Inversion and normalization of time-frequency transform

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.