Impact of User Mobility on Optimal Linear Receivers in Cellular Networks

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by  Anastasios K. Papazafeiropoulos

Impact of User Mobility on Optimal Linear
Receivers in Cellular Networks
Anastasios K. Papazafeiropoulos
Communications and Signal Processing Group, Imperial College London, London, U.K.
Email: [email protected]
Abstract—We consider the uplink of non-cooperative multi- limited prior work on massive MIMO has considered the
cellular systems deploying multiple antenna elements at the effect of channel aging due to user mobility. Specifically,
base stations (BS), covering both the cases of conventional in [8], Truong et. al derived deterministic equivalents for
and very large number of antennas. Given the inevitable pilot
the MRC (uplink) and maximum-ratio-transmission (MRT)
contamination and an arbitrary path-loss for each link, we
5 address the impact of time variation of the channel due to the for the downlink case, while in [9], [10], the more complex
1 relative movement between users and BS antennas, which limits deterministic equivalents for MMSE (uplink) and regularized
0 system’s performance even if the number antennas is increased, ZF (downlink) were obtained and a comparison among the
2 as shown. In particular, we propose an optimal linear receiver various strategies was provided. In the context of channel
(OLR) maximizing the received signal-to-interference-plus-noise
aging,theexactandapproximatesum-rateshavebeenobtained
n (SINR). Closed-form lower and upper bounds are derived as
a well as the deterministic equivalent of the OLR is obtained. in [9] for finite and large number of antennas, respectively,
J Numerical results reveal the outperformance of the proposed when zero-forcing (ZF) decoder is applied. The arising need
OLR against known linear receivers, mostly in environments istoderiveanOLRthataddressestheinevitablechannelaging
0
with high interference and certain user mobility, as well as occurring in time-varying channels.
2 that massive MIMO is preferable even in time-varying channel
conditions. Motivated by the above discussion, we hereafter derive the

] OLR in a cellular scenario subject to time-varying channel
T I. INTRODUCTION conditions, exploiting the correlation between the channel
I Multiple-input multiple-output (MIMO) systems have ap- estimates and the interference from other cells. Furthermore,
.
s peared as a method to increase the data rate and the spec- weelaborateonacomparisonamongvariousknownreceivers
c
tral efficiency over wireless links through multiplexing gain (MRC, ZF, and MMSE) to depict the outperformance of the
[
and spatial diversity [1]. However, the evolution of cellular proposedOLRandshedlightonhowtheusermobilityaffects

1 networks has resulted to a greedy need for higher data rates, the performance in such case. In fact, we obtain lower and
v achievedbymeansofmulti-userMIMO(MU-MIMO)capable upper bounds of the sum-rate, accounting for channel aging,
4 of exploiting multi-user diversity [2]. Despite that the channel with an OLR assuming a finite number of BS antennas. In
8 state information at the transmitter (CSIT) contributes little to case of large number of antennas, we derive the deterministic
8 the multiplexing gain in single-user scenarios, its knowledge equivalentapproximationoftheSINRthatgivesustheability
4
in MU-MIMO is crucial [3]. of avoiding lengthy Monte-Carlo simulations. Notably, it is
0
Nevertheless,anewbreakthroughtechnique,calledmassive of high importance to massive MIMO that in time-varying
.
1 MIMO or very large MIMO, makes it possible to apply low- channels they are still favorable, but the performance of the
0 complexity linear decoders to maximize system capacity by system is limited, even if the number of antennas is cranked
5 vanishingthermalnoise,intra-cellinterference,andestimation up, as shown by our results.
