The Channel Between Randomly Oriented Dipoles:Statistics and Outage in the Near and Far Field

Gregor Dumphart and Armin WittnebenWireless Communications Group, D-ITET, ETH Zurich

Zurich, SwitzerlandEmail: {dumphart, wittneben}@nari.ee.ethz.ch

Abstract—We consider the class of wireless links whose prop-agation characteristics are described by a dipole model. Thiscomprises free-space links between dipole antennas and magneto-inductive links between coils, with important communication andpower transfer applications. A dipole model describes the channelcoefficient as a function of link distance and antenna orientations.In many use cases the orientations are random, causing a randomfading channel. This paper presents a closed-form description ofthe channel statistics and the resulting outage performance forthe case of i.i.d. uniformly distributed antenna orientations in 3Dspace. For reception in AWGN after active transmission, we showthat the high-SNR outage probability scales like pe ∝ SNR−1/2

in the near- or far-field region, i.e. the diversity exponent is just1/2 (even 1/4 with backscatter or load modulation). The diversityexponent improves to 1 in the near-far-field transition due topolarization diversity. Analogous statements are made for thepower transfer efficiency and outage capacity.

Index Terms—fading, outage, polarization diversity, dipoleantennas, loop antennas, coil misalignment, backscatter

I. INTRODUCTION

Wireless engineers often rely on statistical channel modelsto describe complicated propagation environments and theirfading characteristics. Fading occurs when (for a narrowbandchannel) the random channel coefficient h ∈ C is close tozero, i.e. h ≈ 0, which can cause a link outage [1]. Astatistical channel model allows to study the performanceimplications of fading analytically. For example, for a digitalmodulation scheme with transmit power P , Rayleigh fadingh ∼ CN (0, σ2) due to rich multipath propagation, and additivewhite Gaussian noise (AWGN), the bit error rate is asymptot-ically proportional to P−L with L = 1. The small diversityexponent L means that Rayleigh fading hinders reliable low-power communication [1, Cpt. 3]. The existing literature statesmany such performance results for various models, e.g. see [2].

Fading events h ≈ 0 can be caused by mechanismsother than multipath propagation or shadowing; they caneven occur in free space in the case of inopportune antennaorientations. For example, the transmitter-to-receiver (TX-to-RX) direction may coincide with a zero of the TX-antennaradiation pattern or the RX antenna may be misaligned withthe incident field [3]. To that effect, non-isotropic antennaswith random orientations can give rise to a fading channelwith random channel coefficient h, even in free space [4],[5]. Such random antenna orientations are to be expected forwireless applications with high mobility or application-specific

node locations. Associated fading channels have been studiedin [4] for mobile radio devices and in [6], [7] for magneticinduction links between randomly arranged coils.

In this paper we study fading due to random antenna orien-tations for the class of wireless links that can be adequatelymodeled by a free-space link from a TX dipole to a RX dipole.This comprises:• Magnetic induction links between two weakly coupled

coils (loop antennas).• Capacitive links between small electric dipole antennas.• Links between λ/2-length dipole antennas in free space.• Links from a magnetic dipole to a small loop or from an

electric dipole to a pair of terminals with small separation.Those have important applications in wireless power transferand data communication, either with an active TX or a passivetag (RFID load modulation or backscatter modulation). [8]

The statistics and communication-theoretic performance as-pects of this fading channel are, to the best of our knowledge,not covered by existing work. The need for an appropriatestatistical channel model was highlighted in [9] where, becauseof the lack of a better model, a Rayleigh fading model wasassumed for RFID links. Similarly, the heuristic assumptionof a Gaussian-distributed data rate was made in [10] for arandomly arranged magnetic induction link.

Contribution: This paper contains the following novel re-sults for links between dipoles with uniformly distributed 3Dorientations, presented in communication-theoretic parlance.• We derive the channel statistics for the near- and far-field

region. We show that the outage behavior is characterizedby a diversity exponent of just 1

2 .• We derive the channel statistics in the near-far-field

transition and demonstrate a diversity exponent of 1.• An outage analysis demonstrates the severity of this

fading channel in terms of the behavior of the outagePTE, outage capacity, and bit error probability.

