Symbol-level Precoding for the Non-linear Multiuser MISO Downlink Channel

16 Pages·2017·3.42 MB·English

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSP.2017.2781647, IEEE
Transactions on Signal Processing
Symbol-level Precoding for the Non-linear
Multiuser MISO Downlink Channel
Danilo Spano, Student Member, IEEE, Maha Alodeh, Member, IEEE, Symeon Chatzinotas, Senior Member, IEEE,
Bjo¨rn Ottersten, Fellow Member, IEEE
Abstract—This paper investigates the problem of the interfer- in the interference between the simultaneous transmissions
enceamongmultiplesimultaneoustransmissionsinthedownlink towards the different co-channel users. As a consequence,
channel of a multi-antenna wireless system. A symbol-level
advanced signal processing techniques are required to handle
precoding scheme is considered, in order to exploit the multi-
themulti-userinterference(MUI),inordertoboosttheoverall
user interference and transform it into useful power at the
receiver side, through a joint utilization of the data information performance of full frequency reuse systems.
and the channel state information. In this context, this paper Linear precoding schemes have been shown to be an ef-
presents novel strategies which exploit the potential of symbol- fective way to manage the MUI, while guaranteeing specific
levelprecodingtocontroltheper-antennainstantaneoustransmit system performance requirements. Precoding techniques can
power. In particular, the power peaks amongst the transmitting
be classified in block-level and symbol-level. In the block-
antennas and the instantaneous power imbalances across the
level precoding, the transmitted signal is precoded in order to
differenttransmittedstreamsareminimized.Theseobjectivesare
particularlyrelevantwithrespecttothenon-linearamplitudeand mitigate the MUI using the knowledge of the channel state
phase distortions induced by the per-antenna amplifiers, which information (CSI). In this class of techniques, the generic
are important sources of performance degradation in practical scheme relies on the design of a precoding weight matrix (or
systems. More specifically, this work proposes two different
precoder),whichdependsonlyontheCSI.Asaconsequence,
symbol-level precoding approaches. A first approach performs
the precoder remains constant for a whole block of symbols
a weighted per-antenna power minimization, under Quality-
of-Service constraints and under a lower bound constraint on whose length is related to the coherence time of the channel.
the per-antenna transmit power. A second strategy performs a In this framework, several strategies have been considered
minimization of the spatial peak-to-average power ratio, evalu- fortheprecoderdesign[2]–[6],includingpowerminimization
ated amongst the transmitting antennas. Numerical results are
schemes with Quality-of-Service (QoS) costraints, as well as
presented in a comparative fashion to show the effectiveness
max-minfairapproaches.Thelatteronesaimatincreasingthe
of the proposed techniques, which outperform the state of the
art symbol-level precoding schemes in terms of spatial peak-to- fairnessofthesystem,bymaximizingtheminimumsignal-to-
average power ratio, spatial dynamic range, and symbol-error- interference-plus-noise ratio (SINR) across the users.
rate over non-linear channels. On the other hand, the transmitted signals in symbol-level
precoding are designed based on the knowledge of both the
CSI and the data information, constituted by the symbols to
I. INTRODUCTION
be delivered to the users [7]–[13]. In this approach, the aim
Amajorchallengefornextgenerationwirelesscommunica- is not to cancel the interference, but rather to control it so to
tion systems is the increasing demand for higher capacity, to have a constructive interference effect at each user. In [7] the
be provided utilizing the available wireless spectrum, which classification of the interference as constructive or destructive
is a scarce resource becoming more and more congested. A was given, and a selective channel inversion scheme was
possible solution relies on the use of multi-antenna transmit- proposed in order to eliminate the destructive interference. A
ters, which allow aggressive reuse of the frequency spectrum more advanced symbol-level precoding scheme was proposed
by exploiting the additional degree of freedom given by the in[8],basedontherotationofthedestructiveinterferencesoto
spatial dimension. This way, different users can be served by transform it into useful power. Similarly to the channel-level
thetransmittersharingthesametimeandfrequencyresources, case, also in this approach different optimization strategies
throughaspacedivisionmultipleaccessscheme[1].However, have been considered in the literature. In [9] the sum power
full frequency reuse architectures have a crucial limitation minimization and the max-min fair problem were solved
for M-PSK modulations. Extensions of such works include
Danilo Spano, Maha Alodeh, Symeon Chatzinotas and Bjo¨rn optimization strategies for multi-level modulations [10]- [11]
Ottersten are with Interdisciplinary Centre for Security Reliability
and more flexible approaches for exploiting the constructive
and Trust (SnT) at the University of Luxembourg, Luxembourg. E-
mails:{[email protected], [email protected], symeon.chatzinotas interference [12]. Furthermore, symbol-level precoding has
@uni.lu,[email protected]}. been considered also in relation to physical layer multicasting
ThisworkissupportedbyH2020projectSANSA(SharedAccessTerrestrial-
[12], and taking into account the imperfect knowledge of the
Satellite Backhaul Network enabled by Smart Antennas), FNR projects
SeMiGod (Spectrum Management and Interference Mitigation in Cognitive CSI [13].
Radio Satellite Networks) and SATSENT (SATellite SEnsor NeTworks for In the context of symbol-level precoding, this work copes
spectrummonitoring),FNR-EPSRCprojectCI-PHY(Exploitinginterference
with the following problems:
forphysicallayersecurityin5Gnetworks),andFNR-AFRprojectBroadSat.
PartofthisworkhasbeenacceptedinSPAWC2017. • Inrealsystemsitiscommonthateachindividualantenna
1053-587X (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSP.2017.2781647, IEEE
Transactions on Signal Processing
hasadedicatedamplifier,resultinginareducedflexibility framework.
in the power allocation amongst the different RF chains The main novelty of the proposed techniques with respect
of the transmitter. This dictates the need to consider to the state of the art on symbol-level precoding [7]–[13]
power limitations independently for each antenna, hence is their ability to exploit the potential of the symbol-level
to move to a per-antenna based framework. design for producing more robust waveforms to the harmful
• Thepoweramplifiersusuallyintroducenon-lineareffects effects of non-linear channels. It should be noted that a
which can degrade the transmitted waveform [14], [15]. symbol-level precoding scheme accounting for the channel
Therefore, good dynamic properties of the per-antenna non-linearities has already been proposed by the authors in
transmitpowerarerequiredtolimitthedistortioneffects. [19],[20].Nevertheless,theschemethereinproposedperforms
For single-user links, predistortion techniques are used just a peak power minimization, without imposing any lower
to deal with problem [16]. However, their extension to bound nor optimizing the SPAPR. On the other hand, the
multi-user systems relying on precoding is not straight- techniques proposed herein directly aim at a reduction of the
forward, due to the complex nature of the constellations powerimbalancesacrossthedifferentantennas,thereforethey
produced by the precoding operation. are able to tackle the problem of differential phase shift and
• With respect to the non-linear effects of the per-antenna toachieveenhancedperformance.Theschemeof[19],[20]is
amplifiers, an additional degradation is the phase distor- considered as a benchmark in the numerical evaluation of the
tion which applies independently to the signals trans- proposed techniques, in Section V-C, in order to quantify the
mitted by each antenna and depends on the transmit- relative gains.
