Generalized spectrogram

Generalized Spectrogram, also called "Two-window Spectrogram", is the generalized application of spectrogram.

Definition

$S{P_{x,{w_{1}},{w_{2}}}}(t,f)={G_{x,{w_{1}}}}(t,f)G_{_{x,{w_{2}}}}^{*}(t,f)$,
where
${G_{x,{w_{1}}}}\left({t,f}\right)=\int _{-\infty }^{\infty }{{w_{1}}\left({t-\tau }\right)x\left(\tau \right)\,{e^{-j2\pi \,f\,\tau }}d\tau }$
${G_{x,{w_{2}}}}\left({t,f}\right)=\int _{-\infty }^{\infty }{{w_{2}}\left({t-\tau }\right)x\left(\tau \right)\,{e^{-j2\pi \,f\,\tau }}d\tau }$

Here, ${G_{x,{w_{1}}}}$ denotes the Gabor Transform of $x(t)$.
For $w_{1}(t)=w_{2}(t)=w(t)$, it reduces to the classical spectrogram:
$S{P_{x,w}}(t,f)={G_{x,w}}(t,f)G_{_{x,w}}^{*}(t,f)=|{G_{x,w}}(t,f)|^{2}$ The feature of Generalized spectrogram is that the window sizes of $w_{1}(t)$ and $w_{2}(t)$ are different. Since the time-frequency resolution will be affected by the window size, if one choose a wide $w_{1}(t)$ and a narrow $w_{1}(t)$ (or the opposite), the resolutions of them will be high in different part of spectrogram. After the multiplication of these two Gabor transform, the resolutions of both time and frequency axis will be enhanced.

References

• Class notes of Time frequency analysis and wavelet transform -- from Prof. Jian-Jiun Ding's course website
• P. Boggiatto, G. De Donno, and A. Oliaro, “Two window spectrogram and their integrals," Advances and Applications, vol. 205, pp. 251-268, 2009.