Related Work: Regarding misaligned magnetic-inductionlinks, most studies focused on small lateral or angular de-viations in the regime of short-range power transfer [11], [12]where the specific coil geometries must be considered. Theconcept of outage probability, diversity exponent and outagecapacity in relation to fading is well-established for multipathradio channels [1]. Likewise, polarization diversity is a well-established concept [13, Sec. 2.5]. The work in [10] identified

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the outage capacity as a meaningful performance measure ofrandomly arranged magneto-inductive communication links.The distribution in (13) and various results on diversity com-bining appeared in our paper [6]. The contents of this paperare also contained in the dissertation of the first author [14].

Paper Structure: Sec. II describes the dipole model indetail. The rather technical Sec. III then derives the channelstatistics between randomly oriented antennas, which enablesthe subsequent outage and diversity analysis in Sec. IV. Aftercommenting on the implications for RFID and backscattersystems in Sec. V, we conclude the paper in Sec. VI.

II. DIPOLE CHANNEL MODEL

We consider a narrowband wireless link from a transmittingdipole, driven by a TX amplifier, to a receiving dipole whichfeeds a low-noise amplifier or tank circuit. The link geometryis shown in Fig. 1 and is described by the link distance rand three unit vectors: the TX and RX dipole axis directionsoT,oR ∈ R3 and the TX-to-RX direction u ∈ R3.

transmit dipoleorientation

oT

receive dipoleorientation

oR

TX-to-RXdirection

u

linkdistance

r

Fig. 1. Geometry of a link between dipoles with arbitrary orientationsoT,oR ∈ R3 (unit vectors) and distance r. The model serves as a descriptionof unaligned links between weakly coupled coils or between dipole antennas.

The channel coefficient h ∈ C is given by [3], [14]

h = α

((1

(kr)3+

j

(kr)2

)JNF +

1

2krJFF

)(1)

where k = 2πλ = 2πf

c is the wavenumber and j the imaginaryunit. The prefactor α is of no formal importance for this paper.It is given by α = α e−jkr where α ∈ C subsumes technicalparameters (e.g., coil diameters) which are described in theappendix together with the detailed model conditions.

The RX may be located in the near-field region (kr � 1) orthe far-field region (kr � 1), or in the transition in between.The formula (1) uses the near- and far-field alignment factors,given by the inner products

JNF = oTRβNF , JNF ∈ [−1, 1] , (2)

JFF = oTRβFF , JFF ∈ [−1, 1] . (3)

They account for signal attenuation due to suboptimal nodeorientations (misalignment). The formulas use unitless fieldvector quantities βNF and βFF, which we call the scaled nearfield and the scaled far field, respectively. They are given by

βNF =1

2(3uuT − I3)oT , (4)

βFF = (I3 − uuT)oT (5)

and illustrated in Fig. 2. The formulas use a convenient linear-algebraic formalism, which has been derived in our previous

work [14, App. A] from an existing trigonometric descriptionof the dipole field [3]. The field magnitudes are given by

βNF = ‖βNF‖ =1

2

√1+3(uToT)2 , 1

2 ≤ βNF ≤ 1 , (6)

βFF = ‖βFF‖ =√

1−(uToT)2 , 0 ≤ βFF ≤ 1 . (7)

The far-field magnitude βFF can fade to zero but βNF can not,as can be seen in Fig. 2.

oT

βNF

radial distance

axia

ldist

ance

(a) scaled near field βNF

oT

βFF

radial distance

axia

ldist

ance

(b) scaled far field βFF

Fig. 2. Scaled near and far field around a transmitting dipole with verticalaxis orientation (unit vector oT). By definition these fields do not comprisepath loss; the maximum magnitude is 1. In particular, βNF = 1, βFF = 0 holdon the dipole axis while βNF = 1

2, βFF = 1 hold in the perpendicular plane.

