Transmit Beamforming for MISO Broadcast Channels with Statistical and Delayed CSIT

0.34 MB·English

by  Mingbo Dai

1
Transmit Beamforming for MISO Broadcast
Channels with Statistical and Delayed CSIT
Mingbo Dai, Student Member, IEEE, Bruno Clerckx, Member, IEEE
Abstract—This paper focuses on linear beamforming design SBF strategy with Weakest Eigenvectoras SWEBF. Recently,
and power allocation strategy for ergodic rate optimization in SBFmethodwithGeneralizedEigenvector(SGEBF)hasbeen
a two-user Multiple-Input-Single-Output (MISO) system with
shown to maximize the ergodic sum-rate at high SNR for
5 statistical and delayed channel state information at the trans-
M-user M-transmit-antenna MISO BC when M = 2 [8] or
1 mitter (CSIT). We propose a transmission strategy, denoted
0 as Statistical Alternative MAT (SAMAT), which exploits both M → ∞ [9]. However, the optimal precoder for the general
2 channel statistics and delayed CSIT. Firstly, with statistical M > 2 case is still unknown due to a lack of closed-
CSIT only, we focus on statistical beamforming (SBF) design form ergodic sum-rate expression. In [10], the generalized
n that maximizes a lower bound on the ergodic sum-rate. Sec-
eigenvector (GE) solution is arrived based on the ergodic
a ondly, relying on both statistical and delayed CSIT, an iterative
J signal-to-leakage-and-noiseratio(SLNR),whichleveragesin-
algorithm is proposed to compute the precoding vectors of

8 Alternative MAT (AMAT), originally proposed by Yang et al., dependencebetweenthenumeratoranddenominatorofSLNR.
whichmaximizesanapproximationoftheergodicsum-ratewith However, the optimal solution to the true problem (ergodic

] equal power allocation. Finally, via proper power allocation, sum-rate) is not apparent. In this paper, we focus on a lower
T
the SAMAT framework is proposed to softly bridge between
bound on the ergodic sum-rate and address the problem for
I SBF and AMAT for an arbitrary number of transmit antennas
. two-user, arbitrary M-transmit-antenna case.
s and signal-to-noise ratio (SNR). A necessary condition for the
c power allocation optimization is identified from the Karush- Whenever the feedback delay is larger than the channel’s
[
Kuhn-Tucker (KKT) conditions. The optimum power allocation coherence time, the CSIT is delayed but was proved to still
1 to maximize an ergodic sum-rate approximation is computed benefit the Degree-of-Freedom (DoF) of MISO BC [6]. In
using Sequential Quadratic Programming (SQP). Simulation
v the example of two-transmit-antenna, two-user channel, the
resultsshowthattheproposedSAMATschemeyieldsasignificant
4 maximum sum DoF of 4 can be obtained by retransmitting
6 sum-rate enhancement over both SBF and AMAT. 3
theoverheardinterferenceanddoinginterferencecancellation.
8 Index Terms—MISO systems, statistical CSIT, delayed CSIT,
ThisstrategyisreferredtoasMAT.Thework[11]generalized
1 power allocation, statistical beamforming, SQP.
0 the finding of [6] as GMAT and achieved a higher data
. rate at finite SNR by constructing precoders which strike a
1
0 I. INTRODUCTION balance between desired signal enhancementand interference
5 IN multi-user (MU) MISO Broadcast Channel (BC), alignment. Moreover, an alternative MAT (AMAT) transmis-
1 sion strategy [12] was introduced to achieve a sum DoF
schemesthatachievethesum-ratecapacityandthecapacity
v: region have been studied in [1–4]. The performance of these between [43, 2] by utilizing both imperfect current CSIT and
Xi optimized approaches heavily depends on the acquisition of perfectdelayed CSIT. In time correlated channel,the solution
proposedin [12] smoothly bridgesbetween MAT and PZFBF
accurateandinstantaneousChannelStateInformation(CSI)at
r
in terms of sum DoF. Inspired by [12], we aim to bridge
a thetransmitter,whichisnotfeasibleinpracticeduetochannel
between AMAT and SBF in terms of ergodic sum-rate based
estimation errors, limited feedback resources and delay [5],
on statistical and delayed CSIT. Although our framework is
[6]. Moreover, obtaining perfect CSIT can incur unaffordable
similar to [12], the channel models are completely different.
feedback overhead [7].
Inourscenario,spatiallycorrelatedchannelmodelisassumed.
In practice, statistical CSI and/or perfect but outdated CSI
Statistical CSIT (full-rank channel covariance matrix) is use-
is only available at the transmitter side. The former term
less for DoF enhancement or equivalent transmission slot
whichischaracterizedbythechannelcovariancematrixvaries
reduction, since the channel estimation error based only on
slowly and thereby can be easily and accurately acquired
this statistical information does not scale down with SNR.
throughlong-termfeedback.OnesimpleStatisticalBeamform-
Nevertheless, statistical CSIT is useful for boosting the sum-
ing(SBF)approachistoprecodethetransmittedsymbolalong
rate at finite SNR.
the weakest eigenvector of the channel covariance matrix of
the unintended user. Such a scheme generalizes the idea of With both channel statistics and delayed CSIT at hand,
the Perfect CSIT-aided Zero-Forcing Beamforming (PZFBF) authorsin [13] developedanenhancedMATstrategy,denoted
to the statistical CSIT only environment.Thus, we denote the as VMAT, yielding a higher sum-rate than the original MAT
at finite SNR. However, in highly-correlated channel, the
M.DaiandB.ClerckxarewiththeDepartmentofElectricalandElectronic rate performance of VMAT is still inferior to SBF1 which
Engineering,ImperialCollegeLondon,UK,SW72AZUK(e-mail:{m.dai13, exploits only statistical CSIT [13]. In a nutshell, statistical
b.clerckx}@imperial.ac.uk). B. Clerckx is also with the School ofElectrical
Engineering,KoreaUniversity,Seoul,Korea.B.Clerckxisthecorresponding
author. 1Throughoutthepaper,SBFreferstoeitherSWEBForSGEBF.
2
channel information is not fully exploited. In [14], authors is the expectation operator. λ () and λ () indicate the
max min
· ·
analyze the error rate performance at finite SNR and the largest and smallest eigenvalues of a matrix and their cor-
diversity-multiplexing trade-off at infinite SNR of a space- responding eigenvectors are denoted by u () and u (),
max min
time encoded transmission with delayed and statistical CSIT. respectively. N() stands for the null space· of a matri·x.
·
So far, there has been no investigation on how to further Operators Tr() and det() refer to the trace and determinant
· ·
enhance the finite SNR sum-rate performance beyond that of a matrix. We denote Exp(c) as the exponentialdistribution
achievable with either statistical CSIT strategies (e.g., SBF) with parameter c and U(a,b) as the uniform distribution. Let
or delayed CSIT strategies (e.g., AMAT). In this paper, we Φ = R CM×M R is positive definite .
