Reassignment method: Difference between revisions

Revision as of 12:45, 10 April 2009

Revision as of 16:04, 23 November 2009

Line 140:

transform. The coefficients in this decomposition are defined

<center><math>\begin{matrixalign}

\epsilon( t, \omega )

&= \int x(\tau) h( t - \tau ) e^{ -j \omega \left[ \tau - t \right]} d\tau \\

Line 146:

&= e^{ j \omega t} X(t, \omega) \\

&= X_{t}(\omega) = M_{t}(\omega) e^{j \phi_{\tau}(\omega)}

\end{matrixalign}</math></center>

where <math>M_{t}(\omega)</math> is the magnitude, and

Line 156:

<math>x(t)</math> can be reconstructed from the moving window coefficients by

<center><math>\begin{matrixalign}

x(t) & = \iint X_{\tau}(\omega) h^{*}_{\omega}(\tau - t) d\omega d\tau \\

& = \iint X_{\tau}(\omega) h( \tau - t ) e^{ -j \omega \left[ \tau - t \right]} d\omega d\tau \\

&= \iint M_{\tau}(\omega) e^{j \phi_{\tau}(\omega)} h( \tau - t ) e^{ -j \omega \left[ \tau - t \right]} d\omega d\tau \\

&= \iint M_{\tau}(\omega) h( \tau - t ) e^{ j \left[ \phi_{\tau}(\omega) - \omega \tau+ \omega t \right] } d\omega d\tau

\end{matrixalign}</math></center>

For signals having magnitude spectra,

Line 177:

or equivalently, around the point <math>\hat{t}, \hat{\omega}</math> defined by

<center><math>\begin{matrixalign}

\hat{t}(\tau, \omega) & = \tau - \frac{\partial \phi_{\tau}(\omega)}{\partial \omega} =

- \frac{\partial \phi(\tau, \omega)}{\partial \omega} \\

\hat{\omega}(\tau, \omega) & = \frac{\partial \phi_{\tau}(\omega)}{\partial \tau} =

\omega + \frac{\partial \phi(\tau, \omega)}{\partial \tau} .

\end{matrixalign}</math></center>

This phenomenon is known in such fields as optics as the

Line 227:

only makes sense when there is energy to attribute, so the

method of reassignment has no meaning at points where the

spectrogram is zero-valued.

== Efficient Computation of Reassigned Times and Frequencies ==