arX

iv:1

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0v1

[cs.

IT]

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1

Distributed Channel Estimation and PilotContamination Analysis for Massive MIMO-OFDM

SystemsAlam Zaib, Mudassir Masood, Anum Ali, Weiyu Xu and Tareq Y. Al-Naffouri

Abstract—Massive MIMO communication systems, by virtueof utilizing very large number of antennas, have a potentialtoyield higher spectral and energy efficiency in comparison withthe conventional MIMO systems. In this paper, we consideruplink channel estimation in massive MIMO-OFDM systemswith frequency selective channels. With increased numberof antennas, the channel estimation problem becomes verychallenging as exceptionally large number of channel parametershave to be estimated. We propose an efficient distributed linearminimum mean square error (LMMSE) algorithm that canachieve near optimal channel estimates at very low complexityby exploiting the strong spatial correlations and symmetry oflarge antenna array elements. The proposed method involvessolving a (fixed) reduced dimensional LMMSE problem at eachantenna followed by a repetitive sharing of information throughcollaboration among neighboring antenna elements. To furtherenhance the channel estimates and/or reduce the numberof reserved pilot tones, we propose a data-aided estimationtechnique that relies on finding a set of most reliable datacarriers. We also analyse the effect of pilot contaminationonthe mean square error (MSE) performance of different channelestimation techniques. Unlike the conventional approaches,we use stochastic geometry to obtain analytical expressionfor interference variance (or power) across OFDM frequencytones and use it to derive the MSE expressions for differentalgorithms under both noise and pilot contaminated regimes.Simulation results validate our analysis and the near optimalMSE performance of proposed estimation algorithms.

Index Terms: Channel estimation, massive MIMO, stochasticgeometry, OFDM, LMMSE.

I. I NTRODUCTION

In wireless communications, the demand for higher datarates has been dramatically increasing mostly owing to the un-precedented usage of data-hungry devices e.g., smart-phones,super-phones, tablets etc., for wireless multimedia applications[1]. Over the years, the MIMO technology (that exploitsmultiple antennas at the transmitter and/or receiver) has playeda pivotal role in sustaining the increased data rates. Installingmultiple antennas offers key advantages such as multiplexinggain and diversity gain due to increased spatial reuse [2],[3]. The MIMO technology has already been incorporatedinto many wireless products and standards such as WiFiIEEE802.11n [4], WiMAX IEEE 802.16e [5], LTE (4G) [6].

Recently, it was established that the use of very largeantenna arrays, typically of the order of few hundreds, atthe base station (BS) can potentially provide huge gains insystem throughput, energy efficiency, security and robustnessof wireless communication systems [7]. Such systems, known

as massive MIMO or large scale MIMO systems [8]–[10],overcome many limitations of traditional MIMO systems.Massive MIMO increases system capacity by simultaneouslyserving tens of users using the same time-frequency resources.Moreover, the large number of low power active antennasallows to focus energy in a small spatial region by forminga sharp beam towards desired users. This additionally impliesthat there will be little intra-cell interference [9]. Because ofthese vital advantages, massive MIMO has attracted a lot ofresearch interest and is envisioned as an enabling technologyfor next generation (5G) wireless communications [11].

Hand in hand with the advantages are entirely new researchchallenges that need to be tackled for massive MIMO. Thebottleneck in achieving the full advantages of massive MIMOis the accurate estimation of the channel impulse response(CIR) for each transmit-receive antenna pair. Having a verylarge number of antennas means that a significant number ofchannel coefficients need to be estimated− far more than thatcould be handled by traditional pilot-based MIMO channelestimation techniques (see [12] and references therein). Inthis regard, Bayesian minimum mean square error (MMSE)estimator provides an optimal estimate in the presence of ad-ditive white Gaussian noise (AWGN). The method is complexand therefore, a number of approaches have been developedto reduce its complexity such as those proposed in [13]–[17]. Unlike the least squares (LS) or interpolation basedtechniques [18], the MMSE estimation has a clear edge inthat it can effectively utilize the channel statistics to improvethe estimation accuracy. However, the direct generalization ofthese techniques to massive MIMO has some drawbacks. Inparticular, they suffer from huge complexity due to matrixinversion of very large dimensionality, making it impractical.Some methods to reduce the complexity of MMSE estimatorin massive MIMO have also been proposed e.g., [19]–[24].It is important to note that most of the existing methodsmake assumptions that are not always true. For example,many methods deal with flat fading channels only while othersassume that the channels are sparse. Therefore, low complexitychannel estimation approaches suited to multi-cell and multi-carrier massive MIMO systems need further investigations.

In this paper we propose a distributed algorithm for theestimation of correlated Rayleigh fading channels in massiveMIMO-OFDM systems. The novel distributed LMMSE algo-rithm significantly reduces the computational complexity whileattaining near optimal CIR estimates. The distributed approachis inspired by our previous work in [25] (where channels are

2

assumed to be sparse and exhibit common support with theneighboring antennas). Furthermore, in order to enhance theestimation performance, we also propose a data-aided estima-tion technique that relies on finding a set of most reliable datacarriers to increase the number of measurements, instead ofincreasing the reserved pilot tones [26]. Equivalently, byusingthe data-aided technique, the number of reserved pilot tonescan be reduced to attain a performance that is comparable topilot-based estimation, thus increasing the spectral efficiency.

In a multi-cell setting, allocation of orthogonal pilot se-quences for all users cannot be guaranteed due to finite coher-ence time of the channel and the limited available bandwidth[7]. Therefore, it is inevitable to reuse the pilot sequencesacross the cells. One of the major consequences of pilots reuseis that when the BS in a cell is performing channel estimationvia uplink training, the channel estimates will be severelydistorted (contaminated) by the pilots of the neighboring cellusers. The impact of pilot contamination on channel estimationis far greater than AWGN. In fact, it was shown in [27] thatthe effect of uncorrelated interference and fast Rayleigh fadingdiminishes as the number of BS antennas increase while theeffect of pilot contamination is not eliminated. Hence, it isimportant to investigate the effect of pilot contaminationonMSE performance of different channel estimation techniques.Although the effect of pilot contamination on system per-formance has been analysed by many researches e.g., [28],[29], only few studies have analysed its impact on channelestimation performance [20]. Moreover, in these works, theanalysis is carried out for fixed locations of (interference)users. Also it can be seen from analytical expressions derivedin these works, that the pathloss, which is determined by user'slocations, plays an important role in MSE performance eval-uation. As such, the above works cannot analytically answerhow the randomness of users’s locations would effect MSEperformance under pilot contamination. In contrast to existingstudies, we approach the problem by using concepts fromstochastic geometry. By assuming that the interfering users aredistributed according to homogeneous poisson point process(PPP), we derive analytical expressions for MSE of LS andLMMSE based channel estimation algorithms in the presenceof both AWGN and pilot contamination. The analytical resultsare validated by simulations. The results clearly show thedependence of important massive MIMO network parameters,such as pathloss and user's density, on the MSE performanceand give clue to mitigate the effect of pilot contamination.Itis shown that the increasing the number of pilots does notimprove the estimation performance in the presence of pilotcontamination. Moreover, the dependence of MSE on antennaspatial correlations suggests that the massive antenna arraystructure could be optimized to slightly improve the estimationperformance under pilot contamination.

The remainder of the paper is organized as follows. SectionII describes the system and spatial channel correlation model.In Section III, we present the MMSE and LS based chan-nel estimation in the presence of AWGN only and discusstheir limitations for massive MIMO. The proposed distributedLMMSE algorithm is presented in Section IV. To enhanceestimation performance, the data-aided approach is consid-

ered in Section V. Section VI describes the effect of pilotcontamination on channel estimation, and the expression forinterference correlation is presented. Based on this, the MSEexpressions for different algorithms are derived under AWGNand pilot contamination. Simulation results are presentedinSection VII and finally we conclude in Section VIII.

A. Notations

We use the lower case lettersx and lower case boldfacelettersx to represent the scalar and the (column) vector re-spectively. Matrices are denoted by upper case boldface lettersX whereas the calligraphic notationX is reserved for vectorsin the frequency domain. Theith entry ofx is represented byx(i), the element ofX in ith row andjth column is denotedby xi,j and the vectorxk represents thekth column ofX. Weusex(P) to denote a vector formed by selecting the entriesof x indexed by setP andX(P) to denote a matrix formedby selecting the rows ofX indexed byP . We also useXij torefer to the(i, j)th block entry of a block matrix. Further,(.)T,(.)∗ and (.)H represent transpose, conjugate and conjugatetranspose (Hermitian) operations respectively. We usediag(x)to transform a vectorx into a diagonal matrix with the entriesof x spread along the diagonal.〈X (k)〉 denotes the harddecoding i.e., maximum likelihood (ML) decision ofX (k).E. represents the statistical expectation. The discrete Fouriertransform (DFT) and inverse DFT (IDFT) matrices are repre-sented byF andFH respectively, where we the(l, k)th entry ofF is defined asfl,k=N−1/2e−2πlk/N , l, k=0, 1, 2, · · · , N−1for anN -dimensional Fourier transform. Finally, the weightednorm of a vectorx is given by‖x‖2

A, xHAx.