1 errors [4]. In massive MIMO cellular networks, a base station
v: (BS) deploys a large number of antennas, and multiple users II. SYSTEMMODEL
i canbeservedbythesametime-frequencyresource,sincemore
X degrees of freedom are provided. We consider a cellular uplink scenario, where the system is
r Unfortunately,thereuseof pilotsequencesinadjacentcells composed of L cells with one multi-antenna BS (N antennas)
a causespilotcontamination[4],[5],whichdegradesthequality andK single-antennausers(N >K)percell.Allusersshare
of CSIT. In addition, the commonly present mobility of the the same time-frequency resource, while we assume non-
users,implyingtime-varyingchannelconditions,inducesextra cooperative BSs. Taking into account a quasi-static block fad-
inaccuracy at the CSIT. As a consequence, the current CSIT ingmodel,wherethechannelsareconstantduringthesymbol
cannot be perfectly known but only estimated. period but vary from symbol to symbol, the channel vector
It has been shown that maximum-ratio combining (MRC), ggg [n] ∈ CN×1 between the kth user in the ith cell and the
lik
zero-forcing (ZF), and minimum mean-square error (MMSE) BSincelll atthenthtime-slotisdecomposedasindependent
perform the same in single-cell systems, when the throughput small-scale fading by means of hhh ∈ CN×1 ∼ CN (000,I )
lik N
peruseris1bitperchanneluse,butMMSEisingeneralopti- andlarge-scalefading(shadowfadingandpathloss)bymeans
mal [6]. The behavior of MMSE has been studied extensively of β , given that the antennas are sufficiently well separated
lik
in multi-cell networks, and in fact, in [7] an optimal linear re- in realistic systems. More concretely, we have
ceiver (OLR), achieving the maximum signal-to-interference-
(cid:112)
plus-noiseratio(SINR),hasbeenobtained.Nevertheless,some ggg [n]= β hhh [n]. (1)
lik lik lik
ThisresearchwassupportedbyaMarieCurieIntraEuropeanFellowship
In simple mathematical terms, the N × 1 frequency-flat
withinthe7thEuropeanCommunityFrameworkProgrammeforResearchof
theEuropeanCommissionundergrantagreementno.[330806],IAWICOM. received signal vector at the lth BS (l=1,2,...L) at the nth
time-slot is as follows wherevec(cid:16)EEE†[n](cid:17)∼CN (cid:0)000,I ⊗(cid:0)1−α2(cid:1)DDD (cid:1)andGGG [n−1]
li N li li
L are uncorrelated. They model the stationary Gaussian channel
√ (cid:88)
yyy [n]= p GGG [n]xxx [n]+zzz [n] error vector due to the time variation of the channel and the
l li i l
channelattheprevioussymbolduration.Theα=J (2πf T )
i=1 0 D s
parameter, where J (·) is the zeroth-order Bessel function of
L 0
=√p(cid:88)HHH [n]DDD1/2xxx [n], (2) the first kind, fD and Ts are the maximum Doppler shift and
li li i the channel sampling period, expresses the two-dimensional
i=1
isotropic scattering. It is worthwhile to mention that the
√
where pxxxi[n]∈CK×1 is the zero-mean stochastic transmit maximum Doppler shift fD equals fD = vcfc, where v (in
signal vector of K users allocated in the ith cell with p > 0 m/s) is the relative velocity of the user, c=3×108m/s is the
beingtheaveragetransmittedpowerperuser,nnnl istheadditive speed of light, and fc is the carrier frequency.
white Gaussian noise (AWGN) vector at the receiver (BS), The combination of (3) in (5) allows the characterization
(cid:2) (cid:3)
modeledasCN (000,IN),andGGGli[n]= gggli1[n],...,gggliK[n] ∈ of both effects at the same time instance, i.e., it provides the
CN×K denotes the complex combined matrix between all channelattime-slotn,accountingforpilotcontaminationand
users in cell i and BS l. Note that in (2), we have applied (1) time variation of the channel. Specifically, we have
in matrix form, i.e., where HHH [n] ∈ CN×K is the fast
li
fading matrix and DDD is a K × K diagonal matrix with GGG [n]=αGGG [n−1]+EEE [n]
li li li li
[DDDli]kk =βlik1. =αGGGˆ [n−1]+EEE˜ [n], (6)
The detection of the signals transmitted from K users by li li
theirassociatedBSnecessitatestheknowledgeofchannelstate where GGGˆ [n − 1] and EEE˜ [n] = αGGG˜ [n − 1] + EEE [n] ∼
information (CSI), which can be obtained by using uplink (cid:16) li (cid:16) li(cid:17)(cid:17) li li
pilots. Unfortunately, in realistic systems, the CSI at the BS CN 000,IN ⊗ DDDli−α2DDDˆli are mutually independent.
