On the Performance of Zero-Forcing Processing in Multi-Way Massive MIMO Relay Networks

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by  Chung Duc Ho

IEEECOMMUNICATIONSLETTERS,ACCEPTEDFORPUBLICATION 1
On the Performance of Zero-Forcing Processing in
Multi-Way Massive MIMO Relay Networks
Chung Duc Ho, Student Member, IEEE, Hien Quoc Ngo, Member, IEEE, Michail Matthaiou, Senior Member,
IEEE, and Trung Q. Duong, Senior Member, IEEE
Abstract—We consider a multi-way massive multiple-input should be taken into consideration. In [7], the authors analyze
multiple-outputrelaynetworkwithzero-forcingprocessingatthe the performance of multi-way massive MIMO systems with
relay. By taking into account the time-division duplex protocol imperfect CSI and MR processing at the relay. To the best of
with channel estimation, we derive an analytical approximation
the authors’ knowledge, there is no work on ZF processing
of the spectral efficiency. This approximation is very tight and
simple which enables us to analyze the system performance, as with imperfect CSI in literature, partially due to the difficulty
well as, to compare the spectral efficiency with zero-forcing and in manipulating products of Wishart matrices.
maximum-ratio processing. Our results show that by using a In this paper, we investigate a multi-way massive MIMO
7 very large number of relay antennas and with the zero-forcing relay network with ZF processing and imperfect CSI. The
1 technique, we can simultaneously serve many active users in the
relayestimatesthechannelsviauplinkpilotsandtheminimum
0 sametime-frequencyresource,eachwithhighspectralefficiency.
mean square error (MMSE) scheme. Then, it uses these chan-
2
I. INTRODUCTION nel estimates and the ZF technique to combine and beamform
n thesignalstoallusers.Wederiveanapproximateclosed-form
Multi-way relay networks are relevant for many applica-
a expression for the spectral efficiency. This approximation is
J tions, such as data transfer in multimedia teleconference and
data exchange between sensor nodes and data fusion centers very tight and enables us to further analyze the performance
3
of the considered system.
in wireless communications [1]. Due to the multiplexing

] gain, the spectral efficiency of multi-way relay networks is Notation: The superscripts (·)T, (·)∗, and (·)H stand for
T much larger than that of two-way or one-way relay networks. the transpose, conjugate, and Hermitian, respectively. The
I Therefore, during the past years, multi-way relay networks notationsE{·}andVar{·}aretheexpectationandthevariance
s. have attracted considerable research interest [2]. On a parallel operators, respectively. Furthermore, [A]k or ak denotes the
c k-th column of matrix A.
avenue, massive multiple-input multiple-output (MIMO) has
[
also attracted a significant amount of research interest from
II. MULTI-WAYMASSIVEMIMORELAYMODEL
1 both academia and industry [3]. In massive MIMO, hundreds
v ofantennasaredeployedatthebasestationtoservesimultane- We consider a multi-way relaying massive MIMO network
5 ously tens of users. With simple linear processing techniques, which includes one relay station and K users.1 The relay
4 station is equipped with M antennas, and each user has a
suchasmaximum-ratio(MR)orzero-forcing(ZF)processing,
6 single antenna (M > K). In this system, each user wants to
massive MIMO can offer huge spectral and energy efficiency
0 communicatewithK−1otheruserswiththeaidoftherelay.
[4]. Thus, massive MIMO combined with multi-way relaying
0 We assume that the direct links (user-to-user links) are absent
. techniqueisastrongcandidateforthenext-generationwireless
1 due to large path loss and/or severe shadowing.
communication systems.
0 Let G ∈ CM×K be the channel matrix from the K users
Recently,therehavebeensomeworksinmulti-waymassive
7 to the relay, which includes the small-scale fading and the
MIMO relay systems [5], [6]. These systems can offer all
1 large-scale fading and is modeled as
benefits of both massive MIMO and multi-way relaying tech-
:
v nologies, and hence, are expected to offer very high spectral G=HD1/2, (1)
i
X efficiency. In particular, in [5], the authors show that by
using very large antenna arrays at the relay together with ZF where H∼CN(0,IM) represents the small-scale fading, and
ar processing,thesystemperformancecanimprovesignificantly. D ∈ CK×K is a diagonal matrix containing the large-scale
Furthermore, [6] shows that the transmit power of each user fadingcoefficientswhose k-thdiagonalelementis denoted by
and/or the relay can be made inversely proportional to the βk.
