Novel Scheme for Spatial Extrapolation of Multipath Michael I. Y. Williams, Glenn Dickins, Rodney A. Kennedy, Tony S. Pollock and Thushara D. Abhayapala Abstract—It has been shown that the effects of multipath prop- agation in a mobile wireless communications system can be miti- gated if the receiver can make predictions about the multipath fad- ing. In this paper, we introduce a novel scheme for extrapolating multipath fields outwards in space, given field observations within a limited region. Whereas previous work has concentrated on sim- ple multipath propagation with a finite number of plane wave scat- terers, we use a less restrictive continuous model of scattering. The extrapolation scheme is based on a information-weighted modal expansion of the field, where modes containing too little informa- tion are penalized to minimize the extrapolation error. The per- formance of this scheme is shown to be far better than pessimistic error bounds derived in previous work. I. I NTRODUCTION Multipath fading is a major factor limiting the performance of wireless communications systems. Although progress has been made in exploiting the properties of multipath fields us- ing spatial diversity of antennas [1], there is still considerable interest in multipath prediction schemes, particularly for mo- bile applications. These schemes allow a mobile receiver to extrapolate a multipath field outwards in space or time, allow- ing improved power control schemes to be negotiated with the transmitter based on future fading. In this paper, we propose a novel spatial extrapolation scheme. Traditional approaches to the extrapolation problem have been based on simple models of multipath fading where a small number of plane wave scatterers are used to represent multiple paths [2]. In [3] and [4], extrapolation schemes were developed which recover this plane wave model using direction-of-arrival (DOA) algorithms. Other approaches to extrapolation have in- volved building adaptive auto-regressive models of the field [5] [6]. The problem with these schemes is that these simplistic scat- tering models cannot represent more complex multipath fields, such as fields generated from continuous scattering distribu- tions [7] [8]. This means that these simpler schemes may give overly optimistic estimates of performance. In [9], Teal et al. take a different approach based on a more complex physical model of wave propagation. Taking a fairly pessimistic approach, they demonstrate that wave equation con- straints cause a worst case extrapolation error which grows rapidly with distance, and conclude that extrapolation beyond a wavelength is not practically feasible. In this paper we use a similar physical model of multipath scattering, and show that the limitations imposed by the wave The authors are with National ICT Australia (NICTA) Limited, and the De- partment of Information Engineering, Research School of Information Sciences and Engineering, The Australian National University, Canberra ACT 0200, Australia. email: firstname.lastname@anu.edu.au National ICT Australia is funded through the Australian Government’s Back- ing Australia’s Ability initiative, in part through the Australian Research Coun- cil. effective farfield distribution scattering represented by F (φ) small ring field observed on field extrpolated to φ r R r S f (x) some larger ring transmitter x s Fig. 1. The Geometry of the Multipath Extrapolation Problem equation can actually be used to improve extrapolation perfor- mance. We introduce an improved scheme for multipath extrap- olation based on a modal expansion, which allows us to repre- sent multipath fields using an infinite set of modal coefficients. Given basic statistics of the signal and noise, we reconstruct the entire field from these modes, penalizing ‘noisy’ coefficients which we expect to contain little information about the field. This modal technique generalizes similar work on the problem of bandlimited extrapolation [10]. We show that the expected error of our extrapolation scheme is far lower than the worst case error predicted by [9]. In section II, we introduce a physical model of multipath scattering, and show that the resulting wavefields can be repre- sented by an infinite modal expansion. In section III, we exam- ine the effect of observing this field on a ring of radius r S in the presence of spatially white noise. Importantly, the modal ex- pansion coefficients can only be recovered approximately from the noisy field. In section IV, we demonstrate a scheme for op- timally recombining the recovered modal coefficients to mini- mize the extrapolation error. The optimal combination involves imposing an exponential penalty on modes containing too little information about the multipath field. II. MULTIPATH FIELDS Consider a narrowband multipath field, f (x), in two- dimensional space. As this field is a valid wavefield, it must be a solution to the Helmholtz equation [11], ∇ 2 f (x)+ k 2 f (x)=0, (1) where k  2π/λ is the wavenumber, ∇ 2 is Laplacian operator, and λ is the wavelength.