Fourier Transform vs Short Time Fourier Transform...

[fog37]

TL;DR Summary

Fourier Transform vs Short Time Fourier Transform: why no always use the STFT?

Hello,

I understand how the FT and the STFT work. The STFT provides time-frequency localization, i.e. it can tell us when the spectral components are acting in the time-domain signal...The STFT is also useful for non-stationary signals which are signals whose statistical characteristics are changing in time...

That said, why don't we always use the STFT given its extra benefits?

I guess that if a deterministic signal is stationary, it means that it looks "the same" over different intervals and the STFT will output identical results for each different window...

Thanks!

 

[osilmag]

To put it simply edge effects at the limit of each section of time makes the output inaccurate there. I think that is called the uncertainty principle.

 

[fog37]

To put it simply edge effects at the limit of each section of time makes the output inaccurate there. I think that is called the uncertainty principle.

Yes, the bigger is the window the lesser the edge effects. But regardless of these effects, the STFT at least gives time localization while in the case of the FT the spectral components are global and spread across the entire duration of the signal in the time domain....

 

[hutchphd]

Ya pays your money and you makes your choices. If you want an exact answer you need to use a complete set of functions. The full space FT is, in principle, exact.

 

[Baluncore]

It only takes a few hundred pixels to display a dynamic spectrogram. Frequency resolution is the reciprocal of the data block acquisition time. STFT is a compromise that provides both a time and a frequency distribution, in one dynamic display, with only the pixel resolution and computation required.

Maybe only 90% of the FFT processing time is needed for the STFT, but:
Frequency resolution is reduced in the STFT.
Phase information is ignored by the STFT.
Signal-to-noise ratio is greatly reduced in the STFT.

The STFT can be extended in time by power spectrum accumulation, PSA, to recover a broad signal channel, with discontinuous phase, from the noise.

 

[boneh3ad]

There are numerous trade-offs for one over the other. Edge effects are obviously one that was already cited, but also frequency resolution. By limiting the size of your window, you are sacrificing frequency resolution, which can be important at the lower end of your spectrum.

Another thing to consider is that, when calculating something like a power spectrum, it's not typical to use FFT on the whole signal anyway. Instead you segment it (much like the STFT) and apply a window function to reduce edge effects, then average all of the windowed segments together to reduce the variance of the estimate (Welch's method). You can't do that with a STFT.

In short, it's context-specific. You have to consider the trade-offs of each method against the needs of the analyst.