Question

Which of the following is satisfied by a linear system?

• A. Reciprocal property
• B. Time invariance
• C. Zero initial condition
• D. Homogeneity

Hint:

Linearity requires two conditions to be met: 1. Additivity (sum of inputs equals sum of outputs), 2. Homogeneity (scaling the input scales the output by the same factor).

Answer 1

The Correct Option is D

Concept: For any mathematical or physical system to be classified as a linear system, it must satisfy the foundational Principle of Superposition. The principle of superposition consists of two core mathematical properties: Additivity and Homogeneity.
• Additivity Property: If an independent input signal \(x_1(t)\) produces an output response \(y_1(t)\), and a second input \(x_2(t)\) produces response \(y_2(t)\), then applying the combined sum of the inputs must yield the sum of the individual outputs: \[ x_1(t) + x_2(t) \longrightarrow y_1(t) + y_2(t) \]
• Homogeneity (Scaling) Property: If an input \(x(t)\) produces an output \(y(t)\), scaling that input by an arbitrary constant factor \(\alpha\) must scale the resulting output response by that same exact factor: \[ \alpha \cdot x(t) \longrightarrow \alpha \cdot y(t) \]

Step 1: Combining the properties into a unified condition.
We can combine the additivity and homogeneity criteria into a single operational check. A system characterized by the functional transformation operator \(T\{\cdot\}\) is linear if and only if: \[ T\{\alpha \cdot x_1(t) + \beta \cdot x_2(t)\} = \alpha \cdot T\{x_1(t)\} + \beta \cdot T\{x_2(t)\} = \alpha \cdot y_1(t) + \beta \cdot y_2(t) \] Where \(\alpha\) and \(\beta\) are arbitrary scalar constants.

Step 2: Evaluating the alternate options.

• Time Invariance (Choice B): This is an independent system property. A system can be linear but time-varying (e.g., \(y(t) = t \cdot x(t)\)), or non-linear but time-invariant. It does not define linearity on its own.
• Zero Initial Conditions (Choice C): While a system must have zero initial conditions to be modeled by a standard s-domain transfer function, this is a state requirement rather than a primary defining property of linearity itself. Since homogeneity is one of the two core mathematical requirements for linearity, Option (D) is the correct choice.