Transformation between distributions in time–frequency analysis: Difference between revisions

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In the field of [[time–frequency analysis]], several signal formulations are used to represent the signal in a joint time–frequency ___domain.<ref>L. Cohen, "Time–Frequency Analysis," ''Prentice-Hall'', New York, 1995. {{ISBN |978-0135945322}}</ref>.

There are several methods and transforms called "time-frequency distributions" (TFDs), whose interconnections were organized by Leon Cohen.<ref>[[Leon Cohen|L. Cohen]], "Generalized phase-space distribution functions," ''JourJ. Math. Phys.'', vol.'''7,''' (1966) pp.&nbsp;781–786, 1966[https://dx.doi.org/10.1063/1.1931206 doi:10.1063/1.1931206] </ref>

In the field of [[time–frequency analysis]], several signal formulations are used to represent the signal in a joint time–frequency ___domain.<ref>L. Cohen, "Time–Frequency Analysis," ''Prentice-Hall'', New York, 1995. ISBN 978-0135945322</ref>.

<ref>L. Cohen, "Quantization Problem and Variational Principle in the Phase Space Formulation of Quantum Mechanics," ''JourJ. Math. Phys.'', vol.'''7,''' pp.&nbsp;1863–1866, 1976.</ref><ref name="Janssen">A. J. E. M. Janssen, "On the locus and spread of pseudo-density functions in the time frequency plane," ''Philips Journal of Research'', vol. 37, pp.&nbsp;79–110, 1982.</ref><ref>E. Sejdić, I. Djurović, J. Jiang, “Time-frequency feature representation using energy concentration: An overview of recent advances,” ''Digital Signal Processing'', vol. 19, no. 1, pp. 153-183, January 2009.</ref>

The most useful and popular methods form a class referred to as "quadratic" or [[bilinear time–frequency distribution]]s. A core member of this class is the [[Wigner–Ville distribution]] (WVD), as all other TFDs can be written as a smoothed or convolved versions 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 also 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".<ref>B. Boashash, “Theory of Quadratic TFDs”, Chapter 3, pp. 59–82, in B. Boashash, editor, Time-Frequency Signal Analysis & Processing: A Comprehensive Reference, Elsevier, Oxford, 2003; {{ISBN|0-08-044335-4}}.</ref>

The scope of this article is to illustrate some elements of the procedure to transform one distribution into another. The method used to transform a distribution is borrowed from the [[phase space formulation]] of [[quantum mechanics]], even though the subject matter of this article is "signal processing". Noting that a signal can be recovered from a particular distribution under certain conditions, given a certain TFD ''ρ''₁(''t'',''f'') representing the signal in a joint time–frequency ___domain, another, different, TFD ''ρ''₂(''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 in Cohen's book.

There are several methods and transforms called "time-frequency distributions" (TFDs), whose interconnections were organized by Leon Cohen.<ref>[[Leon Cohen|L. Cohen]], "Generalized phase-space distribution functions," ''Jour. Math. Phys.'', vol.7, pp.&nbsp;781–786, 1966.</ref>

<ref>L. Cohen, "Quantization Problem and Variational Principle in the Phase Space Formulation of Quantum Mechanics," ''Jour. Math. Phys.'', vol.7, pp.&nbsp;1863–1866, 1976.</ref><ref>A. J. E. M. Janssen, "On the locus and spread of pseudo-density functions in the time frequency plane," ''Philips Journal of Research'', vol. 37, pp.&nbsp;79–110, 1982.</ref>

The most useful and popular methods form a class referred to as "quadratic" or [[bilinear time–frequency distribution]]s. A core member of this class is the [[Wigner–Ville distribution]] (WVD), as all other TFDs can be written as a smoothed or convolved versions 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 also 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 illustrate some elements of the procedure to transform one distribution into another. The method used to transform a distribution is borrowed from the [[phase space formulation]] of [[quantum mechanics]], even though the subject matter of this article is "signal processing". Noting that a signal can recovered from a particular distribution under certain conditions, given a certain TFD ''ρ''₁(''t,f'') representing the signal in a joint time–frequency ___domain, another, different, TFD ''ρ''₂(''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 in Cohen's book.

