Antenna Grouping based Feedback Reduction for FDD-based Massive MIMO Systems.pdf

8/11/2019 Antenna Grouping based Feedback Reduction for FDD-based Massive MIMO Systems.pdf

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a r X i v : 1 4 0 8 . 6 0 0 9 v 1 [ c s . I T ] 2

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Antenna Grouping based Feedback Reduction forFDD-based Massive MIMO SystemsByungju Lee, Junil Choi, Ji-yun Seol, David J. Love, and Byonghyo Shim

Abstract Recent works on massive multiple-inputmultiple-output (MIMO) have shown that a potentialbreakthrough in capacity gains can be achieved bydeploying a very large number of antennas at thebasestation. In order to achieve optimal performance of massive MIMO systems, accurate transmit-side chan-nel state information (CSI) should be available at thebasestation. While transmit-side CSI can be obtainedby employing channel reciprocity in time division du-plexing (TDD) systems, explicit feedback of CSI fromthe user terminal to the basestation is needed forfrequency division duplexing (FDD) systems. In thispaper, we propose an antenna grouping based feedbackreduction technique for FDD-based massive MIMOsystems. The proposed algorithm, dubbed antennagroup beamforming (AGB), maps multiple correlatedantenna elements to a single representative value usingpre-designed patterns. The proposed method intro-duces the concept of using a header of overall feedbackresources to select a suitable group pattern and thepayload to quantize the reduced dimension channel vec-tor. Simulation results show that the proposed methodachieves signicant feedback overhead reduction overconventional approach performing the vector quanti-zation of whole channel vector under the same targetsum rate requirement.

Index Terms Massive multiple-input multiple-output, antenna group beamforming, feedback

reduction, vector quantization, Grassmanniansubspace packing.

I. Introduction

Multiple-input multiple-output (MIMO) systems withlarge-scale transmit antenna array, often called massiveMIMO, have been of great interest in recent years becauseof their potential to dramatically improve spectral effi-ciency of future wireless systems [ 2], [3]. By employing tensor hundreds of antennas at the basestation, massive MIMOsystems can control intra-cell interference and thermalnoise by simply using linear precoding in the downlinkand receive ltering in the uplink [2]. Additionally, massiveMIMO can improve the power efficiency by scaling down

B. Lee and B. Shim are with Dept. of Electrical and ComputerEngineering, Seoul National University, Seoul, Korea, J. Choi andD. J. Love are with School of Electrical and Computer Engineering,Purdue Univ., West Lafayette, IN, USA, and J. Seol is with SamsungElectronics Co., Ltd., Suwon, Korea.

This paper was presented in part at the International Conferenceon Communications (ICC), 2014 [1].

This work was sponsored by Communications Research Team(CRT), DMC R&D Center, Samsung Electronics Co. Ltd, the MSIP(Ministry of Science, ICT & Future Planning), Korea in the ICTR&D Program 2013 (KCA-12-911-01-110) and the NRF grant fundedby the Korea government (MEST) (No. 2012R1A2A2A01047510).

the transmit power of each terminal inversely proportionalto the number of basestation antennas [3].

Presently, standardization activity for massive MIMOhas been initiated [ 4], [5], and there is on-going debateregarding the pros and cons of time division duplexing(TDD) and frequency division duplexing (FDD). In ob-taining the CSI, FDD requires the CSI to be fed backthrough the uplink [6] while no such procedure is requiredfor TDD systems owing to channel reciprocity [7]. Infact, under the assumption that RF chains are properlycalibrated [8], the CSI of the downlink can be estimated

using the pilot signal in the uplink so that the CSI feedbackinformation from the user terminal to the basestation isunnecessary. Due to this benet, most of the massiveMIMO works in the literature have focused on TDD [ 9](possible exceptions are [10], [11]). However, since FDDdominates current cellular networks and offers many ben-ets over TDD (e.g., small latency, continuous channelestimation, backward compatibility), it is important toidentify and develop solutions for potential issues arisingfrom FDD-based massive MIMO techniques.

One well-known problem of FDD system is that theamount of CSI feedback must scale linearly with thenumber of antennas to control the quantization error [ 12]

[14]. Therefore, it is not hard to convince oneself that theoverhead of CSI feedback is a serious concern when thenumber of transmit antennas is large. Needless to say, atechnique that efficiently reduces the feedback overheadwhile affecting minimal impact on system performance iscrucial to the success of massive MIMO systems.