PD
{ ∈ | }
design a spatial precoding AMAT-style transmission scheme,
denoted as statistical AMAT (SAMAT), that softly bridges II. SYSTEM MODEL
the gap between AMAT and SBF at any SNR. With this
Consider a MISO broadcast channel where the transmitter
background, the main focus of this paper is to investigate
equipped with M antennas (M 2) wishes to send private
power allocation and beamforming optimization. Specifically, ≥
messagestotwouserseachwithasingleantenna.PerfectCSI
the main contributions are listed as follows:
is instantaneously available at the receiver side whereas the
• With either statistical or delayed CSIT, the proposed transmitter acquires this information with a delay larger than
SAMAT strategy can directly boil down to SBF or the coherencetime of the channel(such that the reportedCSI
AMAT. In the former case, we show that SGEBF is knownatthetransmitterisuncorrelatedwiththecurrentCSI).
optimal to maximize a lower bound on the ergodic sum- We assume perfect statistical CSIT which is characterized by
rate at high SNR. Under both CSIT but equal power the spatialcorrelationmatrix.Thisis a reasonableassumption
allocation, statistical CSIT-aided AMAT can achieve a because channel statistics are more related to the scattering
significant boost of rate performance relative to the environmentand independent of the transmission period.
original AMAT. More specifically, an efficient iterative Rayleighfadingchannelmodelisconsidered,whichimplies
algorithm is developedto compute the optimal statistical that the spatial statistics can be completely depicted by the
precoders to maximize an approximation of the ergodic second-order moments of the channel [8]. Specifically, we
sum-rate. Monotonic convergence of the algorithm is denotethechannelbetweenthetransmitteranduserAintime
proved.For two-user two-transmit-antennacase, anytwo slot j as h and similarly g for user B:
j j
beamforming vectors constituting a unitary matrix are
certified to be optimal. h =R1/2h
j A w,j (1)
• The power allocation can be further optimized to max- gj =R1B/2gw,j,
imize the ergodic sum-rate. A SAMAT transmission
where h and g are M 1 vectors with independentand
strategy is proposed to bridge the gap between SBF w,j w,j ×
identical distribution (i.i.d) (0,1) entries. They are as-
and AMAT at any SNR. Due to the complexity of CN
sumedconstantwithinonetimeslotandvaryingindependently
deriving a closed-form expression for the ergodic sum-
across time slots. R and R are full rank positive definite
rate, a tractable approximation needs to be computed. A A B
covariancematrices2 foruserA andBrespectively,whichcan
SequentialQuadraticProgramming(SQP)algorithm[15]
bedecomposedasR =V Λ VH, k =A,B.V CM×M
is implemented to solve the consequent non-linear non- k k k k k ∈
is a unitary matrix whose columns are eigenvectors of R ,
convex constrained optimization problem. The necessary k
while the diagonal Λ that contains the eigenvalues of R
condition on power allocation optimization is further k k
is normalized as Tr(Λ ) = M. Λ = I indicates the k-th
identified. In low spatial correlation channels, the pro- k k
channel is spatially uncorrelated while rank(Λ )=1 implies
posed SAMAT scheme boils down to AMAT. In highly k
it is fully correlated [16].
correlatedchannels,SAMATbehavesasSBFatlowSNR
The proposed SAMAT framework is shown in Fig. 1.
while performing as AMAT at high SNR. In general,
It contains two stages/three time slots. In the 1st slot, the
SAMAT enables a significantly higher sum-rate than
transmitter superposes four private symbols s ,s ,s ,s
both SBF and AMAT. This achievementstems from two A1 A2 B1 B2
and sends them to both users. Denote u = WP1/2s
aspects:1) the transmissionof extraprivate messages;2) A A A
and u = QP1/2s as the encoded symbols with statistical
the optimized power allocation for SAMAT. B B B
beamformerandpowerallocation,whereW=[w w ],Q=
1 2
The rest of this paper is organized as follows. Section II [q q ],P = diag(P , P ),P = diag(P , P ). s =
1 2 A 1 2 B 3 4 k
introducesthe system model. Section III and IV elaborate the [s ,s ]T represents the Gaussian symbols intended to user
k1 k2
optimal precoder design for SBF and statistical CSIT-aided k and E s sH = I. At the end of this stage, each user
AMAT, respectively.In Section V, we formulateSAMAT and receivesit{sdkeskir}edsignalaswellastheoverheardinterference
maximize the achievable ergodic sum-rate by optimizing the due to the superposed transmission. Denote η =hHu and
A 1 B
power allocation. Numerical results are shown in Section VI η = gHu as the interference overheard by user A and B,
B 1 A
while Section VII concludes the paper. respectively. In stage II, the transmitter has access to h and
1
Notations: Bold lower case and upper case letters denote
vectors and matrices, respectively. The superscripts ()T and 2Forrankdeficientcase,thesymbolintendedtouseriissimplyprecoded
()H represent the transpose and conjugate transpos·e. The by a column vector in N(Rj). By doing this, the overheard interference of
· eachsymbolcanbecompletelyremoved.Thus,wecantransmittwosymbols
notation diag() stands for a diagonal matrix whereas E() atonetimeinstant, achieving asumDoFof2asifwehaveperfectCSIT.
· ·
3
Fig. 1: Block diagram of the proposed SAMAT scheme.
g (delayedCSIT).Then,η andη canbereconstructedand power budget of the transmitter for each time slot.
1 A B
broadcastviaa singleantennainthe followingtwo slots. This Althoughour frameworkis similar to [12], there are essen-
stagehelpsbothuserseliminatetheoverheardinterferenceand tial distinctions between them. A first distinction lies in the
reinforcethedesiredsignals.Inaddition,newprivatemessages channelmodel. We exploitspatial correlationto compressthe
spA1,spA2,spB1,spB2aresenttobothusersinasuperposedfashion interference and make room for extra symbols transmission
andthisextratransmissionmakesuseofstatisticalCSITonly. whiletheymakeuseoftimecorrelation.Thepowerallocation
Pk 0,k = 1,...,10 indicate the power allocated to each in [12] depends on SNR and quality of current CSIT while
≥
symbol. wk and qk,k = 1,...,3 denote M 1 unit-norm our power allocation strategy relies on SNR, precoder design
×
precoders which depend only on statistical CSIT. as well as spatial correlation.
