Sampling theorem for the complex spectrogram, and Gabor's expansion in Gaussian elementary signals

The complex spectrogram of a signal is defined as the Fourier transform of the product of the signal and the shifted and complex conjugated version of a so-called window function; it is thus a function of time and frequency, simultaneously, from which the signal can be reconstructed uniquely. It is shown that the complex spectrogram is completely determined by its values on the points of a certain time-frequency lattice. This lattice is exactly the one suggested by Gabor in 1946; it arose in connection with Gabor's suggestion to expand a signal into a discrete set of Gaussian elementary signals. Such an expansion is a special case of the more general expansion of a signal into a discrete set of properly shifted and modulated window functions. It is shown that this expansion exists. Furthermore, a set of functions is constructed, which is bi-orthonormal to the set of shifted and modulated window functions. With the help of this bi-orthonormal set of functions, the expansion coefficients can be determined easily