Position Aided Beam Alignment for Millimeter Wave Backhaul Systems with Large Phased Arrays

0.17 MB·

by  George C. Alexandropoulos

Position Aided Beam Alignment for Millimeter
Wave Backhaul Systems with Large Phased Arrays
George C. Alexandropoulos
Mathematical and Algorithmic Sciences Lab, France Research Center, Huawei Technologies France,
20 Quai du Point du Jour, 92100 Boulogne-Billancourt, France
email: [email protected]
Abstract—Consensus has been reached between wireless in- beamforming (BF) supporting long outdoor links. To achieve
7
dustry bodies and academia on the indispensability of network the full benefit from BF in a communication link between
1
densification in meeting the throughput and coverage demands
0 multi-antennanodes,theentirechannelstateinformationneeds
for fifth generation (5G) wireless networks. One of the major
2 to be available at both communication ends. However, this
challenges with the current trend of dense multi-tier network
b deployments is the wireless backhauling of the various small information is hard to acquire in mmWave systems due to
e cells.Recently,wirelessbackhaulcommunicationhasbeenlargely the low coherence time, the radio frequency (RF) hardware
F realized with large antennas operating in the millimeter wave limitations (a phased antenna array has a large number of
(mmWave) frequency band and implementing highly directional
8 antenna elements attached to one RF chain), and the small
beamforming.Inthispaper,wefocusonthealignmentproblemof
1 signal-to-noise ratio (SNR) before BF. Although very recent
narrowbeamsbetweenfixedpositionnetworknodesinmmWave
backhaul systems that are subject to small displacements dueto theoretical works [5], [9]–[11] capitalized on the spatial spar-
]
T windfloworgroundvibration.Weconsidernodesequippedwith sity of mmWave channels [12] to estimate portions of the
I antennaarraysthatarecapableofperformingonlyanalogbeam- channelgainmatrix,thepresentedapproachesrequiredlengthy
s. formingandcommunicatethroughwirelesschannelsincludinga training phases to estimate the channel in multiple directions
line-of-sight component. Aiming at minimizing the time needed
c usingcomplexcompressedsensingalgorithms.Anotherfamily
[ to achieve beam alignment, we present an efficient method that
capitalizes on the exchange of position information between the of approaches (e.g., [13]–[18]) for efficient BF is based on

2 nodes to design their beamforming and combining vectors. Our beamswitchingbetweenthecommunicatingnodesin orderto
v representative numerical results on the outage probability with find a pair of beams from their available codebooks meeting
1 the proposed beam alignment method offer useful preliminary
a predefined performance threshold. When such a beam pair
9 insightsontheimpactofsomesystemandoperationparameters.
is found,beam alignmentis consideredto be achievedand no
2
3 IndexTerms—Analogbeamforming,antennaarrays,backhaul further beam searching is needed. The two-stage exhaustive
0 systems, beam alignment, combining, tracking, millimeter wave. search BF protocol for medium access control adopted in
. the IEEE 802.11ad standard [4] is mainly based on the
1
0 multi-sector BF technique of [13]. In [14], a two-sided beam
7 I. INTRODUCTION alignment method that is based on multi-round exchanges of
1 Mobile communication in the millimeter wave (mmWave) control information between the communicating nodes was
:
v frequency band [1] is a promising technology for addressing presented. Multi-round beam searching techniques for initial
i thehighthroughputrequirement(atleast1Gbpsoutdoors)for channelaccess in mmWave systems have been also presented
X
the fifthgeneration(5G)mobilecommunicationnetworks[2], in [15]–[17] primarily intending at alleviating the exhaustive
r
a [3]. Short-range mmWave communication at the unlicensed beamsearchdelayissue.In[18],abeamsearchingmethodfor
band of 60 GHz is already standardized in IEEE 802.11ad mmWave backhaul systems with phased antenna arrays that
[4] and initial theoretical investigations on mmWave cellular is robust to random movements of the arrays was presented.
