Energy efficiency maximization for 5G multi-antenna receivers · range communications [4]. For...

TRANSACTIONS ON EMERGING TELECOMMUNICATIONS TECHNOLOGIESTrans. Emerging Tel. Tech. 2015; 26:3–14

Published online 27 October 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/ett.2892

SPECIAL ISSUE PAPER

Energy efficiency maximization for 5Gmulti-antenna receiversQ. Bai* and J. A. NossekInstitute for Circuit Theory and Signal Processing, Technische Universität München, Theresienstr. 90, 80333 Munich, Germany

Abstract

In a digital communication system, the analog signal that the receiver receives with its radio frequency front end is con-verted into digital format by using the analog-to-digital converter (A/D converter, ADC). Quantisation takes place duringthe conversion from continuous amplitude signal to discrete amplitude signal, leading inevitably to losses in informationwhich are dependent on the number of bits that is used to represent each sample. Although employing a higher bit resolu-tion reduces the quantisation error, a higher power dissipation of the ADC is incurred at the same time. This trade-off isessential to the energy efficiency of the receiver, which is commonly measured by the number of information bits conveyedper consumed Joule of energy. We investigate, in this work, the adaptation of ADC resolutions of a multi-antenna receiverbased on instantaneous channel knowledge, with the goal of maximising receiver energy efficiency. The formulated opti-misation is a combinatorial problem, and we propose several algorithms which yield near-optimal solutions. Results fromnumerical simulations are presented and analysed, which provide guidelines to operation and deployment of the system.Copyright © 201 John Wiley & Sons, Ltd.

*Correspondence

Q. Bai, Institute for Circuit Theory and Signal Processing, Technische Universität München, Theresienstr. 90, 80333 Munich, Germany.E-mail: [email protected]

Received 14 July 2014; Accepted 14 September 2014

1. INTRODUCTION

The next generation of wireless communication technolo-gies, that is, 5G, is best known for its prediction andpromise of supporting 1000 times data traffic as today be-yond the year 2020. The energy consumption of wirelesssystems and networks, from an operation point of view,cannot and should not increase with the same pace. There-fore, improving the energy efficiency of wireless systemsand networks has also become a key target of 5G [1].Driven by economical interest and environmental responsi-bility, green communication has already drawn significantresearch and industrial attention over the past years, for ex-ample, [2][3]. For portable devices and wireless sensors,prolonging the lifetime of batteries is a major concern.For base stations in cellular networks, minimising the en-ergy consumption while satisfying the quality-of-servicerequirements of the end users is desirable in reducing theoperation and maintenance costs. In both scenarios, it iscritical to consider and appropriately model the circuitpower consumed by the hardware, which comes to thesame order of magnitude as the radiated power for short-range communications [4]. For multi-antenna systemssuch as the multiple-input multiple-output (MIMO) sys-tem, while a larger channel capacity and higher spectrum

efficiency can be achieved, the increased number of radiofrequency (RF) chains lead also to higher power dissipa-tion, which could become dominant and unaffordable if avery large antenna array is deployed. This is to say, for thekey enabling technologies of 5G such as femtocells [5] andmassive MIMO [6], circuit power plays an important rolein the improvement of system energy efficiency.

The focus of this work is on the impact of analog-to-digital converter (ADC) on the energy efficiency of amulti-antenna receiver. In modern receiver design, moreand more receive functions are implemented by digitalhardware due to its high speed and low cost. Therefore, theimportance of ADCs, which enable the subsequent digitalsignal processing by sampling and converting the receivedanalog signal into digital format, is evident and has longbeen realised [7]. However, the ADC is expected to be alimiting factor of the system as it consumes a significantamount of power when operating at high sampling rateand resolution. It was reported in [8] that the power dis-sipation of an eight-bit ADC with sampling rate 20 GS/sreaches as much as 10 Watt, which is obviously imprac-tical for most mobile devices. In the recent paper [9], theauthors pointed out that the ADC is one of the major chal-lenges for millimetre wave communication when hardwareconstraints are concerned. From a communication point

Copyright © 2014 John Wiley & Sons, Ltd. 3

4

Q. Bai and J. A. Nossek

of view, this motivates the employment of low ADC res-olution, because high sampling frequency is required bymany applications such as cognitive radio. In recent years,there have been various works investigating the perfor-mance limit of communications over quantised channels,for example, [10][11], where the focus was on the deriva-tion of capacity loss when low ADC resolution is used, aswell as design of the quantiser. As the ADC naturally linksthe system performance in terms of capacity or achievablerate to the power consumption, that is, based on the trade-off between quantisation error and power dissipation, wepropose to treat the ADC resolution as an adaptable pa-rameter and attempt to maximise the energy efficiency ofthe system, which is quantified as the number of informa-tion bits conveyed per Joule of energy consumption [12].Significant gain over systems employing fixed, althoughlow, ADC resolution has been observed with the simulationresults provided in the paper.

