Question

The Impulse Response of an initially relaxed linear system is \( e^{-2t}u(t) \). To produce a response of \( te^{-2t}u(t) \), the input should be:

• A. \( 2e^{-t}u(t) \)
• B. \( 0.5 e^{-2t}u(t) \)
• C. \( e^{-2t}u(t) \)
• D. \( e^{-t}u(t) \)

Hint:

The input required to produce a scaled and time-multiplied response in an LTI system can be determined using Laplace domain division.

Answer 1

The Correct Option is B

Step 1: The response of a linear time-invariant (LTI) system is given by the convolution of the input \( x(t) \) with the impulse response \( h(t) \): \[ y(t) = x(t) * h(t) \] Step 2: Given:
- Impulse response: \( h(t) = e^{-2t} u(t) \)
- Desired response: \( y(t) = te^{-2t} u(t) \) 
Step 3: The Laplace Transform of the given impulse response is: \[ H(s) = \frac{1}{s + 2} \] And for the desired response: \[ Y(s) = \frac{1}{(s+2)^2} \] Step 4: The required input is: \[ X(s) = Y(s) / H(s) = \frac{1}{(s+2)^2} \times (s+2) = \frac{1}{s+2} \] which corresponds to: \[ x(t) = 0.5 e^{-2t} u(t) \]