A Multicast Approach for Constructive Interference Precoding in MISO Downlink Channel

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by  Maha Alodeh

A Multicast Approach for Constructive Interference
Precoding in MISO Downlink Channel
Maha Alodeh Symeon Chatzinotas Bjo¨rn Ottersten
SnT-Interdisciplinary Centre for Security, Reliability and Trust, University of Luxembourg
4, rue Alphonse Weicker, L-2721 Luxembourg
e-mail:{maha.alodeh, symeon.chatzinotas, bjorn.ottersten}@uni.lu
Abstract—This paper studies the concept of jointly utilizing • As preliminary step, we develop a new constructive
the data information (DI) and channel state information (CSI) interference algorithm, called constructive interference
5
in order to design symbol-level precoders for a multiple input
1 maximum ratio transmissions (CIMRT). This technique
andsingleoutput(MISO)downlinkchannel.Inthisdirection,the
0 is shown to outperform the constructive rotation zero
interferenceamongthesimultaneousdatastreamsistransformed
2 to useful signal that can improve the signal to interference forcing precoding (CRZF) in [7].

y noise ratio (SINR) of the downlink transmissions. We propose • We illustrate the relation between the constructive inter-
a a maximum ratio transmissions (MRT) based algorithm that ferenceprecodingandconstrainedconstellationmulticast
M jointlyexploitsDIandCSItogainthebenefitsfromtheseuseful
precoding and as a result we establish a suitable upper-
signals.Inthiscontext,anovelframeworktominimizethepower
consumption is proposed by formalizing the duality between the bound for constructive interference precoding systems.
2
constructive interference downlink channel and the multicast To characterize the multicast performance, we derive the
1
channels. The numerical results have shown that the proposed minimum power that is required to satisfy the quality of

] schemes outperform other state of the art techniques. service of multicast MISO for MPSK inputs.
T
• We verify the convexity of the power minimization
I I. INTRODUCTION with the additional constraints. Therefore, we formulate
.
s
the optimal precoders for power minimization for con-
c Inthelastdecade,multiuserMISOtechniqueshaveattracted
[ strainedconstellationmulticastchannel.Finally,weshow
a lot of attention due to their capability of serving co-channel

that there is a duality between constructive interference
3 usersinthesamefrequencyandtimeslots.Theapplicationsof
downlink channel (CIDC) and constrained constellation
v these techniques vary according to the requested service. The
0 first service type is known as broadcast in which a transmitter multicast channel (CCMC). Based on this duality, we
8 derivetheoptimalprecodingforconstructiveinterference
has a common message to be sent to multiple receivers. In
5 downlink channel.
physical layer research, this service has been studied under
6
Notation: We use boldface upper and lower case letters
. the term of multicasting [1]. Since a single data stream is
1 formatrices andcolumn vectors,respectively. (·)H, (·)∗ stand
sent to all receivers, there is no need to combat the interuser
0 for Hermitian transpose and conjugate of (·). E(·) and (cid:107)·(cid:107)
4 interference. In the remainder of this paper, this case will be
denote the statistical expectation and the Euclidean norm, ⊗
1 referred to as multicast. The second service type is known
: as unicast, in which a transmitter has an individual message denotes the kronecker product, and A(cid:23)0 is used to indicate
v the positive semidefinite matrix. ∠(·), |·| are the angle and
i for a receiver. Due to the nature of the wireless medium and
X magnitude of (·) respectively. R(·), I(·) are the real and the
the use of multiple antennas, multiple simultaneous unicast
r transmissions are possible in the downlink of a base station imaginary part of (·). Finally, the vector of all zeros with
a length of K is defined as 0K×1.
