On-off Analog Beamforming for Massive MIMO

0.42 MB·

by  Shengli Zhang

1
On-off Analog Beamforming for Massive MIMO
Shengli Zhang, Member, IEEE, Chongtao Guo, Member, IEEE, Taotao Wang, Member, IEEE, and Wei Zhang,
Fellow, IEEE
Abstract—This paper investigates a new analog beamforming (DAC) or analog-to-digital converter (ADC). The implemen-
architectureformassivemultiple-inputmultiple-output(MIMO) tation of a large amount of RF chains and DACs/ADCs
systems,whereeachofthemultipletransmitantennasisswitched
for digital beamforming is rather costly in massive MIMO
to be on or off to form a beam according to the channel state
systems.
information at transmitters. This on-off analog beamforming
(OABF) scheme has the advantages in terms of both hardware InordertosavethenumberofRFchainsandDACs/ADCs,
complexitiesandalgorithmiccomplexities.OABFcancompletely analog beamforming is proposed and attracting more atten-
7 removethehigh-cost,power-consumingandbulkyanalogphase- tions.AnalogbeamformingusuallyemploysRFphaseshifters,
1 shifters that are extensively employed by traditional analog variablegainamplifiers(VGAs)intheanalogdomaintoform
0 beamforming schemes; it only requires the deployment of low-
signalbeams.Theworksonanalogbeamformingshowthatthe
2 cost analog switches that are easy to implement. Moreover, we
showthatthebeamsformedbysuchsimpleantennaon-offswitch performance gap of analog beamforming with respect to the
n
operations can achieve rather good performances with low- conventional digital beamforming is justified by the reduction
a
complexitybeamformingalgorithms.Specifically,wefirstpropose in its hardware costs [5], [9]–[21].
J
two optimal signal-to-noise ratio maximization algorithms to Although analog beamforming has power and cost advan-
1
determine the on-off state of each switch under the per-antenna
3 tages over digital beamforming, the used RF phase shifters
power constraint and the total power constraint, respectively.
After that, we theoretically prove that OABF can achieve the raise tremendous challenges and reliability issues for the
] full diversity gain and the full array gain with complexities up design of RF hardwares, especially for millimeter wave
T
to a polynomial order. Numerical results are consistent with our (mmWave) carrier signals [22]. First, RF phase shifters do
I
. theoreticalanalysis.WebelievethatthesimplestructureofOABF not achieve fine enough phase resolutions, which limit the
s makesmassiveMIMOmucheasiertoimplementinrealsystems.
c beamforming accuracy. Moreover, shifting signal phases in
[ the RF domain degrades noise figures along with RF chains.

1 Therefore, RF phase shifters make the implementation of
v I. INTRODUCTION analog beamforming still challenging and costly.
6 Among the potential solutions for 5G cellular communi- In this paper, we propose to further reduce the hardware
0
cations, massive multiple-input multiple-output (MIMO) has complexity of analog beamforming by replacing all the bulky
9
8 been shown to be able to increase the system spectral ef- and costly RF phase shifters with RF switches. The RF
0 ficiency by several times [1], [2]. Recently, extremely high switches are employed to control the on-off state of each
. frequency (EHF) bands from 30 to 300 GHz are designated transmit antenna according to the channel state information
1
0 for future wireless communications [3], since the available (CSI)attransmitters.Werefertothisnewanalogbeamforming
7 bandwidths over EHF are much wider than current cellular architecture as on-off analog beamforming (OABF). The aim
1 networks. For the deployment of wireless systems over fre- of OABF is to form beams over the air via these transmit
v: quencies higher than 30 GHz, a large amount of antennas are antennas activated to be on state, without any phase and am-
i not only possible but also necessary to compensate the small plitude pre-processing neither in the analog domain nor in the
X
array gain due to the compact antenna size [4], [5]. digital domain. Actually, commercial RF switches have been
ar A typical application of massive MIMO in cellular com- widely used in wireless transceivers, and they possess very
munications is that a base station (BS) equipped with a large attractive properties such as cheap, compact size, fast speed,
number of antennas simultaneously serves mobile users using almost no power consumption, linear for wide bandwidth and
beamforming [6]. Digital beamforming is a well-developed high frequency [23].1 Therefore, the OABF architecture is
technique for MIMO systems, because of its advantages of much easier for implementations than the RF phase shifters
flexibility, adaptability, and performance optimality [7], [8]. based analog beamforming.
However,digitalbeamformingistooexpensiveandtoopower- Intuitively, we can determine the on-off status of the
consuming to be applied in massive MIMO systems, since switchesoftransmitantennastomaximizethereceivedsignal-
it requires each antenna to be connected with one expensive to-noise ratio (SNR) — we select the subset of antennas that
radio frequency (RF) chain and a digital-to-analog converter havebetterchannelconditionsandsimilarphasesfortransmit-
tingsignals.Atthefirstsight,theproblemonfindingthesubset
S. Zhang and C. Guo are with the Faculty of Information Engineering, of transmit antennas that maximizes SNR looks like a com-
ShenzhenUniversity,Shenzhen,China(email:{zsl,ctguo}@szu.edu.cn).
binatorial optimization problem, which is NP-hard in general.
