Channel Pre-Inversion and max-SINR Vector Perturbation for Large-Scale Broadcast Channels

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by  David A. Karpuk

1
Channel Pre-Inversion and max-SINR Vector
Perturbation for Large-Scale Broadcast Channels
David A. Karpuk, Member, IEEE, and Peter Moss
Abstract—We study channel pre-inversion and vector pertur- A. Background and Related Work
bation (VP) schemesfor large-scale broadcast channels,wherein
6 a transmitter has M transmit antennas and is transmitting to We studyalinearfadingchannelconsistingofatransmitter
1 K single-antenna non-cooperating receivers. We provide results with M transmit antennas transmitting data to K single-
0 which predict the capacity of MMSE pre-inversion as K →∞. antenna,non-cooperatingreceivers,whereK M. Thebasic
2 We construct a new VP strategy, max-SINR vector perturbation model we consider is ≤
(MSVP), which maximizes a sharp estimate of the signal-to-
p interference-plus-noise ratio. We provide results which predict y=Hs+w (1)
e the performance of MSVP and demonstrate that MSVP out-
S
performsotherVPmethods.Lastly,wecombineMSVPwiththe where s CM is an encoded data vector, H CK M is
9 low-complexitySortedQRPrecodingmethodtoshowthatMSVP the chann∈el matrix, w CK is additive noise,∈and t×he ith
2 ohfasutsherespaottecnltoisaelttooecfhfiacniennetllycadpelaicvietry.datatoaverylargenumber Wcoeoradsisnuamteeocfhyan∈neClsKtatie∈sionbfosermrvaetdiobny(CreScIe)iviseravia=ila1b,le..a.t,tKhe.
]
Index Terms—Channel Pre-inversion, MMSE Inverse, Vector
T transmitter,inwhichcasethetransmittercanwritetheencoded
Perturbation,SINR,RandomMatrixTheory,SortedQRPrecod-
.I ing, Broadcast Channels data vector as s = Au where u is the uncoded data vector
s and A is a precoding matrix depending on the channel.
c
[ As was observed in [1], the zero-forcing inverse of H is a
I. INTRODUCTION poor choice for A when M = K, as the sum capacity does
2
notscale linearlywiththenumberofusersK.Instead,setting
v Successful implementation of next-generation (e.g. 5G)
2 mobile broadband internet will require the delivery of high- A to be a regularizedinverse results in superior performance,
1 scaling the sum capacity linearly with the number of users.
volume and high-fidelity data (e.g. streaming video) simulta-
1 However, regularized inversion still suffers from a large gap
neouslytoalargenumberofusers.Therapidincreaseofboth
8
to channel capacity when the ratio K/M is close to unity.
0 the number of mobile devices and the volume of data to be
The methods of [1] were improved upon by the vector
. delivered is putting heavy demands on broadcast networks.
1 perturbation (VP) method of [2], in which a perturbation
The algorithms underlying data delivery in such networks
0
vectorisaddedtotheuncodeddatavector.Vectorperturbation
6 must evolve along with the networks themselves, to meet the
closes the gap to channelcapacity substantially, but the trans-
1 demands of the ever-increasing number of end users.
v: Effectivelydeliveringalargeamountofdatatoalargenum- mitter is now burdened with solving a closest vector problem
in an arbitrary lattice. While algorithms such as the sphere
i berofuserssimultaneouslyimposestwomajorandseemingly
X decoder[3], [4] exist to tackle such problems,the complexity
contradictory demands on any system. First, the transmission
r offindingthemaximum-likelihood(ML)solutionpreventsVP
schememustbescalablewiththenumberofusersK,ormore
a
from being scalable to a large number of users [5]. Lattice
precisely,theencodingoperationmusthavelowcomplexityin
reductionalgorithmssuchastheLLLalgorithm[6]havebeen
termsofthenumberofusers. Secondly,the system musthave
usedin VP systems[7], [8], butforverylargedimensionsthe
little to no performance degradation as the number of users
LLL algorithm itself can be prohibitively complex.
increases. That is, we wish to deliver data at rates close to
Traditionally,the perturbationvectoris chosen to minimize
channel capacity even as K .
→∞ thepowerrenormalizationconstantγ (seeequation(4)below)
With the goalofmeetingtheabovetwo demands,we study
required at the transmitter [2], [9], [10], [11]. A notable
channel pre-inversion [1] and vector perturbation (VP) [2]
exception is [12], wherein the perturbation vector is chosen
methodsforGaussianbroadcastchannelsasK .Among
→∞ to minimize the mean square error (MSE) of the system
other results, the main contribution is our max-SINR vector
and is shown to have superior performance compared to
perturbation (MSVP) scheme, which when combined with a
the ‘minimize γ’ approach. However, such ‘minimize MSE’
low-complexity encoding algorithm has the potential to meet
schemesseem largelyunstudied,with most authorspreferring
the demands of next-generation broadcasting networks.
tosettheprecodingmatrixtobethezero-forcinginverseofthe
channelmatrix,despitepoorperformanceforsquaresystemsat
D.Karpukiswith theDepartment ofMathematics andSystemsAnalysis,
AaltoUniversity, Espoo,Finland. P.MossisformerlyofBBCResearch and lowersignal-to-noise-ratio(SNR).TheMSEofthesystemwas
Development, London, United Kingdom, and is currently an independent also studied in [13] when VP is used in conjunction with the
consultant. emails:david.karpuk@aalto.fi, [e-mail].
blockdiagonalizationtechnique[14].VPtechniqueshavealso
D. Karpuk is supported by Academy of Finland Postdoctoral Researcher
grant268364. been studied in channels where users have multiple antennas,
2
i.e. MU-MIMO channels,in [15], [16], [17], thoughwe focus II. SYSTEMMODEL
on the single-antenna receiver case.
