Research Article Limited Feedback Precoding for Massive MIMOLimited Feedback Precoding for Massive...

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2013, Article ID 416352, 9 pageshttp://dx.doi.org/10.1155/2013/416352

Research ArticleLimited Feedback Precoding for Massive MIMO

Xin Su, Jie Zeng, Jingyu Li, Liping Rong, Lili Liu, Xibin Xu, and Jing Wang

Tsinghua National Laboratory for Information Science and Technology, Tsinghua University, Beijing 100084, China

Correspondence should be addressed to Jie Zeng; [email protected]

Received 11 January 2013; Revised 18 July 2013; Accepted 7 August 2013

Academic Editor: Wei Xiang

Copyright © 2013 Xin Su et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The large-scale array antenna system with numerous low-power antennas deployed at the base station, also known as massivemultiple-input multiple-output (MIMO), can provide a plethora of advantages over the classical array antenna system. Precodingis important to exploit massive MIMO performance, and codebook design is crucial due to the limited feedback channel. In thispaper, we propose a new avenue of codebook design based on a Kronecker-type approximation of the array correlation structure forthe uniform rectangular antenna array, which is preferable for the antenna deployment of massive MIMO. Although the feedbackoverhead is quite limited, the codebook design can provide an effective solution to support multiple users in different scenarios.Simulation results demonstrate that our proposed codebook outperforms the previously known codebooks remarkably.

1. Introduction

High-rate data demand increases faster and faster with thenew generation of devices (smart phones, tablets, netbooks,etc.). However, the huge increase in demand can be hardlymet by current wireless systems. As is known to us, MIMOchannels, created by deploying antenna arrays at the trans-mitter and receiver, promise high-capacity and high-qualitywireless communication links by spatial multiplexing anddiversity. Basically, the more antennas the transmitter orthe receiver equipped with, the more degrees of freedomthat the propagation channel can provide, and the higherdata rate the system can offer. Therefore, there is significanteffort within the community to research and developmassiveMIMO technology, which is a hot topic nowadays [1].

FormultiuserMIMO systems, we can utilize precoding toexplore massive MIMO potentials. The essence of precodingtechniques is to mitigate the interuser interference and toimprove the effective received SNR. Herein, channel stateinformation at the transmitter (CSIT) is an essential compo-nent when trying to maximize massive MIMO performancevia precoding. In time division duplexing (TDD) system,channel reciprocity can be utilized for pilot training in theuplink to acquire the complete CSIT, but the pilot contam-ination and imperfect channel estimation based on uplink

pilots would lead to inaccuracy of the CSIT. In frequencydivision duplexing (FDD) system, the CSIT shall be acquiredvia the feedback channel, which is usually limited in practice.Hence, a finite set of precoding matrices, named codebook,known to both the receiver and the transmitter shouldbe predesigned. The receiver selects the optimal precodingmatrix from the codebook according to the channel stateinformation (CSI) and reports the precodingmatrix indicator(PMI) to the transmitter via the limited feedback channel [2].With this mechanism, the system can obtain performanceimprovement by employing a well-designed codebook.

For traditional MIMO systems, several codebooks havebeen proposed, such as Kerdock codebook [3], codebooksbased on vector quantization [4], Grassmannian packing [5],discrete Fourier transform (DFT) [6], and quadrature ampli-tude modulation [7]. Codebooks based on vector quantiza-tion have taken the channel distribution into account buthave to be redesigned when the channel distribution changes.For uncorrelated channels, the Grassmannian is nearly theoptimal codebook, but its construction requires numericaliterations, and with high storage requirement. The Kerdockcodebook has simple systematic construction, significantlylow storage, and selection computational requirements dueto finite quaternary alphabet. However, the Grassmanniancodebook and the Kerdock codebook are not optimized for

2 International Journal of Antennas and Propagation

ŝN𝑢

ŝ1d s x

P

HStream

mapping

Multi-userscheduling

Codewordlookup

Precodergeneration

Uplink feedback

Demodulating

Demodulating

Channel estimation

Channel estimation

Codeword selection

Codebook

Codebook

Codebook

Codeword selection

PMI1

PMIN𝑢

y1

UE1

UEN𝑢

yN𝑢

......

......

...

