Coupled smoothing If we look at the Moyal's formula (property 9. see page ), it is easy to express the spectrogram as a smoothing of the WVD: $$S_x(t,\nu)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} W_h(s-t,\xi-\nu)\ W_x(s,\xi)\ ds\ d\xi.$$ (4.7) Thus, the spectrogram is the element of the Cohen's class for which $\Pi(s,\xi)$ is the WVD of the window $h$. This new formulation provides us with another interpretation of the embarrassing trade-off between the time and frequency- resolutions of the spectrogram: if we choose a short window $h$, the smoothing function will be narrow in time and wide in frequency, leading to a good time resolution but bad frequency resolution; and vice-versa.
© GdR ISIS - Contact