Question

If the input $x(t)$ and output $y(t)$ of a system are related as $y(t) = \max(0, x(t))$, then the system is

• A. linear and time-variant
• B. linear and time-invariant
• C. non-linear and time-variant
• D. non-linear and time-invariant

Answer 1

The Correct Option is D

The given system is defined by the equation \(y(t) = \max(0, x(t))\), which means the output \(y(t)\) is equal to the input \(x(t)\) if \(x(t) \geq 0\), otherwise \(y(t) = 0\).

Let's analyze the system's properties:

1. Linearity: A system is linear if it satisfies both the superposition (additivity) and homogeneity (scaling) properties.
   • Superposition: If we have two inputs \(x_1(t)\) and \(x_2(t)\), the superposition property states that the output for the input \(x_1(t) + x_2(t)\) should be equal to the sum of the outputs for the individual inputs:
   • Homogeneity: This property states that if the input is scaled by a constant \(k\), the output should be scaled by the same constant:
2. For the given system, neither of these properties hold true:
   • If \(x_1(t) = 2\) and \(x_2(t) = -1\), then:
     • \(y(x_1(t) + x_2(t)) = y(1) = \max(0, 1) = 1\)
     • \(y(x_1(t)) + y(x_2(t)) = \max(0, 2) + \max(0, -1) = 2 + 0 = 2\)
   • Thus, \(y(x_1(t) + x_2(t)) \neq y(x_1(t)) + y(x_2(t))\), violating the superposition property.
   • Similarly, check for homogeneity using \(k = 2\) and \(x(t) = -1\):
     • \(y(kx(t)) = y(-2) = \max(0, -2) = 0\)
     • \(k y(x(t)) = 2 \times \max(0, -1) = 0\)
   • So, \(y(kx(t)) = k y(x(t))\) is accidentally satisfied here because both are zero.
3. Time-Invariance: A system is time-invariant if the behavior and characteristics of the system are unaltered over time.
   • If \(y(t) = \max(0, x(t))\), then for an input \(x(t - t_0)\), the output should be \(y(t - t_0) = \max(0, x(t - t_0))\).
   • The output remains consistent with time shifts, proving time-invariance.

In conclusion, the system described by \(y(t) = \max(0, x(t))\) is indeed non-linear due to the failure of the superposition property, and time-invariant as it does not explicitly depend on time.

Therefore, the correct answer is: non-linear and time-invariant.