On Dimensionality for Sparse Multipath Leif W. Hanlen, Roy Timo and Rasika Perera Australian National University and National ICT Australia Leif.Hanlen@nicta.com.au, {Roy.Timo,Rasika.Perera}@anu.edu.au Abstract— We give a 2WT style result for the degrees of freedom of multipath signals that pass through spatially lim- ited (sparse) scattering environments. The dimensionality scales with the circumference of the scattering region, and the total communications path length. We provide a direct comparison to the time-frequency case, where space replaces time. This is a rigorous wavefield examination of previous heuristic geometric arguments. I. I NTRODUCTION MIMO wireless systems famously require dense scattering (rich) environments [1–3] in order to achieve their spatial multiplexing of data streams. Intuitively, with a “sufficiently random” channel environment, and a “sufficiently well be- haved” group of antenna elements, a MIMO system perceives an independent channel for every transmit antenna. Much work has been devoted to the concept of “sufficiently dense.” Much of this falls into two arenas: 1) (Upper) bounds [4–6] on the number of spatial modes (dimensionality) caused by local effects, such as antenna array length, geometry and local scatter angular restric- tions and 2) (Lower) bounds [7, 8] on the mutual information be- tween (well separated) antenna arrays, due to scattering geometries in space. The work of [7] considered mutual information as a function of scattering radius at the transmitter and the receiver, the distance between the transmit and receive arrays, and the antenna beamwidths and spacing. The pin-hole effect is shown in [7] where spacial fading is uncorrelated and yet the channel has low rank. The mutual information curve is parameterised by [8]: η = 2πR/(λD) where R was the scatter-ring radius, D the distance between transmit and receive arrays and λ the wavelength. The intuition given by [7, 8] is that the scatterers act as a lens. Geometric arguments cannot bound the dimensionality of the received signal space due to scattering. However, most dimensionality results [5, 6, 9] which are not geometric in na- ture focus on local effects – such as Angle-of-arrival/departure statistics, antenna length. These works have assumed that scattering may be everywhere in the environment. A theoretically derived antenna saturation point is shown to exist in MIMO systems [9, 10] and mutual information increases linearly with the radius of the region containing the receiver antennas and independent of the number of anten- nas. Further work [4, 5, 11] has shown that spatially diverse wireless systems may be modelled using continuous spatial techniques, which focus on the continuous nature of space, rather than individual antenna elements. A new scattering model was recently introduced [12] using a continuous array model which considered the scatterer distribution as being clustered around several angular intervals. For this paper we shall not address capacity per se, we will instead focus on dimensionality. Our reason is that capacity requires a valid noise model, whilst dimensionality allows estimation of the number of parallel channels over which data might be multiplexed in a noiseless environment. It is clear that capacity is a function of dimensionality. The remainder of this paper is arranged as follows: Section II reviews bandlimiting mechanics and dimensionality results. Section III provides dimensionality bounds for the transmis- sion of signals via limited scattering regions. Section IV gives a simple example of the bandlimited nature of received signals. Conclusions are in section V. All proofs are in the appendix. II. THEORY Consider transmitting via an over-the-horizon radio system. An example might be sending UHF TV over non-line-of-sight channels 1 . We ask “how is the dimensionality of our signal increased by using scattering?” If we transmit through a cloud, how rich is the environment? How many independent channels can we use? We examine the spatially-limited scattering region depicted in Figure 1, and assume all transmission to be via the “cloud” of scattering objects (eg. microwaves through rain shadow). The transmitter creates a symbol (a current distribution) along a closed interval I X of a curve in space. The transmitter is constrained to act only within this particular region. I.e. the symbol is zero outside I X . The curve on which I X lies is assumed to be an analytic manifold M X . Physically, we are ignoring edge effects in our manifold, which removes many of the more esoteric problems associated with dimensionality [6]. We assume the transmit symbol is defined by a real function J (r X ) for r X ∈ M X and is of bounded spatial energy ‖J (r X )‖ 2 =  r X ∈M X |J (r X )| 2 dr X =  r X ∈I X |J (r X )| 2 dr X < ∞ The symbol propagates from M X as an electromagnetic wave- field and is scattered by B. The incident wavefield E XB (·) 1 eg. Through bouncing off the reflective surface of the ionosphere. The ionospheric reflection/refraction occurs around the frequency range 3 KHz - 300 MHz.