is degraded by pilot contamination. This effect arises due to We focus on the study of linear detection by the receiver
frequency reuse across a multiplicity of cells. According to (BS) by means of a linear N×K matrixWWW [n] dependent on
l
MMSE channel estimation, the channel can be written as [8] thechannelestimateGGGˆ [n]AfterapplyingthedetectorWWW [n]
li l
GGG [n]=GGGˆ [n]+GGG˜ [n], (3) to its received signal yyyl[n] at time-slot n, the lth BS obtains
li li li
where vec(cid:16)GGGˆ†li[n](cid:17) = HHHˆli[n]DDDˆl21i∼CN (cid:16)000,IN ⊗DDDˆli(cid:17) and rrrl =WWWHl[n]yyyl =√pWWWHl[n](cid:88)L GGGli[n]xxxi[n]+WWWHl[n]zzzl[n]
vec(cid:16)GGG˜†[n](cid:17)∼CN (cid:16)000,I ⊗(cid:16)DDD −DDDˆ (cid:17)(cid:17) are the indepen- i=1
li N li li L
dent channel estimate and channel estimation error matrices, =α√pWWWH[n](cid:88)GGGˆ [n−1]xxx [n]
respectively.NotethatHHHˆ [n]isaN×N matrixwithGaussian l li i
entries having zero meanliand unit variance as well as DDDˆ is i=1
li L
a(cid:80)KLj=1×ββll2jiKkk+1d/ipapg,ownahlermeaptrpixiswthitehteralenmsmenittspo[DDDwˆelir]kokf =eacβhˆlipkilo=t +√pWWWHl[n](cid:88)i=1EEE˜li[n]xxxi[n]+WWWHl[n]zzzl[n] (7)
symbol (pp =τp). Nevertheless, an important relationship, where in (7), we have used (6). Hence, the kth element ofrrrl,
going to help our derivations, concerns the estimated channel corresponding to the signal from the kth user, is given by
between the lth BS and the users in cell i. It is given by [7]
√
r =α pwwwH [n]gggˆ [n−1]x [n]
GGGˆ [n]=GGGˆ [n]RRR , (4) lk lk llk lk
li ll li
K L
where GGGˆli[n] (cid:44) [gggˆli1[n],...,gggˆliK[n]] ∈ CN×K is the esti- +α√p(cid:88)wwwHlk[n]gggˆllj[n−1]xlj[n]+α√p(cid:88)wwwHlk[n]GGGˆli[n−1]xxxi[n]
mated combined channel matrix from all users in cell i to BS j(cid:54)=k i(cid:54)=l
l and RRR =diag(cid:8)βli1,βli2,...,βliK(cid:9). L
Besidleis pilot coβnltla1mβinlla2tion, iβnllKany common propagation +√p(cid:88)wwwHlk[n]EEE˜li[n]xxxi[n]+wwwHlkzzzl[n],
scenario, a relative movement takes place between the anten- i=1
nas and the scatterers that degrades more channel’s perfor-
where x and www are the kth element and column of xxx
mance.Underthesecircumstances,thechannelistime-varying lk lk l
and WWW , respectively. Taking into account the independence
and can be modeled by the famous Gauss-Markov block l
between GGGˆ [n−1] and EEE˜ [n], the achievable uplink SINR
fadingmodel[12],whichisbasicallyanautoregressivemodel li li
SINR of the kth user can be written as in (8)2. Note that
of certain order that incorporates two-dimensional isotropic k
L
scattering(Jakesmodel).Morespecifically,ouranalysistargets RRR =(cid:88)E(cid:110)EEE˜ EEE˜H(cid:111), i.e.,
to relate the current channel state with its past samples. E˜ li li
For reasons of simplicity and computational complexity, we i=1
consider the following autoregressive model of order 1 [8]
L L K
RRR =(cid:88)e˜ I =(cid:88)(cid:88)(cid:104)β −α2βˆ (cid:105)I .
GGGli[n]=αGGGli[n−1]+EEEli[n], (5) E˜ li N lik lik N
i=1 i=1k=1
1Regarding shadowing, i.e., βlik, it is independent of symbol’s index n
becauseweassumethatthecoherencetimeislargerelativetoanylarge-scale 2Weassumeencodingofthemessageovermanyrealizationsofallsources
delayconstraintofthechannel. ofrandomnessinthemodelincludingnoiseandchannelestimateerror[7].
α2p|wwwH [n]gggˆ [n−1]|2
SINR = lk llk . (8)
k α2p(cid:80)K |wwwH [n]gggˆ [n−1]|2+p(cid:80)L wwwH [n]RRR www [n]+α2p(cid:80)L (cid:107)wwwH [n]GGGˆ [n−1](cid:107)2+(cid:107)wwwH [n](cid:107)2
j(cid:54)=k lk llj i=1 lk E˜ lk i(cid:54)=l lk li lk
III. OPTIMALLINEARRECEIVER Applicationoftheeigenvaluesdecompositiontothefollow-
ing N ×N matrix
In this section, following a similar procedure as in [7], we
derive the OLR maximizing the received SINR. Specifically, L
we have SSS[n−1]=(cid:88)GGGˆli[k][n−1]GGGˆHli[k][n−1]
i=1
SINRk = www|Hwww[Hlnk][ΞΞΞn]gggˆ[lnlk−[n1−]www1]|[2n] (=4)GGGˆll[k][n−1]RRRl[k]GGGˆHll[k][n−1]
(cid:13)(cid:13)wwwH [n]ΞΞΞl1k/2[n−k1](cid:13)(cid:13)2(cid:13)(cid:13)ΞΞΞl−k1/2[n−1]gggˆ [n−1](cid:13)(cid:13)2 (=1)HHHˆll[k][n−1]DDDˆl[k]HHHˆHll[k][n−1]
(cid:13) lk k (cid:13) (cid:13) k lllk (cid:13)
≤ (9)
wwwHlk[n]ΞΞΞk[n−1]wwwlk[n] with RRRl[k] = (cid:80)Li=1RRR2li[k] and DDDˆl[k] = (cid:80)Li=1DDDˆ2li[k] provides
=gggˆH [n−1]ΞΞΞ−1[n−1]gggˆ [n−1], (10) the eigenvalues in terms of a diagonal matrix BBB[n−1] and
llk k llk
thecorrespondingeigenvectorsbymeansofthecolumnsofthe
unitary matrix UUU as SSS[n−1]=UUUH[n−1]BBB[n−1]UUU[n−1].