A. Vector Perturbation Channel Model
Consider the M K MIMO channel where the transmitter
×
has M antennas and is communicating to K M non-
B. Summary of Main Contributions ≤
cooperating users, each with a single antenna. The intended
InSectionIIwereviewtheregularizedVPsystemmodel.In data u = [u ,...,u ]T is a length K column vector of
1 K
SectionIIIwestudythesignal-to-interference-plus-noiseratio information symbols (e.g. QAM symbols) with u intended
i
(SINR) of the system and derive a useful approximation of for receiver i, normalized so that
this quantity.In Section IV we use RandomMatrix Theoryto
predicttheSINRandergodiccapacityofregularizedinversion Eu ui 2 =c, c=K/M. (3)
| |
for large systems with no perturbation. The approximation is
TheentriesoftheK M channelmatrixHarei.i.d.circularly
shown to be accurate through simulations, and generalizes a ×
symmetric complex random Gaussian with variance 1/K per
theorem by the current authors for square systems (K = M)
complexdimension.ThechannelHisassumedtobeknownat
given in [18].
thetransmitter,whichcomputesaM K precodingmatrixA
In Section V we study VP and construct a scheme, which ×
andanoffsetperturbationvectorx. Thevectorxisafunction
we deem max-SINR vector perturbation (MSVP), which
of both H and u and belongs to a scaled integer lattice; the
provably maximizes our estimate of the SINR when any
precise nature of x will be made clear in the next subsection.
regularizedchannelinverseisemployed.Thisschemeisshown
The transmitter computes an encoded data vector
to outperform the Wiener Filter VP introduced in [12] which
itselfimplicitlymaximizesadifferentnotionofSINR.Weuse s=A(u+x)/√γ, where γ =Eu A(u+x) 2/K (4)
|| ||
RandomMatrixTheorytoestimatetheperformanceofMSVP
to within 0.5-1 dB. is a power renormalization constant. The encoded data then
In Section VI we focus on VP for large systems, where we satisfies the power constraint Eu s 2 = K which allows for
|| ||
fair comparison when we fix K and vary M.
use the sub-ML Sorted QR method of [19] to solve for the
perturbation vector. We show that for small K, the resulting The ith receiver observes the ith coordinate yi of the total
performance is very close to the performance of the ML length K received vector
solution, and is essentially the same as that of the lattice-
y=Hs+w =HA(u+x)/√γ+w (5)
reduction-aided broadcast precoding of [7], even though the
SQR method offers less complexity. Lastly we show that for from which they attempt to decode u . Here w =
i
large K, MSVP outperforms the zero-forcing VP method of [w ,...,w ]T is a length K column vector of additive noise
1 K
[2].Weendthepaperbyprovidingconclusionsanddiscussing whoseentriesarei.i.d.circularlysymmetriccomplexGaussian
potential future work. with Ew wi 2 = σ2. We define ρ = 1/σ2, and will often
| |
measure system performanceas a function of ρ or the system
size K. Following convention, we set ρ (dB) = 10log (ρ)
10
C. Notation and usually measure ρ in dB.
ThesymbolsZ,R,andCdenotetheintegers,realnumbers,
and complex numbers, respectively. Capital boldface letters B. Choosing the Perturbation Vector
suchasAdenotematrices,andlowercaseboldfaceletterssuch
TheoffsetvectorxischosenfromascaledGaussianinteger
as x denote vectors. We write A† for the conjugate transpose lattice τZ[i]K for some τ > 0, and may depend on both the
of the complex matrix A, and AT for the (non-conjugate)
given channel matrix H and given data vector u. Following
transpose. If A is rectangular, its pseudo-inverse is denoted
[2],thescalarτ ischosensothatifthecoordinatesofthedata
by A+. If A is square, its trace and determinant are denoted
vectors u are N-QAM constellation points, then the set
by tr(A) and det(A), respectively. The squared Frobenius
noArm2o=f Atr(=A(Aai)j)=is denoated2b.yT|h|Ae |i|d2Fe,ntaitnydmisatdriexfinoefdsibzye u+x∈CK | ui ∈N-QAM and x∈τZ[i]K (6)
K|| i|s|FdenotedI† . Foranyi,jsq|uiajr|ematrixB=(b ), we define is a tra(cid:8)nslated lattice in CK. In other words, τ is ch(cid:9)osen so
K ij
a square matrix dg(B)Pof the same size by that the various translates of the set of all u are “spaced out
evenly”throughouttheEuclideanspaceCK.Onecancompute
b if i=j easily thatforunscaled,standardN-QAMsignalingthe value
dg(B)ij = 0ii if i=j (2) of τ is 2√N. For our scaling, we have
(cid:26) 6
√c N
so that dg(B) has the entries of B on the diagonal and τ =2√N = 6c (7)
zvearroiasbleelsXewhaerree.deTnhoeteedxpbeyctEat(iXon)aannddvVaarira(nXce),orfesapercatnivdeolmy. 23(N −1) r N −1
q
The Gaussian integers Z[i] are defined to be Z[i] = a + where 2(N 1) is the average per-symbol energy of an
bi a,b Z C where i2 = 1. { unscaled3 N-Q−AM constellation.
| ∈ }⊂ −
3
Following [2], we assume that the ith receiver has knowl- interferenceandsignaltermsin(9).However,the interference
edge of dg(HA)iiτ. The receivers model their observation as terms is dwarfed by the noise term in practice, thus one can
√γ
usually safely ignore this apparent correlation. Secondly, the
u+x u+x
y=dg(HA) +(HA dg(HA)) +w (8) correlationbetweenthesetermsatanygivenreceivervanishes
√γ − √γ as K , as the effect of any single u on how we choose
i
→∞
andsincetheithreceiverknows dg(HA)iiτ,theycanreducey the total perturbation vector x becomes insignificant.
