The Channel Between Randomly Oriented Dipoles: Statistics and Outage in the Near and Far Field
TThe Channel Between Randomly Oriented Dipoles:Statistics and Outage in the Near and Far Field
Gregor Dumphart and Armin Wittneben
Wireless Communications Group, D-ITET, ETH Zurich
Z¨urich, 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 p e ∝ SNR − / 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. I
NTRODUCTION
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 ≈ , 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 fading h ∼ CN (0 , σ ) 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 ≈ 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 coupledcoils (loop antennas). • Capacitive links between small electric dipole antennas. • Links between λ/ -length dipole antennas in free space. • Links from a magnetic dipole to a small loop or from anelectric 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-fieldregion. We show that the outage behavior is characterizedby a diversity exponent of just . • We derive the channel statistics in the near-far-fieldtransition and demonstrate a diversity exponent of . • An outage analysis demonstrates the severity of thisfading 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 a r X i v : . [ c s . I T ] F e b he 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. D
IPOLE C HANNEL M ODEL
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 r and three unit vectors: the TX and RX dipole axis directions o T , o R ∈ R and the TX-to-RX direction u ∈ R . transmit dipoleorientation o T receive dipoleorientation o RTX-to-RXdirection u linkdistance r Fig. 1. Geometry of a link between dipoles with arbitrary orientations o T , o R ∈ R (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 (cid:18)(cid:18) kr ) + j ( kr ) (cid:19) J NF + 12 kr J FF (cid:19) (1)where k = πλ = πfc 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 (cid:28) ) orthe far-field region ( kr (cid:29) ), or in the transition in between.The formula (1) uses the near- and far-field alignment factors,given by the inner products J NF = o T R β NF , J NF ∈ [ − , , (2) J FF = o T R β FF , J FF ∈ [ − , . (3)They account for signal attenuation due to suboptimal nodeorientations (misalignment). The formulas use unitless fieldvector quantities β NF and β FF , which we call the scaled nearfield and the scaled far field, respectively. They are given by β NF = 12 (3 uu T − I ) o T , (4) β FF = ( I − uu T ) o T (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 β NF = (cid:107) β NF (cid:107) = 12 (cid:113) u T o T ) , ≤ β NF ≤ , (6) β FF = (cid:107) β FF (cid:107) = (cid:113) − ( u T o T ) , ≤ β FF ≤ . (7)The far-field magnitude β FF can fade to zero but β NF can not,as can be seen in Fig. 2. o T β NF radial distance a x i a l d i s t a n ce (a) scaled near field β NF o T β FF radial distance a x i a l d i s t a n ce (b) scaled far field β FF Fig. 2. Scaled near and far field around a transmitting dipole with verticalaxis orientation (unit vector o T ). By definition these fields do not comprisepath loss; the maximum magnitude is . In particular, β NF = 1 , β FF = 0 holdon the dipole axis while β NF = , β FF = 1 hold in the perpendicular plane. We shall point out two specific dipole arrangements: • Dipoles in coaxial arrangement o T = o R = u :in this case J NF = 1 , J FF = 0 , and thus h = h coax with h coax = α (cid:18) kr ) + j ( kr ) (cid:19) . (8) • Dipoles in parallel arrangement o T = o R with u T o T = 0 :in this case J NF = − , J FF = 1 and thus h = h para with h para = α (cid:18) − kr ) − j ( kr ) + 1 kr (cid:19) . (9)We note that η = | h | is the power transfer efficiency (PTE)over the link. An important quantity is the maximum PTEgiven kr and α , denoted as η opt = | h opt | . We find that η opt = | h opt | = max o T , o R | h | = (cid:26) | h coax | if kr ≤ kr th | h para | if kr > kr th (10)whereby the threshold fulfills | h coax | = | h para | . It is given by kr th = (cid:115) √
37 + 52 ≈ . . (11) To prove the statement (10), we write h as bilinear form h = o T R Ao T anddeduce A = α (( kr ) + j ( kr ) )( uu T − I )+ kr ( I − uu T )) ∈ C × from (1) to (5). We find that h coax is an eigenvalue by verifying Au = h coax u .Furthermore, h para is a double eigenvalue because Au ⊥ = h para u ⊥ for anyvector u ⊥ that is orthogonal to u . Therefrom, the statement (10) followsfrom basic linear algebra. The threshold kr th in (11) is found by solving theequation | h coax | = | h para | for kr , using the definitions (8) and (9). II. C
HANNEL S TATISTICS
Our starting point is the assumption that the TX and RXantenna orientations (unit vectors) are random and statisticallyindependent, with uniform distributions o T , o R i.i.d. ∼ U ( S ) (12)on the unit sphere S ⊂ R . The quantities α, k, r, u areconsidered non-random throughout. Hence, the statistics of h in (1) are determined by the joint statistics of J NF , J FF . A. In the Near-Field Region or Far-Field Region
First, we address the important marginal distributions of J NF and J FF which describe the statistics of h in the near-field region ( kr (cid:28) kr th ) and the far-field region ( kr (cid:29) kr th ),respectively. Proposition 1.