The proposed SAMAT scheme facilitates a smart use of The secondary distinction lies in the encoding/decoding
statisticaland/ordelayedCSIT.With statisticalCSIT only,we strategy (and hence the transmission protocol). More specif-
simply put P1 = P3 = P5 = P8 = 0 and then SAMAT ically, interference quantization is crucial for [12], where
boilsdowntoSBFineachtimeslot.WithdelayedCSITonly, the overheard interference symbol with a reduced power is
we simply put P6 = P7 = P9 = P10 = 0. Then, SAMAT transmitted with full power in order to save channel re-
becomes the AMAT scheme and enables a sum DoF of 4 at sources. Interference quantization is proposed to solve the
3
high SNR. If the transmitter has both statistical and delayed consequent problem of power mismatch (which scales with
CSIT, proper power allocation and statistical precoding can transmit power). By decoding the interference symbols first,
make room for extra symbols transmission. The benefits of [12]equivalentlyobtainsoneAMATtransmissionplustwoZF
transmitting new symbols overcome the loss caused by the transmissions.DoFgainathighSNRcanbeobtainedoverthe
interferenceitcreatestothe mainAMATtransmission.Inthis original AMAT scheme.
case, the proposedSAMAT frameworkallows for the parallel Incontrastwith[12],theoverheardinterferenceismulticast
transmission of SBF on top of AMAT while outperforming
by analog transmission in our scenario and the reason is
AMATandSBFatanySNR.Morespecifically,thetransmitted
threefold. First, we retransmit the interference symbols after
signals are written as scaling them by constant (i.e., not scaling with the trans-
x1 = uA+uB mit power) factors P5 and P8. P1,2,3,4 in the main AMAT
x = √P η , 0 T + √P w sp + √P q sp transmission scale with the transmit power at high SNR to
2 5 A 6 3 A1 7 3 B1 achieve the DoF of 4. However, to guarantee the power
x = √P η , 0 T + √P w sp + √P q sp . 3
3 8(cid:2) B (cid:3) 9 3 A2 10 3 B2(2) constraint, the multiplication terms P5(λA1P3 +λA2P4) and
P (λ P + λ P ) in eq. (3) limit P and P to some
For (A)MAT-b(cid:2)ased s(cid:3)chemes, the transmit power in stage II 8 B1 1 B2 2 5 8
constants.Second,interferencequantizationwouldpreventthe
isinherentlydependentonthechannelrealizationthatchanges
proposed SAMAT scheme from bridging SBF at low SNR.
rapidly. The power consumption in each transmission period More specifically, in the 2nd/3rd slot, SAMAT should behave
hardly keeps constant. Thus, a less restrictive metric is the
as SBF at low SNR and should therefore allocate most of
long-term average power constraint
the transmit power to the extra symbols and only a little
powertotheoverheardinterferencesymbol.Following[12],if
P¯ = E[Tr(x xH)]+E[Tr(x xH)]+E[Tr(x xH)]
c 1 1 2 2 3 3 interference quantization is applied, the digitized interference
= P +P +P +P +P +P +P +P + shouldbedecodedfirstbytreatingtheextrasymbolsasnoise.
1 2 3 4 6 7 9 10
P (λ P +λ P )+P (λ P +λ P ) In this case, however, the decoding would fail because the
5 A1 3 A2 4 8 B1 1 B2 2
noisepowerwouldoverwhelmthedesiredsignalpower.Third,
3P, (3)
≤ due to the inherent properties of the channel model (full-
where λ = qHR q , λ = qHR q , λ = rank channel covariance matrix), a sum DoF strictly larger
A1 1 A 1 A2 2 A 2 B1
wHR w , λ = wHR w . The expectation is taken over than 4 cannot be achieved in our case (contrary to [12]).
1 B 1 B2 2 B 2 3
the input signals and the channels and P denotes the average Hence the SAMAT transmission and reception strategies are
4
not motivated by a DoF maximization. With SAMAT, a sum
DoF of 4 is achieved where the extra private symbols are
3 w =u (R−1R ), q =u (R−1R ). (7)
not used to increase the DoF at high SNR (contrary to [12]) GE max B A GE max A B
but to boost the sum-rate at low/finite SNR. This implies that The corresponding lower bound of the ergodic sum-rate is
the retransmitted overheard interference does not have to be
decoded first in SAMAT (contrary to [12]) but can simply
R = log χ R−1R =log χ R−1R ,
be aligned and cancelled so as to decode the private and sum,lb 2 B A 2 A B
(8)
extra symbols. Recall again that [12] relies on interference
where χ()= λmax(·) (cid:0)is t(cid:0)he condit(cid:1)i(cid:1)on numbe(cid:0)r.(cid:0) (cid:1)(cid:1)
quantization to decode first the overheard interference and · λmin(·)
then the private messages in order to increase the sum DoF Proof: A detailed proof is relegated in Appendix B.
beyond 4. The detailed description of our decoding strategy A special case3 of (8) is confirmed by [10, Corollary 2]
3
is provided in section V. by noting that χ() > 1. In the low SNR region where the
·
interference can be completely ignored, the optimal choice
is to send along the dominant statistical eigen-mode of the
III. STATISTICAL BEAMFORMING
user’s own channel [17]. At intermediate SNR, however, [8]
Let us first focus on the scenario with statistical CSIT has shown the difficulty of finding a closed-form expression
only. As mentioned before, the proposed SAMAT scheme of the optimal precoders even for M = 2 case. Instead, it
boils down to SBF, i.e., the transmitter sends two statistically is solved by an exhaustive search operated upon a linearly
precodedsymbols,eachintendedtooneuser.Sincethesystem combined high- and low-SNR solution. In the general M >
model for the transmission in each slot is identical, we 2 case, we compute only a high-SNR solution and avoid the
can focus on one slot and omit the subscript. Assume the line search method. The simulation results show that it works
transmitter equally allocates its power budget to both users, well at practical SNR.
the delivered signal described by (2) becomes Remark 1: The closed-form precoders that maximize the
ergodic sum-rate of SBF is difficult to compute due to the
x=√ρws + √ρqs , (4)
A B coupled nature in SINR expression. To solve this problem,
where ρ = P. For simplicity, we will look at the rate we can use an alternative SLNR metric, which is defined as
performance of2user A only and a similar derivation can be SLNR = ρ|hHw|2 . At high SNR, the maximization of
A 1+ρ|gHw|2
easily extended to user B. The received signal at the receiver a lower bound on E[SLNR ] also leads to the solution (7).
A
side is given as y = √ρhHws +√ρhHqs +n , where Similarly, the effectiveness of the SLNR metric in designing
A B A
h CM is the channel vector and n (0,1) is the multi-user transmit beamforming vectors has been examined
∈ A ∼ CN
standardcomplexadditivewhiteGaussiannoise(AWGN).The in [8, 10, 18].
achievable ergodic rate of user A is given by In contrast with SGEBF, the precoding vectors of
SWEBF in correlated channel can be written as w =
WE
RA =E[log2(1+SINRA)], (5) umin(RB), qWE =umin(RA). However, the rate performance
where SINR = ρ|hHw|2 is the instantaneous signal-to- ofSWEBFisunfavourableinthescenariowherebothchannels
A 1+ρ|hHq|2 of user A and B have similar weakest eigen-direction (e.g.,
interference-and-noiseratio (SINR) by treating the multi-user
co-locatedusers). To be specific, the precodingvectorswhich
interference as noise. Consequently, the ergodic sum-rate of
are designed to remove the interference also cancel out the
the system with linear beamforming is R ,R +R .