√γ i We briefly point out that in [2, Equation (25)], the channel
modulothelattice dg(HA)iiτZ[i] toremovetheith coordinate
√γ modelaftersuccessfulreductionmodulotheappropriatelattice
of the offset vector x. We assume that the modulo operation is given by
always decodes the offset vector x correctly, when in fact
it may not if, for example, the noise vector w is very y=u/√γ+(HA IK)(u+x)/√γ+w (12)
−
large. However, this assumption allows for clean analysis, is
which would result in
pervasivein the literature, and furthermoreseems to affectall e
^
VP strategiesin questionapproximatelyequally.So while our SINR=
capacityplotswillslightlyoverestimateabsoluteperformance, Kc (13)
they remain useful when comparing VP strategies to each Eu( (HA IK)(u+x) 2+ A(u+x) 2σ2)
|| − || || ||
other.WerestrictourVPsimulationstoρ 10dBtomitigate asadefinitionoftheSINRforregularizedperturbation.How-
≥
the effect of this potential decoding error.
ever, this model overestimates the overall signal strength and
therefore the capacity of the scheme, especially at low values
III. SIGNAL-TO-INTERFERENCE-PLUS-NOISERATIOOF ofρwheretheentriesofdg(HA)maybesubstantiallysmaller
VECTORPERTURBATION than unity. We will return to this point in Section V when
In this section we discuss the signal-to-interference-plus- we define our max-SINR vector perturbation strategy and
noise ratio (SINR) of VP systems. After providing the basic compare it with the Wiener Filter vector perturbation method
definition of the SINR for regularized VP systems, we show of [12], whichselects the perturbationvectorto maximizethe
howitdiffersfrompreviouslyconsiderednotionsofSINRfor MSE associated with (13).
such systems (as in [1], [2], [12]), briefly explain connections
with mean square error (MSE) and capacity, and provide B. Connection to Mean Square Error and Capacity
a simple approximations of the SINR and capacity when Theconnectionbetweenthe SINRin equation(10) andthe
employing a certain class of precoding matrices.
mean square error (MSE) of the system is as follows. Let us
fix a data vector u and corresponding offset x. The relevant
A. Basic Definition estimate of dg(HA)u at the receivers is uˆ =√γy′ where y′
is as in (9). The resulting MSE for the fixed data vector u is
After successful reduction modulo the various lattices
dg(√HγA)iiτZ[i], the receivers model the resulting vector y′ MSEu =Ew uˆ dg(HA)u 2 (14)
obtained from y by || − ||
= (HA dg(HA))(u+x) 2+ A(u+x) 2σ2
u u+x || − || || || (15)
y =dg(HA) +(HA dg(HA)) + w (9)
′
√γ − √γ
so that
noise
signal interference dg(HA) 2c
|{z} SINR= || ||F , MSE=Eu(MSEu) (16)
and treat|the i{nzterfer}enc|e as noise {wzhen decodi}ng. Modeling MSE
thereceivedsignalasdg(HA)u/√γ accountsforthefactthat Theexpression(9)allowsonetowritetheresultingchannel
whenAischosentobedifferentfromthezero-forcinginverse capacity for user i, and the average per-user capacity, for a
of H, the diagonal gains of the effective channel matrix HA fixed channel H as
need not be unity. Eu dg(HA)iiui 2
From (9) we derive, for a fixed channel H and precoding Ci,H =log2 1+ Eu (HA dg|(HA))(u+|x) 2+γσ2
matrix A, the signal-to-interference-plus-noise (SINR) ratio (cid:18) | − i| (cid:19)
K
of the system to be 1
CH = Ci,H
Eu dg(HA)u 2/γ K i=1
SINR= Eu (HA dg(H||A))(u+x|)| 2/γ+Ew w 2 X (17)
|| − || || || respectively. As in [1, Equation (32)], we make the mild
(10)
assumptionthatthesignalandinterferencepowersareapprox-
dg(HA) 2c
= || ||F imately uniformly distributed across all users. This allows us
Eu(||(HA−dg(HA))(u+x)||2+||A(u+x)(|1|21σ)2)mtoeaapspurroexoimf caatepaCcHity≈ElHo(gC2(H1)+bSyINR) andhence the ultimate
Herewehaveimplicitlyassumedaslowfadingmodel,wherein
thechannelHstaysconstantforalargenumberoftransmitted C :=EH(CH)≈EH(log2(1+SINR)) (18)
data vectors u. Note that the perturbation vector x depends Theapproximation(18)isgenerallyagoodnumericalestimate
on u, and thus there may be some correlation between the for the vector perturbation strategies under consideration.
4
C. Tikhonov Pre-Inversion where d and T are as in Theorem 1. The above implicitly
contains the approximation
We will consider precoding matrices of the form
A=Hα =H†(αIK +HH†)−1 (19) M[SEu :=||T(u+x)||2 (28)
forsome(small)constantα 0,whichistheTikhonovinverse ofthemeansquareerrorforagivendatavectoru(andafixed
of the channel matrix H w≥ith regularization parameter α. channel H).
Whenα=0,theTikhonovinversereducestothezero-forcing
inverse, which we will denote IV. CAPACITY OFMMSE PRE-INVERSION FORLARGE
SYSTEMS
HZF =H†(HH†)−1 (20) In this section we fix the offset vector x to be x = 0;
When we set the regularization parameter α = σ2, we will thatis, we are presentlyonlyconcernedwith the performance
referto the correspondinginverseof H as the MMSE inverse, of linear precoding strategies with no vector perturbation.
which we will denote by Furthermore, we fix the precoding matrix to be A=H .
MMSE
The goal of this section is to obtain explicit approximations
HMMSE =Hσ2 =H†(σ2IK +HH†)−1 (21) for EH(SINR) and the capacity C for MMSE pre-inversion
Theoptimalregularizationparameterαwasfoundin[1]tobe to measure performance of large systems.
approximatelyKσ2forsquaresystems.Theapparentdisparity
with the above matrix HMMSE is a consequence of how we A. Predicting SINR and Capacity of Large Systems
have normalized the channel matrix and the transmit power.