We shall point out two specific dipole arrangements:

• Dipoles in coaxial arrangement oT = oR = u:in this case JNF = 1, JFF = 0, and thus h = hcoax with

hcoax = α

(1

(kr)3+

j

(kr)2

). (8)

• Dipoles in parallel arrangement oT = oR with uToT = 0:in this case JNF = − 1

2 , JFF = 1 and thus h = hpara with

hpara =α

2

(− 1

(kr)3− j

(kr)2+

1

kr

). (9)

We note that η = |h|2 is the power transfer efficiency (PTE)over the link. An important quantity is the maximum PTEgiven kr and α, denoted as ηopt = |hopt|2. We find that1

ηopt = |hopt|2 = maxoT,oR

|h|2 =

{|hcoax|2 if kr ≤ krth

|hpara |2 if kr > krth(10)

whereby the threshold fulfills |hcoax| = |hpara|. It is given by

krth =

√√37 + 5

2≈ 2.3540 . (11)

1To prove the statement (10), we write h as bilinear form h = oTRAoT and

deduce A = α(( 1(kr)3

+ j(kr)2

)( 32uuT− 1

2I3)+ 1

2kr(I3−uuT)) ∈ C3×3

from (1) to (5). We find that hcoax is an eigenvalue by verifying Au = hcoaxu.Furthermore, hpara is a double eigenvalue because Au⊥ = hparau⊥ for anyvector u⊥ that is orthogonal to u. Therefrom, the statement (10) followsfrom basic linear algebra. The threshold krth in (11) is found by solving theequation |hcoax|2 = |hpara|2 for kr, using the definitions (8) and (9).

III. CHANNEL STATISTICS

Our starting point is the assumption that the TX and RXantenna orientations (unit vectors) are random and statisticallyindependent, with uniform distributions

oT,oRi.i.d.∼ U(S) (12)

on the unit sphere S ⊂ R3. The quantities α, k, r,u areconsidered non-random throughout. Hence, the statistics of hin (1) are determined by the joint statistics of JNF, JFF.

A. In the Near-Field Region or Far-Field Region

First, we address the important marginal distributions ofJNF and JFF which describe the statistics of h in the near-field region (kr � krth) and the far-field region (kr � krth),respectively.

Proposition 1. Assume (12). Then the near-field alignmentfactor JNF has the marginal probability density function (PDF)

fJNF(JNF) =1

2βNF·

1 |JNF| ≤ 1

2

1− arcosh(2|JNF|)arcosh(2)

12 < |JNF| < 1

0 1 ≤ |JNF|(13)

with βNF =√3

2 arcosh(2) . The far-field alignment factor exhibits

fJFF(JFF) =1

2

(π2− arcsin |JFF|

)· 1[−1,1](JFF) . (14)

Thereby, 1[−1,1] is the indicator function for this interval. ThePDFs are shown in Fig. 3a and 3b.

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

fJNF

(a) PDF of alignment factor JNF

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

fJFF

(b) PDF of alignment factor JFF

0 0.25 0.5 0.75 10

1

2

3

4

fβNF

(c) PDF of magnitude βNF

0 0.25 0.5 0.75 10

1

2

3

4

fβFF

(d) PDF of magnitude βFF

Fig. 3. Marginal PDFs arising from random antenna orientations on bothends with uniform distributions in 3D.

Proof. We will heavily use the fact that, for a random constant-length vector in R3 with uniform distribution on a sphere, anyprojection has uniform distribution. This fact is a corollaryof Archimedes’ hat-box theorem or of the fact that thelateral surface area of a sphere cap is linear in its height(which implies a linear CDF for a projection, cf. [6]). A firstimplication to our formalism is that the TX-side projectionuToT, which determines the magnitudes βNF and βFF, hasuniform distribution uToT ∼ U(−1, 1) due to oT ∼ U(S).

Consequently, with a basic change-of-variables calculation weobtain from (6) and (7) the PDFs of the field magnitudes

fβNF(βNF) =4√3

βNF√4β2

NF − 1· 1[ 12 ,1]

(βNF) , (15)

fβFF(βFF) =βFF√

1− β2FF

· 1[0,1](βFF) (16)

which are shown in Fig. 3c and 3d. The random RX orientationoR ∼ U(S) in (2) and (3) results in conditional distributionsJNF |βNF ∼ U(−βNF, βNF) and JFF |βFF ∼ U(−βFF, βFF). Thejoint PDFs fJNF|βNF · fβNF and fJFF|βFF · fβFF yield fJNF and fJFF

via marginalization integrals (the steps are omitted).