==General class==

If we use the variable {{math|1=''ω'' = 2''πf''}}, then, borrowing the notations used in the field of quantum mechanics, we can show that time–frequency representation, such as [[Wigner distribution function]] (WDF) and other [[bilinear time–frequency distribution]]s, can be expressed as

: {{NumBlk||<math display="block">C(t,\omega) = \dfrac{1}{4\pi^2}\iiint s^*\left(u-\dfrac{1}{2}\tau\right) s\left(u+\dfrac{1}{2}\tau\right)\phi(\theta,\tau)e^{-j\theta t-j\tau\omega+j\theta u}\, du\,d\tau\,d\theta ,</math> (|{{EquationRef|1)}}}}

: <math>C(t,\omega) = \dfrac{1}{4\pi^2}\iiint s^*\left(u-\dfrac{1}{2}\tau\right)s\left(u+\dfrac{1}{2}\tau\right)\phi(\theta,\tau)e^{-j\theta t-j\tau\omega+j\theta u}\, du\,d\tau\,d\theta ,</math> (1)

where <math>\phi(\theta,\tau)</math> 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 theThe kernel of the [[Wigner distribution function]] (WDF) is one. However, it is no particular significance should be attached to that, since it is possible 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.

==Characteristic function formulation==

The characteristic function is the double [[Fourier transform]] of the distribution. By inspection of Eq. ({{EquationNote|1}}), we can obtain that

: {{NumBlk||<math display="block">C(t,\omega) = \dfrac{1}{4\pi^2}\iint M(\theta,\tau)e^{-j\theta t-j\tau\omega}\, d\theta\,d\tau</math> (|{{EquationRef|2)}}}}

: <math>C(t,\omega) = \dfrac{1}{4\pi^2}\iint M(\theta,\tau)e^{-j\theta t-j\tau\omega}\, d\theta\,d\tau</math> (2)

where

: {{NumBlk||<math display="block">\begin{alignat}{2}

: <math>M_2 M(\phitheta,\tau) & = \phi_2phi(\theta,\tau)\int s^*\left(u-\dfrac{1}{2}\tau\right)s\left(u+\dfrac{1}{2}\tau\right)e^{j\theta u}\, du</math> (5)\\

: <math>\begin{alignat}{2}

M(\theta,\tau) & = \phi(\theta,\tau)\int s^*\leftA(u-\dfrac{1}{2}theta,\tau\right)s\left(u+\dfrac{1}{2}\tau\right)e^{j\theta u}\,du \\

\end{alignat}</math> | ({{EquationRef|3)}}}}

& = \phi(\theta,\tau)A(\theta,\tau) \\

\end{alignat}</math> (3)

and where <math>A(\theta,\tau)</math> is the symmetrical ambiguity function. The characteristic function may be appropriately called the generalized ambiguity function.

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To obtain that relationship suppose that there are two distributions, <math>C_1</math> and <math>C_2</math>, with corresponding kernels, <math>\phi_1</math> and <math>\phi_2</math>. Their characteristic functions are

{{NumBlk||<math display="block">M_1(\phi,\tau) = \phi_1(\theta,\tau)\int s^*\left(u-\tfrac{\tau}{2}\right)s\left(u+\tfrac{\tau}{2}\right)e^{j\theta u}\, du</math> | {{EquationRef|4}}}}

: {{NumBlk||<math display="block">M_1M_2(\phi,\tau) = \phi_1phi_2(\theta,\tau)\int s^*\left(u-\dfractfrac{1\tau}{2}\tau\right)s\left(u+\dfractfrac{1\tau}{2}\tau\right)e^{j\theta u}\, du</math> | (4){{EquationRef|5}}}}

: <math>M_2(\phi,\tau) = \phi_2(\theta,\tau)\int s^*\left(u-\dfrac{1}{2}\tau\right)s\left(u+\dfrac{1}{2}\tau\right)e^{j\theta u}\, du</math> (5)