At the receiver side, diversity can be achieved via thedeployment of multiple antennas. With a single transmitantenna, independent signal paths and maximal ratio com-bining (MRC), the gain in receive signal-to-noise ratio(SNR) equals the number of receive antennas [13]. Withmultiple transmit antennas, spatial multiplexing techniquescan be realised to provide linear growth of the channelcapacity with the minimum of transmit and receive an-tenna numbers [14]. However, the hardware complexityand power consumption of the receiver scale as well withthe number of RF chains, which include, besides the ADC,also the low noise amplifier, the mixer, the automatic gaincontrol and some filters. As a result, energy efficiency anal-ysis for massive MIMO systems has put its emphasis oncircuit power [15][16]. In both papers, however, the depen-dency of ADC power on the bit resolution employed is nottaken into account.

A common and long-developed technique to alleviatethis burden of affording too many RF chains, known asantenna selection [17], is to choose signals received by asubset of antennas based on the channel conditions for fur-ther processing, thus reducing the necessary RF chains andthe associated power consumption. When jointly optimis-ing the ADC resolutions for all receive antennas, antennaselection can be performed in an implicit way by allow-ing the 0 bit resolution, which suggests that the signalfrom the corresponding antenna is not processed. Or, wecan explicitly perform antenna selection and use positivebit resolutions for chosen active antennas. This motivatesthe proposal of suboptimal algorithms, which interactivelychooses antennas and their bit resolutions. As processingpower also accounts for part of the circuit power, we are inparticular interested in low-complexity algorithms, whichdo not require large amount of online computations. Be-sides the design of ADC resolution adaptation algorithms,we also aim at gaining insight into the energy efficientoperation of the system, for instance, which parametersinfluence the optimal bit resolution that should be em-ployed, and how are the dependencies characterised. Dueto the lack of analytical solutions, we mainly illustrate

these results via numerical simulations. Part of this workhas been published in conference contributions [12] and[18], where the idea of ADC resolution adaptation was pro-posed and studied for SIMO systems. We summarise andrefine the concept as well as algorithms here, and more im-portantly, extend the investigation to the MIMO case wherethe task of antenna selection becomes nontrivial.

The rest of the paper is organised as follows. InSection 2, we introduce the system model and, in particu-lar, give expressions of a lower bound on channel capacityand the total power dissipation as functions of the ADCresolution vector. The problem of maximising energy ef-ficiency is then formally given. The application of theparticle swarm optimisation technique to tackle the combi-natorial problem at hand is described in Section 3, whereits effectiveness is shown in comparison to global optimalsolutions obtained with exhaustive search. We propose sev-eral antenna selection-based ADC adaptation schemes inSection 4, followed by simulation results and analysis ex-hibited in Section 5. Section 6 concludes the paper with asummary of ideas, methods and results, which have beenpresented, and shortly discusses related open issues as wellas possible future works.

Notations: we use boldfaced letters to represent vectorsand matrices throughout the paper. The operator .�/H standsfor the Hermitian of a matrix, the symbol 1M denotes theidentity matrix of dimension M �M, and jAj and diag.A/denote the determinant of A and the diagonal matrix withthe same diagonal elements as A, respectively.

2. SYSTEM MODEL ANDPROBLEM FORMULATION

We consider the point-to-point communication between atransmitter with N > 1 antennas and a receiver with M > 1antennas over a flat fading channel. Throughout the paper,it is assumed that the receiver has perfect channel state in-formation, which is an idealised condition for evaluatingthe performance limit of the system. In practice, a trainingsequence for which both the transmitter and the receiverhave a priori knowledge is sent at the beginning of thecommunication, based on which the receiver can performchannel estimation [19]. It should be noted that the ob-tained channel knowledge is imperfect due to the additivenoise of the channel, as well as quantisation of the receivedtraining symbols. This problem is interesting and importantin itself but is beyond the scope of this paper.

The communication scenario under investigation is de-picted in Figure 1. The vector channel output y 2 CM

before quantisation is given as

y Dp˛Hx C �,

where ˛ is the average power gain of the wireless chan-nel, H 2 CM�N is the matrix containing i.i.d. complexGaussian distributed channel coefficients with zero meanand unit variance, x 2 CN is the vector of transmitted sym-bols and � 2 CM is the i.i.d. zero-mean complex circular

4 Trans. Emerging Tel. Tech. 26:3–14 (2015) © 2014 John Wiley & Sons, Ltd.DOI: 10.1002/ett


Figure 1. Communication over a quantised channel.

