<aside> 💡

Key takeaway: If you know the output response to an impulse input (this output is known as the impulse response), you can find the output response to any input by calculating the convolution integral. We assume all initial conditions are zero.

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What does this mean?

Let’s say we know the unit impulse response as $h(t)$.

If an arbitrary input $x(t)$ is given to our system,

We can find the output $y(t)$. $y(t)=(x*h)(t)$.

$**$ is the symbol for the convolution operation. You should have learned convolution in 공학수학3. In this course, we skip over calculating the convolution integral so that students don’t drop out after the first week…*

However, a (unit) impulse input is very hard to apply in real life.

Impulse input means it has to occur within 0 seconds by definition (very close to 0 seconds). To work around this problem, we can apply a unit step input (the integral of a unit impulse function) to the system and get an output response.

Then, to get the impulse response, we can compute the time derivative of this output response. This is thanks to the properties of linear systems.

But what if the output response contains noise?

The reason we computed the time derivative of the output response was to get the impulse response from experimental data.

However, if the output response contains noise, the derivative will have very high or low values.

We express the unit impulse function as $\delta(t)$

The “unit” in unit impulse function means that the area beneath this function is equal to 1.

We only apply convolution to linear systems.

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