Generalized spectrogram

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

Definition

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 f in time by multiplying it with translations of a window function [math]\displaystyle{ w(t) }[/math].

The definition of spectrogram is

[math]\displaystyle{ S{P_{x,w}}(t,f) = {G_{x,w}}(t,f)G_{_{x,w}}^*(t,f)=|{G_{x,w}}(t,f)|^2 }[/math],

where [math]\displaystyle{ {G_{x,{w_1}}} }[/math] denotes the Gabor Transform of [math]\displaystyle{ x(t) }[/math].

Based on the spectrogram, the generalized spectrogram is defined as:

[math]\displaystyle{ S{P_{x,{w_1},{w_2}}}(t,f) = {G_{x,{w_1}}}(t,f)G_{_{x,{w_2}}}^*(t,f) }[/math],

where:

[math]\displaystyle{ {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]

[math]\displaystyle{ {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 } }[/math]

For [math]\displaystyle{ w_1(t) = w_2(t)=w(t) }[/math], it reduces to the classical spectrogram:

[math]\displaystyle{ 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]\displaystyle{ w_1(t) }[/math] and [math]\displaystyle{ w_2(t) }[/math] are different. Since the time-frequency resolution will be affected by the window size, if one choose a wide [math]\displaystyle{ w_1(t) }[/math] and a narrow [math]\displaystyle{ 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

[math]\displaystyle{ \mathcal{SP}_{w_1,w_2}(t,f)(x,w) = Wig (w_1', w_2')*Wig (t,f)(x, w), }[/math]

where [math]\displaystyle{ w_1'(s):=w_1(-s), w_2'(s):=w_2(-s) }[/math]

Time marginal condition

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

where [math]\displaystyle{ \delta }[/math] denotes the Dirac delta function

Frequency marginal condition

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

where [math]\displaystyle{ \delta }[/math] denotes the Dirac delta function

Conservation of energy

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

Reality analysis

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

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.