Generalized spectrogram: Difference between revisions

Revision as of 05:20, 7 June 2021

Revision as of 05:21, 7 June 2021

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==Properties==

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

:<math>\mathcal{SP}_{w_1,w_2}(t,f)(x,w) = Wig (w_1', w_2')*Wig (t,f)(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>.