Question

Transfer function is valid only for:

• A. Linear time-invariant systems
• B. Linear time-variant systems
• C. Non-linear time-invariant systems
• D. Non-linear time-variant systems

Hint:

Transfer functions are derived using Laplace or Z-transforms applied to constant-coefficient linear differential equations. This algebraic approach is uniquely valid for Linear Time-Invariant (LTI) systems.

Answer 1

The Correct Option is A

Concept: A transfer function (\(H(s)\) or \(H(z)\)) is a frequency-domain mathematical model that defines the input-output relationship of a system under zero initial conditions. It is defined as the ratio of the Laplace transform (or Z-transform) of the output signal to the Laplace transform of the input signal: \[ H(s) = \frac{Y(s)}{X(s)} \] This algebraic framework relies fundamentally on two core system assumptions: Linearity and Time-Invariance.

Step 1: Analyzing the requirement for Linearity.
The derivation of a transfer function relies on the ability to apply integral transformations linearly across terms. In a linear system, the differential equations describing system behavior contain no product terms or non-linear powers of the dependent variables (such as \(y^2(t)\) or \(\sin(y)\)). This linearity allows the system response to be separated from its scaling factors via superposition, enabling individual terms to be transformed independently.

Step 2: Analyzing the requirement for Time-Invariance.
Time-Invariance ensures that the coefficients of the system's differential equations remain constant over time. Let us look at a standard linear ordinary differential equation with constant coefficients: \[ a_n \frac{d^n y(t)}{dt^n} + \cdots + a_0 y(t) = b_m \frac{d^m x(t)}{dt^m} + \cdots + b_0 x(t) \] Applying the differentiation property of the Laplace transform (\(\mathcal{L}\{\frac{dy}{dt}\} = sY(s)\)) under zero initial conditions gives: \[ (a_n s^n + \cdots + a_0)Y(s) = (b_m s^m + \cdots + b_0)X(s) \] This allows us to factor out \(Y(s)\) and \(X(s)\) algebraically to find the transfer function: \[ H(s) = \frac{Y(s)}{X(s)} = \frac{b_m s^m + \cdots + b_0}{a_n s^n + \cdots + a_0} \] If the coefficients were time-varying (\(a_n(t)\)), this algebraic factoring would be impossible because the Laplace transform of a product requires complex frequency-domain convolution. Therefore, transfer functions are uniquely valid only for Linear Time-Invariant (LTI) systems, matching Option (A).