... x = Ps

BS

Precoding

Γ = {W}

PMIi(i = 1, 2, . . . , Nu)

Wi , i = 1, 2, . . . , Nu

Figure 1: The feedback precoding model of the multiuser MIMO system.

correlated channels when closely spaced (𝜆/2) antenna arraysare employed. This case with the substantial correlation canbe better reflected by DFT codebooks.

Traditionally, the uniform linear array (ULA) setup isadopted in a MIMO system. But for the consideration ofconstrained array aperture and aesthetic factor, the ULAsetup is not suitable for massive MIMO. Besides, the lineararray with antenna elements of identical gain patterns (e.g.,isotropic elements) brings the problem of front-back ambigu-ity and is also unable to resolve signal paths in both azimuthand elevation [8]. For these reasons, the massive MIMOmight employ two-dimensional array structures, such as theuniform rectangular array (URA). However, the current pro-posals ofDFT codebooks fail toworkwell on the new antennalayout, since the codebook construction is aimed at the lineararrays, such asULA.Thus,we need a new scheme to constructthe codebook to better reflect the channel properties.

The rest of the paper is outlined as follows. In Section 2,we introduce the system model. In Section 3, we present thecodebook designs including the novel codebook design forURA purpose. Then, we analyze the channel capacity andevaluate the performance of different codebooks in Section 4.Finally, we give some conclusions in Section 5.

2. System Model

In this paper, we focus on the downlink transmission of amassive MIMO system, that is, the base station (BS) withmassive antennas as the transmitter and the user equipment(UE) as the receiver, and we adopt the multiuser MIMOtransmission architecture with feedback precoding mecha-nism, as shown in Figure 1.The BS is equipped with𝑁

𝑡trans-

mit antennas, and 𝑁𝑢UEs with 𝑁

𝑟receive antennas, each is

served by the BS in the cell; and the maximum number ofspatial multiplexing UEs is 𝐾. In this paper, we assume that𝑁𝑟= 1,𝑁

𝑡≫ 𝐾, and𝐾 ≥ 1.

In order to report the CSI from the receiver to the trans-mitter via a limited feedback channel, we need to predesign acodebook. Given the codebook containing a list of codewordsW𝑖at both the transmitter side and the receiver side, each of

which reflects one state of the channel at a certain time, thereceiver can only report an index within the codebook, calledPMI, to the transmitter; then the transmitter obtains thePMI and queries the codebook to accomplish the precodingprocess.

For the codeword selection at the receiver, we can adoptthe capacity selection criterion [5] which is to maximize thechannel gain given the channel matrix Ĥ by traversing thewhole codebook Γ:

Wopt𝑘

= WΓ (Ĥ𝑘) = max

W∈Γ

WĤ𝑘

2

2. (1)

For a multiuser MIMO system, we need to select 𝐾 UEsto serve simultaneously from a larger number of UEs. Thus,a scheduling module is necessary to select proper UEs. AllUEs shall report their own requested codewords W

𝑖; and

then we can find out the UE set to schedule according tocertain scheduling criteria. After determining the scheduledUEs, the transmitter starts the following data transmissionprocedure: firstly, multiplexing streams denoted by the vectors are selected from the user data stream d inputting inthe system according to the multiuser scheduling module;secondly, the data is preprocessed by a precoder P which isformed by combining the UEs’ requested codewords columnby column, and we get the transmit signal x:

x = Ps, (2)

where s = [𝑠1, 𝑠2, . . . , 𝑠

𝐾], P = [W

𝑠(1),W𝑠(2)

, . . . ,W𝑠(𝐾)

], andW𝑠(𝑖)

denotes the codeword for scheduled UE 𝑠(𝑖). For thesake of clarity without misunderstanding, we use W

𝑖instead

of W𝑠(𝑖)

in the following expressions.

International Journal of Antennas and Propagation 3

After x passing the channel and being added the noise, wewill get the received signal y:

y = Hx + z = HPs + z, (3)

where H denotes the channel matrix of size (𝐾 ×𝑁𝑡) with its

entry 𝐻𝑖𝑗denoting the complex channel response from the

𝑗th transmit antenna to the 𝑖th UE’s receive antenna, and z isthe additive white complex Gaussian noise (AWGN) vectorwith covariance matrix 𝜎2I

𝐾. For UE 𝑘, the received signal 𝑦

𝑘

is

𝑦𝑘= H𝑘x𝑘+ ∑

𝑖 ̸= 𝑘

H𝑘x𝑖+ 𝑧𝑘,

= H𝑘W𝑘𝑠𝑘+ ∑

𝑖 ̸= 𝑘

H𝑘W𝑖𝑠𝑖+ 𝑧𝑘.