Proposition 2. Consider the cumulative distribution function(CDF) F|h|2(s) = P[ |h|2 ≤ s ]. In the near-field region kr �krth (described by J∗ = JNF) or the far-field region kr � krth

(described by J∗ = JFF), the approximation

F|h|2(s) ≈ 2 · fJ∗(0)

|hopt|√s (17)

applies under assumption (12). It is accurate for s� |hopt|2.

Proof. Either case fulfills |h|2 ≈ |hopt|2J2∗ . We calculate

F|h|2(s) ≈ FJ2∗

( s

|hopt|2)

= P

[|J∗| ≤

√s

|hopt|

]= 2

ˆ √s

|hopt|

0

fJ∗(x) dx ≤ 2 · fJ∗(0)

ˆ √s

|hopt|

0

dx . (18)

This bound is tight for small integration intervals because theintegrand is continuous, as seen in Fig. 3a and 3b.

The CDF behavior F|h|2(s) ∝√s for small s hints that

fading events |h|2 ≈ 0 occur with significant probability. Thisis due to the probability densities fJNF(0) > 0, fJFF(0) > 0.

B. Near-Far Transition with Random Receiver Orientation

We consider the statistics of h ∈ C when both near- andfar-field propagation make significant contributions. First, weconsider the case of a random RX orientation oR ∼ U(S)while oT and u are fixed. This interesting case will serve aspreparation for the fully random case.

We start our mathematical approach by observing from (1)to (3) that the channel coefficient is an inner product

h = oTRv , oR ∈ R3, v ∈ C3 (19)

of the random oR and a unitless, complex-valued field vector

v = α

((1

(kr)3+

j

(kr)2

)βNF+

1

2krβFF

). (20)

This field vector is non-random in this context because itis determined by the non-random α, kr,u,oT. We considervRe = Re(v) and vIm = Im(v) and note that these two vectorsare linearly independent unless uToT = 0 or uToT = ±1; thesimple proof thereof is omitted.

The random channel coefficient is expressed as

h = Re(h) + j · Im(h) = oTRvRe + j · oT

RvIm , (21)

which exhibits a statistical dependence between the real andimaginary part because the random oR affects both. In thefollowing, we specify the statistics of h in terms of theconditional PDF f(h |oT) = f(h |v).

Proposition 3. Consider a random unit vector oR ∼ U(S),i.e. with uniform distribution on the unit sphere in R3, anda non-random vector v = vRe + j · vIm ∈ C3 with linearlyindependent vRe,vIm ∈ R3. Let vRe = ‖vRe‖, vIm = ‖vIm‖,and ρ =

vTRevIm

vRevIm(correlation coefficient). Then the joint PDF

of the projections Re(h) = oTRvRe and Im(h) = oT

RvIm is

f(h |v) = f(Re(h), Im(h) |vRe,vIm

)= (22)

1

vRevIm√

1−ρ2ψ

(∥∥∥∥[ 1 0

ρ√

1−ρ2

]−1[Re(h)/vRe

Im(h)/vIm

]∥∥∥∥2)

with ψ(x) = 12π√1−x1[0,1](x). The uniform marginal distri-

butions hRe ∼ U(−vRe, vRe) and hIm ∼ U(−vIm, vIm) apply.

Proof. We apply the Gram-Schmidt process to vRe,vIm toobtain orthonormal vectors m = vRe

vRe, n = (I3−mmT)vIm

‖(I3−mmT)vIm‖ .They fulfill vRe = vRem and vIm = vImρm + vIm

√1− ρ2 n.

Written as linear map, [vRe vIm] = [m n]ET. The projectionsof oR thus fulfill [Re(h) Im(h)] = oT

R [m n]ET = [mo no]ET.

The joint PDF fmo,nois given by Lemma 1 below. We sub-

sequently obtain the PDF of Re(h), Im(h) with a change-of-variables argument: for random mo, no with PDF fmo,no andan invertible linear map E, the PDF f(Re(h), Im(h) |v) =

1det(E)fmo,no

(E−1[Re(h) Im(h)]T) applies.

Lemma 1. Consider orthonormal vectors m,n ∈ R3 and arandom unit vector o ∼ U(S). The joint PDF of mo = oTm,no = oTn is then given by fmo,mo

(mo, no) = ψ(m2o + n2o).