II. SYSTEM MODEL

We consider a multi-cell massive MIMO-OFDM wirelesssystem as shown in Fig. 1, where the BS in each cell isequipped with uniform planar array (UPA) consisting of alarge number of antennas. Moreover, we assume that each BSserves a number of single antenna user terminals. The antennason UPAs are distributed acrossM rows andG columns withhorizontal and vertical spacing ofdx anddy respectively. Wedefine the(m, g)th antenna as the antenna element inmthrow andgth column which corresponds tor=m+M(g−1)thantenna index where1 ≤ m ≤ M , 1 ≤ g ≤ G and1 ≤ r ≤ R,whereR=MG is the total number of antennas in a UPA. Fig.2 shows an example of aM×G UPA structure with antennaindexing. Note that, depending on values ofG and M , theantennas could have linear or a rectangular configuration. Wehowever, confine our attention to rectangular UPA structurewhich is a viable configuration in deployment scenarios formassive MIMO [9].

Each user communicates with the BS using OFDM andtransmits uplink pilots for channel estimation. We assume thatall users in a particular cell are assigned orthogonal frequencytones so that there is no intra-cell interference. However,dueto necessary reuse of pilots, there are users in the neighboringcells that transmit pilots at the same frequency tones, resultingin an inter-cell interference or pilot contamination. Sinceonly the user in a particular cell of interest will experience

3

Cell 1

Cell 2

Pilot

contamination

Massive UPA

antenna array

UE 3

UE 2

UE 1

UE 1

UE 2

Cell 3UE 3

UE 2

UE 1

Pilot

contamination

Figure 1: Multi-cell massive MIMO system layout.

1

2

3

M+1

M+2

M+3

M 2M

4 M+3

Central antenna r

Up neighbor rU

Right neighbor rR

Left neighbor rL

Down neighbor rD

GM

Linear indices of

antennas

dx

dy

Figure 2: An example ofM ×G UPA structure with antennaindexing.

interference from the users of neighboring cells that sharepilots at the same frequency tones, hence without loss ofgenerality, it suffices to consider one user per cell with allusers transmitting pilots at same OFDM frequency tones.

A. Channel Model

In the discussion that follows, we assume that there is nointer-cell interference and thus focus on a single-cell single-user scenario (the case of multi-cell will be treated in sectionVI further ahead). Further, we assume a multi-path channelbetween user and receive antennar modeled by a GaussianL-tap CIR vector. Specifically, the channel between user andantennar is defined byhr, [hr(0), hr(1), · · · , hr(L − 1)]

T

wherehr(l) ∈ C represents thelth tap complex channel gain.We append all the CIR vectors from a user to theR antennasof the BS to form anRL dimensional composite channelvectorh,

[

hT1 ,h

T2 , · · · ,hT

R

]T. Further, we collect thelth tap

of all transmit-receive pairs to form anR dimensionallth tapvectorh(l), [h1(l), h2(l), · · · , hR(l)]

T. Then, theRL × RLdimensional composite channel correlation matrix can bewritten as,

Rh , EhhH = Rarray ⊗Rtap , (1)

which is the kronecker product (⊗) of two components: (i)The R×R dimensional antenna spatial correlation matrix,Rarray=Eh(l)h(l)H, ∀l=0, 1, · · · , L − 1, which representsthe correlation among thelth taps across the array and(ii) The L×L dimensional channel tap correlation matrix,Rtap=Ehrh

Hr , ∀r=1, 2, · · · , R, which represents the cor-

relation among the CIR taps that depends on channel power

delay profile (PDP). As manifested by (1),Rarray is assumedto be identical across thel taps whileRtap is assumed to beidentical across the array. For the spatial correlation matrixRarray, we adopt a ray-based 3D channel model from [30]which is more appropriate for rectangular arrays. Accordingly,the spatial correlation between array elementsr=(m, g) andr′=(p, q) is given by,

[Rarray]r,r′ =D1√D5

e−D7+(D2(sinφ)σ)2

2D5 eD2D6D5 , (2)

where theDi's are defined as,

D1 = e2πdx

ν(p−m)cos(θ)e−

12 (ξ

2πdxν

)2(p−m)2sin2θ ,

D2 =2πdxν

(q − g)sin(θ) ,

D3 = ξ2πdxν

(q − g)cos(θ) ,

D4 =1

2

(

ξ2π

ν

)2

(p−m)(q − g)sin(2θ) ,

D5 = (D3)2(sin(φ)σ)2 + 1 ,

D6 = D4(sin(φ)σ)2 + cos(φ) ,

D7 = (D3)2cos2φ− (D4)

2(sin(φ)σ)2 − 2D4cosφ .

Here, ν is the carrier-frequency wavelength in meters,φand θ are the mean horizontal angle-of-departure (AoD) andthe mean vertical AoD in radians respectively,σ and ξ arethe standard deviation of horizontal AoD and the standarddeviation of vertical AoD respectively. As shown in [30], thespatial correlation matrix can be well approximated as,

Rarray ≈ Raz ⊗Rel , (3)

whereRaz and Rel are the correlation matrices in azimuth(horizontal) and elevation (vertical) directions, havingdimen-sions(M×M) and (G×G) respectively and are defined as,

[Rel]m,p = e2πdx

ν(p−m)cos(θ)e−

12 (ξ

2πdxν

)2(p−m)2sin2θ ,

[Raz]g,q =1√D5

e−

D23cos2φ

2D5 eD2cosφ

D5 e− 1

2(D2σ)2

D5 .

B. Signal model

We assume that there areN OFDM sub-carriers and letX represent theN -dimensional information symbol whoseentries are drawn from a bi-dimensional constellation e.g.,Q-QAM. The equivalent time-domain symbol is obtained bytaking inverse Fourier transform i.e.,x=FHX . The time-domain symbol is then transmitted after inserting a cyclicprefix (CP) of length at leastL−1 to avoid inter-symbol-interference (ISI). After removing the CP at the receiver,the frequency-domain OFDM symbol atrth antenna can berepresented as,

Yr = diag(X )Hr +Wr , (4)

where,Wr is frequency domain AWGN vector of zero meanand covarianceRw=σ2

wIN andHr is the channel frequency

4

response between the user and receive antennar i.e.,

Hr =√NF

[

hr

0N−L×1

]

=√NFhr . (5)

Here F is truncated Fourier matrix formed by selecting thefirst L columns ofF. Using (5), we can re-write (4) as,

Yr =√Ndiag(X )Fhr +Wr = Ahr +Wr , (6)

whereA,√Ndiag(X )F and the noise vectorWr is assumed

to be uncorrelated with the channel vectorhr. We assumethatK sub-carriers are reserved for pilots and the remainingN −K for the data transmission. Further, it is best to allocatethe pilots uniformly as shown in [31]. Hence, for a set of pilotindices denoted by vectorP , the system equation (6) reducesto,

Yr(P) = A(P)hr +Wr(P) , (7)

whereYr(P) andWr(P) are formed by selecting the entriesof Yr andWr indexed byP while A(P) is aK ×L matrixformed by selecting the rows ofA indexed byP .

We can now collect the pilot measurements (7) received byall antennas into a single system of equations as follows,

Y(P) = [IR ⊗A(P)]h+W(P) , (8)

where, Y(P)=[

YT1 (P), · · · ,YT

R(P)]T

, W(P) =[

WT1 (P), · · · ,WT

R(P)]T

, IR represents anR × R identitymatrix andh, as defined earlier, represents the compositechannel vector from user to the BS. For convenience, weassume the noise variance to be identical across the array sothat W(P) ∼ CN (0,Rw=σ2

wIRK). Note that the numberof unknown channel coefficients in (8) areRL whereas thetotal number of equations areRK. Therefore, a necessarycondition to solve (8) forh (and also (7) forhr) using leastsquares, is that the number of pilots be at least equal toLi.e., K ≥ L. However,K could be reduced if we utilize thecorrelation information. With the models defined above, weare ready to estimate the CIRs between the user and each BSantenna. We pursue different approaches that can be adoptedfor channel estimation in massive MIMO setup depending onwhether the information processing takes place independentlyat each antenna element or jointly at a centralized processor.We start with naive LMMSE and LS based techniques anddiscuss their limitations, and then propose a new distributedapproach in section IV which is further extended in sectionV with the help of data-aided approach.