(4)

After UE 𝑘 obtains the estimated channel matrix Ĥ𝑘

through the channel estimation, it demodulates the receivedsignal 𝑦

𝑘:

𝑠𝑘= 𝑑𝑘𝑦𝑘. (5)

If the interference is unaware to the receiver, whichmeansthat it is treated as part of the noise, the matched filter (MF)is usually adopted:

𝑑𝑘= (Ĥ𝑘W𝑘)∗

. (6)

3. Codebook Design

The codebook design is a quantization problem, in which weshould balance the accuracy and the overhead of bits. TheGrassmannian codebook would offer the optimal solutionfor fully uncorrelated channels [5], but it is not practical formassive MIMO systems due to its difficulty of constructionfor higher-dimensional space. Kerdock can be easily extendedto massive transmit antennas due to its systematic construc-tion and low codeword selection complexity. However, likethe Grassmannian codebook, the Kerdock codebook is onlysuitable for uncorrelated channels. For high-correlated chan-nels, a DFT codebook able to respond the channel correlationprovides a good fit. Since the channels are likely to be highlycorrelated due to the closely spaced arrays probably utilizedby massive MIMO, we focus on the DFT codebook and itsextension for massive MIMO with the URA deployment.

3.1. The Traditional DFT Codebook. Closely spaced arrays,including both crosspolarized and copolarized ones, implycertain spatial correlation structures which may be utilizedto compress the channel into an effective channel of lowerdimensionality. With such setup, the channel is spatiallycorrelated, and the spatial covariance matrix can be approx-imated using its eigenvectors with the closely spaced (𝜆/2)array. The linear DFT codebook design targets at the linearclosely spaced array, which has two separate parts repre-senting long-term and short-term channel states, respectively[6, 9]. The first codeword W1 in the long-term feedback partconsists of multiple beams such that the beams cover the full

signal space over the wideband with closely-spaced arrays,hence capturing the correlation properties of the channel.The second codeword W2 in the short-term feedback partcombines the beams to capture short-term variations. Thefinal precoder W is given by

W = W1W2. (7)

This is the DFT codebook design for the ULA [10]. It isnoted that the design essentially comes from the concept ofthe adaptive codebook [11]. Under the assumption that thechannel correlation is both known by the transmitter withmultiple antennas and the receiver with single antenna, theprecoder can be computed by

W = R1/2V, (8)

whereR is the spatial covariancematrix of the channelH, thatis, R = 𝐸[H𝐻H], and V is a 𝑁

𝑡× 1 codeword of a uniform

codebook. Adaptive codebook is well known to provide goodperformance in correlated channels, especially for multiuserMIMO. In practical application, a double codebook is utilizedto minimize the feedback overhead. The transmitter andthe receiver could share the knowledge of the matrix Rthrough the feedback of first codeword W1. This feedbackoverhead is low even though the information of R is vast,since the feedback interval could be very long due to the stablechannel correlation. Besides, the second codeword W2 =V is reported more frequently to represent the short-termvariations.TheprecoderW = W1W2 can adapt to the angularspread of the channel by covering the instantaneous subspaceover the entire band.

For the copolarized arrays of closely spaced (𝜆/2) antennaelements, the codeword W1 in the long-term feedback code-book is expressed as

W1 (𝑔) = D (𝑔) , (9)

where D(𝑔) (𝑔 = 0, 1, . . . , 𝐺 − 1) are DFT rotation matriceswith the size𝑁 ×𝑀, 𝐺 is the total number of DFT matrices,and each element can be expressed as

[D (𝑔)]𝑛,𝑚

=1

√𝑁exp(𝑗2𝜋

𝑀𝑛(𝑚 +

𝑔

𝐺)) , (10)

where 𝑛 = 0, 1, . . . , 𝑁 − 1 and 𝑚 = 0, 1, . . . ,𝑀 − 1. Here, wehave 𝑁 = 𝑁

𝑡and 𝐺 = 2𝐵/𝑀, where 𝐵 is the codebook size,

that is, the feedback overhead (in bits).While the first codeword W1 describes the correlation

property of the channel, the second codeword W2 consistsof beam selection vectors. Since the size and the energy ofthe UEs are usually limited, frequently only one antenna isemployed in practical. As a result, rank-1 channels are essen-tial for current multiuser MIMO. For the rank-1 channel, ithas the form

W2 (𝑖2) = ^𝑖2

, (11)

where ^𝑘is a𝑀× 1 selection vector that has 1 on the 𝑘th row

and 0 elsewhere and 𝑖2indicates the index ofW2 in the second

part of the codebook.