For the proof of Lemma 1 we refer to [14, Lemma 4.5].We note that the distribution h |oT is equivalent to h |uToT

because oR ∼ U(S) has rotational invariance. Thus,

f(h |oT) = f(h |v) = f(h |uToT) . (23)

An evaluation of this conditional PDF is shown in Fig. 4 forthe exemplary value uToT = 0.3.

Proposition 4. Let h be distributed according to Prop. 3. Thenthe CDF F|h|2 is within the bounds

s

2b≤ F|h|2|v(s|v) ≤ s

2b

(1− s

s0

)−1/2(24)

if s < s0, whereby s0 = a−√a2 − b2 with a = 1

2 (v2Re + v2Im)

and b = vRevIm√

1− ρ2.

Proof. The joint PDF of [Re(h) Im(h)] = oTR [vRe vIm] is

given by Prop. 3. There, a linear map E ∈ R2×2 mapsfrom the closed unit disk to the ellipse that is the supportof f(h|v). Let s0 be the smaller eigenvalue of ETE; thestated formula is obtained from the characteristic polynomial.Now s < s0 guarantees that f(h|v) < ∞ because then h

is in the interior of supp f(h|v). In particular, f(h|v) =ψ(‖E−1[Re(h) Im(h)]T‖2)

b ≤ ψ(s/s0)b . We find the upper bound

via P[ |h|2 ≤ s|v ] =´|h|2≤s f(h|v)dh ≤ ψ(s/s0)

b

´|h|2≤s dh =

ψ(s/s0)·πsb = s

2b/√

1− s/s0. Analogously, the lower bound isdue to f(h|v) ≥ ψ(0)

b = 12πb for s < s0.

In essence, Prop. 4 states that F|h|2(s) ∝ s for small s inthe transition region. In contrary, the near- and far-field regionbehavior F|h|2(s) ∝

√s from Prop. 2 exhibits a larger con-

centration of probability mass near |h|2 = 0. The advantageof the transition region is caused by the sum of phase-shiftedfield vectors in (20) providing polarization diversity: a deepfade h = 0 can only occur if oR is orthogonal to both vRe

and vIm. In other words, the field vector Re(vej2πft) nowoscillates on an ellipse, not on a line [13, Sec. 2.5].

-2 -1 0 1 2

10-3

-1

0

1

10-3

0

high

cond

ition

alPD

Ff(h|u

To

T=

0.3)

Im(h

)

Re(h)

Fig. 4. Conditional PDF of the channel coefficient h ∈ C for random RXorientation oR, described by Prop. 3 in closed form. This evaluation assumesthe values uToT = 0.3, kr = 2, α = 10−2.

C. Near-Far Transition, Random Orientations at Both Ends

Finally, we address the distribution of h in the fully-randomcase oT,oR

i.i.d.∼ U(S), again under consideration of all termsin (1). Fig. 5 shows scatter plots of h for various kr values.

-100 -50 0 50 100

10-3

-20

0

2010-3

(a) kr = 0.5

-20 -10 0 10 20

10-3

-4-2024

10-3

(b) kr = 1

-4 -2 0 2 4

10-3

-2

-1

0

1

2

10-3

(c) kr = 2

%-0.4 -0.2 0 0.2 0.4

10-3

-0.2

-0.1

0

0.1

0.2

10-3

(d) kr = 5π

Fig. 5. Scatter plots of the random channel coefficient h ∈ C between twodipoles with random orientations oT,oR ∼ U(S) for different regions. Forkr � krth or kr � krth, all samples lie on a line. The plots were obtainedwith random sampling and α = 10−2 was assumed.

Proposition 5. Under (12) and based on the conditional PDFf(h|uToT) = f(h|v) from Prop. 3, the PDF of h is given by

f(h) =1

2·ˆ +1

−1f(h |uToT = x) dx . (25)

Proof. We find f(h) =´ +1

−1 f(h|uToT = x)f(uToT = x)dxfrom (23) and marginalization. With oT ∼ U(S) while uis non-random, the uniform distribution uToT ∼ U(−1, 1)applies (see the proof of Prop. 1). Thus f(uToT) = 1

2 .