III. LMMSE AND LS BASED CHANNEL ESTIMATION

In this section, we present three different techniques forchannel estimation in massive MIMO-OFDM based on thewell-known LMMSE and LS estimators and discuss theirlimitations. For now, we assume that estimates are corruptedonly by the white noise. Hence, without loss of generality, weconsider a single-cell single-user scenario for the approachespresented below.

A. The Localized LMMSE (L-LMMSE) estimation

In this approach, all CIRs are estimated independentlybased on the observations received at each antenna elementby using the classical LMMSE estimation. Using the linearsystem model in (7), the LMMSE estimate ofhr is obtainedby minimizing the (local) MSE,E‖hr−hr‖2, over hr asfollows [32]

hr =(

R−1tap +AHR−1

w A)−1

AHR−1w Yr , (9)

where we drop the index vectorP for convenience. Similarly,it follows that the (minimum) MSE is,

mser = trace(

R−1tap +AHR−1

w A)−1

. (10)

The overall global MSE is obtained by taking summationover all array elements i.e.,MSE(L)=

∑Rr=1mser, which after

simplifying (10), can be expressed as,

MSE(L) = R

L∑

i=1

(

δi1 + ρKδi

)

, (11)

where δiLi=1 are eigenvalues ofRtap, ρ , Ex/σ2w is the

SNR with Ex representing the average signal energy persymbol and the superscript(L) indicates L-LMMSE. Observefrom (11) that channel delay spreadL, has an adverse effecton MSE performance, which can be reduced by increasingthe number of pilot tones. The computational complexity of L-LMMSE is of the orderO

(

RL3)

(see Table I), which increaseslinearly with the number of BS antennas. However, the CIRestimates are not optimal in the sense of minimizing the overallor global MSE. The estimates would have been optimal, hadthe antennas been placed sufficiently apart so that the channelvectors were effectively uncorrelated. But for massive MIMOwith extremely large number of antennas, it is expected thatantennas are located in close proximity, so the channel vectorsare highly likely to be correlated with each other.

B. The Optimal LMMSE (O-LMMSE) Solution

In this strategy all the channel vectors are estimated simulta-neously by minimizing the global MSE,E‖h−h‖2 over thecomposite channel vectorh. This could be realized by sendingall observations to a central processor and then invoking theLMMSE estimation based on the composite system model in(8). The solution to this problem is given by,

h =(

R−1h

+ AHR−1w A

)−1

AHR−1w Y , (12)

where,A=IR ⊗A, Rh is as given in (1) and for notationalconvenience we dropped the indexP . The corresponding MSEis,

MSE(O) = trace(

R−1h

+ AHR−1w A

)−1

, (13)

which can be simplified to yield,

MSE(O) =

R∑

j=1

L∑

i=1

ηjδi1 + ρKηjδi

, (14)

where, ηj and δi are eigenvalues ofRarray and Rtap re-spectively. By comparing (14) with (11), we conclude that

5

in presence of spatial correlation, the optimal solution yieldsbetter MSE performance than the localized strategy, however,it has the following two major drawbacks:

1) Realization of optimal strategy requires global sharingof information to/from the central processor that resultsin communication overhead (as it requires complexsignalling which can be very expensive).

2) As evident from (12), the computation of optimalLMMSE requires inverting a non-trivial matrix of veryhigh dimension (RK×RK) that leads to computationalcomplexity of orderO

(

R3L3)

, which is cubic in numberof BS antennas.

In massive MIMO scenario whereR is of the order of fewhundreds, both of the above mentioned operations are veryexpensive and possibly impractical.

C. Estimation using Least Square (LS)

If the channel statistics are unknown, one can employ simpleLS based estimation. In the absence of correlation, we canlet the inverse of channel correlation matrix go to zero, i.e.,R−1

tap → 0, thereby ignoring the channel statistics. Therefore,the localized LS solution from(9) is,

hlsr =

(

AHA)−1

AHYr , (15)

and the resulting MSE is given by,

mselsr = trace(

AHR−1w A

)−1. (16)

In this case, the overall MSE simplifies to,

MSE(LS) =

R∑

r=1

mselsr =RL

ρK. (17)

Comparing (17) with (11), we conclude that LS has poorperformance in comparison with the LMMSE as it does notutilize the channel statistics. It is for this reason that thecentralized LS (C-LS) solution would achieve the same MSEperformance as the localized one as shown below.

MSE(C−LS)=trace(

(IR ⊗A)H(IR ⊗Rw)

−1(IR ⊗A)

)−1

= trace(

IR ⊗AHR−1w A

)−1,

=

R∑

r=1

trace(

AHR−1w A

)−1,

= MSE(LS) ,

where we have used the Kronecker product identities,(A ⊗B)(C⊗D)=AC⊗BD and (A⊗B)−1=A−1 ⊗B−1.

In short, the L-LMMSE estimation has the advantage oflow complexity (and better performance than LS) but it isunable to exploit the strong spatial correlation among antennaelements which is inevitable in massive MIMO systems. Onthe other hand, O-LMMSE exploits the spatial correlations butat a significantly higher computational cost. This motivatesus to propose a method that can overcome the shortcomingsof aforementioned techniques without affecting the estimationquality. Specifically, we propose a distributed estimationofCIRs based on antenna coordination that attains near optimal

performance with tractable complexity. The proposed dis-tributed LMMSE estimation is described below and is furtherextended in section V via a data-aided technique.

IV. T HE PROPOSED DISTRIBUTEDLMMSE (D-LMMSE)ESTIMATION

It is well known from equivalence results in linear esti-mation theory [33] that the O-LMMSE solution (12) couldbe alternatively obtained by solving anRL dimensional opti-mization problem,

argminh

‖Y −A′h‖2R

−1w

+ ‖h‖2R

−1h

, (18)

where all the variables are as defined earlier. Instead ofsolving (18) globally (as done earlier), we aim to solve itin a distributed manner overR antennas in which therthantenna has access toYr only. Moreover, the antennar isinterested only in determining its own CIR (i.e.,hr) withoutworrying about otherhj 's. Here, we would like to mention thatthis problem is fundamentally different from those consideredin the context of adaptive networks [34]. Also, most of theexisting distributed estimation techniques in adaptive networksdeal with single task problems in which all nodes in thenetwork estimate a single common parameter of interest.Furthermore, they rely on full cooperation between the nodes,i.e., exchanging both the estimates and the observations withthe neighbors. Although, the distributed recursive least squares(RLS) algorithm of [35] might be adopted to solve (18), itwould be gravely complex in number of dimensions (due tolarge R in massive MIMO and large channel delay spread)and hence might suffer from convergence issues. Our proposedsolution, the distributed LMMSE (D-LMMSE) algorithm, aswill become clear, is much simpler in that it exploits thestructure of spatial correlation matrixRarray and relies onlyon exchanging the (partial) weighted estimates of CIRs withimmediate neighbors, thus reducing the communication andcomputational cost significantly. The proposed D-LMMSEalgorithm is composed of three main steps namely the esti-mation, sharing and update, as explained below.

A. Estimation

In the estimation step, each antenna acting as a centerantennarC , estimates not only its own CIR but also the CIRsof its neighborhood. The neighborhood ofrC consists of 4-direct neighbors represented by the setN=rL, rR, rU , rD1

on the left, right, top and bottom positions respectively asshown in Fig. 3(a). Also, let the corresponding channel vectorsbe represented byhC , hL, hR, hU andhD respectively andlet hc represent|N+|L × 1 dimensional composite channelvector of the central antenna and its|N | direct neighbors (i.e.,hc=

[

hTC ,h

TL,h

TR,h

TU ,h

TD

]T). During the estimation process,

each antenna acting as a central antenna computes the estimateof hc by solving a reduced dimensional weighted least squares

1Note that for elements lying at the edges of a UPA, the number ofneighbors are different, so that2 ≤ |N | ≤ 4. The set of neighbors includingthe central antenna is represented byN+.

