1

Information-Theoretic Pilot Design for DownlinkChannel Estimation in FDD Massive MIMO

SystemsYujie Gu, Senior Member, IEEE and Yimin D. Zhang, Fellow, IEEE

Abstract—Massive multiple-input multiple-output (MIMO) isone of the most promising techniques for next generation wirelesscommunications due to its superior capability to provide highspectrum and energy efficiency. Considering the very large num-ber of antennas employed at the base station, however, the pilotoverhead for downlink channel estimation becomes unaffordablein frequency division duplex (FDD) multi-user massive MIMOsystems. In this paper, we propose an information-theoreticmetric to design the pilot for downlink channel estimation inFDD multi-user massive MIMO systems. By exploiting the low-rank nature of the channel covariance matrix, we first derive theminimum number of pilot symbols required to ensure perfectchannel recovery, which is much less than the number of antennasat the base station. Further, under a general channel modelthat the channel vector of each user follows a Gaussian mixturedistribution, the pilot symbols are designed by maximizing theweighted sum of the Shannon mutual information between themeasurements of the users and their corresponding channelvectors on the complex Grassmannian manifold. Simulationresults demonstrate the effectiveness of the proposed information-theoretic pilot design for the downlink channel estimation in FDDmassive MIMO systems.

Index Terms—Channel estimation, frequency division duplex(FDD), Gaussian mixture distribution, Grassmannian manifold,information-theoretic metric, massive multiple-input multiple-output (MIMO), pilot design.

I. INTRODUCTION

MASSIVE multiple-input multiple-output (MIMO) be-comes a key enabling technology for next generation

wireless communications benefited from its promising systemcapacity, energy efficiency, security and robustness [2–6]. Inmassive MIMO systems, the very large number of antennas atthe base station simultaneously serve a much smaller numberof users in the same radio frequency (RF) channel, whereeach user is usually equipped with a single antenna due tothe physical size limitation. With the very large number ofantennas, the number of degrees-of-freedom is high enoughto eliminate multiuser interference using transmit precodingor receive beamforming. In addition, massive MIMO alsoimproves the detection and estimation performance in wirelesssensor networks [7–14].

Similar to general wireless communication systems, thereare two commonly used duplex modes in massive MIMO

Part of this work was presented at the 2018 IEEE International Conferenceon Acoustics, Speech and Signal Processing [1].

The authors are with the Department of Electrical and Computer Engi-neering, Temple University, Philadelphia, PA, 19122 USA (e-mail: guyu-jie@hotmail.com; ydzhang@temple.edu).

systems, i.e., time division duplex (TDD) and frequencydivision duplex (FDD). Among them, the TDD is popular formassive MIMO [2, 15–18], where the principle of reciprocitycan be leveraged, that is, the downlink channel vector (ormatrix) is simply the transpose of the uplink channel vector(or matrix). As such, the number of required pilot symbolsfor the downlink channel estimation (via the uplink channelestimation) is independent of the number of antennas at thebase station, which is very large in massive MIMO systems.It is worth noting that the accurate channel state informationis essential in wireless communications for reliable signaltransmission and efficient resource allocation.

Despite the attractiveness of the TDD mode in channel es-timation, many current wireless cellular systems are primarilydominated by the FDD mode. Unlike the TDD mode, thechannel reciprocity property does not apply for the FDD modewhere the downlink and uplink channels occupy differentfrequency bands. In the FDD mode, each user estimates itsown downlink channel from the received pilot symbols trans-mitted from the base station, and feeds the estimated downlinkchannel information back to the base station for subsequentsignal transmission and resource allocation. Considering thatthe number of required pilot symbols for downlink channelestimation in the FDD mode is proportional to the number ofantennas at the base station, there is a huge pilot overhead formassive MIMO systems equipped with a very large number ofantennas. Hence, there is a more urgent requirement in FDDmassive MIMO systems to use much less pilot symbols fordownlink channel estimation.

Reducing the downlink pilot overhead in FDD massiveMIMO systems has been the subject of recent studies dueto its importance. These techniques are mainly developed byexploiting the sparsity of the channel on the virtual angulardomain [19, 20], the low-rank nature of the channel covariancematrix [21–26], or joint sparse and low-rank structures [27].Both the channel sparsity and the low-rank structure aredue to the narrow angular spread of the incoming/outgoingrays at the base station in typical cellular systems, whichsubsequently leads to a high correlation between differentpaths that link the base station and the user. The sparsity and/orthe low rankness enable the estimation of the downlink channelwith much less pilot symbols than the number of antennasemployed at the base station in a sparse reconstruction manner.Specifically, the distributed compressive channel estimationin [19] and the compressive sensing based adaptive channelestimation and feedback scheme in [20] exploit the sparsity

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of massive MIMO channels to reduce the pilot overheadsignificantly. By reducing the dimensionality of the effectivechannels via the correlated channel covariance matrix, thejoint spatial division and multiplexing scheme in [21] and thebeam division multiple access transmission scheme in [24]enable significant savings in both the downlink pilot and theuplink feedback. By assuming that each training signal has amuch lower rank than the number of antennas, open-loop andclosed-loop training frameworks in [22] reduce the overheadof the downlink training phase by exploiting prior channelinformation such as the long-term channel statistics. In [23],the rank-deficient channel covariance matrices were exploitedto design efficient downlink pilot symbols with dimensionalityreduction. By exploiting the low-rank property of the massiveMIMO channel matrix caused by correlation among users, thejoint channel estimation for all users is performed at the basestation based on the matrix completion [25], based on whichthe overhead of downlink channel training as well as uplinkchannel feedback can be reduced. By exploiting the low-rankproperty of the channel covariance matrices, the required num-ber of pilot symbols was proved to be significantly reduced[26].

In addition, other researchers tried to utilize additionalknowledge beyond the sparsity in order to further reduce thepilot overhead. For example, by exploiting the reciprocity ofthe angular scattering function [28] and the uplink-downlinkcovariance interpolation [29], the downlink channel estimationin FDD massive MIMO systems performs well even whenthe pilot dimension is less than the inherent dimension ofthe channel vectors. By estimating the directions-of-arrival(DOAs) of the users, the pilot symbols can then be designedsuch that the transmit energy concentrates over the knownDOAs to achieve a beamforming gain [30–32]. Specifically,a two-stage compressive sensing scheme was proposed forchannel estimation in [30], where the first stage randomlygenerates the pilot to coarsely estimate candidates for DOAswhereas the second stage adaptively designs the pilot to refinethe candidates exhaustively. A joint RF training and compres-sive channel estimation scheme was proposed in [31], whichachieves a better tradeoff between pilot overhead reductionusing random RF training vectors and beamforming gain usingnarrow-beam RF training vectors. In [32], the out-of-bandspatial information extracted from a sub-6 GHz channel isexploited to design a structured random codebook (pilot) andthe associated weighted sparse channel recovery algorithm.

Typical pilot design criteria include maximizing the systemspectral efficiency [21], minimizing the mean squared error(MSE) [22, 26], maximizing the average received signal-to-noise ratio (SNR) [22], maximizing the summation of theconditional mutual information [23], maximizing the sum-rateupper bound [24], minimizing the sum of MSEs [26], andminimizing the weighted sum MSE [33]. However, they allassume that the channel vector can be modeled as a singlesmooth Gaussian variable, which limits their applicationsin more complex environments. In [34], the uplink channelcomponent in the beam domain is modeled to a more generalGaussian mixture, i.e., a weighted sum of multiple Gaussiandistributions with different variances. To the best of our

knowledge, however, there is no pilot design for the downlinkchannel estimation in massive MIMO systems based on theGaussian mixture distribution.

In this paper, in addition to exploiting the sparsity and thelow-rank nature, we further model the downlink channel inFDD massive MIMO systems to follow a general Gaussianmixture distribution for the pilot design [34, 35]. First, westudy the asymptotic behavior of the minimum mean-squarederror (MMSE) estimator. It reveals that a perfect channelrecovery can be asymptotically reached in the high-SNRregime, provided that the number of pilot symbols is no lessthan the maximum rank of the channel covariance matricesof all Gaussian components of all users. Then, we adoptthe information-theoretic approach to optimize the downlinkpilot symbols for the estimation of Gaussian mixture chan-nels in FDD massive MIMO systems. More specifically, theallocation of pilot symbols is optimized by maximizing theweighted sum of the Shannon mutual information betweenthe measurements of multiple users served in massive MIMOsystems and their corresponding downlink channel vectors. Itis noted that the weighted sum of the mutual information isinvariant when the pilot matrix undergoes an arbitrary unitaryrotation, i.e., the weighted sum of the mutual informationis a function of the subspace spanned by the pilot symbolvectors. Consequently, it enables us to optimize the pilotmatrix on the complex Grassmannian manifold. Unlike thegeneral sparse reconstruction problem, there is a closed-formMMSE solution for the downlink channel estimation under theGaussian mixture assumption. Simulation results demonstratethe effectiveness of the proposed pilot design approach forthe downlink channel estimation for an arbitrary number ofusers served in the massive MIMO system by exploiting theavailable a priori knowledge of the desired channel vector.

