Generalized spectrogram explained

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

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 in time by multiplying it with translations of a window function

w(t)

.

The definition of spectrogram is

S{Px,w

}(t,f) = (t,f)G_^*(t,f)=|(t,f)|^2,where
{G
x,{w1
}} denotes the Gabor Transform of

x(t)

.

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

S{P
x,{w1

,{w2}}}(t,f)=

{G
x,{w1
}}(t,f)G_^*(t,f),where:
{G
x,{w1
}}\left(\right) = \int_^\infty
{G
x,{w2
}}\left(\right) = \int_^\infty

For

w1(t)=w2(t)=w(t)

, it reduces to the classical spectrogram:

S{Px,w

}(t,f) = (t,f)G_^*(t,f)=|(t,f)|^2The feature of Generalized spectrogram is that the window sizes of

w1(t)

and

w2(t)

are different. Since the time-frequency resolution will be affected by the window size, if one choose a wide

w1(t)

and a narrow

w1(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.

Properties

Relation with Wigner Distribution
l{SP}
w1,w2

(t,f)(x,w)=Wig(w1',w2')*Wig(t,f)(x,w),

where

w1'(s):=w1(-s),w2'(s):=w2(-s)

Time marginal condition
  • The generalized spectrogram
    l{SP}
    w1,w2

    (t,f)(x,w)

    satisfies the time marginal condition if and only if

    w1w2'=\delta

    ,

    where

    \delta

    denotes the Dirac delta function
    Frequency marginal condition
  • The generalized spectrogram
    l{SP}
    w1,w2

    (t,f)(x,w)

    satisfies the frequency marginal condition if and only if

    w1w2'=\delta

    ,

    where

    \delta

    denotes the Dirac delta function
    Conservation of energy
  • The generalized spectrogram
    l{SP}
    w1,w2

    (t,f)(x,w)

    satisfies the conservation of energy if and only if

    (w1,w2)=1

    .
    Reality analysis
  • The generalized spectrogram
    l{SP}
    w1,w2

    (t,f)(x,w)

    is real if and only if

    w1=Cw2

    for some

    C\in\R

    .

    References

    This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Generalized spectrogram".

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