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.

## Best 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)