By generalizing the downlink channels in FDD massiveMIMO systems to follow Gaussian mixture distributions, themain contributions of this paper can be summarized as follows:• We analyze the asymptotic behavior of the MMSE esti-

mator under Gaussian mixture distribution, and present theminimum number of pilot symbols required for Gaussianmixture channel estimation.• We perform the Taylor series expansion to approximate

the weighted sum of the mutual information associated withall users, and optimize the pilot matrix on a complex Grass-mannian manifold.

The rest of the paper is organized as follows. In SectionII, we describe the multi-user model in massive MIMO sys-tems, and present the MMSE channel estimator based on theGaussian mixture assumption. In Section III, we analyze theasymptotic behavior of the MMSE estimator in the high-SNRregime. In Section IV, we propose an information-theoreticpilot optimization on the Grassmannian manifold. In SectionV, we compare the proposed pilot symbols with random pilotsymbols in terms of channel estimation performance. We makeour conclusions in Section VI.

II. SYSTEM MODEL

Assume that N ≫ 1 transmit antennas equipped at the basestation serve M ≥ 1 single-antenna users in the FDD mode,

3

ymhm

Frontend circuit #1

Frontend circuit #2

Frontend circuit #3

Frontend circuit #4

Frontend circuit #5

Frontend circuit #6

Frontend circuit #N

Frontend circuit #N-1

Frontend

circuit

User #m

y1

Frontend

circuit

User #1

yM

Frontend

circuit

User #M

h1

hM

Fig. 1. Pilot demo in the FDD multi-user massive MIMO system.

as shown in Fig. 1. The downlink channel vector of the m-th user, hm ∈ CN , is estimated at the m-th user terminalfrom the noisy measurements of the known pilot symbolstransmitted from the base station. The estimated downlinkchannel information is then sent back to the base station forsubsequent scheduling, precoding and transmission.

Without loss of generality, the downlink channel of eachuser is assumed to be a frequency-flat fading channel undera narrowband assumption. This assumption can be easilyextended to wideband frequency-selective channels in the con-text of multi-carrier transmission schemes such as orthogonalfrequency-division multiplexing (OFDM).

Assume that the base station transmits a set of pilot symbols{ϕ(l), l = 1, 2, · · · , L}, where L is the length of the pilot (i.e.,the number of pilot symbols in time) transmitted from eachantenna. The l-th baseband signal received at the m-th userterminal is expressed as

ym(l) = ϕT(l)hm + nm(l), ∀ m ∈M, (1)

where M = {1, 2, · · · ,M} is the set denoting the indexesof users served in the massive MIMO system, ϕ(l) ∈ CN isthe l-th pilot symbol vector transmitted from N base stationantennas, and nm(l) ∼ CN (0, σ2

nm) denotes the zero-mean

additive white Gaussian noise with variance σ2nm

. Here, ( · )Tdenotes the transpose operator. Note that, in this signal model,the inter-cell interference leaked from neighboring frequency-reuse cells is ignored for the simplicity and clarity of theproblem.

Stacking the received signals of the m-th user terminal overall L pilot symbols as ym = [ym(1), ym(2), · · · , ym(L)]

T ∈CL gives

ym = Φhm + nm, (2)

where Φ = [ϕ(1),ϕ(2), · · · ,ϕ(L)]T ∈ CL×N is the pilotsymbol matrix, and nm = [nm(1), nm(2), · · · , nm(L)]

T ∼CN (0, σ2

nmI) is the zero-mean additive Gaussian noise vec-

tor with I denoting the identity matrix with an appropriatedimension. The least squares (LS) estimate of hm is given by

hLSm =

[ΦHΦ

]−1ΦHym, (3)

where ( · )H denotes the Hermitian transpose operator. In orderto perform the matrix inversion in the above expression, thenumber of pilot symbols must not be less than the numberof antennas, i.e., L ≥ N . Unfortunately, in massive MIMOsystems, N , the number of antennas at the base station, istypically very large, which means that such a pilot overheadrequirement becomes unaffordable in the FDD mode.

In order to guarantee the transmission efficiency, the numberof pilot symbols in FDD massive MIMO systems should bekept much less than the number of antennas at the base station,i.e., L ≪ N . In such a case, the above downlink channelestimation in (3) becomes ill-conditioned because the numberof unknowns is much larger than the number of measurements.As a result, the least squares method is no longer applicable.

On the other hand, because of the compact configurationof the antennas in massive MIMO systems, the channelis a correlated random vector depending on the scatteringgeometry, which leads to a low-rank channel covariance matrix[23, 26]. In addition, the channel can be regarded as sparseor approximately sparse in some suitable representation bases,where the channel only contains a small number of dominantcomponents while the others are negligible [34]. When thechannel vector is sparse or approximately sparse, it can beestimated in the framework of compressive sensing with lessmeasurements than the number of unknowns [20].

In addition to the sparsity and low-rank property, in thispaper, we further model the channel vector in massive MIMOsystems as a Gaussian mixture distribution. This mixturedistribution is very popular and well verified in practice todescribe the real environment signals with high flexibilityand tractability (see, for example, [36, 37] and the referencestherein). Let the probability density function (pdf) of thechannel vector hm be modeled by a mixture of Gaussiandistributions given by

f(hm) =∑

k∈Km

p(m)k f (k)(hm)

=∑

k∈Km

p(m)k CN

(u(k)hm

,R(k)hmhm

), (4)

where Km = {1, 2, · · · ,Km} has a cardinality Km, and∑k∈Km

p(m)k = 1. The Gaussian mixture distribution of the

channel vector hm implies that it contains Km Gaussiancomponents, and the k-th Gaussian component is activatedwith probability p

(m)k > 0 and, when activated, that component

generates a complex-valued Gaussian vector with distribu-tion f (k)(hm) = CN

(u(k)hm

,R(k)hmhm

). The parameter set{

p(m)k ,u

(k)hm

,R(k)hmhm

; k ∈ Km

}defines the Gaussian mixture

distribution of the channel vector hm. The parameters char-acterizing the Gaussian mixture distribution can be estimatedby using, e.g., the expectation maximization (EM) algorithm[38, 39], the sparse Bayesian learning method [40, 41], or thepiecewise-Gaussian approximation method [1, 36].

Under the channel distribution model (4), it can be shownthat the measurement ym in (2) also follows the Gaussianmixture distribution with Km components, and its pdf is

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expressed as

f(ym) =∑

k∈Km

p(m)k f (k)(ym), (5)

where the k-th component f (k)(ym) = CN(u(k)ym

,R(k)ymym

)is

complex Gaussian distributed with mean vector and covariancematrix given by

u(k)ym

= Φu(k)hm

,

R(k)ymym

= ΦR(k)hmhm

ΦH + σ2nm

I. (6)

The MMSE estimate of the channel vector hm, defined by

minhm

E{∥∥hm − hm

∥∥22

}, (7)

is given by [42]

hMMSEm = E{hm|ym} =

∑k∈Km

pk|ymu(k)hm|ym

, (8)

where E{ · } denotes the statistical expectation operator,

u(k)hm|ym

= u(k)hm

+R(k)hmhm

ΦH[R(k)

ymym

]−1(ym −u(k)

ym

)(9)

is the k-th component of the MMSE estimator of hm giventhe measurement ym, and

pk|ym=

p(m)k f (k)(ym)

f(ym)(10)

is the corresponding posterior probability [43]. When thedownlink channel estimate h

MMSEm is obtained, it will be sent

back to the base station.Note that, although the MMSE estimator for hm has an

analytical form, its performance measure, the MMSE itself,does not have such a closed form because of the Gaussianmixture distribution. According to [42], the MSE of hm isupper and lower bounded as (11), shown at the bottom of thepage, where Tr[ · ] denotes the trace of a matrix, and

Rhmhm=∑

k∈Km

p(m)k

[R

(k)hmhm

+u(k)hm

(u(k)hm

)H]−u(m)h

(u(m)h

)H(12)

is the covariance matrix of the channel vector hm with themean vector given by

u(m)h =

∑k∈Km

p(m)k u

(k)hm

. (13)

It is indicated in [42] that the upper and lower bounds approacheach other as the SNR increases, and they must coincide asthe SNR tends to infinity.