Assume (12) . Then the near-field alignmentfactor J NF has the marginal probability density function (PDF) f J NF ( J NF ) = 12 ¯ β NF · | J NF | ≤ − arcosh(2 | J NF | )arcosh(2) 12 < | J NF | <
10 1 ≤ | J NF | (13) with ¯ β NF = √
32 arcosh(2) . The far-field alignment factor exhibits f J FF ( J FF ) = 12 (cid:16) π − arcsin | J FF | (cid:17) · [ − , ( J FF ) . (14)Thereby, [ − , is the indicator function for this interval. ThePDFs are shown in Fig. 3a and 3b. -1 -0.5 0 0.5 100.20.40.60.8 f J NF (a) PDF of alignment factor J NF -1 -0.5 0 0.5 100.20.40.60.8 f J FF (b) PDF of alignment factor J FF f β NF (c) PDF of magnitude β NF f β FF (d) PDF of magnitude β FF 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 R 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 projection u T o T , which determines the magnitudes β NF and β FF , hasuniform distribution u T o T ∼ U ( − , due to o T ∼ U ( S ) . Consequently, with a basic change-of-variables calculation weobtain from (6) and (7) the PDFs of the field magnitudes f β NF ( β NF ) = 4 √ β NF (cid:112) β NF − · [ , ( β NF ) , (15) f β FF ( β FF ) = β FF (cid:112) − β FF · [0 , ( β FF ) (16)which are shown in Fig. 3c and 3d. The random RX orientation o R ∼ U ( S ) in (2) and (3) results in conditional distributions J NF | β NF ∼ U ( − β NF , β NF ) and J FF | β FF ∼ U ( − β FF , β FF ) . Thejoint PDFs f J NF | β NF · f β NF and f J FF | β FF · f β FF yield f J NF and f J FF via marginalization integrals (the steps are omitted). Proposition 2.
Consider the cumulative distribution function(CDF) F | h | ( s ) = P[ | h | ≤ s ] . In the near-field region kr (cid:28) kr th (described by J ∗ = J NF ) or the far-field region kr (cid:29) kr th (described by J ∗ = J FF ), the approximation F | h | ( s ) ≈ · f J ∗ (0) | h opt | √ s (17) applies under assumption (12) . It is accurate for s (cid:28) | h opt | .Proof. Either case fulfills | h | ≈ | h opt | J ∗ . We calculate F | h | ( s ) ≈ F J ∗ (cid:16) s | h opt | (cid:17) = P (cid:20) | J ∗ | ≤ √ s | h opt | (cid:21) = 2 ˆ √ s | h opt | f J ∗ ( x ) dx ≤ · f J ∗ (0) ˆ √ s | h opt | 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 | ( s ) ∝ √ s for small s hints thatfading events | h | ≈ occur with significant probability. Thisis due to the probability densities f J NF (0) > , f J FF (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 o R ∼ U ( S ) while o T 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 = o T R v , o R ∈ R , v ∈ C (19)of the random o R and a unitless, complex-valued field vector v = α (cid:18)(cid:18) kr ) + j ( kr ) (cid:19) β NF + 12 kr β FF (cid:19) . (20)This field vector is non-random in this context because itis determined by the non-random α, kr, u , o T . We consider v Re = Re( v ) and v Im = Im( v ) and note that these two vectorsare linearly independent unless u T o T = 0 or u T o T = ± ; thesimple proof thereof is omitted.The random channel coefficient is expressed as h = Re( h ) + j · Im( h ) = o T R v Re + j · o T R v Im , (21)hich exhibits a statistical dependence between the real andimaginary part because the random o R affects both. In thefollowing, we specify the statistics of h in terms of theconditional PDF f ( h | o T ) = f ( h | v ) . Proposition 3.