In this paper, we provide a novel framework for FDD-based massive MIMO systems that achieves a reductionin the CSI feedback overhead by exploiting the spatialcorrelation among antennas. The proposed algorithm,henceforth dubbed antenna group beamforming (AGB),maps multiple correlated antenna elements to a singlerepresentative value using properly designed grouping pat-terns. When the antenna elements are correlated, the losscaused by the grouping antenna elements is shown to besmall, meaning that grouping of antenna elements withcorrelated channels is an effective means to reduce thedimension of the channel vector to be quantized. In fact,by allocating a small portion of the feedback resourcesto represent the grouping pattern, the number of bitsrequired for channel vector quantization can be reducedsubstantially, thereby achieving signicant reduction infeedback overhead. In order to support the antenna group-ing operation, the proposed AGB algorithm divides thefeedback resources into two parts: a header to indicate
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Antenna grouping Vectorquantization Expansion

Original channel vectorReduced dimension

channel vector Quantized vector Expanded quantized vectorT

r,4r,4r,4r,4r,1r,1r,1r,1 ]h h h h h h h h[~ L=

r h

Tr,4r,3r,2r,1 ]hhh[h=r h

4hhhh

h 4321r,1+++

=

Tr,4r,3r,2r,1 ]h h h h[ =r hT161514134321 ]hhhh hhh[h L=h

Pattern 1

Pattern N P

Fig. 2. Illustration of the AGB algorithm for N t = 8 , N g = 4. The reduced dimension channel vector h r is obtained by mapping antennaelements of a group as a representative value. Note that h r is the quantized version of h r and h (

i )r is expanded version of h r .

Channel vector quantization (B bits)

Feedback packet

(a)

Pattern selection (Bp bits) Channel vector quantization (B-Bp bits)

Feedback packet

(b)

Fig. 3. Feedback packet structure: (a) conventional method and (b)proposed method.

each group. This approach, although it may offer betterquantization error, will increase the feedback overhead dueto the additional weight information.

Once the reduced dimension channel vector h ( i )r is ob-tained, h ( i )r is quantized by a B B p bit codebook C ={c 1 , , c2B Bp }. We note that since a codebook designedfor i.i.d channels is not a proper choice for correlatedchannels, we consider a channel statistic-based codebookfor channel vector quantization [20] (see Section III.C fordetails). The codeword h ( i )r maximizing the absolute innerproduct with h ( i )r is chosen as

h ( i )r = argmaxc C |h( i )H r c|2 , i = 1 , , N P (11)

where h ( i )r = h ( i )

rh ( i )r

is the direction of the reduced di-mension channel vector for the i-th pattern. This process

is repeated for each group pattern and N p candidatecodewords h ( i )r , i = 1 , , N P , are chosen in total.

Once N P candidate codewords are obtained, the code-word minimizing the distortion between h and h ( i )r ischosen. We note that the comparison between h and h ( i )ris not possible since the dimension of h ( i )r C N g is smallerthan that of the original channel vector h C N t . Thus, weuse h ( i )r C N t , an expanded version of h ( i )r , to computethe distortion ( D (h , h ( i )r ) = E [ h 2(1| h H h ( i )r |2)]) causedby the grouping and quantization. The expansion process,which essentially is done by copying each element in h ( i )r

to N tN g elements in h

( i )r , is performed by multiplying an

expansion matrix E ( i ) R N t N g to h ( i )r . The expandedquantized vector h ( i )r is expressed as

h ( i )r = E( i ) h ( i )r , i = 1 , , N P (12)

where E ( i ) = G ( i )T

and satises G ( i ) E ( i ) = I N g ( =N tN g ). For example, for the grouping matrix in ( 10), E

( i ) =

G ( i )T

= 1 1 0 00 0 1 1

T

and the expanded quantizedvector is

h ( i )r = E( i ) h ( i )r = (G

( i ) )T h ( i )r = h( i )r, 1

h( i )r, 1 h( i )r, 2

h( i )r, 2T

.

