Impulse response in time domain

Thread Starter


Joined Feb 25, 2010
Hello All,

I experiencing difficulty understanding how to determine the impulse response for a given function. I think I understand it intuitively, that is the impulse response is a unit impulse function applied to a given system e.g. \( h(t) = x(t) \delta (t) \) .

This is where my confusion is:

If \( x(t) = e^{-\alpha*t}\: u(t) \) then if the unit impulse is applied then \( h(t) = \delta (t)(e^{-\alpha*t}\: u(t)) = u(t) \)

I can understand that but then I see an example where \( x(t) = u(t) \) and its impulse response is \( h(t) = e^{-\alpha*t}\: u(t) \)

How is this possible, I am missing a vital piece of this jigsaw. Could someone please advice.


Joined Jan 3, 2011
Can you show the example from whatever book or website you found it? u(t) is a stimulus signal (step function) so it seems weird to me to see it as part of a transfer function that is then excited by a different stimulus signal (impulse function).


Joined Nov 30, 2013
I may be way off base because it's been a while since I was doing this stuff in school...but...

You can find an impulse response to a system, not to a function. You can represent a system as a transfer function in the Laplace domain, but your notations all indicate time domain. If you want to work in the time domain, you can convolve a system's impulse response with an input function to get the resulting time domain response.

If you mean to be in the Laplace domain, then you are correct to multiply a transfer function with the stimulus, in this case an impulse function, to get the resulting response. The impulse function in the Laplace domain is just 1, so you just take the inverse Laplace transform of the transfer function to see the impulse response in the time domain.

I think you may need to clarify yourself a bit. If x(t) is the impulse response already, that may make more sense.

Thread Starter


Joined Feb 25, 2010
Thanks to both of you.

I was extremely confused, the problem was I thought that the impulse response can be determined without the system. The input is x(t), the impulse response (t) is the delta function applied to the system equation. The impulse response is then convolved with the input.

The first derivative of the output y'(t) is the step response
The second derivative of the output y''(t) is the impulse response.

I think this is right now, makes sense in my head now.

Thanks again.