T.WangiswiththeInstituteofNetworkCoding,TheChineseUniversityof
HongKong,HongKong,China,andtheFacultyofInformationEngineering, Incontrasttothisintuition,wefindtwooptimalbeamforming
ShenzhenUniversity,Shenzhen,China(email:[email protected]).
W. Zhang is with the School of Electrical Engineering and Telecommu- 1Forexample,accordingtothepowermodelderivedin[24]fromtherecent
nications, the University of New South Wales, Sydney, Australia (email: progressesonmmWaveRFcircuits,typicallythepowerofaRFphaseshifter
[email protected]). isround30mWandthatofaRFswitchisonlyaround5mW.
2
algorithms that determine the on-off status of each antenna RF RF
DAC ADC
Chain Chain
for OABF with only a linear complexity and a polynomial
complexity,respectively.Moreover,wetheoreticallyprovethat RF RF
DAC ADC
our OABF scheme can achieve the full array gain and the . Chain Chain .
diversitygainwiththeproposedbeamformingalgorithms.The . .
. .
main contributions of this paper are summarized as follows. RF RF
DAC ADC
1) We propose OABF — a new analog beamforming ar- Chain Chain
chitecture that only relies on RF switches — for mas- (a) digital beamforming
sive MIMO systems. OABF can achieve beamsforming
without any phase or amplitude pre-processing neither
in the analog domain nor in the digital domain. This
simple OABF architecture further reduce the hardware
RF . . RF
complexity of analog beamforming. DAC Chain . . Chain ADC
2) We develop two optimal beamforming algorithms for . .
OABF.Thebeamformingalgorithmsdeterminestheon-
off statues of the transmit antennas for maximizing the
(b) analog beamforming
received SNR. One algorithm is developed under the
per-antenna power constraint and it only has a linear
complexity; the other is developed on the total power
constraint and it only has a polynomial complexity.
3) We perform theoretical analysis to show that the simple RF . . RF
DAC ADC
OABF architecture can achieve the full array gain and Chain . . Chain
. .
the full diversity gain with the proposed beamforming
algorithms; the achievable rate gap between the optimal
phase-aligned beamforming and OABF is a constant (c) antenna selection
2log π = 3.3 bits/symbol that dose not scale with
2
the number of antennas and SNR. Simulations are per-
formed to validate our results.
RF . . RF
A. Related Works DAC ADC
Chain . . Chain
The analog beamforming algorithms and their correspond- . .
ing analog RF architectures have been extensively studied
in many works [5], [9]–[21]. For these analog beamforming (d) on-off analog beamform
schemes, the implementations of RF phase shifters and/or
VGAs are necessary. Fig.1. Blockdiagramsforthearchitecturesof(a)digitalbeamforming;(b)
In[24],themassiveMIMOtransceiverarchitecturethatonly analog beamforming; (c) antenna selection; (d) on-off analog beamforming.
Theleftisthetransmitterandtherightisthereceiver.
employs RF switches is proposed. Its hardware structure is
very similar to our OABF. However, there is no optimal solu-
tiontothebeamformingproblemfoundin[24].Itclaimedthat
The rest of this paper is organized as follows. Section
the optimal beamforming problem has a complexity growing
II gives the architecture of OABF and the exploited system
exponentially with the number of antennas, and suboptimal
model. The beamforming algorithm under per-antenna power
solutionwasgivenbyusingsimultaneousorthogonalmatching
constraint and its performance analysis are presented in Sec-
pursuit strategy [16].
tion III. The beamforming algorithm under total power con-
Our OABF is related to the subset antenna selection
straint and its performance analysis are presented in Section
schemes, where the best K antennas are selected from the
IV.InsectionV,numericalsimulationsareperformedtoverify
total N antennas [25]–[27]; however, OABF is substantially
the performance of OABF. Section VI concludes the paper.
different from antenna selection. For the traditional antenna
selectionschemes,eachselectedantennaisconnectedwithone
II. OABFARCHITECTUREANDSYSTEMMODEL
RF chain (and followed by a phase shifting operation in the
A. OABF Architecture
digital domain) to achieve a coherent combination. Therefore,
K is usually determined by the number of available RF In this part, we present the architecture of the proposed
chains,andbeamformingeffectcomesfromthephaseshifting OABF. For a purpose of comparison, we first review three
processing at the digital domain. For OABF, all the selected typicalexistingmulti-antennabeamformingarchitectures,i.e.,
antennas are directly connected to one RF chain without any digital beamforming, phase-aligned analog beamforming and
RFphaseshiftersintheanalogdomainandanyphaseshifting antenna selection [28].
operations in the digital domain; the beamforming effect is In most current wireless systems, fully optimal beamform-
simply achieved by selecting a subset of the antennas (see the ing is implemented in the digital domain — the signal phases
later part for details). (and amplitudes) are accordingly adjusted inside the digital
3
signalprocessing(DSP)unit.Thearchitectureofdigitalbeam- showninFig.1(d)),whereasubsetoftheantennasisselected
forming is illustrated in Fig. 1 (a)2, where the components to connected to one RF chain and these antennas are used to
highlighted by red are phase shifters. transmit signals without any analog signal processing; other
The advantage of the digital beamforming architecture is antennas are disconnected. In our OABF, the RF switches
thatthephaseshifterscanbeeasilyimplementedinthedigital on the selected antennas are kept on while others are kept
domain. This saves the expenses on RF circuits, especially off. Therefore, we refer to it as on-off analog beamforming
for wireless systems over low frequency bands where phase (OABF).