ted instantaneous power. A high variation between the Itshouldbealsoremarkedthatanumberofworksavailable
instantaneous power transmitted by different antennas in the literature have proposed precoding techniques suitable
determines different phase shifts in the amplification for non-linear channels, especially in the context of massive
stages, and this differential effect is a further source of MIMO systems [21]–[23]. In particular, these works aim
performance degradation. at reducing the peak-to-average power ratio (PAPR) of the
transmittedwaveforms,orevenatdesigningconstant-envelope
To tackle the aforementioned problems, this work intro-
signals. A fundamental novelty of this work with respect to
duces novel symbol-level precoding schemes for multi-level
[21]–[23]liesinthefactthattheproposedtechniquesareable
modulations aimed at exploiting the constructive interference
to leverage the constructive interference effect, while at the
effectand,atthesametime,controllingtheinstantaneousper-
same time addressing the problem of non-linearities. As a
antenna power levels. It should be mentioned that this is not
consequence,theexploitationoftheMUIasabeneficialfactor
possible in the conventional block-level approach, where the
constitutes an inherent advantage of the proposed schemes.
precoderisdesignedforanentirecodeword,includingseveral
Another relevant difference is that the techniques proposed
symbols, hence the transmitted power can be controlled only
herein focus solely on a reduction of the power imbalances
in average and not symbol by symbol. As a consequence,
in the spatial dimension, i.e., among the different antennas,
the precoded waveforms can demonstrate unsuitable dynamic
while the schemes of [21]–[23] allow to improve the dynamic
propertiesintermsofpowerpeaks[17].Inparticular,themain
properties of the signals (e.g. the PAPR) also in the temporal
contributions of this paper can be summarized as follows:
dimension. This is due to the fact that the symbol-level pre-
• The non-linear effects, and specifically the differential coding formulation with QoS constraints, which is considered
phase shift effect, are introduced and investigated in the inthiswork,performstheoptimizationonlyinaspatialsense,
context of precoding. withoutadirectcontrolofthepowervariationinthetemporal
• Two different symbol-level precoding strategies are pro- dimension. Overall, the benefit of the techniques proposed
posed, with the objective of reducing the power imbal- hereinintermsofconstructiveinterferenceiscounterbalanced
ances between the multiple transmit antennas, in order by their lack of control of the power dynamic in the temporal
to have improved performance over non-linear channels dimension. A performance comparison between the proposed
withrespecttothesymbol-levelschemesoftheliterature. schemes and the approach of [21], which designs a constant
In both the proposed approaches QoS constraints are envelope precoder, is discussed in Section V-C.
considered to guarantee a per-user required SINR. Finally, it should be highlighted that this work relies on
• The first presented algorithm minimizes the per-antenna the assumption of a frequency flat block fading channel. In
transmit power, and imposes a lower bound to the power fact, the proposed precoding techniques are not designed to
carried by each transmitted signal. In this approach, the handle frequency selective channels, as they do not optimize
imbalances between the different RF chains are reduced the waveforms in the temporal dimension. This does not
by constraining the per-antenna transmit power within a allow them to compensate for the inter-symbol interference
specific range. that arises in frequency selective scenarios. Although a first
• Thesecondproposedschemedirectlytacklesaminimiza- step towards a spatio-temporal generalization of symbol-level
tion of the spatial peak-to-average power ratio (SPAPR) precoding (for linear channels) has been carried out by the
amongst the multiple transmit antennas. A simplified authors, and presented in [24], this extension is out of the
version of this SPAPR minimization approach has been scope of the present contribution, which is focused on non-
presented in [18] for a multi-beam satellite scenario, linear systems.
while herein it is addressed in general multiuser MISO The remainder of the paper is organized as follows. In
1053-587X (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSP.2017.2781647, IEEE
Transactions on Signal Processing
Section II, the system and signals communication model is In the conventional block-level precoding schemes, the
delineated.InSectionIII,theproblemofweightedper-antenna transmitted vector x[n] can be written as (cid:80)K w [n]d [n],
j=1 j j
power minimization with lower bound constraints is proposed where wj ∈ CNt×1 represents the precoding vector for the
and solved. In Section IV, the problem of SPAPR reduction user j. Alternatively, by defining a compact precoding matrix
is formulated and solved. In Section V the proposed approach W = [w1...wK] ∈ CNt×K, we can write x[n] = Wd[n].
is validated through simulation results. Finally, in Section VI The matrix W is designed by the precoding module based on
conclusions are drawn. an optimization algorithm depending only on the CSI and not
Notation: We use upper-case and lower-case bold-faced onthedatainformation,andforthisreasontheseconventional
letters to denote matrices and vectors, respectively. (·)T de- schemes are also referred to as channel-level precoding. As a
notes the transpose of (·), while and (·)∗ and (·)† denote consequence, the optimized precoder W changes only when
the conjugate and the conjugate transpose of (·), respectively. the CSI changes, staying constant for several symbol slots,
| · | and ∠(·) denote the amplitude and the phase of (·), and the relation between x[n] and d[n] is linear.