A closed-form solution of the integral (25) is unavailable,but an evaluation obtained with numerical integration is shownin Fig. 6a. The results are supplemented by the geometricexplanation of the rhombus-shaped support of the distributionin Fig. 6b.

-2 -1 0 1 2

10-3

-2

-1

0

1

2

10-3

0

high

chan

nel

coef

f.PD

Ff(h

)

Re(h)

Im(h

)

(a) fully-random-case PDF f(h) for......kr = 2 (near-far-field transition)

-2 -1 0 1 2

10-3

-2

-1

0

1

2

10-3

hpara

−hpara

hcoax

−hcoax

Re(h)

Im(h

)

(b) rhombus-shaped support of.....the PDF f(h) for kr = 2

Fig. 6. PDF of the random channel coefficient h ∈ C for random dipole orien-tations oT,oR ∼ U(S), evaluated here for kr = 2 and α = 10−2. The PDFwas computed by solving (25) numerically. Fig. 6b shows how the rhombus-shaped support supp f(h) arises from a union of ellipses supp f(h |uToT),shown for uToT ∈ {0, .1, .3, .5, .7, .9, 1}. For uToT ∈ {0, 1} theellipse becomes a line. The ellipses for uToT ≈ 0 cause a concentrationof probability mass between ±hpara.

-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 010-5

10-4

10-3

10-2

10-1

100

kr = 0.1

kr = 100

kr = krth

= 2.354

kr = 1

kr = 10

kr = 1000 (towards far-field region)

misalignment loss |h|2/

|hopt|2 [dB]

empi

rical

CD

F

Fig. 7. Statistics of the channel attenuation due to random TX and RXorientations, computed via Monte Carlo simulation for i.i.d. uniform distribu-tions in 3D. We observe that severe misalignment loss occurs with significantprobability, especially in the near-field region and the far-field region. Thenear-far-field transition features a beneficial polarization diversity effect.

We argue that, via (25), the beneficial property F|h|2(s) ∝ sfor small s carries over from F|h|2|v(s |v) ∝ s in Prop. 4. Thisis supported by the Monte-Carlo simulation in Fig. 7, but arigorous argument is unavailable. A non-rigorous argument isthat polarization diversity (i.e. linearly independent vRe,vIm)occurs with probability 1. Put differently, a problematic caseuToT = 0 or uToT = ±1, where the boundary ellipse ofsupp fh|uToT degenerates to a line, occurs with probability 0.

IV. OUTAGE ANALYSIS

A. Outage Power Transfer Efficiency ηεThe power transfer efficiency (PTE) η = |h|2 is a random

variable in the context of this paper. Analogous to the conceptof outage capacity [1], we consider the outage PTE, defined asthe PTE value ηε for which an outage event |h|2 < ηε occurswith a certain probability ε. The CDF of |h|2 describes thisvery dependence: ε = F|h|2(ηε).

Proposition 6. Assume (12) and that the RX is in the near-field region (J∗ = JNF) or the far-field region (J∗ = JFF).Then, a target PTE ηε results in an outage probability

ε = F|h|2(ηε) ≈ 2 · fJ∗(0)

√ηεηopt

. (26)

Vice versa, a target outage probability ε yields the outage PTE

ηε = F−1|h|2(ε) ≈ε2 ηopt

(2 · fJ∗(0))2. (27)

The approximations are accurate for ηε � ηopt.

Proof. The statements follow directly from Prop. 2.

From (26) and (27) we observe the proportionality ε ∝ η−12

optas well as ηε ∝ ε2. This scaling behavior demonstrates thedrastic fading effect due to random antenna orientations. Onthe one hand, increasing ηopt (e.g., by improving technicalparameters or reducing the distance) is not an efficient meansfor reducing ε. On the other hand, requiring some degree ofreliability (i.e. a small ε) is associated with an extremely smallPTE ηε. For example, aiming for a factor-10 improvement ofε demands a 20 dB loss for ηε.

The situation improves in the near-far-field transition: there,F|h|2(s) ∝ s holds for small s (cf. Prop. 4), which results inthe more beneficial proportionalities ε ∝ η−1opt and ηε ∝ ε. Theimprovement stems from polarization diversity.