shifters are difficult for design and very expensive for inte- The intuition for our OABF is that a subset of the N
gration [22]. The disadvantage of digital beamforming is that antennas, with better channel conditions and similar phases,
each antenna requires a RF chain and an ADC/DAC. This is selected to connect to the RF chain and thus can form a
makes digital beamforming not suitable for massive MIMO signalbeamwithlargeSNR.TheconnectionsbetweenOABF
systems. Moreover, for wireless systems with high frequency andanalogbeamforming/antennasectionareasfollows.Ifthe
and wide bandwidth (e. g., mmWave systems), the costs of cardinalityofthesubsetislimitedto1,OABFdegradestothe
a RF chain, especially the price, power consumption and antenna selection scheme; if the analog coefficients in analog
space cost, are much higher than that of a antenna [22]; beamforming is limited to be 0 or 1 — RF phase shifters are
the implementation of the low power, high solution and fast replaced by RF switches, it reduces to our OABF.
DACs/ADCs are also very challenging [22]. Therefore, we
needalternativebeamformingarchitecturesformassiveMIMO
B. Baseband System Model
to save RF chains, DACs/ADCs and can also maintain the
Thispartintroducesthebasebandsystemmodelanditsrela-
gains of multiple antennas.
tivenotationsforOABF.Withoutlossofgenerality,thispaper
One typical such scheme is the analog beamforming archi-
considersapoint-to-pointtransmissionwithN antennasatthe
tecture that fulfills the beamforming operation in the analog
transmitter and one antenna at the receiver. The extension to
domain, as illustrated in Fig. 1 (b). For analog beamforming,
multiple receive antennas is straightforward. We assume that
each antenna is connected with one RF phase shifter to rotate
there is only one data stream, hence one RF chain at each
the signal phase in the analog domain. This analog beam-
side. The channel between the j-th transmit antenna and the
forming is also called phase-aligned analog beamforming.
receiveantennaisdenotedbyh ,whichareperfectlyknownat
The signal amplitudes can also be controlled in the analog j
the transmitter by some feedback schemes or the reciprocity
domainusingRFVGAs.Inthispaper,wejustconsidertheRF
of the channel. We further assume that all h ,1 ≤ j ≤ N
phase shifter based phase-aligned analog beamforming as the j
are independent identically distributed variables with complex
benchmark to our OBFA. It is shown in [28] that the same as
Gaussian distribution, i.e., h ∼ CN(0,1) for all j. The
digital beamforming, the phase-aligned analog beamforming j
channel coefficients keep constant during one packet trans-
can achieve the full diversity gain and the full array gain,
mission and change independently between different packet
which equal to the antenna number N.
transmissions.Thisimpliesablockfadingenvironmentthatis
However,analogRFphaseshifters,especiallytheseworking
valid for indoor channels and many outdoor channels [29].
with wide bandwidth and high frequency, are also very costly
A more detailed OABF transmit architecture is illustrated
and bulky [22]. A further lower complexity/cost architecture
in Fig. 2. We denote the transmit power for the j-th antenna
is antenna selection, where only one antenna is selected to
by P . At the transmitter side, some antennas that forms a set
connect to one RF chain, as shown in Fig. 1 (c). Antenna j
T are selected for transmissions. Then, the baseband received
selection schemes can also obtain the full diversity gain [28].
But, its array gain is only (cid:80)N 1/i, which is much smaller signal can be expressed as
i=1
than the full array gain of N. Antenna selection is then y = (cid:88) (cid:112)P h x+n
j j
extended to select K antennas from all the antennas if there
hj∈T
are K available RF chains. For a multiple-input single-output
N
transmission, simply selecting the K channels with largest (cid:88)(cid:112)
= P I h x+n
i i i
channelgainsisalreadytheoptimalantennaselectionscheme.
i=1
With a comparison between analog beamforming and an- s.t. I =1 if h ∈T, else I =0 (1)
i i i
tenna selection, we can find that analog beamforming utilizes
alltheantennastoalignsignalphases;whileantennaselection where n ∼ CN(0,σ2) is the receiver side additive white
only utilizes one antenna (connected with one RF chain) Gaussian noise and Ii is an indicate variable, T is a subset of
without any further analog signal processing. In this paper, the set {h1,h2···hN}. By intuition, select the best subset T
we propose a new low complexity analog beamforming archi- to optimize the received SNR is a combination optimization
tectureinbetweenantennaselectionandanalogbeamforming. problem and has an exponential complexity with respect to
The proposed analog beamforming architecture is OABF (as the number of antennas, as discussed in [24].
In this paper, we consider two cases of power constraint,
2Forsimplicity,thispaperonlyconsidersthecaseofonlyonedatastream the per-antenna power constraint, i.e., Pj ≤ Po,∀hj ∈ T,
transmitted from the BS to the mobile user. The extension to multiple and the total transmit power constraint i.e., (cid:80) P ≤ P .
data streams is straightforward. Without losing generality, we only consider hj∈T j t
For both cases, we develop beamforming algorithms with low
beamformingbyshiftingsignalphases,andweomittheamplitudeadjusting
operations. complexities.
