Impulse Response and Step Response

The impulse response of a linear time-invariant (LTI) system is the output of the system when the input is an impulse function. It describes how the system responds to a sudden change or impulse in the input signal. The step response, on the other hand, is the output of the system when the input is a unit step function. It describes how the system responds to a sudden change from zero to a constant value in the input signal.

Given Impulse Response

In this question, the given impulse response of the LTI system is h(t) = δ(2)(t), where δ(2)(t) represents a delta function shifted by 2 units to the right.

Impulse Response and Step Response Relationship

The step response of an LTI system can be obtained by integrating the impulse response. In other words, the step response is the integral of the impulse response.

Integration of Impulse Response

To find the step response, we need to integrate the impulse response h(t) = δ(2)(t).

Since the impulse response is a delta function, its integral will result in a step function. However, due to the shift in the delta function by 2 units to the right, the resulting step function will also be shifted by the same amount.

Analysis of Options

a) Option '1': This option suggests that the step response is equal to 1. This is incorrect because the impulse response is not a constant value.

b) Option 'u(t)': This option suggests that the step response is a unit step function. This is incorrect because the impulse response is not a unit step function.

c) Option 'δ(3)(t)': This option suggests that the step response is a delta function shifted by 3 units to the right. This is incorrect because the given impulse response is a delta function shifted by 2 units to the right, not 3 units.

d) Option 'δ(t)': This option suggests that the step response is a delta function. This is incorrect because the given impulse response is not a delta function.

Correct Answer

The correct answer is option 'A': 1. This means that the step response of the LTI system is a constant value of 1. The impulse response, being a delta function shifted by 2 units to the right, does not affect the step response. The step response remains constant regardless of the impulse response shape or shift.

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