Our strategy is to compute lim EH(d) explicitly, and
We prefer the given normalization, since the regularization K
combinetheresultwith(27)tomeas→u∞relarge-scalesystemper-
parameter of interest is now independent of the size of the
formanceofMMSEpre-inversion.Thisrequiresthefollowing
system.
two lemmas,the firstof whichisan elementarysimplification
of our expressionsfor the SINR and capacity,and the second
D. Estimating SINR and Capacity
of which is a technical lemma which essentially validates
WhenwesettheprecodingmatrixtobeaTikhonovinverse, replacing d with lim EH(d) in the SINR and capacity
theresultingSINR(andthusthecapacityC)canbeestimated approximations. K→∞
by a simple, compact expression. To begin, let Lemma 1: Suppose that A=H and that x=0. Then
MMSE
\
the approximations SINR and C from (27) are given by
d=tr(HH )/K =tr(dg(HH ))/K (22)
α α
The following theorem is the basis for our approximation of S\INR= 1 d d and C b=−EHlog2(1−d). (29)
the SINR. −
Theorem 1: For a fixed channel matrix H and Tikhonov where d=tr(HHMMSE)/K.b
parameter α, define Proof:Sincex=0andEu ui 2 =c,asimplecalculation
ε1 =||dg(HHα)−dIK||2F (23) fgoivreasnyEum||aTtr(ixuB+,xw)|e|2se=e e||aTsi|l|y2F|fcr.|oSmin(c2e7)||tBha||t2F = tr(B†B)
ε2 =Eu ||(HHα−dg(HHα))(u+x)||2 \ d2K
(cid:0) (HHα dIK)(u+x) 2 (24) SINR= tr(T T) (30)
−|| − || †
where d is as in (22). Let T be any matrix such(cid:1)that d2K
= (31)
d2K+(1 2d)tr(HH )
T T=d2I 2dHH +HH H+ H (25) − MMSE
† K − α α MMSE α d
= (32)
Then we can bound the SINR by 1 d
−
d2Kc d2Kc+ε where T is as in Theorem 1. The statement about C is
SINR 1 (26)
Eu T(u+x) 2+ε2 ≤ ≤ Eu T(u+x) 2 immediate from (27).
|| || || ||
and furthermore, we have Klim K1EH(ε1)=0. ranLdeommmvaar2ia:blLeestsu{cXhKth}a∞Ktt=h1efboelloawisnegqutherneceecoonfdirteiaoln-svahblouledd:
Proof: See the Appendi→x.∞ (i) 0 < X < 1 for all K, (ii) 0 < lim E(X ) < 1, and
K K
While we are unable to prove that lim K1EH(ε2) = 0, (iii) lim Var(X )=0. Then K→∞
K K
simulation results suggest that this is the→c∞ase, and that even K
→∞
for small K the quantity ε2 is very small relative to the other X lim E(XK)
terWmseinnotwheolbotwaienr obuorunadp.proximations of the SINR and the Kl→im∞E(cid:18)1−KXK(cid:19)= 1−K→Kl∞im E(XK). (33)
capacity C by ignoring the error terms ε and ε , and setting →∞
1 2 Proof: See the Appendix.
S\INR:= Eu||Td(2uK+cx)||2, C :=EHlog2(1+S\INR()27) cirWTcuheleaocrrlayenmsny2om:wmLseetttarHtiecctbhoeemKmpl×aeixnMrtahnmedoaortmerimxGwoafuhsothssiieasnesnevtcartriieioasnba.leresiw.ii.tdh.
b
5
variance 1/K per complex dimension, let H = H (αI + 25
α † K
HH†)−1 be the Tikhonov inverse with parameter α>0, and
let d=tr(HH )/K as in (22). Then we have 20
α
lim EH(d)=d(c,α) (34) 15
K B)
→∞ d
where R) ( 10
N
SI
1+c+cα 1+2c( 1+α)+c2(1+α)2 (H
d(c,α):= − − . E 5
2c
p (35) ρρ == 05 ddBB
ρ = 10 dB
Furthermore, if we fix A=HMMSE, then 0 ρ = 15 dB
ρ = 20 dB
ρ = 25 dB
lim EH(S\INR)= d(c,σ2) (36) -5 ρ = 30 dB
K 1 d(c,σ2) 5 10 15 20 25 30 35 40
→∞ − Value of K in K× K MIMO System
35
and
lim C log (1 d(c,σ2)) (37) 30
K ≥− 2 −
→∞
25
forsystemswhichempbloyMMSEpre-inversionwithnovector
B)
pertuPrbraotoiof:nS.ee the Appendix. R) (d 20
N
Theorem 2 provides the approximations SI 15
(H
E
d(c,σ2) 10 ρ = 0 dB
EH(SINR)≈EMMSE(c,σ2):= 1 d(c,σ2), (38) ρρ == 51 0d BdB
C log (1 d(c,σ2)−) (39) 5 ρρρ === 122505 dddBBB
≈− 2 − ρ = 30 dB
0
for large systems which employ MMSE pre-inversion with 5 10 15 20 25 30 35 40
Value of K in 2K × K MIMO System
α=σ2 and no vector perturbation.
Fig.1. Ontop,EH(SINR)forK×K systems(c=1)employingMMSE
pre-inversion, forvarious values ofρ=1/σ2.Onbottom,thesameplotfor
B. Simulation Results 2K×K systems.
Inthissubsectionwe collectsimulationresultswhichstudy
the accuracy and predictive ability of the approximation(38), log (1 d(c,σ2)) in a Taylor series as σ2 0 to obtain
for channel pre-inversion with A = HMMSE and no vector −(recal2l tha−t ρ=1/σ2) →
perturbation.
1) Approximating SINR: In Fig. 1 we plot EH(SINR) as C &−log2(1−d(c,σ2))
a function of K for systems with M = K (top), and M = = log2(ρ)+log2(1−cc)+O(1) c<1 (40)
2K (bottom), for various values of ρ = 1/σ2. In both plots, (cid:26) 12log2(ρ)+O(1) c=1.
the solid curves represent experimentally measured values of The experimentalresults of the previoussubsection show that
EH(SINR), and the dashed lines the correspondingvalues of the leading term of the series approximates C quite well for
EMMSE(c,σ2). We see thatEMMSE(c,σ2) predictsthe limiting ρ>15dBorso.Ifoneacceptsthat log (1 d(c,σ2))isan
valueofEH(SINR)verywell,forallvaluesofρ.Ontheother accuratepredictorofC forlargeK,t−hent2hea−boveshowsthat
hand, note that the error introduced by applying the large-K the qualitative performanceof square and non-squaresystems
limit to small-K systems may be non-negligible when c=1. is different.