Gaussian noise vector. We assume that ˛ and H are per-fectly known to the receiver, and their values stay constantover the period of time that the data transmission takesplace. The covariance matrix of y is computed as

Ryy D EhyyH

iD R�� C ˛HRxxH

H,

where Rxx and R�� are the covariance matrices of thetransmitted signal and the additive noise, respectively. As-suming uniform power allocation at the transmitter anduncorrelated transmit symbols, we have that Rxx is equalto a scaled identity matrix given by

Rxx DPtx

N� 1N ,

where Ptx stands for the total transmit power. Letting �2

denote the noise power at each receive antenna, we furtherhave R�� D �2 � 1M . Consequently, the average receiveSNR at each antenna is given by

� D˛Ptx

�2,

which allows us to write Ryy as

Ryy D �2�1M C

�

N�HH H

�.

The ADC at each receive antenna samples the receivedsignal and uses a finite number of bits to represent eachsample. We assume here that the M A/D converters all actas scalar quantisers, and let b 2 f0, 1, : : : , bmaxg

M be thevector of resolutions employed by each ADC, where bmaxis the maximal number of bits that an ADC could use fora single sample. Given the instantaneous channel state in-formation, the vector of bit resolutions b can be adaptedaccordingly in order to optimise certain performance met-rics, for example, the mutual information between channelinput and the quantised output, the mean squared error afterreceive filtering and so on. The associated cost to accountfor is the power or energy consumption of the receivercircuitry. We introduce in the sequel the computation ofa capacity lower bound of the MIMO quantised channeland the power consumption of the receiver, both as func-tions of the vector b. Note that the capacity lower boundwe derive is more general than [21] in that non-uniformADC resolutions are allowed and has a simpler form than

the expression given in [12]. This means, the followingderivation is a new contribution of the paper.

2.1. Capacity lower bound of thequantised channel

The quantisation operation is in general nonlinear, and theresulting quantisation error is correlated with the inputdata. The Bussgang theorem [20][21] suggests a decom-position of the output of the nonlinear quantiser into adesired signal part and an uncorrelated distortion, whichprovides us a convenient analytical approach to formulat-ing the quantisation process. To this end, the quantisedoutput vector r can be written as

r D Fy C e,

where the noise vector e is uncorrelated with y , and thelinear operator F is taken as the MMSE estimator of rfrom y:

F D EhryH

iEhyyH

i�1D RryR

�1yy .

Consequently, we have

r D F .p˛Hx C �/C e D

p˛FHx C F�C e

D H 0x C �0,

where the effective channel H 0, the effective noise �0 andits covariance matrix are given respectively by

H 0 Dp˛FH ,

�0 D F�C e,

R�0�0 D FR��FH CRee.

Note that the effective noise �0 is not necessarily Gaussian.Therefore, if we define a new MIMO Gaussian channelwith the input-output relation rG D H

0xC�G and assumethat E

��G�

HG

�D R�0�0 , the capacity of the new channel

provides a lower bound on that of the quantised channel,for Gaussian distributed noise minimises the mutual infor-mation [22]. Based on this observation and assuming thatthe channel input x is Gaussian distributed, we have

I.x; r/ > log2

ˇ̌̌1M CR

�1�0�0H

0RxxH0Hˇ̌̌. (1)

Given Gaussian distributed channel input x, a determin-istic channel matrixH , and uncorrelated Gaussian noise �,the channel output y is also Gaussian. For such a quantisa-tion source and a distortion-minimising scalar non-uniformquantiser, the distortion factor �, which is the inverse of thesignal-to-quantisation-noise ratio, attains the values givenin Table I [23] with respect to different b. For b > 5,

the asymptotic approximation � D �p

32 � 2�2b can be

used [24].

Trans. Emerging Tel. Tech. 26:3–14 (2015) © 2014 John Wiley & Sons, Ltd. 5DOI: 10.1002/ett


Table I. Distortion factor � for different analog-to-digital con-verter resolutions b.

b 1 2 3 4

� 0.3634 0.1175 0.03454 0.009497

b 5 6 7 8

� 0.002499 0.0006642 0.0001660 0.00004151

Let �i denote the distortion factor of the ADC associatedwith antenna i, and � 2 RM�M denote the diagonal matrixcontaining all M distortion factors. Analytical expressionsforH 0 and R�0�0 have been derived in [21][12] as

H 0 Dp˛RryR

�1yy H D

p˛.1M � �/H ,

R�0�0 D .1M � �/��2.1M � �/C �diag.Ryy/

�,

which enable the calculation of the capacity lower bound(1). With the denotation of hi, i D 1, : : : , M as the Mcolumns ofH H, we have

diag.Ryy/ D �2 � diag

n1C

�

N� hH

i hi

oM

iD1

4D �2 � diag f1C ˇig

MiD1 ,

where ˇi D�N � jjhijj

22, i D 1, : : : , M. The inverse of R�0�0

is then given by

R�1�0�0 D

1

�2diag

�1

1C �iˇi

�M

iD1.1M � �/

�1,

which leads to

I.x; r/ > log2

ˇ̌̌1M CR

�1�0�0H

0RxxH0Hˇ̌̌

D log2

ˇ̌̌ˇ1N C

Ptx

N�H 0

HR�1�0�0H

0

ˇ̌̌ˇ

D log2

ˇ̌̌ˇ̌1NC

�

N�H Hdiag

�1 � �i

1C �iˇi

�M

iD1H

ˇ̌̌ˇ̌ .