Divide one equation by the other to obtain

: {{NumBlk||<math display="block">M_1(\phi,\tau) = \dfrac{\phi_1(\theta,\tau)}{\phi_2(\theta,\tau)}M_2(\phi,\tau)</math> (|{{EquationRef|6)}}}}

: <math>M_1(\phi,\tau) = \dfrac{\phi_1(\theta,\tau)}{\phi_2(\theta,\tau)}M_2(\phi,\tau)</math> (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 transform]] of both sides and use Eq. ({{EquationNote|2}})

: {{NumBlk||<math display="block">C_1(t,\omega) = \dfrac{1}{4\pi^2}\iint \dfrac{\phi_1(\theta,\tau)}{\phi_2(\theta,\tau)}M_2(\theta,\tau)e^{-j\theta t-j\tau\omega}\, d\theta\,d\tau</math> (|{{EquationRef|7)}}}}

: <math>C_1(t,\omega) = \dfrac{1}{4\pi^2}\iint \dfrac{\phi_1(\theta,\tau)}{\phi_2(\theta,\tau)}M_2(\theta,\tau)e^{-j\theta t-j\tau\omega}\, d\theta\,d\tau</math> (7)

Now express <math>M_2</math> in terms of <math>C_2</math> to obtain

: {{NumBlk||<math display="block">C_1(t,\omega) = \dfrac{1}{4\pi^2}\iiiint \dfrac{\phi_1(\theta,\tau)}{\phi_2(\theta,\tau)}C_2(t,\omega^')e^{j\theta(t^'-t)+j\tau(\omega^'-\omega)}\, d\theta\,d\tau\,dt^'\,d\omega^'</math> (|{{EquationRef|8)}}}}

: <math>C_1(t,\omega) = \dfrac{1}{4\pi^2}\iiiint \dfrac{\phi_1(\theta,\tau)}{\phi_2(\theta,\tau)}C_2(t,\omega^')e^{j\theta(t^'-t)+j\tau(\omega^'-\omega)}\, d\theta\,d\tau\,dt^'\,d\omega^'</math> (8)

This relationship can be written as

: {{NumBlk||<math display="block">C_1(t,\omega) = \iint g_{12}(t^'-t,\omega'-\omega)C_2(t,\omega')\,dt^'\,d\omega'</math> (|{{EquationRef|9)}}}}

: <math>C_1(t,\omega) = \iint g_{12}(t^'-t,\omega'-\omega)C_2(t,\omega')\,dt^'\,d\omega'</math> (9)

with

: {{NumBlk||<math display="block">g_{12}(t,\omega) = \dfrac{1}{4\pi^2}\iint \dfrac{\phi_1(\theta,\tau)}{\phi_2(\theta,\tau)}e^{j\theta t+j\tau\omega}\, d\theta\, d\tau</math> (|{{EquationRef|10)}}}}

: <math>g_{12}(t,\omega) = \dfrac{1}{4\pi^2}\iint \dfrac{\phi_1(\theta,\tau)}{\phi_2(\theta,\tau)}e^{j\theta t+j\tau\omega}\, d\theta\, d\tau</math> (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. ({{EquationNote|9}}), both <math>C_1</math> to be the spectrogram and <math>C_2</math> to be arbitrary are set. In addition, to simplify notation, <math>\phi_{SP} = \phi_1</math>, <math>\phi = \phi_2</math>, and <math>g_{SP} = g_{12}</math> are set and written as

: {{NumBlk||<math display="block">C_{SP}(t,\omega) = \iint g_{SP} \left (t^'-t,\omega^'-\omega \right)C \left (t,\omega^' \right )\,dt^'\,d\omega^'</math> (|{{EquationRef|11)}}}}

: <math>C_{SP}(t,\omega) = \iint g_{SP}(t^'-t,\omega^'-\omega)C(t,\omega^')\,dt^'\,d\omega^'</math> (11)

