Question

Given \( h(t) = \delta(t-1) + \delta(t-3) \), find step response at \( t = 2 \).

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

Impulse response integrates to step response.

Answer 1

The Correct Option is B

To find the step response of the given impulse response \( h(t) = \delta(t-1) + \delta(t-3) \), we need to understand the relationship between these signals. The step response \( s(t) \) is the integral of the impulse response:

\[ s(t) = \int_{-\infty}^{t} h(\tau) \, d\tau \] 

Since \( h(t) = \delta(t-1) + \delta(t-3) \), theorem dictates that the step response would change by 1 unit at each impulse:

• The impulse \( \delta(t-1) \) causes a unit step change of magnitude 1 at \( t=1 \).
• The impulse \( \delta(t-3) \) causes another unit step change of magnitude 1 at \( t=3 \).

Therefore, the step response \( s(t) \) can be expressed piecewise:

• For \( t < 1 \), \( s(t) = 0 \)
• For \( 1 \leq t < 3 \), \( s(t) = 1 \)
• For \( t \geq 3 \), \( s(t) = 2 \)

We need to evaluate the step response at \( t = 2 \). Since \( 1 \leq 2 < 3 \), we have:

\[ s(2) = 1 \]

Thus, the step response at \( t = 2 \) is:

1