what is matlab function is taking as sign of second order IIR filter?

Question

I am getting confused from the implementation regarding matlab function filter([b0,b1,b2],[a1,a2],x) and other function similar as freqz. I do not understand which sign I must put in a coefficients, since there is two different way of writing the transfer function for the same second order filter as you see on the these pic below

when for example the besself function generate the coefficients, does it generate them as I write them $H(z)$ or I must write $1-a_0z^{-1}-a_2z^{-2}$?

Answer 1

It looks like your source us using a pretty unusual convention. Most books or courses use the "+" sign in the denominator and that's the convention Matlab follows. See for example https://ccrma.stanford.edu/~jos/filters/Z_Transform_Difference_Equations.html

The reason for that is simple: This way zeros are roots of the numerator polynomial and the poles are the roots of the denominator polynomial.

Answer 2

The filter function assumes this form (for your case):

$$ \tilde{H}(z) = \frac{b_0+b_1z^{-1}+b_2z^{-2}}{1+a_1z^{-1}+a_2z^{-2}} $$

where the $1$ on the bottom is from the assumption that $a_0=1$, and call it $\tilde{H}(z)$ so not to be confused with your $H(z)$. Since you have written the denominator with the signs of $a_1$ and $a_2$ inverted, then you just have to fix that before using filter.

Suppose someone wrote the transfer function from your question down $\big(H(z)\big)$, handed you the coefficients, and said use MATLAB to find the output. The only thing you need to do before is to invert $a_1$ and $a_2$: filter(..., [a0, -a1, -a2], ...).

When using besself, it returns the coefficients in the form according to the transfer function $\tilde{H}(z)$, and you can then directly use those coefficients.