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Optimal Solution for Higher Order Time-Frequency Uncertainty
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DOI：10.1016/j.sigpro.2026.110944.png)
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Abstract
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The time–frequency uncertainty principle plays a fundamental role in the design and analysis of signal processing and communication systems, as it governs the joint localization of signals in time and frequency. Different multicarrier waveforms employ different prototype filters, leading to varying spectral and localization characteristics. In this paper, the classical time–frequency uncertainty principle is generalized to higher-order moments, allowing a more comprehensive characterization of time–frequency localization. The minimization of the higher-order uncertainty product is reduced to a single eigenvalue problem in the Gauss–Hermite basis. For the classical case m = 1 , the optimal function is the Gaussian and is obtained in closed form. For m ≥ 2, the optimal functions and the corresponding lower bounds are obtained semi-analytically, by solving the eigenvalue problem numerically after truncation in the Hermite basis. Furthermore, a discrete-time formulation of the higher-order uncertainty principle is developed, and its optimizer is connected to samples of the continuous-time optimal function as the sampling rate increases. Simulation results are presented to compare the optimal solutions associated with different orders of time–frequency moments, highlighting their localization behavior and validating the theoretical analysis.
Keywords:
Higher-order moments
Uncertainty
Time–frequency localization
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