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

Revision as of 00:26, 11 January 2013

Revision as of 18:00, 17 September 2013

Line 3:

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><ref>B. Boashash, “Heuristic Formulation of Time-Frequency Distributions”, Chapter 2, pp. 29–58, in B. Boashash, editor, Time-Frequency Signal Analysis and Processing: A Comprehensive Reference, Elsevier Science, Oxford, 2003; ISBN 008044335.</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 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.

Line 11:

==General class==

If we use the variable ''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

: <math>C(t,\omega) = \dfrac{1}{4\pi^2}\iiint s^*(u-\dfrac{1}{2}\tau)s(u+\dfrac{1}{2}\tau)\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 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.

Line 24:

: <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

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

Line 41:

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

Divide one equation by the other to obtain

: <math>M_1(\phi,\tau) = \dfrac{\phi_1(\theta,\tau)}{\phi_2(\theta,\tau)}M_2(\phi,\tau)</math> (6)

Line 59:

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

with

: <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)

Line 81:

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

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

and

Line 100:

{{Reflist}}

{{DEFAULTSORT:Transformation between distributions in time-frequency analysis}}

[[Category:Time–frequency analysis]]