numberofrelayantennas,whilemaintainingarequiredquality The transmission protocol is the same as the one in [7].
ofservice.However,theseworksassumeperfectchannelstate More precisely, the data exchange between all the K users
information(CSI)attherelayandusers.Inpractice,especially is done via time-division duplex (TDD) operation. With TDD
in massive MIMO systems, the impact of channel estimation operation,eachcoherenceintervalisdividedintothreephases:
channel estimation, multiple-access, and broadcast phases.
Manuscript received November 26, 2016; accepted January 2 2017. The
A. Channel Estimation Phase
associate editor coordinating the review of this paper and approving it for
publicationwasC.Masouros.TheauthorsarewiththeSchoolofElectronics, All the K users simultaneously send pilot sequences to the
Electrical Engineering and Computer Science, Queens University Belfast,
relay.Therelaythenestimatesthechannelstoallusersthrough
Belfast, U.K. (e-mail:{choduc01, m.matthaiou, trung.q.duong}@qub.ac.uk,
receivingpilots.LetT andτ bethelengthsofeachcoherence
[email protected]). H. Q. Ngo is also with the Department of Electrical
Engineering(ISY),Linko¨pingUniversity,Linko¨ping,Sweden. interval and the training duration (in symbols), respectively,
This work was supported by project no. 3811/QD-UBND, Binh Duong
government,Vietnam.TheworkofH.Q.NgowassupportedbytheSwedish 1 It would be more practical to consider multi-cell setups. Unfortunately,
ResearchCouncil(VR)andELLIIT.TheworkofM.Matthaiouwassupported ifweconsidermulti-cellsetups,thesystemmodelbecomestoocomplicated
in part by the EPSRC under grant EP/P000673/1. The work of T. Q. toanalyze.Notethatourresultscanberegardedasanupperboundofwhat
DuongwassupportedbytheU.K.RoyalAcademyofEngineeringResearch is actually achieved in multi-cell setups. If a pilot reuse scheme is applied,
FellowshipunderGrantRF1415\14\22. thenthisupperboundisverytight[8].
2 IEEECOMMUNICATIONSLETTERS,ACCEPTEDFORPUBLICATION
with T > τ. We assume that the pilots used by the K users With the transmitted signal given in (9), the K users receive
are pairwisely orthogonal. This requires τ ≥ K. We denote
y (t) =GTs(t)+n(t)
by P the normalized transmit signal-to-noise ratio (SNR) per u R u
p (cid:112) (cid:112)
pilot symbol. Then, the MMSE channel estimate of G can be = α(t)P GTB(t)x+ α(t)GTC(t)n +n(t), (14)
u r u
represented as [7]
Gˆ =G−E, (2) where nu ∼CN(0,IK) is the AWGN vector at the users.
whereEistheestimationerrormatrix,whichisindependentof III. SPECTRALEFFICIENCYANALYSIS
Gˆ. Furthermore, Gˆ ∼CN(0,Dˆ) and E∼CN(0,D ), where We derive a closed-form expression for the spectral effi-
E
Dˆ andD arediagonalmatriceswhose(k,k)-thelementsare ciency of the transmission in the first time-slot. The same
E
σ2 = τPpβk2 , and σ2 =β −σ2, respectively. analysis can be done for other time-slots. Note that, hereafter,
k τPpβk+1 e,k k k we set k+1=1 if k =K and set k−1=K if k =1.
B. Multiple-Access Phase By using the bounding technique in [9], the received signal
After sending the pilot sequences’ phase, all the users at the k-th user y(1) is expressed as:
u,k
sEim(cid:110)u|xltka|n2e(cid:111)ou=sly1,siesndthethseiigrndalattaratnosmthitetedrelfaryo.mLtehtexkk-,thwuhseerre. yu(1,k) =(cid:112)α(1)PuE(cid:110)gkTb(k1+)1(cid:111)xk+1+ N(cid:124)˜(cid:123)k((cid:122)1(cid:125)) , (15)
(cid:124) (cid:123)(cid:122) (cid:125)
Then, the relay sees effectivenoise
desiredsignal
(cid:112)
y = P Gx+n, (3) where
R u r
wherePuisthenormalizedtransmitSNR,x(cid:44) [x1,...,xK]T, N˜k(1) (cid:44)(cid:112)α(1)Pu(cid:16)gkTb(k1+)1−E(cid:110)gkTb(k1+)1(cid:111)(cid:17)xk+1
and n ∼ CN(0,I ) is the AWGN vector at the relay.
