where $\Phi (T)=e^{AT}$ and $\displaystyle \theta(T)=\int_{kT}^{(k+1)T} \Phi ((k+1)T-\tau) B d\tau$ .

If $t'=\tau -kT$ , we can rewrite $\theta(T)$ as $\displaystyle \theta(T)=\int_{0}^{T} \Phi (T-t') B dt'$ .

Equation (4) has a similar form as that of equation (3) if we consider $\phi(T)=\bar A$ and $\theta(T)=\bar B$ . Similarly by setting t = kT , one can show that the output equation also has a similar form as that of the continuous time one.

When T = 1,

2.2 State Equations of Inherently Discrete Systems

When a discrete system is composed of all digital signals, the state and output equations can be described by