Above are the constructions for the linear closely spacedantenna elements. Under such scenarios with the substan-tially correlated channels, the DFT-based codebook is ableto respond the correlation. However, for two-dimensionalantenna arrays, the long-term statistical properties of thechannel cannot be directly reflected by DFT vectors, sinceevery DFT vector can only represent the beams emitted bythe linear closely spaced antenna elements. Hence, we need toextend the DFT codebook to adapt to two-dimensional arraystructures, like the URA.

3.2. The Proposed Codebook

3.2.1. Kronecker-Type Approximation of Correlation. In thispaper, we consider a URA lying on the 𝑋𝑌 plane with 𝑥-axis parallel to one edge of the URA and 𝑦-axis parallelto the other vertical edge, as shown in Figure 2 taking 64copolarized antennas as an example.

We assume that the correlation between the antennaelements along 𝑥 does not depend on the antenna elementsalong 𝑦 and its correlation matrix is described as matrixR𝑥; the correlation along 𝑦 does not depend on the antenna

elements along 𝑥 and its correlation matrix is describedas matrix R

𝑦. Thus, we have the following Kronecker-type

approximation for the URA correlation matrix [12]:

R = R𝑥⊗ R𝑦, (12)

where ⊗ denotes the Kronecker product.Formula (12) indicates that the URA correlation matrix

R is the Kronecker product of two ULA correlation matricesR𝑥and R

𝑦.This approximationmodel is reasonably accurate,

allowing the well-developed theory of Toeplitz matrices forthe analysis of multidimensional antenna arrays. In thefollowing, we take the Kronecker-type approximation modelas the theoretical basis of the first codeword construction.

3.2.2. Codebook Construction

Theorem 1. If X and Y are diagonalizable square matrices,then

(X ⊗ Y)1/2 = X1/2 ⊗ Y1/2. (13)

Proof. Firstly, we prove that (D𝑥⊗ D𝑦)1/2

= D1/2𝑥

⊗ D1/2𝑦

fordiagonal matrices D

𝑥and D

𝑦.

Suppose thatD𝑥= diag(𝜆

1, 𝜆2, . . . , 𝜆

𝑛),D𝑦= diag(𝜇

1, 𝜇2,

. . . , 𝜇𝑛), then

D1/2𝑥

⊗ D1/2𝑦

= diag (𝜆1/21, 𝜆1/2

2, . . . , 𝜆

1/2

𝑛)

⊗ diag (𝜇1/21, 𝜇1/2

2, . . . , 𝜇

1/2

𝑛)

= diag ((𝜆1𝜇1)1/2

, (𝜆2𝜇1)1/2

, . . . , (𝜆𝑛𝜇1)1/2

,

(𝜆1𝜇2)1/2

, . . . , (𝜆𝑛𝜇2)1/2

, . . . , (𝜆𝑛𝜇𝑛)1/2

)

𝜆/2

𝜆/2

Figure 2: An example of the URA deployment for 64 antennas.

= (diag (𝜆1𝜇1, 𝜆2𝜇1, . . . , 𝜆

𝑛𝜇1, 𝜆1𝜇2,

𝜆2𝜇2, . . . , 𝜆

𝑛𝜇2, . . . , 𝜆

𝑛𝜇𝑛))1/2

= (D𝑥⊗ D𝑦)1/2

.

(14)

An 𝑛 × 𝑛matrix A is diagonalizable if there is a matrix V anda diagonal matrix D such that A = VDV−1. In this case, thesquare root of A is R = VD1/2V−1. With this rule, we canprove the equation for any diagonalizable square matrices Xand Y.

