Generalized spectrogram: Difference between revisions

Revision as of 18:18, 18 April 2020

Revision as of 05:20, 7 June 2021

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The definition of the spectrogram relies on the Gabor transform (also called short-time Fourier transform, for short STFT), whose idea is to localize a signal {{math|''f''}} in time by multiplying it with translations of a window function <math>w(t)</math>.

The definition of spectrogram is<br>

:<math>S{P_{x,w}}(t,f) = {G_{x,w}}(t,f)G_{_{x,w}}^*(t,f)=|{G_{x,w}}(t,f)|^2</math>,<br>

where <math>{G_{x,{w_1}}}</math> denotes the [[Gabor transform|Gabor Transform]] of <math>x(t)</math>.

Based on the spectrogram, the '''generalized spectrogram''' is defined as<br>:

:<math>S{P_{x,{w_1},{w_2}}}(t,f) = {G_{x,{w_1}}}(t,f)G_{_{x,{w_2}}}^*(t,f)</math>,<br>

where:

where <math>{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 }</math>,<br>

and :<math>{G_{x,{w_2w_1}}}\left( {t,f} \right) = \int_{ - \infty }^\infty {{w_2w_1}\left( {t - \tau } \right)x\left( \tau \right)\,{e^{ - j2\pi \,f\,\tau }}d\tau }</math>

where :<math>{G_{x,{w_1w_2}}}\left( {t,f} \right) = \int_{ - \infty }^\infty {{w_1w_2}\left( {t - \tau } \right)x\left( \tau \right)\,{e^{ - j2\pi \,f\,\tau }}d\tau }</math>,<br>

For <math>w_1(t) = w_2(t)=w(t)</math>, it reduces to the classical spectrogram:<br>

:<math>S{P_{x,w}}(t,f) = {G_{x,w}}(t,f)G_{_{x,w}}^*(t,f)=|{G_{x,w}}(t,f)|^2</math>

The feature of Generalized spectrogram is that the window sizes of <math>w_1(t)</math> and <math>w_2(t)</math> are different. Since the time-frequency resolution will be affected by the window size, if one choose a wide <math>w_1(t)</math> and a narrow <math>w_1(t)</math> (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.