# Why is window flipped in Short-time Fourier transform?

Using the formulation of the short-time Fourier transform where the signal slides behind the fixed window. When looking at it from the perspective of the "filtering view", I don't understand why the window needs to be flipped. Here's my slides from class showing the flipped window (2nd slide):

• The filter-bank point of view is that $y_f(n) = X(n,f) = x \ast w_f(n)$ where $w_f(n) = w(n) e^{2i \pi f n}$ is some band-pass filter around the frequency $f$, while the short-time Fourier transform point of view is that $Y_n(f) = X(n,f) = FT[x(.) w(n-.)]$ where $w(n-.)$ isolates 20ms of the signal around the $n$th time sample – reuns Nov 7 '16 at 3:36
• In the first case you look at one row of the STFT, in the other case at one column – reuns Nov 7 '16 at 3:38
• But what I'm wondering is from the filter bank perspective why does the second slide flip the window to be w[-n] rather than w[n]] is it required/more useful to do that for some reason? – Austin Nov 7 '16 at 3:56
• Because usually the window is not symmetric and has its support on $[0,N-1]$, for example the Hanning window $\tilde{w}(n) = (1-\cos(\pi n / N)) 1_{n \in [0,N-1]}$ in your notation and $w(n) = (1-\cos(\pi n / N)) 1_{n \in [-N+1,0]}$ in mine – reuns Nov 7 '16 at 4:08
• And note how from the filtering point of view, the filter isn't causal with such a window (the most convenient when dealing with a numerical signal already recorded) – reuns Nov 7 '16 at 4:09