where we have denoted that ΞΞΞ [n − 1] =
k
(cid:80)K gggˆ [n − 1]gggˆH [n − 1] + (cid:80)L GGGˆ [n − 1]GGGˆH[n − Given that the rank ofSSS[n−1] is K−1, the matrixBBB has
j(cid:54)=k llj llj i(cid:54)=l li li the form
(cid:16) (cid:17)
1]+ 1 (cid:80)L e˜ + 1 I as well as in (9) we have used
α2 i=1 li p N
N−K+1
Cauchy-Schartz’s inequality. According to its property, the
(cid:2) (cid:122) (cid:125)(cid:124) (cid:123)(cid:3)
equality holds whenwwwlk[n]=cΞΞΞ−k1[n−1]gggllk[n−1] for any BBB[n−1]=diag λ1,...,λK−1,0,...,0 [n−1],
nonzero c∈C. It is important to mention that ΞΞΞ [n−1] can
k
be also written as wherethefirstK−1termsrepresenttherespectiveeigenvalues
of SSS[n − 1] and the subscript [n − 1] in front of a matrix
L denotes its instance at the (n−1)th slot. Taking advantage of
ΞΞΞk[n−1]=(cid:88)GGGˆli[k][n−1]GGGˆHli[k][n−1]+σ2IN, (11) the eigenvalues decomposition and (11), we can express (10)
i=1 as a sum of two independent terms as follows
whereGGGˆli[k][n−1]isthematrixGGGˆli[n−1]withitskthcolumn (cid:16) (cid:17)H(cid:20) 1(cid:32)(cid:88)L 1(cid:33) (cid:21)−1
removed and σ2 = 1 (cid:16)(cid:80)L e˜ +1(cid:17). SINRk= UUU[n−1]gggˆllk[n−1] BBB[n−1]+α2 e˜li+p IN
α2 i=1 li p i=1
Remark 1: Given that the typical MMSE receiver is ×UUU[n−1]gggˆ [n−1]
llk
=(cid:2)g¯∗ ,...,g¯∗ (cid:3)
(cid:18) 1 (cid:18) 1(cid:19) (cid:19)−1 llk1 llkN [n−1]
wwwlk[n]=αGGGˆll[n−1]GGGˆHll[n−1]+ α2 ZZZl+p IN gggˆllk[n−1], ×(cid:20) g¯llk1 ,..., g¯llkK−1 ,g¯llkK,...,g¯llkN(cid:21)T
λ +σ2 λ +σ2 σ2 σ2
1 K−1 [n−1]
where the matrix ZZZ is deterministic and equals to
l =K(cid:88)−1 |g¯llkj|2 + 1 N−(cid:88)K+1|g¯ |2, (12)
λ +σ2 σ2 llkj+K−1
ZZZl =E(cid:104)EEE˜llEEE˜Hll(cid:105)+(cid:88)L E(cid:2)GGGliGGGHli(cid:3) (cid:124)j=1 (cid:123)j(cid:122) (cid:125) (cid:124) j=1 (cid:123)(cid:122) (cid:125)
i(cid:54)=l I1 I2
 
K L (cid:2) (cid:3)T
(cid:88) (cid:88) where UUU[n − 1]gggˆ [n − 1] = g¯ ,...,g¯ . The
= e˜li+ βlikIN, llk llk1 llkN [n−1]
multiplication ofgggˆ [n−1] with the unitary matrixUUU[n−1]
k i(cid:54)=l llk
preservesthestatisticalpropertiesoftheformerone.Moreover,
λ for j ∈ [1,K −1] and |g¯ | are independent, since λ
and the expression of the typical MRC iswww [n]=αgggˆ [n−1], j llkj j
lk llk representstheeigenvaluesofSSS[n−1],i.e.,GGGˆ [n−1]GGGˆH[n−1]
the SINR for the OLR and MMSE are expected to behave li li
with the kth column removed.
similarly in low interference conditions, while the proposed
OLR achieves higher sum-rate in interference-limited scenar- The sum-rate Rk of the kth user is written as
ios, as expected [7]. However, the main contribution rests on
the exhibition of the effect of the user mobility on the OLR, Rk =E[log2(1+SINRk)]. (13)
which is clearly described in (11). As can be seen, the SINRs
corresponding to all receivers under investigation coincide Becauseofthearisingdifficultytoobtaintheexactdistribution
after a certain value of α that makes the term multiplied by of the SINR , we resort to derive bounds of the sum-rate that
k
1/α2 negligible. prove to be very tight, as shown.
