Discrete Fourier-Invariant Signals: Design and Application

In this paper, two methods for the design of discrete Fourier-invariant signals are proposed. The direct design method provides splitting between independent and dependent signal parts and calculation of the dependent part for any given independent part. The iterative design method generates a family of discrete Fourier-invariant signals by a successive approach. Further we show how the proposed direct design method can be combined with the Gabor uncertainty principle to generate discrete Fourier-invariant signals with the minimum product of their bandwidth (B) and their time-width (T). We show that these signals as well as signal families generated with the iterative design method achieve the theoretical lower BT bound. Also, it is shown that the BT product of discrete Hermite-Gauss signals converges to the theoretical lower bound. Finally, possible applications are illustrated in the case of time-frequency spectral analysis using the obtained discrete Fourier-invariant signals as the window that provides isoresolution.