B. Outage Capacity CεWe shift our focus to narrowband data communication over

this fading channel, with transmit power PT and reception inadditive white Gaussian noise (AWGN) of power PN. Thesignal-to-noise ratio SNR = |h|2PT/PN is random and theinstantaneous channel capacity C = log2(1+SNR), measuredin bit/s/Hz, is thus also random and can fade to zero.

A well-established measure for the communication perfor-mance of a fading channel is the outage capacity [1, Eq. 5.57]

Cε = log2

(1 +

F−1|h|2(ε) · PT

PN

)(28)

for which, by definition, the event log2(1+SNR) < Cε occurswith probability ε. We argue that Cε ∝ F−1|h|2(ε) for small εbecause the bound Cε ≤ log2(e) · F−1|h|2(ε) · PT/PN, which isobtained through log-linearization, is tight for low SNR or fora small target ε. Hence, by (27), the near- and the far-fieldregions exhibit the scaling behavior

Cε ∝ ε2. (29)

This means that a target outage probability ε � 1 can onlybe achieved with an extremely small data rate. In contrary,the near-far-field transition exhibits the more beneficial scalingbehavior Cε ∝ ε due to polarization diversity (by Prop. 4).

C. Bit Error Rate pe

Another popular measure of the communication perfor-mance over a fading channel is the bit error rate pe. Forantipodal modulation (BPSK) and reception in AWGN, itsvalue given h is Q(

√2 SNR) whereby SNR = |h|2PT/PN

is random and subject to fading. [1, Eq. 3.13]

Proposition 7. Assume (12), AWGN, and either the near-fieldregion (J∗ = JNF) or the far-field region (J∗ = JFF). Then,the bit error rate of BPSK modulation has the upper bound

pe <fJ∗(0)√π · SNRopt

(30)

which becomes tight for large SNRopt = |hopt|2PT/PN.

Proof. SNR ≈ J2∗ SNRopt applies in the near- or far-

field region. For the bit error rate we calculate pe =

E[Q(√

2J2∗ SNRopt

)] =´ 1−1 fJ∗(J∗)Q(

√2J2∗SNRopt)dJ∗ ≤

2 · fJ∗(0)´ 10Q(√

2J2∗SNRopt)dJ∗ = 2 · fJ∗(0)

(1−e−SNRopt√

4π·SNRopt+

Q(√

2SNRopt ))and apply Q(x) < 1

x1√2πe−x

2/2.

The large-SNR description in Prop. 7 has the standardform pe ∝ SNR−Lopt from [1, Eq. 3.158]. We deduce that thediversity exponent is L = 1

2 for the near- and far-field regions,associated with catastrophic fading (worse than L = 1 ofRayleigh fading). In a similar fashion, it can be argued thatpe ∝ SNR−1opt in the near-far-field transition for large SNRopt,i.e. L = 1 (like Rayleigh fading). This is a direct consequenceof f|h|(x) ∝ x for small x; the details are omitted.

V. IMPLICATIONS FOR RFID AND BACKSCATTER

So far, the results concerned links with an active TXequipped with a TX amplifier, where SNR ≈ SNRoptJ

2∗ with

J∗ ∈ {JNF, JFF} in the near- or far-field region. For a passiveRFID tag that uses load modulation or for backscatter com-munication, the fading channel applies twice and the relationchanges to SNR ≈ SNRoptJ

4∗ , cf. [6], with the following

severe consequences. The misalignment losses double in termsof dB value (e.g., the abscissa of Fig. 7). Likewise, the bit errorrate pe ∝ SNR

−1/4opt with a diversity exponent of only 1/4.

VI. SUMMARY & CONCLUSIONS

We provided an analytical description of the statistics ofthe fading channel between two randomly oriented dipoles.Our outage analysis revealed that drastic signal losses are verylikely to occur, especially in the near- and far-field regions.This emphasizes the importance of diversity concepts for thisclass of links, e.g., the use of a rotating source, appropriateantenna polarization, or antenna arrays and beamforming. Theresults also suggest that, for special applications, it can befavorable to design a link such that the RX will typically belocated in the transition region between near and far field.

