What is the difference between impulse, step and sum of sinusoids signals in terms of their spectra and their degrees of excitation ?


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An impulse has a flat/constant power density for all frequencies. The impulse response of a system can also be used to characterize a system, namely convolution of the input with impulse response, will yield the system response (however usually you would do this in the frequency domain, since then convolution becomes simple multiplication). The downsides of using an impulse as a way to excite a system is that the amount energy you can input into the system will be limited, since a too big impulse might damage a physical system. This might mean that you need a sensor with a higher accuracy, to measure lightly damped modes with (very) small gains, otherwise this might go unnoticed due to the resolution of the sensor.

A step has a spectrum with a slope of minus one. So low frequency dynamics will get a higher gain than high frequency dynamics. And again a step can only supply a limited amount of energy into the system.

If you actually mean just a sum of a couple of sinusoidal signals, then you will mainly see the system response of the system at those frequencies, but also some leakage to other frequencies, which could excite eigenmodes of the system. After these eigenmodes have died out (assuming the system is stable and sufficiently damped) then you should only see the response of the system at those frequencies. This can be useful if you are only interested in a specific frequency range or to see if the system can be modeled as a linear time independent system, because nonlinear or time variant systems will most likely also contain different frequencies in the response after the transient has died out.

Another good options for signals are sine sweep, by applying a single "sinusoidal" signal whose frequency changes (relatively slowly) over time, and (white) noise. The sine sweep is similar to the sum of sinusoidal signals, but it might be harder to be used to detect non-LTI system behavior. White noise, similar to the impulse, also has a flat spectral power density, but has the advantage that it continuously adds energy to the system and keeps exciting all modes. But it would be even harder to detect non-LTI system behavior.


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