The uncertainty principles of random signals related to the linear canonical transform

In this paper, we investigate uncertainty principles for random signals associated with the linear canonical transform (LCT). First, the LCT of random signals is formulated on the probability space. Based on this representation, the Heisenberg uncertainty principle is established to characterize the relationship between the expectations in the time and frequency domains. Furthermore, the Donoho-Stark uncertainty principle, developed from a measure theoretic perspective, reveals that a random signal cannot be simultaneously concentrated on arbitrarily small sets in both the time and frequency domains. The bounds obtained in these two uncertainty principles explicitly depend on the LCT parameters, indicating that the LCT offers greater flexibility than the Fourier transform (FT). The corresponding results in the fractional Fourier transform and FT domains are also given as special cases.