# Impulse and unit step response

I'm trying to understand impulse and unit step responses, but I'm not sure if my understanding is correct. In general, are the following statements true:

a) The impulse response for a system $x(t) \mapsto y(t)$ is $y(t)$ when $x(t) = \delta(t)$ where $\delta$ is the Dirac delta, and

b) The unit step response for a system $x(t) \mapsto y(t)$ is $y(t)$ when $x(t) = u(t)$ where $u$ is the unit step (Heaviside step function)

• That is correct. – Brian Sep 22 '15 at 14:12
• Wow, I'm almost shocked - all the resources I consulted made this sound so much more complicated than it actually is. – user2398029 Sep 22 '15 at 15:22
• I'm not sure this is correct (it may be just that I'm misunderstanding the notation). If the input to an LTI system is $x(t)$, then the output is $y(t)=x(t) \ast h(t)$, where $h(t)$ is the system's impulse response. To me this is not the same as statement a) in the question. – MBaz Sep 22 '15 at 15:39

"the function $h(t)$ such that the mapping $x(t) \mapsto y(t)$ is computed as $y(t) = x(t) * h(t)$"
then notice that if $x(t) = \delta(t)$ you get $y(t) = \delta(t) * h(t) = h(t)$.