(13)The group pattern index i minimizing the distortionbetween h and h ( i )r is

i = arg mini =1 , ,N P

D (h , h ( i )r ). (14)

Once this pattern index i is obtained, we deliver thisindex and the corresponding codeword index to the bases-tation. After receiving the pattern index and codewordindex of all user terminals, the basestation decompressesthe reduced dimension channel vector via the expansion(h = E ( i ) h ( i )r ) and then performs the beamforming using


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h 9h 1 h 5 h 13

h 10h 2 h 6 h 14

h 11h 3 h 7 h 15

h 12h 4 h 8 h 16

Pattern 1 Pattern 3Pattern 2

Fig. 4. Example of antenna group patterns ( N t = 16 , N g = 8 , N P = 3). Antenna elements belonging to the same pattern are mapped toone representative value.

the composite channel matrix H . A block diagram of theproposed AGB algorithm is depicted in Fig. 5.

B. Antenna Group Pattern Generation

Since multiple correlated antenna elements are mapped

to a single representative value, the AGB algorithm issensitive to the choice of the antenna group pattern. With-out doubt, selecting the best pattern among all possiblecombinations would be the ideal option. However, sincethe number of patterns increases exponentially with thenumber of transmit antennas, it is not possible to investi-gate all possible patterns for the massive MIMO systems.This observation tells us that a simple yet effective patterndesign is crucial to the success of the AGB algorithm.

One easy and intuitive way to construct an antennagroup pattern G ( i ) is to group highly correlated antennaelements together. Typically, adjacent antenna elementsare highly correlated so that the grouping of nearbyantenna elements may be a desirable option in prac-tice (see the example in Fig. 4). Alternatively, one canconsider Grassmannian subspace packing in the designof the antenna group patterns [21]. The main goal of Grassmannian subspace packing is, when the subspacedistance metric and the number of feedback bits B areprovided, to nd a set of 2 B subspaces in 2 G (N t , m) thatmaximizes the minimum subspace distance between anypair of subspaces in the set. The chordal distance metrichas been popularly used in generating the beamformingcodebook [21]. Our task of generating the pattern set issimilar in spirit to the Grassmannian subspace packingbased codebook generation in the sense that we constructa pattern set (containing 2 B p patterns) from all possiblepattern candidates ( E ( i ) R N t N g ) using a distancemetric exploiting the spatial correlation among antennaelements.

In the rst step of the pattern set design, we computethe quasi-correlation matrix norm R ( i )t F to measurethe spatial proximity of the antenna elements in theantenna group. The quasi-correlation matrix R ( i )t , denedas R ( i )t = R

1/ 2t E

( i ) , captures the actual inuence of thepattern E ( i ) on the transmit correlation matrix R t . In

2 G(N t , m ) is the set of m -dimensional subspaces in C N t (or R N t ).

general, a pattern generated by grouping closely spacedantenna elements tends to have a higher quasi-correlationmatrix norm than that generated by grouping antennaelements apart. Thus, one can expect that a pattern witha large-correlation matrix norm exhibits lower groupingloss than that with a small quasi-correlation matrix norm.For patterns with high quasi-correlation matrix norm,we perform subspace packing to generate 2 B p patterns(expansion matrices) maximizing the minimum distancemetric between any pair of subspaces. To measure the dis-tance, we use the correlation matrix distance dcorr (A , B )between two matrices A and B [22]

dcorr (A , B ) = 1 tr( A H B )

A F B F. (15)

Note that dcorr (A , B ) measures the orthogonality betweentwo correlation matrices A and B . When the correlationmatrices are equal up to a scaling factor, dcorr is minimized

(dcorr = 0). On the other hand, when the inner productbetween the vectorized correlation matrices is zero (i.e.,vec(A ) and vec( B ) are orthogonal), dcorr is maximized(dcorr = 1). In our numerical simulations, we show thatthe proposed subspace packing approach achieves a sub-stantial gain over an approach using randomly selectedpatterns (see Section IV.B). We summarize the antennagroup pattern generation procedures in Table I.

C. Quantization Distortion Analysis

We now turn to the performance analysis of the AGBalgorithm. In our analysis, we analyze the distortion Dinduced by the quantization of the channel direction vectorh = hh , which is dened as

D = E h 2 | h H h r |2

= E h 2 1 | h H h r |2 (16)

where h r is the expanded version of the quantized vectorh r (see (12)).