A. Upper Bound Substitution of (19) and (20) into (15) concludes the proof.
Proposition 1: The upper bound of R for user k in the
k B. Lower Bound
reference cell j with OLR, accounting for user mobility, is
Proposition 2: The lower bound of R for user k in the
k
(cid:18) K−1K−1
R ≤log 1+ 1 (cid:88)(cid:88) Dlvu reference cell j with OLR, accounting for user mobility, is
k 2 (cid:81)K−1(t −t ) Γ(N−K+u+1) given by
i<j lj li v=1u=1
×(cid:18)(−1)N−K+u−1(cid:0)σ2(cid:1)N−K+uetσl2vEi(cid:18)−tσlv2(cid:19) Rk≥log2(cid:18)1+2(K−1)exp(cid:104)−2γ− 2(cid:81)iK<−j11(tlj −tli)
+N−(cid:88)K+(ur−1)!(cid:0)−σ2(cid:1)N−K+u−r(cid:19)tr+K−N−2+(N−K−1)(cid:19), ×K(cid:88)−1K(cid:88)−1DlvutlKv−N−2(N−K+u)!
lv σ2 Γ(N −K+u+1)
r=1 v=1 u=1
(14)
where Ei(z) = −(cid:82)∞ e−t/tdt is the exponential integral ×(cid:18)ln(cid:0)σ2(cid:1)(cid:18)tσ2(cid:19)N−K+u+1etσl2v+N−(cid:88)K+uEir+1(cid:18)tσ2(cid:19)(cid:19)(cid:105)(cid:19), (21)
−x lv lv
function [16, Eq. (06.35.07.0002.01)]. In addition, t is the r=0
lv
vth element of the diagonal matrix DDDˆ and D is the where γ is Euler’s constant with numerical value γ (cid:39)
(v,u)th cofactor of a (K−1)×(K−l1[k)]matrix DDDlvu whose 0.577216 and Ei (z) = (cid:82)∞e−zt/tndt, n = 0,1,2,...,
l n 1
(i,j)th entry is {DDD } =tj−1. Re(z) > 0, is the exponential integral function of order
l i,j li
Proof: From (13), we have n [16, Eq. (06.34.02.0001.01)], which can be related to the
incomplete gamma function (cid:0)Ei (z)=xn−1Γ(1−n,x)(cid:1).
R ≤log (1+E[SINR ]) n
k 2 k Proof:Theproofreliesonthegeneralboundingtechnique
(=12)log (1+E[I +I ]), (15) according to which we can re-express (13), according to
2 1 2
where we have applied Jensen’s inequality, since log2(·) is a Rk =E(cid:2)log2(cid:0)1+exp(cid:0)ln(I1+I2)(cid:1)(cid:1)(cid:3)
conTchaeveexfupneccttiaotino.n in (15) can be calculated by taking the ≥E(cid:20)log2(cid:16)1+exp(cid:16)(cid:0)ln(2I1)+ln(2I2)(cid:1)(cid:17)/2(cid:17)(cid:21)
expectation of each term separately. Hence, the first term can
be written as ≥log (cid:18)1+exp(cid:16)E(cid:2)(cid:0)ln(2I )+ln(2I )(cid:1)(cid:3)/2(cid:17)(cid:19),(22)
2 1 2
(cid:20) |g¯ |2 (cid:21)
E[I ]=(K−1)E llk (16)
1 λ+σ2 where first we have used the arithmetic mean-geometric
(cid:20) (cid:21) mean inequality, and then we have exploited the fact that
1
=(K−1)E (17) log (1+exp(x)) is convex in x and thereafter applying
λ+σ2 2
Jensens inequality. The last expression is constituted by two
= 1 K(cid:88)−1K(cid:88)−1 Dlvu parts. Regarding the first part, we have
(cid:81)Ki<−j1(tlj −tli) v=1 u=1 Γ(N −K+u+1) E(cid:2)ln(2I )≥E(cid:104)K(cid:88)−1ln(cid:18)2(K−1)|g¯llkj|2(cid:19)(cid:46)(K−1)(cid:105) (23)
×(cid:90) ∞λN−K+ue−tλlvtKlv−N−2dλ, (18) 1 j=1 λj +σ2
0 λ+σ2 =E(cid:104)ln(cid:18)2(K−1)|g¯llk|2(cid:19)
= (cid:81)Ki<−j1(1tlj −tli)K(cid:88)v=−11Ku(cid:88)=−11Γ(N −DKlv+u u+1) =E(cid:104)ln(cid:0)2(K−λ+1)σ|g¯2llk|2(cid:1)−E(cid:104)ln(cid:0)λ+σ2(cid:1)(cid:105)