The kernel for the spectrogram with window, <math>h(t)</math>, is <math>A_h(-\theta,\tau)</math> and therefore

: <math display="block">\begin{alignat}{3align}

g_{SP}(t,\omega) & = \dfrac{1}{4\pi^2}\iint \dfrac{A_h(-\theta,\tau)}{\phi(\theta,\tau)}e^{j\theta t+j\tau\omega}\, d\theta\,d\tau \\

& = \dfrac{1}{4\pi^2}\iiint \dfrac{1}{\phi(\theta,\tau)}h^*(u-\dfractfrac{1\tau}{2}\tau)h(u+\dfractfrac{1\tau}{2}\tau)e^{j\theta t+j\tau\omega-j\theta u}\, du\,d\tau\,d\theta \\

& = \dfrac{1}{4\pi^2}\iiint h^*(u-\dfractfrac{1\tau}{2}\tau)h(u+\dfractfrac{1\tau}{2}\tau)\dfrac{\phi(\theta,\tau)}{\phi(\theta,\tau)\phi(-\theta,\tau)}e^{-j\theta t+j\tau\omega+j\theta u}\, du\,d\tau\,d\theta \\

\end{alignatalign}</math> (12)

If takingwe theonly consider kernels for which <math>\phi(-\theta,\tau)\phi(\theta,\tau) = 1</math>, <math>g_{SP}(t,\omega)</math>holds is just the distribution of the window function, except that it is evaluated at <math>-\omega</math>. Therefore,then

<math display="block">g_{SP}(t,\omega) = \dfrac{1}{4\pi^2}\iiint h^*(u-\tfrac{\tau}{2})h(u+\tfrac{\tau}{2}) \phi(\theta,\tau) e^{-j\theta t+j\tau\omega+j\theta u}\, du\,d\tau\,d\theta = C_h(t,-\omega)</math>

and therefore

: <math>g_{SP}(t,\omega) = C_h(t,-\omega)</math> (13)

: <math display="block">C_{SP}(t,\omega) = \iint C_s(t^',\omega^')C_h(t^'-t,\omega^'-\omega)\,dt^'\,d\omega^'</math> (14)

for kernels that satisfy <math>\phi(-\theta,\tau)\phi(\theta,\tau) = 1</math>

and

: <math>C_{SP}(t,\omega) = \iint C_s(t^',\omega^')C_h(t^'-t,\omega^'-\omega)\,dt^'\,d\omega^'</math> (14)

for kernels that satisfy <math>\phi(-\theta,\tau)\phi(\theta,\tau) = 1</math>

This was shown by Janssen[4]. For the case where <math>\phi(-\theta,\tau)\phi(\theta,\tau)</math> does not equal one, then

: <math>C_{SP}(t,\omega) = \iiiint G(t^{''},\omega^{''})C_s(t^',\omega^')C_h(t^{''}+t^'-t,-\omega^{''}+\omega-\omega^')\,dt^'\,dt^{''}\,d\omega^\,d\omega^{''}</math> (15)

This was shown by Janssen[4].<ref Forname="Janssen"/> the case whereWhen <math>\phi(-\theta,\tau)\phi(\theta,\tau)</math> does not equal one, then

: <math display="block">C_{SP}(t,\omega) = \iiiint G(t^{''},\omega^{''})C_s(t^',\omega^')C_h(t^{''}+t^'-t,-\omega^{''}+\omega-\omega^')\,dt^'\,dt^{''}\,d\omega^'\,d\omega^{''}</math> (15)

where

: <math display="block">G(t,\omega) = \dfrac{1}{4\pi^2}\iint \dfrac{e^{-j\theta t-j\tau\omega}}{\phi(\theta,\tau)\phi(-\theta,\tau)}\, d\theta\,d\tau</math> (16)

: <math>G(t,\omega) = \dfrac{1}{4\pi^2}\iint \dfrac{e^{-j\theta t-j\tau\omega}}{\phi(\theta,\tau)\phi(-\theta,\tau)}\, d\theta\,d\tau</math> (16)

==References==