Question

The response of a system to a unit impulse input is given by What would be the response of the same system to a unit step input?

Answer 1

Since the unit step function, γ(t), is closely related to the unit impulse, δ(t), it should not be surprising that the unit impulse response (the response of a system to a unit impulse) is also closely related to the unit step response. To develop this relationship, consider first the unit step response of a system.

Input, Unit Step, y(t) Output, Unit Step Response, y,(t) System

In this diagram the input is the unit step function, γ(t);, and the output is the unit step response, γ(t). If we delay the step, we simply delay the response:

Input, Delayed Unit Step, y(t T) Output, Delayed Unit Step Re sponse, y^(t-T) System

If we scale the step (multiply by a constant), we simply scale the response

Output, Scaled Unit Step Response, yy(t) Input, System Y(t)

By linearity, if we apply the sum of two inputs, the output is simply the sum of the individual outputs:

Input, Rec tangular Pulse, Y(t)-y(t T) Output System

Now if we take T→0, the input is an impulse (the derivative of a step function),

limT→0γ(t)−γ(t−T)T=dγ(t)dt=δ(t)limT→0⁡γ(t)−γ(t−T)T=dγ(t)dt=δ(t)

so the output is the impulse response (the derivative of the unit step response).

limT→0yγ(t)−yγ(t−T)T=dyγ(t)dt=yδ(t)limT→0⁡yγ(t)−yγ(t−T)T=dyγ(t)dt=yδ(t)

or

Input, Output, Unit ImpulseS SystemUnit Impulse Response 6 (t)

It is important to keep in mind that the impulse response of a system is a zero state response (i.e., all initial conditions equal to zero at t=0^-). If the problem you are trying to solve also has initial conditions you need to include a zero input response (i.e., the response due to initial conditions) in order to obtain the complete response