r M K
Then,therelayusesthechannelestimateandZFtechniqueto +(cid:112)α(1)P (cid:88)gTb(1)x +(cid:112)α(1)gTC(1)n +n(1). (16)
combine the received signals from all M antennas as u k i i k r u,k
i=1
y˜ =WTy , (4) i(cid:54)=(k+1)
R R
The worst-case Gaussian noise yields an achievable spectral
where WT is the ZF receiver given by [4] efficiency for the k-th user, which is given as
WT =(cid:16)GˆHGˆ(cid:17)−1GˆH. (5) (cid:18) (cid:19)(cid:18) (cid:19)  α(1)P (cid:12)(cid:12)E(cid:110)gTb(1) (cid:111)(cid:12)(cid:12)2
SE(1)= T−τ K−1 log 1+ u(cid:12) k k+1 (cid:12) .
C. Broadcast Phase k T K 2 Var(cid:16)N˜(1)(cid:17) 
k
TosendallsignalstoK users,therelayspendsK−1time-
(17)
slots. In the t-th time-slot, the relay aims to send x to user
k+t
k (if k+t > K, then x is set to be x ). Thus, the To derive the spectral efficiency in closed-form, we need to
k+t k+t−K (cid:110) (cid:111) (cid:16) (cid:17)
transmit signal vector at the relay for the t-th time-slot is compute E gTb(1) and Var N˜(1) . From the indepen-
k k+1 k
s(t) =(cid:112)α(t)AΠ(t)y˜ , t=1,2,...,K, (6) dence between Gˆ and E, we have
R R (cid:110) (cid:111) (cid:110) (cid:111)
where Π(t) ∈ CK×K is the permutation matrix at the t-th E gkTb(k1+)1 =E gˆkTAΠ(1)WTgˆk+1 =1. (18)
time-slotgivenby[5,Eq.(17)],AistheZFprecodingmatrix
expressed as Since N˜k(1) has a complicated form which includes matrix
(cid:16) (cid:17)−1 inversions and multiplications of Wishart matrices, we cannot
A=Gˆ∗ GˆTGˆ∗ , (7) (cid:16) (cid:17)
obtain an exact closed-form of Var N˜(1) . However, thanks
k
andα(t) ischosentosatisfythepowerconstraintattherelay, to the law of large numbers (for large M), we can obtain the
(cid:26)(cid:13) (cid:13)2(cid:27) following approximation.
E (cid:13)s(t)(cid:13) =P. (8)
(cid:13) R (cid:13) r Theorem 1: As M → ∞, the spectral efficiency (17) can
be approximated as
Denote B(t) (cid:44) AΠ(t)WTG, and C(t) (cid:44) AΠ(t)WT. Then
 
s(t) in (6) can be rewritten as
R s(Rt) =(cid:112)α(t)PuB(t)x+(cid:112)α(t)C(t)nr. (9) SE(k1)→(cid:18)TT−τ(cid:19)(cid:18)KK−1(cid:19)log21+α(1)P (cid:80)K Iα(1)+Puα(1)J +1,
u k,i k
Plugging (9) into (8), we have i=1
(32)
P
α(t) = r , (10)
P Q(t)+P Q(t)+Q(t) where
u 1 u 2 3 M(M −K)P
where α(1) (cid:44) r , (33)
(cid:26) (cid:20)(cid:16) (cid:17)(cid:16) (cid:17)H(cid:21)(cid:27) MPu(cid:80)Kk=1 σ12 +Pu(cid:80)Kk=1σe2,k(cid:37)+(cid:37)
Q(t) (cid:44)E Tr AΠ(t)WTGˆ AΠ(t)WTGˆ , (11) k
1 Mσ2 σ2 +Mσ2 σ2 +σ2 σ2 σ2 σ2 (cid:37)
I (cid:44) i−1 e,i k+1 e,k k+1 i−1 e,k e,i , (34)
(cid:26) (cid:20)(cid:16) (cid:17)(cid:16) (cid:17)H(cid:21)(cid:27) k,i M(M −K)σ2 σ2
Q(t) (cid:44)E Tr AΠ(t)WTE AΠ(t)WTE , (12) i−1 k+1
2
Q(3t) (cid:44)E(cid:26)Tr(cid:20)(cid:16)AΠ(t)WT(cid:17)(cid:16)AΠ(t)WT(cid:17)H(cid:21)(cid:27). (13) Jk (cid:44) MM(M+σ−k2+K1σ)σe2,k2k+(cid:37)1, and (cid:37)(cid:44)k(cid:88)(cid:48)K=1σk2(cid:48)σ1k2(cid:48)+1. (35)
Proof: See Appendix VI-B.