Defining the real-valued diagonal matrix

D D diag

�1 � �i

1C �iˇi

�M

iD1,

which depends both on the employed bit resolutions andthe channel coefficients, we write the capacity lower boundCL (in bit/s) in a very similar form with the well-knownMIMO capacity formula as

CL D B log2

ˇ̌̌1N C

�

N�H HDH

ˇ̌̌, (2)

where B is the signal bandwidth. As it has been shown in[21] that this lower bound on channel capacity is quite tight

especially in the low SNR regime, we employ it in our per-formance measure of energy efficiency for the achievablerate of the system.

2.2. Power consumption model

Power dissipation of the ADC associated with the ithantenna can be calculated as [25]

PADC,i D

(c0 � �

2 � 2bi , bi > 0,

0, bi D 0,(3)

where c0 is a constant determined by the specific designof the ADC. When positive bit resolution is employed,power dissipation of the ADC grows exponentially withbi. Otherwise, the corresponding RF chain is considered tobe turned off, and no power consumption is incurred. Wemodel the total power consumption of the receiver by

P D c0 � �2X

i:bi>0

2bi C c1, (4)

where c1 is a constant accounting for the power consump-tion of baseband components of the receiver. This means,we roughly model the power consumption of the RF chainas equal to the power consumption of the ADC and assumethat the processing power at the baseband is independentof data rate. In practice, the specific design of the receiver,especially the complexity of decoding, determines whetherthe power consumption of A/D converters is the dominantpart in the total power consumption [26]. For now, we set-tle for the generic and general model (4) to have a clearerstructure of our problem. For the simulations, we havetaken B D 1 MHz, bmax D 8, c0 D 1 � 10�4=�2 Watt andc1 D 0.02 Watt with reference to [27].

2.3. Maximisation of energy efficiency

Motivated by the demand to increase the lifetime of mo-bile terminals and other communication devices poweredby batteries, as well as by the desire to conduct greencommunications, energy efficiency has become anotherimportant performance metric for communication systemsover the past years. For different applications, one mightwant to minimise the energy needed to transmit/receive acertain amount of data or to maximise the operation timegiven fixed available energy while a constant data rate isprovided. In this work, we focus on the unconstrained max-imisation of the bit per Joule metric at the receiver withthe ADC resolution vector b as the optimisation variable,which is defined as

maxb2f0,1,:::,bmaxgM

CL.b/

P.b/4D � (5)

where the expressions for CL.b/ and P.b/ are given by (2)and (4), respectively. This problem is of particular interest



as it potentially provides insight to the solutions of the con-strained optimisations we mentioned earlier. We denote theoptimisation objective, that is, energy efficiency of the re-ceiver, with � and its optimal value with ��. The optimalbit resolution vector is denoted with b� accordingly.

The optimisation problem (5) with respect to integer-valued bit resolutions is a combinatorial problem with asearch space growing exponentially with M and bmax C

1, leading to prohibited computational complexity if theglobal optimum is to be found. For a practical ADC, themaximal bit resolution bmax is typically a rather smallnumber. Therefore, when the number of receive anten-nas M is also small, an exhaustive search for the optimalb is possible. Yet for a large receive antenna array, wewould need more effective search algorithms. Commontechniques for tackling integer programming, such as thebranch and bound method, often requires the computationof upper bounds on the optimal objective, which is usuallyachieved via solving a relaxation of the original problem[28]. In our case, this does not seem an option due to thenon-linearity and discontinuity of the objective function,that is, even if we allow bi to take real values on Œ0, bmax�,it is still very hard to solve (5) to optimality. We proposeto apply the particle swarm optimisation (PSO) technique[29], which serves as a good candidate here because it israther independent of the problem structure and can beimplemented easily and is able to produce near-optimalsolutions for systems of small-to-medium scale. For large-scale systems, however, more effective suboptimal algo-rithms need to be designed to concur the problem ofcomputational complexity.