When the Gaussian mixture distribution degrades into asingle Gaussian distribution (i.e., Km = 1), the upper and

lower bounds of the MSE of the MMSE estimator for thechannel vector hm in (11) become identical, i.e., the MSE hasa closed-form expression, thereby facilitating the pilot designvia minimizing the MSE [26] or minimizing the weighted sumMSE [33]. However, the existing pilot design methods underthe MMSE criterion for the Gaussian channel are not directlysuitable for the Gaussian mixture channel which does nothave an analytic MSE expression. Considering the relationshipbetween mutual information and MMSE [44], we propose aninformation-theoretic pilot design approach for the Gaussianmixture channel estimation in FDD massive MIMO systems.In the sequel, we first prove the asymptotic behavior ofthe MMSE estimator for the Gaussian mixture channel inSection III, and then propose the information-theoretic pilotoptimization on the Grassmannian manifold in Section IV.

III. ASYMPTOTIC BEHAVIOR OF THE MMSE ESTIMATOR

In this section, we analyze the behavior of the MMSEestimator in the asymptotic high-SNR regime. The asymptoticanalysis verifies the possibility of perfect channel recoveryfrom a small number of pilot symbols, because of the low rankof the channel covariance matrix in massive MIMO systems.

Theorem: Let r(k)m be the rank of the channel co-

variance matrix of the k-th Gaussian component in theGaussian mixture distribution of the m-th user, R

(k)hmhm

,i.e., r

(k)m = rank

(R

(k)hmhm

), where rank( · ) denotes

the rank of a matrix. Let V (k)m Γ(k)

m

(V (k)

m

)Hdenote the

eigen-decomposition of(R

(k)hmhm

) 12ΦHΦ

(R

(k)hmhm

) 12 , where

V (k)m =

[v(k)m1 , · · · ,v

(k)mN

]is a unitary matrix consisting

of the eigenvectors of V (k)m Γ(k)

m

(V (k)

m

)H, and the diagonalmatrix Γ(k)

m = diag(γ1, γ2, · · · , γr(k)

m, 0, · · · , 0

)consists of the

corresponding eigenvalues with γ1 ≥ γ2 ≥ · · · ≥ γr(k)m

> 0.Assume that the number of randomly generated pilot symbols,L, is no less than the maximum of r

(k)m of all Gaussian

components for each user, i.e., L ≥ maxm∈M maxk∈Km r(k)m ,

then the lower bound of the MSE of the MMSE estimate ofhm is given by

ε(m)Lower ≤ E

{∥∥∥hm − hMMSEm

∥∥∥22

}

=∑

k∈Km

p(m)k

r(k)m∑i=1

(1 +

γiσ2nm

)−1(v(k)mi

)HR

(k)hmhm

v(k)mi

,

∀ m ∈M, (14)

which approaches zero in the asymptotic low-noise regime,i.e., σ2

nm→ 0. Considering that the upper and lower bounds

∑k∈Km

p(m)k Tr

[R

(k)hmhm

−R(k)hmhm

ΦH(ΦR

(k)hmhm

ΦH + σ2nm

I)−1

ΦR(k)hmhm

]≤ E

{∥∥∥hm − hMMSEm

∥∥∥22

}≤ Tr

[Rhmhm −RhmhmΦH

(ΦRhmhmΦH + σ2

nmI)−1

ΦRhmhm

], (11)

5

approach each other with increasing SNR and they mustcoincide as the SNR tends to infinity [42], we have

limSNRm→∞

E{∥∥∥hm − h

MMSEm

∥∥∥22

}= 0, ∀ m ∈M, (15)

where SNRm = ∥hm∥2/σ2nm

denotes the SNR of the m-thuser’s channel. That is, perfect channel recovery is possiblefor each user.

Proof : See Appendix A. �The above asymptotic analysis proves that

an accurate channel estimation from as few asL = maxm∈M maxk∈Km rank

(R

(k)hmhm

)pilot symbols

is possible in the asymptotic regime. Nevertheless, it doesnot tell us how to design pilot symbols in the non-asymptoticregime. In the following section, we propose a novelinformation-theoretic pilot design method for the channelestimation in FDD massive MIMO systems serving for anarbitrary number of users.

IV. INFORMATION-THEORETIC PILOT OPTIMIZATION ONTHE GRASSMANNIAN MANIFOLD

In this section, we adopt the maximum mutual informationcriterion to optimize the pilot symbols on the Grassmannianmanifold for the estimation of Gaussian mixture channels.Define I(ym;hm) as the Shannon mutual information be-tween the measurement vector ym at the m-th user and thecorresponding channel vector hm, i.e., [45]

I(ym;hm) = h(ym)− h(ym|hm), (16)

where h(ym) = −E {log[f(ym)]} denotes the differentialentropy of the measurement vector ym, and h(ym|hm) =−E {log[f(ym|hm)]} denotes the differential entropy of themeasurement vector ym conditioned on the channel vectorhm. Note that, the mutual information I(ym;hm) is animplicit function of the pilot matrix Φ via the measurementequation (2). It is difficult, if not impossible, to analyticallyderive the differential entropy even for a simple estimationproblem, let alone the parameter estimation problem with thehigh dimensionality and non-Gaussianity.

In massive MIMO systems, the pilot symbols transmittedfrom the base station are common to all served users andshould be optimized to maximize the mutual informationassociated with all these users. It is assumed that the channelsof different users are independent to each other because of theseparation between the users and the rich multipath. Let

J(Φ) =∑

m∈MwmI(ym;hm) (17)

denote the weighted sum of the mutual information associatedwith all users served in the massive MIMO system, where wm

is the weight assigned to the m-th user with∑

m∈M wm = 1according to different criteria, such as the commonly usedequal weights and different fairness weights (e.g., max-minfairness, weighted fairness, and proportional fairness) [46].Unlike in (16), here we express the weighted sum of the mutualinformation as an explicit function of Φ.

The information-theoretic design of pilot symbols can beformulated as an optimization problem to maximize the

weighted sum of the mutual information associated with allusers, i.e.,

maxΦ

J(Φ)

subject to ΦΦH = I, (18)

where the orthonormal constraint ΦΦH = I is introduced toallocate the equal power to each pilot symbol so as to avoidincreasing the mutual information by simply scaling Φ to belarger because scaling Φ only affects the channel rather thanthe noise. Another commonly adopted pilot constraint is thetotal transmit power constraint [23, 26, 33], i.e., Tr[ΦΦH] ≤ L.It is noted that the orthonormal constraint always satisfies thepower constraint, but not vice versa. That is to say, the feasibleset with orthonormal constraint is a subset of the feasible setwith power constraint, i.e, {Φ|ΦΦH = I} ⊂ {Φ|Tr

[ΦΦH

]≤

L}.Note that the objective function in (18) is non-convex with

respect to the optimization variable Φ, which belongs tothe L-dimensional subspace in CN . In general, the entropycalculation does not have a closed-form solution for mostpdf’s of interest, including the Gaussian mixture model usedin this paper. In order to obtain a feasible solution, therefore,we perform a Taylor series expansion of the logarithm of theGaussian mixture pdf required in the definition of the entropy,which enables a gradient-based search method.

By performing the first-order Taylor series expansion of thelogarithm of the Gaussian mixture distribution in the definitionof the differential entropy of the measurement vector ym, weobtain the approximated differential entropy as [1, 36]

h(ym) ≈ −log

[ ∑k∈Km

p(m)k f (k)

(y(m)0

)], (19)

where y(m)0 = E{ym} is the mean value of the measurement

ym. The derivation of the above approximated differentialentropy can be found in Appendix B. Following the zero meanassumption of the channel vector [21, 23, 26, 33, 34, 47],we have u

(k)ym

= Φu(k)hm

= 0 for all individual Gaussiancomponents of ym. In this case, it is natural to set the Taylorseries expansion point to y

(m)0 = 0, resulting in

f (k)(y(m)0

)=

1

πL∣∣∣R(k)

ymym

∣∣∣ , (20)

where | · | denotes the determinant of a matrix. We now havethe approximated differential entropy of ym expressed as

h(ym) ≈ −log

[ ∑k∈Km

p(m)k

∣∣∣R(k)ymym

∣∣∣−1]+ Llogπ. (21)

Because the additive white Gaussian noise vector nm ∼CN (0, σ2

nmI) is independent of the channel vector hm, the

conditional differential entropy is given by

h(ym|hm) = h(nm) = Llog(πeσ2nm

). (22)

Hence, the mutual information I(ym;hm) is approximated as

I(ym;hm) ≈ −log

[ ∑k∈Km

p(m)k

∣∣∣R(k)ymym

∣∣∣−1]− Llog(eσ2

nm).

