Question

Shifting property of impulse ∂(t) is given by ______

I have been asked this question in an internship interview.

My enquiry is from Elementary Signals in chapter Signals and Systems Basics of Signals and Systems

Answer 1

The shifting property of the continuous-time unit impulse function, \( \delta(t) \), is given by:

\[

\delta(t - t_0) =

\begin{cases}

0, & \text{for } t \neq t_0 \\

\infty, & \text{for } t = t_0

\end{cases}

\]

and satisfies:

\[

\int_{-\infty}^{\infty} x(t) \delta(t - t_0) \, dt = x(t_0)

\]

This property essentially "samples" the function \( x(t) \) at \( t = t_0 \), isolating the value of \( x(t) \) at that point. It’s a fundamental concept used to simplify expressions and analyze systems, as it allows shifting the impulse function to any desired point \( t_0 \) in time.