Dimensionality of Spatio-Temporal Broadband Signals Observed Over Finite Spatial and Temporal Windows

Dimensionality of spatio-temporal signals is fundamental to the performance limits of any wireless communications system with multiple antennas. In this paper, we use the spherical harmonic analysis of wave propagation to derive a closed-form expression for the dimensionality of band-limited wireless signals observed over a finite region of space within a finite time window. Specifically, we decompose spatio-temporal signals into independent and orthogonal spatial modes (waveforms) and apply Shannon's classical signal dimensionality result, namely, time-frequency bandwidth product to each spatial mode. Our analysis shows that: 1) the effective observation time of all the spatial modes is the sum of the finite time window and the time it takes a wave front to propagate across the entire spatial region and 2) though the effective bandwidth at lower spatial modes is equal to the bandwidth of the underlying transmitted signal, the effective bandwidth decreases as the mode index increases depending on the acceptable signal-to-noise ratio at each spatial mode. Thus, only a finite number of spatial modes (independent channels) carry information with its own time-frequency bandwidth product. These findings show that the classical signal dimensionality result of time-bandwidth product does not extend directly to the product of spatial degrees of freedom and time-bandwidth product. Simulation result indicates that increasing the radius of the region leads to a sub-quadratic growth in the derived signal dimensionality irrespective of the bandwidth or the observation time.