Lattice Filter Structures

Single Lattice Stage

General Form

\begin{align*} y[n] &= K v[n] + u[n-1] \\ v[n] &= x[n] - K u[n-1] \\ \end{align*} \begin{align*} Y(z) &= K V(z) + U(z) z^{-1} \\ V(z) &= X(z) - K U(z) z^{-1} \\ \end{align*} or \begin{align*} Y(z) &= K V(z) + U(z) z^{-1} \\ X(z) &= V(z) + K U(z) z^{-1} \\ \end{align*}

All-Pass Filter Form

\begin{align*} \frac{Y(z)}{X(z)} = \frac{K V(z) + U(z) z^{-1}}{V(z) + K U(z) z^{-1}} \end{align*} if we connect $U(z)$ to $V(z)$, we get \begin{align*} \frac{Y(z)}{X(z)} &= \frac{V(z)[K + z^{-1}]}{V(z) [1 + K z^{-1}]} \\ &= \frac{K + z^{-1}}{1 + K z^{-1}} \end{align*}

FIR Filter Form

\begin{align*} Y(z) &= K V(z) + U(z) z^{-1} \\ X(z) &= V(z) + K U(z) z^{-1} \\ \end{align*} Notice that \begin{align*} U(z) &= [Y(z) - K V(z)]z^{+1} \\ \end{align*} If we connect $X(z)$ and $Y(z)$ together, \begin{align*} Y(z) &= V(z) + K U(z) z^{-1} \\ Y(z) &= V(z) + K [Y(z) - K V(z)] \\ Y(z)[1 - K] = ... \end{align*} I think I should be gettin V(z)/X(z) = A(z) S(z) See here: https://web.ece.ucsb.edu/Faculty/Rabiner/ece259/digital%20speech%20processing%20course/lectures_new/Lecture%2014_winter_2012_6tp.pdf

IIR Filter Form

\begin{align*} X(z) &= V(z) + K U(z) z^{-1} \end{align*} if we connect $U(z)$ to $V(z)$, we get \begin{align*} X(z) &= V(z)[ 1 + K z^{-1} ] \\ \frac{V(z)}{X(z)} = \frac{1}{1 + K z^{-1}} \end{align*}

Multi-Lattice Stage