Commutation in linear and nonlinear systems

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    4 Citations (Scopus)

    Abstract

    In this paper we consider the commutability of linear and nonlinear blocks. We show that linearity is not necessary nor sufficient requirement for the commutability since matrix multiplication is not in general commutative although scalar multiplication is commutative. For linear time-invariant systems following the superposition principle we can use the product of two transfer functions, essentially a set of scalar multiplications, and therefore the blocks commute. On the other hand, some blocks are represented by a matrix, for example linear time-variant blocks and blocks describing I/Q imbalance. Such blocks do not in general commute unless the matrices have some special properties. Furthermore, nonlinear systems are not in general commutative. The commutability is not valid unless there is a special reason for that. Typical examples for commutability include (1) the systems are combined through a commutable operation, or (2) one of the systems is the inverse of the other system.
    Original languageEnglish
    Pages (from-to)92-94
    Number of pages3
    JournalFrequenz
    Volume60
    Issue number5-6
    DOIs
    Publication statusPublished - 2006
    MoE publication typeA1 Journal article-refereed

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    Electric commutation
    Linear systems
    Nonlinear systems
    Transfer functions

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    Mämmelä, Aarne. / Commutation in linear and nonlinear systems. In: Frequenz. 2006 ; Vol. 60, No. 5-6. pp. 92-94.
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    abstract = "In this paper we consider the commutability of linear and nonlinear blocks. We show that linearity is not necessary nor sufficient requirement for the commutability since matrix multiplication is not in general commutative although scalar multiplication is commutative. For linear time-invariant systems following the superposition principle we can use the product of two transfer functions, essentially a set of scalar multiplications, and therefore the blocks commute. On the other hand, some blocks are represented by a matrix, for example linear time-variant blocks and blocks describing I/Q imbalance. Such blocks do not in general commute unless the matrices have some special properties. Furthermore, nonlinear systems are not in general commutative. The commutability is not valid unless there is a special reason for that. Typical examples for commutability include (1) the systems are combined through a commutable operation, or (2) one of the systems is the inverse of the other system.",
    author = "Aarne M{\"a}mmel{\"a}",
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    Commutation in linear and nonlinear systems. / Mämmelä, Aarne.

    In: Frequenz, Vol. 60, No. 5-6, 2006, p. 92-94.

    Research output: Contribution to journalArticleScientificpeer-review

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