In order to view a signal (taken to be a function of time) represented over both time and frequency axis,
time–frequency representation
A time–frequency representation (TFR) is a view of a signal (taken to be a function of time) represented over both time and frequency. Time–frequency analysis means analysis into the time–frequency domain provided by a TFR. This is achieved b ...
is used.
Spectrogram
A spectrogram is a visual representation of the spectrum of frequencies of a signal as it varies with time.
When applied to an audio signal, spectrograms are sometimes called sonographs, voiceprints, or voicegrams. When the data are represen ...
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 in time by multiplying it with translations of a window function
.
The definition of spectrogram is
:
,
where
denotes the
Gabor Transform The Gabor transform, named after Dennis Gabor, is a special case of the short-time Fourier transform. It is used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. The function to be transf ...
of
.
Based on the spectrogram, the generalized spectrogram is defined as:
:
,
where:
:
:
For
, it reduces to the classical spectrogram:
:
The feature of Generalized spectrogram is that the window sizes of
and
are different. Since the time-frequency resolution will be affected by the window size, if one choose a wide
and a narrow
(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
:
:where
;Time marginal condition
:The generalized spectrogram
satisfies the time marginal condition if and only if
,
:where
denotes the
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
;Frequency marginal condition
:The generalized spectrogram
satisfies the frequency marginal condition if and only if
,
:where
denotes the
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
;Conservation of energy
:The generalized spectrogram
satisfies the conservation of energy if and only if
.
;Reality analysis
:The generalized spectrogram
is real if and only if
for some
.
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, â
Two window spectrogram and their integrals" Advances and Applications, vol. 205, pp. 251–268, 2009.
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Time–frequency analysis