(BS). In these cases, multiple streams are simultaneously
sent, which motivates precoding techniques that mitigate the II. SYSTEMANDSIGNALMODELS
interuserinterference.Ininformationtheoryterms,thisservice We consider a single-cell multiple-antenna downlink sce-
type has been studied using the broadcast channel [2]. In nario,whereasingleBSisequippedwithnttransmitantennas
the remainder of this paper, this case will be referred to as that supports data traffic to K user terminals, each one of
downlink. them is equipped with a single receiving antenna. We assume
The concept of jointly using the DI and CSI in symbol a quasi static block fading channel hj ∈ C1×nt between the
level was originally proposed in [6]- [7]. Taking into account BSantennasandthejthuser.Letwk betheCnt×1normalized
the data information, the cross correlations among the users’ precodingvectorfortheuserj.Thereceivedsignalatjth user
subspaces can be optimized. The contributions of this paper yj is given by
can be summarized in the following points: y =√p h w d +(cid:88)√p h w d +z (1)
j j j j j k j k k j
k(cid:54)=j
This work was supported by the National Research Fund (FNR) of where z denotes the noise at jth receiver, which is assumed
LuxembourgundertheAFRgrant(reference4919957)fortheprojectSmart j
ResourceAllocationTechniquesforSatelliteCognitiveRadio. independent and identically distributed (i.d.d) complex Gaus-
sian distributed variable CN(0,1). A more compact system IV. CONSTRUCTIVEINTERFERENCEPRECODINGFOR
formulation is obtained by stacking the received signals and MISODOWNLINKCHANNELS
the noise components for the set of K selected users as
y = HWP12d + z, with H = [h1,...,hK]T ∈ CK×nt, In the remainder of this paper, it is assumed that the
W = [w ,...,w ] ∈ Cnt×K as the compound channel transmitter is capable of designing precoding on symbol level
1 K
and precoding matrices. Notice that the transmitted signal for each time instance utilizing both CSI and DI, so that the
d ∈ CK×1 includes the uncorrelated data symbols d (t) for createdinterferenceamongthesimultaneousspatialstreamsis
k
all users with√E[|dj|2]√=1, P12 is the power allocation matrix constructive.
P21 =diag( p ,..., p ). The CSI and DI are available at
1 K
the transmitter side. A. Correlation Rotation Zero Forcing Precoding (CRZF)
The precoder aims at minimizing the mean square error
III. CONSTRUCTIVEINTERFERENCEPRECODING while it takes into the account the rotated constructive inter-
ference [7]. The optimization problem can be formulated as
The interference among the simultaneous spatial streams
leadstodeviationofthereceivedsymbolsfromtheirdetection
region.However,insomecases(e.g.M-PSK)thisinterference J =min E{(cid:107)Rφd−(HWd+z)(cid:107)2}.
W
pushes the received symbols further into the correct detection
region and, as a consequence it enhances the system per- The solution can be easily expressed as
formance. Therefore, the interference can be classified into
constructive or destructive based on whether it facilitates or W =γHH(HHH)−1R (5)
CRZF φ
deteriorates the correct detection of the received symbol.
(cid:114)
where γ = (cid:0) P (cid:1) ensures the power normal-
A. Constructive Interference Definition tr RHφ(HHH)−1Rφ
ization. The cross correlation factor between the jth user’s
Assuming both DI and CSI is available at the transmitter, channel and transmitted kth data stream can be expressed as
the normalized created interference from the kth data stream
h hH
on jth user can be formulated as: ρ = j k . (6)
jk (cid:107)h (cid:107)(cid:107)h (cid:107)
k j
h w
ψ = j k . (2)
jk (cid:107)hj(cid:107)(cid:107)wk(cid:107) The relative phase φij that grants the constructive simultane-
ous transmissions can be expressed as
Since the adopted modulations are M-PSK ones, a definition
φ =∠d −∠(ρ .d ). (7)
for constructive interference can be stated as ij j ij i
Lemma 1: ForanyM-PSKmodulatedsymbold issaidto
k The corresponding rotation matrix can be implemented as:
receive constructive interference from another simultaneously
transmittedsymbold whichisassociatedwithw ifandonly R (j,k)=ρ exp(φ i), (8)
j j φ jk jk
if the following inequalities hold
and the received signal at jth user can be expressed as
(cid:32) (cid:33)
∠dj − Mπ ≤tan−1 RI{{ψψjjkkddkk}} ≤∠dj + Mπ (3) yj =a γ(cid:107)hj(cid:107)((cid:88)K ρjkdk)=b γ(cid:107)hj(cid:107)((cid:88)K εjk)d, (9)
k=1 k=1
R{dk}.R{ψkjdj}>0,I{dk}.I{ψkjdj}>0. (4) where εjk has the same magnitude as ρjk but with different
phase, and d : d ∈ C1×1,|d| = 1,∠d = θ,θ ∈ [0,2π]. By
Proof: For any M-PSK modulated symbol, the region of taking a look at (9-b), it has a multicast formulation since it
correct detection lies in φ ∈[∠d − π ,∠d + π ]. In order seems for each user that BS sends the same symbol for all
j j M j M
for the interference to be constructive, the received interfering users.
