Question

A.C. Fundamentals - Relation between step response and impulse response

Answer 1

In the context of linear time-invariant (LTI) systems, the step response and impulse response are two important concepts that provide insight into the behavior and characteristics of a system.

Impulse Response:
The impulse response of a system is the output of the system when an ideal impulse (also known as a Dirac delta function) is applied to its input. Mathematically, if the impulse response of a system is denoted by "h(t)," and the input impulse is denoted by "δ(t)" (Dirac delta function), then the output response "y(t)" can be expressed as:

y(t) = h(t) * δ(t)

In practice, the Dirac delta function is used to represent a very short-duration, high-amplitude pulse. The impulse response characterizes how the system responds to this instantaneous change and provides information about the system's behavior over time.

Step Response:
The step response of a system is the output of the system when a unit step function (also known as the Heaviside step function) is applied to its input. Mathematically, if the step response of a system is denoted by "s(t)" and the input step is denoted by "u(t)" (unit step function), then the output response "y(t)" can be expressed as:

y(t) = s(t) * u(t)

The unit step function represents a sudden change in the input from zero to one at a specific time. The step response illustrates how the system settles and reaches a steady-state value in response to this change.

Relation between Step Response and Impulse Response:
The relationship between the step response and impulse response is given by the integral of the impulse response. Specifically, the step response of a system is obtained by integrating the impulse response over time. Mathematically, if "h(t)" is the impulse response of a system, and "s(t)" is its step response, then:

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s(t) = ∫[0 to t] h(τ) dτ


In words, the step response at a given time "t" is the accumulated effect of the impulse response up to that time. This relationship provides a way to connect the transient behavior of a system (characterized by the impulse response) with its steady-state behavior (characterized by the step response).

In summary, the impulse response provides information about how a system responds to an instantaneous change, while the step response illustrates how the system settles and reaches a steady-state value after a change. The step response can be obtained by integrating the impulse response over time.

Answer 2

In the context of linear time-invariant (LTI) systems in electrical engineering, both the step response and the impulse response are important concepts that describe how a system behaves in response to different input signals. Let's explore their relationship:

Impulse Response (h(t)):
The impulse response of a system is the output of the system when an ideal impulse (also known as a Dirac delta function) is applied as the input. Mathematically, if the input signal is δ(t) (the Dirac delta function), then the output signal is given by the impulse response: y(t) = h(t) * δ(t), where "*" represents the convolution operation.

Step Response (s(t)):
The step response of a system is the output of the system when a unit step function (also known as a Heaviside function) is applied as the input. Mathematically, if the input signal is u(t) (the unit step function), then the output signal is given by the step response: y(t) = s(t) * u(t), where "*" represents the convolution operation.

The relationship between the impulse response and the step response is straightforward:

The step response can be obtained by integrating the impulse response with respect to time. Mathematically, if h(t) is the impulse response, then the step response s(t) is given by: s(t) = ∫[0, t] h(τ) dτ, where the integral is taken from 0 to t.

In the frequency domain, the step response is the integral of the system's frequency response over frequency. The frequency response is the Fourier transform of the impulse response.

In summary, the step response is the cumulative effect of the impulse response over time, and it describes how the system responds when a unit step input is applied. The impulse response, on the other hand, represents the system's behavior when an instantaneous impulse input is applied. The step response is obtained by integrating the impulse response, which means the step response captures the overall response of the system to a continuous change in input over time.

Keep in mind that these concepts are fundamental in the study of linear systems, and they are used to analyze and design various types of systems, including electronic circuits and control systems.