# Dimension of impulse response

asmani
Hi all

The impulse response h(t) of an electric circuit (maybe in some special cases) is the derivative of the step response s(t) of the same circuit. right?
So does it mean they have different dimension, namely if the dimension of s(t) is X, then the dimension of h(t)=ds/dt is x over second?

rbj
The impulse response h(t) of an electric circuit (maybe in some special cases) is the derivative of the step response s(t) of the same circuit. right?
So does it mean they have different dimension, namely if the dimension of s(t) is X, then the dimension of h(t)=ds/dt is x over second?

okay, let's say that your impulse response is for a device in which the dimension of the output is the same as the dimension of the input. like voltage-in, voltage-out (but it could be current in/out or something else).

then, for the convolution integral to work

$$y(t) = \int_{-\infty}^{+\infty} h(t-u) x(u) du = \int_{-\infty}^{+\infty} h(u) x(t-u) du$$

the dimension for $h(t)$ must cancel the dimension of the $du$ which we normally attach to "time". so the dimension of $h(t)$ is the reciprocal of time.