(23)

6

Accordingly, the weighted sum of the mutual informationassociated with all served users is approximated as

J(Φ) ≈ −∑

m∈M

wmlog

[ ∑k∈Km

p(m)k

∣∣∣R(k)ymym

∣∣∣−1]−Llog(eσ2

nm),

(24)where the second term is a constant independent of the pilotmatrix Φ. It is noted that J(Φ), the objective function in (18),remains unchanged when the pilot matrix Φ undergoes anarbitrary unitary rotation, i.e.,

J(QΦ) = J(Φ), (25)

where Q ∈ CL×L is a unitary matrix satisfying QHQ =QQH = I . Hence, J(Φ) is a function of the subspace spannedby the rows of the pilot matrix, or equivalently, the row spaceof Φ, because the row spaces of Φ and QΦ are the same.

Based on (25), we can optimize the mutual informationmaximization problem (18) over the complex Grassmannianmanifold G(N,L), which composes of all L-dimensionalsubspaces in CN . The Grassmannian is a compact Rieman-nian manifold, and its geodesics can be explicitly computed[48, 49]. Following the framework in [50], we derive theGrassmannian gradient ascent algorithm to search the pilotmatrix Φ on the Grassmannian manifold G(N,L) such that theweighted sum of the mutual information J(Φ) is maximized.

To compute the gradient of J(Φ) on the Grassmannianmanifold, we first need to compute the derivative of J(Φ) withrespect to Φ. Taking the derivative of the weighted sum of theapproximated mutual information in (24) with respect to thepilot matrix Φ, we have dJ(Φ)

dΦ in (26), shown at the bottom ofthe page, where the first denominator is a positive real numberthat scales the assigned weight of the mutual information ofa specific user, which, in turn, affects the derivative direction,and the second denominator affects the derivative direction byscaling the activated probabilities of the Gaussian componentsin the mixture.

For the special case that there is only a single user in themassive MIMO system, by ignoring the index m in (4), theGaussian mixture distribution of the channel vector h is givenby

f(h) =∑k∈K

pkf(k)(h) =

∑k∈K

pkCN(u(k)h ,R

(k)hh

), (27)

where K denotes the cardinality of the index set K ={1, 2, · · · ,K}. Accordingly, the derivative of the weighted

sum of the approximated mutual information degenerates intothe approximated mutual information gradient as [1]

d

dΦI(y;h) ≈

∑k∈K

pk

∣∣∣R(k)yy

∣∣∣−1 [R(k)

yy

]−1

ΦR(k)hh∑

k∈Kpk

∣∣∣R(k)yy

∣∣∣−1 , (28)

where the denominator does not affect the derivative directionbut the convergence rate.

According to [48], for the function J(Φ) defined on theGrassmannian manifold, the gradient of J(Φ) at Φ is definedto be the tangent vector ∇ΦJ(Φ) such that

Tr

[(dJ(Φ)

dΦ

)H

∆

]≡ Tr

[(∇ΦJ(Φ))H∆

](29)

for all tangent vectors ∆ at Φ. Solving (29) for ∇ΦJ(Φ) suchthat (∇ΦJ(Φ))ΦH = 0 yields the Grassmannian gradient as

∇ΦJ(Φ) =dJ(Φ)

dΦ

(I −ΦHΦ

), (30)

which is the steepest ascent direction on the manifold. Usingthe steepest ascent method on the Grassmannian manifoldG(N,L), the pilot matrix is updated according to

Φ = Φ+ γ∇ΦJ(Φ), (31)

where γ > 0 is a small step size. The updated pilot matrixis a linear combination of the current pilot matrix and theGrassmannian gradient of the weighted sum of the approx-imated mutual information with respect to the pilot matrix,which generally lies outside the manifold G(N,L).

To project the updated pilot Φ back onto G(N,L), theorthonormal constraint in (18) is enforced by seeking the closetrow-orthonormal matrix to the updated pilot matrix Φ, whichis a orthogonal Procrustes problem [51]. The orthonormal pilotmatrix closest to Φ is given by

Φ = DIL×NGH, (32)

where DΣGH = Φ is the singular value decomposition(SVD) of Φ with D ∈ CL×L and G ∈ CN×N being theunitary matrices, IL×N =

[IL×L,0L×(N−L)

]is a rectangular

diagonal matrix composed by an identity matrix and a zeromatrix. Namely, the orthonormal pilot matrix Φ has the sameunitary matrices as the updated pilot matrix Φ, but with allone singular values. Then, Φ is used to substitute Φ in thenext iteration in calculating (26) and then updated using (31)

dJ(Φ)

dΦ≈ −

∑m∈M

wmd

dΦ

{log

[ ∑k∈Km

p(m)k

∣∣∣R(k)ymym

∣∣∣−1]}

= −∑

m∈M

wm∑k∈Km

p(m)k

∣∣∣R(k)ymym

∣∣∣−1

∑k∈Km

p(m)k

d

dΦ

{∣∣∣R(k)ymym

∣∣∣−1}

=∑

m∈M

wm∑k∈Km

p(m)k

∣∣∣R(k)ymym

∣∣∣−1

∑k∈Km

p(m)k∣∣∣R(k)ymym

∣∣∣[R(k)

ymym

]−1

ΦR(k)hmhm

, (26)

7

Algorithm 1 : Information-theoretic pilot optimization on the GrassmannianInitialize: Random pilot matrix Φ;Repeat

Step 1: Compute the derivative of the weighted sum of the approximated mutual information dJ(Φ)dΦ via (26);

Step 2: Compute the Grassmannian gradient ∇ΦJ(Φ) via (30);Step 3: Update the pilot matrix Φ in the steepest ascent direction on G(N,L) via (31);Step 4: Enforce the orthonormal constraint to obtain Φ via (32), and let Φ← Φ. Go back to Step 1.

Until convergenceOutput: Information-theoretic pilot matrix Φ.

to achieve an iterative search procedure. The convergencecriterion of the iterative process requires that the gradientof the weighted sum mutual information with respect to thepilot matrix converges to zero. Algorithm 1 summarizes theproposed information-theoretic pilot design for the downlinkchannel estimation in FDD massive MIMO systems.

From (26), the computational complexity of calculatingthe derivative of the weighted sum of the approximatedmutual information with respect to the pilot matrix isO(LN2

∑m∈M Km

)because L ≪ N . Hence, the overall

computational complexity of the proposed pilot optimizationmethod is O

(TLN2

∑m∈M Km

), where T denotes the num-

ber of iteration.For effective channel estimation at the user terminal, the

base station is required to send the optimized pilot symbols toall users served in the massive MIMO system. The signalingoverhead of the optimized pilot matrix is generally very high.To reduce the actual overhead required in practical systemimplementations, one can predesign a set of sequences whichare shared by the base station and the users [22, 33]. As such,the base station only needs to transmit the indexes of thechosen sequences, thus greatly relaxing the required signalingoverhead.

V. SIMULATION RESULTS

In this section, we carry out simulations to demonstrate theperformance advantages of the proposed pilot design methodfor downlink channel estimation in FDD massive MIMOsystems. Throughout the simulations, we assume that the basestation is equipped with a uniform linear array (ULA) withN = 100 omnidirectional antennas spaced a half wavelengthapart. It is worth noting that there is no limit on the arraygeometry for the proposed pilot design to apply. That is, itcan be applied to an arbitrary array geometry, such as two-dimensional array geometry [41].

In order to remove the power differences among differentuser channels and the effects of user channel SNR scaling onabsolute error levels, we utilize the normalized MSE (NMSE),defined as

NMSE(hm)=1

NMC

NMC∑q=1

∥∥∥hm(q)−hMMSEm (q)

∥∥∥2∥hm(q)∥2

, ∀ m ∈M,

(33)to evaluate the channel estimation performance, whereh

MMSEm (q) is the MMSE estimate of hm(q), i.e., the m-th

user’s channel obtained in the q-th Monte Carlo trial. Here,

NMC = 5, 000 is the number of Monte-Carlo trials. The stepsize for the iterative search for the optimal pilot is set asγ = 0.1.

