Convolution Input/output Units
impulse responses, $h(t)$, of continuous-time systems (those with output that is the same species of animal of the input) have dimensions 1/time. 

consider a wire which has impulse response

$$ h(t) = \delta(t) $$

the area of the $\delta(t)$ is equal to the dimensionless $1$.  the width of the nascent delta is in dimension of time, so the height must be in dimension of 1/time.

the impulse response for a simple RC low-pass filter is

$$ h(t) = \tfrac{1}{RC} e^{-^{\, t}/_{RC}} \ u(t) $$

the dimension of $h(t)$ is in 1/time because $RC$ is in units of time.