systems [1], [5] have identified their potentials together with Althoughthemethodavoidsthecostlyexhaustivesamplingof
theirkeychallenges.ThemmWavefrequencieshavebeenalso all pairs of transmit and receive beams using the hierarchical
recently considered as a possible solution for the wireless codebooks proposed therein, multi-round control information
backhaul communication of small cells [6], [7], which are exchange for beam alignment is still needed.
expected to be densely deployed as an efficient means for In this paper, we focus on the mmWave backhaul system
increasing the geographic spectrum reusability [8]. of [18] and present a robust beam alignment method for
Reliable mmWave backhaulingdependson verydirectional wireless channels including a line-of-sight(LOS) component.
transmissions and receptions,which are implementedin prac- Since reducing the time needed to find an acceptable pair
tice either with antennas having large apertures or with large of beams frees up time for data transmission and reduces
phased antenna arrays [6]. By exploiting the fact that the the probability of interfering to adjacent links, the proposed
wavelengthatveryhighfrequenciesisverysmall,largephased method intends at achieving beam alignment in at most two
antennas can be cheaply packed into small form factors and, rounds of control information exchange. The core idea of the
thus,havebeeneffectivelyusedin realizinghighlydirectional method lies on the control information, which is considered
to be the new position of the nodes after each declared beam
y
(B)
misalignment event, on its exchange, and on its utilization M
d t
in designing the BF and combining technique. By assuming t θ(t+1)
1
that each node is equipped with a position sensor (various
d-y(B)
high sensitivity displacement sensors are available in civil t t+1 M(B)
engineering), a set of actions are first performed by one of t+1
the nodes serving as the master node. If needed, a set of
(A)
actions follows by the other node that plays the role of the Mt+1 φ(t+1) y(A) O
slave node, and finally, a second set of actions by the master 1 t+1 t+1,t+1
node might be used. It is noted that positioning sensors have (A)
M
beenalsoconsideredin[19],[20],however,theywereadopted t
-x(A) 0 x(B) x
for identifying and circumventingbeam misalignment only at t+1 t+1
thetransmitnode.Byintroducingasimplemodelthatcaptures
thesmalldisplacementsoftheantennaarraysofthenodesdue
to wind flow or ground vibration, we present representative Fig. 1. The considered point-to-point wireless backhaul communication
system.ThepositionsofthenodesAandBatthetimeinstanttaredenoted
performanceevaluationresultsfortheoutageprobability(OP)
bythepointsM(A)andM(B),respectively,anddefineaCartesiancoordinate
with the proposedbeam alignmentmethodfor varioussystem t t
system.Atthetimeinstantt+1bothnodesAandBmovetonewpositions
and operation parameters. representedbythepointsM(A)andM(B),respectively.Thesenewpositions
t+1 t+1
Notation: Vectors and matrices are denoted by boldface determine thecoordinates ofthepointOt+1,t+1;its firstsubscriptrefers to
lowercase letters and boldface capital letters, respectively. Mt(+A1) andthesecondtoMt(+B1).
The transpose and Hermitian transpose of a matrix A are
denoted by AT and AH, respectively, diag a denotes a
squarediagonalmatrixwitha’selementsinits{ma}indiagonal, A is assumed to have a large uniform linear antenna array
whereas In (n 2) is the n n identity matrix. The ith (ULA)withNA elements,whereasnodeB isequippedwitha
element of a an≥d the (i,j)th e×lement of A are denoted by largeNB-elementULA. We hereinafterassume for simplicity
[a] and [A] , respectively, and A gives the Frobenius that the positions of the nodes coincide with the positions of
i i,j F
norm of A. represents the com|p|le|x| number set, card( ) the centers of their antenna arrays.