signalshouldlieintheregionofthetargetsymbol.Forthefirst Remark 1: Itcanbenotedthatthissolutionincludesazero
condition, the arctan(·) function checks whether the received forcing step and a correlation step Rφ. The correlation step
interferingsignaloriginatingfromthed th transmitsymbolis aims at making the transmit signals constructively received at
k
located in the detection region of the target symbol. However, each user. Unfortunately, this design fails when we deal with
the trigonometric functions are not one-to-one functions. This co-linearusersρjk →1.However,intuitivelyhavingco-linear
means that it manages to check the two quadrants which the users should create more constructive interference and higher
interfering symbol may lie in. To find which one of these gain should be anticipated. It can be easily concluded that
quadrants is the correct one, an additional constraint is added the source of this contradiction is the zero forcing step. In an
to check the sign compatibility of the target and received effort to overcome the problem, we propose a new precoding
interfering signals. technique in the next section.
B. ProposedConstructiveInterferenceMaximumRatioTrans- theremainingbeamformingvectors.Therefore,itjustmodifies
mission (CIMRT) the value of ξ , and the precoder reads as
kk
The maximum ratio transmissions (MRT) are not suitable W = DHV(cid:48)B(cid:48). (17)
CIMRT
forsimultaneousdownlinktransmissionsinMISOsystemdue
to the intolerable amount of the created interference. On the To grant constructive interference, we need to rotate the
other hand, this feature makes it a good candidate for con- (b ,b )-planebyformulatingtherotationasasetofnon-linear
k j
structive interference. The naive maximum ratio transmission equations as
(nMRT) can be formulated as ξ(cid:48) d = ξ cos(α)d −ξ sin(α)e−jδd
(cid:104) (cid:105) kk k kk k kj j
WnMRT = (cid:107)hh11H(cid:107),(cid:107)hh22H(cid:107),...,(cid:107)hhKKH(cid:107) . (10) ξj(cid:48)jdj = ξjksin(α)ejδdk+ξjjcos(α)dj (18)
A new look at the received signal can be viewed by exploit-
Since the set of non-linear equations can have different roots,
ing the singular value decomposition of H = SVD, and
thefunctionneedstobeevaluatedattheobtainedrootinorder
W =DHV(cid:48)SH as follows
MRT to find the optimal ones. The optimal solution can be found
y = HWd=SVDDHV(cid:48)SHP1/2d (11) when solving for ξ(cid:48) = (cid:113)ξ2 +ξ2 , ξ(cid:48) = (cid:113)ξ2 +ξ2 .
kk kk kj jj jj jk
Sometimes it is not feasible to solve for ξ∗ and ξ∗ , and
G = SVV(cid:48), B=SH (12) theirvaluesneedtobereducedcorrespondingklyk.Theprjojposed
algorithm can be illustrated in the following table:
where S ∈ CK×K is a unitary matrix that contains the left-
singular vectors of H, the matrix V is an K ×nt diagonal A1:ConstructiveInterferenceRotationforCIMRTAlgorithm
matrix with nonnegative real numbers on the diagonal, and 1) FindPassumingalltheusershaveconstructiveinterference.