In our signal model, both the pilot matrix optimization andthe channel vector estimation depend on the a prior knowledgeof the downlink channel vector. In practical applications, thisknowledge is unavailable and needs to be estimated beforeperforming the pilot optimization and channel estimation. Con-sidering the time-varying nature of wireless communications,it is more appropriate to model the downlink channel vectoras a mixture distribution, e.g., Gaussian mixture distributionconsidered in this paper, rather than a single, smooth Gaussiandistribution despite that the latter offers a closed-form solution[26].

A. Gaussian mixture approximation

It is well known that the EM algorithm performs maximumlikelihood estimation of Gaussian mixture distribution [38],and there are some approximations (see [39] and referencestherein). However, the EM algorithm may not be the bestchoice for the channel characterization in massive MIMOsystems due to not only the high dimensionality of channelvectors but also the limited training samples especially atthe beginning of cell handover. Here, we adopt a piecewise-Gaussian approximation to model the Gaussian mixture dis-tribution of channel vectors in massive MIMO.

In typical massive MIMO cellular systems, the channelvector from the base station to the user terminal is highlycorrelated due to the narrow angular spread [21, 47], and canbe modeled by

h = R12

hhv, (34)

where Rhh ≽ 0 denotes a positive semi-definite channelcovariance matrix, and v ∼ CN (0, I) denotes a zero-meancomplex-valued Gaussian random vector. In this paper, as wemodel the channel vector as a Gaussian mixture vector withKm components, the channel associated with the m-th user isexpressed as

hm =(R

(k)hmhm

) 12v, ∀ m ∈M, (35)

with an activation probability of p(m)k , where k ∈ Km. Then,

the pdf of channel vector is equivalent to the expression off(hm) in (4).

A Gaussian mixture model for channel vectors of users islearned from the a priori power azimuth spread of the user

8

SNR (dB)-20 -10 0 10 20 30 40

NM

SE

(dB

)

-50

-40

-30

-20

-10

0

Random Pilot (L = 6)Random Pilot (L = 8)Random Pilot (L = 10)Optimized Pilot (L = 6)Optimized Pilot (L = 8)Optimized Pilot (L = 10)

Fig. 2. NMSE performance comparison of the channel estimation versus theinput SNR of the channel for different numbers of pilot symbols.

channel. Specifically, the channel covariance matrix in (35) isgenerated as

R(k)hmhm

=

∫A(k)

σ2hm

a(θ)aH(θ)dθ (36)

according to the piecewise-Gaussian approximation, wherea(θ) =

[1, e−jπ sin θ, · · · , e−jπ(N−1) sin θ

]T is the steeringvector of the ULA, σ2

hmis the channel power of the m-th

user, and A(k) denotes the k-th observation region at thebase station (e.g., A(k)

∩A(k

′) = ∅,∀ k, k

′ ∈ Km and∪k∈Km

A(k) = (−π/2, π/2] for the ULA). In the simulations,the observation regions are assumed to have the same value ofA(k) = 1◦,∀ k ∈ Km. The corresponding probability of thek-th Gaussian component, which reflects the power azimuthspread, can be modeled by a Laplacian distribution as [52–54]

p(m)k =

1√2σ

(m)AS

e−

√2|θk−θm|σ(m)AS , (37)

where θm and σ(m)AS respectively denote the mean DOA and

the azimuth spread of downlink channel associated with them-th user. Although the Laplacian distribution is the mostpopular distribution in the typical outdoor propagations, thereare other classes of distributions that are applicable in certaincircumstances [54].

B. Single-user scenarios

In the first example, a simple scenario with a single useris considered, where the mean DOA is randomly distributedas a uniform distribution over the interval (−90◦, 90◦], i.e.,θ ∼ U(−90◦, 90◦], and the azimuth spread is set as σAS = 3◦.Both the mean DOA and the azimuth spread are assumed tobe known at the base station.

Numerical results show that the maximum rank of thechannel covariance matrices over different Gaussian compo-nents is 8, i.e., maxk∈K r(k) = maxk∈K rank

(R

(k)hh

)= 8. To

understand the impact of the number of pilot symbols on the

SNR (dB)-20 -10 0 10 20 30 40

NM

SE

(dB

)

-50

-40

-30

-20

-10

0

Random Pilot (σAS

= 1.5°)

Random Pilot (σAS

= 3°)

Random Pilot (σAS

= 6°)

Optimized Pilot (σAS

= 1.5°)

Optimized Pilot (σAS

= 3°)

Optimized Pilot (σAS

= 6°)

Fig. 3. NMSE performance comparison of the channel estimation versus theinput SNR of the channel for different practical azimuth spread.

channel estimation performance, we consider three differentchoices of the number of pilot symbols compared with themaximum rank of the channel covariance matrices, i.e., L =10 > maxk∈K rank

(R

(k)hh

), L = 8 = maxk∈K rank

(R

(k)hh

),

and L = 6 < maxk∈K rank(R

(k)hh

), respectively. In Fig.

2, we depict the NMSEs of the channel estimation versusthe input SNR of the channel with different lengths of pilotsymbols, where the optimized pilot symbols and the randompilot symbols are compared. The NMSE performance is clearlya function of the input SNR for both the optimized andrandom pilot symbols, where the channels are scaled by√

SNR to model varying the quality of channel. From Fig.2, it is observed that the channel estimation performance canbe greatly improved by using the proposed pilot symbols ascompared to the random generated pilot symbols for the fixednumber of pilot symbols. The performance advantage becomesmore pronounced as the input SNR increases. It is alsoobserved that, when the number of the optimized pilot symbolsreaches the maximum rank of the channel covariance matricesover different Gaussian components, the channel estimationperformance cannot be further improved by increasing thenumber of pilot symbols. On the contrary, when the numberof pilot symbols is less than the maximum rank of the channelcovariance matrices, the channel estimation performance withthe proposed pilot is unstable, and is worse than that of theproposed pilot which number is no less than the maximumrank of the channel covariance matrices.

As the approximated mutual information gradient (28)shows, the proposed pilot optimization depends on the statisti-cal information of the channel vectors to be estimated. Hence,it is necessary to study the robustness of the proposed pilotoptimization for channel estimation against the perturbationof the channel knowledge. In Fig. 3, we consider a situationwhere the assumed azimuth spread in the a priori channeldistribution remains 3◦, whereas the actual azimuth spreadvaries between 1.5◦ (smaller than the assumed value) and 6◦

(larger than the assumed value). It is clear that there is no

9

SNR (dB)-20 -10 0 10 20 30 40

NM

SE

(dB

)

-50

-40

-30

-20

-10

0

Random Pilot (User 1)Random Pilot (User 2)Optimized Pilot (User 1)Optimized Pilot (User 2)

(a)

SNR (dB)-20 -10 0 10 20 30 40

NM

SE

(dB

)

-50

-40

-30

-20

-10

0

Random Pilot (User 1)Random Pilot (User 2)Optimized Pilot (User 1)Optimized Pilot (User 2)

(b)

SNR (dB)-20 -10 0 10 20 30 40

NM

SE

(dB

)

-50

-40

-30

-20

-10

0

Random Pilot (User 1)Random Pilot (User 2)Optimized Pilot (User 1)Optimized Pilot (User 2)

(c)

SNR (dB)-20 -10 0 10 20 30 40

NM

SE

(dB

)

-50

-40

-30

-20

-10

0

Random Pilot (User 1)Random Pilot (User 2)Optimized Pilot (User 1)Optimized Pilot (User 2)

(d)

Fig. 4. NMSE performance comparison of the channel estimation versus the input SNR of the channels. (a) Equal user weight (w1 = w2 = 0.5); (b)Different user weight (w1 = 0.7, w2 = 0.3); (c) Different user weight (w1 = 0.9, w2 = 0.1); (d) Extreme user weight (w1 = 1, w2 = 0).

obvious performance loss in the estimated channel when theactual azimuth spread is smaller than the assumed one, whilea significant performance loss is observed when the actualazimuth spread is higher than the assumed value. Hence, forthe proposed pilot optimization to achieve effective channelestimation, the a priori distribution should at least cover theactual spreading of the channel to be estimated.

C. Multi-user scenarios

In the second example, we examine the multi-user case. Asystem with two separated users is considered as an examplewithout loss of generality. It is assumed that the two usersrespectively have the mean DOAs of θ1 = 0◦ and θ2 = 50◦,but with a same azimuth spread σ

(m)AS = 3◦,m = 1, 2. The

mean DOAs and the azimuth spread are assumed to be knownat the base station. The two user channels are further assumedto have the same power.