2) ApproximatingCapacity: In Fig. 2 we plot the capacity
C =EH(CH)asafunctionofρforK =8(top)andK =64 V. MAX-SINRVECTORPERTURBATION
(bottom),forc=1,2/3,1/3,1/6.Thesolidmarkedlinesare
In this and all subsequent sections we turn our attention
the experimentally measured values of C, the dashed lines
towardsschemeswhichemploynon-trivialvectorperturbation,
are the values of the approximation −log2(1−d(c,σ2)) of thatis,choosex τZ[i]K accordingtosomealgorithmwhich
C from (38), and the solid unmarked lines are the channel ∈
is intendedto optimizesystem performance.In [2] andnearly
capacity, computed numerically using the results of [20]. We
all subsequent literature, the authors fix the precoding matrix
seethattheapproximation(38)ofthecapacityisverygoodin
tobeA=H forα 0,andforafixeddatavectoruchoose
α
allscenarios,exceptforsmallsquaresystemswhenρislarge. ≥
the offset vector x to be
x= argmin γ = argmin A(u+x) 2 (41)
′
C. Qualitative Behavior of Capacity of MMSE Pre-Inversion x′ τZ[i]K x′ τZ[i]K || ||
∈ ∈
To study the qualitative behavior of the capacity of which is a closest-vector problem in a lattice and hence
Tikhonov pre-inversion with A = H , we expand solvable with a sphere decoder.
MMSE
6
14 systems with K =M =12 employing 16-QAM modulation.
E (C ), M = 8
H H We see a consistent gain of approximately 0.5 dB over the
12 EH(CH), M = 12
E (C ), M = 24 WFVP strategy.
H H
E (C ), M = 48
z) 10 H H
H 5.5
) (bits/sec/H 68 5 MWpchlaSFaiVVnn PnPTeikl hcoanpoavc itinyversion
(CH Hz) 4.5
E 4 c/
se 4
s/
2 bit
) (H3.5
C
00 5 10 15 20 25 30 E(H 3
ρ = 1/σ2 (dB)
14
E (C ), M = 64 2.5
H H
12 EH(CH), M = 96
E (C ), M = 192 2
H H 10 12 14 16 18 20
E (C ), M = 384
z) 10 H H ρ = 1/σ2 (dB)
H
c/
se 8 Fig. 3. Capacity of the WFVP strategy of [12] and the MSVP defined by
s/ (42),forasystemwithK=M =12.
bit
) (H 6
C
(H
E 4
B. Estimating the Performance of max-SINR Vector Pertur-
2 bation
In this subsection we will estimate the performance of
0
0 5 10 15 20 25 30 MSVP when using the regularization parameter α = σ2.
ρ = 1/σ2 (dB)
The regularization parameter α = σ2 is not known to be
Fig. 2. Capacity of MMSE pre-inversion for K = 8 (top) and K = the optimal regularization parameter for the MSVP strategy,
64 (bottom). The marked lines are the experimentally computed capacity and without knowledge of the optimal α choosing α=σ2 is
C of MMSE pre-inversion, the dashed lines represent our approximation simply convenient. The main result of this subsection is an
−log2(1−d(c,σ2))ofthisquantity,andthesolidlinesthechannelcapacity. approximation of EH(SINR) which can be used to predict
system performance to within about 0.5-1 dB, which we
A. Max-SINR Vector Perturbation demonstrate through numerous simulations.
The results of [9], specifically [9, Lemma 1 and Corollary
Ratherthanchoosingxtominimizeγ,weinsteadchoosex
1], estimate the power renormalization constant γ for the
to minimizethe meansquareerrorofthesystem. Specifically,
precoding matrix A = H ; Jensen’s Inequality can then be
for a fixed channel matrix H and a fixed data vector u, we ZF
used to estimate the expected SINR for such a ‘zero-forcing’
choose the perturbation vector x according to
strategy.HowevertheMSVPstrategy,whichsetsA=H
MMSE
x= argmin MSEu = argmin T(u+x′) 2 (42) and chooses x according to (42), has qualitatively different
x′ τZ[i]K x′ τZ[i]K || || performanceatlowervaluesofρwhencomparedtothe‘zero-
∈ ∈
forcing’ strategy. Thus a new predictor of performance is
whereT satisfies T T=d2I 2dHH +HH H+ H
† K− α α MMSE α required.
and d = tr(HH )/K. Note that this provides a VP strategy
α While the estimate of γ for the ‘zero-forcing vector per-
for any regularization parameter α 0 whatsoever, not just
≥ turbation’ method of [9] does not provide a useful predictor
the MMSE parameter α = σ2. We will refer to this strategy
for the SINR of max-SINR vector perturbation, the general
as max-SINR vector perturbation, or MSVP.
strategy therein remains applicable. In particular, we let
We emphasizethatthis isnot the WienerFilter VP strategy
(WFVP) of [12], which chooses the offset vector x to min- = hypercube in CK, with side length
K
imize ||L(u+x′)||2, where L†L = (σ2IK + HH†)−1. An H N (43)
argument similar to the proof of our Theorem 1 shows that lim τ = lim 6c =√6c
L(u+x) 2 is the denominatorof the alternative expression N→∞ N→∞r N −1
|| ||
(13) of the SINR. Thus WFVP attempts to maximize the and we consider data vectors u chosen from the uniform
SINR, but does not account for the diagonal entries of HH distribution on . Heuristically, we are approximating the
α K
H
being less than unity. discreteN-QAMdistributionbytheuniforminputdistribution
To demonstrate the improvement offered by our MSVP on the minimal hypercube surrounding the constellation as
method over the WFVP strategy of [12], we plot the capacity N . One can check that our energy constraint is
C =EH(CH)asdefinedin(18)ofbothschemesinFig.3for prese→rved∞, in other words that for such uniform inputs u, we
7
have We can complete our approximation of EH(SINR) by
1 performing the following series of approximations:
Eu u 2 = t 2dt=Kc (44)
and hence Eu||ui||2 = cvosli(nHceKt)heZHenKtr|i|es||of the data vector u EH(SINR)≈ dE(cH,(σM[2)S2EK)c
| |
are assumed i.i.d. d(c,σ2)2Kc
H,Roeucralelsftirmomate(1M[6)SEanodf(t2h8e)mtheaatnfsoqruaarefixeerdrocrhoafntnheelsmysatterimx ≤ 6πcK(KK+!)11/KEH(det(T†T)1/K) (50)
π K+1 d(c,σ2)2
is given by
M[SE=Eu||T(u+x)||2 (45) ≈ π6(KK!)+1/1K (EHKid=e0t(TKi†T(M)M)1!i/)K!β2i 1/K
ownhe[r9e,TLeimsmasain1]T,hweoilrlemall1ow. Tuhse btoeloewstimpraotpeoEsiHtio(nM[,SmEo)dealnedd ≈ 6(K!)1/K "PKi=0(cid:0)Ki(cid:1)(MM−−!i)!β1i#
thus provide a useful estimate of EH(SINR). where β1 and β2 are as in equPation (cid:0)(49(cid:1)). We now have
Proposition 1: Suppose an M×K MIMO system employs EH(SINR) Evp(K,M,σ2) (51)
MSVP with a fixed channel matrix H and data vectors u ≈
where
uniform on , with any regularization parameter α 0.
