Frequency Response Function Estimate via FFT Question
In general, the discrete Fourier transform is supposed to be applied to periodic signals.

The FRF of a perfectly periodic signal only has energy content in discrete frequencies (at multiples of the signal's frequency and DC). If the signal is not periodic, then the Fourier transform will have spectral content for all continuous frequencies.

So when executing an experiment where you wish to measure the FRF of an LTI system, you can think of the output as a sum of the periodic signal (that remains indefinitely) and a second part that is not periodic (you could call them "transients"). The periodic signal will give you contributions at discrete frequencies, while the transient term will add stuff that you may not have wanted to include in your transfer function. This insight does not depend on the type of excitation signal. If you apply a chirp, impulse, multisine or any other (periodic) excitation signal, you should make sure that the output signal is periodic as well before measuring. To do so you can:


measure using an excitation signal with a period that is long enough such that the signal has completely died out before the end of the period.
measure a few periods before actually storing data to allow the system to first converge to a steady-state solution.


If you are not in a position to wait for long times, then there are ways to estimate and/or partly cancel transient effects on the FRF. The idea is to deliberately not excite some frequency bins. If the output signal has spectral content in those "empty" frequency bins, then you know that they have to be due to transient effects (assuming the system is LTI) allowing you to estimate the effect of them. You can then go on and cancel these effects by using the property that the FRF of a transient is a continuous complex function by approximating the "extra" contributions caused by transients in the excited bins by interpolation of the values in the non-excited bins.