In Fig. 4, we compare the NMSE of the channel estimation

versus the input SNR of the channel with different userweights (wm,m ∈ M), where the length of pilot symbolsis fixed to be the maximum rank of the channel covari-ance matrices of all Gaussian components of all users, i.e.,L = 8 = maxm∈M maxk∈Km rank

(R

(k)hmhm

). It is clear

from Fig. 4(a) that the two users with equal weights havealmost identical channel estimation performance. From Fig.4(a) to Fig. 4(d), we observe that, as the user weight increases,the corresponding user achieves a better channel estimationperformance while the performance of the other user degrades.In the extreme scenario where one user occupies the wholeweight (i.e., wm = 1), as Fig. 3(d) shows, the correspondinguser has the same channel estimation performance as that inthe first example with a single-user scenario, while the channelestimation performance of the other user is even worse thanthat with random pilot symbols. Hence, in order to providethe desired quality of service (QoS) to all users served in thesame massive MIMO system, we prefer to assign the same

10

SNR (dB)-20 -10 0 10 20 30 40

Ave

rage

NM

SE

(dB

)

-50

-40

-30

-20

-10

0

Random Pilot (5 users)Optimized Pilot (5 users)Random Pilot (7 users)Optimized Pilot (7 users)Random Pilot (10 users)Optimized Pilot (10 users)Random Pilot (15 users)Optimized Pilot (15 users)Random Pilot (18 users)Optimized Pilot (18 users)

Fig. 5. Average NMSE performance comparison of the channel estimationversus the number of served users.

user weight in the proposed pilot design.To further evaluate the channel estimation performance with

a higher number of served users, we compare the averageNMSEs of the channel estimation versus the input SNR of thechannel with different numbers of the simultaneously servedusers. Here, we define the average NMSE over different usersas

Average NMSE =1

M

M∑m=1

NMSE(hm). (38)

In the simulations, the number of simultaneously served usersincreases from 5 to 7, 10, 15 and 18, which mean DOAs arerespectively uniform distributed over [−30◦, 30◦], [−36◦, 36◦],[−60◦, 75◦], [−84◦, 84◦], and [−85◦, 85◦]. All served users areassumed to have the same channel quality (i.e., SNR) withthe same azimuth spread σ

(m)AS = 3◦,∀m ∈ M and the same

weights wm = 1M ,∀m ∈ M (e.g., wm = 1

5 for 5 users). Itis observed from Fig. 5 that the average channel estimationperformance of the proposed pilot design method degrades

as the number of served users increases. Nevertheless, theperformance advantage of the proposed pilot design methodover the random pilot design remains significant, especiallyfor high-gain channels. An interesting observation is that,when we continue to increase the number of served users,the average channel estimation performance of the proposedpilot will no longer degrade.

VI. CONCLUSION

Pilot overhead for FDD downlink channel estimation isa challenging problem in massive MIMO systems equippedwith a very large number of antennas at the base station.By modeling the channel vector as a flexible and tractableGaussian mixture distribution, we first proved that the chan-nel can be perfectly recovered in the asymptotic high-SNRregime, when the number of pilot symbols is not less thanthe maximum rank of the channel covariance matrices ofall Gaussian components of all users. Then, we proposed aninformation-theoretic pilot design by maximizing the weightedsum of the mutual information between the measurements ofall served users and their corresponding channel vectors on theGrassmannian manifold. Hence, the proposed pilot can servean arbitrary number of users in FDD massive MIMO systemsby exploiting the a priori knowledge of the channel vectors tobe estimated. With the available Gaussian mixture distribution,there is a closed-form solution to the underdetermined channelestimation problem under the MMSE criterion. Simulationresults demonstrated that the proposed pilot outperforms therandom pilot in terms of the NMSE of the channel estimationversus the input SNR of the channel.

APPENDIX AProof: Similar to the proof in [26], the lower bound of

the MSE of the MMSE estimator of the channel vector hm

can be written as (39), shown at the bottom of the page, wherethe first term vanishes as σ2

nm→ 0.

Now, let us examine the second term. Because the rank ofR

(k)hmhm

is r(k)m , the eigen-decomposition of R

(k)hmhm

can bedenoted as

R(k)hmhm

= U (k)m Λ(k)

m

(U (k)

m

)H, (40)

ε(m)Lower =

∑k∈Km

p(m)k Tr

[R

(k)hmhm

−R(k)hmhm

ΦH(ΦR

(k)hmhm

ΦH + σ2nm

I)−1

ΦR(k)hmhm

]=

∑k∈Km

p(m)k Tr

[(R

(k)hmhm

) 12

(I −

(R

(k)hmhm

) 12ΦH

(ΦR

(k)hmhm

ΦH + σ2nm

I)−1

Φ(R

(k)hmhm

) 12

)(R

(k)hmhm

) 12

]=

∑k∈Km

p(m)k Tr

[(R

(k)hmhm

) 12

(I + σ−2

nm

(R

(k)hmhm

) 12ΦHΦ

(R

(k)hmhm

) 12

)−1(R

(k)hmhm

) 12

]=

∑k∈Km

p(m)k Tr

[(R

(k)hmhm

) 12V (k)

m

(I + σ−2

nmΓ(k)m

)−1(V (mk)

)H(R

(k)hmhm

) 12

]=

∑k∈Km

p(m)k Tr

[R

(k)hmhm

V (k)m

(I + σ−2

nmΓ(k)m

)−1(V (mk)

)H]

=∑

k∈Km

p(m)k

r(k)m∑i=1

(1 + γi/σ

2nm

)−1 (v(k)mi

)HR

(k)hmhm

v(k)mi

+∑

k∈Km

p(m)k

N∑i=r

(k)m +1

(v(k)mi

)HR

(k)hmhm

v(k)mi

, (39)

11

where U (k)m ∈ CN×r(k)

m and Λ(k)m ∈ Cr(k)

m ×r(k)m . We can write(

R(k)hmhm

) 12ΦH = U (k)

m

(Λ(k)

m

) 12(U (k)

m

)HΦH

= U (k)m C(k)

m , (41)

where C(k)m =

(Λ(k)

m

) 12(U (k)

m

)HΦH ∈ Cr(k)

m ×L. WhenL ≥ maxm∈M maxk∈Km r

(k)m and the pilot symbols of Φ

are randomly generated according to some distribution, thematrix C(k)

m has a full row rank with probability one, i.e.,rank

(C(k)

m

)= r

(k)m . Hence, let Range( · ) denote the range of

a matrix, i.e., the column space spanned by its column vectors.Then,

Range((

R(k)hmhm

) 12ΦH

)= Range

((R

(k)hmhm

) 12ΦHΦ

(R

(k)hmhm

) 12

)= Range

(U (k)

m

), ∀ k ∈ Km,m ∈M. (42)

Therefore, we have(u(k)mi

)HR

(k)hmhm

=(v(k)mi

)HU (k)

m

= v(k)mi

(R

(k)hmhm

) 12ΦHΦ

(R

(k)hmhm

) 12

= 0, ∀ i = r(k)m + 1, · · · , N. (43)

Hence, the second term in (39) disappears when the number ofpilot symbols is no less than the maximum rank of the channelcovariance matrices of all Gaussian components of all users,i.e., L ≥ maxm∈M maxk∈Km r

(k)m . That is to say, the lower

bound of the MSE of the MMSE estimate of hm approacheszero in the limit of vanishing noise,

limSNRm→∞

ε(m)Lower = 0. (44)

In the Gaussian mixture distribution, the upper and lowerbounds of the MSE of the MMSE estimate approach each otherwith increasing SNR, and they must coincide as the SNR tendsto infinity, i.e., [42],

limSNRm→∞

ε(m)Upper = lim

SNRm→∞ε(m)Lower = 0, (45)

which implies that

limSNRm→∞

E{∥∥∥hm − h

MMSEm

∥∥∥22

}= 0. (46)

APPENDIX B

For completeness, we briefly recall the result in [36] for thederivation of the approximated differential entropy (19). Per-forming the first-order Taylor series expansion of log [f(ym)]

around y(m)0 = E[ym] = uym

, i.e., the mean value of ym, wehave

log [f(ym)] ≈ log[f(y(m)0

)]+ gH

(y(m)0

) [ym − y

(m)0

],

(47)where log

[f(y(m)0

)]= log

[∑k∈Km

p(m)k f (k)

(y(m)0

)], and

g(y(m)0

)= d

dymlog [f(ym)]

∣∣ym=y

(m)0

denotes the first deriva-

tive of log [f(ym)] evaluated at the point y(m)0 . Substituting

(47) into the definition of differential entropy, we obtain theapproximated differential entropy (48), shown at the bottomof the page.

