Impulse Response

An impulse response of a dynamical system is the output of the system as a function of time when the system is subjected to an impulse. A perfect impulse has an amplitude of infinity at t=0 and a zero amplitude elsewhere. Therefore, it does not exist in practical situations, however, one can come close by hitting a system briefly with a hammer.

In the theory, it can be modelled with a Dirac delta, $\delta(t)$, which is in fact a unit impulse. A visualization of the Dirac delta is shown in Figure 1.

The Dirac delta has the nice property that the Laplace and Fourier transform are constant. In essence, this means when the Dirac delta is injected into the system, all frequencies are evaluated. Which results in being able to characterize the system at hand.

The result of this for an LTI-system is that the output becomes the convolution between the input and the impulse response.

As explained in the article on the transfer function, we can compute a convolution in the time domain, by multiplication in the Laplace domain. Therefore, it is often easier to convert the above equation into the s-domain by taking the Laplace Transform. Subsequently, when $Y(s)$ is computed, the Inverse Laplace Transform should be taken to obtain $y(t)$ again. This procedure is visualized in the schematics of Figure 2.

In a mathematical way, this is performed by;

With $y(t)$ being the desired output in the time-domain.

Now there is a more basic understanding of the impulse response and the transfer functions there will be more explained about stability in the upcoming articles.