Consider a random unit vector o R ∼ U ( S ) ,i.e. with uniform distribution on the unit sphere in R , anda non-random vector v = v Re + j · v Im ∈ C with linearlyindependent v Re , v Im ∈ R . Let v Re = (cid:107) v Re (cid:107) , v Im = (cid:107) v Im (cid:107) ,and ρ = v T Re v Im v Re v Im (correlation coefficient). Then the joint PDFof the projections Re( h ) = o T R v Re and Im( h ) = o T R v Im is f ( h | v ) = f (cid:0) Re( h ) , Im( h ) | v Re , v Im (cid:1) = (22) v Re v Im (cid:112) − ρ ψ (cid:32) (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) ρ (cid:112) − ρ (cid:21) − (cid:20) Re( h ) /v Re Im( h ) /v Im (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) (cid:33) with ψ ( x ) = π √ − x [0 , ( x ) . The uniform marginal distri-butions h Re ∼ U ( − v Re , v Re ) and h Im ∼ U ( − v Im , v Im ) apply.Proof. We apply the Gram-Schmidt process to v Re , v Im toobtain orthonormal vectors m = v Re v Re , n = ( I − mm T ) v Im (cid:107) ( I − mm T ) v Im (cid:107) .They fulfill v Re = v Re m and v Im = v Im ρ m + v Im (cid:112) − ρ n .Written as linear map, [ v Re v Im ] = [ m n ] E T . The projectionsof o R thus fulfill [Re( h ) Im( h )] = o T R [ m n ] E T = [ m o n o ] E T .The joint PDF f m o ,n o is given by Lemma 1 below. We sub-sequently obtain the PDF of Re( h ) , Im( h ) with a change-of-variables argument: for random m o , n o with PDF f m o ,n o andan invertible linear map E , the PDF f (Re( h ) , Im( h ) | v ) = E ) f m o ,n o ( E − [Re( h ) Im( h )] T ) applies. Lemma 1.
Consider orthonormal vectors m , n ∈ R and arandom unit vector o ∼ U ( S ) . The joint PDF of m o = o T m , n o = o T n is then given by f m o ,m o ( m o , n o ) = ψ ( m o + n o ) . For the proof of Lemma 1 we refer to [14, Lemma 4.5].We note that the distribution h | o T is equivalent to h | u T o T because o R ∼ U ( S ) has rotational invariance. Thus, f ( h | o T ) = f ( h | v ) = f ( h | u T o T ) . (23)An evaluation of this conditional PDF is shown in Fig. 4 forthe exemplary value u T o T = 0 . . Proposition 4.
Let h be distributed according to Prop. 3. Thenthe CDF F | h | is within the bounds s b ≤ F | h | | v ( s | v ) ≤ s b (cid:18) − ss (cid:19) − / (24) if s < s , whereby s = a − √ a − b with a = ( v Re + v Im ) and b = v Re v Im (cid:112) − ρ .Proof. The joint PDF of [Re( h ) Im( h )] = o T R [ v Re v Im ] isgiven by Prop. 3. There, a linear map E ∈ R × mapsfrom the closed unit disk to the ellipse that is the supportof f ( h | v ) . Let s be the smaller eigenvalue of E T E ; thestated formula is obtained from the characteristic polynomial.Now s < s guarantees that f ( h | v ) < ∞ because then h is in the interior of supp f ( h | v ) . In particular, f ( h | v ) = ψ ( (cid:107) E − [Re( h ) Im( h )] T (cid:107) ) b ≤ ψ ( s/s ) b . We find the upper boundvia P[ | h | ≤ s | v ] = ´ | h | ≤ s f ( h | v ) dh ≤ ψ ( s/s ) b ´ | h | ≤ s dh = ψ ( s/s ) · πsb = s b / (cid:112) − s/s . Analogously, the lower bound isdue to f ( h | v ) ≥ ψ (0) b = πb for s < s .In essence, Prop. 4 states that F | h | ( s ) ∝ s for small s inthe transition region. In contrary, the near- and far-field regionbehavior F | h | ( s ) ∝ √ s from Prop. 2 exhibits a larger con-centration of probability mass near | h | = 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 o R is orthogonal to both v Re and v Im . In other words, the field vector Re( v e j πft ) nowoscillates on an ellipse, not on a line [13, Sec. 2.5]. -2 -1 0 1 210 -3 -101 10 -3 c ond iti on a l P D F f ( h | u T o T = . ) I m ( h ) Re( h ) Fig. 4. Conditional PDF of the channel coefficient h ∈ C for random RXorientation o R , described by Prop. 3 in closed form. This evaluation assumesthe values u T o T = 0 . , kr = 2 , ¯ α = 10 − . C. Near-Far Transition, Random Orientations at Both Ends
Finally, we address the distribution of h in the fully-randomcase o T , o R 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 10010 -3 -20020 10 -3 (a) kr = 0 . -20 -10 0 10 2010 -3 -4-2024 10 -3 (b) kr = 1 -4 -2 0 2 410 -3 -2-1012 10 -3 (c) kr = 2 % -0.4 -0.2 0 0.2 0.410 -3 -0.2-0.100.10.2 10 -3 (d) kr = 5 π Fig. 5. Scatter plots of the random channel coefficient h ∈ C between twodipoles with random orientations o T , o R ∼ U ( S ) for different regions. For kr (cid:28) kr th or kr (cid:29) kr th , all samples lie on a line. The plots were obtainedwith random sampling and ¯ α = 10 − was assumed. Proposition 5.
