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

The complex spectrogram of a signal f(t) is defined by ¿f(t)g*(t-to)exp[-iwot]dt ; it is, in fact, the Fourier transform of the product of the signal and the complex conjugated and shifted version of a so-called window function g(t). From the complex spectrogram the signal can be reconstructed uniquely. It is shown that the complex spectrogram is completely determined by its values on the points (to=mT, wo=nO), where OT= 2tt and m and n take all integer values. The lattice of points (mT,nO) is exactly the lattice suggested by Gabor as early as 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 general expansion ¿¿mn amng(t-mT)exp[inOt] of a signal into a discrete set of 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 amn can be determined easily.