For now I use FFT to analyze the response of an electrical system to some transient signal. The transient signal is $x(t)$, which translates to $X(w)$ in the frequency domain. On the other hand I have $H(w)$, the response of my system to a unit input signal. Then to get the response of the system to the input $X(w)$ I simply do $X(w)*H(w)$; I then get $Y(w)$ and finally $y(t)$ follows from IFFT.
My question is, would wavelet be a better way of doing this? I know that wavelet analysis is usually very good for transient signal analysis, but I wonder if they apply when one want to get the response of a system to an transient input?
I'm very curious about your application of this, the reason for this is because I think I should remind you the conditions for the use of the Fourier transform.
Fourier transform is suitable for square integrable signals (finite energy) and also that are absolutely summable, and with this are my worries, when we deal with transient signals the system usually does not behave stationary, hence Fourier transform is not a suitable tool, and also is very common that the system is unstable and does not converge, that's the main reason for the use of the Z-transform, in order to analyze the stability of the system and once the system is stable then the frequency analysis is made by evaluating the Z-transform over the unit circle.
Wavelet sound like a suitable tool for doing the analysis but again, first you need to warranty the convergence if the signal in order to justify the use of the tool.