State-based sequence processors

Sequence processors may be built around a state; state-based sequence processors maintain and update at each time $t$ a state $x[t]$ which stores the information about the input sequence they have seen so far ( $u[1], \ldots, u[t]$) which is necessary to compute the current output $y[t]$ or future outputs. State is recursively computed: the state at time $t$, $x[t]$, is computed from the state at time $t-1$, $x[t-1]$, and the current input $u[t]$ using a suitable next-state function:

$$x[t]=f(x[t-1],u[t]).$$ (4.1)

The output is then computed using an output function, usually from the current state,

$$y[t]=h(x[t]),$$ (4.2)

but sometimes from the previous state and the current input, like current state itself

$$y[t]=h(x[t-1],u[t]).$$ (4.3)

Such a state-based sequence processor is therefore defined by the set of available states, by its initial state $x[0]$, and by the next-state ($f$) and output ($h$) functions (the nature of inputs and outputs is defined by the task itself). For example, Mealy and Moore machines (sections 2.3.1 and 2.3.2) and deterministic finite-state automata (section 2.3.3) are sequence processors having a finite set of available states. As will be seen in the following section, neural networks may be used and trained as state-based adaptive sequence processors. .