Transformation between distributions in time-frequency analysis
Encyclopedia
In the field of time–frequency analysis, the goal is to define signal formulations that are used for representing the signal in a joint time–frequency domain (see also time–frequency representations). There are several methods and transforms called "time-frequency distributions" (TFDs). The most useful and used methods form a class referred to as "quadratic" or bilinear time–frequency distributions. A core member of this class is the Wigner–Ville distribution (WVD), as all other TFDs can be written as a smoothed version of the WVD. Another popular member of this class is the spectrogram
which is the square of the magnitude of the short-time Fourier transform
(STFT). The spectrogram has the advantage of being positive and is easy to interpret, but has disadvantages like being irreversible which means that once the spectrogram of a signal is computed, the original signal can't be extracted from the spectrogram. The theory and methodology for defining a TFD that verifies certain desirable properties is given in the "Theory of Quadratic TFDs" .
The scope of this article is to outline some elements of the procedure to transform one distribution into another. The method used to transform a distribution is borrowed from quantum mechanics
, even though the subject matter of the article is "signal processing". Noting that a signal can recovered from a particular distribution under certain conditions, given a certain TFD ρ1(t,f) representing the signal in a joint time–frequency domain, another different TFD ρ2(t,f) of the same signal can be obtained to calculate any other distribution, by simple smoothing or filtering; some of these relationships are shown below. A full treatment of the question can be given from a signal processing perspective..
(WDF) and other bilinear time–frequency distributions, can be expressed as
where
is a two dimensional function called the kernel, which determines the distribution and its properties (for a signal processing terminology and treatment of this question, the reader is referred to the references already cited in the introduction).
For the kernel of the Wigner distribution function
(WDF) is one. However, it is no particular significance should be attached to that since it is to write the general form so that the kernel of any distribution is one, in which case the kernel of the Wigner distribution function
(WDF) would be something else.
of the distribution. By inspection of Eq. (1), we can obtain that
where
Spectrogram
A spectrogram is a time-varying spectral representation that shows how the spectral density of a signal varies with time. Also known as spectral waterfalls, sonograms, voiceprints, or voicegrams, spectrograms are used to identify phonetic sounds, to analyse the cries of animals; they were also...
which is the square of the magnitude of the short-time Fourier transform
Short-time Fourier transform
The short-time Fourier transform , or alternatively short-term Fourier transform, is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time....
(STFT). The spectrogram has the advantage of being positive and is easy to interpret, but has disadvantages like being irreversible which means that once the spectrogram of a signal is computed, the original signal can't be extracted from the spectrogram. The theory and methodology for defining a TFD that verifies certain desirable properties is given in the "Theory of Quadratic TFDs" .
The scope of this article is to outline some elements of the procedure to transform one distribution into another. The method used to transform a distribution is borrowed from quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, even though the subject matter of the article is "signal processing". Noting that a signal can recovered from a particular distribution under certain conditions, given a certain TFD ρ1(t,f) representing the signal in a joint time–frequency domain, another different TFD ρ2(t,f) of the same signal can be obtained to calculate any other distribution, by simple smoothing or filtering; some of these relationships are shown below. A full treatment of the question can be given from a signal processing perspective..
General class
If we use the variable ω=2πf, then, borrowing the notations used in the field of quantum mechanics, we can show that time–frequency representation, such as Wigner distribution functionWigner distribution function
The Wigner distribution function was first proposed to account for quantum corrections to classical statistical mechanics in 1932 by Eugene Wigner, cf. Wigner quasi-probability distribution....
(WDF) and other bilinear time–frequency distributions, can be expressed as
-
(1)
where
is a two dimensional function called the kernel, which determines the distribution and its properties (for a signal processing terminology and treatment of this question, the reader is referred to the references already cited in the introduction).For the kernel of the Wigner distribution function
Wigner distribution function
The Wigner distribution function was first proposed to account for quantum corrections to classical statistical mechanics in 1932 by Eugene Wigner, cf. Wigner quasi-probability distribution....
(WDF) is one. However, it is no particular significance should be attached to that since it is to write the general form so that the kernel of any distribution is one, in which case the kernel of the Wigner distribution function
Wigner distribution function
The Wigner distribution function was first proposed to account for quantum corrections to classical statistical mechanics in 1932 by Eugene Wigner, cf. Wigner quasi-probability distribution....
(WDF) would be something else.
Characteristic function formulation
The characteristic function is the double Fourier transformFourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of the distribution. By inspection of Eq. (1), we can obtain that
-
(2)
where
-
(3)
and where
is the symmetrical ambiguity function. The characteristic function may be appropriately called the generalized ambiguity function.
Transformation between distributions
To obtain that relationship suppose that there are two distributions,
and 
, with corresponding kernels, 
and 
. Their characteristic functions are
-
(4)
-
(5)
Divide one equation by the other to obtain
-
(6)
This is an important relationship because it connects the characteristic functions. For the division to be proper the kernel cannot to be zero in a finite region.
To obtain the relationship between the distributions take the double Fourier transformFourier transformIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of both sides and use Eq. (2)
-
(7)
Now express
in terms of 
to obtain
-
(8)
This relationship can be written as
-
(9)
with
-
(10)
Relation of the spectrogram to other bilinear representations
Now we specialize to the case where one transform from an arbitrary representation to the spectrogram. In Eq. (9), both
to be the spectrogram and 
to be arbitrary are set. In addition, to simplify notation, 
, 
, and 
are set and written as
-
(11)
The kernel for the spectrogram with window,
, is 
and therefore
-
(12)
If taking the kernels for which
, 
is just the distribution of the window function, except that it is evaluated at 
. Therefore,
-
(13)
for kernels that satisfy
and
-
(14)
for kernels that satisfy
This was shown by Janssen[4]. For the case where
does not equal one, then
-
(15)
where
-
(16)
-
-