailo R. Jovanovic, Bassam Bamieh

Research output: Contribution to journalArticlepeer-review

20 Scopus citations


We investigate several fundamental aspects of the theory of linear distributed systems with spatially periodic coefficients. We develop a spatial-frequency domain representation analogous to the lifted or frequency response operator representation for linear time periodic systems. Using this representation, we introduce the notion of the H2 norm for this class of systems and provide algorithms for its computation. A stochastic interpretation of the H2 norm is given in terms of spatially cyclostationary random fields and spectral-correlation density operators. When the periodic coefficients are viewed as feedback modifications of spatially invariant systems, we show how they can stabilize or destabilize the original systems in a manner analogous to vibrational control or parametric resonance in time periodic systems. Two examples from physics are provided to illustrate the main results.

Original languageEnglish (US)
Pages (from-to)2266-2279
Number of pages14
JournalIEEE Transactions on Automatic Control
Issue number10
StatePublished - 2008


  • Cyclostationary random fields
  • Frequency domain lifting
  • Frequency response opertators
  • H2 norm
  • Partial differential equation (PDE) with periodic coefficients
  • Spatially periodic systems

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering


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