Suppose X = V𝑥

D𝑥

V−1𝑥

and Y = V𝑦

D𝑦

V−1𝑦, then

X1/2 ⊗ Y1/2 = (V𝑥

D1/2𝑥

V−1𝑥) ⊗ (V

𝑦D1/2𝑦

V−1𝑦)

= (V𝑥⊗ V𝑦) (D1/2𝑥

⊗ D1/2𝑦) (V−1𝑥⊗ V−1𝑦)

= (V𝑥⊗ V𝑦) (D1/2𝑥

⊗ D1/2𝑦) (V𝑥⊗ V𝑦)−1

,

(15)

X ⊗ Y = (V𝑥

D𝑥

V−1𝑥) ⊗ (V

𝑦D𝑦

V−1𝑦)

= (V𝑥⊗ V𝑦) (D𝑥⊗ D𝑦) (V−1𝑥⊗ V−1𝑦)

= (V𝑥⊗ V𝑦) (D𝑥⊗ D𝑦) (V𝑥⊗ V𝑦)−1

,

(16)

(X ⊗ Y)1/2 = (V𝑥⊗ V𝑦) (D𝑥⊗ D𝑦)1/2

(V𝑥⊗ V𝑦)−1

= (V𝑥⊗ V𝑦) (D1/2𝑥

⊗ D1/2𝑦) (V𝑥⊗ V𝑦)−1

.

(17)

The formulae (15) and (16) utilize the Kronecker productproperties includingmixed-product property and the inverseproperty [13].

Thus, (X ⊗ Y)1/2 = X1/2 ⊗ Y1/2.

Since first codeword W1 reflects the channel correlation,we shall design it to satisfy

W1 = R1/2. (18)


With formula (12) andTheorem 1, we have

W1 = R1/2 = (R𝑥⊗ R𝑦)1/2

= R1/2𝑥

⊗ R1/2𝑦. (19)

Hence, assuming D𝑥(𝑔𝑥) and D

𝑦(𝑔𝑦) are two DFT rota-

tionmatrices designed for two orthogonal ULAs, we can con-structW1 for the URAwith the Kronecker product of the twomatrices:

W1 (𝑖1) = D

𝑥(𝑔𝑥) ⊗ D𝑦(𝑔𝑦) ,

[D𝑥(𝑔𝑥)]𝑛,𝑚

=1

√𝑁𝑥

exp(𝑗 2𝜋𝑀𝑥

𝑛(𝑚 +𝑔𝑥

𝐺𝑥

)) ,

[D𝑦(𝑔𝑦)]𝑛,𝑚

=1

√𝑁𝑦

exp(𝑗 2𝜋𝑀𝑦

𝑛(𝑚 +𝑔𝑦

𝐺𝑦

)) ,

(20)

where 𝑖1indicates the index of W1 in the long-term feedback

codebook, and 𝑔𝑥and 𝑔

𝑦denote the indexes of the rotation

DFT matrices for two directions, respectively. We have thefollowing:

𝑖1= 𝐺𝑦𝑔𝑥+ 𝑔𝑦, 𝑔𝑥= 0, 1, . . . , 𝐺

𝑥− 1,

𝑔𝑦= 0, 1, . . . , 𝐺

𝑦− 1.

(21)

Take the URA-64 in Figure 2 as an example: we have𝑁𝑥= 8,

𝑁𝑦= 8, and W1 with the size of 64 ×𝑀

𝑥𝑀𝑦.

The construction of the𝑀𝑥𝑀𝑦× 1W2 is the same as the

one for the ULA purpose (see formula (11)).

4. Evaluation

4.1. Channel Coverage. In this paper, for the rank-1 codebookwhich means that the number of spatial streams for a useris 1, we define the channel coverage as the gain using thecodebook relative to the gain obtained byMF precoding withperfect CSIT:

𝐶 = 𝐸H [

[

HWΓ(H)

2

|HWopt(H)|2]

]

, (22)

whereWΓ(H) indicates the codeword selected from the code-book Γ and Wopt(H) = H∗/‖H‖

2is the optimal precoding

matrix with perfect CSIT. The channel coverage can be usedas a metric for the quality of a quantized codebook.