respectively,whileRe(·)andIm(·)aretherealandimaginary In the proposed symbol-level approach the precoding op-
parts of (·), and ι is used to denote the imaginary unit. (cid:107)·(cid:107) eration uses both the CSI and the data information in order
and (cid:107)·(cid:107) represent the Euclidean norm and the l norm of to design the transmitted vector. Thus, this scheme optimizes
∞ ∞
(·), respectively. Moreover, diag(·) denotes a diagonal matrix directly the vector x[n] for every symbol slot without any
whosediagonalentriesaretheelementsof(·),while◦isused intermediate steps (such as the design of W), aiming at
fordenotingtheelement-wiseHadamardoperations.(cid:68)isused constructively exploit the interference. This implies that the
as a generalized inequality for the optimization constraints, to relation between the input symbol vector d[n] and the output
be read as ≥ or as = depending whether the constraint is vector x[n] cannot be explicitly described as in conventional
referred to a boundary symbol or to an internal symbol of linear precoding, but lies into the optimization design of the
the constellation, respectively. Finally, ∂ and ∂ denote the precoding module1. The data symbols are assumed to be
∂x ∂x∗
gradient operator with respect to x and x∗, respectively. uncorrelatedandtakenfromagenericmulti-levelconstellation
represented by the symbol set D, having unit average power,
II. SYSTEMANDSIGNALSMODEL i.e., ED[|dj|2]=1.
We consider a single-cell multiple-antenna downlink sce-
A. Non-linear Amplification Stage
nario,whereasinglebase-stationisequippedwithN transmit
t
antennas serving K user terminals, with N ≥ K, each one Thesystemmodeldescribedin(2)isalinearone.However,
t
equipped with a single receiving antenna. We assume a block as already mentioned, it should be considered that the intro-
fading channel hj ∈C1×Nt between the transmit base-station ducedsystemmodelisactuallycorruptedbythenon-linearef-
antennasandthej-thuser.Thereceivedsignalatthej-thuser fects introduced in the per-antenna power amplification stages
in the symbol slot n can be written as: [14], [15], which affect both the amplitude and the phase of
the transmitted waveforms. The input-output characteristics
y [n]=h x[n]+z [n], (1) of several typologies of high-power amplifiers (HPAs) are
j j j
available in the literature [15], [17], [25]–[28], including the
where x[n] ∈ CNt×1 represents the transmitted signal vector amplitude-to-amplitude(AM-AM)andtheamplitude-to-phase
fromtheNt transmitantennas,andzj[n]isarandomvariable (AM-PM) effects. Moreover, analytical models are available
distributed as CN(0,σz2), modeling the zero mean Additive for describing the distortion effects of the HPAs, such as
White Gaussian Noise (AWGN) measured at the j-th user’s Saleh model [29] for traveling-wave-tube amplifiers (TWTAs)
receiving antenna. and a modified version [30] for solid state power amplifiers
Bycollectingthereceivedsignalsbyalltheusersinavector (SSPAs). To model such non-linear effects, we can write in
y[n]∈CK×1, the above model can be rewritten in a compact polar coordinates the input signal to the HPA on the generic
form as: i-th RF chain of the transmitter as2:
y[n]=Hx[n]+z[n], (2) x =ρ exp(ιθ ), (3)
i i i
where H = [hT1 ...hTK]T ∈ CK×Nt represents the system where ρi and θi are the amplitude and the phase of xi,
channel matrix, and z[n] ∈ CK×1 collects the AWGN com- respectively.Then,theoutputsignaloftheHPAcanbewritten
ponents for all the users. as:
The transmitted signal vector x[n] is obtained as output of
a precoding module, which takes as input the CSI, which is xˆ =f (ρ )exp(ιf (ρ ))exp(ιθ ), (4)
i A i P i i
an estimate of H, and the data information d[n] ∈ CK×1,
namely the data symbols to be conveyed to the users. In the 1In general, it is still possible to formalize the symbol-level precoding
scheme in order to design a precoder W[n], such that x[n] = W[n]d[n].
transmission scheme we assume a framing structure including
However,unliketheblock-levelschemes,inthiscasetheprecoderitselfwould
apreambleofpilotsymbols.Suchpilotsareexploitedbyeach depend on the data information, thus such intermediate step would actually
user to estimate the related channel vector, and the resulting add a redundant layer of complexity. Therefore, in this work we choose to
performthedirectoptimizationofx[n].
CSI is fed back to the base-station in order to be available for
2In order to ease the notation, hereafter the time index n is omitted in
the precoding operation. formulas.
1053-587X (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSP.2017.2781647, IEEE
Transactions on Signal Processing
• In other applications the power scarcity is not a crucial
issue, therefore stretching the operating region of the
HPAs close to saturation is not necessitated. However,
there might be the need to employ cheap hardware
components for the RF chains, and specifically cheap
amplifiersshowingmoreseverenon-idealities.Thisisfor
example the case of massive-MIMO [31].
In this framework, it is important to control the instantaneous
transmitted power and to minimize its peaks, in order to
mitigate the performance degradation due to the AM-AM
distortion. Furthermore, the use of separate per-antenna HPAs
Figure 1: Block scheme of the transmitter relying on symbol-level
comeswithanadditionalimpairment.Infact,asclearfromthe
precoding, for a generic symbol slot.
phase characteristic of the example in Fig. 2, the per-antenna
amplifiers introduce a phase shift which is considerably dif-
ferent for different instantaneous powers feeding the HPAs.
0
60 As a consequence, the precoded data streams transmitted on
the separate RF chains will experience a different phase shift
-2
50 through the amplification stages, due to the variable power
-4 eg] carried out by the symbols. This specific issue, which will be
er [dB] -6 40Shift [D rinefeprrreevdiotuosalsitderifafteurreen,tiaanldphcaosnestsithuitfet,shaansandodtibtieoennalcosonusirdceereodf
utput Pow -8 30ut Phase dcoegpTrehaedwaptiitrohonptohosefetdhtwesoyomvpebrrooablll-llesemyvsestleampbroepvceeor,fdoinnrmagmaaneplcype.rothaechseastutrraytioton
O 20p
-10 ut effect and the differential phase shift, by reducing the power
O
peaks among the different antennas of the transmitter3. In
-12 AM-AM Characteristic (left y-axis) 10 the following sections, these novel precoding strategies are
AM-PM Characteristic (right y-axis) further explained, and the related optimization problems are
-14 0
-20 -15 -10 -5 0 5 formalized and solved.
Input Power [dB]
Figure 2: An example of non-linear amplifier: normalized AM-AM III. SYMBOL-LEVELPRECODINGFORWEIGHTEDPEAK
andAM-PMcharacteristicsofthenon-linearizedTWTAof POWERMINIMIZATIONWITHLOWERBOUND(WPPMLB)
[27].
