Degrees of freedom of band limited signals measured over space

Abstract

In this paper, we provide the degrees of freedom of a band limited signal observed over a finite spatial window at a given time instant. The limit is based on Claude E. Shannon's sampling theorem which is also known as Whittaker-Nyquist-Kotelnikov-Shannon sampling theorem. The result is derived by using a series expansion of the solution to the Helmholtz wave equation. The expanded terms of wave equation are then bounded exponentially to zero beyond a threshold. The derived result is more general compared to the existing ones and explores the effects of spatial information on the degrees of freedom of the signal observed. In addition, we derive a new expression to calculate the degrees of freedom of a signal observed over band limited and spatially constrained channel at a constant time in terms of wavelengths and fractional bandwidth. We show that by increasing the size of the three dimensional spatial region, the degrees of freedom of the observable signal over the region can be increased for a given signal bandwidth.