Is a Fourier transform a sound way to analyse a transient signal?
If the signal is zero-valued outside the time window being analyzed (&quot;analysis window&quot;), then the discrete Fourier transform (DFT) exactly samples the discrete-time Fourier transform (DTFT) of the signal at harmonic frequencies. DTFT implies no time-domain periodicity. The sampling of DTFT by DFT is evident from definitions of the two transforms, if we apply a finite support signal constraint:
$$\text{DTFT:}\quad X(\omega) = \sum_{n=-\infty}^{\infty} x[n] \,e^{-i \omega n}$$
If $x[n] = 0$ for all $n$ such that $n &lt; 0$ or $n \ge N$ (finite support signal constraint), then:
$$= \sum_{n=0}^{N-1} x[n] \,e^{-i \omega n}$$
The frequency variables of the two transforms are related by $\omega = \frac{2\pi}{N}k$, so:
$$= \sum_{n=0}^{N-1} x[n] \,e^{-\frac {i 2\pi}{N}kn}$$
This is identical to the definition of DFT:
$$\text{DFT:}\quad X[k] = \sum_{n=0}^{N-1} x[n] e^{-\frac {i 2\pi}{N}kn}$$
Choosing a longer window (larger $N$) enables DFT to sample DTFT more densely.
For a signal that has non-zero values outside the analysis window, doing the analysis on just the window is equivalent to first multiplying the signal by a rectangular window, and then sampling the DTFT of this windowed signal using DFT. There are other window functions that may have more favorable characteristics for the type of signal and for the particular analysis task, allowing to isolate a transient of interest without too much distortion in time domain or smearing in frequency domain.
Fast Fourier Transform (FFT) is often used because it is very fast to compute, even though a decomposition into harmonic sinusoids might not always be the best way to describe a signal.