Example 1 These time and frequency localizations can be evaluated thanks to the M-files loctime.m and locfreq.m of the Toolbox. The first one gives the average time center ($t_m$) and the duration ($T$) of a signal, and the second one the average normalized frequency ($\nu_m$) and the normalized bandwidth ($B$). For example, for a linear chirp with a gaussian amplitude modulation, we obtain (see fig. 2.1): >> sig=fmlin(256).*amgauss(256); >> [tm,T]=loctime(sig) ---> tm=128 T=32 >> [num,B]=locfreq(sig) ---> num=0.249 B=0.0701 Figure 2.1: Linear chirp with a gaussian amplitude modulation One interesting property of this product $T\times B$ is that it is lower bounded : $$T \times B \geq 1.$$ This constraint, known as the Heisenberg-Gabor inequality, illustrates the fact that a signal can not have simultaneously an arbitrarily small support in time and in frequency. This property is a consequence of the definition of the Fourier transform. The lower bound $T\times B = 1$ is reached for gaussian functions : $$x(t) = C \exp{[-\alpha(t - t_m)^2 + j2\pi \nu_m(t-t_m)]}$$ with $C \in {\cal R}$, $\alpha \in {\cal R}_{+}$. Therefore, the gaussian signals are those which minimize the time-bandwidth product according to the Heisenberg-Gabor inequality.
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