C F
is the cardinality of set , denotes the amplitude of To establish wireless communicationbetween nodes A and
a complex number, and FE | ·i|s the expectation operator. B, i.e., meeting a minimum required performance threshold
{·}
d(M,N) denotes the length of the line segment connecting forreliable informationexchange,a controlphasefor channel
the points M and N. Notation x 0,σ2 indicates that accessisadopted.Inparticular,weassumethatcommunication
∼CN
xisacircularly-symmetriccomplexGaussianrandomvariable is realized in discrete time instants each including a control
(cid:0) (cid:1)
withzeromeanandvarianceσ2,whilex (α,β)represents and a data phase. The control phase constitutes of several
∼U
a uniformly distributed random variable in [α,β]. time slots during which data communication is being set
up. We assume that during each discrete time instant the
II. SYSTEMAND CHANNEL MODELS channelremainsconstant,butitmaychangebetweendifferent
The system model under investigation and the considered instances.Duringthecontrolphaseforchannelaccess,relevant
mmWave channel model are presented in the following. In information needs to exchanged between nodes A and B.
this work, we assume that the system nodes are deployed in Thiscontrolinformationexchangerequiresin generala much
theplane,andweleavetheextensionforthethreedimensions less stringent performance threshold than data exchange, and
for the extended version of this paper. can be in principle handled by the consideredcommunication
systems. Another option would be to equip both nodes with
A. System Model
a dedicated robust low frequency communication system in-
Suppose the wireless backhaul communication system of tended exclusively for reliable control information exchange.
Fig.1operatinginthemmWavefrequencybandandconsisting It is adequate for this system to operate at very low data rate,
of two half duplex multi-antenna transceiver nodes A and and hence, it can be narrowband and realized with a single
B. As shown in the figure, the fixed positions of the nodes antenna at each node. Very recently, a method for mmWave
defines a Cartesian coordinate system, according to which beam alignment assisted by a radar system was presented in
the positions of nodes A and B at the time instant t are [21] for vehicular-to-infrastructurecommunication.
representedbythepointsM(A) andM(B),respectively.Their Before signal transmission from node A, the unit power
t t
respectivecoordinatesare (0,0) and(0,d ), whered denotes data stream s (chosen from a discrete modulation set)
t t
the physical distance between the nodes at this time instant. is processed by∈aCBF vector v NA×1, and upon signal
A
Regarding the mmWave access, each node is assumed to be reception at node B, the combini∈ngCvector u NB×1 is
B
∈ C
equipped with one RF chain and is capable of realizing only used for processing the received signal. Similarly, node B
analog transmit BF when being in transmit mode, and only makes use of the BF vector v NB×1 when transmitting,
B
analog receive combining when being in receive mode. Node and node A utilizes the combi∈niCng vector u NA×1
A
∈ C
whenbeingin receivemode.Due to practicallimitationswith mounted on outdoor structures that are exposed to wind flow
the considered analog antenna arrays [5], [18], the analog and gusts. For example, picocell units are expected to be
antenna weights at both nodes are supposed to take values mounted on road signs and lampposts. These structures are
from discrete sets. In particular, we assume that v and u susceptible to movement (or sway) due to wind or ground
A A
at node A are chosen from the finite set of predefined vibration, which might cause unacceptable OP if beam align-
A
N -elementvectors.Thesame holdsfor noFdeB;v andu mentis notfrequentlyperformed[18]. To capturethe random
A B B
belongto the finite set of predefinedN -elementvectors. movements of the antenna arrays at both nodes A and B in
B B
Without loss of generalFity, it is assumed that for any f Fig. 1, we model the displacements at the x-axis and y-axis
A
with f NA×1 and any z with z NB×1 ∈hoFlds for both nodes A and B as
B
[f] 2 ,∈ NC −1 with i = 1,2∈,.F..,N and∈[zC] 2 , N−1
| i| A A | j| B x(A) ( x(A),x(A)), y(A) ( y(A),y(A)), (4a)
with j = 1,2,...,NB. When node A transmits information ∼U − w e ∼U − s n
to node B, the output signal of the combiner at node B can x(B) ( x(B),x(B)), y(B) (y(B),y(B)). (4b)
be mathematically expressed as ∼U − w e ∼U s n
Inthelatterexpressions,x(A),x(A),x(B),andx(B) revealthe
yB =√puHBHBAvAs+uHBnB, (1) position limits in the x-axiws foreboth nwodes, whiele y(A), y(A),
s n
where p is the transmit power, HBA NB×NA denotes the ys(B), and yn(B) represent their position limits in the y-axis.