6

(WLS) optimization problem,

hc = argminhc

‖YC(P)−A(P)hC‖2R−1w

+ ‖hc‖2R

−1hc

, (19)

where YC(P) represents pilot observations at the centralantenna,Rhc is channel correlation matrix defined asRhc ,

Ehc(hc)H andRw=σ2wIK is the noise covariance matrix

at the central antenna. From (19) it is clear that informationis processed locally at each antenna as each antenna usesonly its own observations and interacts with its neighborhoodonly through Rhc (it is assumed that the central antennahas available correlation information of its neighborhoodtoconstructRhc). The solution to the above WLS minimizationproblem can be obtained by first re-writing (19) explicitly interms ofhc as,

hc = argminhc

∥

∥Y − Ahc∥

∥

2

R−1w

+ ‖hc‖2R

−1hc

, (20)

where,Y=YC(P) andA=[

A(P) 0K×L|N |

]

. Then, by in-voking the equivalence between LMMSE and WLS estimationproblems we obtain,

hc =(

R−1hc + AHR−1

w A)−1

AHR−1w Y . (21)

We definePc , (Cce)

−1 as the inverse of error covariancematrix at the central element, which is given by the expression,

Pc = R−1hc + AHR−1

w A . (22)

Then, by using (22) into (21), the weighted estimate ofcomposite channel at each antenna is simply,

hcw = Pchc = AHR−1

w Y . (23)

This weighting of the estimates asserts that we put moreconfidence into the estimates which are more reliable and viceversa. The estimation step is non-recursive and is computedonce for all antennas in the array. Thus, having found theP matrices in (22) and the weighted estimates in (23), eachantenna is ready to initiate sharing.

B. Sharing

The sharing step is the key to the proposed distributedalgorithm where the information is shared through collabo-ration between antennas. Let us define the sub-vectorhwj

of composite vectorhk

w as a (weighted) CIR estimate ofantennaj (i.e. the vectorhj) computed by the antennak.In sharing step, each antenna acting as a central element,shares only the partial information with its neighbors such

that the antennak, with composite vectorhk

w, would shareonly the selected components; its own (weighted) estimatehwk and the (weighted) estimatehwj, j ∈ N , with its jthneighbor. Henceforth, the shared vectors will be termed aspartial vectors and represented by an underlined notation.Anexample of how this sharing takes place is also depicted in Fig.3(b) for a3× 4 array with central elementrC=1 having onlytwo neighbors;N=rR=4, rD=2. As shown, each of theneighboring element shares only two sub-vectors (i.e., partialinformation) of its composite vector with the central antenna.The collaboration between the rest of the array elements takes

place in a similar fashion.As a result of information sharing, each antenna acting as

a central noderC receives|N | partial vectors,hj

w, j ∈ N ,from its neighbors, each of dimension|N+|L× 1 and havingonly two non-zero components;hwj andhwc. For the examplein Fig. 3(b), the composite vector of the central node and thepartial vectors received from its neighbors are given as follows,

h1w =

hw1

hw4

hw2

, h4

w =

hw1

hw4

0

and h2

w =

hw1

0

hw2

. (24)

Note that the estimates which are not shared have beenassigned as null vectors.

C. Update

Having received the (partial) LMMSE estimates from theneighboring elements, each antenna acting as the centralelement updates its estimate and error covariance matrix. Theupdate rule is based on the optimal combining of estimators,a standard result in LMMSE estimation theory. The result issummarized in the following lemma,

Lemma 1. Let y1 and y2 be two separate observations ofa zero mean random vectorh, such thaty1=A1h + w1

andy2=A2h+w2, where we assume thath is uncorrelatedwith both w1 and w2. Let h1 and h2 denote the LMMSEestimates ofh and C1 and C2 be the corresponding errorcovariance matrices in two experiments. Then, the optimalLMMSE estimator and the error covariance matrix ofh givenboth the observations are,

C−1h = C−11 h1 +C−1

2 h2 , (25)

and

C−1 = C−11 +C−1

2 +R−1h −R−1

1 −R−12 , (26)

where,Rh=EhhH andR1 andR2 are covariance matricesof h in two experiments.

Proof: See [33].Aforementioned lemma can be easily extended to more thantwo observations. The lemma suggests an optimal way ofcombining the individual estimates obtained by independentobservations. We use this lemma at each antenna to improvethe initial channel estimate by combining it with the estimatescomputed and shared by|N | neighbors. Consequently, bytreating each antenna as a central elementrC , the update ruleis given by following equations,

hc(i)w = hc(i−1)

w +∑

j∈N

hj(i−1)

w , (27)

andPc(i) = Pc(i−1) +

∑

j∈N

(

Pj(i−1) −R−1hj

)

, (28)

where Pj and Rhj represent the partial (inverse) error

covariance and correlation matrices associated with thepartial estimatesh

j

w and i represents the iteration index.Note that in the update equations, we employed the weighted

7

rU

rC rR

rD

rL

rI rJ

rP rQ

rA rB rE rF rG

rH rK

rM rN

rO rS

rT rV rW rX rY

(a) Information diffusion process

4

5 8

6

2

1 7

3 9

h2

=

h2

h5

h1

h3

h1=

h1

h4

h2

h5=

h5

h2

h8

h4

h6

h4

h8

h7

h3h

2=

h2

h5

h1

h3

h6=

h6

h3

h9

h5

h5=

h5

h2

h8

h4

h6

h9

h8

h1=

h1

h4

h2

h4=

h4

h1

h7

h5

h4

h7

h2

h5

h5=

h5

h2

h8

h4

h6

h8=

h8

h5

h7

h9

h3

h6=

h6

h3

h9

h5

h6

h9

(b) Information sharing process

Figure 3: (a) During the first iterationrC (blue antenna) receives information from its 4-direct neighbors (pink antennas). Inthe second iteration, the information from next nearest neighbors (green antennas) also comes in and so on. (b) An exampleof a 3× 4 antenna array where the neighboring antennas (indices4 and2) share the selected estimates (highlighted) with thecentral antenna (index1).

estimates and inverse error covariance matrices to minimizethe computational requirements. The recursions in the updateequations are initialized by (23) and (22) respectively, whichare available after the estimation step. In the subsequentiterations, each antenna would also require the partial matrices,Pj 's andR

hj 's, for each of its|N | neighbors. Fortunately,they can be obtained fromPc andRhc respectively (whichare available at the central antenna) by exploiting thesymmetrical structure ofRarray. Thus, there is no need toshare them across the neighboring elements, that in turn savesa significant amount of communication burden. Specifically,the matricesRhc andPc exhibit the following two properties:2

Property 1: The matrixRhc is identical for all elements inthe neighborhood ofrC i.e., Rhc=Rhj , ∀j ∈ NProperty 2:The matrixPc is identical for all elements in theneighborhood ofrC i.e., Pc=Pj , ∀j ∈ N

Property 1 is attributed to the symmetric nature of the spatialcorrelation matrixRarray, which implies that the spatialcorrelation between any two antennas, placed equidistant apart,is the same. Therefore, it is not difficult to see that property1 holds exactly under the Kronecker model and our earlierassumption of identical tap correlation across the antennaarrayin section II. Property 2 is the consequence of property 1 whenincorporated into (22).

Hence, to obtain the patrial correlation matrices,Rhj , j ∈

N , we use property 1 to first setRhj=Rhc and then mod-ify the off-diagonal block entries corresponding to the nullvectors of partial estimates asRij=0 if any hwi, hwj=0 andthe diagonal block entries asRii=IL if hwi=0, where thesubscriptij denotes the(i, j)th block. The matricesPj 's areobtained in the similar fashion except that the diagonal block

2These properties are generally satisfied as the spatial correlation matrix isusually symmetric, if not, then the antennas can share thesematrices as well.

entry corresponding to null vectors is replaced byaI where0 < a ≪ 1 is a small positive number, which indicatesvery low weight or confidence in null estimates (that are notshared). In essence, the central element has the full informationneeded to constructPj 's andR

hj 's corresponding to shared

estimateshj

w. We illustrate how these matrices could beobtained for the example in Fig. 3(b). Consider the centralantennarC=1, its |N |=2 direct neighbors with (shared) partialestimates given in (24). The partial correlation and errorcovariance matrices associated with those estimates (shownunderlined) along with that of central element are given in(29) and (30) respectively.

Based on above steps and procedures, the proposed D-LMMSE algorithm is summarized in Algorithm 1.

