Commutation in linear and nonlinear systems

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. Index Terms – commutability, linear systems, nonlinear systems, time-invariant systems, time-variant systems, inverse systems, predistortion, postdistortion, I/Q imbalance

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