4
Image
Data RF Chain Sector 3 Sector 2
Splitter
Information Transmitter PA 1
Receiver LNA Conme/rbi Sector 4 2 Sector
1
Real
Channel
Control 4
Information O
OABF Algorithm Information Sector 5 3 Sector 8 1
Fig.2. Thesystemstructureforon-offanalogbeamformingatthetransmitter.
Sector 6 Sector 7 O
2
Forperformancecomparisons,weanalyzetwobasicasymp- O
3
totic gains of beamforming: the array gain and the diversity O4
gain. As defined in [30], the array gain is referred to as the
V {h,h ,h} V {h,h} V {h,h ,h} V {h,h}
increase of the average SNR achieved by using multiple an- 1 1 2 3 2 1 2 3 1 2 4 4 1 4
tennas, with respect to the single-input single-output case; the V {h} V {h,h} V {h} V {h ,h}
diversitygainisreferredtoasthedecreaseratioofthebiterror 5 4 6 3 4 7 3 8 2 3
rateP averagedoverthefading,d=−lim logPe .
e SNR→∞ log(SNR) Fig.3. Thecomplexplanewherethefourcomplexchannelcoefficientsare
denoted by the blue vectors; their orthogonal lines are denoted by red dash
linesandtheydividethewholeplaneintoeightsectorregions.
III. OABFWITHSEPARATEPOWERCONSTRAINT
In this section, we consider the scenario that the power
of each transmit antenna is separately constrained by Po At a first glance, the problem looks like a combinatorial opti-
and there is no constraint on the total transmit power. This mization problem, which is NP hard in general and the high
assumption results from the case that each antenna in a complexity is not affordable for massive MIMO. However,
practical transmitter is driven by a separate power amplifier we will show that this optimization problem in (4) is actually
that operates properly only when its transmit power is below not NP hard by proposing an optimal algorithm only with
a predesigned threshold. For current analog beamforming a linear complexity. An intuitive explanation is that channel
schemes, the separate transmit power constraint is sometimes coefficients h in T∗ must have close phases to contribute
j
more relevant than the total power constraint [31]. to each other, and we only need to consider the sets whose
Then, each selected antenna will transmit with the maxi- elements have adjacent phases. The detailed algorithm is
mized power Po and the system model in (1) can be rewritten elaborated as follows.
as
We first plot all the N complex channel coefficients h
(cid:112) (cid:88) j
y = Po hjx+n. (2) (j = 1,2,··· ,N) on a two-dimensional complex plane,
j∈T where the horizontal and vertical axes correspond to their real
Therefore, the received SNR at the receiver can be expressed and imaginarycomponents, respectively.Since everycomplex
as number can be mapped to a vector on the complex plane, we
SNR = Po|(cid:80)j∈T hj|2. (3) use a complex number and its vector (on the complex plane)
s σ2 interchangeably in our descriptions, when ambiguity is not
caused.
In this section, we first introduce an optimal and linear
For each vector that represents a complex channel coef-
complexity algorithm to determine the set T that maximizes
ficient h , we plot its orthogonal line O passing through
SNR fortheseparatepowerconstraintcase.Thatbeamform- j j
s
the origin on the complex plane, j = 1,2,··· ,N. Then, the
ing algorithm is referred to as OABF-s. After that, we prove
complexplaneisdividedinto2N sectorsbytheN orthogonal
the full diversity gain and the full array gain of OABF-s.
lines. An example is shown in Fig. 3, where there are 4
complex channel coefficients (their vectors are denoted by
A. Optimal OABF-s Algorithms blue solid lines) and their orthogonal lines (denoted by red
With the separate power constraint on each antenna, maxi- dot lines) divide the complex plane into 8 sectors.
mizingSNR isequivalenttomaximizingthereceivedsignal AsshownintheexampleofFig.3,eachorthogonallineO
s j
power|(cid:80) h |2 in(3).Specifically,wecanfindtheoptimal bisectsthecomplexplaneintotwohalves;foranyvectorlying
j∈T j
subset T∗ by inthesamehalfplaneasachannelcoefficienth ,thechannel
j
(cid:88)
T∗ =argmax| hj| (4) coefficient hj has positive projection on this vector, and vice
T versa. Since each sector must be in one of the two half planes
j∈T
5
divided by each O , we can observe an important property of 1) Complexity: As shown in Algorithm 1, each channel
j
the sectors: the projections of a channel coefficient h on any coefficient is added for one time and minus for another time
j
vectorsthatfallinthesamesectoralwayshavethesamesign, (only a part of overlapped channel coefficients in V are
1
i.e., minus or plus. For sector k (k = 1,2,··· ,2N), we use involved in the calculation four times), and the complexity
V to denote the set of complex channel coefficients where is linear with the number of antenna elements. Specifically,
k
each element h lies in the same half plane divided by O as the algorithm can lead to the global optimality with a linear
j j
sector k — all the channel coefficients in set V have positive complexity O(N).
k
projections on the vectors falling in sector k. In Fig. 3, we 2) Diversity Gain: Our OABF-s scheme performs better
also show these sets with respect to each sector. than the antenna selection scheme, since antenna selection is
We define that f∗ is the summation of all the elements in only a special case of our OABF-s. Since antenna selection
the optimal set T∗, i.e., f∗ =(cid:80) h . Then, we have the can achieve the full diversity gain, OABF-s must also be able
following result for f∗, which whijl∈lTh∗elpjus to determine the to achieve the full diversity order N.