ACKNOWLEDGMENT

The authors would like to thank the Associate Editor Prof.Bruno Clerckx and the anonymous reviewers for their helpfulcomments and suggestions.

REFERENCES

[1] Y. Gu and Y. D. Zhang, “Pilot design for Gaussian mixturechannel estimation in massive MIMO,” in Proc. IEEE Int. Conf.Acoust., Speech, Signal Process., Calgary, Canada, Apr. 2018,pp. 3266–3270.

[2] F. Rusek, D. Persson, Buon K. Lau, E. G. Larsson, T. L.Marzetta, O. Edfors, and F. Tufvesson, “Scaling up MIMO:Opportunities and challenges with very large arrays,” IEEESignal Process. Mag., vol. 30, no. 1, pp. 40–60, Jan. 2013.

[3] R. C. de Lamare, “Massive MIMO systems: Signal processingchallenges and future trends,” URSI Radio Sci. Bull., vol. 2013,no. 347, pp. 8–20, Dec. 2013.

[4] E. G. Larsson, F. Tufvesson, O. Edfors, and T. L. Marzetta,“Massive MIMO for next generation wireless systems,” IEEECommun. Mag., vol. 52, no. 2, pp. 186–195, Feb. 2014.

[5] L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhmin, andR. Zhang, “An overview of massive MIMO: Benefits andchallenges,” IEEE J. Sel. Topics Signal Process., vol. 8, no.5, pp. 742–758, Oct. 2014.

[6] Z. Gao, L. Dai, D. Mi, Z. Wang, M. A. Imran, and M. Z. Shakir,“MmWave massive-MIMO-based wireless backhaul for the 5Gultra-dense network,” IEEE Wireless Commun., vol. 22, no. 5,pp. 13–21, Oct. 2015.

h(ym) = −∫ ∑

k∈Km

p(m)k f (k)

(ym

)log [f(ym)] dym

≈ −∑

k∈Km

p(m)k

∫f (k)

(ym

){log

[f(y(m)0

)]+ gH

(y(m)0

) [ym − y

(m)0

]}dym

= −∑

k∈Km

p(m)k

{log

[f(y(m)0

)]+ gH

(y(m)0

) [u(k)ym− y

(m)0

]}= −log

[f(y(m)0

)]− gH

(y(m)0

) [ ∑k∈Km

p(m)k u(k)

ym− y

(m)0

]= −log

[f(y(m)0

)]. (48)

12

[7] D. Ciuonzo, P. S. Rossi, and S. Dey, “Massive MIMO channel-aware decision fusion,” IEEE Trans. Signal Process., vol. 63,no. 3, pp. 604–619, Feb. 2015.

[8] X.-C. Gao, J.-K. Zhang, Z.-Y. Wang and J. Jin, “Optimalprecoder design maximizing the worst-case average receivedSNR for massive distributed MIMO systems,” IEEE Commun.Lett., vol. 19, no. 4, pp. 589–592, Apr. 2015.

[9] A. Shirazinia, S. Dey, D. Ciuonzo, and P. S. Rossi, “MassiveMIMO for decentralized estimation of a correlated source,”IEEE Trans. Signal Process., vol. 64, no. 10, pp. 2499–2512,May 2016.

[10] Y. Gu, Y. D. Zhang, and N. A. Goodman, “Optimized compres-sive sensing-based direction-of-arrival estimation in massiveMIMO,” in Proc. IEEE Int. Conf. Acoust., Speech, SignalProcess., New Orleans, LA, Mar. 2017, pp. 3181–3185.

[11] F. Zhu, N. Wu, and Q. Liang, “Channel estimation for massiveMIMO with 2-D nested array deployment,” Phys. Commun.,vol. 25, pp. 432–437, Dec. 2017.

[12] Z. Shao, R. C. de Lamare, L. T. N. Landau, “Iterative detectionand decoding for large-scale multiple-antenna systems with 1-bit adcs,” IEEE Wirelesss Commun. Lett., vol. 7, no. 3, pp.476–479, June 2018.

[13] C. Zhou, Y. Gu, X. Fan, Z. Shi, G. Mao, and Y. D. Zhang,“Direction-of-arrival estimation for coprime array via virtualarray interpolation,” IEEE Trans. Signal Process., vol. 66, no.22, pp. 5956–5971, Nov. 2018.

[14] C. Zhou, Y. Gu, Z. Shi, and Y. D. Zhang, “Off-grid direction-of-arrival estimation using coprime array interpolation,” IEEESignal Process. Lett., vol. 25, no. 11, pp. 1710–1714, Nov.2018.

[15] T. L. Marzetta, “Noncooperative cellular wireless with unlim-ited numbers of base station antennas,” IEEE Trans. WirelessCommun., vol. 9, no. 11, pp. 3590–3600, Nov. 2010.

[16] H. Yin, D. Gesbert, M. Filippou, and Y. Liu, “A coordinatedapproach to channel estimation in large-scale multiple-antennasystems,” IEEE J. Sel. Areas Commun., vol. 33, no. 2, pp. 264–273, Feb. 2013.

[17] R. R. Muller, L. Cottatellucci, and M. Vehkapera, “Blind pilotdecontamination,” IEEE J. Sel. Topics Signal Process., vol. 8,no. 5, pp. 773–786, Oct. 2014.

[18] D. Mi, M. Dianati, L. Zhang, S. Muhaidat, and R. Tafazolli,“Massive MIMO performance with imperfect channel reci-procity and channel estimation error,” IEEE Trans. Commun.,vol. 65, no. 9, pp. 3734–3749, Sept. 2017.

[19] X. Rao and V. K. N. Lau, “Distributed compressive CSITestimation and feedback for FDD multi-user massive MIMOsystems,” IEEE Trans. Signal Process., vol. 62, no. 12, pp.3261–3271, Jun. 2014.

[20] Z. Gao, L. Dai, Z. Wang, and S. Chen, “Spatially commonsparsity based adaptive channel estimation and feedback forFDD massive MIMO,” IEEE Trans. Signal Process., vol. 63,no. 23, pp. 6169–6183, Dec. 2015.

[21] A. Adhikary, J. Nam, J.-Y. Ahn, and G. Caire, “Joint spatialdivision and multiplexing – The large-scale array regime,” IEEETrans. Inf. Theory, vol. 59, no. 10, pp. 6441–6463, Oct. 2013.

[22] J. Choi, D. J. Love, and P. Bidigare, “Downlink trainingtechniques for FDD massive MIMO systems: Open-loop andclosed-loop training with memory,” IEEE J. Sel. Topics SignalProcess., vol. 8, no. 5, pp. 802–814, Oct. 2014.

[23] Z. Jiang, A. F. Molisch, G. Caire, and Z. Niu, “Achievablerates of FDD massive MIMO systems with spatial channelcorrelation,” IEEE Trans. Wireless Commun., vol. 14, no. 5,pp. 2868–2882, May 2015.

[24] C. Sun, X. Q. Gao, S. Jin, M. Matthaiou, Z. Ding, and C.

Xiao, “Beam division multiple access transmission for massiveMIMO communications,” IEEE Trans. Commun., vol. 63, no.6, pp. 2170–2184, Jun. 2015.

[25] W. Shen, L. Dai, B. Shim, S. Mumtaz, and Z. Wang, “JointCSIT acquisition based on low rank matrix completion for FDDmassive MIMO systems,” IEEE Commun. Lett., vol. 19, no. 12,pp. 2178–2181, Dec. 2015.

[26] J. Fang, X. Li, H. Li, and F. Gao, “Low-rank covariance-assisteddownlink training and channel estimation for FDD massiveMIMO systems,” IEEE Trans. Wireless Commun., vol. 16, no.3, pp. 1935–1947, Mar. 2017.

[27] X. Li, J. Fang, H. Li, and P. Wang, “Millimeter wave channelestimation via exploiting joint sparse and low-rank structures,”IEEE Trans. Wireless Commun., vol. 17, no. 2, pp. 1123–1133,Feb. 2018.

[28] M. B. Khalilsarai, S. Haghighatshoar, X. Yi, and G. Caire,“FDD massive MIMO via UL/DL channel covariance extrapo-lation and active channel sparsification”, arXiv:1803.05754.