In this section a novel symbol-level precoding scheme
is presented, which performs a weighted per-antenna power
minimization and imposes a lower bound on the transmit
with f (·) and f (·) denoting the AM-AM and the AM-
A P
power on each RF chain. According to the general framework
PM conversions, respectively. The resulting system model is
ofsymbol-levelprecoding,themainobjectiveoftheproposed
shown in Fig. 1. A practical example of non-linear HPA is
scheme is to design the transmitted vector x by assuring a
giveninFig.2,wherethe(normalized)AM-AMandAM-PM
constructive interference effect at the users’ side, following
characteristics of the non-linearized TWTA model of [27] are
the definition in [9]. In other words, x should be optimized
represented. Such characteristics clearly show the saturation
so that the superposition of the multiple streams through the
effect and the introduced phase distortion, respectively. The
channel forces the received signal to the detection region
model of Fig. 2 is used as a practical reference in this work,
of the desired symbol, for each user. The novelty of the
in particular for the numerical results section, as it constitutes
proposed scheme, with respect to the work on constructive
ahighlynon-linearmodel(especiallywithrespecttotheAM-
interference carried out in [9], [10], lies in the different
PM curve) allowing a proper validation of this contribution.
optimization of the transmitted power. While the available
However, it should be stressed that all schemes proposed in
literature focuses on minimizing the total transmitted power
the remainder of this work are applicable to any non-linear
while guaranteeing some QoS targets at the users, here the
AM-AM/AM-PM characteristic, modeled in the form of (4).
focus is on the per-antenna transmitted power. Taking into
Theimportanceoftakingintoaccountthenon-lineareffects
account the non-linear effects of the channel, the goal is to
of the amplification stages in the precoding design lies in the
minimize the maximum power among the different antennas
two following considerations.
and, at the same time, to put a lower bound constraint on
• In some applications, such as satellite communications such power. This would guarantee a reduction of the power
[17], [26], the power is a scarce resource that has to be
efficiently exploited. As a consequence, the HPAs need 3It should be noted that the inter-symbol interference created by the
to be operated as close as possible to their saturation non-linear amplification stages is not modeled in the proposed optimization
schemes,astheydesignthetransmitsignalsonlyinthespatialdimension,for
point, and the consequent AM-AM distortion cannot be
eachsymbolslot.However,thiseffectistakenintoaccountinthenumerical
neglected. evaluationstageofSectionV-C.
1053-587X (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSP.2017.2781647, IEEE
Transactions on Signal Processing
peaks and a reduced variation of the instantaneous power The problem (5) appears to be complex and hard to tackle.
transmittedbythedifferentantennas,grantingbetterproperties However, it is possible to reformulate it in a more tractable
with respect to the non-linear amplification stages and, in form. More specifically, the following theorem holds, whose
particular, limiting the differential phase shift. Lastly, the per- proof is reported in the Appendix A:
antennatransmitpowerisconsideredinaweightedfashion,so
Theorem 1. Theoptimizationproblem(5)isequivalenttothe
astoaccountpossibleasymmetriesinthedifferentRFchains.
following one:
The resulting problem, referred to as weighted peak power
minimization with lower bound (WPPMLB), can be written
as follows4: x˜(d,H,γ,p)=argmin r
r,x˜
s.t. C1: (cid:107)B x˜(cid:107)≤r, i=1,...,N ,
i t
x(d,H,γ,p)=argmin r
r,x C2: x˜†(Ai)x˜≤−αr2, i=1,...,Nt, (6)
s.t. C1: αr2 ≤ |xi|2 ≤r2, i=1,...,N , C3: Re(D)H1x˜(cid:68)βR,
pi t (5) C4: Im(D)H x˜(cid:68)β ,
2 I
C2: |h x|2 (cid:68)κ2γ σ2, j =1,...,K,
j j j z C5: (TH −H )x˜=0,
1 2
C3: ∠h x=∠d , j =1,...,K,
j j where the optimization variable x˜ is a stacked version of
where r is a non-negative slack variable used for bounding x, namely x˜ = [Re(x)T,Im(x)T]T ∈ R2Nt×1, and the
the power, α is a parameter determining the lower bound other introduced quantities are defined as functions of the
(which is better characterized afterwards), γj is the target input parameters only5, with the matrices A being negative
i
SINR that should be granted for the j-th user, pi is the power semidefinite (NSD) ∀i=1,...,N .
weight for the i-th antenna, and κj = |dj|/(cid:112)ED[|dj|2] is a In particular, the sets of consttraints C1 and C2 in (6)
magnitude scaling factor for the symbol dj, which allows to correspond to the upper and lower bounding of the per-
account the different amplitudes of the symbols in the multi- antenna transmit power, respectively, while the remaining
level constellation D. The assumption to have symbols with constraints are related to the attained SINR and the received
unit average power implies that κj = |dj|. Moreover, the symbol phases at the different users6.
vector γ = [γ ...γ ]T ∈ CK×1 stacks the target SINR (cid:3)
1 K
for all the users, while p = [p1...pNt]T ∈ CNt×1 stacks
the power weights for all the antennas. Further, the notation
(cid:68) represents a generalized inequality: it shall be read as In the problem in (6), it can be seen how all the constraints
are convex, with the exception of C2, imposing the lower
≥ or = depending whether the constraint is referred to a
bound. Indeed, the upper bound constraint is a second order
boundary symbol or to an inner symbol of the constellation
cone (SOC) constraint, the QoS constraints and the phase
D,respectively(generalizedinequalitiesrelatedtothedifferent
constraints are affine, whilst the lower bound constraint is a
detection regions can be also found in [10]).
The set of constraints C1 in (5) gathers two different kinds non-convex, since the matrices Ai are NSD.
Hereafteranapproachtosolvethenon-convexproblem(6),
of constraints. In particular, such constraints impose an upper
basedonasuccessiveconvexapproximation(SCA)procedure
bound on the per-antenna weighted transmit power (through
the slack variable r, so to have a peak power minimization), [32], [33], is proposed.
and a lower bound at the same time. The lower bound is
defined through the design parameter α. This parameter shall A. FPP-SCA Algorithm
be chosen such that 0 ≤ α ≤ 1 and, if considered in dB, The main idea of the SCA algorithm is to iteratively
representsthewidthoftheregionwherethetransmitpoweris approximate the non-convex problem at hand into a convex
constrained.Thecloserto1isα,themorethepowervariations one, so to converge to the solution of the original problem.
will be limited. Nonetheless, the choice of a high value for More specifically, assuming a random point z ∈R2Nt×1, it is
α comes with a reduction of the degrees of freedom of the always true that (x˜−z)†(A )(x˜−z) ≤ 0, being A NSD.
i i
optimization problem, whose feasibility is not guaranteed, as Hence, the following inequality holds:
discussed afterwards. Concerning the power weights, they are
positive parameters affecting the loading of the different RF x˜†(A )x˜≤2z†(A )x˜−z†(A )z, (7)
i i i
chains: a higher value for p implies as higher power loading
i which represents a linear restriction of x˜†(A )x˜ around the
for the i-th antenna with respect to the other ones. The set of i
point z. By using the above inequality, the non-convex con-
constraints C2 represents a QoS constraint for each user. The
straint can be replaced by the following, which is affine in
set of constraints C3 represents the constructive interference
x˜:
condition,guaranteeingthateachuserreceivesthedesireddata
symbol with the correct phase.
