# Cascading LP filter transfer function

1. Oct 29, 2013

### stiive

Hi All,
I'm new to this forum. Its been awhile since university, so i've unfortuntely forgotten most of my teachings on transfer functions it seems!

Basically im trying to design a 3 stage cascading digital low pass filter. I am sampling an AC waveform and need to integrate the signal. I'm getting some DC offset error from pure integration (trapezoidal method), so have decided to use a cascading filter with an adjustable cut-off freq (set just above the current variable AC electrical freq), and gain multiplication to compensate for attenuation and for the integration.

I have been told that the transfer function for the 3 identical stage filter of a sine would be
H(s)=$G*(\frac{1}{\tau*s+1})^{3}$

Obviously as i am sampling discretely, I need to be in Z domain
So I think i'd apply tunstin transformation(?) ;

s=$\frac{2(z-1)}{T(z+1)}$

therefore,

H(Z) = G*$(\frac{1}{\tau[\frac{2(z-1)}{T(z+1)}]+1})^{3}$

where $\tau$= cut-off freq, G = gain, T = Δtime

But from here i'm kinda stuck, and would appreciate any prompting/help in the right direction. I'm guessing i'll need to do partial fractions? I have tried this, but the answer I got I'm fairly sure is wrong as most of the terms are future sample input/outputs (ie y[n+3], x[n+3]). Is perhaps the z term meant to be $z^{-1}$? For instance, I have seen in MATLAB simulink the trapezoidal transfer function as;

$\frac{K*Ts(z+1)}{2(z-1)}$

whereas i think it should be;

$\frac{K*Ts(z^{-1}+1)}{2(z^{-1}-1)}$

Perhaps MATLAB uses geophysical(?) definition?

Perhaps also instead of tunstin transformation, i should just use $s=z^{-1}$??

Thanks for any help in advance!!