Lattice Structure Realization:

Lattice filters are used in implementation of adaptive filter $ y(n) = x(n) + a_1x(n-1) $ as a filter of order 1, then is only one filter coefficient. The filter coefficient $a_1$ is also called the predictor coefficient, as it is used for predicting the current output samples knowing the currents and previous inputs samples. Here, both the inputs are excited by x(n) and the output is taken top of branch. We can write the expression for $ f_1(n) $ and $ g_1(n) $ $$ f_1(n) = x(n) + k_1x(n-1) \\ g_1(n) = kx(n) + x(n-1) $$

*Note that $ f_0(n) $ and $ g_0(n) $ are both are equal to x(n). $k_1$ is known as the reflection coefficient for stage 1 as $ f_0(n) $ and $ g_0(n) $ are reflected by factor $k_1$ in the other branch

$$ y(n) = x(n) + a_1x(n-1) + a_2x(n-2) $$