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?
Here's the outline of the argument, feel free to fill in the details.
The averaging operator is like a convolution with a "square" pulse of height $1/T_s$ supported on the interval $[-T_s/2, T_s/2]$.
You can express the square pulse as a sum of two heaviside step functions.
Finally, recall the Laplace transform of a step function $\mathcal{L} \{H(t-T_s/2)\}(s) = \frac{e^{-sT_s/2}}{s}.$
$int_{t-T_s/2}^{t+T_s/2}=int_{-\infty}^{t+T_s/2}-int_{-\infty}^{t-T_s/2}$
Take the Lapalce transform of each term
$exp^{+sT_s/2}/s-exp^{-sT_s/2}$ since the Lapalce transform of $int_{-\infty}^{t}$ is $1/s$ and the time shifting operation $x(t+-T_s/2)$----> $exp^{+-T_s/2}$