sum A B intended signal. By contrast, the GE beamforming approach
In the spatially correlated channels, the role of spatial
obtains a balance between interference cancellation and de-
statistics on the rate performance of a linear beamformer has
sired signal enhancement. In other words, SGEBF exhibits
been well studied in [8]. They consider a MISO broadcast
robustness with respect to different channels compared to
setting with M antennas at the transmitter and M single-
SWEBF.Moreover,wecanobservefromthenumericalresults
antennausers.WhenM =2,optimalprecodersthatmaximize
that SGEBF outperforms SWEBF further as M increases.
the ergodic sum-rate were developed at extremely high/low
SNR.Meanwhile,thebestprecodingvectorsarestillunknown
IV. ALTERNATIVE MAT
for general case (M > 2) due to the difficulty of computing
a closed-form expression for the ergodic sum-rate. In this Letusfocusonthescenariowithbothdelayedandstatistical
paper,we show by the followingtheoremthat the generalized CSIT. Under equal power allocation, [13] has shown that
eigenvector (GE) is still valid for the two-user, M-transmit- additional channel statistics enable a higher achievable sum-
antenna scenario. rate relative to the original MAT. However, there are big
Theorem 1: For any M, the ergodic sum-rate of SBF at differences between this section and [13]. First, they released
high SNR can be lower bounded by the power constraint in the interference retransmission phase
(stageII),whichleadstoavariationofthetotaltransmitpower.
wHR wqHR q We here controlthe power consumptionby using a long-term
R log A B , (6)
sum ≥ 2 wHR wqHR q power constraint. Second, an efficient iterative algorithm is
(cid:18) B A (cid:19)
where γ is the Euler constant. The precoders that maximize 3When the two users share the same set of statistical eigenmodes but
the lower bound in (6) are given by orthogonal dominanteigenvectors andM =2.
5
TABLE I: Algorithm 1&2: Precoder Optimization for AMAT
developedtocomputethestatisticalprecoderstomaximizean
approximationoftheergodicsum-rate.Particularly,monotonic 1: Initialize: Set iteration index m=0, randomly generate
convergence of the algorithm is proved. This section reveals and then normalize w(0), w(0)
howmuchtheergodicsum-ratecanbeimprovedbystatistical 1 2
2: Repeat
precoderonly.Itwillbeusedasabaselineforthenextsection
3: m m+1
where the rate performance is further enhanced by additional ←
4: Update w(m) with GradAct [Algorithm 1], or
power allocation. 1
with Max-Eig [Algorithm 2]
5: Update w(m) with GradAct or with Max-Eig
2
A. Rate Approximation 6: Until Θ(m) Θ(m−1) ǫ
| A − A |≤
The proposed SAMAT scheme becomes statistical CSIT-
aided AMAT by simply letting P = P = P = P =
6 7 9 10
0,P = P = 1 in (2) and the transmitter distributes equal where
5 8
power to four symbols, denoted as ρ. The transmitted signals Θ = Tr(WHR W)Tr(WHR W)
A A B −
can be expressed as Tr(WHR WWHR W) (14)
A B
x1 = √ρWsA + √ρQsB and a=eEi( 1) 2γ, Ei(x)= ∞ e−tdt is the exponen-
x2 = √ρ hH1 QsB, 0 T (9) tial integral. − − −R−x t
x3 = √ρ(cid:2)g1HWsA, 0(cid:3)T . Proof: See Appendix C.
Then, we can obtain that R = R + R
With delayed CSIT only,(cid:2)W = Q =(cid:3)I and the proposed 2log 1+ρ√eaΘ + 2log sum1+ρ√eaΘA , Bwhe≈re
SAMAT scheme further boils down to the original AMAT 3 2 A 3 2 B
scheme. For simplicity, we focus on the performance of user ΘB =(cid:16)Tr(QHRAQ)T(cid:17)r(QHRBQ)−(cid:16)Tr(QHRAQ(cid:17)QHRBQ).
It shows that the AMAT scheme exploiting delayed CSIT
A and similar results can be symmetrically applied to user B.
enables a DoF of 4 at high SNR, while the beamforming
The signal vector received by user A is given by 3
based on statistical CSIT makes no contribution to the DoF
gain. However,the ergodic rate performanceat practicalSNR
hHW hHQ n
1 1 A1 benefits from such spatial correlation information.Observe in
y =√ρ 0 s +√ρ h∗ hHQ s + n ,
A   A  21 1  B  A2 (13) and (14) that the ergodicrate relies on the precodersand
h∗ gHW 0 n
31 1 A3 thespatialcorrelationmatrices.Thelattertermsareinvariable
     (10)
where y , [y ,y ,y ]T denotes the received signals in a certain environment.Therefore, RA and RB only depend
A A1 A2 A3 on Θ (W) and Θ (Q), respectively. To maximize the
over three time slots and hjm denotes the channel coefficient A B
ergodic sum-rate performance, the precoders W and Q can
between m-th transmit antenna and user A in time slot j.
be independently designed. Let us focus on Θ only and
n (0,1) is the normalized complex AWGN. After A
Aj ∼ CN optimize W. Similarly, we can obtain the optimal Q that
further interference elimination
maximizes Θ .
B
h∗ n n
y =√ρHs + 21 A1− A2 , (11)
A A n B. Precoder Design
(cid:20) A3 (cid:21)
1) Multi-antenna case (M > 2): It is difficult to obtain
where H = [e(h∗21hH1 We)T, (h∗31g1HW)T]T. hjm denotes the a closed-form expression of the beamforming vectors that
channel coefficient between m-th transmit antenna and user
maximize (14) and further (13). For such a problem where
A in timee slot j. By using a Minimum Mean-Square Error
joint optimization is difficult but the objective function is
(MMSE) receiver with Successive Interference Cancellation
convex in each of the optimization variables w and w , an
(SIC), the ergodic rate achieved per slot by user A is written 1 2
alternating algorithm, also known as Block Coordinate De-
as:
scent method, has been widely used in optimization [19, 20].
R = 1E log det I+ρHHK−1H , (12) Morespecifically,wemaximize(14)bysequentiallyfixingone
A 3 2 vector and updating the other. Fix w and focus on w (vice
2 1
h (cid:16) (cid:17)i
where K is the covariance matrix of the noise vector in (11) versa,thefollowingderivationsstillhold).Wecanreformulate
e e
and given by diag(1+ h 2,1). It is challenging to obtain the subproblem as
21
| |
the closed-form expression of the ergodic rate, especially for
M > 2 case. Hence, we optimize the linear beamforming maxΘ (w )=wHR w wHR w +wHR w wHR w
A 1 1 A 1 2 B 2 1 B 1 2 A 2
vectorsbased on an analyticalapproximationof R , which is wHR w wHR w wHR w wHR w
given by the following proposition. A s.t. w =−1. 1 A 2 2 B 1− 1 B 2 2 A 1
1
Proposition 1:InspatiallycorrelatedRayleighfadingchan- k k (15)
nel,theergodicrateofuserAforAMATcanbeapproximated Since it is convex in w 4, the classical gradient ascent
1
as (GradAct) method can be used to determine the optimal
2 4Theconvexitycanbeeasilyprovedwiththesecondordercondition,which
R log 1+ρ√eaΘ , (13)
A ≈ 3 2 A isomitted hereforconciseness.
