control - What's the Laplace transfer function of a moving average?

What is the Laplace transfer function of a moving average?

$$y_k=\frac{x_k+x_{k-1}+x_{k-2}+x_{k-3}+...+x_{k-N+1}}{N}$$ I tried to get it from the z-domain transfer function using conversion tables: $$\frac{y_k}{x_k}=\frac{1+z^{-1}+z^{-2}+z^{-3}+...+z^{-N+1}}{N}$$

But unless I've read them wrong they don't have the "bricks" I need to get me anywhere.

Answer

$$\frac{y_k}{x_k}=\frac{1+z^{-1}+z^{-2}+z^{-3}+...+z^{-N+1}}{N}$$

can be re-written as

$$\frac{y_k}{x_k}=\frac{1}{N}\frac{1-z^{-N}}{1-z^{-1}}$$

That should be straightforward to model in the s-domain by replacing z by

$$e^{sT}$$

i.e.

$$H(s)=\frac{1}{N}\frac{1-e^{-sTN}}{1-e^{-sT}}$$

This is a SINC function in the frequency domain whose magnitude versus frequency is of the form:

$$\dfrac{sin(\pi fN)}{Nsin(\pi f)}$$

(source)