K
Then we havM[HeSE≥ 6πcKK(K+!)11/K det(T†T)1/K ≥(46) Evp(K,M,σ2):= π6(KK!)+1/1K "PKiKi==00(cid:0)KKii(cid:1)((MMMM−−!!ii))!!ββ21ii#1/(K52)
P (cid:0) (cid:1)
for max-SINRvectorperturbationwith α=σ2. The approxi-
where T is as in Theorem 1.
mationsin(50)allstemfromreplacingaquantitybyitslarge-
Proof: The proof is the same as for [9, Lemma 1],
K limit, or from an application of Jensen’s Inequality. We
[
wherein MSE is expressed as the second moment of the
omit further details in favor of demonstrating the validity of
VoronoicellofthelatticegeneratedbyTinCK,andisrelated
the approximation through simulations.
to the second moment of the unit sphere in R2K. We omit
further details.
C. Simulation Results
ToapplytheaboveresulttoapproximatetheexpectedSINR
In this subsection we empirically demonstrate the accuracy
of the system, we will need to estimate EH(det(T†T)). To of (51) for MSVP with α = σ2. We fix the signaling
that end, we recall a result from Random Matrix Theory. Let
alphabet to be a 16-QAM constellation for all experiments.
W=KHH† be takenfromthe complexWishartdistribution The solid curves represent experimentally measured values
WK(M,IK) [21, Section 2.1.3], and let β be constant with of EH(SINR) or the capacity C, and the dashed curves the
respect to H. Then [21, Theorem 2.13] states that
resulting approximations.
1) SINR as a function of K: We study (51) in Fig. 4
K
K M!
EH(det(IK +βW))= βi. (47) for M = K (top) and M = 2K (bottom). We see that the
i (M i)! approximationis accurate to within about 1 dB when K 4.
Xi=0(cid:18) (cid:19) − ≥
Implicit in our approximation (51) is (38) which essentially
Whenα=σ2 wecanrewriteT†Tasaproductofmatricesof replaces the value d by its large-K limit, hence one should
theformIK+βWandthenapplytheaboveresulttoestimate expect (51) to also be more accurate for larger K.
EH(det(T†T)). Straightforward computation gives 2) Approximating Capacity: In Fig. 5 we plot, for K = 8
andc=1,8/9,and4/5,theergodiccapacityC ofmax-SINR
2 1
1 d 1 1 − vector perturbation as well as the corresponding estimate ob-
T†T=d2 IK +(cid:18) −d (cid:19) Kσ2W!(cid:18)IK + Kσ2W(cid:19) tainedbycombining(51)and(18)toobtaintheapproximation
C log (1+E (K,M,σ2)).Again,weseethatthisestimate
≈ 2 vp
Replacingdwithitslarge-K limitd(c,σ2),againapproximat- predicts the expected capacity well.
ing the expectation of a ratio by the ratio of the expectations,
and using (47) we arrive at VI. MAX-SINRVECTORPERTURBATION FORLARGE
SYSTEMS
EH(det(T†T))≈d(c,σ2)2KEEHH((ddeett((IIKK ++ββ12WW)))) sioVne,cbtuotrcpoemrtpuurbtiantgiotnheofofeprtsimlaarlgoeffbseentevfietcstoorvxericnh(a4n2n)emlianyvebre-
K K M! βi (48) prohibitivelycomplexforlarge K. In[7] the authorsusedthe
=d(c,σ2)2K i=0 i (M−i)! 1 LLL lattice-reduction algorithm to achieve this goal, but for
PK (cid:0)K(cid:1) M! βi
i=0 i (M i)! 2 verylargeK thisreductionitselfcanbeprohibitivelycomplex.
−
where P (cid:0) (cid:1) We will use a method with even smaller complexity, which
we show approaches the performance of the ML solution for
1 d(c,σ2) 2 1 1 small K, and slightly outperforms LLL-based precoding for
β1 = −d(c,σ2) Kσ2, β2 = Kσ2. (49) large K. For all experiments 16-QAM modulation was used.
(cid:18) (cid:19)
8
35 fora squareK K matrixT wherex rangesoveran integer
′
ρ = 10 dB ×
ρ = 15 dB lattice. The SQRP algorithm is a modified Gram-Schmidt
30 ρ = 20 dB procedurewhichdecomposestheK KmatrixTasaproduct
ρ = 25 dB
ρ = 30 dB ×
T=QRP (54)
25
R) where Q is K K unitary, R = (rij)1 i,j K is K K
SIN 20 upper-righttrian×gular,and P is a K K p≤erm≤utationma×trix,
(H to attemptto maximizethe diagonal×entriesr of R, in order
E ii
15 as i=K,...,1. Substituting into (53) we obtain
argmin y Tx′ 2 =argmin y QRPx′ 2 (55)
10 x′ || − || x′ || − ||
=argmin y˜ Rz 2 (56)
z || − ||
5
2 3 4 5 6 7 8
Value of K in K× K MIMO System where y˜ =Q†y and z=Px′.
