Week 3 - Basic Impulse and Unit Functions

The Unit Step Function

The continuous-time unit step function $u(t)$ is defined as:

\[u(t)={\begin{cases}1,&t\geq 0\\0,&otherwise\end{cases}}\]

• Discontinuous at $t=0$
• Differentiation:

\[\frac{du}{dt}={\begin{cases}0,&t\neq 0\\\infin,& t=0 \end{cases}}\]

Operations with the Unit Step Function

• Time Shift:

\[u(t-t_0) = {\begin{cases}1,&t\geq t_0\\0,&otherwise\end{cases}}\]

• Time-limited Pulse

\[p(t)=u(t)-u(t-t_0)\]

• • This pulse has finite energy
• • It has many applications, e.g. binary transmission