ONTHEPERFORMANCEOFZERO-FORCINGPROCESSINGINMULTI-WAYMASSIVEMIMORELAYNETWORKS 3
Fig.1. SumspectralefficiencyversusthenumberofusersK.Wechoose Fig.2. Cumulativedistributionofthesumspectralefficiency.HereM =
T =200,Pu=Pp=0dB,Pr=10dB,andβk=1. 100,K=20.
IV. NUMERICALRESULTS a new tractable approximate closed-form expression for the
spectral efficiency. For a large number of relay antennas,
Weconsiderthesumspectralefficiencyasourperformance
the inter-user interference and noise reduces significantly,
metric. The sum spectral efficiency is defined as
and hence, the system can deliver a substantial sum spectral
SE =(cid:88)K SE(1) bit/s/Hz. (36) efficiency.Furthermore,weshowedthat,formostofthecases
sum k (particularly at high SNRs), ZF processing offers a higher
k=1 spectral efficiency than MR processing does.
For the first example, we assume that β = 1, and choose
k VI. APPENDICES
T = 200,P = P = 0 dB, P = 10 dB. Figure 1 compares
u p r
the performance of multi-way massive MIMO systems for ZF A. Preliminary Results
and MR processing with different K, while the ratio M/K Lemma 1: Let X ∈ CM×K, M > K. Each row of X is
is kept fixed. For MR processing, we used the results in CN(0,D), where D is a diagonal matrix. Furthermore, let
[7, Eq. (26)]. Clearly, the simulated spectral efficiency and Dˆ ∈CK×K be another diagonal matrix. Then, we have
the approximate one match perfectly. At a small K (low
inter-user interference) and large K (large channel estimation E(cid:110)Tr(cid:104)Dˆ (cid:0)XHX(cid:1)−1(cid:105)(cid:111)= 1 (cid:88)K [Dˆ]kk. (37)
overhead), the spectral efficiencies of ZF and MR processing M −K [D]
kk
arecomparable.However,whenK =20–180,ZFsignificantly k=1
outperforms MR processing. Interestingly, regardless of the Proof: By expressing Tr(cid:104)Dˆ (cid:0)XHX(cid:1)−1(cid:105) =
ratio M/K, the sum spectral efficiency is maximum when
K is around 100. Furthermore, when M/K increases, the (cid:80)K [Dˆ]kk[W−1] , where W is a K × K central
k=1 [D]kk kk
inter-user interference reduces, and hence, the sum spectral Wishart matrix of M degrees of freedom, and using [10,
efficiency increases. Lemma 2.10], we obtain (37).
We next consider a more practical scenario where users are Lemma 2: Let A∈CM×M, and x∼CN(0,I ). Then,
M
l1o0c0a0temd.unTihfoerlmarlgyea-tscraalnedofamdiinngsiidsemaoddisekllewditahstβhekd=iam1+e(ztdkekr)νof, E(cid:110)(cid:12)(cid:12)xTAx(cid:12)(cid:12)2(cid:111)=Tr(cid:0)AAH(cid:1)+Tr(AA∗). (38)
where z is the log-normal random variable with standard
deviationk of 8 dB, ν = 4 denotes the path-loss exponent, Proof: To obtain (38), we first express xTAx as
Fanudrthdekrmisorteh,ethdeisntoanrmcealbizeetdwetreannstmheitkS-NthRsusPerr,PanudanthdePrpelcaayn. (cid:80)E(cid:8)Mm|x=m1(cid:80)|4(cid:9)Mm=(cid:48)=12aamndm(cid:48)Exm(cid:8)|xxmm(cid:48)x,ma(cid:48)|n2d(cid:9)=the1n, foursme (cid:54)=them(cid:48)i.dentities
becalculatedbydividingthesepowersbythenoisepowerN .