×(cid:18)(−1)N−K+u−1(cid:0)σ2(cid:1)N−K+uetσl2vEi(cid:18)−σ2(cid:19) =(cid:90) ∞ln(cid:0)2(K−1)xe−x(cid:1)dx− 1
N−K+u (cid:33) tlv 0 (cid:81)Ki<−j1(tlj −tli)
+ (cid:88) (r−1)!(cid:0)−σ2(cid:1)N−K+u−r tlrv+K−N−2, (19) ×K(cid:88)−1K(cid:88)−1 DlvutKlv−N−2
r=1 Γ(N −K+u+1)
v=1 u=1
where in (16), we have used that the terms of the sum are (cid:90) ∞
i.i.d. Given the independence between |g¯llk| and λ and that × ln(cid:0)λ+σ2(cid:1)λN−K+ue−tλlvdλ (24)
|g¯ |2 is an exponential variable with unit mean and unit 0
llk
variance,weleadto(17).KeepinmindthattheK−1nonzero =ln(2(K−1))−2γ
eeiiggeennvvaalluueess ooff SSSSSS[Hn[n−−1]1]h.aSvepetchieficsaalmlye, sduisbtsrtiibtuuttiionng athsethPeDNF − 1 K(cid:88)−1K(cid:88)−1DlvutlKv−N−2(N−K+u)!
of the unordered eigenvalue of SSSH[n−1], obtained by [13] (cid:81)Ki<−j1(tlj −tli) v=1 u=1 Γ(N −K+u+1)
iuwnsetionhg(a1v[71e)5,,,f(oE1lql8o.)w(3iis.n3go5b3tth.a5ei)n]se.adCm.oeTnhwceearynin,inttehggarttahleisnec(o1n8d) tiesrmsolivne(d15b)y, ×(cid:18)ln(cid:0)σ2(cid:1)(cid:18)tσlv2(cid:19)N−K+u+1etσl2v+N−(cid:88)K+uEir+1(cid:18)tσlv2(cid:19)(cid:19).(25)
r=0
E[I ]= (N −K−1)E(cid:2)|g¯ |2(cid:3)= (N −K−1). (20) Note that in (23), we have used again the arithmetic mean-
2 σ2 llk σ2 geometric mean inequality and in (24) the fact that |g¯ |2
llk
is exponential with unit mean as well as that λ is an spectral efficiency in bits/s/Hz is investigated. In particular,
unordered eigenvalue of semi-correlated Wishart distribution we study the total spectral efficiency given by
according to [13]. The last expression is obtained after us-
ing [15, Eq. (4.332.1), (3.351.3)] and [14, Eq. (47]. Similarly, (cid:16) τ (cid:17)(cid:88)K
E(cid:2)ln(I2)(cid:3) is obtained as R= 1− T Rk,
E(cid:2)ln(2I )(cid:3)=−2γ+ln(2(K−1)), k=1
2
where R is the sum-rate of user k in cell 1 and T is the
k
which when substituted with (25) into (22), returns (21). coherence time interval (in symbols). Note that D is given
1i
in terms of β expressing the effect of the interference from
IV. DETERMINISTICEQUIVALENTOFSINR
othercells.Ithastobestressedthatinallcasestheoutperfor-
The emerging technology of very large MIMO leads us to mance of the OLR is depicted. In addition, the deterministic
study the effect of finite Doppler shift on their performance. approximation of the sum-rate is indistinguishable comparing
Specifically, the following proposition quantifies the degrada- to the simulation results. It is worthwhile to mention that the
tion of the SINR due to the relative movement of users. sum-rate saturates for high SNR values (interference-limited
Proposition 3: The deterministic equivalent of SINRk for system), i.e., the system performance cannot be improved by
user k in the reference cell j with OLR, accounting for user usingmorepower,sinceincreaseofthetransmitpowerresults
mobility, is given by also to increase of the interference from other cells.