2z†(A )x˜−z†(A )z ≤−αr2. (8)
i i
4Itshouldbementionedthatthisformulationoftheoptimizationproblem
isreferredtocircularconstellations,suchasM-PSKandM-APSK.Asimilar 5Thedefinitionoftheintroducedvectorsandmatricescanbefoundinthe
formulationforrectangularconstellations,suchasM-QAM,canbestraight- proof,andisnotreportedhereforthesakeofbrevity.
forwardlygivenbutitisnotshownforthesakeofbrevity.Thisconsideration 6Thegeneralizedinequalities(cid:68)appliedtovectorsintheconstraintsshall
appliestotheoptimizationproblemsformulatedintheremainderofthispaper. beconsideredelement-wise.
1053-587X (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSP.2017.2781647, IEEE
Transactions on Signal Processing
By using the above linear restriction, the problem (6) can context of non-linear systems, is the peak-to-average power
be written in the following approximated form, to be tackled ratio (PAPR) evaluated over time. For example, the temporal
iteratively: PAPR has been tackled for OFDM systems [22], [35], [36].
Here we focus instead on the SPAPR which, in light of the
impairments described in Section II-A, is also important in
x˜(d,H,γ,p)=argmin r
r,x˜ ordertoutilizethemultipleamplifiersinamorehomogeneous
s.t. C1: (cid:107)B x˜(cid:107)≤r, i=1,...,N , way. The SPAPR can be defined as:
i t
C2: 2z†(A )x˜−z†(A )z ≤−αr2,
i=k1,.i..,Nt,k i k (9) SPAPR= (cid:107)x(cid:107)2∞ . (11)
(cid:107)x(cid:107)2/N
C3: Re(D)H x˜(cid:68)β , t
1 R
In the direction of designing the transmit waveforms con-
C4: Im(D)H x˜(cid:68)β ,
2 I trolling their instantaneous power, the opportunity to have
C5: (TH1−H2)x˜=0, low SPAPR allows a reduction of the per-antenna power
imbalancesacrossthedifferentHPAs,thuslimitingtherelated
where z is the introduced auxiliary variable at the k-th
k
differential phase shift.
iteration, which is updated as z = x˜ until convergence,
k+1
i.e., until (cid:107)x˜−z (cid:107) is smaller than a predefined threshold.
k
TheintroducedapproachresortstotheSCAprocedure[32]– A. Spatial PAPR Optimization
[34],whichrequirestheinitialpointz0 tobeafeasibleonefor The SPAPR minimization (SPAPR-Min) problem can be
the original problem. However, since the feasibility of (6) is formulated as non-linear fractional program, as:
notguaranteed,itisnoteasytofindsuchfeasibleinitialpoint.
To solve the issue, we can rely on the feasible point pursuit (cid:107)x(cid:107)2
SCA(FPP-SCA)algorithm[33].Inparticular,theproblemcan x(d,H,γ)=argmin ∞
x (cid:107)x(cid:107)2
be made always feasible by introducing an additional slack (12)
s.t. C1: |h x|2 (cid:68)κ2γ σ2, j =1,...,K,
penalty term s as follows: j j j z
C2: ∠h x=∠d , j =1,...,K.
j j
x˜(d,Hˆ,γ)=arg min r+λ(cid:107)s(cid:107) It is possible to reformulate the problem (12), as shown
r,x˜,s in the following theorem, whose proof is reported in the
s.t. C1: (cid:107)Bix˜(cid:107)≤r+si, i=1,...,Nt, Appendix B:
C2: 2z†(A )x˜−z†(A )z ≤−αr2+s ,
k i k i k i+Nt Theorem 2. The optimization problem (12) is equivalent to
i=1,...,N , the following one:
t
C3: Re(D)H x˜(cid:68)β ,
1 R
C4: Im(D)H2x˜(cid:68)βI, x(d,H,γ)=argmin (cid:107)x(cid:107)2∞
C5: (TH −H )x˜=0, x (cid:107)x(cid:107)2
1 2
with s∈R2Nt×1, and λ a trade-off term between the orig(i1n0a)l s.t. C1: Re(dj)hjx+2x†h†j (cid:68)σz√γjRe2(dj),
objective function and the new penalty one. The modified j =1,...,K,
problem in (10) is always feasible for any choice of z and
theconvergenceisguaranteed[32],[33].Therefore,thei0nitial C2: Im(d )hjx−x†h†j (cid:68)σ √γ Im2(d ),
point z can be randomly chosen. If the converged slack j 2ι z j j
0
penalty variables turn out being all zero, then the related j =1,...,K,
solution solves the original problem (6). In general, the FPP- C3: t (ιt −1)h x+(ιt +1)x†h† =0,
j j j j
SCA algorithm can be applied by using different starting
j =1,...,K.
points z0, and then choosing the best solution, namely the (13)
oneresultinginthelowest-normpenaltyterm.Concerningthe where t =tan(∠d ).
j j
trade-off term λ, in the fashion of [33] we consider λ(cid:29)1, in (cid:3)
order to force the penalty terms toward zero, hence pushing
theiteratestowardsthefeasibleregionoftheoriginalproblem
In the reformulated problem (13), the challenging part is
(6), when it is non-empty.
in the non-linear fractional objective function. Dinkelbach
suggests a parametric way of solving the non-linear fractional
IV. SPATIALPEAK-TO-AVERAGEPOWERRATIO problems[37],[38],whosebasicideaistotacklethefractional
REDUCTION problem by solving a sequence of easier problems which
Inthissection,adifferentsymbol-levelprecodingschemeis converges to the global solution. Nevertheless, Dinkelbach’s
proposed,withtheobjectiveofdirectlyminimizingthespatial algorithmcanbeappliedonlyifthenumeratoranddenomina-
peak-to-average power ratio (SPAPR) amongst the transmit tor are convex and concave, respectively. Therefore, it cannot
antennas. A metric usually considered in the literature, in the bedirectlyappliedtotheproblemathand,sincethenumerator
1053-587X (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSP.2017.2781647, IEEE
Transactions on Signal Processing
(cid:107)x(cid:107)2 and denominator (cid:107)x(cid:107)2 are both convex functions. In Table I: Proposed Successive Linear Approximation for Non-linear
∞ Fractional Programming
order to solve the problem, we can again resort to a SCA
approach [32]–[34]. In particular, we can approximate the
quadratic function around a certain vector z ∈ CNt×1 by a 1) Initialization:Set(cid:15),k=0,η=0in(16),whichresultsinsolving
concave (affine) function as: F(0).