(cid:16) (cid:17)
6
It implies that the optimal beamforming vectors are always
10
orthogonally chosen (w w ). For the special M = 2
1 2
9 case, since w is uniquely⊥defined in N(w ) and vice versa,
1 2
any two beamforming vectors constituting a unitary matrix
8
are optimal. Moreover, eq. (14) becomes constant Θ =
7 A
e Tr(R )Tr(R ) Tr(R R ).
valu 6 ThiAs propoBsit−ion reveAalsBthat any orthogonal beamforming
ve 5 vectors with equal power allocation achieve the same ergodic
Objecti 4 ssuecmti-ornateV.peTrhfoenrm,laentcues, cwohmicphutiesthveereifiqeudalbpyowtheeraFlilgo.ca4t(iobn) iρn.
3
According to (3), the long-term average power consumption
2 for AMAT is represented by
Algorithm 1
1
Algorithm 2
P¯ =4ρ+ρTr(QHR Q)+ρTr(WHR W)
0 c B A
1 2 3 4 5 6 7 8 9 10
iterative index =ρ(4+qH1 RBq1+qH2 RBq2+w1HRAw1+w2HRAw2)
ρ (4+2M). (17)
Fig. 2: Convergence of iterative algorithm 1&2. ≤
where the inequality (17) is obtained by using xHRx +
1 1
xHRx λ (R)+ λ (R) Tr(R), where unit-norm x
2 2 ≤ 1 2 ≤ i
solution (step 4 of Table I). Once the optimal w is obtained are mutually orthogonal and λ (R) corresponds to the i-th
1 i
in terms of certain w , the process is repeated the other largest eigenvalue [21], [22]. In order to maintain the power
2
way around (step 5), leading to an iterative algorithm. Since constraint, equal power allocation is calculated as ρ = 3P
4+2M
the steepest ascent direction acts as the best direction to (e.g., ρ = 3P for two transmit antennas). Equality in (17)
8
increase the objective function, a proper step size can be holds for M = 2 case, which also justifies proposition 2 in
computedfor a non-decreasingobjectivevalue, i.e., Θ(m,4) thesensethatorthonormalprecoders(optimally)useupallthe
Θ(m,5) Θ(m+1,4) where Θ(m,4) refers to the obAjectiv≤e power budget.
A ≤ A A
value at step 4 in the m-th iteration in Table I (Algorithm
1). Thus, the convergence of Algorithm 1 is ensured, since
Θ is monotonically increased (non-decreased) after each V. STATISTICAL AMAT
A
iterationandupperbounded.Eventhoughtheoptimalsolution
is obtained for each subproblem, the iterative algorithm can Inthelastsection,weexploredhowthestatisticalCSITaids
not guarantee the global optimal beamforming vectors. in AMAT under equal power allocation. The corresponding
Alternatively, (15) is quadratic in w1 and the op- rate performance is superior to the original AMAT, how-
timal solution can be obtained by eigen-decomposition. ever, still inferior to SBF in highly correlated channels at
Rewrite (15) as ΘA(w1) = w1HM(w2)w1, where low/intermediate SNR regime. In fact, the ergodic sum-rate
M(w2) = w2HRBw2RA + w2HRAw2RB − RAw2w2HRB canbefurtherenhancedbypowerallocationoptimization.The
−RBw2w2HRA. The closed-form solution is the maximum proposedSAMAT scheme in Fig. 1 is developedto overcome
eigenvector (Max-Eig), bothSBFandAMATatanySNRinarbitraryspatialcondition.
Proper power allocation and statistical precoder design can
w1 =arg maxΘA(w1)=umax(M(w2)). (16) compress the received interference ηA(ηB) in 1st slot (as well
kw1k=1 as common symbols in 2nd/3rd slot). Meanwhile, it makes
With (16) at hand, we can easily compute the optimal room for new symbol transmission in stage II which brings
precoders by the proposed iterative approach. In Fig.2, we more rate benefits. To achieve this, the power allocation is
show by two cases (M = 4,8) that the iterative algorithm optimized based on a tractable approximation of the ergodic
converges very fast, where the covariance matrices are ran- sum-rateofSAMAT.Thepowerallocationstrategythatmaxi-
domly generated. mizes the rate performancedependson SNR, precoderdesign
2) Two-antennacase (M = 2): A special case of consider- as well as spatial correlation.
ableinterestisthetwotransmitantennascenario.Theoptimal Hereafter, we focus on user A and similar results can be
precoderscanbeeasilyobtainedbythefollowingproposition. derivedforuserB.ThereceivedsignalofuserAcanbewritten
Proposition 2: For M = 2 MISO BC with spatially as
correlated Rayleigh fading, any unitary beamforming matrix
is optimal to maximize (14) and further (13). yA =H1P1A/2sA+H2P1B/2sB+H3spA+H4spB+nA, (18)
Proof: For arbitrary M and w2, it is easy to verify where
that wHM(w )w = 0, i.e., M(w )w N(w ). The
wombeasxefiirxmvawi2ztia1otniaonntdhauo2tpftdhΘae1teAo(wpwt2i1m,)wume=hwawv1e1H∈thMeNo(2(pwwti22m))1.uwmS∈1imwlei2laadrlsy2N,t(owwht1eh)ne. H1 ,√P8hhH1∗301Wg1HWH3 ,√P6h00H2 w3 √P9h00H3 w3, (19)
∈
7
hHQ 0 0
H2 ,√P5h10∗21hH1 QH4 ,√P70hH2 q3 √P100hH3 q3, (20) RsA ≈log2(cid:16)1++δδA1δ(τAΘ1P1P+PτA2P2)+δA2(λB1P1+λB(22P52))
A1 A2 A 1 2
   
and y , [y ,y ,y ]T, sp , [s ,s ]T, sp , (cid:17)
A A1 A2 A3 A A3 A4 B
[s ,s ]T.n ,[n ,n ,n ]T with n (0,1).The τ P τ P
deBc3odiBn4g procAedureAth1atAm2ainAl3y uses inteArjfe∼reCncNe alignment RspA ≈log2(cid:18)1+ 1+P5A+3 λ6A3P7(cid:19)+log2(cid:18)1+ 1+Aλ3A39P10(cid:19).