[29] S. Haghighatshoar, M. B. Khalilsarai, and G. Caire, “Multi-band covariance interpolation with applications in massiveMIMO,” in Proc. IEEE Int. Symp. Inf. Theory, Vail, CO, June2018, pp. 386–390.

[30] Y. Han and J. Lee, “Two-stage compressed sensing for mil-limeter wave channel estimation,” in Proc. IEEE Int. Symp.Inf. Theory, Barcelona, Spain, July 2016, pp. 860–864.

[31] A. Liu, V. Lau, M. L. Honig, and L. Lian, “Compressive rftraining and channel estimation in massive mimo with limitedrf chains,” in IEEE Int. Conf. Commun., Paris, France, May2017.

[32] A. Ali, N. Gonzalez-Prelcic, and R. W. Heath, “Millimeter wavebeam-selection using out-of-band spatial information,” IEEETrans. Wireless Commun., vol. 17, no. 2, pp. 1038–1052, Feb.2018.

[33] S. Bazzi and W. Xu, “Downlink training sequence design forFDD multi-user massive MIMO systems,” IEEE Trans. SignalProcess., vol. 65, no. 18, pp. 4732–4744, Sept. 2017.

[34] C.-K. Wen, S. Jin, K.-K. Wong, J.-C. Chen, and P. Ting, “Chan-nel estimation for massive MIMO using Gaussian-mixtureBayesian learning,” IEEE Trans. Wireless Commun., vol. 14,no. 3, pp. 1356–1368, Mar. 2015.

[35] K. Lemke-Rust and C. Paar, “Gaussian mixture models forhigher-order side channel analysis,” in Proc. Workshop Cryp-tograph. Hardw. Embedded Syst., Vienna, Austria, Sept. 2007,pp. 14–27.

[36] Y. Gu, N. A. Goodman, and A. Ashok, “Radar target profilingand recognition based on TSI-optimized compressive sensingkernel,” IEEE Trans. Signal Process., vol. 62, no. 12, pp. 3194–3207, June 2014.

[37] Y. Gu and N. A. Goodman, “Information-theoretic compressivesensing kernel optimization and Bayesian Cramer-Rao boundfor time delay estimation,” IEEE Trans. Signal Process., vol.65, no. 17, pp. 4525–4537, Sept. 2017.

[38] A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximumlikelihood from incomplete data via the EM algorithm,” J. Roy.Statist. Soc., vol. 39, no. 1, pp. 1–38, 1977.

[39] J. P. Vila and P. Schniter, “Expectation-maximization Gaussian-mixture approximate message passing,” IEEE Trans. SignalProcess., vol. 61, no. 19, pp. 4658–4672, Oct. 2013.

[40] M. E. Tipping, “Sparse Bayesian learning and the relevancevector machine,” J. Mach. Learn. Res., vol. 1, pp. 211–244,June 2001.

[41] J. Dai, A. Liu, and V. Lau, “FDD massive MIMO channelestimation with arbitrary 2D-array geometry,” IEEE Trans.Signal Process., vol. 66, no. 10, pp. 2584–2599, May 2018.

13

[42] J. T. Flam, S. Chatterjee, K. Kansanen, and T. Ekman, “OnMMSE: A linear model under Gaussian mixture statistics,”IEEE Trans. Signal Process., vol. 60, no. 7, pp. 3840–3844,Jul. 2012.

[43] S. M. Kay, Fundamentals of Statistical Signal Processing:Estimation Theory. Upper Saddle River, NJ, USA: Prentice-Hall, 1993.

[44] D. Guo, S. Shamai (Shitz), and S. Verdu, “Mutual informationand minimum mean-square error in Gaussian channels,” IEEETrans. Inf. Theory, vol. 51, no. 4, pp. 1261–1282, Apr. 2005.

[45] T. M. Cover and J. A. Thomas, Elements of Information Theory,2nd Edition. Hoboken, NJ, USA: John Wiley & Sons, 2006.

[46] J.-Y. Le Boudec, “Rate adaptation, congestion control andfairness: A tutorial,” Ecole Polytechnique Federale de Lau-sanne (EPFL), Switzerland, 2003, Tech. Rep. Available athttp://ica1www.epfl.ch/PS files/LEB3132.pdf.

[47] D. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, “Fadingcorrelation and its effect on the capacity of multi-elementantenna systems,” IEEE Trans. Commun., vol. 48, no. 3, pp.502–513, Mar. 2000.

[48] A. Edelman, T. Arias, and S. Smith, “The geometry of algo-rithms with orthogonality constraints,” SIAM J. Matrix AnalysisApplicat., vol. 20, no. 2, pp. 303–353, 1998.

[49] T. E. Abrudan, J. Eriksson, and V. Koivunen, “Steepest descentalgorithms for optimization under unitary matrix constraint,”IEEE Trans. Signal Process., vol. 56, no. 3, pp. 1134–1147,Mar. 2008.

[50] J. H. Manton, “Optimization algorithms exploiting unitaryconstraints,” IEEE Trans. Signal Process., vol. 50, no. 3, pp.635–650, Mar. 2002.

[51] P. H. Schonemann, “A generalized solution of the orthogonalProcrustes problem”, Psychometrika, vol. 31, no. 1, pp. 1–10,Mar. 1966.

[52] K. I. Pedersen, P. E. Mogensen, and B. H. Fleury, “Powerazimuth spectrum in outdoor environments,” Electron. Lett.,vol. 33, no. 18, pp. 1583–1584, Aug. 1997.

[53] L. M. Correia, Wireless Flexible Personalized Communications.Hoboken, NJ, USA: John Wiley & Sons, 2001.

[54] S. Saunders and A. Aragon-Zavala, Antennas and Propagationfor Wireless Communication Systems, 2nd Edition. Hoboken,NJ, USA: John Wiley & Sons, 2007.

Yujie Gu (SM’16) received the Ph.D. degree inelectronic engineering from Zhejiang University,Hangzhou, China, in 2008. After graduation, heheld multiple research positions in China, Canada,Israel, and the USA. He is currently a Postdoc-toral Research Associate with Temple University,Philadelphia, PA, USA. His research interests arein statistical and array signal processing includingadaptive beamforming, compressive sensing, MIMOsystems, radar imaging, target localization, wave-form design, etc.

Dr. Gu is currently a Subject Editor for Electronics Letters, an AssociateEditor for Signal Processing, IET Signal Processing, Circuits, Systems andSignal Processing, and EURASIP Journal on Advances in Signal Processing.He is also the Lead Guest Editor of special issue on Source Localization inMassive MIMO for the Digital Signal Processing. He has been a memberof the Technical Program Committee of multiple international conferencesincluding IEEE ICASSP. He is also a Special Sessions Co-Chair of the 2020IEEE Sensor Array and Multichannel Signal Processing Workshop.

Yimin D. Zhang (F’19) received the Ph.D. degreefrom the University of Tsukuba, Tsukuba, Japan, in1988. He is currently an Associate Professor withthe Department of Electrical and Computer Engi-neering, College of Engineering, Temple University,Philadelphia, PA, USA. From 1998 to 2015, he wasa Research Faculty with the Center for AdvancedCommunications, Villanova University, Villanova,PA, USA. His research interests lie in the areasof statistical signal and array processing, includingcompressive sensing, machine learning, convex op-

timization, time-frequency analysis, MIMO systems, radar imaging, directionfinding, target localization and tracking, wireless and cooperative networks,and jammer suppression, with applications to radar, wireless communications,and satellite navigation. He has authored/coauthored more than 340 journaland conference papers and 14 book chapters in these areas.

Dr. Zhang is an Associate Editor for the IEEE Transactions on SignalProcessing and an Editor for the Signal Processing journal. He was an Asso-ciate Editor for the IEEE Signal Processing Letters during 2006–2010, and anAssociate Editor for the Journal of the Franklin Institute during 2007–2013.He is a member of the Sensor Array and Multichannel Technical Committeeand the Signal Processing Theory and Methods Technical Committee of theIEEE Signal Processing Society. He was the Technical Co-Chair of the 2018IEEE Sensor Array and Multichannel Signal Processing Workshop. He wasthe recipient of the 2016 IET Radar, Sonar & Navigation Premium Awardand the 2017 IEEE Aerospace and Electronic Systems Society Harry RoweMimno Award, and coauthored a paper that received the 2018 IEEE SignalProcessing Society Young Author Best Paper Award.