D ∈ Cnt×nt contains right-singular vectors of H. V(cid:48) is the 2) FindsingularvaluedecompositionforH=SVD.
power scaled of V to normalize each column in W to 3) ConstructB,G.
MRT 4) Select(bk,bj)-planeforalluserspair.
unit. The received signal can be as 5) Findtheoptimalrotationparametersα,δfor(bk,bj)consid-
(cid:88)K √ eringPbysolving(18).
yj =(cid:107)hj(cid:107)( pkρjkdk). (13) 6) UpdateB=BRkj(α,δ)
k=1
V. MULTICASTMISOSYSTEMS
Utilizing the reformulation of y in (11), the received signal
The multicast channel is defined as the channel in which
can be written as
a multiantenna transmitter sends a single message to multiple
K K
(cid:88)√ (cid:88)√ single antenna users [1]- [5].
y =(cid:107)g (cid:107) p ξ d =(cid:107)g (cid:107) p ξ exp(θ )d (14)
j j k jk k j k jk k
k=1 k=1
A. Constrained M-PSK Multicast Transmissions
wheregj isthejth rowofthematrixG,ξjk = g(cid:107)jgbj(cid:107)k.SinceB The optimal input covariance for power minimization in
is a unitary matrix, it can have uncoupled rotations which can multicast system can be found as a solution of the following
granttheconstructivityofinterference.LetRkj betherotation optimization
matrix in the (b ,b )-plane, which performs an orthogonal
k j min tr(Q) s.t. h QhH ≥ζ ,∀j ∈K.(19)
rotation of the kth and jth columns of a unitary matrix while Q:Q(cid:23)0 j j j
keeping the others fixed, thus preserving unitarity. Assume
where ζ is the required SNR threshold for jth user. Different
without loss of generality that k > j. The rotation matrix in j
solutions in the literature were proposed as in [1]. However,
the (b ,b )-plane can be written as
k j
forM-PSKinputsweshoulddesignthemulticastprecodersso
A1:ConstructiveInterferenceRotationforCIMRTAlgorithm thatthereceivedsignalfallsintothedetectionregionofdesired
1) FindPassumingalltheusershaveconstructiveinterference. symbol.Assumingaunit-ranksolutionforQ,theoptimization
2) FindsingularvaluedecompositionforH=SVD. problem can be written as follows
3) ConstructB,G. (15)
4) Select(bk,bj)-planeforalluserspair. wCMC(d,H)= argmin tr(wwH)
5) eFriinndgtPheboyptsimolavlinrogta(1ti8o)n.parametersα,δfor(bk,bj)consid- s.t.w ∠(hjw)=∠(d) ∀j ∈K
6) UpdateB=BRkj(α,δ) h wwHhH ≥ζ ∀j ∈K.(20)
j j j
where the non trivial entries appear at the intersections of kth
and jth rows and columns. Hence, any unitary matrix B(cid:48) can Here, we have 2K additional constraints which limit the
performance of (20) in comparison to (19). These constraints
be expressed using the following parameterization
grantthereceptionofthesymboldwiththetargetSNRζ for
j
(cid:48) (cid:89)K (cid:89)K the jth user. More flexible constraints for the detection region
B =B R . (16)
kj ofthetargetsymbolsarediscussedin[8].Theminimumtrans-
j=1k=j+1 mit power in (19)-(20) occurs when the inequality constraints
It can be seen from the structure of the matrix in (15) that are replaced by equality (i.e. all users should achieve their
rotationinthe(b ,b )-planedoesnotchangethedirectionsof target threshold SNR). We can reformulate the constraint as
k j
min (wHw) B. From Multicast to Constructive Interference
w
(cid:112) In multicast, the cross correlations among users’ channel
s.t. I{h w}= ζ I{d} ,∀j ∈K
j j are exploited to aid the transmission of the data symbols.