3. ADAPTING ANALOG-TO-DIGITALCONVERTER RESOLUTIONS VIAPARTICLE SWARM OPTIMISATION

The PSO method, although originally proposed for solv-ing unconstrained optimisations with real-valued variables[29], can be applied to tackle integer programming with-out any complicated modification [30]. It is a stochasticoptimisation technique based on the social behaviour ofa population of individuals. Each individual, termed as aparticle, moves in the feasible region of the optimisationproblem probing for good solutions and shares informa-tion with the whole set of particles, called the swarm. Themovement of the particles in each generation of the algo-rithm is random but also depends on the memory of theindividual particles as well as of the swarm. Let the swarmcontain S particles. For initialisation of the algorithm, Sfeasible solutions of the optimisation are generated at ran-dom, which are denoted with b0

1,b02, : : : ,b0

S. In the .kC1/thgeneration, the particle s evolves according to the formula

vkC1s D wvk

s C s1r1

�pk

s � bks

�C s2r2

�pk

g � bks

�,

bkC1s D bk

s C vkC1s ,

that is, the position of particle s is incremented by a veloc-ity vector vkC1

s , which is computed based on the velocityvector vk

s in the previous generation, the displacement be-tween the particle and the best self-found solution pk

s �bks ,

and the displacement between the particle and the globalbest solution found by the swarm pk

g � bks . The parame-

ter w is called the inertia weight and is usually chosen asa decreasing function in the generation index, facilitatingglobal search in early generations of the algorithm and lo-cal search in later generations. The weights s1 and s2 arecalled the cognition and social learning rates, and they areusually kept constant throughout the generations. On theother hand, the uniformly distributed random numbers r1and r2 on Œ0, 1�, which are renewed in every generation,add randomness into the trajectory of each particle. In ournumerical studies, the parameter settings s1 D s2 D 1,w D 1 � 0.02k are used.

In the application of the PSO method to solving (5),the particle positions correspond to the ADC resolutionvectors b, hence each fractional valued particle positionobtained from the update needs to be mapped to a fea-sible solution by rounding and fitting the values into theset f0, : : : , bmaxg. After the mapping and fitting, the objec-tive function is evaluated at each new particle position, andthe local best solutions as well as the global best solutionneed to be updated. The algorithm terminates when a max-imal number of generations G is reached. Depending onthe dimension of the optimisation, the values of S and Gcan be increased to improve the performance of the PSO.One could also repeat the algorithm for several runs andpick the best solution among all obtained global best so-lutions. From simulations, we have found that combiningboth schemes yields better performance than relying solelyon either one of them. For the simulation results shownin the sequel, we take G D 20 and run the algorithm 20times for each given array size and channel realisation, andchoose S to be dependent on the scale of the system.

To verify the effectiveness of the PSO algorithm, we setup a test scenario with N D 2, � 2 f0, 30g dB and restrictM to small numbers such that an exhaustive search for b� ispossible. The successful rate, that is, the ratio that the PSOmethod finds the global optimum, and the averaged valueof �PSO=�

� are used as performance measures, which arecomputed after 1000 simulation trials and listed in Table II.One immediately sees that the PSO method is able to findsolutions very close to the global optimum, even though, insome cases, the unsuccessful rate is not trivial. Expandingthe swarm further helps improve the performance of thealgorithm, yet it should be noted that the number of costfunction evaluations scales linearly with the swarm size.

The PSO method requires S � G evaluations of thecost function per run. According to (2), this involves thecalculation of the determinant of an N � N matrix andsome matrix multiplications with complexity O.N2M/.Therefore, the complexity order of the PSO algorithm isapproximately O.N3M C N2M2/, if S is chosen to be lin-ear in M. From Table II, we see that the performance ofthe algorithm deteriorates even though S increases linearly



Table II. Effectiveness of the particle swarm optimisationmethod.

� (dB) M �PSO=�� (%) Succ. rate (%)

S DM

0

3 100 1004 100 99.805 100 98.016 99.97 86.50

30

3 99.96 99.704 99.87 84.905 99.82 79.306 99.78 73.20

S D 4M

0

3 100 1004 100 1005 100 1006 99.99 99.99

30

3 100 1004 100 1005 100 1006 99.99 99.90

with M, which implies that the PSO might not be suitablefor systems of a large M. To this end, we discuss severalalternative suboptimal algorithms in the next section.

4. ANTENNA SELECTION-BASEDRECEIVE STRATEGIES

Both from intuitions and results of the PSO method, wefind that in order to have better system energy efficiency,a number of receive antennas could be switched off. Theseare usually antennas with weaker channel conditions, theusage of which does not lead to a capacity incrementthat pays off the boost in power consumption of the cor-responding RF chains. In other words, we can considerthe optimisation problem (5) as the selection of a numberof active antennas, and the choice of positive bit reso-lutions for each selected antenna. Antenna selection is awell-investigated method for reducing the hardware costand computational complexity of MIMO systems, whilethe channel capacity can be preserved to a large por-tion [31][32]. Existing receive antenna selection schemesmostly employ the channel capacity of the selected subsys-tem as performance measure and assume that the receiverhas infinite precision to access the received signal [33][34].We shall investigate in this section, the influence of quanti-sation by A/D conversion on the antenna selection process.Moreover, we propose different schemes for finding the bitresolutions of the selected active antennas.

