Is "Windowed Fourier Transform" a synonym for "STFT"?

Sometimes I find the notation WFT (Windowed Fourier Transform) while other times I see STFT (Short Time Fourier Transform). Are they the same?

Answer

In Chapter 2.4 Previous Work of The Short Time Fourier Transform and Local Signals, S. Okamura, 2011, one reads:

The STFT is also known under many names such as the windowed Fourier Transform, the Gabor transform, and the local Fourier transform.

Later, one discovers that the definition may slightly vary with different authors, but, at first glance, they are the same. Be careful though. For some persons, a "Windowed Fourier Transform" is a mere Fourier transform of a windowed signal. Because there is no sliding nor shift involved. For some, "short" implies "finite support".

Leaving theses versions aside, my personal interpretation and use is that a “Windowed Fourier Transform” is a special case of a slightly more generic “STFT” acceptation, for two (mild) reasons:

• "windowed" somehow conveys the idea that only one window is used. But you can easily use different windows for STFT, long ones for low frequency, shorter ones for high frequencies.



• the concept behind "what a window is" is not evident. However, for most people, a window often has a non-zero sum (or unit), and often is either increasing/decreasing (unimodal, flat being a trivial version) or monotonic (like exponential window or weighting) for causal implementations. For short-term, it only needs to be "short", or well-concentrated in time, and could be more complicated.