SC · LTI and convolution

The Step Response

Feed a system a unit step and the output is the step response. See that it is the running integral of the impulse response, and that in a stable system it settles to a finite final value.

Does s settle to its final value

The top is the impulse response h(τ); the green area up to the slide time t is the step response s(t) at that instant. Below is that s(t) curve. Increase t and watch s overshoot, then settle onto the dashed final value.

impulse response h(τ)

step response s(t) = running areafinal value s(∞)

time t0.30

Current and final value

s(0.30) = 0.105  s(∞) = 0.311

It is still early. The running area has just begun to build, so s is well below its final value. This is the transient where the output is responding.

Not settled

A step makes the impulse response accumulate

The unit step u(t) was the impulse accumulated from the left up to t (A2). Because the system is LTI, an integral of the input is answered by an integral of the output. So the step response is the impulse response accumulated the same way, s(t) = ∫₀ᵗ h(τ) dτ. The green area on screen draws exactly this value at each instant.

Observes(t) = ∫ h(τ) dτ

The step response is the impulse response accumulated.

Chooseh(t) = ?

Differentiate the step response for the impulse response.

Fill ins(∞) = ?

The final value is the total area of h.

On your owns = h ?

The step response is h convolved with the step.

s and h are an integral-derivative pair

Since accumulation is integration, differentiating the step response gives the impulse response back: h(t) = ds/dt. In practice a clean impulse is hard to make but a step is easy, so engineers measure the step response and differentiate it to obtain the impulse response. Just as impulse and step were a pair in A2, the two responses are a pair too.

The final value is the DC gain

In a stable system h decays to 0, so the running area stops and s settles at a finite final value. That value s(∞) = ∫₀^∞ h(τ) dτ is the output to a constant (DC) input — the DC gain. Rise time, overshoot, and settling time on the way there are all practical metrics read from this step response, revisited in detail in control design.

Back to the first screen

As h’s positive humps added in, s rose fast and overshot the final value; then negative dips added and it rang, and once h decayed to 0 it settled. The green area up to the slide time was exactly s(t) at every moment. The step response is the impulse response integrated once, and differentiating returns h. That settled value is the DC gain. With one impulse response, the time-domain picture of an LTI system is complete.

The step response s(t) is the output an LTI system gives to the unit step u(t), and is the running integral of the impulse response. s(t) = ∫₀ᵗ h(τ) dτ = (h ∗ u)(t), and conversely h(t) = ds/dt. In a stable system s settles to a finite final value s(∞) = ∫₀^∞ h(τ) dτ, the DC gain. Rise time, overshoot, and settling time are practical metrics all read from s.

On to the next chapter

We have now drawn an LTI system completely in the time domain: one impulse response and convolution give the output for any input. But convolution is laborious. The next chapter, Fourier, opens a new view in which a signal is a sum of sinusoids. Feed a sinusoid into an LTI system and out comes a sinusoid of the same frequency, changed only in amplitude and phase — so the cumbersome convolution of the time domain becomes a simple multiplication in the frequency domain. The starting point is the next unit, the Fourier series.