Rh1=

R11 R14 R12

R41 R44 R42

R21 R24 R22

,Rh4=

R44 R41 0

R14 R11 0

0 0 IL

and Rh2 =

R22 0 R21

0 IL 0

R12 0 R11

(29)

P1=

P11 P14 P12

P41 P44 P42

P21 P24 P22

, P4=

P44 P41 0

P14 P11 0

0 0 aI

and P2 =

P22 0 P21

0 aI 0

P12 0 P11

(30)

Remarks:

1) The information sharing and update take place duringeach iteration of the algorithm such that after few itera-tions the information diffuses across the whole antennaarray. This concept of sharing is depicted in Fig. 3(a)

8

Algorithm 1 Distributive LMMSE (D-LMMSE) algorithm

1) (Estimation) Each antenna acting as a central elementrC computes h

c

w and Pc by using (23) and (22)respectively.

2) (Sharing) Each antenna acting as a central elementrCshares partial estimates,h

c

w with its |N | neighbors asdescribed in IV-B.

3) (Pre-processing)Using Rhc , Pc from step 1 and the

received (partial) informationhj

w|N |j=1 in step 2, each

antenna, acting as a central elementrC , constructsR−1

hj , Pj, j ∈ N .4) (Update) Each antenna acting as a central elementrC ,

updates its weighted estimate and error covariance using(27) and (28) respectively.

5) (Iterate) Repeat steps 2-4D times, whereD representsmaximum number of iterations.

6) (Output) Computehc=(Pc)−1h

c

w and output the esti-mated CIRhC .

which shows that information diffusion process growsexponentially, resulting in fast convergence.

2) The repetitive sharing enables each antenna in the arrayto utilize the observations from distant elements, therebyimproving its estimate in each iteration till it convergesto near optimal solution.

3) As opposed to the central processing mechanism, theproposed sharing step is more convenient and computa-tionally more efficient as all antennas do not commu-nicate with each other. The collaboration takes placeonly among the neighboring antennas. Therefore, thecomplexity of proposed algorithm is significantly lessthan the centralized approach.

4) Note that, the antennas share only the partial informationbecause only selected vectors are transmitted to theneighbors which save significant amount of communi-cation. Also, estimation step and the repetitive sharing,pre-processing and update steps require simple linearblock processing and have a fixed size data structurewhich is well suited for real implementations. In con-trast, the memory and processing requirements for thecentralized approach are even more challenging withlarge array dimensions.

D. Complexity Analysis

In Table I, we compare the computational complexity ofproposed D-LMMSE algorithm with LS, L-LMMSE and thecentralized O-LMMSE algorithm in terms of multiply andadd operations. The figures indicate that complexity of pro-posed algorithm is slightly higher (linear with BS antennas)than L-LMMSE but is significantly less than the centralizedapproach. For the proposed D-LMMSE algorithm, it is alsoworth mentioning here that, theP matrices in (22) can becomputed off-line and in parallel at all antennas as theydo not depend on observations. Moreover, the computationof weighted estimates in (23) does not involve any matrixinversion. Further, the update in (27) requires simple addition

Table I: Computational Complexity

Algorithm Multiplications ( ×) Additions (+) Complexity

LS RK(L+ 1) R(KL− 1) O(RLK)

L-LMMSER[

2L3 + L2 +K(L+ 1)

] RL[L+K−1] O(RL3)

O-LMMSER[

(L3 + 1)R2 +RL(L+K)+K

]

+L3 R2LK O(R3L3)

D-LMMSER[

(53+1)L3+2(5L)2

+ L(K+1) + 53]

R[D(5L)3+(5L)2

+L(K−1)−D]O(R53L3)

during each step of iteration, while (28) needs one timecomputations of inversionsR−1

hj as they do not depend oniteration index. Finally, the computation of inverse,(Pc)−1 isalso required only after the convergence when each antennaoutputs its final estimate.

E. Choice ofD

The choice of parameterD i.e., maximum number ofrequired iterations, has a great influence on computationalcomplexity and convergence of the proposed D-LMMSE al-gorithm. A trivial choice forD is that it can be set to thelargest dimension of the array i.e.,D=max(M,G), whichwill ensure that each antenna receives information from everyother antenna in the array. The aforementioned choice ofDwould guarantee the convergence of the proposed D-LMMSEalgorithm but such a high value ofD is very inefficient fromthe computational complexity point of view, particularly if thearray dimensions are large. We therefore, derive a simple looseupper bound on maximum number of iterationsD that is muchbetter than the trivial choice. To this end, we first note thatthetotal number of antennas sharing information inD iterationsof algorithm are2D(D+1)+1. Hence, in order to ensure thateach antenna receives information from every other antennainthe array, we should have2D(D + 1) + 1 ≤ R. Solving thisinequality we get,

D ≤√

R

2− 1

4− 1

2. (31)

It must be emphasised here, that the actual value ofD alsodepends on the spatial correlations among antennas. If theantennas are not very strongly correlated, then we might notgain from sharing and a small number of iterations might besufficient. In fact, if the antennas are completely uncorrelated,then the sharing cannot improve the channel estimates as theO-LMMSE solution will converge to the L-LMMSE solution(see Section III).

F. Convergence of D-LMMSE Algorithm

The notion of convergence of D-LMMSE algorithm isattributed to the fact that every iteration of algorithm succes-sively bringsnew informationfrom the neighboring tiers tothe central antenna.

For example, consider the antenna array depicted in Fig.3(a) and focus on the central antennarC . Let A(i) andR(i)

represent the extended data matrix and channel correlation

9

matrix respectively, during thei-th iteration. Then by definingI1 , 1, we can write,

A(i) = Ii+1 ⊗A and R(i) = R(i)array ⊗Rtap (32)

where,R(i)array represents the spatial correlation matrix of the

central antenna and all its neighbors up toi-th tier. Furtherassume that,δlLi=1 and ηjRj=1 are eigenvalues ofRtap

and Rarray respectively, arranged in decreasing order ofmagnitude. Then forD=0 (i.e., no sharing case), the resultingMSE at the central element is,

mse(0)rC = trace(

R−1(0) +AH

(0)R−1w A(0)

)−1

=

L∑

l=1

(

δl1 + ρKδl

)

(33)

which is obviously the MSE of L-LMMSE in (10). Similarly,for D = 1 (i.e., sharing up to the first tier),rC receivesinformation from its|N | neighbors (e.g., red antennas of the1st tier) and updates its estimate by optimal combining ofneighboring estimates as described in IV-C. Therefore, theresulting MSE atrC can be written as,

mse1rC =1

|N+| trace(

R−1(1) +AH

(1)R−1w A(1)

)−1

,

=1

|N+|

|N+|∑

j=1

L∑

l=1

ηjδl1 + ρKηjδl

. (34)

Comparing (33) with (34), we note thatmse1rC≤mse0rC ,where the equality holds only ifηj=1, ∀j (i.e., spatiallyuncorrelated channels). Proceeding similarly, it can be shownthat mseDrC≤mseD−1

rC , so that the MSE during each iterationdecreases monotonically till it converges after utilizingobser-vations from all antennas in the array.

V. DATA -AIDED CHANNEL ESTIMATION

The basic idea of data-aided channel estimation is to exploitthe data sub-carriers in order to improve the initial channelestimates obtained using only the pilots. As the data aidedtechnique does not require additional pilots, it is spectrallymore efficient. Here, the pilot-based channel estimate is usedfor data detection, which along with the reserved pilots cansignificantly enhance the channel estimation. It is possiblethat some of the data-pilots be erroneous due to noise andchannel estimation errors, while some of the other data-carriersare reliable i.e., they are likely to be decoded correctly. Animportant problem is how to down-select a subset of the mostreliable data-carriers to be used as data-pilots.

A. Reliable Carriers Selection

Consider the received OFDM symbol at any antenna asshown in (4), and leth andH be the CIR and CFR estimatesobtained using pilots. Then, the tentative estimates of thedata symbols are obtained by equalizing the received OFDM

×××

×××

×××

×××

Xc Xb

Xa XX1

X2

Figure 4: Concept of reliable carriers selection. Here,X (2)has higher probability of decoding correctly thanX (1).

symbol using zero-forcing (ZF) as follows,

X (k) =Y(k)

H(k), k ∈ 1, 2, · · · , N \ P

≈ X (k) +W(k)

H(k)= X (k) +Z(k), (35)

where, Z(k) represents the distortion onk-th data-carrierdue to noise and channel estimation error. Given the CFRestimate,Z(k) can be modelled as Gaussian with zero meanand varianceσ2

z=H(k)−2σ2w. The recovery of data symbols is

then performed by simple hard decisions on estimated symbolsX (k) denoted by〈X (k)〉. Clearly, the errors in the decodingprocess occur due to noise as well as inaccurate channel esti-mates. Hence, some data-carriers would be severely effectedby noise and channel perturbation errors i.e.,Z(k) and falloutside their correct decision regions, while for some otherdata-carriers the distortion is not strong enough and they aredecoded correctly. All those data carriersX (k) which satisfythe condition〈X (k)〉=X (k) with high probability, are termedreliable carriers.

