On the rank of the discrete Gabor transform matrix

By using the discrete Gabor transform or expansion, the time domain sequences are mapped into joint time-frequency domain matrices or vice versa. In many applications, it is more effective to process signals, i.e., two-dimensional matrices, in the joint time-frequency domain than in the time or frequency domain alone. From the mathematical point of view, the processing of the discrete Gabor coefficients is no more than matrix computation. So it is beneficial to understand the properties of the Gabor coefficient matrix. In this letter, we shall investigate the rank of the Gabor coefficient matrix of a one-dimensional time domain signal, which is one of the most important matrix properties.