Band Limited Signals Observed Over Finite Spatial and Temporal Windows: An Upper Bound to Signal Degrees of Freedom

TL;DR

This paper investigates the fundamental limits on the degrees of freedom of signals in broadband multipath fields within finite observation regions, revealing a complex coupling between space, time, and frequency that affects signal capacity.

Contribution

It introduces a new characterization of degrees of freedom in multipath fields, accounting for spatial, temporal, and frequency coupling, extending classical models.

Findings

Effective observation time is independent of spatial modes.

Effective bandwidth varies with spatial mode and differs from the actual bandwidth.

Degrees of freedom depend on the time-bandwidth product extended across multiple spatial modes.

Abstract

The study of degrees of freedom of signals observed within spatially diverse broadband multipath fields is an area of ongoing investigation and has a wide range of applications, including characterising broadband MIMO and cooperative networks. However, a fundamental question arises: given a size limitation on the observation region, what is the upper bound on the degrees of freedom of signals observed within a broadband multipath field over a finite time window? In order to address this question, we characterize the multipath field as a sum of a finite number of orthogonal waveforms or spatial modes. We show that (i) the "effective observation time" is independent of spatial modes and different from actual observation time, (ii) in wideband transmission regimes, the "effective bandwidth" is spatial mode dependent and varies from the given frequency bandwidth. These findings clearly indicate the strong coupling between space and time as well as space and frequency in spatially diverse wideband multipath fields. As a result, signal degrees of freedom does not agree with the well-established degrees of freedom result as a product of spatial degrees of freedom and time-frequency degrees of freedom. Instead, analogous to Shannon's communication model where signals are encoded in only one spatial mode, the available signal degrees of freedom in spatially diverse wideband multipath fields is the time-bandwidth product result extended from one spatial mode to finite modes. We also show that the degrees of freedom is affected by the acceptable signal to noise ratio (SNR) in each spatial mode.