optimal set T. 3) ArrayGain: Forthearraygainanalysis,wefirstpresent
Theorem 1: If f∗ is located in sector k, then we have the a sub-optimal algorithm, referred to as OABF-b, to bridge the
optimal set T∗ =V . optimal OABF-s and the antenna selection scheme. We will
k
Proof: It can be proved by contradictions to the assump- thentheoreticallyprovethatOABF-bcanachievethefullarray
tionT∗ (cid:54)=V .Ononehand,ifthereisanycomplexnumberh gain. The sub-optimal OABF-b algorithm is given as follows.
k i
inV butnotinT∗,wecanincreasethevalueof|(cid:80) h |
by akdding h to T∗, since h has a positive projehcjt∈ioTn∗ ojn Algorithm 2 OABF-b Algorithm
i i
f∗ (f∗ falls in sector k). On the other hand, if there is any 1: Select the channel coefficient with maximum amplitude
cthhoaems vpaalelnuxeegnaoutfimv|eb(cid:80)eprhrojh∈jieTcn∗tioohtnj|inobnyVfkd∗eb.leuTttihniegnrehTfio∗rf,er,owmbeotTcha∗nc,asisniencscreefianhsdei 23:: dMAedandkoetaeladlesaleestmhRemn=,tsmh{hi=mtoa}Rr.gimfiatsxip=ro1,j2e,c··t·i,oNn|ohni|hmispositive,
contradictions to the assumption that Vk is not the optimal i.e., R={R,hi} if (hi∗hm+h∗i ∗h∗m)≥0.
set. Hence our theorem is proved.
Due to the fact that f∗ must fall in one of the 2N sectors,
the optimal set T∗ must equal one of the 2N sets V for With reference to Fig. 3, we can see that the OABF-b
k
k = 1,2,··· ,2N. In other words, the optimal set T∗ can be algorithm simply divides the complex plane into two halves
AdentdermaicnoendseaqsuTen∗c=e oVfkT∗,hweohreermek1∗is=thaergstmataexmke|n(cid:80)t inhjt∈hVeknhejx|t. bchyatnhneelorctoheofgfiocnieanltlsinaetothfehsma;mtehehnalift spelalencetsasallhtmh.eWcoemhpalvexe
corollary. the following theorem for OABF-b.
Corollary 1a: When changing from one set V to the Theorem 2: The OABF-b algorithm can achieve the full
k−1
next set V , at least one elements in V is removed or at diversity gain and the full array gain under the per-antenna
k k−1
least one new element is added. (The proof of this corollary power constraint.
is straightforward and we omit it here. ) Proof: We prove the full array gain by proving that the
According to the above discussion, we can summarize expected received SNR of OABF-b is larger than that of
the algorithm for obtaining the optimal T∗ as the following the phase-aligned analog beamforming scheme divided by a
algorithm. constant factor π2. The detailed proof can be found in the
appendix.
Algorithm 1 OABF-s Algorithm Since the optimal OABF-s algorithm performs better than
OABF-b, OABF-s can also achieve the full array gain with a
1: Partition the whole space in to 2N sectors based on the
linear complexity.
orthogonal lines of the N complex channel coefficients.
Since the SNR loss of OABF-b is less than 1/π2 compared
2: For each sector k, we determine the associated set Vk.
to the optimal phase-aligned analog beamforming, we can
Particularly, if h has a positive projection on sector k,
i
directlyobtaintheupperboundoftheratelossfortheOABF-s
then we have h ∈V ; otherwise, we have h ∈/ V .
i k i k
(cid:80) algorithm. We state it in the following corollary.
3: Calculate f1 = j∈V1h(cid:80)j. (cid:80) Corollary 2a: Compared to phase-aligned analog beam-
4: Calculate fk =fk−1− j∈Vk−1\Vkhj + j∈Vk\Vk−1hj forming,theachievableratelossofOABF-sisupperbounded
for all k =2,3,··· ,2N.
5: Obtain k∗ = arg max |fk| and obtain the corre- by 2log2π, which is independent of the antenna number.
k=1,2,···,2N 4) Robustness: Although we assume perfect CSI at the
sponding set T∗ by T∗ =V .
k∗ transmitter, our OABF-s algorithm is robust to inaccurate
CSI. First, OABF-s only depends on the relative phase of
the channel coefficients, the channel amplitude errors does
B. Performance Analysis
not affect the function of OABF-s. Moreover, since we have
Inthispart,weshowthattheOABF-salgorithmcanleadto proved that OABF-b can also achieve the full diversity gain
the global optimality with linear-time complexities. We also andthefullarraygain,asuboptimalalgorithm(canbetreated
prove that the OABF architecture with the OABF-s algorithm as a version of the optimal algorithm with errors) can also
can achieve the full diversity gain and the full array gain. achieve a near optimal performance.
