Question

The input \( x(t) \) and the output \( y(t) \) of a system are related as \[ y(t) = e^{-t} \int_{-\infty}^t e^t x(\tau) d\tau, \quad -\inftyLt;tLt;\infty. \] The system is:

• A. Nonlinear
• B. Linear and time-invariant
• C. Linear but not time-invariant
• D. Noncausal

Hint:

A system is causal if the output at any time \( t \) depends only on the present and past values of the input.

Answer 1

The Correct Option is B

Step 1: Check for linearity. The system involves an integral and an exponential scaling. Both operations satisfy the principle of superposition. Thus, the system is linear. Step 2: Check for time-invariance. Let the input be \( x(t - t_0) \). The output becomes: \[ y(t) = e^{-t} \int_{-\infty}^t e^\tau x(\tau - t_0) d\tau. \] Substituting \( \nu = \tau - t_0 \), the limits of integration remain unchanged, and the time-shifted output matches the shifted input. Hence, the system is time-invariant. Step 3: Check for causality. The integration limit depends only on the past values (\(-\infty\) to \(t\)) of the input. Thus, the system is causal.