2) Evaluateη0= (cid:107)(cid:107)xx(cid:107)(cid:107)2∞2 ,z0=x.
3) Solvethefollowingoptimization:
(x−z)†(x−z) = x†x−2Re(z†x)+z†z ≥0
x†x≥2Re(z†x)−z†z x=argmxin (cid:107)x(cid:107)2∞−ηk(2Re(zk†x)−zk†zk)
x†x≈2Re(z†x)−z†z. (14) s.t. C1: Re(dj)hjx+x†h†j (cid:68)σz√γjRe2(dj), j=1,...,K,
2
Ucasninbgetrheiwsrliottwenerabs:ound approximation, the problem in (12) C2: Im(dj)hjx−x†h†j (cid:68)σz√γjIm2(dj), j=1,...,K,
2ι
x(d,H,γ)=argmin (cid:107)x(cid:107)2∞ C3: tj(ιtj−1)hjx+(ιtj+1)x†h†=0, j=1,...(1,K7).
x (2Re(z†x)−z†z) 4) Evaluate|F(ηk)|and(cid:107)x−zk(cid:107);if|F(ηk)|≥(cid:15)or(cid:107)x−zk(cid:107)≥(cid:15)
s.t. C1: Re(dj)hjx+2x†h†j (cid:68)σz√γjRe2(dj), 5) gSoettηoks+te1p=5.(cid:107)(cid:107)xx(cid:107)(cid:107)2∞2 ,zk+1=x,k=k+1,gotostep3.
j =1,...,K,
C2: Im(d )hjx−x†h†j (cid:68)σ √γ Im2(d ),
j 2ι z j j provided that the approximation is a concave lower bound
j =1,...,K, having the same first order behavior of the original function.
C3: t (ιt −1)h x+(ιt +1)x†h† =0, This is the case of the approximation in (14), as stated in the
j j j j
following proposition.
j =1,...,K.
(15) Proposition 1. Given the convex function f(x) = (cid:107)x(cid:107)2,
Now the problem can be solved by applying parametric and its concave (affine) approximation around z f˜(x,z) =
programming on the approximated formulation in (15). To 2Re(z†x)−z†z, the following properties hold:
this aim, we should define the optimization function F(η) =
min{(cid:107)x(cid:107)2 −η(2Re(z†x)−z†z )}, where S represents the
x∈S ∞ k k k f˜(x,z)≤f(x), (18)
setsofconstraintsC ,C andηisanauxiliaryvariabletoapply
parametric program1min2g techniques. Therefore, the problem f˜(x,x)=f(x), (19)
(cid:12)
can be formulated as: ∂ f˜(x,z)(cid:12)(cid:12) = ∂ f(x), (20)
x(d,H,γ)=argmin (cid:107)x(cid:107)2 −η(2Re(z†x)−z†z) ∂x (cid:12) ∂x
∞ z=x
x,η (cid:12)
s.t. C1: Re(dj)hjx+2x†h†j (cid:68)σz√γjRe2(dj), ∂∂x∗f˜(x,z)(cid:12)(cid:12)(cid:12)z=x = ∂∂x∗f(x), (21)
where the gradient is considered with respect to x and x∗,
j =1,...,K,
and these variable are treated as independent, in the fashion
C2: Im(d )hjx−x†h†j (cid:68)σ √γ Im2(d ), of [39].
j 2ι z j j
Proof. The properties (18) and (19) come straightforwardly
j =1,...,K,
from(14).Inordertoprove(20)and(21),wederivehereafter
C3: tj(ιtj −1)hjx+(ιtj +1)x†h† =0, the gradient of f(x) and f˜(x,z) with respect to x and x∗,
j =1,...,K. based on [39]:
(16)
Ultimately, to efficiently solve (13) using the formulation ∂ ∂
in (16), we propose the algorithm in Table I. The theoretical f(x)= {x†x}=x†, (22)
∂x ∂x
lowerboundoccurswhenalltheantennashavethesamepower ∂ ∂
(i.e., (cid:107)x(cid:107)2/N ), with an achieved unit SPAPR. f(x)= {x†x}=xT, (23)
t ∂x∗ ∂x∗
∂ ∂
f˜(x,z)= {z†x+x†z}=z†, (24)
B. Convergence of the Algorithm ∂x ∂x
∂ ∂
Itisprovenin[37]thattheparametricprogrammingscheme f˜(x,z)= {z†x+x†z}=zT. (25)
applied to concave/linear fractional programs converge to a ∂x∗ ∂x∗
global optimum (if the objective is to maximize the con- By evaluating (24) and (25) in z = x, the properties (20)
cave/linearfunction).Inourcasethisholds,sincetheobjective and (21) follows, and the proposition is proved. (cid:3)
is to minimize a convex/linear function.
On the other hand, considering the SCA approach, it is Although the parametric programming and the SCA ap-
provenconvergent[32]toaKarush-Kuhn-Tucker(KKT)point, proaches have been shown convergent individually, it shall be
1053-587X (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSP.2017.2781647, IEEE
Transactions on Signal Processing
600 realizations. Moreover, the noise variance σ2 is assumed
z
4.5 unitary. Finally, the target SINR is assumed the same for all
the users for the sake of simplicity, and it is fixed to 12 dB
4 SCA Error with K=5, N=10
Parametric Program Etrror with K=5, N=10 for all the results7.
3.5 t
SCA Error with K=5, Nt=8 In the following, the performance of the WPPMLB and
3 Parametric Program Error with K=5, Nt=8 SPAPR-Min schemes are analyzed with respect to some pa-
etric2.5 rameters, such as the number of transmit antennas and the
M input α (for the WPPMLB case). Then, some comparative
Error 2 results are presented, in order to compare the proposed tech-
1.5 niques to the benchmarks, i.e., the sum power minimization
approachof[9],[10],thepeakpowerminimizationschemeof
1
[19], [20], and the constant envelope precoder of [21].
