Exercise 18.2 - Seasonal LG-SSM model in standard form

From Fig.18.6(a), we have:

where $\lambda \to \infty$. The trick here (by using $c_t=-{\sum <!--\; FUNCTION\; APPLICATION\; -->}_{s=1}^{S-1}{c}_{t-s}$ to preserve periodicity) is widely adopted in circuit design and embedded systems. The latent state has to include $S-1$ states as the memory for the last $S-1$ states that have appeared.