The proposed strategy for selecting the subsetR of themost reliable data-carriers, motivated by [26], is based onthecriteria,

R(k)=fz

(

Z(k)=X (k)− 〈 ˆX (k)〉)

∑Mm=1,Am 6=〈 ˆ

X(k)〉fz (Z(k)=X (k)−Am)

, (36)

where,fz(.) is the pdf of Z(k) and Am represents the setof constellation alphabets. Note that the numerator in (36)isthe probability thatX (k) will be decoded correctly while thedenominator sums the probabilities of all possible incorrectdecisions due to distortionZ(k). The subsetR is formed byselecting only those data-carriers for whichR(k) > 1 i.e.,

R = k | R(k) > 1 . (37)

The metric (37) is intuitively appealing as it selects only thosesub-carriers which are likely to be decoded correctly with highprobability. Fig 4 further elaborates this idea; that even thoughX (1) and X (2) have the same distance fromX , X (2) ismore likely to be decoded correctly thanX (1), as it is fartherfrom the nearest neighbours and therefore is less likely to bedecoded as any other constellation point.

10

B. Revisiting the Estimation Step

We now revisit the estimation step of the proposed Algo-rithm 1 using both the pilots and reliable carriers in order toenhance the initial estimates. LetRr be the set of indices ofreliable data carriers for antennar, obtained in reliable carriersselection process. Each antenna could revisit the estimationstep by solving (19) using an extended set of indices,P ∪Rr

corresponding to pilots and reliable data carriers. To makeit computationally efficient, we instead proceed by exploitingthe block form of RLS to update the pilot-based estimates.Skipping the derivation, the update equations for data-aidedestimation at antennar are given by,

hr

d = hr+Cr

eAHd G

(

Yd − Adhr)

, (38)

Cred = Cr

e −CreA

Hd GAd , (39)

G =(

Rw + AdCreA

Hd

)−1, (40)

where, Yd=Yr(P ∪ Rr) is extended set of observations,Ad=

[

A(P ∪Rr) 0|P∪Rr|×|N|L

]

is the extended data ma-trix, G represents the gain matrix andh

randCr

e are respec-tively the estimate and error covariance matrix obtained usingonly the pilots during the estimation step. The complete data-aided approach is described in Algorithm 2.

Algorithm 2 Data-aided Distributive LMMSE (DAD-LMMSE) Algorithm

1) Run step 1 of Algorithm 1 to gethr

andCre at each

antenna indexr.2) Each antenna uses its CIR estimate,h

rto form the

subsetRr of the most reliable data-carriers.3) Update the estimates and error covariance in step (1)

using (38)-(39).4) Run steps (2)-(6) of Algorithm 1, withPr=(Cr

e)−1 and

hr

w=Prhr.

VI. EFFECT OFPILOT CONTAMINATION

So far, we assumed single-cell scenario where all the usershave been allocated orthogonal resources for uplink channelestimation, thus the pilot observations are corrupted onlybyAWGN. In a multi-cell scenario, predominated by pilot con-tamination due to aggressive reuse of the pilots, the knowledgeof the interference statistics is critical in studying the effect ofpilot contamination on channel estimation techniques. Unlikethe existing pilot contamination analyses, we take a stochasticgeometry based approach to derive analytical expressions forinterference correlation.

A. Modified Network Model

To characterise the inter-cell interference resulting frompilot contamination, we modify our previous 2-D networkmodel of Fig. 1 by introducing interferes that are assumedto be distributed according to a PPP. Due to its simplicityand tractability, the PPP has been widely used in stochasticgeometry for modelling of the interference in cellular networks

Interferers

Base Station

γo

γm

Figure 5: Realization of interferes distributed accordingto PPPof λ=0.3, γo=2m andγm=5m with BS at the origin.

(see [36] and references therein). Specifically, without lossof generality, we assume a single user in a reference cell ofradiusγo, communicating with the BS located at the originO in a 2-D plane. The interfering users (outside radiusγo)are distributed over a circular region of radiusγm accordingto a homogenous PP, denoted byΨ and having intensityλ.Thus, the interfering space is an annular region with radiiγo and γm and where the distance ofith interferer fromBS satisfiesγo < γi < γm. Fig. 5 shows a realizationof interferes distributed according to homogeneous PP ofλ=0.3 with γo=2m andγm=5m. Further, from [37]–[39], weconclude that the interference itself is not correlated acrossOFDM frequency tones. This makes the analysis considerablysimple and tractable because each OFDM frequency tone canbe treated as an independent narrow-band channel. Hence, itsuffices to characterize the interference at single OFDM tone.

Consider the complex received interference at any givensub-carrier (at the BS antennar) due to all interfering users,which can be represented as [39],

I =∑

i∈Ψ

√

Exxihi (41)

where, xi=aiexpjθi is the interfering symbol,hi=γ−β

i αiexpjφi is the interfering channel, whereβ > 1 is the pathloss exponent,αi is an independentRayleigh distributed random variable withΩ=Eα2

i =1and φi is independent random variable that is uniformlydistributed over[0, 2π). The symbolsxi are generated froma general bi-dimensional constellation withM equiprobablesymbolsAm=a(m)expjθ(m), m=1, 2, · · · ,M . We assumethat all interfering users transmit with the same averageenergy per symbolEx and that the transmission constellationis normalized so thatE|xi|2=1. Therefore, (41) can beexpressed as,

I=∑

i∈Ψ\O

√Exaiαiexpj(θi + φi)

γβi

=∑

i∈Ψ\O

√Exzi

γβi

(42)

where,zi = aiαiexpj(θi + φi).

B. Interference characterization

AlthoughI can be completely characterized, to simplify ouranalysis of pilot contamination, we assumeI to be Gaussianand thus require only the first two moments, i.e., the meanand the variance. They are given in the following lemma.

11

Lemma 2. Using the network model of VI-A, the mean andvariance of interferenceI is,

µI = EI = 0 (43)

and

σ2I = E|I|2

= πλ(β − 1)−1E|x|2ExΩ

(

1

γ2β−2o

− 1

γ2β−2m

)

(44)

respectively.

Proof: See Appendix A.Although (44) is derived by considering that the interference

space is annular, it can be extended for an infinite interferencespace with a protection region ofγo by taking the limit asγm → ∞ yielding,

σ2I = πλγ2

o(β − 1)−1E|x|2

(

ExΩ

γβ−1o

)

. (45)

C. Effect of PC on MSE Performance

The knowledge of interference statistics at single OFDMfrequency tone, obtained through lemma 2, allows us toevaluate the aggregate interference correlation over all OFDMtones and/or across the whole BS antenna array using knownchannel statistics. Consider the received OFDM symbol atrthBS antenna, after omitting the indexP ,

Yr = Ahr + Ir +Wr

= Ahr + Er (46)

where, Ir is the interference at antennar of BS due topilot contamination andEr is the interference term whichcaptures the effect of both pilot contamination and the noise.Due to independence of noise and pilot contamination terms,each having a zero mean, the correlation matrix ofEr isREr

=RIr+Rw. Now, using the interference power (or vari-

ance) at each OFDM sub-carrier from lemma 2, the inter-ference correlation matrixRIr

across OFDM tones can beeasily obtained asRIr

=σ2IARtapA

H, where we assumed thatall user channels (both desired and interfering) have identicalcorrelations (as in section II) and use the same pilots, which isthe worst case scenario from pilot contamination perspective.Similarly, in the multi-antenna case, based on system modelof(8), the interference correlation matrix for the whole BS arraycan be obtained asRE=RI+Rw, with RI=σ2

IARhAH.

Using these interference correlations, we can derive the MSEexpressions for LS, L-LMMSE and O-LMMSE algorithms inthe presence of noise and pilot contamination by replacingthe noise covariance matrixRw=σ2I with matrix REr

or RE

in the MSE expressions already obtained in section III. Theresults are presented in following theorems.

Theorem 1. For the system model described in section II andpilot contamination as characterised in section VI, the MSEexpression for LS estimation algorithm of section III-C underboth AWGN and pilot contamination is given by,

MSE(LS) =RL

ρK+Rσ2

I trace(Λ) , (47)

where,σ2I is given in (44) andΛ is a diagonal matrix with

eigenvalues ofRtap spread along the diagonal and all usersare assumed to have similar channel characteristics.