40 Let us recall the definition of the Babai point zB =
ρρ == 1105 ddBB [z1B,...,zKB]T, an estimate of the solution to (56) given
35 ρ = 20 dB recursively by
ρ = 25 dB
ρ = 30 dB
c =y˜ /r , zB =[c ] (57)
30 K K KK K K
K
NR) c =(y˜ r zB)/r , zB =[c ], (58)
SI 25 i i− ij j ii i i
E(H j=Xi+1
for i=K 1,...,1
20
−
where [] denotes rounding to the nearest element of the
·
15 underlying per-coordinate constellation. The final estimate of
the ML solution x is obtained by computing P 1zB. The
−
10 modifiedSQR algorithmof [19] increases the probabilitythat
2 3 4 5 6 7 8
Value of K in 2K × K MIMO System P 1zB istheMLsolutionto(53).Wereferto[19]forfurther
−
details.
FSiIgN.R4.veOcntotroppe,rEtuHrb(aStiIoNnRw)itfhorαa=Kσ×2,Kforsvyasrtieomus(vca=lue1s)oefmρp=loy1i/nσg2m.aOxn- ToapplythisalgorithmtotheVPprocedure,werewritethe
bottom,thesameplotfor2K×K systems. argmin problem (42) as
x= argmin T(u+x′) 2 = argmin y Tx′ 2 (59)
11 x′ Z[i]K || || −x′ Z[i]K || − ||
K = 8, M = 8 ∈ ∈
10 K = 8, M = 9 where y = Tu is the ‘received’ vector. We then apply the
K = 8, M = 12
decomposition (54) and compute the estimate of x, namely
9
Hz) 8 P−1zB,asabove.WerefertotheprocessasSortedQRvector
c/ perturbation, or just SQR vector perturbation.
e
s 7
s/
bit B. Comparison with ML and Lattice-Reduction-AidedBroad-
) (H 6 cast Precoding
C
(H 5 InthetopplotofFig.6wecompareSQRMSVPforK =8
E
4 and c = 1, 4/5 to ML MSVP wherein (42) is solved using
a sphere decoder. We also plot the performance of lattice-
3
reduction-aidedbroadcastprecoding[7] appliedto ourMSVP
2
10 15 20 25 30 method,whichusesamatrixdecompositionofTbasedonthe
ρ = 1/σ2 (dB) LLL lattice reduction algorithm [6] and similarly computes
a Babai estimate of the optimal perturbation vector. In the
Fig. 5. The experimentally measured capacity C of MSVP for systems
with K = 8 and M = 8, 9, and 12, versus the approximation log2(1+ bottom plot of Fig. 6 we repeat the experiment for K = 80
Evp(K,M,σ2))ofthisquantity. and c = 1, 4/5, omitting the performance of ML MSVP as
using a sphere decoder for such a large system is infeasible.
As we see from the plots, the performance degradation of
A. Sorted QR max-SINR Vector Perturbation using the SQR method instead of an ML method to find
WeemploytheSortedQRPrecoding(SQR)methodof[19], the perturbation vector x is minimal. Surprisingly, the SQR
asub-MLalgorithmfordecodingspace-timecodeswhichcan methodoffersamarginalbutconsistentimprovementoverthe
be summarized as follows. For our purposes it suffices to LLL method at high values of ρ. This is especially notable
consider the problem of computing since, as we discuss further in Section VI-D, computing the
SQRmatrixdecompositioncanbedonewithlowercomplexity
x=argmin y Tx′ 2 (53) than computing the LLL reduction.
x′ || − ||
9
10 9
M = 8, ML ZFVP
9 M = 8, SQR 8 MSVP
M = 8, LLL plain Tikhonov inversion
M = 10, ML channel capacity
8 M = 10, SQR 7
Hz) M = 10, LLL Hz)
C) (bits/sec/H 567 C) (bits/sec/H456
E(H 4 E(H3
3 2
2 1
10 15 20 25 30 10 15 20 25 30
ρ = 1/σ2 (dB) ρ = 1/σ2 (dB)
10
9
M = 80, SQR
ZFVP
M = 80, LLL
9 MSVP
M = 100, SQR 8
M = 100, LLL plain Tikhonov inversion
channel capacity
8
7
Hz) z)
ec/ 7 c/H 6
s e
E(C) (bits/HH 56 (C) (bits/sHH45
4 E
3
3
2
2
10 15 20 25 30 1
ρ = 1/σ2 (dB) 10 15 20 25 30
ρ = 1/σ2 (dB)
Fig. 6. On top, capacity of MSVP systems with K = 8 and c = 1, 4/5
employingML,SQR,andLLLmethodsforcomputingtheperturbationvector Fig. 7. On top, the value of C = EH(CH) for the MSVP and ZFVP
x.Onbottom,thesameplotforK=80,omitting theMLstrategy. strategies with K = M = 256 employing 16-QAM signaling, using the
SQRdecompositiontosolvefortheperturbationvector.Onbottom,thesame
plotforK=M =1024.
C. Comparison with Zero-Forcing Vector Perturbation
D. Remarks on Complexity
TodemonstratethatSQRMSVPcanbeusedinpractice,we
In this subsection we compare MSVP (with α = σ2) with
nowbrieflydiscussthecomplexityoftheinvolvedalgorithms.
the zero-forcing strategy in which A = H and the offset
ZF Solving for the perturbation vector x is the main bottleneck
vector x is chosen using the SQR algorithm of Section VI-A
to implementing VP systems, as it must be done multiple
to minimize γ = A(u+x) 2/K as in [2], [9]. We denote
times per channel realization and finding the ML solution
|| ||
this strategy ZFVP. Channel pre-inversion with A = H
MMSE is notoriously complex. The preprocessing performed on the
and no perturbation is also shown as a helpful basis for
channel matrix H (e.g. computing the matrix T) must only
comparison.
be doneonceper channelrealizationand thereforehas less of
In Fig. 7 we show the performance of ZFVP and MSVP animpactontotalcomputation.Nevertheless,wediscussboth
for K = M = 256 (top) and K = M = 1024 (bottom) aspects below.
when the SQR algorithm is employed. When K =M =256, 1) Preprocessing: During SQR MSVP, the preprocessing
MSVP offers a steady improvement of approximately 1 dB consistsof two parts,namelycomputingthe K K matrix T
×
over ZFVP between ρ = 10 dB and 20 dB, though for large asinTheorem1,andthenperformingtheSQRdecomposition
valuesofρ ZFVPslightlyoutperformsMSVP.ForK =M = toT.TheformercanbedonewithaCholeskydecomposition,
1024 the performance of ZFVP degrades to the point where the computation of which requires O(K3) operations. The
we see that MSVP outperforms it at all values of ρ under latter can be done using a modified Gram-Schmidt algorithm
consideration. On the contrary, the performance of MSVP is (see [19]), the complexity of which is easily be seen to be
apparently constant with increasing system size. To simplify O(K3), and therefore all preprocessing can be performed in
presentationweomittedtheperformanceofSQRWFVPfrom O(K3) operations. If LLL reduction is used instead of the
these plots, but the behavior is essentially identical to that SQR method, then between O(K4) and O(K5) operations
already depicted in Fig. 3. Specifically, MSVP outperforms are needed [6].