channel gain matrix between nodes∈BCand A, and n These limits disclose the tolerances of the structures hosting
B
NB×1 represents the zero-mean additive white Gaussia∈n theantennaarraysanditisassumedthattheirvaluesaremuch
Cnoise (AWGN) vector with covariance matrix σ2I . The smaller than the physical distance of the nodes. Note that,
B NB
reverse link with node B transmitting and node A receiving in practice, the distribution of a node’s position in certain
can be similarly defined, and we assume that the forward time periods depends on the characteristics of the underlying
channel H and the reverse channel are reciprocal, i.e., phenomenonthat caused the node’s displacementswithin this
BA
H =HH . period. Althoughthe modelin (4) is quite general, describing
AB BA
the worst case scenario of independent displacements at con-
B. Channel Model secutive time instants, more sophisticated modeling of wind
A geometric channel model with L scatterers is adopted induced displacement (e.g., as in [18, Sec. III-B]) is left for
similar to [5], [12], where each scatterer contributes a single future work.
propagationpathin the communicationlink betweennodesB
III. POSITION AIDEDBEAM ALIGNMENT
and A. We assume that during each discrete time instant the
wirelesschannelremainsconstant,butitmaychangebetween In this section, we first summarize the considered beam
different instances. According to this channel model, H alignment design objective and then present a robust method
BA
included in (1) is expressed as suitable for mmWave backhaul communication systems oper-
ating over fading channels including a LOS component.
H =A (θ)diag a AH(φ), (2)
BA B { } A
A. Design Objective
where matrices A (φ) NA×L, with φ , [φ φ φ ],
A 1 2 L
and A (θ) NB×L, ∈wiCth θ , [θ θ θ ], are·d·e·fined Suppose that at the time instant t + 1 both nodes A
B 1 2 L
∈ C ··· and B move to new positions represented by the points
as follows
M(A) and M(B), respectively, with respective coordinates
t+1 t+1
AA(φ),[aA(φ1) aA(φ2) aA(φL)], (3a) (x(A),y(A)) and (x(B),d y(B)), as shown in Fig. 1.
··· t+1 t+1 t+1 t − t+1
A (θ),[a (θ ) a (θ ) a (θ )]. (3b) According to the presented channel model in Sec. II-B, the
B B 1 B 2 B L
··· channel between the receive node B and the transmit node
In (3), variable φℓ [0,2π] with ℓ = 1,2,...,L denotes the A at the discrete time instant t + 1 can be expressed as
∈
ℓthpath’sangleofdeparture(AoD)fromnodeAandvariable H (t+1)=A (θ(t+1))diag a(t+1) AH(φ(t+1)),
θnℓod∈e[B0,.2Fπo]lrloepwriensgenthtsethinevℓetshtigpaattiho’nssainngl[e22o]f,awrreivaasls(uAmoeAt)haatt wθ(iBtthA+φ1()t,+ [1θ)(,tB+[φ11)(θt (+t+1)1φ)2(t{+θ1()t·+··}1φ)L]A(dte+not1in)]gatnhde
1 2 L
···
the 1st channel path is a LOS one with energy much larger AoDs and AoAs, respectively, resulting from the new node
thaneachoftherestL 1paths.Inaddition,aA(φℓ) NA×1 positions, while a(t + 1) includes their channel gains. The
and aB(θℓ) NB×1−are the array response vectors∈atCnodes objective for beam alignment in this time instant is to design
∈C
A and B, respectively (for ULAs, these vectors are given by the BF vector at node A and the combiningvector at node B
[5, eq. (5)]). In (2), a L×1 includes the path channel as follows
∈ C
gains α ℓ = 1,2,...,L. We further assume that each
ℓ ∀ v (t+1),u (t+1) = max zHH (t+1)f 2.