andcancellation(similarto[6])isdescribedasfollows.Denote
(26)
y as the received signal after subtracting √P h∗ y from
ybeAyA2reagnadrdreintagintihnegeyxAtr1a5.sFyimrsbt,odlsec(sopAd,estpBh)eapsriivnatte5erfsey2r1me·nbcoAel1:s(sA) Similarly, we have RB , 31(RsB +RspB) and
R log 1+δ (τ P +τ P )+δ (λ P +λ P )
sB ≈ 2 B1 B1 3 B2 4 B2 A1 3 A2 4
yA =H1P1A/2sA+H2P1B/2sB+H3spA+H4spB+nA, (21) (cid:16) +δB1δB2ΘBP3P4 (27)
z (cid:17)
wHehere=He1[(=hH[(QhH1)TW,|e)T0,T−,(√0PT5]Th∗21ahn{H1dzWn)T,=(√P[n8he∗31,}gn1HW)T]TR,spB ≈log2(cid:18)1+ 1+τBλ3PB37P6(cid:19)+log2(cid:18)1+ 1+PτB83+P1λ0B3P9(cid:19),
2 1 A A1 A2 − (28)
√P h∗en ,n ]T. K is the covariance matrix of the
5 21 A1 A3
ienterferenceplusnoisevectorz.Itisgiveenbydiag(k1,k2,k3), where
wherek =1+ hHQP 2,k =1+P h 2+P hHw 2+
P hHq1 2,k =| 11+PB|hHw2 2 +P5| h21H|q 2.6|To2 res3o|lve δ = 1 + P5
7| 2 3| 3 9| 3 3| 10| 3 3| A1 1+λ P +λ P 1+P +τ P +λ P
s , MMSE-SIC receiver is applied on (21) and the ergodic A1 3 A2 4 5 A3 6 A3 7
A 1 P
sum-rate of s can then be written as δ = + 8
A B1 1+λ P +λ P 1+P +λ P +τ P
B1 1 B2 2 8 B3 9 B3 10
P P
δ = 8 ,δ = 5 (29)
R =E log det I+P HHK−1H P . (22) A2 1+τ P +λ P B2 1+λ P +τ P
sA 2 A 1 1 A A3 9 A3 10 B3 6 B3 7
h (cid:16) (cid:17)i
Once sA is obtained,we can suebtract it ferom yA. Then, we λA1 =qH1 RAq1, λA2 =qH2 RAq2, λB1 =w1HRBw1
can decode the extra symbols by taking the second and third λ =wHR w , τ =wHR w , τ =wHR w
B2 2 B 2 A1 1 A 1 A2 2 A 2
entries of yA as yˆA: e τ =qHR q , τ =qHR q , λ =qHR q
B1 1 B 1 B2 2 B 2 A3 3 A 3
λ =wHR w , τ =wHR w , τ =qHR q (30)
e yˆ =Hˆ sp +Hˆ sp +nˆ , (23) B3 3 B 3 A3 3 A 3 B3 3 B 3
A 3 A 4 B A
where Hˆ3 = diag(√P6hH2 w3, √P9hH3 w3), ΘA=Tr(WHRAW)Tr(WHRBW)−Tr(WHRAWWHRBW)
Hˆ = diag(√P hHq , √P hHq ) and nˆ = Θ =Tr(QHR Q)Tr(QHR Q) Tr(QHR QQHR Q). (31)
4 7 2 3 10 3 3 A B A B − A B
[n √P h∗ n ,n ]T. The covariance matrix of
nˆ A2is−given5by21NA1=Ad3iag(1 + P h 2, 1). The ergodic Proof: Refer to Appendix D for proof.
suAm-rate of sp is given by 5| 21| Remark 2: Compared to the interference quantization ap-
A proach in [12], the analog transmission induces a noise en-
hancement. Namely, interference alignment cancels the over-
RspA =E log2det I+HˆH3 (N+Hˆ4HˆH4 )−1Hˆ3 . (24) heard interference while scaling up the noise by P5 (P8).
h (cid:16) (cid:17)i This noise enhancement can be observed in (21) ∼ (28). At
Itischallengingtoobtaintheclosed-formexpressionforthe
low SNR, the proposed SAMAT scheme behaves as SBF in
ergodicrate. Instead, we derivea tractable approximationand
each time slot. The scaling factors are small and therefore
optimize the power allocation based on such approximation.
the effect of noise enhancement is negligible. The gain over
Proposition 3: The achievable ergodic sum-rate per slot at AMAT mainly comes from extra symbol transmission and
userAwithlinearbeamformingcanbeapproximatedasR , statistical precoding. At high SNR, the proposed SAMAT
A
13(RsA +RspA) where schemebehavesasAMAT,achievingaDoFof 43.Inthiscase,
the ergodicrates of extra symbols can be eliminated by noise
enhancement. Namely, we have little benefit by transmitting
5A(MAT)-based schemes use one observation to completely remove the extra symbols. However, the proposed SAMAT scheme still
overheard interference and two independent observations remain to resolve achieves significant gain over SBF and AMAT by power
two symbols. By contrast, we cancel the overheard interference while we allocation optimization and statistical precoding.
maintain all three observations. The reasons are explained as follows: 1) In
With predefined beamforming vectors, the proposed
some cases (e.g., highly correlated channel) where SAMAT boils down to
SBF in each time slot, symbols need to be decoded slot by slot. However, SAMAT scheme softly bridges between SBF and AMAT by
conventional decoding strategy causes rate loss because one observation is power control. Let us concentrate on two cases:
dropped. 2) The ergodic rate of the proposed decoding method is slightly
case 1: bridge between SWEBF and AMAT, w =
better than the conventional one, due to one more observation even with 1
stronginterference. umax(RB), q1 = umax(RA), w2 = w3 = wWE =
8
u (R ), q =q =q =u (R ); Proof: At high SNR, problem (32) can be rewritten as
min B 2 3 WE min A
(a)
case 2: bridge between SGEBF and AMAT, w = maxR log (δ δ Θ P P )+log (δ δ Θ P P )
uummainx((RR−B−B11RRAA)),,qq12 ==qu3m=in(qRG−AE1=RBu)m,awx(2R−A=1RwB3).= wG1E =  {Pi} s++umll≈oogg2(cid:16)112++A11+τBPAτ352AP+37λPAA63P1+7(cid:17)2lo+glog122+(cid:16)1B+1τ1BB+32τPλA1A3B03PP9130(cid:17)4
case 1 is used to show the efficacy of the power alloca- 2 1+λB3P6 2 1+P8+λB3P9
tsicohnemopeticmanizastoioftnlytebcrhidngiqeubeebtwyewenhiSchWtEhBeFpraonpdosAedMSAAT.MBAeT-  s.t.(a)P¯c−3P =(cid:16)0, Pi ≥0 (cid:17)i=1,...(cid:16),10 (34(cid:17))
where comes from the fact that the last terms in (25) and
yond this, case 2 makes better use of statistical CSIT in
≈
(27) are dominant at high SNR. Based on KKT necessary
the sense that SGEBF exhibitshigher robustness comparedto
SWEBF. Instead of using the optimized AMAT precoders in conditions, there exist multipliers λ and µ1,µ2 such that
transmission stage I (w ,w ,q ,q as developed in Section
1 2 1 2 R (P )=λ(1+λ P )+µ
IV), we use the precoders above (w1,w2,q1,q2 as WE/GE ∇Rsum(P1)=λ(1+λB1P8)+µ1 (35)
precoder) and the motivations are explained as follows. First,  ∇ sum 2 B2 8 2
µ P =0, µ P =0,
theoptimalprecodersinSectionIVthatmaximizetheergodic  1 1 2 2
sum-rate of AMAT under equal power allocation are not where P = 0,P = 0, otherwise DoF loss occurs due to
1 6 2 6
necessarily optimal for SAMAT with power control. Second, R = 0. Therefore, we have µ = µ = 0 and the first
sA 1 2
SAMAT boils down to SBF at low to intermediate SNR equation in (33) can be computed from (35). Likewise, the
in highly correlated channel, where the optimized AMAT second equation can be obtained.