In order to evaluate the distortion D, we use thequantization cell upper bound (QUB) [23]. As mentioned,since a codebook designed for the i.i.d channels is notthe right choice for correlated channels, we employ achannel statistic-based codebook obtained by applying


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Antenna grouppattern mapping

Vectorquantization

Patternselection

User k

Antenna grouppattern expansion

)(~ ir

h

x

Codeword index(B-Bp bits/user)

Nt

Beamforming

1

Nt-1

2

Antenna grouppattern expansion

Pattern index(Bp bits/user)

H

Basestation

W

s1s2

sK

yk )(i

r h )( i

r h

*i *c

*i

Fig. 5. A block diagram of the AGB algorithm in the multi-user downlink system.

the transmit correlation matrix R 1/ 2t to the codebookgenerated from the GLP. Let f i C r be the i-th unit

norm vector generated from the GLP, then the set of B-bit codewords for the channel statistic-based codebook is[20]

C = {c1 , , c2B } = R 1/ 2t f 1R 1/ 2t f 1

, , R 1/ 2t f 2B

R 1/ 2t f 2B.

(17)When the channel statistic-based codebook is used, thenormalized distortion DE [ h 2 ] = 1 | h

H c i |2 between thechannel direction vector h and codebook vector c i can beupper bounded as [15]

R i { h : 1 | h H c i |2 } (18)

where = 22

212

Br 1 (i is the i-th largest singular value of

the transmit correlation matrix R t C r r ) and B is thenumber of quantization bits.

In our analysis, we restrict our attention to the scenariowhere two antenna elements are mapped to a single rep-resentative value for mathematical tractability. Neverthe-less, since the key factor affecting the quantization distor-tion is the transmit correlation coefficient (see ( 29)), ourresults can be readily applied to the general scenario wheremore than two antenna elements are grouped together.The minimal set of assumptions used for the analyticaltractability are as follows:

A-i) The channel vector h is partitioned into twosubvectors h A and h B (h = [h T A h T B ]T ). h A isa vector composed of N g antenna elements andeach entry of h A is the rst element in the group.Therefore, there are N g antenna groups and eachgroup consists of two antenna elements.

A-ii) The reduced dimension channel vector h r is de-signed such that h r = h A . For example, if h =[h1 h2 h3 h4]T , then

G = 1 0 0 00 0 1 0 (19)

and hence h r = h A = [h1 h3 ]T (h B = [h2 h4]T ).A-iii) Antenna elements in a group are highly corre-

lated. That is, E [|h

H A

h r |

2

] E [|h

H B

h r |

2

] whereh r is the quantized vector of h r and generatedfrom the channel statistic-based codebook.

The following theorem provides an approximate upperbound of the quantization distortion D under these as-sumptions.

Theorem 3.1: The quantization distortion D of theAGB algorithm under the channel statistic-based code-book satises

D N t + N t (1 (1 )2 ) (20)where =

22

212

B B pN g 1 is an upper bound of the normalized

distortion between h A and

h r (i is the i-th largestsingular value of R t, A )3 and is the correlation coefficient

between two random variables h 2 and 1 | h H h r |2 .Proof: Using (16), we have

D = E h 2 1 | h H h r |2 (21)= E h 2 1 E |h H h r |2 + Cov( h 2 , 1 | h H h r |2)

(22)= E h 2 1 E |h H h r |2

+ V ar ( h 2) V ar 1 | h H h r |2 (23) E h 2 1 E |h H h r |2+ V ar [ h

2] 1 E |h

H h r |

2 2

(24)where (22) is because 4 Cov(X, Y ) = E [XY ] E [X ]E [Y ],

3 For example, if R t =

1 2 3 1 2

2 1 3 2 1

and A = {1, 3}, then

R t, A =1 2

2 1 .4 Note that the covariance becomes zero if X and Y are statistically

independent. For i.i.d channel, h w 2 and |h w w |2 are independentwhere h w =

h wh w

and w is quantized vector [14]. However, due tothe spatial correlation, C ov ( h 2 , 1 | h H h r |2 ) cannot be zero in ourproblem.


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TABLE ISummary of the antenna group pattern generation.

Initialization B p : the number of bits for the pattern setS : the set of patterns to be selected

Main operation 1) Initialize the index set = {1, . . . , N max } where

N max =N g 1n =0

N t n

N g ! . = N tN g is the number of elements in an antenna group.

2) For each pattern i , calculate the Frobenius norm of the quasi-correlation matrix

r i = R ( i )

t F .Without loss of generality, assume r 1 r 2 r N max .3) Choose L ( 2B p ) patterns T = {r 1 , . . . , r L }.4) Apply the subspace packing to T to generate the pattern set S .

