On dimensionality of multipath fields: Spatial extent and richness

Abstract

We establish that an arbitrary narrowband multipath field in any circular region in two dimensional space has an intrinsic functional dimensionality of (πe)R/λ ≈ 8.54 R/λ that scales only linearly with radius R/λ in wavelengths. This result implies there is no such thing as an arbitrarily complicated multipath field. That is, a field generated by any number of nearfield and farfield, specular and diffuse multipath reflections is no more complicated than a field generated by a limited number plane waves. As such, there are limits on how rich multipath can be. This result has significant implications including means: i) to determine a parsimonious parameterization for arbitrary multipath fields, ii) of synthesizing arbitrary multipath fields with arbitrarily located nearfield or farfield, spatially discrete or continuous sources. We give examples of multipath field analysis and synthesis.

On Finite Dimensional Approximation in MIMO

“…For a truncated representation, to (N + 1) 2 terms, we may perform the truncation bound developed in [12]. The truncated representation of an arbitrary field with (N + 1) 2 terms is given by,…”

“…where j n (k|r|) are the spherical Bessel functions of integer order n and Y m n ( r) ≡ Y m n (θ, φ) the spherical harmonics [10][11][12].…”

“…The motivation for this paper arises by noting that there is a wide gap between simple point-like MIMO models (vector channels) [9] and the continuous model used in formulation of the bounds in [2,3]. Clearly the continuous model should be an asymptotic limit to the point-like antenna model in the limit of large numbers of small elements.…”

“…We shall draw on the recent work [2,3], which has provided estimates of richness in multipath settings. This work approaches the problem of multipath richness from a continuous perspective.…”

“…This work approaches the problem of multipath richness from a continuous perspective. The bounds developed in [2,3] of spatial richness allow an infinite dimensional problem to be modelled rigourously -within arbitrary bounds of error -to be considered in terms of a much simpler finite dimensional approximation. This result has reduced problems of MIMO modelling [4,5], MIMO capacity [6,7] and direction of arrival estimation [8] to finite dimensional problems.…”