4.2. Sum Rate. In order to evaluate the precoding perform-ance in a multiuser MIMO system, we can use the sum ratemetric. For the downlink of the system, the optimal sum ratecan be achieved by the interference presubtraction codingtechnique called dirty-paper coding (DPC), as long as thetransmitter has perfect side information about the additiveinterference at the receiver [14]. The optimal DPC sum ratefor the multiuser case is given as follows [15]:

𝑅DPC = 𝐸H [maxΛ

(log2det (I

𝑁𝑡

+ 𝛾H𝐻ΛH))] , (23)

101 102 103

K = 1

K = 2K = 4

K = 6

K = 8

K = 10

0

10

20

30

40

50

60

70

80

90

Number of transmit antennas

Achi

evab

le su

m ra

te (b

ps/H

z)

Figure 3: Optimal sum rate curves of different 𝐾 values for 𝛾 =10 dB.

whereΛ is a𝐾×𝐾 diagonal matrix for power allocation withfactor 𝜆

𝑖on its main diagonal and ∑𝜆

𝑖= 1.

Figure 3 depicts the optimal sum rate curves achieved byDPC for multiuser MIMO with 𝐾 scheduled UEs equippedsingle-receive antenna under 10 dB SNR setting. Since thecurves for differentNLOS scenarios defined by [16], includingurban macrocell (UMa), urban microcell (UMi), and indoorhotspot (InH) of the URA deployment are very close, we onlydraw the curves for the UMa scenario in this figure. Fromthe result, we can see that the optimal sum rate increaseslinearly with the increase of the logarithm of 𝑁

𝑡that is,

we need to double the number of antennas in order toimprove the capacity by roughly 𝐾 bps/Hz. However, forlarge numbers of transmit antennas, the signal processingcomplexity, scheduling algorithm complexity for numeroususers, and the CSI feedback overhead are significantly high,which may overshadow the capacity gain.

While the optimal sum rate can be used as the upperbound for the limited feedback precoding, the achievable sumrate of the precoding system using a quantized codebookcan be used as an important indicator of the quality ofthe codebook [17]. For the multiuser MIMO system, if thechannel estimation is considered to be ideal (i.e., Ĥ

𝑘=

H𝑘), the power of each transmit antenna is uniform, and

the MF receiver is adopted (as shown in formula (6)) theinstantaneous achievable sum rate can be expressed as

𝑅 (H) =𝐾

∑

𝑘=1

log2(1 +

H𝑘W𝑘2

2

𝐾/𝛾 + ∑𝑖 ̸= 𝑘

H𝑘W𝑖2

2

) , (24)

where 𝐾 is the number of scheduled users, 𝛾 = 𝑃𝑡/𝜎2 is

the signal-to-noise ratio (SNR) with 𝑃𝑡as the total transmit


power, H𝑘, the column vector of H denotes the channel

matrix for UE 𝑘, and W𝑘is its requested codeword.

Derivation.The received signal for UE 𝑘 can be expressed as

𝑦𝑘= H𝑘W𝑘𝑠𝑘+ ∑

𝑖 ̸= 𝑘

H𝑘W𝑖𝑠𝑖+ 𝑧𝑘. (25)

With formula (6), the demodulated signal can be writtenas

𝑠𝑘= (H𝑘W𝑘)∗

𝑦𝑘=H𝑘W𝑘

2

𝑠𝑘

+ (H𝑘W𝑘)∗

∑

𝑖 ̸= 𝑘

H𝑘W𝑖𝑠𝑖+ (H𝑘W𝑘)∗

𝑧𝑘.

(26)

And the power of the demodulated signal is

𝐸 [𝑠∗

𝑘𝑠𝑘] =

H𝑘W𝑘4

𝐸 [𝑠∗

𝑘𝑠𝑘] +

H𝑘W𝑘2

𝐸 [𝑧∗

𝑘𝑧𝑘]

+ ∑

𝑖 ̸= 𝑘

(H𝑘W𝑘)∗H𝑘W𝑖

2

𝐸 [𝑠∗

𝑖𝑠𝑖]

=H𝑘W𝑘

2

(𝑃𝑡

𝐾

H𝑘W𝑘2

+ 𝜎2+𝑃𝑡

𝐾∑

𝑖 ̸= 𝑘

H𝑘W𝑖2

2) .

(27)

Noting that the first term is the power of thewanted signalwhile the second and the third ones denote the power ofthe noise and the interference, we can compute the SINR asfollows:

SINR=(𝑃𝑡/𝐾)

H𝑘W𝑘2

𝜎2 + (𝑃𝑡/𝐾)∑

𝑖 ̸= 𝑘

H𝑘W𝑖2

2

=

H𝑘W𝑘2

2

𝐾/𝛾+∑𝑖 ̸= 𝑘

H𝑘W𝑖2

2

.