6
IV. OABFWITHTOTALPOWERCONSTRAINT iii) for any two adjacent sectors, their corresponding sets only
differ by one element. These conditions can be explained
So far, we have studied the OABF algorithm subject to the
by noting that the values of the N channel coefficients are
per-antennapowerconstraintatthetransmitter.Inthissection,
we study the case where a total power constraint P is evenly discrete. Therefore, there must be a small sector with angles
t
allocatedtoalltheK selectedantennas.Thispowerconstraint [θ−δl,θ+δr], determined by two small values δl,δr, such
that T(θ,K) is the same for on any θ in this sector, T(θ,K)
means that we just need to deploy one power amplifier at the
changes when θ is outside this range. Since the change of θ,
RF chain.
With the total power constraint P , the transmit power of as well as hj’s projection on it, is continuous, there must be
t
each antenna is Pt/K and the system model in (1) can be an adjacent sector, [θ+δr,θ+δr1], such that for any θ(cid:48) in it
we will generate a new set T(θ(cid:48),K) with only one different
rewritten as
y =(cid:112)P /K (cid:88) h x+n. (5) element from T(θ,K).
t j
Third, we divide the whole complex plane into such
hj∈T
proper sectors by finding the boundaries between adjacent
where K is the cardinality of the set T. The received SNR
sectors. At a boundary, the two different elements of the
can then be expressed as
two corresponding sets should have the same projections.
SNR = Pt|(cid:80)j∈T hj|2. (6) In particular, for a given θ which belongs to one sector,
t Kσ2 T(θ,K) = {hπ1,hπ2,··· ,hπK} is the set formed with the
largest K projections, and TC = {h ,h ,··· ,h }
From (6), we can see that maximizing SNRt is equivalent to πK+1 πK+2 πN
maximizing |(cid:80)j∈Thj|2.Differentfromthecaseofper-antenna is the complementary set formed by the other channel coeffi-
K cients. The different element, belonging to the set of the next
power constraint, we need to find a set not only with a large
sector,mustalsobelongtoTC.Tofindtheboundary,weneed
summation but also with a small cardinality.
to find a minimum phase shift δ >0, such that
In this section, we first introduce the optimal algorithm,
referred to as OABF-t, to determine the optimal set T∗ that
maximizes |(cid:80)j∈KThj|2. After that, we prove the full diversity hmi∈inT|hi|cos(θ+δ−θi)=hmj∈aTxC|hj|cos(θ+δ−θj), (7)
gain and the full array gain of OABF-t by showing its
superiority over OABF-s. where θi and θj are the phase of hi and hj, respectively.
Therefore, δ can be calculated as
A. Optimal OABF-t Algorithm (cid:18) (cid:19)
|h |cos(θ−θ )−|h |cos(θ−θ )
For the previous OABF-s algorithm, the essential idea is δ =arctan min j j i i .
to combine a number of complex channel coefficients hj to hi∈T,hj∈TC |hi|sin(θ−θi)−|hj|sin(θ−θj)(8)
obtain one set Vk such that all the elements of this set have Then, θ+δ is a boundary of the sector. In a similar way, we
positive contributions (projections) to any vectors in sector can find all the boundaries so as to determine all the sectors.
k. And all the elements outside this set have non-positive As discussed above, we can generate the set TK for each
contributions to any vectors in sector k. In other words, given (cid:80)
sector and find the maximum value | h | of all the
one sector k, there is an unique such set V . Extending this hj∈T(K) j
k generated sets. The detailed algorithm of OABF-t is given as
idea, we obtain the effective part of OABF-t algorithm, which
Algorithm 3.
(cid:80)
maximizes| h |undertheconditionthatthecardinality
hj∈T j
of the set T is given. Then, the OABF-t algorithm, which
maximizes |(cid:80)j∈Thj|2, is finally obtained by checking all
K B. Performance Analysis
possible cardinalities of the sets.
First, we use the obtained CSI to find the set T that 1) Complexity: We first analyze the complexity of the
(cid:80)
maximizes | hj∈T hj| and the cardinality of set T (the OABF-t algorithm. In this algorithm, the number of the for
number of selected antennas) is K. We use T(K) to denote iterations is less than N; the number of the while iterations is
(cid:80)
this set with K elements that maximizes | h |. We of the order N since one element of the set T is changed
use θ(K) to denote the phase of f(K) =(cid:80) hj∈T(Kh) .jThen, during one iteration; inside the while iteration, the most
we sort the N channel coefficients h inhj∈thTe(Kd)esjcending complex processing is the calculation of δ in (8), whose
j
order according to their projections on f(K), denoted as h . complexity is less than N2/4. Therefore, the complexity of
Then T(K) must be formed by the first K coefficients, i.πej., the OABF-t algorithm is of the order N4, i.e., a polynomial
h ,h ,··· ,h . We refer to this process as the generation complexity.