APPENDIX: PHYSICAL CONDITIONS & DETAILS

The channel coefficient h in (1) relates the power waveemitted by a power-matched TX amplifier to the power waveinto a power-matched RX amplifier or tank circuit. A neces-sary condition for (1) to apply is that h (obtained with thisformula) fulfills |h|2 � 1, i.e. the dipoles are weakly coupled.

The prefactor α = α e−jkr in (1) comprises technical linkparameters in α. For loop antennas (i.e. coils), the valueα = jµ0ATNTARNRfk

3

√4RTRR

applies if the coils are electricallysmall, the turn pitch angle is small, and the coil diametersare significantly smaller than r. We use the permeability µ0,TX- and RX-side number of turns NT and NR, the coil areasAT and AR, carrier frequency f , wavenumber k, and the TX-and RX-side antenna resistances RT and RR. [3, Sec. 5.2]

We note that µ0

2πATNTARNR r−3JNF is the mutual induc-

tance M . Furthermore, a small power-matched TX coil is de-scribed by a magnetic dipole moment phasor m = ATNTiToT

with current phasor iT =√PT/RT and TX power PT.

Between dipole antennas without ohmic losses, α is givenby the antenna directivity: α = 1.5 for electrically small dipoleantennas and α ≈ 1.64 for the λ/2-length case [3, Cpt. 4].

ACKNOWLEDGEMENT

We would like to thank Robin Kramer for valuable inputsand Bharat Bhatia for helping with the appendix.

REFERENCES

[1] D. Tse and P. Viswanath, Fundamentals of Wireless Communication.Cambridge University Press, 2005.

[2] E. Biglieri, J. Proakis, and S. Shamai, “Fading channels: Information-theoretic and communications aspects,” IEEE Transactions on Informa-tion Theory, vol. 44, no. 6, pp. 2619–2692, 1998.

[3] C. A. Balanis, Antenna Theory: Analysis and Design. Wiley, 2005.[4] D. Cox, “Antenna diversity performance in mitigating the effects of

portable radiotelephone orientation and multipath propagation,” IEEETransactions on Communications, vol. 31, no. 5, pp. 620–628, 1983.

[5] C. B. Dietrich, K. Dietze, J. R. Nealy, and W. L. Stutzman, “Spatial,polarization, and pattern diversity for wireless handheld terminals,” IEEETransactions on Antennas and Propagation, vol. 49, no. 9, pp. 1271–1281, 2001.

[6] G. Dumphart and A. Wittneben, “Stochastic misalignment model formagneto-inductive SISO and MIMO links,” in IEEE PIMRC, Sep. 2016.

[7] Z. Zhang, E. Liu, X. Qu, R. Wang, H. Ma, and Z. Sun, “Connectivityof magnetic induction-based ad hoc networks,” IEEE Transactions onWireless Communications, vol. 16, no. 7, pp. 4181–4191, Jul 2017.

[8] K. Finkenzeller, RFID Handbook. Carl Hanser, 2015.[9] C. Angerer, R. Langwieser, and M. Rupp, “RFID reader receivers for

physical layer collision recovery,” IEEE Transactions on Communica-tions, vol. 58, no. 12, pp. 3526–3537, 2010.

[10] Z. Sun, I. F. Akyildiz, S. Kisseleff, and W. Gerstacker, “Increasingthe capacity of magnetic induction communications in RF-challengedenvironments,” IEEE Transactions on Communications, vol. 61, no. 9,pp. 3943–3952, 2013.

[11] M. Soma, D. C. Galbraith, and R. L. White, “Radio-frequency coils inimplantable devices: misalignment analysis and design procedure,” IEEETransactions on Biomedical Engineering, no. 4, pp. 276–282, 1987.

[12] K. Fotopoulou and B. W. Flynn, “Wireless power transfer in loosely cou-pled links: Coil misalignment model,” IEEE Transactions on Magnetics,vol. 47, no. 2, pp. 416–430, 2011.

[13] S. J. Orfanidis, Electromagnetic waves and antennas. Rutgers Univer-sity, 2002.

[14] G. Dumphart, “Magneto-inductive communication and localization: Fun-damental limits with arbitrary node arrangements,” Ph.D. dissertation,ETH Zurich, 2020.