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.

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September 29, 2026 1 min read

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