Cascading LP filter transfer function

[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 I am 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!

 

[anorlunda]

You leaped ahead and made a number of assumptions. S to z domain transformations are not that simple. I recommend you spend time studying the tutorial.

https://www.mathworks.com/help/signal/examples/practical-introduction-to-digital-filter-design.html