In the Clifford algebra setting of Euclidean spaces, monogenic signals are naturally defined as the boundary limit functions of the associated monogenic functions in the related domain. In an earlier paper, we defined a scalar-valued phase derivative as a candidate of instantaneous frequency of a multivariate signal. In this paper, we obtain fundamental relations between such defined phase derivative and the Fourier frequency. The results generalize the latest results of quadrature phase derivative in one-dimensional to multi-dimensional cases in the Clifford algebra setting. We also prove two uncertainty principles in higher dimensions of which one is for scalar-valued signals and the other is for vector-valued signals with the axial form.