path’samplitudeisRayleighdistributedand,inparticular,that A B BA
{ } f∈FA,z∈FB
each αℓ (0,NANBPL), where PL denotes the average (cid:12) (cid:12) (5)
pathloss∼betCwNeen nodes B and A. If perfect knowledge of H (t + 1) w(cid:12)as available at (cid:12)both
BA
In practical deployments of wireless backhaul systems, nodes,itiswellknown[23]thatnodesAandB shouldusethe
the antenna arrays of the communicating nodes are usually principal left and right singular vectors of H (t+1) as the
BA
BF and combining vectors, respectively, to maximize the BF In mathematical terms, node A designs its BF vector at each
gain.However,neitherH (t+1)foranyt canbeestimated time instant t+1 as
BA
at any of the nodes due to the assumed hardware limitations
2
nor the nodes are capable of realizing any arbitrary vector. vA(t+1)= min f aA φˆ1(t+1) . (10)
f∈FA − F
Therefore,latestapproaches[13]–[18]relyonbeamsearching (cid:13) (cid:16) (cid:17)(cid:13)
withintheavailablesets Aand B,aimingatachievingbeam Similarly,receivenode B us(cid:13)(cid:13)esits available estim(cid:13)(cid:13)ate θˆ1(t+1)
F F
alignmentwith as much as possible reduced controlsignaling ateachtimeinstantt+1torealizeabeamascloseaspossible
overhead. tothe directionofthe nodeA. Therefore,thisnodeconstructs
its combining vector as
B. Proposed Method
2
u (t+1)= min z a θˆ (t+1) . (11)
Let us consider that, upon installation of the mmWave B z∈FB − B 1 F
backhaul system of Fig. 1 at the initial discrete time instant (cid:13) (cid:16) (cid:17)(cid:13)
By using the vectors given(cid:13)by (10) and (11) at n(cid:13)odes A and
t, both nodes A and B are aware of the coordinate system (cid:13) (cid:13)
B, respectively, the instantaneous received SNR at node B is
defined by the coordinates of the points M(A) and M(B). In
t t calculated as
the next discrete time instant t+1, suppose that both nodes γ = p µ 2, (12)
move to the new position points M(A) and M(B), as shown t+1 σ2 | t+1|
t+1 t+1 B
in the same figure. Each node’s displacement in the x and
where µ is defined as
t+1
y axes is assumed to be available to the node (through, for ∈C
example,dedicatedhighsensitivitydisplacementsensors).The L
µ = α (t+1)uH(t+1)a θˆ(t+1)
latterindicatesthateachnodeisawareofthecoordinatesofits t+1 ℓ B B ℓ
new position, i.e., node A learns the coordinates(x(A),y(A)) Xℓ=1 (cid:16) (cid:17) (13)
t+1 t+1
and B obtains (x(t+B1),dt − yt(+B1)). If also the coordinates of ×aHA φˆℓ(t+1) vA(t+1).
the position of node B become available to A (through, for (cid:16) (cid:17)
Note that for the special case of L = 1, perfectly estimated
example,adedicatedcontrolchannel),thenthelatternodemay
AoD and AoD for the LOS path, and infinite resolutionbeam
estimatetheAoDoftheLOSchannelpathforitstransmission
codebooks at both nodes yields µ = α (t + 1). This
at this time instant as t+1 1
indicatesthat for this ideal case the vectorsgivenby (10) and
arcsin(g ), x(B) x(A) (11) maximize the BF gain described in (5), and equivalently
φˆ1(t+1)=(π−arcsitn+1(gt+1), xt(t++B11) ≥<xtt(++A1)1 , (6) theTShNeRprgoipvoesnedbybe(1a2m).alignment method is summarized in
where the positive real g is given by Algorithm 1. Although the method is presented with node
t+1
A being the transmitter and node B the receiver, it can
d O ,M(B) be evenly used for the case where nodes exchange roles.