Construct N c = L2B p candidate sets {S k }N ck =1 where S k = {R

k, 1t , R

k, 2t , , R

k, 2B pt }.

R k,it is the i -th quasi-correlation matrix from the k-th candidate set.

5) Calculate the minimum dcorr of a S kdk, min (S k ) = min 1 m n 2B p dcorr ( R

k,mt , R

k,nt ).

Decide the pattern set S S = arg max k =1 , ,N c dk, min (S k ).

(23) is because Cov(X, Y ) = V ar(X ) V ar(Y ) and(24) is because V ar 1 | h H h r |2 = E |h H h r |4 E |h H h r |2 2 1 E | h H h r |2 2 .

The normalized distortion term 1 E |h H h r |2 in theright-hand side of ( 24) is approximately upper bounded as

1 E |h H h r |2

(a )= 1 E |h H A h r + h H B h r |2

= 1 E [|h H A h r |2 + | h H B h r |2 + ( h H A h r )H (h H B h r )+ ( h H B h r )

H (h H A h r )]

= 1 E | h H A h r |2 E |h H B h r |2

2E Re( h H A h r )H (h H B h r )(b) 1 E | h H A h r |2 E |h H B h r |2

(c) 1 2E | h H A h r |2

(d) 1 2(1 )E h A 2

(e)= (25)

where (a) is because |h H h r |2 = |h H A h r + h H B h r |2 ,and (b) follows from E [Re(( h H A h r )H (h H B h r ))] =

i Re(E [hH B,i ]E [h H r h A h r,i ] 0 (since E [h H B ,i ] = 0),

and (c) follows from A-iii), and (d) follows from the QUB

in5

(18), and (e) follows from E h A

2

= N gN t =

12 .Plugging ( 25) into ( 24), we have

D E h 2 + V ar [ h 2] 1 (1 )2= N t + N t (1 (1 )2)where the last equality follows from E [ h 2] = N t andV ar[ h 2 ] = N t , which is the desired result.

5 Plugging h = h A

h A, C =

R 1 / 2t, A

f 1

R 1 / 2t, A

f 1, ,

R 1 / 2t, A

f 2 B

B p

R 1 / 2t, A

f 2 B

B p , and

= 22 21

2

B B pN g 1 into ( 18), we get E | h H A h r |

2 (1 )E h A 2 .

We note that the relationship between the quantizationdistortion D and the transmit antenna correlation is not

clearly shown in (20). When a specic correlation modelis used, however, we can observe the relationship betweentwo. For example, if the exponential correlation modelis employed, then the transmit correlation matrix R t isexpressed as [24]

R t =

1 N t 1H 1 N t 2

......

. . . ...

(N t 1) H (N t 2) H 1

(26)

where = e j is the transmit correlation coefficient, and is the magnitude of correlation coefficient, and is the

phase of the user. When the number of transmit antennasN t is large, (non-ordered) singular value i of R t becomesapproximately [25]

i N t 1

k= (N t 1)

| k | ej2 ik

N t

1 2

1 + 2 2 cos( 2iN t ), i = 1 , . . . , N t . (27)

Using the rst and second largest singular values (i.e.,N t , N t 1) of (27), we have

2

1

1 + 2 2

1 + 2 2 cos(2 N t 1N t ). (28)

Noting that = 22

212

B B pN g 1 in Theorem 3.1, we have

D N t + N t (1 (1 )2 ) (29)where 1 +

2 21 + 2 2 cos(2 N t 1N t )

2

2B B pN g 1 .

In ( 29), we can observe that the quantization distortion Ddecreases with the correlation coefficient . Fig. 6 plots the


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0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

Correlation coefficient | |

N o r m a

l i z e

d q u a n

t i z a

t i o n

d i s t o r t

i o n

Analysis upper bound (AGB algorithm)Numerical evaluation (AGB algorithm)Analysis upper bound (Conventional vector quantization)Numerical evalution (Conventioanl vector quantization)

Fig. 6. Normalized quantization distortion as a function of thecorrelation coefficient ( N t = 16 , N g = 8 , B = 16 , B p = 8).

normalized quantization distortion DE [ h 2 ] as a function of the correlation coefficient . We observe that if || > 0.3,the quantization distortion D of the AGB algorithm issmaller than that of conventional vector quantization. Wecan also observe that the analysis matches well with thesimulation results when the transmit antennas are highlycorrelated ( || > 0.6). However, when the magnitude of is small, the assumption in A-iii) is violated so that theproposed bound is invalid.