(28)

Thus, we have the instantaneous achievable sum rateexpression given by formula (24).

The ergodic achievable rate can be expressed as

𝑅 = 𝐸H (𝑅 (H)) . (29)

In order to evaluate the codebooks with the achievablesum rate metric, we adopt the Monte-Carlo simulationmethod and take the arithmetic mean of the instantaneousvalues under different channel realizations as the output met-ric. The simulation flow is described by the pseudocode inAlgorithm 1.

4.3. Results. Based on the evaluation metrics, we presentsome simulation results in this section. The Winner II chan-nel model [18] can be used for the ULA deployment in thesimulation. But it needs to be modified to support theURA deployment. Thus, we extend the model by associatingelevation angles to paths generated by the original WinnerII model and correlating elevation statistics with other large-scale fading parameters. Without loss of generality, we takethe case of 64 transmit antennas as an example for theperformance evaluation of different codebook designs underdifferent scenarios. For the sake of clarity, the conventional

// L: the total number of channel realizations//𝑁𝑢: the total number of UEs

// K: the number of scheduled UEsInitializationfor l = 1 : L

generate the channel matrix Hfor u = 1:𝑁

𝑢

compute the requested codeword W𝑢

report W𝑢to the BS

end forscheduling: select K UEs from the𝑁

𝑢UEs

form the precoder Pcompute the instantaneous rate 𝑅

𝑙(formula (23))

end for

Compute the metric 𝑅 = 1𝐿

𝐿

∑

𝑙=1

𝑅𝑙

output 𝑅

Algorithm 1: Simulation flow.

ULA DFT codebook and the proposed URA DFT codebookare termed as DFT-ULA and DFT-URA, respectively.

Table 1 gives the parameter configuration for the fol-lowing simulations. As we are discussing the capability ofcodebooks to reflect the channel correlation, we focus on thedesign of the first codeword W1 for DFT codebooks; hence,we set𝑀 = 1, whichmeans that the codebooks have only onesecond codeword; that is, W2 = 1.

4.3.1. Channel Coverage. With the fixed number of transmitantennas, the codebook can be enlarged in order to improvethe quantization accuracy, but the feedback overhead wouldlimit the codebook size and codeword selection complexitywould become a significant bottleneck. Consequently, it isnecessary for us to investigate the performance of codebookswith different feedback overhead limits. Figure 4 presents thechannel coverage defined by formula (22) as a function of thefeedback overhead (𝐵) for the DFT-ULA, the DFT-URA, andthe Kerdock codebook under different NLOS scenarios withthe URA deployment. It is noted that the maximum numberof codewords in the Kerdock codebook for 64-dimensionalspace is 4096, and thus, the curve ends at the point of 12-bit feedback overhead. From the curves, we can see thatwhen the feedback overhead becomes larger than 10 bits, theimprovement becomes insignificant. In conclusion, the 8∼10bits of feedback is adequate for 64 transmit antennas withthe URA. This feedback overhead is close to that of LTE-Advanced Release 10 in which the 8-bit rank-1 codebookfor eight crosspolarized antennas is utilized [19]. Besides,effective approaches [20] can be used to further reduce thefeedback overhead.

4.3.2. Achievable Sum Rate. We have run the multiuserMIMO simulation described in Algorithm 1 with 10-bit feed-back overhead (i.e., 𝐵 = 10), and the results are shown inFigures 5 and 6.


Table 1: Simulation configuration.

Parameter ValueChannel model UMa NLOS; UMi NLOS; InH NLOSBS antenna setup URA, 64 copolarized antennas, and 𝜆/2 spacingUE antenna setup Single antennaSystem frequency (GHz) 2.1Number of channel realizations 1000Number of UEs 10Scheduling criteria Minimizing the codeword correlation coefficientNumber of DFT matrices

DFT-ULA 𝐺 = 2𝐵

DFT-URA 𝐺𝑥= 𝐺𝑦= 2𝐵/2

Size of DFT matricesDFT-ULA 𝑁 = 64, 𝑀 = 1DFT-URA 𝑁

𝑥= 𝑁𝑦= 8, 𝑀

𝑥= 𝑀𝑦= 1

4 6 8 10 12 14Feedback overhead (bit)

Chan

nel c

over

age

Kerdock, UMaDFT-ULA, UMaDFT-URA, UMaKerdock, UMiDFT-ULA, UMi

DFT-URA, UMiKerdock, InHDFT-ULA, InHDFT-URA, InH

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 4: Channel coverage as a function of the feedback overhead.

