Question

The transfer function of an LTI system is defined as:

• A. Output signal
• B. Frequency response
• C. Impulse response
• D. Ratio of Laplace transform of output to input (zero initial conditions)

Hint:

Always look for the key phrase "zero initial conditions" when defining a transfer function. Without this constraint, the ratio \(\frac{Y(s)}{X(s)}\) is not constant and depends on the system's initial energy state.

Answer 1

The Correct Option is D

Concept: A Linear Time-Invariant (LTI) system can be completely modeled in either the time domain or the complex frequency domain (s-domain). In the complex s-domain, the differential equations governing the input-output relationships of the system are transformed into algebraic linear equations using the tool of the Laplace Transform. The standard canonical definitions are:
• Transfer Function, \(H(s)\): The algebraic ratio of the Laplace transform of the output signal \(Y(s)\) to the Laplace transform of the input signal \(X(s)\), calculated under the condition that all initial conditions within the system's energy-storage components are set to zero. \[ H(s) = \left. \frac{\mathcal{L}\{y(t)\}}{\mathcal{L}\{x(t)\}} \right|_{\text{all initial conditions} = 0} = \frac{Y(s)}{X(s)} \]

Step 1: Analyzing the necessity of zero initial conditions.
If a system has stored energy at \(t = 0^-\) (such as residual voltages on capacitors or currents through inductors), the Laplace transform of its differential equations generates extra algebraic constants or polynomial source terms. These independent source terms prevent the factorization of the input-output ratio as a pure single multiplier, violating the definition of a transfer function. Therefore, initial conditions must be set to zero.

Step 2: Distinguishing between options.

• The impulse response \(h(t)\) is the time-domain output when the input is a Dirac delta function \(\delta(t)\). While related, the transfer function is the Laplace transform of this impulse response, not the impulse response itself.
• The frequency response \(H(j\omega)\) is evaluated along the imaginary axis by substituting \(s = j\omega\), which is a specialized subset of the generalized Laplace transfer function. Thus, Option (D) matches the formal definition.