D. Partial Antenna Grouping

In this subsection, we discuss a partial antenna groupingscheme that applies the antenna grouping to the part of antenna arrays. The basic idea of this scheme is to parti-tion the antenna array into multiple sub-arrays and thenapply the AGB algorithm to the part of sub-arrays havingsmaller channel gain. In doing so, we can invest morefeedback resources to the sub-arrays with higher channelgain. Additionally, the computational complexity requiredfor the pattern generation can be reduced substantially.

Specically, we rst partition the channel vector h into

M sub-arrays h sub ,1 , , h sub ,M (h sub ,i CN tM ). Without

loss of generality, we assume that these vectors are orderedbased on their channel gain ( h sub ,1 2 > > h sub ,M 2).Among these, we choose the M ( M ) sub-arrays withsmallest channel gain h sub ,M M +1 , , h sub ,M , which werefer to these as weak sub-arrays, and then apply theantenna grouping operation to the weak sub-arrays (seeFig. 7). If we denote the number of groups in the selectedsub-arrays as N g , then the dimension of the channel vectorto be quantized becomes N g = M N g +( M M ) N tM whereN tM is the number of antenna elements for the rest sub-arrays and the number of bit per antenna element becomes

1h sub,

Antenna grouping

4

BB qsub,1 =

4

BB q2,sub =

Bsu

Bsu

2

h sub,h sub,

h sub,

3

4

(a)

1h sub,

3

3

BB q2,sub =

2

4

Partial antenna grouping

3

BB q1,sub =

B

B

h sub,

h sub,

h sub,

(b)

Fig. 7. Example of 4 8 planar array when N t = 32 , M = 4 , M = 2(a) AGB algorithm and (b) Partial antenna grouping scheme.

B qN g where Bq is the number of bits for vector quantization.Also, the number of bits assigned to the strong and weaksub-arrays are N tM

B qN g and N

g

B qN g , respectively. By setting

N tM > N

g , we can invest more feedback resources to the

strong sub-arrays. As an example, consider the antennaarray with 64 elements ( N t = 64). If the number of sub-arrays is 4 ( M = 4) and 2 sub-arrays are chosen as weaksub-arrays ( N g = 8), then N g = 48. Now, suppose thetotal number of bits for the channel vector quantization is96 bits, then the proposed approach assigns 32 bits to thestrong sub-array and 16 bits to the weak sub-array whilethe original AGB algorithm assigns 24 bits per each sub-array. By investing 33% more feedback resources into thestrong sub-array, quantization loss of the AGB algorithmcan be alleviated substantially.

IV. Simulation Results and Discussions

A. Simulation Setup

In this section, we compare the sum rate performance of the conventional vector quantization technique using thechannel statistic-based codebook and the proposed AGBalgorithm. While all the feedback resources ( B -bit) areused to quantize the channel vector h k in the conven-tional approach, B -bit feedback resource is divided into Bq(channel vector quantization), B p (pattern selection), andB r (sub-array selection) in the proposed AGB algorithm.To express the feedback allocation, we use the notationB = ( Bq , B p, B r ) in the sequel. As a channel model, weconsider the exponential correlation model in ( 26), one-dimensional uniform linear array (ULA) model [26], andtwo-dimensional uniform planar array (UPA) model [27].As a channel vector quantizer, we use RVQ and nonco-herent trellis coded quantization (NTCQ). While RVQ isa well-known and standard channel vector quantizer inthe literatures, the time to choose the codeword increases


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.98

10

12

14

16

18

20

Transmit antenna correlation coefficient

S u m

r a

t e ( b p s

/ h z

)

Conventional vector quantization (Nt=32)

AGB algorithm (Nt=32)



Fig. 8. Sum rate as a function of the transmit antenna correlationcoefficient .

exponentially with the quantization bits (when a B-bitcodebook is used, we need to check 2 B codewords). Sincethe number of transmit antennas N t in our simulationsis large (i.e, 32 and 64 antennas), we use the analyticalclosed-form expression of RVQ [ 28] instead of simulations.To make sure that the analysis result can be translatedinto a realizable codebook, we also use the recently pro-posed low-complexity codebook called NTCQ [ 10]. Notethat the search complexity of the NTCQ codebook scales

linearly to the number of transmit antennas so that onecan evaluate the performance of NTCQ codebook withreasonable simulation time.