(cid:112)
R{hjw}= ζjR{d} ,∀j ∈K (21) The same cross correlations are translated to interference in
the downlink channel due to the individuality of each user’s
A final formulation can expressed as
message. However, the constructive interference techniques
min (wHw)
in MISO downlink channels exploit these cross correlations,
w
s.t. hjw−(hjw)H =(cid:112)ζ I{d},∀j ∈K which raise the question about their relation with multicast
2i j techniques.
hjw+(hjw)H =(cid:112)ζ R{d},∀j ∈K. (22) Theorem 1: The optimal precoder for CIDC
2 j w (d,H)= argmin tr(wwH)
CIDC
w
It can be viewed that the constraints in (20) are turned from s.t. ∠(h w)=∠(d ) ∀j ∈K
j j
inequality constraints to equality constraint (21)-(22) due to
h wwHhH =ζ ∀j ∈K.(28)
signal aligning requirements. The Lagrangian function can be j j j
derived as follows is given by w (d,A(d )H) in (20), where A(d)
(cid:88) (cid:113) CM(cid:40)C j
L(w) = wHw+ µj(−0.5i(hjw−wHhHj )− ζtihI{d}) A(j,k)= exp((∠d−∠dj)i), j =k (29)
j 0, j (cid:54)=k.
(cid:88) (cid:113)
+ α ((h w+wHhH)− ζi R{d}) (23)
j j j th Proof: We assume that we have the following equivalent
j channel as
where µj and αj are the Lagrangian dual variables. The He =AH (30)
derivative for the Lagrangian function can be written as
The power minimization can be rewritten by replacing H by
dL(w) =w+0.5i(cid:88)µ hH +0.5(cid:88)α hH (24) its equivalent channel He in (20) as
dw∗ j j j j
j j min (weHwe)
w
By equating this term to zero, w can be written as s.t. ∠(h w)=∠(d) ∀j ∈K
e,j
K K
w =−0.5i(cid:88)µ hH −0.5(cid:88)α hH ≡(cid:88)ν hH (25) he,jwwHhHe,j =ζj ∀j ∈K. (31)
CMC j j j j j j
j=1 j j=1 where h is the jth row of the H . The optimal precoder
e,j e
where νj ∈C=−0.5iµj −0.5αj. The optimal values of the we for the equivalent channel can be expressed as
Lagrangianvariablesµj andαj canbefoundbysubstitutingw w =(cid:88)K ν hH . (32)
intheconstraints(22)whichresultinsolvingthesimultaneous e e,j e,j
set of 2K equations (26). The final constrained constellation j=1
multicast precoder can be found by substituting all µ and α Rewriting the first constraints in (31) as
j j
in (25). ∠(d−d )∠(h w)=∠(d)
j j
Corollary 1: The optimal precoding for power minimiza- ≡ ∠(h w)=∠(d ) ∀j ∈K (33)
j j
tion w in CCMCmust spanthe subspacesof each user’s
CMC
channel. shows the equivalence between the constrained constellation
Using (25), we can rewrite the received signal at jth receiver multicastchannelandconstructiveinterferencedownlinkchan-
as nel.
K
(cid:88) Corollary 2: K different M-PSK symbols can be received
y = h w d+z =h ν hHd+z
j j CMC j j k k j correctly at K different users by using a single precoding
k=1 vector w ∈ Cnt×1, designed according to Theorem 1, at the
K
(cid:88) BS if K ≤n .