precoders may cause a poorer rate performance compared to Remark 3: As can be seen from (33), the power allocation
the WE/GE precoders. In order to softly bridge between SBF dependsonthespatialcorrelation,theprecoderdesignaswell
and AMAT, we adopt the precoder design as above. as SNR. Take case 1 as an example, λ = wHR w =
B2 2 B 2
λ (R ) while λ = λ (R ). Then, 1+λB2P8 1
Consequently, Rsum , RA +RB and the ergodic sum-rate immpinliesBthat P1 BP12 and lmikaxewisBe P3 P41.+TλBh1isP8im≤plies
≤ ≤
optimization problem is formulated as: that more power needs to be allocated on the weaker eigen-
mode (w ) to constrain the interference imposed to the other
2
m{Paix} Rsum s.t. P¯c =3P, Pi ≥0 i=1,...,10. (32) dtreasnisremdittseydmbpoolwoerf tihne2snadmaenudse3rr.dAtismmeesnltoiotsneids baeffournec,titohne
of beamformers. Such power allocation method enables to
compress the interference and makes room for delivering two
Itwasshownthatthesum-rateoptimizationgenerallyleads
moreprivatesymbols.Moreover,consideri.i.dRayleighfading
to an intractable NP hard problem [23]. Thus, an algorithm
achieving global optimum cannot be expected. However, Se- channels where RA = RB = I. λB2 = w2HRBw2 = 1 and
likewise λ =λ =λ =1. Based on (33) and symmetry,
quential Quadratic Programming (SQP) algorithm provides B1 A1 A2
an efficient way to solve non-linear constrained optimization the optimal power allocation satisfies P1 = P2 = P3 = P4
and therefore SAMAT boils down to AMAT. It makes sense
problem.AnoverviewonSQPisprovidedin[24–26].Briefly,
because no correlated informationis available to suppress the
a quadraticapproximationofthe Lagrangianfunctionismade
interference.Inthiscase,equalpowerallocationistheoptimal
by applying quasi-Newton updating method. The consequent
choice.
QP subproblemcan be optimallysolved and then the solution
is used as a search direction. With proper line search, an To operate the proposed SAMAT transmission protocol,
estimate of the solution is computed for the next iteration. the signaling and feedback procedure is described as follows.
This SQP algorithm can guarantee a super-linear convergence Using LTE-A framework [27], channel state information ref-
to a local minimum. erence signals(CSI-RS) are transmitted to enable the receiver
tomeasuretheshort-termCSIandthelong-termCSI(channel
In order to get insights into the optimal power allocation, covariancematrix),which are then fed back to the transmitter
a necessary condition for optimality of the constrained prob- via a delayed but assumed perfect feedback link. The long-
lem (32) is identified from the Karush-Kuhn-Tucker (KKT) term CSI only varies at a very slow pace and is therefore not
equations. Thus, the optimum power allocation that depends affected by the delay. However by the time the transmitter
ontheprecodersandthechannelcovariancematricesisstated has acquired the short term CSI, the channel has changed
as follows. and the transmitter only has knowledge of a completely stale
short-term CSIT. Based on the long-term and the short-term
Theorem 2:AthighSNR,theoptimalpowerallocationthat CSIT, the transmitter computes the precoders and the power
maximizes R in (32) satisfies: allocation and constructs the transmitted signals that are then
sum
transmitted using demodulation reference signals (DM-RS)
[28]. As far as the implementation complexity is concerned,
P 1+λ P P 1+λ P
1 = B2 8, 3 = A2 5 (33) the potential challenge of SAMAT lies in numerical power
P2 1+λB1P8 P4 1+λA1P5 allocation computation. Eq. (33) provides a necessary condi-
where λ ,λ ,λ ,λ are defined in (30). tion which reveals the basic relationship between the power
A1 A2 B1 B2
9
AMAT M = 4
12 18
ORG
16 WE
s/Hz)10 s/Hz)14 GOEPT
p p
b 8 b12
ot( ot(
er sl 6 er sl10
p p
e e 8
at at
c r 4 c r 6
di di
Ergo MM == 22 WGEE Ergo 4
2
M = 4 WE 2
M = 4 GE
0 0
0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45
SNR [dB] SNR [dB]
Fig. 3: Rate performance comparison of SBF with WE and (a) M =4.
GE precoder.
AMAT M = 2
15
ORG
WE
allocation and the spatial correlation matrices as well as the z) GE
H
precoders.Itwouldbehelpfultoidentifyasufficientcondition s/ Rnd
p
and further closed-form power allocation. ot(b10
sl
er
p
VI. PERFORMANCE EVALUATION e
at
Weprovidenumericalresultstoshowtheefficacyofthepro- c r 5
di
posedprecoderdesignandpowerallocationstrategy.A single go
parameterexponentialcorrelationmodel[29] is consideredas Er
1 tk ... tMk −1 00 5 10 15 20 25 30 35 40 45
tH 1 ... tM−2 SNR [dB]
Rk = k... ... k  (36) (b) M =2.
 
(tH)M−1 ... tH 1 
 k k  Fig.4:Rate performancecomparisonofAMATwith different
 
where t denotes the transmit correlation coefficient t = precoders.
k k
|tk|ejφk, φk ∈ [0, 2π], k = A,B. Throughout the paper,
weusehigh(low)correlationtoindicatelarge(small)condition
numberofthespatialcorrelationmatrix,whichcorrespondsto
B. Precoders Comparison for Alternative MAT
a large(small) t in theexponentialmodel.A largefamilyof
k
| |
spatial correlation is tested to verify our analysis. With the
In Fig. 4(a), we comparethe ergodicsum-rate performance
help of the optimization tool in Matlab, ‘fmincon’ is used to
of AMAT with different precoding methods. t is randomly
implement the SQP algorithm. k
generated: t U(0,1),φ U(0,2π). ORG denotes
| k| ∈ k ∈
original AMAT that the transmitter sends symbols simply
A. Precoders Comparison for Statistical Beamforming using 2 out of M antennas. WE and GE are statistical
InFig.3,weplottheergodicsum-rateofSBFwithWE/GE precodersdefinedin section III. Theoptimalprecoders(OPT)
precoders, averaged over the randomness in the channels and is computed by the proposed iterative algorithm in Table I.
φ . The amplitudesof channelcorrelationcoefficientsof both We can observe that OPT achieves a better ergodic sum-rate
k
users are given by t =0.95, t =0.9 and the superiority than others.
| A| | B|
of GE over WE precoder is illustrated by two cases (M = Meanwhile, Fig. 4(b) confirms the validity of proposition
2,4). In Fig. 3, GE beamforming vector shows robustness 2. It can be seen that any orthogonal beamforming vectors
for large M as well as varying scattering environment (i.e., constituting a unitary matrix are optimal for M = 2 case.