B. Simulation Results

We rst consider the exponential channel model [29]

r ij = | j i |k i j(| j i |k )

H i > j(30)

where r ij is the (i, j )-th element of R t,k and k = e j kis a transmit correlation coefficient for the k-th user ( is the magnitude of correlation coefficient and k is thephase of the k-th user). Note that the phase of each useris randomly generated from to and independentof other users phases. Note also that all users have thesame transmit correlation coefficient |k | = since isdetermined by the antenna spacing at the basestation. InFig. 8, we plot the sum rate as a function of for themulti-user MISO system with N t = 32 , 64 and K = 4. Inour simulations, we allocate one bit per antenna element(B = N t ). In the AGB algorithm, we set B = (17 , 14, 1)for N t = 32 and B = (46 , 14, 4) for N t = 64, respectively.In this case, N g = 24 for N t = 32 and N g = 56 for

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.912

13

14

15

16

17

18

19

Transmit antenna correlation coefficient

S u m

r a

t e ( b p s

/ h z )

Conventional vector quantization (RVQ)Conventional vector quantization (NTCQ)AGB algorithm (RVQ)AGB algoritm (NTCQ)

Fig. 9. Sum rate of the NTCQ codebook and analytical approxima-tion of RVQ as a function of .

16 24 32 40 48 56 646

8

10

12

14

16

18

20

Number of feedback bits B

S u m

r a t e ( b p s

/ h z

)

Conventional vector quantization ( =0.6)AGB algorithm ( =0.6)Conventional vector quantization ( =0.9)AGB algorithm ( =0.9)

Fig. 10. Sum rate as a function of the number of feedback bits B(N t = 32 , N g = 24).

N t = 64, respectively. When the antenna correlation islow ( < 0.3), we observe that the sum rate of the AGBalgorithm is slightly worse than that of the conventionalvector quantization technique. Whereas, when increases,the antenna grouping operation becomes effective and thusthe sum rate of the AGB algorithm improves drastically.For example, when = 0 .8, the sum rate gains of theAGB algorithm over the conventional vector quantizationtechnique is 33% for N t = 32 and 28% for N t = 64,respectively.


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In order to observe that the analytic result of the RVQis maintained to the real codebook scenario, we comparethe performance between the RVQ and NTCQ codebook.The NTCQ codebook chooses the candidate beamformingvectors using N g -stage trellis search (each stage selects anentry in each of the candidate vectors). In our simulations,we assign 2 bits per each stage of the trellis search and thuswe have Bq = 2 N g . For N t = 32 and N g = 24, the feedbackallocation is set to B = (48 , 14, 1). We observe from Fig. 9that the overall trend of NTCQ is very similar to that of analytical approximation of RVQ, although there exists aperformance gap between two.

We next measure the sum rate as a function of thenumber of feedback bits B. In this case, we set N t = 32and investigate the performance for two scenarios ( = 0 .6and 0 .9). As shown in Fig. 10, the AGB algorithm achievessignicant gain over the conventional vector quantizationtechnique, in particular for the highly correlated scenario( = 0 .9). For example, to achieve 14 bps/Hz, 16 and 48feedback bits are required for the AGB algorithm while

51 and 60 bits are needed for the conventional vectorquantization technique, resulting in 68% and 20% feedbackoverhead reduction for = 0 .9 and = 0 .6.

We also consider the one-dimensional ULA and two-dimensional UPA ( N V N H array) models, which are morerealistic model for massive MIMO scenarios. In the ULAchannel model, R t,k for the k-th user is expressed as

[R t,k ]m,p = 12

+ k

+ ke j 2D (m p) sin( ) d (31)

where is the angular spread, D is the antenna el-ements spacing, and k is the angle of arrival (AoA)for the k-th user. This ULA model can be extended to

the UPA model by combining the vertical correlationmatrix R V C N V N V and the horizontal correlationmatrix R H,k C N H N H using the Kronecker product. Theresulting transmit correlation matrix of the UPA model isexpressed as R t,k = R V R H,k where is the Kroneckerproduct. We summarize the simulation parameters for theULA and UPA model in Table II. In Fig. 11, we plot thesum rate as a function of the number of feedback bits forN t = 32 , 64 and K = 1. For the ULA model, we use R t,kin (31) for N t = 32 and N t = 64. For the UPA model, weset N V = 4 , N H = 8 for N t = 32 and N V = 8 , N H = 8 forN t = 64, respectively. For both ULA and UPA scenarios,the AGB algorithm outperforms the conventional vectorquantization technique with a large margin, resulting inmore than 50% feedback overhead reduction.

