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== Efficient computation of reassigned times and frequencies ==
In digital signal processing, it is most common to sample the time and frequency domains. The discrete Fourier transform is used to compute samples <math>X(k)</math> of the Fourier transform from samples <math>x(n)</math> of a time ___domain signal. The reassignment operations proposed by Kodera et al. cannot be applied directly to the discrete short-time Fourier transform data, because partial derivatives cannot be computed directly on data that is discrete in time and frequency, and it has been suggested that this difficulty has been the primary barrier to wider use of the method of reassignment.
It is possible to approximate the partial derivatives using finite differences. For example, the phase spectrum can be evaluated at two nearby times, and the partial derivative with respect to time be approximated as the difference between the two values divided by the time difference, as in
\frac{\partial \phi(t, \omega)}{\partial t} & \approx \frac{1}{\Delta t} \left[ \phi \left (t + \frac{\Delta t}{2}, \omega \right ) - \phi \left (t - \frac{\Delta t}{2}, \omega \right ) \right] \\
▲\end{matrix}</math></center>
For sufficiently small values of <math>\Delta t</math> and <math>\Delta \omega,</math> and provided that the phase difference is appropriately "unwrapped", this finite-difference method yields good approximations to the partial derivatives of phase, because in regions of the spectrum in which the evolution of the phase is dominated by rotation due to sinusoidal oscillation of a single, nearby component, the phase is a linear function.
Independently of Kodera ''et al.'', Nelson arrived at a similar method for improving the time-frequency precision of short-time spectral data from partial derivatives of the short-time phase
spectrum.<ref name = "crossspectral">{{cite journal |author=D. J. Nelson |date=Nov 2001 |title=Cross-spectral methods for processing speech |journal=Journal of the Acoustical Society of America |volume=110 |issue=5 |pages=2575–2592 |publisher= |doi=10.1121/1.1402616 |url= |accessdate= }}</ref> It is easily shown that Nelson's ''cross spectral surfaces'' compute an approximation of the derivatives that is equivalent to the finite differences method.▼
<ref name = "crossspectral">▼
▲{{cite journal |author=D. J. Nelson |date=Nov 2001 |title=Cross-spectral methods for processing speech |journal=Journal of the Acoustical Society of America |volume=110 |issue=5 |pages=2575–2592 |publisher= |doi=10.1121/1.1402616 |url= |accessdate= }}
Auger and Flandrin showed that the method of reassignment, proposed in the context of the spectrogram by Kodera et al., could be extended to any member of [[Cohen's class]] of time-frequency representations by generalizing the reassignment operations to
▲ \frac{\iint \nu \cdot W_{x}(t-\tau,\omega -\nu) \cdot \Phi(\tau,\nu) d\tau d\nu}
▲\end{matrix}</math></center>
where <math>W_{x}(t,\omega)</math> is the Wigner–Ville distribution of <math>x(t)</math>, and <math>\Phi(t,\omega)</math> is the kernel function that defines the distribution. They further described an efficient method for computing the times and frequencies for the reassigned spectrogram efficiently and accurately without explicitly computing the partial derivatives of
In the case of the spectrogram, the reassignment operations can be computed by
\hat{t} (t,\omega) & = t - \Re \
\hat{\omega}(t,\omega) & = \omega + \Im \
▲\hat{\omega}(t,\omega) & = \omega + \Im \Bigg\{ \frac{ X_{\mathcal{D}h}(t,\omega) \cdot X^*(t,\omega) }
▲\end{matrix}</math></center>
where <math>X(t,\omega)</math> is the short-time Fourier transform computed using an analysis window <math>h(t), X_{\mathcal{T}h}(t,\omega)</math> is the short-time Fourier transform computed using a time-weighted analysis window <math>h_{\mathcal{T}}(t) = t \cdot h(t)</math> and <math>X_{\mathcal{D}h}(t,\omega)</math> is the short-time Fourier transform computed using a time-derivative analysis window <math>h_{\mathcal{D}}(t) = \tfrac{d}{dt}h(t)</math>.
Using the auxiliary window functions <math>h_{\mathcal{T}}(t)</math> and <math>h_{\mathcal{D}}(t)</math>, the reassignment operations can be computed at any time-frequency coordinate
<math>t,\omega</math> from an algebraic combination of three Fourier transforms evaluated at <math>t,\omega</math>. Since these algorithms operate only on short-time spectral data evaluated at a single time and frequency, and do not explicitly compute any derivatives, this gives an efficient method of computing the reassigned discrete short-time Fourier transform.
One constraint in this method of computation is that the <math>| X(t,\omega) |^2</math> must be non-zero. This is not much of a restriction, since the reassignment operation itself implies that there is some energy to reassign, and has no meaning when the distribution is zero-valued.
==Separability==
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