Reassignment method: Difference between revisions

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"cross-terms" that sometimes make other time-frequency
representations difficult to interpret. But the windowing
operation required in spectrogram computation introduces an unsavory tradeoff between time resolution and frequency
unsavory tradeoff between time resolution and frequency
resolution, so spectrograms provide a time-frequency
representation that is blurred in time, in frequency, or in
both dimensions. The method of time-frequency reassignment
is a technique for refocussing time-frequency data in a blurred representation like the spectrogram by mapping the
blurred representation like the spectrogram by mapping the
data to time-frequency coordinates that are nearer to the
true region of support of the analyzed signal.
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where the true Wigner–Ville distribution shows no energy.
 
The spectrogram is a member of Cohen's class. It is a smoothed Wigner–Ville distribution with the smoothing kernel
smoothed Wigner–Ville distribution with the smoothing kernel
equal to the Wigner–Ville distribution of the analysis
window. The method of reassignment smoothessmooths the Wigner–Ville
distribution, but then refocuses the distribution back to
the true regions of support of the signal components. The
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distribution of energy in the analyzed signal.
 
In the classical moving window method, a time-___domain
signal, <math>x(t)</math> is decomposed into a set of
coefficients, <math>\epsilon( t, \omega )</math>, based on a set of elementary signals, <math>h_{\omega}(t)</math>,
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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
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
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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
efficient method for computing the times and frequencies for
the reassigned spectrogram efficiently and accurately
without explicitly computing the partial derivatives of
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transform computed using an analysis window
<math>h(t)</math>, <math>X_{\mathcal{T}h}(t,\omega)</math>
is the short-time Fourier transform computed using a time-weighted anlaysis window <math>h_{\mathcal{T}}(t) = t
time-weighted anlaysis window <math>h_{\mathcal{T}}(t) = t
\cdot h(t)</math> and
<math>X_{\mathcal{D}h}(t,\omega)</math> is the short-time
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The short-time Fourier transform can often be used to
estimate the amplitudes and phases of the individual
components in a ''multi-component'' signal, such as a quasi-harmonic musical instrument tone. Moreover, the time
quasi-harmonic musical instrument tone. Moreover, the time
and frequency reassignment operations can be used to sharpen
the representation by attributing the spectral energy