Impulse response and Transfer function of a LTI system

In order to understand the given answer, we need to first understand the concepts of impulse response and transfer function of a linear time-invariant (LTI) system.

Impulse response: The impulse response of a LTI system is the output of the system when the input to the system is an impulse function (also called delta function). Mathematically, it can be expressed as h(t) = L{δ(t)}, where L{} denotes the Laplace transform and δ(t) is the impulse function.

Transfer function: The transfer function of a LTI system is the ratio of the Laplace transform of the output to the Laplace transform of the input, assuming zero initial conditions. Mathematically, it can be expressed as H(s) = Y(s)/X(s), where Y(s) and X(s) are the Laplace transforms of the output and input, respectively.

Relationship between impulse response and transfer function: The impulse response and transfer function of a LTI system are related to each other through the Laplace transform. Specifically, the transfer function of a LTI system can be obtained by taking the Laplace transform of its impulse response. Mathematically, we have H(s) = L{h(t)}.

Solution to the given problem

Given that the impulse response of a LTI system is a unit step function, we need to find its corresponding transfer function.

Unit step function: The unit step function, denoted by u(t), is defined as u(t) = 0 for t < 0="" and="" u(t)="1" for="" t="" ≥="" />

Impulse response of the system: We are given that the impulse response of the system is a unit step function. Mathematically, we have h(t) = u(t).

Transfer function of the system: Using the relationship between impulse response and transfer function, we can obtain the transfer function of the system as H(s) = L{h(t)} = L{u(t)}. The Laplace transform of the unit step function is given by L{u(t)} = 1/s, for s > 0. Therefore, we have H(s) = 1/s.

Answer: Therefore, the corresponding transfer function of the LTI system whose impulse response is a unit step function is 1/s. The correct answer is option (A).