t+1,t+1 t+1
gt+1 = d(cid:16) M(A),M(B) (cid:17). (7) AccordingtoAlgorithm1,beamalignmentisconsideredtobe
t+1 t+1 achieved when the instantaneous SNR of the communication
(cid:16) (cid:17) link, measured either by node A during its set of actions or
Fromthespecificnodes’positionsinFig.1attimet+1yields
duringtheactionsofnodeB,isgreaterorequaltoaminimum
d y(A) y(B) required SNR γo for data communication. Note also that the
gt+1 = t− t+1− t+1 . (8) methodincludesthreeseparatephases:i)theNodeARecovery
2 2
x(A) +x(B) + d y(A) y(B) Phase 1; ii) the Node B Recovery Phase; and iii) the Node
r t+1 t+1 t− t+1− t+1 A Recovery Phase 2. When beam alignment is achieved with
(cid:16) (cid:17) (cid:16) (cid:17)
In a similar way, if node A shares its coordinatesat the same only the Node A Recovery Phase 1, no control signaling is
timeinstantwithB,thennodeB canestimatetheAoAofthe neededtobeexchangedbetweenthenodes.Onecontrolsignal
LOS path for A’s transmission as is neededto be sent from nodeA to B when beam alignment
isaccomplishedwithintheactionsinsidetheNodeARecovery
π+φ (t+1), x(B) x(A) Phase1andNodeB RecoveryPhase.Finally,whenallphases
θˆ1(t+1)=(2π−φ11(t+1), xt(t++B11) ≥<xt(t++A1)1 . (9) adreescuritbileizdedan,dtwthoecoonnetrofrlosmignnoaldseaBretroeqAu.ired; the previously
TheAoD andAoA ofthe LOSpathcan beobtainedsimilarly
if node B transmits and node A operates in receive mode. IV. NUMERICAL RESULTS ANDDISCUSSION
Capitalizing on (6) and on the considered channel model In this section we evaluate the performance of the pro-
for each time instant t + 1, we propose that the transmit posedbeamalignmentmethodpresentedin Sec.III-Boverthe
node A utilizes its available estimate φˆ (t+1) to realize a presented mmWave channel model in Sec.II-B. In particular,
1
beam that steers towards the direction of the receive node we evaluate the OP performance defined as the probability
B. To accomplish this, it searches inside its available beam that the instantaneous SNR falls below a minimum SNR
codebook for the vector that is closest to a (φˆ (t+1)). threshold γ . To carry out this evaluation, we have simulated
A A 1 o
F
Algorithm 1 Position Aided Beam Alignment 100
Initialization: Construct the coordinate system of Fig. 1
upon the installation of nodes A and B at the initial time
instant t. Determine the minimum required SNR γo for P)
data communication. O
(
y
1: Nfoordne =AtR+ec1o,vter+y2P,h.a.s.edo1: babilit10-1
o
2: if γn γo, then Pr
3: aSne≥dt vtrAan(nsm)i=t dvaAta(.n−1), uB(n)=uB(n−1), Outage NAo1 Action
4: else A1 and B
5: Obtain the coordinates of the new position Mn(A). AO1p,t iBm uamnd BAF2
6: Compute the AOD φˆ1(n) using (6) and the 10-2
coordinatesof the positionsM(A) and M(B), and 1 1.5 2 2.5 3 3.5 4
n n−1 SNRThresholdγoindB
the point On,n−1.
7: ComputeaA(φˆ1(n))anddesignvA(n)using(10). Fig. 2. OP versus the SNR threshold γo in dB for NA = NB = 32,
8: end if p = 5dB, card(FA) = card(FB) = 33, κ = 20dB, and different phases
oftheproposedmethod.CurvesfortheunfeasibleoptimumBFcaseandthe
9: if γn γo then caseofnoactionforbeamalignment areincluded.
10: Se≥t vA(n) as in step 7, uB(n)=uB(n 1),
−
and transmit data.
