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>

<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.

==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> &nbsp; (|{{EquationRef|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).

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==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, \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} \left (t'-t,\omega'-\omega \right)C \left (t,\omega' \right )\,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{align}

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-\tfrac{\tau}{2})h(u+\tfrac{\tau}{2})e^{j\theta t+j\tau\omega-j\theta u}\, du\,d\tau\,d\theta \\

& = \dfrac{1}{4\pi^2}\iiint h^*(u-\tfrac{\tau}{2})h(u+\tfrac{\tau}{2})\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{align}</math>

If we only consider kernels for which <math>\phi(-\theta,\tau)\phi(\theta,\tau) = 1</math> holds then

:<math>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>

If we only consider kernels for which <math>\phi(-\theta,\tau)\phi(\theta,\tau) = 1</math> holds 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 display="block">C_{SP}(t,\omega) = \iint C_s(t',\omega')C_h(t'-t,\omega'-\omega)\,dt'\,d\omega'</math>

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

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

: <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>

This was shown by Janssen[4]. When <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>

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>

: <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>

==References==