Space-Time-Frequency Degrees of Freedom: Fundamental Limits for Spatial Information Leif W. Hanlen † and Thushara D. Abhayapala † National ICT Australia and Australian National University {Leif.Hanlen,Thushara.Abhayapala}@nicta.com.au Abstract— We bound the number of electromagnetic signals which may be observed over a frequency range [F - W, F + W ] a time interval [0,T ] within a sphere of radius R. We show that the such constrained signals may be represented by a series expansion whose terms are bounded exponentially to zero beyond a threshold. Our result implies there is a finite amount of information which may be extracted from a region of space via electromagnetic radiation. I. I NTRODUCTION Wireless communication is fundamentally limited by the physics of the medium. Electromagnetic wave propagation has been given [1,2] as a motivation for developing such limits: information is ultimately carried on electromagnetic waves. Narrowband degrees of freedom (dimensionality) results have been given for dense multipath [3–5] and subsequently ex- tended to sparse systems. Here, the signal bandwidth is neg- ligible: dimensionality results are defined in wavelengths. Narrow-band wavefields were shown to have limited con- centrations [6]. The limit was based upon the free-space Helmholtz (wave) equation – a time independent variation of the electromagnetic wave [7]. Such waves may be represented by a functional series, whose terms are bounded exponentially toward zero beyond some limit. This limit was used to describe a random MIMO channel in dense [8] multipath and provide capacity results. More recent work – including wide-band MIMO motivates analysis of the capability of spatially diverse signals to support multiplexing over significant bandwidths. Given a region, bounded by radius R, centre- frequency F , bandwidth 2W and observation time T , what is the number D of wireless (electromag- netic) signals which may be observed? In [9] an approximate dimensionality result was given. This bound was excessively complex — resulting in a loose over- bound. In this work we provide exponential error bounds — reflecting [3], and provide a tighter bound on the dimension- ality of 3D-spatial broadband signals. The remainder of this paper is arranged as follows: We pro- vide a truncation point and bound the error for electromagnetic signals in space in Section II. This gives our main result in Theorem 1. Section III gives plots of the degrees of freedom, † L. W. Hanlen and T. D. Abhayapala also hold appointments with the Re- search School of Information Sciences and Engineering, ANU. National ICT Australia is funded through the Australian Government’s Backing Australia’s Ability initiative, in part through the Australian Research Council. while Section IV provides an to MIMO mutual information. We draw conclusions in Section V. Proofs are in the Appendix. II. DIMENSIONALITY Existing dimensionality results for signals 3D space, with non-trivial bandwidth are limited by D WT =2WT +1 (1) D space =  eπRF c  +1  2 (2) where (1) is from [10], formalised in [11] and (2) is from [12]. Fundamentally, we seek to develop a result which combines both (1) and (2): broadband, spatially diverse signals. Source-free (propagating) electric fields Ψ(r,t), are solu- tions of the free-space Maxwell wave equation [7]:  △− 1 c 2 ∂ 2 ∂t 2  · Ψ(r,t)=0 (3) where △ =(∂ 2 /∂x 2 ,∂ 2 /∂y 2 ,∂ 2 /∂z 2 ) and c =3 × 10 8 ms -1 is the speed of light. The vector r =(r, θ, φ) with 0 ≤ θ<π, 0 ≤ φ< 2π denotes position. We now formally pose: Problem 1: Given a function in space-time x(r,t) which is non-zero for |r|≤ R and t ∈ [0,T ] and has a frequency component in [F − W, F + W ] and satisfies (3); what number D of signals ϕ(r,t) are required to parameterize x(r,t)? Observe that we may represent any signal which is con- strained to [0,T ] × [0,R] and satisfying (3) by [13]: Ψ k (r,t) = exp (−ι|k|ct − ιk · r) (4) where k is the vector wave-number and ι = √ −1. The magnitude |k| =2πf/c is scalar wave-number. We may then express any radio signal as (an infinite) series expansion (24). The series itself is not important: simply that it exists, and we may truncate the series at some point. This series may be truncated at a point N (r,t; f ) which is an increasing function of frequency, time and space. Our first step is to define the truncation point at which the majority of the signal energy is captured – this will then provide us with our degrees-of-freedom result. ISIT2007, Nice, France, June 24 – June 29, 2007 1-4244-1429-6/07/$25.00 c 2007 IEEE 701