To analyze a filter, different types of analysis are available, depending on the filter's use:

• temporal analysis : a temporal stimulus is injected at the filter input and the filter output is analyzed as a function of time.
• frequency analysis: an alternating signal is injected at the filter input, whose frequency will “sweep” from fmin to fmax, so that amplitude and phase are displayed as a function of frequency at the filter output. These are Bode curves, Nyquist curves, etc.

Temporal analysis

The most commonly used types of time signals are step, impulse and ramp. This short list is not exhaustive. In fact, all kinds of time signals can be used, depending on the filter's analysis requirements (for example, exponential, sine, cosine, triangle, square wave forms, etc.).
Of course, the response is depending of the kind of filter used and its parameters.

1. Step response

A unit step signal, also called Heaviside, is defined by the following equation: $$u(t) = 0$$ if $$t$$ < $$t_0$$ $$u(t) = 1$$ otherwise.

Of course, it is possible to use a step with heigt different from 1. Generally, $$t_0$$ = 0.

The Laplace transform of a unit step starting at 0 is $$\displaystyle \frac{1}{s}$$. For a step of amplitude A, we have $$\displaystyle \frac{A}{s}$$

2. Impulse response (also called pulse response)

For a continuous-time filter, the mathematical model of a pulse is a Dirac distribution. This is an impulse of infinite height at t = 0 and zero everywhere else, and whose integral is equal to 1. In other words, it's a pretty difficult function to model. However, mathematical calculation and simulation software offer this functionality as standard (Matlab, Scilab, Python with packages such as Control or Tbcontrol, etc.). Alternatively, you can use “approximate” functions to obtain very precise results (see https://www.tau.ac.il/~tsirel/dump/Static/knowino.org/wiki/Dirac_delta_function.html and https://fr.wikipedia.org/wiki/Distribution_de_Dirac).

The Laplace transform of a pulse is $$1$$

3. Ramp response

A ramp signal is a straight line with slope $$\alpha$$, for t > $$t_0$$ and zero otherwise.

The Laplace transform of a ramp function starting at 0 is $$\displaystyle \frac{1}{s^2}$$

Frequency analysis

under construction