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I am currently working on a project that involves analysing transient signals from sensors.
While not actually part of the analysis itself, I discussed it with the team, and they are using an fft to analyze the transient measurement.
From what i recall , Fourier Transform is a tool for periodic signals, not transient. Am I right? Am I missing something?
If the signal is zero-valued outside the time window being analyzed ("analysis window"), 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 $n < 0$ 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.
Fourier transforms are multi-purpose tools. While they are well-suited to "stationary" signals, there are several ways to use them in different contexts.
Think about a $2$-point DFT. Up to a scaling factor (for orthonormality), it consists in the matrix $F_2=\begin{bmatrix}1&1\\1&-1\end{bmatrix}$. It computes something that is essentially (thanks to robert bristow-johnson) proportional to $2$-point averages and a $2$-point differences. The former smoothes noisy zones, the latter can be used to differentiate the signal, and detect some types of transients. By combining this with subsampling, you could use it at different scales and obtain a (Haar) wavelet, another tool for transients. I admit this is not a classical use of Fourier. Here are four other ways to use Fourier for transients:
Options are legion. The choice depends on the type of transient you want to analyse and the computational/memory resource you can afford.
You are right. Fourier transforms are not useful for analysing transient signals compared to time-domain analysis or even wavelet analysis.
Transient signal analsysis is a complex endeavour.
However, keep in mind that all the information in the signal is retianed in the Fourier transform, albeit in a quite hard to interpret form.
A better tool should be able to display any transient features more apparently at its output, which Fourier transform will not; unlike the case where it displays the periodicity information so well.
It depends on your application. The term “transient” means different things to different people. Analyzing power line hiccups is different than looking at dolphin chirps.
The start up of a steady state signal can be considered of interest.
You also have complex sound scapes where there are transients in the presence of steady state signals where the differences in DFTs are very pronounced.
The DFT can also be set up so that you have a lot of time bandwidth coverage, which is useful if you don’t know much about the transients a priori.
Some transients, like bat chirps have an almost “fm” harmonic character that have interesting STFT plots.
As mentioned, a FT or DFT can be used to analyse a transient signal but, as a really hand-wavy rule of thumb, the closer the basis vectors resemble the signal itself the less additional terms may be needed to accurately characterise it. This can help with the analysis phase if it leads to simplified data. This is typically where e.g. wavelets and lifting schemes can helpful as some have said - if you're looking at step functions, a Haar wavelet will give you a cleaner 'spectrum' than a FFT for example.