π1 π2 πK
of set T(θK,K) = {h ,h ,··· ,h } based on the vector 2) Diversity gain and array gain: We now analyze the
π1 π2 πK
angle θ(K). diversitygainandarraygainoftheOABF-talgorithm.Firstly,
Second, we see that it is possible to divide the whole com- OABF-t performs better than the antenna selection scheme.
plexplane,anglefrom0to2π,intoM sectors,whichsatisfies Therefore it can achieve the full diversity gain. Secondly,
the following conditions: i) for any θ in the same sector, all OABF-t performs better than OABF-s. Therefore, based on
generated sets T(θ,K) are always the same; ii) for any θ(cid:48) the expected received SNR calculation in (15), the expected
outside this sector, we will generate a different set T(θ(cid:48),K); received SNR of OABF-t denoted by γ can be bounded by
2
7
Algorithm 3 OABF-t Algorithm
1: Calculate the set Ts with OABF-s Algorithm and Ks = 80 OABF−s
|Ts|. Optimal Scheme
2: Initialize K∗ = 1,f∗ = maxi|hi|2 and T∗ = hj where 70 Antenna Selection
j =argmax |h |.
i i
3: for K =2 to Ks do 60
4: Initialize θ =0, f =0. NR
S50
5: hSoπjrt hj according to their projections on θ, denoted as Received 40
76:: wTGheCnile=eraθ{teh<πtKw2+πo1,dshoeπtsK+a2s,·T··=,hπ{Nh}π1,hπ2,··· ,hπK} and Normalized 30
98:: if |f(cid:80)=Ki=|1(cid:80)hKiπ=i1|2h>πi|f2,tThe0n=T 20
10: end if 10
11: Calculate δ as in (8)
0
12: Update θ as θ+δ, 0 5 10 15 20 25 30 35 40 45 50
13: UpdateT andTC byexchangingthetwocorrespond- Number of Antennas
ing elements in (8)
14: end while Fig.4. ThenormalizedreceivedSNRintermsofthenumberofantennas.
15: if f/K >f∗ then
16: Update f∗ =f/K, K∗ =K, T∗ =T0
17: end if 12 OABF−s
18: end for Optimal Scheme
Antenna Selection
10
that of OABF-s
8
P
γ ≥ t (N +N(N −1)π/4) (9) e
2 Ksσ2 Rat
> σP2t (1+(N −1)π/4). hievable 6
Ac
It means that the order of the array gain that is the ratio of γ
2 4
over P /σ2 for large N can also achieve the largest value N.
t
Thus, we have the following theorem.
Theorem 3: The optimal OABF-t algorithm can also 2
achieve the full diversity gain and the full array gain with
the total power constraint and a polynomial complexity.
0
0 10 20 30 40 50
Number of Antennas
V. NUMERICALRESULTS
In this section, we present numerical results to show the
Fig.5. Theachievablerateintermsofthenumberofantennas.
performancesofourOABF.Insimulations,thenoisevariance
at the receiver side is normalized to unit, i.e., σ2 = 1.
For performance comparisons, we evaluate three different receivedSNR, Po|(cid:80)hj∈Thj|2,overthetransmitpower,P |T|.
beamformingschemes:i)ourOABFscheme;ii)theRFphase σ2 o
The results are presented in Fig. 4. We can see that the
shifters based phase aligned analog beamforming scheme that
normalized SNR of both OABF-s and the optimal scheme
is also referred to as the optimal scheme in this section; iii)
increase linearly with the number of antennas, i.e. achieving
the antenna section scheme.
the full array gain. On the other hand, the normalized SNR of
the antenna selection scheme increases in a log way.
A. OABF with Separated Power Constraint
In Fig. 5, the average achievable rate at the receiver is
We first present the numerical results under the per-antenna obtained by averaging over the instantaneous rates for each
power constraint. With a per-antenna power constraint, the channel realizations according to log(1+SNR ). We can see
s
transmit power of each antenna is set to a fixed value of thattheratesofallthreeschemesincreasewiththenumberof
P =1. Each channel coefficient is randomly generated with antennas. The gap between OABF-s and the optimal scheme
o
complex Gaussian distribution of CN(0,1). is upper bounded by a constant as stated in Corollary 2a.
In order to show the array gain of different schemes, we Fig. 6 presents the outage probability, i.e., the probability
simulate the normalized SNRs at the receiver by increasing thattheexactreceivedSNRislessthenagiventhreshold.We
the antenna number. The normalized SNR is defined as the canclearlyseethediversityordersof1,2,3whentheantenna
8
100 7
N=1 OABF−t
Optimal Scheme
Antenna Selection
10−1 6
N=2
10−2 5
ability e Rate
utage Prob10−3 Achievabl4
O 3
10−4
OABF−s,N=1
N=3
Optimal Scheme,N=1 2
10−5 OABF−s,N=2
Optimal Scheme,N=2
OABF−s,N=3
1
10−60 Optimal S5cheme,N=3 10 15 20 0 10 Nu2m0ber of Anten3n0as 40 50
SNR (dB)
Fig.8. Theachievablerateintermsofthenumberofantennas.
Fig.6. TheoutageprobabilityintermsofSNR.
100
80
OABF−t N=1
Optimal Scheme
70 Antenna Selection
10−1
60
N=2
R
d Received SN4500 age Probability10−2
malize30 Out
Nor 10−3 OABF−s,N=1
20 Optimal Scheme,N=1 N=3
OABF−s,N=2
Optimal Scheme,N=2
10
OABF−s,N=3
Optimal Scheme,N=3
10−4
0 0 5 10 15 20
0 10 20 30 40 50
SNR (dB)
Number of Antennas
Fig.9. Theoutageprobabilityintermsofthenumberofantennas.
Fig.7. THEreceivedSNRintermsofnumberofantennas.
number is N = 1,2,3, respectively. When N = 1, OABF-s In Fig. 9, we show the outage probabilities of the optimal
and the optimal scheme have the same performance; when scheme and our OABF-t scheme. As we can see that both
N =2, there is a gap about 2.5 dB, and this gap increases to schemes can achieve the full diversity order of N. The gap
4 dB when N =3. between the two schemes is about 1 dB for N =3, which is
much smaller than the case of per-antenna power constraint.