4.1. Uniform bit resolution forselected antennas

Receive antenna selection is in general a combinatorialproblem which is difficult to solve to optimality. Whenthere is only one transmit antenna, we can simply, and alsooptimally, select a subset of antennas with the largest chan-nel gains. When N > 1, this norm-based selection scheme

is suboptimal because of the correlation between the chan-nel vectors. A number of heuristic algorithms have beenproposed in the literature, for example, [33][34], whichexhibit performance very close to optimal. For our optimi-sation problem, we do not know from the beginning, howmany antennas should be selected. An incremental greedyselection scheme can be a good choice in this case, due toits flexibility and low complexity. To this end, we designthe antenna selection scheme for quantised received signalbased on the algorithm proposed in [33].

On the other hand, as the channel coefficients are i.i.d.,one can think of a receive strategy where uniform bit res-olution is applied to all selected antennas. Let the chosenresolution be b and the corresponding distortion factor be�. The essential idea here is that, in each iteration, the an-tenna that leads to the largest performance gain is selected.As the ADC power consumption of all selected antennasis identical, we simply choose the antenna that results inthe largest capacity increase. The selection process is ter-minated when � decreases on the addition of one moreantenna. Mathematically, let us suppose that antenna k isselected for the .lC1/th iteration, l > 0. At this point, l an-tennas have already been selected, and the correspondingchannel vectors form the matrix Hl 2 Cl�N . The diago-nal matrixDl can also be computed based on the norms ofthe selected channel vectors. After antenna k is chosen, thetwo matrices are updated according to

H HlC1 D

�H H

l hk�

,

DlC1 D

"Dl 0

0T 1��1C�ˇk

#,

which further lead to

H HlC1DlC1HlC1 D H

Hl DlHl C

1 � �

1C �ˇkhkh

Hk .

Define Bl D�1N C

�NH

Hl DlHl

�1. The channel capacity

achieved after iteration l is then given by

Cl D log2

ˇ̌̌1N C

�

NH H

l DlHl

ˇ̌̌D log2

ˇ̌̌B�1

l

ˇ̌̌.

After the selection of antenna k, the increased channelcapacity is computed as

ClC1 D log2

ˇ̌̌1N C

�

NH H

lC1DlC1HlC1

ˇ̌̌

D log2

ˇ̌̌ˇB�1

l C�

N�

1 � �

1C �ˇkhkh

Hk

ˇ̌̌ˇ

D log2

ˇ̌̌B�1

l

ˇ̌̌C log2

ˇ̌̌ˇ1N C

1 � �

1C �ˇkBlhkh

Hk

ˇ̌̌ˇ

D Cl C log2

1C

1 � �

1C �ˇkhH

k Blhk

�

4D Cl C log2

1C

1 � �

1C �ˇkQl,k

�.



Therefore, the selected antenna k satisfies

k D argmaxj2Al

1 � �

1C �ˇj� Ql,j, (6)

where Al is the set of antennas that have not been selectedafter the lth iteration, which should be updated by

AlC1 D Al � fkg.

Obviously, the selection governed by (6) depends not onlyon the channel conditions but also on the chosen ADC res-olution. The update of Bl, by using the matrix inversionlemma, does not require a matrix inversion operation:

BlC1 D�1N C

�

NH H

lC1DlC1HlC1

��1

D

B�1

l C�

N�

1 � �

1C �ˇkhkh

Hk

��1

D Bl �

N

��

1C �ˇk

1 � �C Ql,k

��1

BlhkhHk Bl.

The algorithm can be initialised with B0 D 1N and ter-minated if �lC1 < �l or l D M. The steps of the algorithmare summarised in Algorithm 1. We denote the numberof selected antennas with L. The complexity order of thealgorithm is given by O.MNL/ [33].

Algorithm 1 Antenna selection procedure with given uni-form bit resolution

GivenH , b, �Initialisation: A f1, : : : , Mg, B 1N

ˇi �N � jjhijj

22, Qi jjhijj

22, i D 1, : : : , M

C 0, P c1, � 0while A ¤ ; do

k argmaxj2A

�Qj

1C �ˇj

�

C0 CC log2

1C

1 � �

1C �ˇkQk

�P0 PC c0�

2 � 2b

�0 C0=P0

if �0 < � thenbreak

end if� �0

A A � fkgB B �

�N� �

1C�ˇk1�� C Qk

��1Bhkh

Hk B

for j 2 A doQj hH

j Bhj

end forend whileReturn f1, : : : , Mg �A

The antenna selection method described earlier requiresa given bit resolution for all chosen antennas. One could

employ an outer loop and enumerate all available bit reso-lutions 1, : : : , bmax to determine the best b. We refer to thisscheme with the abbreviation UBA, that is, uniform bit res-olution adaptive scheme. Accordingly, we also propose auniform bit resolution static scheme, termed UBS, in whichthe employed bit resolution only depends on the averagereceive SNR � . More specifically, we first apply the UBAto a large number of channel realisations and obtain theaverage optimal bit resolution. This value is then roundedand given to the UBS, where the subsequent antenna selec-tion process is the same as in UBA. This means, we can setup a lookup table to record the average optimal bit resolu-tions for different � and directly apply the correspondingrounded value to the UBS. The performances of UBA andUBS are presented and compared in the following.