= (cid:107)h (cid:107) (cid:107)h (cid:107)ν ρ d+z t
j k k jk j
VI. NUMERICALRESULTS
k=1
d∗1∠(ν ) The channel between the base station and jth user terminal
1 √
≡ h (cid:2)|ν |∗hH ... |ν |∗hH(cid:3) .. +z . ischaracterizedbyhj = γ◦h(cid:48)j whereh(cid:48)j ∼CN(0,1),andγ◦
j 1 1 K K  .  j istheaveragechannelpower.Inordertocomparealldescribed
d∗1∠(ν )
K techniques, we used an energy efficiency metric as following
(27)
(cid:80)
R
η = j j. (34)
From (27), the constellation constrained multicast can be P
tot
formulated as a constructive interference downlink channel
withsetofprecodershH,...,hH,eachoneoftheseprecoder where Rj is the rate achieved by the jth user and given
1 K by log (1 + ζ ), ζ is selected to satisfy the related M-
is allocated with power |ν | and associated with symbol 2 j j
k
d∗1∠ν . PSK modulation and Ptot is the transmitted power by BS
k
√
0.5(cid:107)h (cid:107)((cid:80) (−µ +α i)(cid:107)h (cid:107)ρ − (cid:80) (−µ +α i)(cid:107)h (cid:107)ρ∗ )= ζ I(d)
1 k k k k 1k k k k k 1k √ 1
0.5(cid:107)h (cid:107)((cid:80) (−µ i−α )(cid:107)h (cid:107)ρ + (cid:80) (−µ i−α )(cid:107)h (cid:107)ρ∗ )= ζ R(d)
1 k k k k 1k k k k k 1k 1
.
. (26)
.
√
0.5(cid:107)h (cid:107)((cid:80) (−µ +α i)(cid:107)h (cid:107)ρ − (cid:80) (−µ +α i)(cid:107)h (cid:107)ρ∗ )= ζ I(d)
K k k k k Kk k k k k Kk √ K
0.5(cid:107)h (cid:107)((cid:80) (−µ i−α )(cid:107)h (cid:107)ρ + (cid:80) (−µ i−α )(cid:107)h (cid:107)ρ∗ )= ζ R(d)
K k k k k Kk k k k k Kk K
tr(WWH).Themotivationofusingthismetricisthefactthat
CRZF and CIMRT do not aim at minimizing the transmitted
power to grant certain quality of service by their design.
Therefore, in some instance one or more users are given rate
whichishigherthanthetargetrate.Howeverforthemulticast
related techniques, the optimality holds when the target rate
canbeachievedexactly.Forthesakeoffairnessincomparison,
we use the metric in (34).
It can be seen in the Fig. (1) that energy efficiency for
multicast with optimal input outperforms all the proposed
techniques which confirms the fact of its optimality. For
the downlink scheme derived from the constrained multicast
takes the second place. It is shown that CRZF has the worst
Fig.2. Transmit power vs target rate.
performance at all SNR values. However, its peer CIMRT has
a superior performance on the expense of higher complexity
due to the need for solving a non-linear set of equations.
Fig.(2)depictsthecomparisonbetweentheoptimalmulticast
Fig.3. Energy efficiency vs target rate
Fig.1. Energy efficiency vs average SNR. ful energy in constructive interference precoding. In these
cases, the precoding design exploits the overlap in users’
subspace instead of mitigating it. Therefore, we proposed a
techniqueandconstrainedconstellationmulticasttransmission
new technique based on MRT to constructively correlate the
from the power minimization perspective. The considered
interference to enure the correct reception of data symbols.
averageSNRis10dB,andallusershavethesametargetrate.
This fact enabled us finding the connection between the
Itcanbeconcludedthattheoptimalmulticastoutperformsthe
constructive interference precoding and multicast precoding
constrained constellation multicast for all target rates, due to
wherein no interference should be mitigated. Therefore, we
thefactthattheconstrainedconstellationmulticastrequiresthe
found the solution for power minimization considering two
phase alignment to ensure the correct reception of the target
inputsscenario:theoptimalinputandtheconstrainedconstel-
symbols.
lation.Fromtheirclosedformulations,weconcludedthattheir
Fig.(3)depictsthecomparisonbetweendifferenttechniques
transmissions should span the subspaces of each user. From
from energy efficiency perspective with increasing the target
the M-PSK constrained constellation multicast, we managed
rates.ItisclearlyillustratedthatCIMRT,constrainedmulticast
tofindthattheoptimalconstructiveinterferenceprecodingcan
and optimal multicast have very close performance at high
be expressed by solving this multicast problem assuming an
target rates. Moreover, it can be concluded that CRZF has
equivalent channel.
inferior performance with respect to the other techniques.
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