Question

Consider the system as shown below \[ y(t) = x(e^t) \] The system is

• A. linear and causal.
• B. linear and non-causal.
• C. non-linear and causal.
• D. non-linear and non-causal.

Hint:

A system is linear if it satisfies superposition and scaling. It is causal if the output depends only on the current and past inputs. Here, the system depends on future input, so it is non-causal.

Answer 1

The Correct Option is B

Step 1: Understanding the system.
The system is described by the relationship \( y(t) = x(e^t) \). We can analyze this system by checking its linearity and causality: - Linearity: A system is linear if it satisfies the principles of superposition and scaling. The system equation \( y(t) = x(e^t) \) is a linear operation as long as \( x(t) \) is a linear function, since the transformation from \( t \) to \( e^t \) is a simple function and does not violate linearity. - Causality: A system is causal if the output at any time \( t \) depends only on the current and past values of the input, i.e., \( y(t) \) should depend only on values of \( x(t') \) for \( t' \leq t \). In this case, \( y(t) \) depends on \( x(e^t) \), which is a future value of \( x(t) \) since \( e^t>t \) for \( t>0 \). Therefore, the system is non-causal. Step 2: Conclusion.
The correct answer is (B) because the system is linear but non-causal.