We first show the energy efficiency performance of theproposed schemes in Figure 2, where the results are the av-eraged outcome of 1 � 104 independent simulation trials.Besides UBS, we also implement the original antenna se-lection scheme from [33], which does not take quantisationinto account, and depict the results for reference with the

Figure 2. Performance of UBA and UBS in terms of energyefficiency (M D 8, unit of � is Mbit/Joule).



label no quan. in Figure 2(a). This means, the algorithmis given the same optimised bit resolution as UBS but pre-sumes � D 0 during the antenna selection process. Fromthe curves, we find that this would lead to a significantperformance loss in the medium SNR range. On the otherhand, the performance gap between UBS and UBA is neg-ligible. In Figure 2(b), we see that increasing the number oftransmit antennas is only helpful to energy efficiency of thereceiver in the high SNR regime, which is in accordancewith the conclusion made with respect to channel capacity.

For the same simulation setup, we illustrate the averagenumber of active antennas and the average bit resolutionfor the active antennas in Figure 3, as dependent on thereceive SNR � . The curves for UBA and UBS have similarshapes, except that those for the UBA are smoother. In thelow SNR regime, most receive antennas are used with lowbit resolution, while in the high SNR regime, a few receiveantennas are used with high bit resolution. In fact, in thelatter case, the number of selected antennas converges tothe number of transmit antennas N.

(a)

(b)

Figure 3. Optimisation results of UBA and UBS (M D 8).

4.2. Non-uniform bit resolution forselected antennas: greedy algorithm

When non-uniform bit resolutions are allowed for the se-lected antennas, we can select one antenna together withits bit resolution in each iteration of the algorithm based onthe improvement of energy efficiency after the selection.To this end, we use the same procedure as the UBA exceptthat in any iteration l, instead of solving (6), we solve thejoint optimisation

.k, b/ D argmax j2Ali2f1,:::,bmaxg

Cl C log2

1C

.1 � �i/Ql,j

1C �iˇj

�Pl C c0�

2 � 2bi,

where Pl indicates the power consumption of the receiverafter l iterations and include the differently chosen bit res-olutions in the diagonal matrix D and the auxiliary matrixB. We term this scheme as NBG, which stands for non-uniform bit resolution greedy algorithm, because now notonly the antennas are chosen in a greedy fashion but alsothe bit resolution. This scheme is of the same complexityorder as the UBA, yet we shall see in the next section thatits performance is not as good.

4.3. Non-uniform bit resolution forselected antennas: decremental algorithm

An undesirable situation in the execution of NBG is thata high bit resolution can be chosen for one of the firstselected antennas, which turns out to be too high andcauses the corresponding antenna to occupy too much re-sources. We therefore propose another algorithm for thenon-uniform bit resolution case, which allows reductionof the bit resolutions of selected antennas and is thereforeless greedy. We call this scheme non-uniform bit resolutiondecremental algorithm, abbreviated as NBD.

In the first iteration, the NBD behaves the same as NBG,that is, it chooses the antenna and the bit resolution thatgive the maximal energy efficiency. In each of the follow-ing iterations, a new antenna is chosen according to (6)where the distortion factor of the last selected antenna isused. Then the resolutions of all selected antennas are ex-amined according to the order they have been selected.From the first chosen antenna, we try to find out whetherreducing its bit resolution by one could lead to a higherenergy efficiency. Note that in this attempt, all subsequentantennas sharing the same bit resolution also reduce their

Table III. Proposed and reference algorithms.

Abbr. Meaning

PSO Particle swarm optimisationUBS Uniform bit resolution static algorithmNBG Non-uniform bit resolution greedy algorithmNBD Non-uniform bit resolution decremental algorithmNAS Without antenna selection



resolution by one. In this way, one antenna cannot usehigher bit resolutions than the antennas that are selectedbefore it, which is consistent with the idea of the greedyantenna selection process. Whether or not the bit resolutionof the first antenna is reduced, the algorithm proceeds to tryreducing the bit resolution of the second antenna, so on andso forth until the antenna chosen in the current iteration. If,

after this decremental process, the energy efficiency of thesystem is not increased, then we resort to the bit resolutionvector of the last iteration and terminate the algorithm; oth-erwise, the current best bit resolution vector is recorded,and the next iteration begins to seek for another antenna.

During each iteration of the NBD, previously chosenbit resolutions can be changed. Therefore, the algorithmic

Figure 4. Performance comparison of proposed schemes with varying � (M D 8).



structure of UBA and UBS does not apply anymore, andwe need to evaluate the capacity lower bound repeat-edly. This suggests that the NBD has a reasonably highercomputational complexity than the UBS, which, from thesimulation time, we have experienced, is still much lowerthan the PSO method.

