· Impulses in h(t) and/or x(t)
In order to represent LTI systems of the form
by a convolution integral
we use impulse functions as follows. Let
Substituting into the convolution expression gives, upon using the sifting property of impulse functions under integral signs,
Notice in particular that if h(t) = d (t), then the output is identical to the input. Naturally enough, this is called the identity system.
For an LTI system described by h(t) with the input signal x(t) = d (t - T), where T is a real constant, the output signal is, by the sifting property, y(t) = h(t - T). In particular if the LTI system is the identity system, h(t) = d (t), the unit-impulse input x(t) = d (t) yields the unit pulse output,
This is a mathematically illegal sifting property, because the impulse does not have the requisite continuity property. This is the kind of symbolism that engineers find convenient despite the often severe reaction of mathematical colleagues!