Definition In order to introduce time-dependency in the Fourier transform, a simple and intuitive solution consists in pre-windowing the signal $x(u)$ around a particular time $t$, calculating its Fourier transform, and doing that for each time instant $t$. The resulting transform, called the short-time Fourier transform (STFT, or short-time spectrum), is $$F_x(t,\nu;h) = \int_{-\infty}^{+\infty} x(u)\ h^*(u-t)\ e^{-j2\pi \nu u}\ du$$ where $h(t)$ is a short time analysis window (see fig. 3.1) localized around $t=0$ and $\nu=0$. Figure 3.1: non-stationary signal $x(u)$ and the short-time window $h^*(u-t)$ centered at time $t$ Because multiplication by the relatively short window $h^*(u-t)$ effectively suppresses the signal outside a neighborhood around the analysis time point $u=t$, the STFT is a "local" spectrum of the signal $x(u)$ around $t$. Provided that the short-time window is of finite energy, the STFT is invertible according to $$x(t) = \frac{1}{E_h}\ \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} F_x(u,\xi;h)\ h(t-u)\ e^{j2\pi t \xi}\ du\ d\xi,$$ with $E_h=\int_{-\infty}^{+\infty} \vert h(t)\vert^2\ dt$. This relation expresses that the total signal can be decomposed as a weighted sum of elementary waveforms $$h_{t,\nu}(u) = h(u-t)\ \exp{[j2\pi \nu u]}$$ which can be interpreted as ``building blocks" or ``atoms". Each atom is obtained from the window $h(t)$ by a translation in time and a translation in frequency (modulation). The corresponding transformation group of translations in both time and frequency is called the Weyl-Heisenberg group. Fig. 3.2 shows two such atoms corresponding to a gaussian window. Figure 3.2: Time-frequency atoms: two atoms corresponding to a gaussian window. The STFT is a projection of the analyzed signal on such atoms which are relatively well localized in time and in frequency The STFT may also be expressed in terms of signal and window spectra: $$F_x(t,\nu;h)=\int_{-\infty}^{+\infty} X(\xi)\ H^*(\xi-\nu)\ \exp{[j\ 2\pi(\xi-\nu)t]}\ d\xi$$ where $X$ and $H$ are respectively the Fourier transforms of $x$ and $h$. Thus, the STFT $F_x(t,\nu;h)$ can be considered as the result of passing the signal $x(u)$ through a band-pass filter whose frequency response is $H^*(\xi-\nu)$, and is therefore deduced from a mother filter $H(\xi)$ by a translation of $\nu$. So the STFT is similar to a bank of band-pass filters with constant bandwidth.
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