WFVP by approximately0.5-1dB atall valuesof ρ when the 2) Solving for the Perturbation Vector: Solving for the
SQR algorithm is employed to find the perturbation vectors. perturbation vector x in (42) using a sphere decoder has
10
complexity which is exponential in the dimension K of the whereΣ hasthesingularvaluess /(s2+α),fori=1,...,K
α i i
lattice [5]. On the other hand, computing the Babai estimate of H along the diagonal. It follows immediately that the
α
xB using (57) and (59) requires only O(K2) multiplications. singular values of H+ are (s2+σ2)/s2 for i=1,...,K.
MMSE i i
The complexity of computing the Babai point is the same, Each singular value s of H gives rise to an eigenvalue λ
regardless of whether we use the SQR or LLL method to of the matrix d2I 2dHH +HH H+ H , which is
compute the perturbation vector. given by K − α α MMSE α
VII. CONCLUSIONS AND FUTURE WORK s2 s2(s2+σ2)
λ = d2 2d + (63)
With the goal of developing scalable and close-to-capacity − s2+α (s2+α)2
data transmission schemes for next-generation broadcast net- s2 s2 2 s2+σ2
works, we have studied channel pre-inversion and vector = d2 2d + (64)
− s2+α s2+α s2
perturbation schemes for a large number K of end users. (cid:18) (cid:19)
To that end, we have provided an explicit, sharp estimate of > d2 2d s2 + s2 2 (65)
the capacity of MMSE channel pre-inversion as K . − s2+α s2+α
→ ∞ (cid:18) (cid:19)
Furthermore, we have proposed a new max-SINR vector s2 2
perturbationscheme which maximizesa sharp estimate of the = d 0. (66)
− s2+α ≥
SINR of the system. Random Matrix Theory was used to (cid:18) (cid:19)
estimate the performance of our vector perturbation scheme,
and the resulting approximation was shown to be accurate. As all λ are obtained this way, the matrix d2IK 2dHHα+
We demonstrated that MSVP outperforms other VP schemes, HHαH+MMSEHα is positive definite. −
such as Wiener Filter VP and zero-forcing VP. Lastly, we Let ε and ε be as in the statementof the theorem.To see
1 2
applied the Sorted QR decomposition method to solve for the bounds for the SINR, note that when A=H we have
α
the perturbation vector, resulting in a scheme which is low-
complexity and close to channel capacity for very large K.
SINR
The low complexity and good performance suggest that our
max-SINR vector perturbationmethod could be implemented = ||dg(HHα)||2Fc
in practice in large broadcast networks. Eu( (HHα dg(HHα))(u+x) 2+ Hα(u+x) 2σ2)
|| − || || ||
Future work will consist of investigations into using fast- (67)
decodable space-time codes [22], [23], [24] at the transmit dg(HH ) 2c
= || α ||F
end, which could naturally reduce the complexity of the ML Eu( (HHα dIK)(u+x) 2+ Hα(u+x) 2σ2)+ε2
search for the perturbation vector. Furthermore, we plan on || − || || || (68)
comparing our method with the Degree-2 Sparse VP method
of [25], which offers comparable complexity. We plan to
Webeginbyboundingthenumeratoraboveandbelow.Bythe
investigatethe performanceof MSVP with imperfectCSI and
triangle inequality, we have
withcorrelatedchannelcoefficients,particularlywhencorrela-
tionoccursbetweenthe transmitantennas.Lastly,preliminary
simulation results suggest that the quantity dg(HA)ii/√γ, ||dg(HHα)||2F = ||dg(HHα)−dIK +dIK||2F (69)
which must be known by the receivers prior to transmission, dI 2 +ε (70)
is nearly constant with respect to H, especially for large ≤ || K||F 1
= d2K+ε (71)
systems.Thusitmaybepossibletoreplacethisquantitywitha 1
simpleconstantatthereceiveend,cuttingdownonpreliminary
communication overhead significantly. This is an avenue of Toseetheotherinequality,leta ,...,a >0beanypositive
1 K
potential future research that deserves investigation. real numbers, and let a = 1 K a be their mean. We
K i=1 i
have ( K a )2 ( K a2)( K 12) by the Cauchy-
i=1 i ≤ i=1 iP i=1
APPENDIX Schwarz Inequality, which is easily seen to be equivalent to
[a,...P,a]T 2 [a ,P...,a ]TP2. Letting a = (HH )
Proof of Theorem 1: To prove that the matrix T exists, || || ≤ || 1 K || i α ii
and a = d = tr(HH )/K we see that d2K = dI 2
HwhHicαhHm+MuMsStEHfirαst isshopwositthivaet dtheefinmitea.triTxodt2hIaKt e−nd2,dlHetHHα =+ ||dg(HHα)||2F. α || K||F ≤
To complete the bounds on the SINR, define
UΣV be a singular value decomposition of the channel,
†
where the diagonal entries of Σ are s ,...,s . The singular
1 K
value decomposition of Hα is M[SEu = (HHα dIK)(u+x) 2+ Hα(u+x) 2σ2
|| − || || ||
H = VΣTU (αI +UΣV VΣTU ) 1 (60) (72)
α † K † † − [
Following [12, Section 4], the idea is to rewrite MSEu as the
= VΣT(αIK +ΣΣT)−1U† (61) normofasinglevector.Toshortennotation,weletz=u+x
[
=:Σα and z′ =U†z. Computing MSEu in terms of the singular
= VΣ U (62) value decompositionsnow gives(notingthatmultiplyingby a
|α † {z }