SINR (cid:16)trTTT βˆ , (26) In Fig. 1, the simulated achievable sum-rate of the OLR
k li llk along with its deterministic equivalent (26) are plotted against
where the average SNR p. In addition, the typical MRC and MMSE
TTT =(cid:32)(cid:88)K βˆlik + 1 (cid:32)(cid:88)L e˜ + 1(cid:33)(cid:33)−1I , dfeerceondceerscoanrfiegsuirmatuiolantsedacarsoswsedlilf.feAredndtrecseslilns,gi.dei.f,feβre=nt1inatenrd-
li 1+δ α2 li p N β =4,weobservethattheOLRpresentsthebestperformance,
lik
k=1 i=1 which increases as shown by the zoomed gap, when the
and δ =lim δ(t), where for t=1,2,... intercell interference increases. The explanation rests on the
lik t→∞ lik fact that the correlation between intercell interference and the
δl(itk) =(cid:88)K 1+βˆδli(jt−1) + α12 (cid:32)(cid:88)L e˜li+ p1(cid:33)−1βˆlikN cthheaInnDnoFelpigpe.lset2irm,sahtthiefetafreeffTeccotnosonidfevrtaherdeioubusysertrhecemesiovtrbeuriclsittuyisrebiloylufsmtthreaeateOndsLbRoyf.
j=1 lij i=1 d s
depicting the variation of the sum-rate with the normalized
with initial values δ(0) =α2(cid:14)(cid:16)(cid:80)L e˜ + 1(cid:17) for all k. Doppler shift. It can be observed that the Doppler shift
lik i=1 li p extensivelylimitstheperformanceofthechannel.Inparticular,
Proof: It is obtained by means of (10) as
the sum-rate decreases from almost 9.2 bits/s/Hz to zero
SINRk =gggˆHllk[n−1]ΞΞΞ−k1[n]gggˆllk[n−1] by following the form of the J0(·), while the Doppler shift
increases. Moreover, we observe that the gap between the
(cid:16)trΞΞΞ−1[n−1]βˆ (27)
k llk performances of the various receivers vanishes to zero for
(cid:16)trTTT βˆ , (28) values of α greater than 0.25. Nevertheless, increasing the
li llk
numberofantennasandkeepingtheotherparametersfixed,the
where the symbol (cid:16) relates two infinite sequences and is sum-rate becomes higher for all values of f T , as expected.
a.s. D s
equivalent to −−−−→ denoting almost sure convergence. As The tightness of the proposed bounds is demonstrated in
N→∞
far as the derivation is concerned, (27) follows from [17, Fig. 3, where (14), (21), and (26) are compared with the
Lemma B.26] as well as in (28) we have applied the rank- simulation results. We can easily observe that the proposed
1 perturbation lemma [18] and then [19, Theorem 1]. approximations are very tight across varying SNR.
The deterministic equivalent uplink sum-rate can be ob-
tained by means of the dominated convergence [20] and the VI. CONCLUSION
continuous mapping theorem [21] as
We have proposed an OLR capable of accounting for
(cid:0) (cid:1) a.s. not only the detrimental effects of pilot contamination and
R −log 1+SINR −−−−→0.
k 2 k path-loss, but also the critical time variation of the channel
N→∞
due to the movement of the users. Taking into account the
V. NUMERICALRESULTS
correlation between the channel estimate and the interference,
Thissectionprovidestherepresentationoftheimpactofthe
we showed the outperformance of the OLR with comparison
usermobility onthesum-rateas wellasthe assessmentofthe
to MRC, ZF, and MMSE by deriving closed-form lower and
proposed bounds and deterministic equivalent. The scenario
upper bounds of the sum-rate for finite N. Moreover, we
under consideration includes L = 7 cells with K = 10 users
obtained a deterministic equivalent of the SINR when both
per cell. For reasons of comparison, the large-scale fading
the numbers of users and antennas go to infinity with a given
matrix D ,i = 1,...,7 and the rest parameters, are chosen
1i ratio under the same channel conditions. Simulations witness
as in [7]3. Specifically, T = 196, τ = 10, p = p, and
p thetightnessoftheboundsaswellasthevalidityoftheresults.
SNR = p. Taking as reference cell 1, the behavior of the
It is worthwhile to mention that the deterministic equivalent
provides a tight approximation of the SINR, allowing us to
3This large-scale matrix models practical channel conditions in hexag-
avoid unbearable time consuming Monte Carlo simulations
onal cells with randomly distributed users under log-normal shadowing
(lnN(0,8))andpathlosswithexponentequalto4. when the number of BS antennas is very large.
Fig. 1. Sum spectral efficiency versus SNR for different interference Fig.3. Comparisonoftheproposedboundsandthedeterministicequivalent
configurations(β=1andβ=4). withthesimulationresults.
[9] A.PapazafeiropoulosandT.Ratnarajah“Linearprecodingfordownlink
massive MIMO with delayed CSIT and channel prediction” in Proc.
IEEEWCNC2014,Apr.2014,pp.821–826.