0
In this example, we choose N0 = −120 dB. We consider 2 B. Proof of Theorem 1
cases: Case-1 corresponds to (P =P =0.2 W, P =1 W),
u p r 1) Derivationofα(1): From(10),tocomputeα(1) weneed
and Case-2 corresponds to (P = P = 0.1 W, P = 0.5 W).
u p r tocomputeQ(1),Q(1),andQ(1).Thesubstitutionof(5)and(7)
Figure2showsthecumulativedistributionofthesumspectral 1 2 3
into (11) yields
efficiency for ZF and MR processing. We can see that, at
hpirgohcestrsainngsmisithpigohweerrth(Canasteh-e1)o,nteheofspMeRctrparloecfefiscsiinengcayndofvZicFe Q(1) =E(cid:26)Tr(cid:16)GˆTGˆ∗(cid:17)−1(cid:27)= 1 (cid:88)K 1 , (39)
1 M −K σ2
versa at low transmit power. Furthermore, compared with MR k=1 k
processing, the spectral efficiency of ZF processing is less
where in the last equality we have used Lemma 1.
concentrated around its median. TocomputeQ(1),wesubstitute(5)and(7)into(12)toobtain
2
V. CONCLUSION
We have investigated a multi-way massive MIMO relay Q(1)=(cid:88)K σ2 E(cid:26)Tr(cid:20)Π(1)(cid:16)GˆHGˆ(cid:17)−1(cid:16)Π(1)(cid:17)H(cid:16)GˆTGˆ∗(cid:17)−1(cid:21)(cid:27).
2 e,k
network with ZF processing and imperfect CSI. We derived
k=1
4 IEEECOMMUNICATIONSLETTERS,ACCEPTEDFORPUBLICATION
From the law of large numbers, we have that GˆHGˆ →MDˆ, Similarly to the derivation of V , we have
1
and hence, Q2 can be approximated as (cid:110) (cid:111) β −σ2 (cid:110) (cid:111) β −σ2
E |I |2 = k k ,E |I |2 = k k .
Q(1)→(cid:88)K σ2 E(cid:26)Tr(cid:20)Π(1)(cid:16)MDˆ(cid:17)−1(cid:16)Π(1)(cid:17)H(cid:16)GˆTGˆ∗(cid:17)−1(cid:21)(cid:27) 1 (M−K)σk2+1 2 (M−K)σk2−1
2 e,k (50)
k=1 (cid:110) (cid:111)
(cid:37) (cid:88)K Next,wecomputeE |I3|2 .Using(38)fromLemma2,
= M(M −K) σe2,k, (40) and the law of large numbers, we obtain
k=1 (cid:110) (cid:111)
E |I |2
whereagainwehaveusedLemma1toobtainthelastequality. 3
Similarly, we obtain (cid:26) (cid:20) (cid:16) (cid:17)−1(cid:16) (cid:17)H(cid:16) (cid:17)−1(cid:21)(cid:27)
=σ4 E Tr Π(1) GˆHGˆ Π(1) GˆTGˆ∗
(cid:37) e,k
Q(1) → . (41)
3 M(M −K) (cid:26) (cid:20) (cid:16) (cid:17)−1(cid:16) (cid:17)∗(cid:16) (cid:17)−1(cid:21)(cid:27)
+σ4 E Tr Π(1) GˆHGˆ Π(1) GˆTGˆ∗
Substituting (39), (40), and (41) into (10), we obtain (33). e,k
(cid:16) (cid:17)
2) Derivation of Var N˜(1) : From (16), we have σ4 (cid:37)
k → e,k . (51)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:26)(cid:12) (cid:12)2(cid:27) M(M −K)
Var N˜(1) =α(1)P Var gTb(1) +α(1)P E (cid:12)gTb(1)(cid:12)
k u k k+1 u (cid:12) k k (cid:12) Substituting (50), and (51) into (49), we get
+α(1)P (cid:88)K E(cid:26)(cid:12)(cid:12)gTb(1)(cid:12)(cid:12)2(cid:27)+α(1)E(cid:26)(cid:13)(cid:13)gTC(1)(cid:13)(cid:13)2(cid:27)+1. (42) E(cid:26)(cid:12)(cid:12)(cid:12)gkTb(k1)(cid:12)(cid:12)(cid:12)2(cid:27)→Ik,k. (52)
