Question

Two systems \( U \) and \( V \) are defined as shown below. Which of the following statement(s) is/are CORRECT? \[ x(t) \quad {System} \, U \quad y(t) = x(t)^2 + 1 \] \[ x(t) \quad {System} \, V \quad y(t) = x(t) + 1 \]

• A. System \( U \) is nonlinear; System \( V \) is linear
• B. System \( U \) is causal; System \( V \) is noncausal
• C. System \( U \) is noncausal; System \( V \) is causal
• D. System \( U \) is nonlinear; System \( V \) is nonlinear

Hint:

A system is linear if the output is directly proportional to the input. A system is causal if the output at any time depends only on the current and past values of the input.

Answer 1

The Correct Option is A, B

Let’s examine the two systems: System U: 
The output of system \( U \) is \( y(t) = x(t)^2 + 1 \). The term \( x(t)^2 \) indicates that the system involves squaring the input, which makes it a nonlinear operation because linear systems must satisfy the principle of superposition.
Therefore, system \( U \) is nonlinear. System V: 
The output of system \( V \) is \( y(t) = x(t) + 1 \), which is a linear operation (it is a simple addition of a constant to the input). Since linear systems satisfy the principle of superposition, system \( V \) is linear. 
Step 1: Causality check. 
A system is causal if the output at any time \( t \) depends only on the input at that time \( t \) or earlier. Both systems \( U \) and \( V \) have outputs that are determined only by the current value of the input, so both systems are causal. 
Step 2: Conclusion. 
The correct answers are (A) and (B): 
System \( U \) is nonlinear; 
System \( V \) is linear, and both systems are causal.