5. SIMULATION RESULTS

We present results obtained with Monte Carlo simulationsin this section, with the purpose to compare proposed algo-rithms as well as to summarise operational guide lines. Theabbreviations used for the algorithms are listed in Table III.Each result, as shown in the figures, is the averaged out-come of 1�104 independent simulation trials. For the PSOmethod, the swarm size is chosen as S D 4M.

We first study the scenario with fixed receive antennanumber M D 8, and varying average receive SNR � inFigure 4. We use the results of UBS as comparison bench-mark, and depict the ratio of achieved energy efficiencyof other proposed algorithms in Figure 4(a) and (b). Themethod termed NAS refers to the scheme of using all re-ceive antennas with one optimised uniform resolution. Itturns out except NAS, the performance of all other pro-posed schemes are rather close. In the low SNR regime,

since almost all receive antennas should be used with a lowbit resolution, the NAS scheme also performs very well.In the high SNR regime, however, the number of activeantennas approaches N, and the NAS scheme exhibits asignificant performance gap. The PSO and NBD schemeshave a very small gain over the UBS, which is slightlylarger at turning SNR values where the optimal bit res-olution changes. The NBG scheme performs fine in lowSNR regime but suffers at high SNR values. Taken thecomputational complexity also into account, we concludethat the UBS method is the most promising solution. TheNBD scheme could potentially be useful for other chan-nel models or communication scenarios, for example, forspatially correlated channel or in a multiuser scenario. ThePSO method can provide some comparison benchmark tosmall-to-medium scale systems but is, in general, ratherexpensive to apply.

With fixed representative receive SNR values of�30, 0, 30 dB and a varying receive antenna number, wealso observe some interesting results in Figure 5. As shownin Figure 5(a), the PSO method fails to outperform otherproposed algorithms for large M, while the NBD schemeconstantly performs a bit better than the UBS. From the re-maining three figures, it can be seen that employing morereceive antennas is more helpful in the low SNR regime

Figure 5. Optimised system performance as dependent on M (N D 2, unit of � is Mbit/Joule).



due to the increased array gain, and a large portion of themshould be active. The optimised bit resolution decreaseswith M, meaning that if the receiver is employed with alarge antenna array, lower bit resolution could be used foreach RF chain, which is a potentially interesting feature formassive MIMO with low-cost RF front ends.

6. CONCLUSION AND OUTLOOK

We have investigated in this work the impact of A/Dconversion at a multi-antenna receiver. Based on the trade-off between quantisation error and power consumption ofthe ADC, an energy efficiency maximisation problem ofthe vector of ADC resolutions is formulated. We proposeseveral optimisation methods for finding near-optimal so-lutions: the PSO method, which is a stochastic optimisationtechnique, and a number of antenna selection based meth-ods. From the results of numerical simulations, we findthat the combination of antenna selection and the offlineoptimisation of a uniform ADC resolution for the activeantennas achieves good balance between performance andcomputational complexity. A decremental resolution algo-rithm also shows potential to suit in other communicationscenarios such as a multi-user situation. Moreover, basedon the optimised solutions, we learn that:

- The optimal number of active antennas and the op-timal bit resolutions depend heavily on the averagereceive SNR;

- With fixed receive antenna number, more antennasshould be selected for reception with a low bit reso-lution in the low SNR regime, while a few antennasshould be active with a high bit resolution in the highSNR regime;

- Employing a large receive antenna array is espe-cially helpful to system performance in the low SNRregime, where lower bit resolution can be used foreach RF chain.

There are many possibilities and directions to extendthis work for future research, and in this short outlook,we try to discuss a few of them. First of all, we haveassumed that the receiver has perfect knowledge aboutthe instantaneous channel state information, which is notexactly realistic especially when we consider the quan-tisation that has to be taken at the receiver. The impactof quantisation of received pilots which in turn influencesthe quality of the channel estimation should be carefullyinvestigated, and the ADC resolution applied for the pi-lot symbols should be chosen appropriately. In our recentwork [35], we have touched upon this issue, but the con-sideration therein is by no means exclusive. Secondly, aninformation theoretic approach has been pursued where weemploy a capacity lower bound to measure the achievablerate of the system. Practical signaling and signal process-ing at the receiver for coarsely quantised data should alsobe investigated, which in turn could provide us with a moredetailed and refined power consumption model. Thirdly, at

the transmitter side, a digital-to-analog converter performsthe inverse function of the ADC at the receiver. There-fore, a similar optimisation problem can be formulatedto improve the energy efficiency of the transmitter. Evenmore interestingly, a joint optimisation of system param-eters at the transmitter and the receiver could potentiallylead to further performance gain, as well as provide in-sight into the energy efficient operation modes of point-to-point communications.

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