# How to calculate IIR Lattice Filter Coefficients for Biquad Filter

If I have a biquad filter with known $$a_1, a_2, b_0, b_1, b_2$$ coefficients, how do I calculate the corresponding lattice/ladder coefficients?

I have found information on deriving the lattice coefficients for all pole filters but not ARMA filters - am I missing something simple?

• Does this help?
– Jdip
Oct 25, 2022 at 14:32

Ok, so in addition to the comment from @Jdip, I've pieced together the method for calculating the lattice $$k$$ and ladder $$v$$ coefficients for a given ARMA filter, given $$M+1$$ normalised pole coefficients $$p$$ (i.e. AR order $$M$$, $$a_0=1$$) and $$B$$ zero coefficients $$b$$.
The first step is to calculate a 'lattice coefficient matrix' (I'm not sure what to call it) defined by $$\begin{equation} \alpha_{m - 1}^{n} = \frac{\alpha_m^{n} - \alpha_m^{m}\alpha_m^{(m - n + 1)}}{1 - (\alpha_m^{m})^2} \end{equation}$$
where $$m = [M - 1, M - 2 ... 0]$$ and $$n = [0, 1, 2, ... m - 1$$]. The matrix must be initialized such that $$\alpha_{M - 1}^n = p^n$$, ignoring $$p^0$$. The lattice coefficients $$k_i$$ are given by $$\alpha_i^i$$ (i.e. the matrix diagonal). Once this matrix is calculated, the ladder coefficients can be calculated using $$\begin{equation} v_m = b_m - \sum_{i=m+1}^B{c_i\alpha_i^{i-m}} \end{equation}$$ where $$B$$ is the number of zeros, $$b$$ is the direct coefficient, and $$\alpha$$ is the previously calculated matrix.