φ ). Interestingly, WE precoded SBF performs even worse More specifically, Rnd indicates that W and Q are randomly
k
for larger M, which is inherently caused by the idea of zero generated unitary matrices. WE precoders, corresponding to
forcing. The precoder is designed to reduce the interference u (R ) and u (R ) k = A,B, also form unitary matri-
max k min k
imposedtotheunintendeduser,butmaycanceloutthedesired ces.ORGbecomesan2 2identitymatrix.Alltheseprecoders
×
signaloftheintendeduser.Inotherwords,asM increases,the show optimality in terms of the ergodicsum-rate whereasGE
M 1WE precoderw = u (R )mayfallintothe (M 1) does not, because GE precoders fail to form a unitary matrix
dim×ensional N(h) with highmeinr proBbability. − (since either R−1R or R−1R is a normal matrix).
A B B A
10
C. Performance of the Proposed Statistical AMAT (SAMAT)
20
AMAT
Fig. 5(a) depicts the achievable ergodic sum-rate perfor- 18 SGEBF
mancesof variousschemeswith two transmitantennas(M = z)16 AMAT_OPT
H VMAT
2):originalAMAT,SBFwithWEprecodersandtheproposed ps/14 SAMAT
SAMAT (case 1). We set t = t = t that varies b
| A| | B| | | ot(12
between 0 and 1, i.e., from uncorrelated to highly correlated sl
channels. Furthermore, φ , φ are randomly generated with per 10
A B e
φ φ π and SNR =20 dB. As t increases, the sum- at 8
|ratAe−of BSB|≥F g2radually goes up while a| |sharp rise occurs at dic r 6
o
veryhighcorrelationlevel.Becauseinhighlycorrelatedspatial Erg 4
channels, linear beamformingbased on statistical information
2
keepstheremaininginterferencesmallenough.Aspecialcase
0
is the fully correlated channel. When t = 1, the overheard 0 5 10 15 20 25 30 35 40 45
| | SNR [dB]
interference can be completely cancelled.
Moreover, the rate performance of original AMAT also Fig. 6: Comparison of the ergodic sum-rate vs. SNR between
dependson the transmit correlationof the channel. In M =2 SAMAT and baselines, t = 0.95, t = 0.9, φ ,φ
| A| | B| A B ∈
case, ORG precoders for original AMAT become W =Q= U(0,2π), M =4.
I . Observe in (13) and (14) that the ergodic rate is a
2×2
function of R and R . More specifically, Θ = Θ =
A B A B
Tr(R )Tr(R ) Tr(R R ). With the correlation model in linear beamforming vectors developed in section IV. SGEBF
A B − A B
(36)andthespecificphasesφ ,φ ,apositive/negativeimpact denotestheSBFschemewithGEprecoders.We plotSAMAT
A B
of transmit correlation amplitude t can be easily computed: (case 2) with the SQP algorithm. Moreover, we compare the
| |
Θ =Θ =2 1 t2 cos(φ φ ) .As t increases,the proposed SAMAT scheme with VMAT [13]. As mentioned
A B −| | |· | A− B| | |
transmit correlation is beneficial when φ φ > π while before,thepowerconstraintofVMATinstageIIwasreleased.
(cid:0) (cid:1)| A− B| 2
it is detrimental when φ φ < π. When φ φ = π, Tomakeafaircomparison,wealsoapplythelong-termpower
| A− B| 2 | A− B| 2
the ergodic rate keeps constant irrespectively of t. constraint for VMAT and scale it down to 3P.
| |
ThecrosspointbetweenSWEBF andAMATis determined In Fig. 6, we observe that AMAT_OPT enables around 5
by the spatial correlation level and SNR. Fig. 5(a) reveals dB enhancement over original AMAT at high SNR. VMAT
that the proposed SAMAT scheme obtains strictly higher rate achieves almost the same ergodic sum-rate as AMAT_OPT,
than SWEBF and AMAT by exploiting both statistical- and sincebothschemesexploitstatisticalCSIT underequalpower
delayed-CSIT. Since the channel statistics includes partial allocation. However, the SBF scheme still outperforms all of
currentCSI,wecan‘virtually’regarditasanimperfectcurrent them in a certain range of low to intermediate SNR. The
CSI. Fig. 5(a) coincides with Fig. 1 in [12] in the sense that proposed SAMAT framework which is precoded by GE with
theproposedschemessoftlybridgebetweenSWEBF(PZFBF) closed-form power allocation outperforms all these schemes.
and AMAT (MAT) in terms of the ergodic sum-rate (DoF). Meanwhile,withtheoptimizedpowerallocationcomputedby
In addition, for given channel covariance matrices, the theSQPalgorithm,theSAMATschememaximizestheergodic
ergodicsum-rateoftheseschemescanbeplottedversusSNR. sum-rate (further 2 dB over AMAT_OPT). The enhancement
ItcanbeobservedfromFig.5(b)thatSAMATachieveshigher over VAMT/AMAT_OPT mainly comes from the optimized
rate than SWEBF as well as AMAT along the entire SNR power allocation.
region. It acts as SWEBF at low SNR while it utilizes the To sum up, SAMAT boils down to AMAT in low-
DoF capability of AMAT in the high SNR regime. As a correlated/uncorrelated channels while for highly correlated
comparison, PZFBF with perfect instantaneous CSIT reaches scenariowhereSBFoutperformsAMAT,itbehavesasSBFin
a sum DoF of 2 at high SNR. A variety of spatial correlation the low to mediate SNR regime and as AMAT at high SNR.
is simulated but omitted here for conciseness. Briefly, when In addition, the optimized power values satisfy Theorem 2.
t , t 0,theSAMATtransmissionprotocolboilsdown
| A| | B|−→
to AMAT since no correlated channel information can be
VII. CONCLUSION
exploitedto enhancethe rate performance.Consider the other
extreme t , t 1 but φ φ 0, it indicates This paper aimed to exploit both statistical and outdated
| A| | B| −→ | A − B| −→
highly correlated channels but their weakest eigen-modes lie CSIT in a MISO broadcast setting to maximize the ergodic
in the similar direction. The rate performance of SWEBF is sum-rate.We consideredtherobustdesignofstatisticalbeam-
unfavorable and therefore SAMAT also behaves as AMAT. forming vectors for arbitrary transmit antennas, showing the
Fig. 6 illustrates the benefits of the proposed scheme with optimality of dominant generalized eigenvectors in maxi-
the power allocation optimization. The transmitter antennas mizing a lower bound of the ergodic sum-rate. Moreover,
M =4androbustGEprecodersareconsidered.φ andφ are the optimal precoders were designed to maximize the rate
A B
randomlygenerated.Specifically,AMATindicatestheoriginal approximation of AMAT under equal power allocation. An
AMAT with equal power allocation only exploiting delayed iterative algorithm was explored to compute these precoders
CSITwhileAMAT_OPTdenotesAMATprecodedbyoptimal with fast convergence.