Uncertainty Principle for the Quaternion Linear Canonical Transform in Terms of Covariance

An uncertainty principle (UP), which offers information about a signal and its Fourier transform in the time-frequency plane, is particularly powerful in mathematics, physics and signal processing community. Under the polar coordinate form of quaternion-valued signals, the UP of the two-sided quaternion linear canonical transform (QLCT) is strengthened in terms of covariance. The condition giving rise to the equal relation of the derived result is obtained as well. The novel UP with covariance can be regarded as one in a tighter form related to the QLCT. It states that the product of spreads of a quaternion-valued signal in the spatial domain and the QLCT domain is bounded by a larger lower bound.