Figure 5 illustrates the curves as a function of the numberof scheduled UEs (𝐾) in the UMa scenario. From the curves,we can see that the DFT-URA outperforms the other code-books for different number of UEs with 𝛾 = 10 dB. Figure 5also shows the asymptotic curves as 𝛾 → ∞, which measurethe impact of the interuser interference.The gap between theDFT-URA and the DFT-ULA indicates that the DFT-URAcan better mitigate the inter-user interference. Besides, in thecapability of the inter-user interference suppression, theDFT-URA defeats the Kerdock when the number of scheduledUEs is relatively large (𝐾 > 5), since the DFT-URA is able

1 2 3 4 5 6 7 8 9 105

10

15

20

25

30

35

40

45

50

Number of scheduled UEs

Achi

evab

le su

m ra

te (b

ps/H

z)

Kerdock SNR = 10dBDFT-ULA SNR = 10dBDFT-URA SNR = 10dB

Kerdock—asymptoticDFT-ULA—asymptoticDFT-URA—asymptotic

Figure 5: Achievable sum rate as a function of 𝐾 for 𝛾 → ∞ and𝛾 = 10 dB.

to provide more beams targeting at more UEs separated indifferent angular positions.

Figure 6 depicts the achievable sum rate of the Kerdock,the DFT-ULA, and the DFT-URA for four scheduled UEs(𝐾 = 4) as a function of SNR under the three scenarios; thetheoretical curves drawn by formula (23) with perfect CSITare presented as the upper bound.The results indicate that theDFT-URA designed to the URA deployment has remarkableperformance gain compared to the DFT-ULA and especiallythe Kerdock codebook under different scenarios. It is becausethe design based on the Kronecker-type approximation of


0 5 10 15 2010

15

20

25

30

35

40

45

SNR (dB)

Achi

evab

le su

m ra

te (b

ps/H

z)

Kerdock DFT-ULA

DFT-URA Perfect CSIT

(a) UMa

0 5 10 15 205

10

15

20

25

30

35

40

45

SNR (dB)

Achi

evab

le su

m ra

te (b

ps/H

z)

Kerdock DFT-ULA

DFT-URA Perfect CSIT

(b) UMi

0 5 10 15 2010

15

20

25

30

35

40

45

SNR (dB)

Achi

evab

le su

m ra

te (b

ps/H

z)

Kerdock DFT-ULA

DFT-URAPerfect CSIT

(c) InH

Figure 6: Achievable sum rate with 𝐾 = 4. (a) Uma, (b) UMi, and (c) InH.

the array correlation structure better reflects the channelproperty with the URA, and thus physically leads to moreaccurate beams, making it suitable for multiuser MIMO inthe case where UEs are separated spatially in angular domain.In addition, the performance in the UMa scenario is betterthan the other two scenarios, since this scenario has strongerchannel correlation, making the DFT vectors better adapt tothe channel.

5. Conclusion

In this paper, we discussed the limited feedback precodingtechniques for the downlink of massive MIMO systems. Onthe theoretical basis of Kronecker-type approximation of thearray correlation structure, we proposed a novel codebook

design for the URA deployment of the numerous closelyspaced antennas, which would be probably adopted by mas-sive MIMO. This codebook design constructs the first code-word representing the whole long-term channel correlationin the plane with the Kronecker product of twoDFTmatricesgenerated for two orthogonalULAs.We proved the validity ofthis construction theoretically, and verified that the proposedcodebook outperforms other kinds of codebooks in termsof the channel coverage and the achievable sum rate undervarious scenarios via simulations. The proposed codebookdesign can contribute to precoding solutions for large-scalearray antenna technologies, which would be probably appliedto future Beyond 4G systems. Our future work will consideradvanced multiuser scheduling algorithms as well as robustreceiving algorithms for massive MIMO.


Acknowledgments

This work was supported by Beijing Natural Science Foun-dation funded project (no. 4110001), National S&T MajorProject (no. 2013ZX03003003-003), and Samsung Company.

References

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