How to calculate IIR Lattice Filter Coefficients for Biquad Filter

Question

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?

Answer 1

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.

Summarized from these slides and this video