Under (12) and based on the conditional PDF f ( h | u T o T ) = f ( h | v ) from Prop. 3, the PDF of h is given by f ( h ) = 12 · ˆ +1 − f ( h | u T o T = x ) dx . (25) roof. We find f ( h ) = ´ +1 − f ( h | u T o T = x ) f ( u T o T = x ) dx from (23) and marginalization. With o T ∼ U ( S ) while u is non-random, the uniform distribution u T o T ∼ U ( − , applies (see the proof of Prop. 1). Thus f ( u T o T ) = .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 1 210 -3 -2-1012 10 -3 c h a nn e l c o e ff . P D F f ( h ) Re( h ) I m ( h ) (a) fully-random-case PDF f ( h ) for...... kr = 2 (near-far-field transition) -2 -1 0 1 210 -3 -2-1012 10 -3 h para − h para h coax − h coax Re( h ) I m ( 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 o T , o R ∼ U ( S ) , evaluated here for kr = 2 and ¯ α = 10 − . 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 | u T o T ) ,shown for u T o T ∈ { , . , . , . , . , . , } . For u T o T ∈ { , } theellipse becomes a line. The ellipses for u T o T ≈ cause a concentrationof probability mass between ± h para . -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 010 -5 -4 -3 -2 -1 kr = 0.1 kr = 100 kr = kr th = 2.354kr = 1kr = 10kr = 1000 (towards far-field region) misalignment loss | h | . | h opt | [dB] e m p i r i c a l C D 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 | ( s ) ∝ s for small s carries over from F | h | | 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 v Re , v Im )occurs with probability . Put differently, a problematic case u T o T = 0 or u T o T = ± , where the boundary ellipse of supp f h | u T o T degenerates to a line, occurs with probability . IV. O UTAGE A NALYSIS
A. Outage Power Transfer Efficiency η (cid:15) The power transfer efficiency (PTE) η = | h | is a randomvariable in the context of this paper. Analogous to the conceptof outage capacity [1], we consider the outage PTE, defined asthe PTE value η (cid:15) for which an outage event | h | < η (cid:15) occurswith a certain probability (cid:15) . The CDF of | h | describes thisvery dependence: (cid:15) = F | h | ( η (cid:15) ) . Proposition 6.
Assume (12) and that the RX is in the near-field region ( J ∗ = J NF ) or the far-field region ( J ∗ = J FF ).Then, a target PTE η (cid:15) results in an outage probability (cid:15) = F | h | ( η (cid:15) ) ≈ · f J ∗ (0) (cid:114) η (cid:15) η opt . (26) Vice versa, a target outage probability (cid:15) yields the outage PTE η (cid:15) = F − | h | ( (cid:15) ) ≈ (cid:15) η opt (2 · f J ∗ (0)) . (27) The approximations are accurate for η (cid:15) (cid:28) η opt .Proof. The statements follow directly from Prop. 2.From (26) and (27) we observe the proportionality (cid:15) ∝ η − opt as well as η (cid:15) ∝ (cid:15) . This scaling behavior demonstrates thedrastic fading effect due to random antenna orientations. Onthe one hand, increasing η opt (e.g., by improving technicalparameters or reducing the distance) is not an efficient meansfor reducing (cid:15) . On the other hand, requiring some degree ofreliability (i.e. a small (cid:15) ) is associated with an extremely smallPTE η (cid:15) . For example, aiming for a factor- improvement of (cid:15) demands a
20 dB loss for η (cid:15) .The situation improves in the near-far-field transition: there, F | h | ( s ) ∝ s holds for small s (cf. Prop. 4), which results inthe more beneficial proportionalities (cid:15) ∝ η − opt and η (cid:15) ∝ (cid:15) . Theimprovement stems from polarization diversity. B. Outage Capacity C (cid:15) We shift our focus to narrowband data communication overthis fading channel, with transmit power P T and reception inadditive white Gaussian noise (AWGN) of power P N . Thesignal-to-noise ratio SNR = | h | P T /P N is random and theinstantaneous channel capacity C = log (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] C (cid:15) = log (cid:18) F − | h | ( (cid:15) ) · P T P N (cid:19) (28)for which, by definition, the event log (1+SNR) < C (cid:15) occurswith probability (cid:15) . We argue that C (cid:15) ∝ F − | h | ( (cid:15) ) for small (cid:15) because the bound C (cid:15) ≤ log ( e ) · F − | h | ( (cid:15) ) · P T /P N , which isobtained through log-linearization, is tight for low SNR or fora small target (cid:15) . Hence, by (27), the near- and the far-fieldregions exhibit the scaling behavior C (cid:15) ∝ (cid:15) . (29)his means that a target outage probability (cid:15) (cid:28) can onlybe achieved with an extremely small data rate. In contrary,the near-far-field transition exhibits the more beneficial scalingbehavior C (cid:15) ∝ (cid:15) due to polarization diversity (by Prop. 4). C. Bit Error Rate p e Another popular measure of the communication perfor-mance over a fading channel is the bit error rate p e . Forantipodal modulation (BPSK) and reception in AWGN, itsvalue given h is Q ( √ whereby SNR = | h | P T /P N is random and subject to fading. [1, Eq. 3.13] Proposition 7.
Assume (12) , AWGN, and either the near-fieldregion ( J ∗ = J NF ) or the far-field region ( J ∗ = J FF ). Then,the bit error rate of BPSK modulation has the upper bound p e < f J ∗ (0) (cid:112) π · SNR opt (30) which becomes tight for large
SNR opt = | h opt | P T /P N .Proof. SNR ≈ J ∗ SNR opt applies in the near- or far-field region. For the bit error rate we calculate p e = E [ Q (cid:16)(cid:112) J ∗ SNR opt (cid:17) ] = ´ − f J ∗ ( J ∗ ) Q ( (cid:112) J ∗ SNR opt ) dJ ∗ ≤ · f J ∗ (0) ´ Q ( (cid:112) J ∗ SNR opt ) dJ ∗ = 2 · f J ∗ (0) (cid:0) − e − SNR opt √ π · SNR opt + Q ( (cid:112) opt ) (cid:1) and apply Q ( x ) < x √ π e − x / .The large-SNR description in Prop. 7 has the standardform p e ∝ SNR − L opt from [1, Eq. 3.158]. We deduce that thediversity exponent is L = 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 that p e ∝ SNR − opt in the near-far-field transition for large SNR opt ,i.e. L = 1 (like Rayleigh fading). This is a direct consequenceof f | h | ( x ) ∝ x for small x ; the details are omitted.V. I MPLICATIONS FOR
RFID
AND B ACKSCATTER
So far, the results concerned links with an active TXequipped with a TX amplifier, where
SNR ≈ SNR opt J ∗ with J ∗ ∈ { J NF , J FF } 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 ≈ SNR opt J ∗ , cf. [6], with the followingsevere consequences. The misalignment losses double in termsof dB value (e.g., the abscissa of Fig. 7). Likewise, the bit errorrate p e ∝ SNR − / opt with a diversity exponent of only / .VI. S UMMARY & C
ONCLUSIONS
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. A
PPENDIX : P
HYSICAL C ONDITIONS & D
ETAILS
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 | (cid:28) , 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 value ¯ α = jµ A T N T A R N R fk √ R T R R applies if the coils are electricallysmall, the turn pitch angle is small, and the coil diametersare significantly smaller than r . We use the permeability µ ,TX- and RX-side number of turns N T and N R , the coil areas A T and A R , carrier frequency f , wavenumber k , and the TX-and RX-side antenna resistances R T and R R . [3, Sec. 5.2]We note that µ π A T N T A R N R r − J NF is the mutual induc-tance M . Furthermore, a small power-matched TX coil is de-scribed by a magnetic dipole moment phasor m = A T N T i T o T with current phasor i T = (cid:112) P T /R T and TX power P T .Between dipole antennas without ohmic losses, ¯ α is givenby the antenna directivity: ¯ α = 1 . for electrically small dipoleantennas and ¯ α ≈ . for the λ/ -length case [3, Cpt. 4].A CKNOWLEDGEMENT
We would like to thank Robin Kramer for valuable inputsand Bharat Bhatia for helping with the appendix.R
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