# Is a Fourier transform a sound way to analyse a transient signal?

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?

• The FT is a tool for analysing both periodic and non-periodic signals. How else would you analyse an audio signal, which is almost always non-periodic? The biggest problem with transients is they're of short duration so may have a difficult time distinguishing them from the background noise. – dsp_user Nov 18 '19 at 9:39
• I think you mean discrete Fourier transform (DFT). Fast Fourier transform (FFT) computes DFT. – Olli Niemitalo Nov 18 '19 at 11:38
• The Laplace transform can be useful to view both pure-sinusoidal and exponential decay (or growth for that matter) behavior of a signal. This tends to lead itself better than using Fourier analysis alone. And as Stanley mentioned below, what it means to be a "transient" signal can be subjective and it may take a bit more work to determine which approach to take. – Envidia Nov 18 '19 at 23:54

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 $$x[n] = 0$$ if $$n < 0$$ 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.

• if i were answering this, i would replace "$X(\omega)$" with "$X(e^{i \omega})$" to make your DTFT definition exactly consistent with the Z transform definition. – robert bristow-johnson Nov 18 '19 at 16:20
• Olli, i did essentially the same thing to a question that more directly asks what the difference is between the DFT and the DTFT. – robert bristow-johnson Nov 18 '19 at 16:26
• The defs here are essentially from Wikipedia. – Olli Niemitalo Nov 18 '19 at 19:22

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:

• Compute interesting transient parameters in the Fourier domain, for instance a Fourier coefficient decay that could be meaningful depending on the transient model, variations in amplitude and phase,
• Track some frequencies using sliding windowed DFT and its avatars,
• Compare spectra in parallel, computed on different lengths,
• Build a time-frequency representation (short-term Fourier transform) to better extract frequency changes over time.

Options are legion. The choice depends on the type of transient you want to analyse and the computational/memory resource you can afford.

• it's a 2-point sum, not a 2-point average. – robert bristow-johnson Nov 18 '19 at 16:28
• Up to a scaling factor, but yes – Laurent Duval Nov 18 '19 at 16:47
• but if you were to put in the same scaling factor, the "difference" would be the "half-difference" – robert bristow-johnson Nov 18 '19 at 17:14
• You win. I changed the explanation. In image processing, where some like to keep with integers, people dare to call a matrix of ones an averaging kernel. Same happens with lifting in wavelets, but yes again – Laurent Duval Nov 18 '19 at 17:52

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.

• i don't think i agree, Fat. the thing that makes the Fourier transform useless will be the presence of non-linear processing. since there is a one-to-one mapping between time and frequency domains, any transient analysis regarding LTI systems that can be done in the time domain can also be done in the frequency domain. – robert bristow-johnson Nov 18 '19 at 16:32
• I didn't want to say useless but not as useful. Indeed my fourth sentence states that it's possible to do it but harder to interpret. I thoughht ttah was enough. – Fat32 Nov 18 '19 at 17:58

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.