One can encounter well-behaved signals x(t) and h(t) for which the convolution integral
simply does not exist! For example take both signals to be identically 1. This is not surprising since the convolution integral is in general an improper integral, that is, integration over an infinite extent. We will not further mention this possibility, leaving it to the reader to perform sanity checks in particular examples. Of course there are mathematically possible signals that, for example, vary so rapidly that the Riemann integral is not well defined. We obviously do not consider these.
Beyond the technical issues, LTI systems that cannot be represented by convolution are weird indeed. A typical example involves a system whose output is a constant that depends only on the infinite past:
where the class of allowable input signals are those that have a limit as t® -¥. You are encouraged to check linearity and time invariance of such a system, but in any case we are happy to leave these out of engineering discussions.
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