0.5
0
1 2 3 4 5 A. Performance of WPPMLB Scheme with respect to the
Iteration Index Parameters
Figure 3: Error metrics of the algorithm of I, versus iteration index,
Hereafter, the performance of the WPPMLB scheme is
for two instances of the algorithm.
investigated with respect to the number of transmit antennas
and to the value of the design parameter α in the problem
(5). Concerning this, it is worth noticing that the value 1/α
noticed how the formulation (16), and accordingly the algo-
represents the imposed spatial dynamic range for the transmit
rithm proposed in Table I, employs these schemes in a joint
signal.Nevertheless,ithasbeenmentionedhowtheimposition
fashion,anditisnotstraightforwardtoprovetheconvergence
of a tight spatial dynamic range (i.e., α close to 1) may
of the final algorithm. However, the proposed scheme has
compromise the feasibility of the problem, since it implies
beenshownconvergentthroughnumericalsimulations.Thisis
showninFig.3fortwoinstancesofthealgorithm,withK =5 a reduction in the degrees of freedom. As a consequence, it is
andN setto10and8,respectively.Thefigureshows,forboth imperative to study to which extent it is possible to constraint
t
the cases, how the SCA error (cid:107)x−z (cid:107) and the parametric theWPPMLBproblem,andhowthisisaffectedbythenumber
k
programming error |F(η )| vary with respect to the iteration of transmit antennas Nt. The presented results are obtained
k
byrunningtheFPP-SCAalgorithmusingtworandomstarting
index, going to zero in a few iterations.
points and then choosing the best solution, as discussed in
SectionIII-A.Moreover,thepowerweightsareassumedequal
V. NUMERICALRESULTS
to one, for simplicity.
Inthissectionsomenumericalresultsarepresented,toshow In Fig. 4 the attained spatial dynamic range is displayed as
the effectiveness of the proposed approaches, in particular a function of the imposed one (i.e. 1/α), in dB, for different
the WPPMLB scheme and the SPAPR-Min sheme. Before valuesofN .Itisapparenthow,whenthenumberoftransmit
t
discussing the results, let us introduce the considered per- antennas is equal to the number of users (fixed to 5), the
formance metrics. The most prominent metrics with respect attained spatial dynamic range is larger than the required one
to the problem of non-linearities, which is the main object for almost all the simulated values. In other words, in this
of this contribution, are the SPAPR and the spatial dynamic casetheWPPMLBproblem(5)turnsoutnottobefeasiblefor
range. The former has been already defined in equation (11), all the symbols and channel realizations8, unless the imposed
while the latter is defined, for a specific symbol slot, as spatial dynamic range is very large (over 7 dB). It can be
the ratio between the maximum and the minimum transmit also noticed how, due to the infeasibility of the optimization
power amongst the antennas, i.e., as m(cid:107)ixn|(cid:107)x2∞i|2. Moreover, the problem, the attained spatial dynamic range shows even a
total transmit power and the average acihieved SINR are also decreasing trend with 1/α when the latter is below 2 dB.
considered for the performance evaluation. The introduced This means that, in the case of 5 antennas, reducing too
quantities, which are symbol dependent by definition, are much the imposed spatial dynamic range can even worsen
considered at a frame level by averaging over a large number the performance. On the other hand, by increasing Nt the
of symbol slots. An additional performance metric used in optimization problem is relaxed, and it is possible to respect
this section is the symbol error rate (SER), which is useful to the imposed constraints also with a reduced spatial dynamic
quantify the effectiveness of the proposed techniques when a range,asitisvisiblefromtheresultsobtainedwithNt =8and
non-linear channel is applied in the communication chain. Nt =10.Remarkably,inthelattercasetheproblemisfeasible
All the results presented in the remainder of this section even when a unit spatial dynamic range (0 dB) is imposed.
are obtained assuming a 16-APSK modulation scheme for the The dependency of the problem feasibility on the imposed
data symbols, while the number of users is fixed to K = 5. spatialdynamicrangeandonthenumberofantennasisfurther
The quasi-static block fading channel coefficients have been
generated, for the generic user j, as h ∼ CN(0,σ2I), 7ThisdoesnotapplyfortheresultsinFigs.8,10,wherethepower-SINR
j h dependenceisstudied.
with σ2 = 1. The results have been obtained averaging over
h 8Thisimpliesthatthelowerboundconstraintsonthepowerarenotmetin
30 fading blocks of 20 symbol slots each, for a total of average,asclearfromthedisplayedresult.
1053-587X (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSP.2017.2781647, IEEE
Transactions on Signal Processing
7 20
6 19 8 Tx Antennas
10 Tx Antennas
e [dB]5 W]18
g B
c Ran4 wer [d17
mi o
atial Dyna23 Total Tx P1156
Sp 5 Tx Antennas
8 Tx Antennas
1 10 Tx Antennas 14
0 13
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
1/α [dB] 1/α [dB]
Figure 4: Attainedspatialdynamicrange,indB,versusimposedone Figure 6: Total transmit power, in dBW, versus imposed spatial
in dB. dynamic range in dB.
1 1.4 40
0.9
1.2 SPAPR (left y-axis) 35
0.8 Total Tx Power (right y-axis) W]
s0.7 1 B
es 30 d
ability of Succ000...456 SPAPR [dB]00..68 25 Tx Power [
Prob0.3 5 Tx Antennas 0.4 20 otal
8 Tx Antennas T
0.2
10 Tx Antennas 15
0.2
0.1
0 0 10
0 1 2 3 4 5 6 7 5 6 7 8 9 10 11
1/α [dB] Number of Tx Antennas
Figure 5: Probability of success versus imposed spatial dynamic Figure 7: Attained SPAPR, in dB, versus number of transmit anten-
range in dB. nas.
shown in Fig. 5, where the probability of success in solving becauseoftheverytightconstraint,especiallyinthecasewith
the WPPMLB problem is shown. This quantity, which is the 8antennas.Nevertheless,theresultshowshowarelaxationof
probability of respecting the imposed constraint on the spatial the constraint on the imposed spatial dynamic range, as well
dynamicrange,iscalculatedthroughaMonteCarlosimulation as an increase in the number of antennas, allows to reduce
overthemultiplechannelandsymbolsrealizations.Thisfigure the transmit power. Such behavior is due to an increase of
shows how the probability of success is low for N =5, and the degrees of freedom of the optimization problem when the
t
how it strongly decreases when 1/α is reduced. This explains constraint is relaxed or more antennas are used.
the decreasing trend observed in Fig. 4 for 1/α below 2 dB.