11: else in the x-axis and y-axis due to wind gusts are obtained
12: Send the coordinates of Mn(A) to Node B. from (4) with x(wA) = x(eA) = x(wB) = x(eB) = 1.5m and
13: end if yw(A) = ye(A) = yw(B) = ye(B) = 1.5m. The beam codebooks
Node B Recovery Phase: and are constructed by quantizing the feasible sets
A B
F F
14: Uponreceptionofacontrolsignalwiththecoordinates ofdepartureandarrivalangles,respectively.Specifically,each
of M(A), obtain the coordinates of M(B). ithBFvectorbelongingin with i card( )isgivenby
n n A A
15: ComputetheAOAθˆ1(n)using(9)andthecoordinates 1 F ∈ F
of the positions Mn(A) and Mn(B), and On,n. fi = √N [1 e−j2πdλcos(χi) ··· e−j2πdλ(NA−1)cos(χi)]T,
16: Compute aB(θˆ1(n)) and design uB(n) using (11). A
17: if γn γo, then where dλ denotes the spacing between adjacent antenna ele-
18: Tr≥igger node A to transmit data. mentsinwavelengthsandthedepartureangleχi takesdiscrete
19: Set vA(n) as in step 7, uB(n) as in step 16, valueswith step size 21−q(χmax−χmin)within[χmin,χmax],
and transmit data. where q represents the number of angle quantization bits.
20: else Each jth combining vector inside B with j card( B)
F ∈ F
21: Send the coordinatesof Mn(B) to Node A and use is constructed in a similar way to FA with each arrival
uB(n) designed in step 16. angle ψj ∈ [ψmin,ψmax] being quantized with q bits. In the
22: end if following plots, we have set χmin = 60o, χmax = 120o,
Node A Recovery Phase 2: ψmin = 240o, and ψmax = 300o with respect to the nodes’
23: Uponreceptionofacontrolsignalwiththecoordinates orientation in Fig. 1.
of M(B), compute the AOD φˆ (n) using (6) and the In Figs. 2 and 3 the OP with the proposedbeam alignment
n 1
coordinates of M(A) and M(B), and O . methodisdepictedasafunctionoftheSNRthresholdγoindB
n n n,n
24: Compute aA(φˆ1(n)) and design vA(n) using (10). forthetransmitpowerp=5dBandtwodifferentscenarios.In
25: Set vA(n) as in step 23, uB(n) as in step 16, the scenario for Fig. 2, it is considered that NA =NB =32,
card( ) = card( ) = 33 resulting from q = 6, and
and transmit data. FA FB
the Ricean factor κ = 20dB. In Fig. 3, we considered a
26: end for
scenariowhereN =N =16,card( )=card( )=17
A B A B
F F
resulting from q = 5, and the Ricean factor κ = 13.2dB. In
bothfigures,the OPcurvesforthe case ofnoactionforbeam
104 channel samples according to (2) with normalized σ2 alignmentandthecaseofoptimumBF[23]whenperfectchan-
B
and P = d−3.75, where each channel sample appears at nelknowledgeisavailableweresketched.Asfortheproposed
L 1
one discrete time instant. We have considered Ricean fading method, the performance for different sequences of phases is
channels with the κ-factor denoting the ratio of the energy also demonstrated. In particular, we provide the performance
in the LOS channel path to the sum of the energies in for the case where only Node A Recovery Phase 1 is used,
the other non LOS paths [22]. The distance of the nodes denoted by A1; the case where Node A Recovery Phase 1
A and B at the initial time instant t = 1 is assumed to followed by Node B Recovery Phase are used, denoted by
be d = 10m and the random displacements of the nodes A1 and B;andthe case whereall phasesare utilized,denoted
1
100 proposed method was highlighted. Future directions include
theperformanceanalysisofthemethodunderrealisticantenna
patterns and codebooks as well as imperfections (e.g., sensor
P) measurement inaccuracies). We also intend at incorporating
(O directionfindingtechniquesintotheproposedmethodinorder
y
bilit to tackle efficient beam alignmentundernon LOS conditions.
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