Generalized spectrogram: Difference between revisions

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In order to view a signal (taken to be a function of time) represented over both time and frequency axis, [[Time–frequencytime–frequency representation]] is used. [[Spectrogram]] is one of the most popular time-frequency representation, and '''Generalizedgeneralized Spectrogramspectrogram''', also called "Twotwo-window Spectrogramspectrogram", is the generalized application of spectrogram.

==Definition==

The definitondefinition 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>.<br>

<br>

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_transformGabor transform|Gabor Transform]] of <math>x(t)</math>.<br>

<br>

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><br>

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>

<br>

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

For <math>S{P_{x,w}}w_1(t,f) = {G_{x,w}}w_2(t,f)G_{_{x,w}}^*(t,f)=|{G_{x,w}}(t,f)|^2</math>, it reduces to the classical spectrogram:

:<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.

==Properties==

* ;Relation with [[Wigner distribution function|Wigner Distribution]]

:<math>\mathcal{SP}_{w_1,w_2}(ft,gf)(x,w) = Wig (w_1', w_2')*Wig (ft,gf)(x, w),</math>

:where <math>w_1'(s):=w_1(-s), w_2'(s):=w_2(-s)</math>

;Time marginal condition

:The generalized spectrogram <math>\mathcal{SP}_{w_1,w_2}(t,f)(x,w)</math> satisfies the time marginal condition if and only if <math>w_1w_2' = \delta</math>,

:where <math>\delta</math> denotes the [[Dirac delta function]]

;Frequency marginal condition

:The generalized spectrogram <math>\mathcal{SP}_{w_1,w_2}(t,f)(x,w)</math> satisfies the frequency marginal condition if and only if <math>w_1w_2' = \delta</math>,

:where <math>\delta</math> denotes the [[Dirac delta function]]

;Conservation of energy

:The generalized spectrogram <math>\mathcal{SP}_{w_1,w_2}(t,f)(x,w)</math> satisfies the conservation of energy if and only if <math>(w_1,w_2) = 1</math>.

;Reality analysis

:The generalized spectrogram <math>\mathcal{SP}_{w_1,w_2}(t,f)(x,w)</math> is real if and only if <math>w_1=C w_2</math> for some <math>C\in \R</math>.

==References==

{{refbegin}}

* [http://djj.ee.ntu.edu.tw/TFW.htm 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, “[httphttps://scholar.google.com.tw/scholar_url?url=http://nozdr.ru/data/media/biblioteka/kolxo3/M_Mathematics/MC_Calculus/MCf_Functional%2520analysis/Schulze%2520B.W.,%2520Wong%2520M.W.%2520(eds.)%2520Pseudo-differential%2520operators..%2520Complex%2520analysis%2520and%2520partial%2520differential%2520equations%2520(Birkhauser,%25202010)(ISBN%25203034601972)(O)(294s)_MCf_.pdf%23page%3D254&hl=zh-TW&sa=X&scisig=AAGBfm3k6US20Q9vL3As15cbbJLAFBdsDA&nossl=1&oi=scholarr&ei=gKW8VLroC5Ht8gWvr4DICw&ved=0CBoQgAMoADAA Two window spectrogram and their integrals,]," Advances and Applications, vol. 205, pp. 251-268&nbsp;251–268, 2009.

{{refend}}

[[Category:Time–frequency analysis]]