[10] A.PapazafeiropoulosandT.Ratnarajah“Uplinkperformanceofmassive
MIMOsubjecttodelayedCSITandanticipatedchannelprediction,” in
Proc.IEEEICASSP2014,May.2014,pp.3186–3189.
[11] A. Papazafeiropoulos, H. Q. Ngo, M. Matthaiou, and T. Ratnarajah
“Uplink performance of conventional and massive MIMO cellular
systems with delayed CSIT,” to appear in Proc. IEEE International
Symposium on Personal, Indoor and Mobile Radio Communications
(PIMRC),Washington,D.C.,September2014.
[12] G.Caire,N.Jindal,M.Kobayashi,andN.Ravindran,“MultiuserMIMO
achievablerateswithdownlinktrainingandchannelstatefeedback,” in
IEEETrans.Inf.Theory,vol.56,no.6,pp.2845–2866,Jun.2010.
[13] S. Jin, M. R. McKay, C. Zhong, and K.K. Wong, “Ergodic capacity
analysis of amplify-and-forward MIMO dual-hop systems,” in IEEE
Trans.Inf.Theory,vol.56,no.5,pp.2204–2224,May2010.
[14] H. Shin and J. Lee, “Capacity of multiple-antenna fading channels:
Spatial fading correlation, double scattering, and keyhole,” in IEEE
Trans.Inf.Theory,vol.49,no.10,pp.2636–2647,Oct.2003.
[15] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and
Products,7thed. SanDiego,CA:Academic,2007.
[16] Wolfram, The Wolfram functions site. [Online]. Available: http://
functions.wolfram.com.
Fig.2. SumspectralefficiencyversusnormalizedDopplershiftfordifferent [17] Z.D.BaiandJ.W.Silverstein, SpectralAnalysisofLargeDimensional
numbersofBSantennas(N =50andN =100). RandomMatrices,2nded.SpringerSeriesinStatistics,NewYork,NY,
USA,2009.
[18] J. W. Silverstein and Z. D. Bai, “On the empirical distribution of
eigenvaluesofaclassoflargedimensionalrandommatrices,”inJournal
REFERENCES ofMultivariateAnalysis,vol.54,no.2,pp.175192,1995.
[19] S.Wagner,R.Couillet,M.Debbah,andD.T.M.Slock, “Largesystem
[1] I.E.Telatar,“Capacityofmulti-antennaGaussianchannels” Eur.Trans. analysis of linear precoding in correlated MISO broadcast channels
Telecommun.,vol.10,no.6,pp.585–595,Nov.-Dec.1999. under limited feedback,” IEEE Trans. Inf. Theory, vol. 58, no. 7, pp.
[2] D.Gesbert,M.Kountouris,R.W.HeathJr.,C.B.Chae,andT.Sa¨lzer, 4509–4537,Jul.2012.
“ShiftingtheMIMOparadigm,” IEEESig.Proc.Mag.,vol.24,no.5, [20] P.Billingsley, “ProbabilityandMeasure,” 3rded.,JohnWiley&Sons,
pp.36–46,2007. Inc.,1995.
[3] N. Jindal, “MIMO broadcast channels with finite-rate feedback,”IEEE [21] A. W. van der Vaart, “Asymptotic Statistics (Cambridge Series in
Trans.Inform.Theoryvol.52,no.11,pp.5045–5059,Nov.2006. StatisticalandProbabilisticMathematics),”CambridgeUniversityPress,
[4] T.L.Marzetta,“Noncooperativecellularwirelesswithunlimitednum- NewYork,2000.
bersofbasestationantennas,” IEEETrans.WirelessCommun.,vol.9,
no.11,pp.3590–3600,Nov.2010.
[5] J. Jose, A. Ashikhmin, T. L. Marzetta, and S. Vishwanath, “Pilot
contaminationandprecodinginmulti-cellTDDsystems,” IEEETrans.
WirelessCommun.,vol.10,no.8,pp.2640–2651,Aug.2011.
[6] H.Q.Ngo,E.G.Larsson,andT.L.Marzetta,“Energyandspectraleffi-
ciencyofverylargemultiuserMIMOsystems,”IEEETrans.Commun.,
vol.61,no.4,pp.1436–1449,Apr.2013.
[7] H.Q.Ngo,M.Matthaiou,andE.G.Larsson,“Performanceanalysisof
largescaleMU-MIMOwithoptimallinearreceivers,”SwedishCommun.
Tech.Workshop(Swe-CTW),pp.59–64,Oct.2012.
[8] K.T.TruongandR.W.Heath,Jr.,“EffectsofchannelaginginMassive
MIMO Systems,” IEEE/KICS J. Commun. Netw., vol. 15, no. 4, pp.
338–351,Aug.2013.