An algebraic aspect of linear system theory

Consideration is given to convolution from an algebraic standpoint and several examples are presented that may be of interest to engineers. In particular, the authors show that convolution need not be associative. This result should cause some concern in the analysis of cascaded systems. Further, it is shown that convolution of nowhere-zero-bounded integrable functions could be everywhere zero. This result should be of interest to those attempting to identify the input to a linear time-invariant system via some operations on the output, such as in deconvolution problems. It shows a problem that may arise in the analysis of cascaded systems in that two linear time-invariant systems each characterized by convolution with a nowhere-zero function can, when cascaded, result in a no-pass filter