Increasing the number of transmit antennas the probability of
success increases considerably, becoming basically 1 for any B. Performance of SPAPR-Min Scheme with respect to the
imposed spatial dynamic range when N =10. Number of Tx Antennas
t
In Fig. 6 the total transmit power, in dBW, is shown as Herein we examine how the performance of the SPAPR-
a function of the imposed spatial dynamic range, for 8 and Min scheme depends on the number of transmit antennas
10 transmit antennas. The case with only 5 antennas is not N considered in the system. This dependency can be seen
t
considered, since the problem is not feasible for basically all in Fig. 7, which displays the attained SPAPR and the total
thevaluesof1/α,aspreviouslydiscussed.Itcanbeseenhow transmit power as functions of N . Similarly to the previous
t
the configuration with α=0 requires a high transmit power9, problem,alsoherewecanobservehowincreasingthenumber
of transmit antennas implies better performance. In particular,
9However, it should be kept in mind that we are assuming a reference theattainedSPAPRdecreaseswhenN increases,reachingthe
t
scenariowithunitnoisevarianceatthereceivers’side,sotheresultsinterms
theoretical lower bound of 0 dB for N ≥ 9. Moreover, the
oftransmitpowershallbeinterpretedmoreinacomparativefashionthanin t
anabsoluteway. total transmit power sensibly decreases when Nt increases.
1053-587X (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSP.2017.2781647, IEEE
Transactions on Signal Processing
Table II: Summary of the considered symbol-level precoding tech- Table III: Comparison of the spatial dynamic range and SPAPR
niques. values for the different schemes.
TechniqueAcronym ExtendedName ProblemRef. SPAPR-Min WPPMLB PPM SPM
WPPMLB Weighted Peak Power Mini- (5) SpatialDynamicRange[dB] 0 1 5.1 14.4
mizationwithLowerBound SPAPR[dB] 0 0.2 0.6 4.7
SPAPR-Min SpatialPeak-to-AveragePower (12)
RatioMinimization
PPM PeakPowerMinimization (27)
SPM SumPowerMinimization (26)
22
20
C. Comparative Performance Analysis
W]
18
Hereafter, some comparative simulation results are dis- B
d
cussed for the proposed techniques, i.e. WPPMLB and er [16
SPAPR-Min. Unless specified otherwise, the number of trans- w
o
mit antennas is fixed to 10, allowing additional degrees of x P14
fimrepedoosemdisnpathtiealodpytinmaimzaictiornanpgreobinlemthsetWo bPePMexLplBoitaepdp,raonadchthies otal T12 WPPPMP MALpBpr oAapcphroach
T SPM Approach
fixed to 1 dB (i.e., α=−1 dB)10. SPAPR-Min Approach
In the context of symbol-level precoding, we consider as 10
benchmarks the sum power minimization (SPM) scheme [9],
8
[10] and the peak power minimization (PPM) scheme [19], 9 10 11 12 13 14 15 16 17 18
SINR Target [dB]
[20]. For the sake of completeness, we provide in the fol-
lowing a formulation for the SPM and the PPM optimization Figure 8: Total transmit power, in dBW, versus target SINR, in dB.
problems, respectively:
In Fig. 8 the total transmit power, in dBW, is shown as
x(d,H,γ)=argmin (cid:107)x(cid:107)2
x a function of the target SINR, in dB, for the considered
s.t. C1: |h x|2 (cid:68)κ2γ σ2, j =1,...,K, (26) techniques. As expectable, the power requirements of the
j j j z
C2: ∠h x=∠d , j =1,...,K, proposed schemes are higher with respect to the benchmarks.
j j
In particular, the SPAPR-Min scheme requires a higher total
x(d,H,γ)=argmin max {|x |2} power than the WPPMLB one. Moreover, Fig. 9 displays the
i
x i=1,...,Nt average power transmitted by each antenna for the different
s.t. C1: |h x|2 (cid:68)κ2γ σ2, j =1,...,K, (27) schemesathand,forafixedchannelrealization.Thisresultal-
j j j z
C2: ∠h x=∠d , j =1,...,K. lowstobettervisualizethetrade-offoftheproposedprecoding
j j
approaches,whichproduceamoreuniformpowerdistribution
For the sake of clarity, all the considered symbol-level
amongst the antennas at the expense of a higher consumed
precoding schemes are summarized in Table II, with their
power. It shall be highlighted how the more uniform power
acronyms and the reference to the respective optimization
distributionattainedbytheproposedapproachesimpliesbetter
problems.
performancewithrespecttothebenchmarks(SPMandPPM),
First of all, we discuss the achieved performance in terms
overagenericnon-linearchannel.Thisisclearlyquantifiedin
of spatial dynamic range and SPAPR, which are summarized
SER analysis presented later on in this section.
in Table III. The displayed values show a substantial gain
In order to give further insights on the proposed symbol-
of the proposed schemes with respect to the benchmarks,
levelprecodingschemes,wealsoconsiderhereinacomparison
especially in terms of spatial dynamic range, which turns out
withtheconstantenvelopeprecodingof[21].Byconstruction,
extremely high when the SPM scheme is used. As already
this approach designs waveforms with 0 dB dynamic range
observed, the SPAPR-Min approach is able to reach the lower
(both in space and in time), but it does not achieve the
boundforthespatialdynamicrangeandtheSPAPR.Itshould
constructiveinterferenceeffectofthesymbol-levelschemesat
be highlighted that this is also possible with the WPPMLB
hand.ThiscanbeseeninFig.10,wheretheachievedaverage
approach by properly setting α to 0 dB, which would come
SINR at the users, in dB, is compared between the different
of course with a higher transmit power required, according
approaches, as a function of the total transmit power in dBW.
to the trade-off shown in Fig. 6. Actually, the strength of the
Infact,wenotethatanincreaseinthetransmitpower,whichin
WPPMLBapproachliesinitsflexibilitywhichallowstocope
turnimpliesanincreaseintheinterferencelevel,enhancesthe
with systems having different requirements, realizing a trade-
constructive interference effect in the symbol-level schemes,
off between imposed spatial dynamic range and consumed
and results in a considerable gain in the attained SINR.
power.
On the other hand, in the scheme of [21] the interference
is harmful, thus the SINR has a slower growth when the
10However,itmaybesettoanyvaluedependingonthespecificnon-linear
scenarioathand. transmit power increases. In particular, in the case of [21] we
1053-587X (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.