Saturday 23 April 2016

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


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