1 On Landaus Eigenvalue Theorem for Line-of-Sight MIMO Channels

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On Landau’s Eigenvalue Theorem for

Line-of-Sight MIMO Channels

Andrea Pizzo Member, IEEE, Angel Lozano, Fellow, IEEE

Abstract—An alternative derivation is provided for the

degrees of freedom (DOF) formula on line-of-sight (LOS)

channels via Landau’s eigenvalue theorem for bandlimited

signals. Compared to other approaches, Landau’s theorem

provides a general framework to compute the DOF in

arbitrary environments, this framework is herein specialized

to LOS propagation. The development shows how the

spatially bandlimited nature of the channel relates to its

geometry under the paraxial approximation that applies to

most LOS settings of interest.

Index Terms— Degrees of freedom, line-of-sight MIMO,

paraxial approximation, Landau’s eigenvalue theorem.

I. INTRODUCTION

The number of distinct waveforms able to transport

information via electromagnetic waves is an inherent

property of a physical channel. It is upper bounded by

the number of degrees of freedom (DOF), a quantity

of interest in information theory [1]–[3], optics [4]–[8],

electromagnetism [9]–[11], and signal processing [12],

[13]. Given the continuous nature of channels, waveforms

span an inﬁnite-dimensional space, yet the noise allows

for a certain error in the representation [10]. Channels are

thus amenable to a discrete representation over a space of

approximately DOF dimensions [14].

There are various ways to compute the number

of DOF in a wireless channel, say by leveraging

diffraction theory [4]–[6], by studying the eigenvalues

of the Green’s operator [3], [7]–[9], or by pursuing a

signal-space approach [1], [13]. This paper provides an

alternative derivation via Landau’s eigenvalue theorem

for multidimensional bandlimited signals (or ﬁelds)

[15]. Analogously to time-domain waveforms of ﬁnite

bandwidth, an electromagnetic channel may be regarded

as spatially bandlimited due to a low-pass ﬁltering

behavior of the propagation [1], [9], [13]. In this

analogy, time is replaced by space and frequency by

spatial-frequency (or wavenumber) [14].

Originally devised for waveform channels [16],

Landau’s theorem has been generalized to electromagnetic

A. Pizzo and A. Lozano are with Univ. Pompeu Fabra

(email: {andrea.pizzo, angel.lozano}@upf.edu). Work supported by

the European Research Council under the H2020 Framework

Programme/ERC grant agreement 694974, by the ICREA Academia

program, by the European Union-NextGenerationEU, and by the

Fractus-UPF Chair on Tech Transfer and 6G.

e(r)

transmit

array j(s)

Fig. 1. LOS communications between continuous arrays.

propagation [15], and lately applied to non-line-of-sight

(NLOS) channels [13]. Prompted by the interest in LOS

multiple-input multiple-output (MIMO) communication

at high frequencies [17], here the connection is drawn

with such channels under the paraxial approximation

that holds when the propagation is focused about the

axis connecting the two arrays [18]. The development

builds on signal theory concepts, without relying on

unconventional mathematics. A bridge between LOS and

NLOS propagation is also uncovered, with implications

for MIMO communication and Nyquist reconstruction at

high frequencies.

Notation: Fnis the n-dimensional Fourier operator,

(Fnh)(f) = RRnh(t)e−j2πfTtdt=g(f), whereas F−1

is its inverse, (F−1

ng)(t) = h(t), with the shorthand

notation F1=Fand F−1

1=F−1. In turn, (Rh)(t) =

R(t)h(t)with R(t)the indicator function of a set

R⊂Rnwhile RAis the set obtained by applying any

invertible linear transform Ato the axes of R, and m(·)

is the Lebesgue measure.

II. PLANE-WAVE REPRESENTATION

OF LOS CHANNELS

Consider two n-dimensional continuous-space arrays

(n= 1 or 2) communicating with scalar electromagnetic

waves at wavelength λin a 3D free-space environment.

We denote by Dthe distance between the array centroids.

Capitalizing on that an arbitrary source can always be

replicated by a ﬂat source on a given plane thanks to

Huygen’s principle [19], we let S⊂Rnand R⊂Rnbe

arXiv:2210.05631v1 [cs.IT] 11 Oct 2022

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1OnLandau'sEigenvalueTheoremforLine-of-SightMIMOChannelsAndreaPizzoMember,IEEE,AngelLozano,Fellow,IEEEAbstractAnalternativederivationisprovidedforthedegreesoffreedom(DOF)formulaonline-of-sight(LOS)channelsviaLandau'seigenvaluetheoremforbandlimitedsignals.Comparedtootherapproaches,Landau'stheorempro...

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1 On Landaus Eigenvalue Theorem for Line-of-Sight MIMO Channels

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