# How to interpret these filter coefficients?

I am trying to make sense of the filter coefficients.

The author claims that they are running a 3rd order lowpass Butterworth filter and were willing to give their design. They are not the most communicative type and it is typically weeks until I am able to get any sort of answer from them.

The coefficients are as follows:

1.22293 1.394312 1.994313
-0.3339159 -0.6572268 -0.9943293


They also provided the transfer function:

$$H_k(z) = \frac{b_{k0}+b_{k1}z^{-1}+b_{k2}z^{-2}}{1-a_{k1}z^{-1}-a_{k2}z^{-2}},\quad k = 1, 2, 3.$$

Whey I plug in these coefficients into the FDA toolbox in MATLAB, I get the following frequency response (sampling rate was set to $20833\textrm{ Hz}$):

What is the correct way to interpret these numbers? I am somewhat out of my element when it comes to the Digital Filtering and could use some help.

• The transfer-function in the equation is only 2nd order... Did they label the coefficients? – Arnfinn Nov 30 '16 at 22:31
• Could you provide a source for the work you are talking about? – Laurent Duval Nov 30 '16 at 23:34
• @Laurent Duval The coefficients and the TF expression are all that is provided to us. We are trying to troubleshoot their system that processes signals generated by our equipment and the author/customer is very secretive regarding their implementation details. – udushu Dec 1 '16 at 1:12
• @Arnfinn the coefficients were not labeled. Just two rows of numbers in the exact same order as shown in the post. – udushu Dec 1 '16 at 1:14
• @udushu Well, I guess you can try various permutations of the coefficients -- trying them out as numerator and denominator and flipping them left-right -- to see if you at the very least can get a stable filter... – Arnfinn Dec 1 '16 at 2:08