Proof: See Appendix B.Theorem 1, shows that MSE is composed of two terms.

The first term due to AWGN can be suppressed by increasingthe number of pilot tones but the second term due to pilotcontamination cannot be reduced by adding more pilots andeven persists at high SNR (i.e.,ρ → ∞).

Theorem 2. For the system model described in section II andpilot contamination as characterised in section VI, the MSEexpression for L-LMMSE estimation algorithm presented insection III-A under both AWGN and pilot contamination isgiven by,

MSE(L) = R

L∑

i=1

δi(

1 + ρKδiσ2I

)

1 + ρKδi + ρKδiσ2I

, (48)

where,σ2I is given in (44),δi are the eigenvalues ofRtap and

all users are assumed to have similar channel characteristics.

Proof: ReplaceRw with Rw +RIrin MSE expression

(10), then invoking the eigenvalue decomposition (EVD) ofRtap, follow the steps of Theorem 1 given in Appendix B.We skip the detailed proof due to its similarity to Theorem 1.

Note that (48) reduces to MSE expression for AWGN (givenin (10)) had there been no pilot contamination. At high SNR(i.e. ρ ≫ 1), when there essentially remains only the effect ofpilot contamination, the MSE expression (48) reduces to,

MSE(L) high SNR−→ R

(

σ2I

1 + σ2I

)

trace(Λ) , (49)

which shows that MSE is independent of number of pilots andthat LMMSE is more robust to pilot contamination comparedto LS.

Theorem 3. For the system model described in section II andpilot contamination as characterised in section VI, the MSEexpression for O-LMMSE estimation algorithm presented insection III-B under both AWGN and pilot contamination isgiven by,

MSE(O) =

R∑

j=1

L∑

i=1

µjδi(

1 + ρKµjδiσ2I

)

1 + ρKµjδi + ρKµjδiσ2I

, (50)

where,σ2I is given in (44),µj and δi are the eigenvalues of

Rarray and Rtap respectively, and all users are assumed tohave similar channel characteristics.

Proof: See Appendix C.Note that (50) reduces to the MSE expression for AWGN

given in (13) in absence of pilot contamination. Again observethat, under the assumption of high SNR, when the effect ofpilot contamination predominates AWGN, the MSE expressionin (50) simplifies to,

MSE(O) high SNR−→(

σ2I

1 + σ2I

)

trace(Rarray)trace(Λ). (51)

12

Table II: Parameters for simulation

Parameter Value

Array Size(M ×G) 10 ×10

Array element spacingdx, dy 0.3ν, 0.5ν

Number of OFDM sub-carriers(N) 256

Number of pilots(K) 32

Signal constellation modulation 4/16/64 – QAM

Channel length(L) 8

This indicates that MSE depends strongly on interferencepower and is independent of number of pilotsK. Sincetrace(Rarray) ≤ R, the O-LMMSE seems to be more robustto pilot contamination compared to both LS and L-LMMSE.The MSE expression also gives us clue that effect of pilotcontamination can be minimized by exploiting the spatialcorrelations and by optimizing the BS antenna array design.

Above theorems quantify the effect of pilot contamina-tion on MSE performance of channel estimation in termsof interference power (or variance) which in turn dependson different parameters described in lemma 2. The MSEperformance against various parameters will be numericallyanalysed through simulations.

VII. S IMULATION RESULTS

We adopt the channel model in (1) with spatial correlationmatrix given in (3) whose parameters are:φ=π/3 (meanhorizontal AoD in radians),θ=3π/8 (mean vertical AoD inradians),σ=π/12 (standard deviation of horizontal AoD) andξ=π/36 (standard deviation of vertical AoD). The channeltap correlation matrix follows an exponentially decaying PDP,E|hr(τ)|2=e−τ , while rest of the parameters are givenin the Table II, whereν represents the carrier frequencywavelength in meters. It is also assumed that receiver has theknowledge of channel correlations.

To assess the performance of different algorithms we usethe following MSE performance criterion:

MSE =1

Θ

Θ∑

i=1

‖hi − hi‖2 (52)

where,hi and hi are true and estimated CIR vectors (at theith trial) respectively, each of sizeRL × 1 andΘ representsthe total number of trials. We usedΘ=100 in our simulations.

We conduct five different experiments to study the perfor-mance of our proposed approach and compare it with the threemethods i.e., LS, L-LMMSE and O-LMMSE described earlierin Section III. We also perform experiments to validate ouranalysis and study the impact of pilot contamination on allthese methods.

A. Experiment 1: How many iterations (D)?

In this experiment we are interested in finding the number ofiterations, required for convergence of the proposed distributedLMMSE algorithm. We plot the MSE of proposed D-LMMSE

0 1 2 3 4 5 6 7 8 9

100

101

# of iterations (D)

MSE

SNR=0 dB

LSL-LMMSED-LMMSEO-LMMSE

(a)

−10 −5 0 5 10 15 20 25 3010−3

10−2

10−1

100

101

SNR (dB)

MSE

D=0D=1D=2D=3

(b)

Figure 6: Number of iterations (D) required to achieve theconvergence of distributed algorithm.

algorithm (red curve) against the parameterD (i.e., number ofiterations) in Fig. 6(a). The SNR was fixed at0 dB. The MSEvalues of other algorithms, which do not depend on parameterD, are also shown. It can be seen that the proposed algorithmconverges very closely to the optimal in 3 iterations. Note that,when the antennas do not collaborate (i.e.,D=0), the MSEof distributed algorithm coincides with that of L-LMMSEbecause no information sharing takes place. As the informationfrom neighbors comes in during the next few iterations, theMSE decays exponentially until it converges to near optimalsolution. Fig. 6(b) also suggests that there would be hardlyany improvement in MSE forD > 3.

B. Experiment 2: MSE Performance in AWGN

In this experiment, we compare the MSE performance ofdifferent algorithms in the presence of AWGN using theparameters in Table II. The results given in Fig. 7, show thatO-LMMSE performs better than both LS and L-LMMSE in termsof MSE as it is able to utilize the antenna spatial correlations.As shown, the proposed D-LMMSE algorithm (Algorithm 1)achieves near optimal results in just 3 iterations. The analyticalMSE expressions given in Section III, for LS, L-LMMSE andO-LMMSE under AWGN are also plotted with legends (Th.),which agree with simulation results.

Fig. 8(a) shows the MSE performance of proposed data-aided algorithm (DAD-LMMSE in Algorithm 2) against otherpilot-based algorithms. It is obvious that data-aided approachhas the best performance compared to all others and that theeffect of using reliable carriers is more pronounced at higherSNR. Fig. 8(b) demonstrates the MSE behaviour of differentalgorithms with varying number of pilotsK with SNR fixedat 20 dB. As is shown, increasing the pilot tones yields betterestimation performance but this comes at the cost of lowerspectral efficiency. The data-aided algorithm however, is ableto achieve the best performance even for a small number ofpilot tones.

C. Experiment 3: Mean and variance of interference

This experiment aims to validate the mean and varianceof the interference given in Lemma 2. In order to mimic thesetup described in Section VI-A, we use single antenna BS andassume that CIRs from each user to the BS has a uniform PDP.Further, we assume that BS is located at the origin, the desired

13

−10 −5 0 5 10 15 20 25 3010−3

10−2

10−1

100

101

102

SNR (dB)

MSE

LS

LS (Th.)L-LMMSE

L-LMMSE (Th.)

D-LMMSE (D=3)O-LMMSE

O-LMMSE (Th.)

Figure 7: MSE performance of different algorithms in whiteGaussian noise.

0 5 10 15 20 25 3010−4

10−3

10−2

10−1

100

101

SNR (dB)

MSE

LSL-LMMSEO-LMMSE

D-LMMSE (D=3)

D-LMMSE (DA)

(a)

0 10 20 30 40 50 60 7010−3

10−2

10−1

100

101

# of pilots (K)

MSE

SNR = 20 dB

LS LS (Th.)

L-LMMSE L-LMMSE (Th.)

D-LMMSE (D=3) O-LMMSE (Th.)

O-LMMSE D-LMMSE (DA)

(b)

Figure 8: MSE performance comparison of data-aided D-LMMSE algorithm with pilot-based techniques in white Gaus-sian noise.