u (cid:12) k i (cid:12) (cid:13) k (cid:13)
i=1 (cid:26)(cid:12) (cid:12)2(cid:27)
i(cid:54)=(k,k+1) c) Compute E (cid:12)gTb(1)(cid:12) , where i (cid:54)= k,k + 1: Fol-
(cid:16) (cid:17) (cid:12) k i (cid:12)
a) ComputeVar gTb(1) :Byexpressingthetruechannel lowing a similar methodology as in the derivation of
k k+1
as the sum of the channel estimate plus the channel E(cid:26)(cid:12)(cid:12)gTb(1)(cid:12)(cid:12)2(cid:27), we obtain
estimation error, we obtain (cid:12) k k (cid:12)
Var(cid:16)gkTb(k1+)1(cid:17)=V1+V2+V3, (43) E(cid:26)(cid:12)(cid:12)(cid:12)gkTb(i1)(cid:12)(cid:12)(cid:12)2(cid:27)→Ik,i. (53)
where
V1 (cid:44)E(cid:26)(cid:12)(cid:12)(cid:12)gˆkTAΠ(1)WTek+1(cid:12)(cid:12)(cid:12)2(cid:27), (44) d) ComputeE(cid:110)(cid:13)(cid:13)gkTC(1)(cid:13)(cid:13)2(cid:111):Byreplacinggk withgˆk+ek
together with Lemma 1 and the law of large numbers, as
(cid:26)(cid:12) (cid:12)2(cid:27) (cid:26)(cid:12) (cid:12)2(cid:27)
V (cid:44)E (cid:12)eTAΠ(1)WTgˆ (cid:12) , (45) in the derivation of E (cid:12)gTb(1)(cid:12) , we obtain
2 (cid:12) k k+1(cid:12) (cid:12) k k (cid:12)
V3 (cid:44)E(cid:26)(cid:12)(cid:12)(cid:12)eTkAΠ(1)WTek+1(cid:12)(cid:12)(cid:12)2(cid:27). (46) E(cid:26)(cid:13)(cid:13)(cid:13)gkTC(1)(cid:13)(cid:13)(cid:13)2(cid:27)= MM(M+σ−k2+K1σ)σe2,2k(cid:37) . (54)
k+1
The term V can be computed as
1 Substituting (47), (52), (53), and (54) into (42) yields (32).
(cid:26)(cid:13) (cid:13)2(cid:27)
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b) Compute E(cid:26)(cid:12)(cid:12)(cid:12)gkTb(k1)(cid:12)(cid:12)(cid:12)2(cid:27): By expressing gk =gˆk+ek, I2E0E13E.Trans. Wireless Commun., vol. 61, no. 12, pp. 4847–4863, Dec.
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E(cid:26)(cid:12)(cid:12)(cid:12)gkTb(k1)(cid:12)(cid:12)(cid:12)2(cid:27)=E(cid:110)|I1+I2+I3|2(cid:111), (48) [7] Cm.asDsi.veHoM,IHM.OQ.wNithgom, aMx.imMumatt-hraatiioou,praoncdesTs.inQg.aDnduoinmgp,e“rMfecutltiC-wSIa.y”
[Online].Available:https://arxiv.org/pdf/1611.01042.pdf.
where I (cid:44) gˆTAΠ(1)WTe , I (cid:44) eTAΠ(1)WTgˆ , [8] T.Marzetta,E.Larsson,H.Yang,andH.Ngo,FundamentalsofMassive
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3 k k 1 2 3
mutually uncorrelated, we obtain tipair full-duplex relaying with massive arrays and linear processing,”
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(cid:26)(cid:12) (cid:12)2(cid:27) (cid:110) (cid:111) (cid:110) (cid:111) (cid:110) (cid:111) [10] A.M.TulinoandS.Verdu´,“Randommatrixtheoryandwirelesscommu-
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