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Inversion and Normalization of Time-Frequency Transform |
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PP: 67S-74S |
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Author(s) |
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Lintao Liu,
Houtse Hsu,
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Abstract |
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| This study defines a time-frequency transform (TFT) covering Gabor transform (GT) and
wavelet transform (WT) and presents two inversion formulas for the TFT. By one of the inversion
formulas, we find a new inversion formula for the WT, which suggests a new more general wavelet
definition free from the conventional admissibility (e.g. zero integration). This study also develops a
concept of normal TFT for precise time-frequency analysis, as we show that the normal TFT is
unique in unbiased explication of the immediate frequency, amplitude and phase of a time harmonic.
The GT should be updated (i.e. normalized) in phase to become a normal TFT, because it does not
explicate the immediate but initial phase of a time harmonic. The WT should be written in L1-norm
to become a normal TFT, as the non-L1-norm WT of a time harmonic yields a frequency-biased
spectrum. Finally, we show how to measure the frequency resolution of a normal TFT. |
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