10−2 10−1 100 1010

1

2

3

4

λ

σ2 I

SimulationTheory

10−2 10−1 100 101−1

−0.5

0

0.5

1

λ

µI

SimulationTheory

Figure 9: Mean and variance of interference at single OFDMsub-carrier as a function ofλ.

user at a distance of 1m from BS while interfering users aredistributed in a region of radius 5m and with a protectionregion of γo=2m according to a PPP with densityλ andpathloss exponentβ=2. All users communicate with BS usingOFDM with N=256, L=8 andK=32 identical pilot symbolsdrawn from a4-QAM constellation. Fig. 9 compares the meanand variance of interference observed on single OFDM carrier(randomly picked) due to simulated sources with expressionsgiven in Lemma 2, as a function ofλ. The results indicate aclose match between simulation and theory.

−10 −5 0 5 10 15 20 25 3010−1

100

101

102

103

SNR (dB)

MSE

LS

LS (Th.)L-LMMSE

L-LMMSE (Th.)

D-LMMSE (D=3)O-LMMSE

O-LMMSE (Th.)

(a)

10−2 10−1 100 101

−10

−6

−2

2

6

10

λ

MSE

(dB)

LS

LS (Th.)L-LMMSE

L-LMMSE (Th.)

D-LMMSE (D=3)O-LMMSE

O-LMMSE (Th.)

(b)

Figure 10: Effect of pilot contamination on MSE performance(a) MSE as a function of SNR forλ=0.1 (b) MSE as a functionof λ for SNR fixed at10 dB.

D. Experiment 4: MSE Performance under AWGN and PilotContamination

In this experiment we study the MSE performance ofdifferent algorithms in presence of both AWGN and pilotcontamination. For simulations, we use the parameters givenin Table II with the interfering users distributed according to aPPP ofλ=0.1 and pathlossβ=2. The desired user is assumed1m away from BS located at origin while the interferingusers are distributed in circular region of radius 5m withprotection region ofγo=2 m. In Fig. 10(a), the simulatedMSE performance of different algorithms is compared overa wide range of SNR with the analytical expressions givenin Theorems 1, 2 and 3 (see Section VI-C). From Fig. 10(a),note that all MSE curves decrease with increasing SNR inlower range but reach an error floor at higher SNR. This isin stark contrast to AWGN case (see Fig. 7), where the MSEalways decreases with increasing SNR. This shows that pilotcontamination persists even at higher SNR and its effect onMSE is more severe than AWGN.

We present similar analysis in Fig. 10(b), where the MSEis plotted as a function ofλ with SNR fixed at10 dB. It isobvious that all algorithms perform well for small values ofλ. However whenλ increases, the interference due to pilotcontamination dominates AWGN, thus severely degrading theperformance as indicated by a sharp increase in MSE curves.Note that LMMSE channel estimation is more robust to pilotcontamination than simple LS based channel estimation. Alsoobserve a close match between simulation and theoreticalanalysis, shown in Fig. 10, over a wide range ofλ.

E. Experiment 5: Computational Complexity

In this experiment we compare the average runtime ofvarious algorithms that can be regarded as a measure ofcomputational complexity. Fig. 11 shows the average runtimewith increasing number of BS antennas under the defaultsimulation parameters of Table II. It is clear that computa-tional requirements for proposed D-LMMSE algorithm, withdifferent values of parameterD, grow at much slower pacethan that of the O-LMMSE algorithm as the number of BSantenna increases. Further, in terms of memory requirementsand communication overhead (not shown here), the advantagesof D-LMMSE are even more tangible.

14

4 16 36 64 10010

−4

10−3

10−2

10−1

100

101

102

No. of BS antennas

Avg

. Run

time

(sec

)

LSL−LMMSED−LMMSE(D=1)D−LMMSE(D=2)D−LMMSE(D=3)O−LMMSE

Figure 11: Average runtime of various algorithms.

VIII. C ONCLUSION

Channel estimation is a challenging problem in massiveMIMO systems as the conventional techniques applicable toMIMO systems cannot be employed owing to an exceptionallylarge number of unknown channel coefficients. We proposeda distributed algorithm that attains near optimal solutionata significantly reduced complexity by relying on coordinationamong antennas. To reduce the pilots overhead, the distributedLMMSE algorithm is extended using data-aided estimationbased on reliable carriers. To gain insight into the effectof pilot contamination on channel estimation performance,we used the stochastic geometry to obtain the aggregatedinterference power and then based on this, we derived MSEexpressions for different algorithms under AWGN and pilotcontaminated scenarios. The derived expressions were verifiedusing simulation results. Extending the obtained results toanalyzing the system throughput under pilot contaminationremains open for future work.

APPENDIX AMEAN AND VARIANCE OF INTERFERENCE

The mean ofI can be determined as follows,

µI = EI = E

∑

i∈Ψ

√Exzi

γβi

= EΨ

∑

i∈Ψ

√Ex Ezzi

γβi

(a)=√

ExEzi∫

R2

1

rβrdrdθ = 0

where,(a)= results from Campbell’s theorem [40] and then the

fact, Ezi = 0 yields the zero mean. Similarly, the variance

of interference can be computed as follows,

σ2I = E|I|2

= EΨ

Ez

∑

i∈Ψ

√Exzi

γβi

∑

j∈Ψ

√Exz

∗j

γβj

(a)= EΨ

∑

i∈Ψ

ExEz|zi|2γ2βi

(b)= λExE|zi|2

∫ 2π

0

∫ γm

γo

1

r2βrdrdθ

(c)= πλ(β − 1)−1ExΩE|x|2

(

1

γ2β−2o

− 1

γ2β−2m

)

where,(a)= is due to the fact thatzi are independent SS random

variables, in(b)= we employed Campbell’s theorem and in

(c)= we

used the resultE|zi|2 = Ea2iα2i = ΩE|x|2, where we

note thatai andαi are independent random variables, whichcompletes the proof.

APPENDIX BPROOF OFTHEOREM 1

By replacingRw with Rw + RIrin MSE expression of

(16), we obtain

mselsr = trace(

AH (Rw +RIr)−1

A)−1

= trace(

AH(

Rw + σ2IARtapA

H)−1

A)−1

(a)= trace

(

AHR−1w A− σ2

IAHR−1

w A(

R−1tap

+ σ2IA

HR−1w A

)−1AHR−1

w A)−1

where,(a)= follows from matrix inversion lemma. Now, using

the EVD of the channel correlation matrixRtap = QΛQH

and the fact thatAHR−1w A = KEx

σ2w

IL we obtain,

mselsr(b)= trace

(

KEx

σ2w

IL − σ2I

(

KEx

σ2w

)2(

Λ−1

+σ2IKEx

σ2w

IL

)−1)−1

=

L∑

i=1

(

KEx

σ2w

−σ2I

(

KEx

σ2w

)2(

δ−1i +

σ2IKEx

σ2w

)−1)−1

(53)

where,(b)= follows from the property thattrace

(

QRQH)

=trace(R) if Q is unitary. After simple algebraic manipulations,the term inside the summation simplifies toσ

2wL

KEx+σ2

I

∑Li=1 δi,

which completes the proof.

15

APPENDIX CPROOF OFTHEOREM 3

Under both AWGN and pilot contamination, we replaceRw

with RE = Rw + σ2IARhA

H to get,

MSE(O) = trace

(

R−1h

+AH(

Rw+σ2IARhA

H)−1

A

)−1

(a)= trace

(

R−1h

+ ARhAH − σ2

IARhAH(

R−1h

+ σ2IARhA

H)−1

ARhAH

)−1

where(a)= follows from matrix inversion lemma. Using

the properties of kronecker product, it can be shown thatARhA

H = KEx

σ2w

(IR ⊗ IL). Further, the channel correla-tion matrix Rh = Rarray ⊗ Rtap can be decomposed asRh = (V ⊗Q)(S ⊗Λ)(V ⊗Q)H, where we introduced theEVDs, Rarray = VSVH andRtap = QΛQH. Incorporating

these results in(a)= yields,

MSE(O) (b)= trace

(

S−1⊗Λ−1+KEx

σ2w

(IR ⊗ IL)−σ2I

(

KEx

σ2w

)2

·(

S−1 ⊗Λ−1+σ2IKEx

σ2w

(IR ⊗ IL)

)−1)−1

(c)=

R∑

j=1

L∑

i=1

(

1

µjδi+

KEx

σ2w

− σ2I

(

KEx

σ2w

)2

·(

1

µjδi+

σ2IKEx

σ2w

)−1)−1

where,(b)= follows from property,trace

(

QRQH)

=trace(R)

whenQ is unitary and(c)= is due to the diagonal nature of the

matrix inside thetrace operator, whereµj andδi represent theeigenvalues of matricesRarray andRtap respectively. After

some algebraic manipulations,(c)= simplifies to the result given

in Theorem 3.

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