Unit Step Function

$$u(t)=\begin{cases} 0, & t<0\\ 1, & t\ge 0 \end{cases}$$

Expressing Piecewise Functions

Consider:

$$f(t)= \begin{cases} f_1(t), & a_1 \le t < a_2\\ f_2(t), & a_2 \le t < a_3\\ \vdots \end{cases}$$

$f(t)$ can be expressed using unit step functions as:

$$f(t) = \sum f_i(t)\,\big[u(t-a_i) - u(t-a_{i+1})\big]$$

$[u(t-a_i) - u(t-a_{i+1})]$ is called a window. Each window ensures the function is active only inside the correct interval.