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 elsewhere
.
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 
, we sum over the set of windows that have all
been moved to the point :
S(t_c,f_c) = _i | C_t_c,f_c g_i \: |. s |^2