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Next: Pyramidal (multiresolution) true-rectangular TF Up: Multiple Mother Chirplets: The Previous: Thomson's method of spectral

True rectangular tiling of the TF plane

While many researchers depict certain tilings of the TF plane (such as given by the STFT), schematically, using rectangular gridsgabor1946, and even refer to them as rectangular tilings, it is important to note that the actual shape of the individual tiles is better described as a tesselation of overlapping ``blobs'', perhaps Gaussian, as was the case with the Gaussian-windowed STFT.

However, the same family of discrete prolate spheroidal sequences (DPSS) used in the Thomson method synthesizes a concentration of energy in the TF plane where the energy is uniformly distributed throughout one small rectangular region, and minimized elsewheregif.

Observing this fact (others have also observed this factshenoyweyl), we now extend the Thomson method to operate in the TF plane. In practice, we calculate a discrete version from the discrete-time signal, simply by partitioning the signal into short segments and applying the Thomson method to each segment. This amounts to a sliding-window spectral estimate where the entire family of windows slides together. As in (12), however, we may write the proposed time-frequency distribution, pointwise. That is, to calculate the energy within a rectangle centered at tex2html_wrap_inline1481, we sum over the set of windows that have all been moved to the point tex2html_wrap_inline1481:

S(t_c,f_c) = _i | C_t_c,f_c g_i \: |. s |^2  

Steve Mann
Thu Jan 8 19:50:27 EST 1998