# Laplace transform of averaging operator

I am studying dc-dc converter now. I got a problem with Laplace transform of the averaging operator as in the image below.

Can anyone help me derive the Laplace transform result $G_{av}(s)$ as in the image?

The averaging operator is like a convolution with a "square" pulse of height $1/T_s$ supported on the interval $[-T_s/2, T_s/2]$.
Finally, recall the Laplace transform of a step function $\mathcal{L} \{H(t-T_s/2)\}(s) = \frac{e^{-sT_s/2}}{s}.$
• You want to go "up" at $-T_s/2$, so that means you need $H(t+T_s/2)$. Then you need to go "down" at $T_s/2$, which means you need to subtract out $H(t-T_s/2)$. – Atul Ingle Dec 6 '16 at 22:03