Finally, in order to observe the effectiveness of thesubspace packing approach discussed in Section III.B, wecompare the proposed approach and the random patterngeneration scheme. In our simulations, we set Bq =16, N t = 16 , N g = 8 and measure the sum rate as afunction of the number of pattern bits B p. To set thesame level of feedback, we set B = Bq + B p bit forthe conventional RVQ based vector quantization. Overall,we observe that the subspace packing approach providesa considerable sum rate gain over the random selection

16 24 32 40 48 56 645.5

6

6.5

7

7.5

8

8.5


S u m

r a

t e ( b p s /

H z )





(a)

16 24 32 40 48 56 644

4.5

5

5.5

6

6.5


S u m

r a

t e ( b p s

/ H z

)

Conventional vector quantization (N t=32)


Conventional vector quantization (N t=64)


(b)

Fig. 11. Sum rate as a function of number of feedback bits (a) ULAand (b) UPA correlation model.

approach as well as the conventional vector quantizationtechnique. For example, to achieve 9 bps/hz, AGB withsubspace packing requires B = 18 bits while AGB withrandom patterns and conventional vector quantizationrequire 21 and 24 bits, respectively.

V. Conclusions

In this paper, we proposed an efficient feedback reduc-tion algorithm for FDD-based massive MIMO systems.Our work is motivated by the observation that the CSI


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TABLE IISimulation parameters for ULA and UPA model.

Transmit correlation model Variables Simulation parametersUniform linear array (ULA) Angular spread = 15

Antenna elements spacing D = 0 .5Angle of arrival for the k -th user k ( , ]

Uniform planar array (UPA) Angular spread (vertical) V = 12 arctan(s + r

u ) arctan(s r

u )Angle of arrival (vertical) V = 12 arctan(

s + ru ) + arctan(

s ru )

Angular spread (horizontal) H = arctan( rs )Angle of arrival (horizontal) H,k ( , ]Elevation of the transmit antenna u = 60 mRadius of the scattering ring for the receiver r = 30 mDistance from the transmitter s = 100 m

2 3 4 5 6 7 8 9 107

7.5

8

8.5

9

9.5

10

10.5

11

Number of pattern bits Bp

S u m

r a

t e ( b p s

/ h z

)

Conventional vector quantizationAGB algorithm (random)AGB algorithm (subspace packing)

Fig. 12. Sum rate as a function of the number of pattern bits ( N t =16, N g = 8 , B q = 16).

feedback overhead scales linearly with the number of transmit antennas so that conventional vector approachperforming the quantization of the whole channel vec-tor is not an appropriate option for the massive MIMOregime. The key feature of the antenna group beamforming(AGB) algorithm is to control relentless growth of theCSI feedback information in the massive MIMO regime bymapping multiple correlated antenna elements to a single

representative value with pre-designed grouping patternsand then choosing the codeword from the codebook gen-erated from the reduced dimension channel vector. It hasbeen shown by distortion analysis and simulation resultsthat the proposed AGB algorithm is effective in achievinga substantial reduction in the feedback overhead in therealistic massive MIMO channels.

Although our study in this work focused on the single-cell scenario, we expect that the effectiveness of theproposed method can be readily extended to multi-cellscenario. In fact, in the multi-cell scenario, more aggres-sive feedback compression is required since the channel

information of the interfering cells as well as the desiredcell may be needed at the basestation to properly controlinter-cell interference. In this scenario, the proposed AGBalgorithm can be used as an effective means to achievereduction in the feedback information. Also, performanceevaluation of the AGB algorithm under channel estima-tion through closed-loop training [30] and investigation of nonlinear transmitter techniques with user scheduling [ 31]would be interesting direction to be investigated. Finally,we note that the proposed method can be nicely integratedinto the dual codebooks structure in LTE-Advanced [32],[33] by feeding back the pattern index for long-term basisand the codebook index for short-term basis. Since themain target of the massive MIMO system is slowly varyingor static channels, dual codebook based AGB algorithmwill bring further reduction in feedback overhead.

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Antenna Grouping based Feedback Reduction for FDD-based Massive MIMO Systems.pdf