B. OABF with Total Power Constraint
Wethenpresentthenumericalresultunderthetotaltransmit VI. CONCLUSIONSANDDISCUSSIONS
power constraint P = 1. In Fig. 7, the received SNR with In this paper, we propose a new analog beamforming
t
respect to the number of antennas is given. As predicted by architecture, on-off analog beamforming (OABF), which only
Theorem 3, OABF-t can achieve better SNR than OABF-s uses simple analog switches to achieve beamforming gains.
doseundertheper-antennapowerconstraint.InFig.8presents To determine the status of each switch with the given channel
the corresponding achievable rates. We can see that the rate information, we propose optimal algorithms to maximize the
gap between our OABF-t scheme and the optimal scheme is receivedSNRundertheper-antennapowerconstraintandtotal
less than 1.5 bits, which is about half of the gap between the power constraint, respectively. With linear/polynomial com-
OABF-sschemeandtheoptimalschemeundertheper-antenna plexities, our OABF can achieve both the full diversity gain
power constraint. and the full array gain. More specifically, the achievable rate
9
gap between the optimal scheme (equal-gain phase-aligned the following inequalities as
beamforming)andourschemeisaconstantof3.3bits/symbol, 1 (cid:88) 1 (cid:88)
regardless the number of antennas and SNRs. Pbri ≥ 2E| hj|2+ 2E| hj|2
This paper only discussed the case of one data stream hj∈R hj∈/R
for a point-to-point transmission. Extending it to the case of  2
1 (cid:88)
multiple users and multiple streams is valuable to investigate, ≥ 2E |hj|cos(θj) +
both in theoretical analyses and practical algorithm designs.
−π/2≤θj≤π/2
More importantly, our on-off analog beamforming is com-  2
patible with other analog beamforming schemes to make up 1 (cid:88)
newhybridarchitectures.Otherbeamformingsystems,suchas 2E |hj|cos(θj)
Radar, can also adopt our OABF scheme to decrease system π/2≤θj≤3π/2
 2
cost.
1 (cid:88) (cid:88)
≥ 4E |hj|cos(θj)− |hj|cos(θj)
−π/2≤θj≤π/2 π/2≤θj≤3π/2
APPENDIX:PROOFOFTheorm 2  2
N
1 (cid:88)
Proof:First,itisobviousthattheOABF-balgorithmper- = 4E |hjcos(θj)| (13)
formsbetterthantheantennaselectionschemethatcanachieve j=1
the full diversity gain of N. Therefore, both the suboptimal wherethelastinequalitycomesfromthefactthat2a2+2b2 ≥
OABF-b and optimal OABF-s algorithms can achieve the full (a+b)2. We can then calculate the expectation by averaging
diversity gain. over all phases as
We now consider the array gain. With per-antenna power
constraint, it is easy to see that the phase-aligned analog P ≥ 1(cid:88)N 2 (cid:90) π/2|h |2cos2(θ )dθ +
beamforming scheme is optimal [32] to make full use of each bri 4 π j j j
j=1 0
antenna’s power. For this optimal phase-aligned analog beam-
forming, the expected total received signal power, averaging 2(cid:88)N (cid:88)N 4 (cid:90) π/2(cid:90) π/2
|h h |cos(θ )cos(θ )dθ dθ
over all possible phases, can be calculated as 4 j=1k=1,k(cid:54)=j π2 0 0 k j j k j k
 
Pop =(cid:88)N |hj|2 =(cid:88)N |hj|2+2(cid:88)N (cid:88)K2 |hkhj|. (10) = 1412(cid:88)N |hj|2+2π42 (cid:88)N (cid:88)N |hkhj|
j=1 j=1k=1,k(cid:54)=j
j=1 j=1 j=1k=1,k(cid:54)=j  
N N N
1 4 (cid:88) (cid:88) (cid:88)
Wethenshowthattheexpectedreceivedsignalpowerofthe ≥ 4π2  |hj|2+2 |hkhj|
bridge algorithm, OFBA-b, is larger than P /π2. According j=1 j=1k=1,k(cid:54)=j
op
to the OFBA- algorithm, the expected received signal power 1
= P (14)
is π2 op
By further averaging over all the amplitude in (10), the
expected SNR of the optimal phase-aligned beamforming
N
(cid:88) (cid:88) scheme can be obtained as (N +N(N −1)π/4) [28]. Then,
P =E| h |2 =E| h |2 (11)
bri j j the expected SNR is lower bounded by
s
hj∈R j=1,−π/2≤θj≤π/2
P
γ =E[SNR ]≥ o EP
where θj is the phase difference between hj and hm. We can 1 s π2σ2 op
decompose the summation vector (cid:80)|jR=|1hj into two compo- = Po (N +N(N −1)π/4) (15)
nents, one align the direction of h and the other orthogonal π2σ2
m
to it. Then, the total power must be larger than or equal to The ratio of γ over the SNR of a single-input single-output
1
that of the component align the direction of hm. Therefore, system, which is NσP2o, is of the order of N. Therefore, our
we have OABF-b algorithm can also achieve the full array gain of N.
 2
N
(cid:88)
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