pca-aidedlinearprecodinginmassivemimosystemswith...

Research ArticlePCA-Aided Linear Precoding in Massive MIMO Systems withImperfect CSI

Van-Khoi Dinh 1 Minh-Tuan Le2 Vu-Duc Ngo3 and Chi-Hieu Ta1

1Le Quy Don Technical University Hanoi Vietnam2MobiFone RampD Center MobiFone Corporation Hanoi Vietnam3Hanoi University of Science and Technology Hanoi Vietnam

Correspondence should be addressed to Van-Khoi Dinh vankhoitcugmailcom

Received 2 October 2019 Revised 22 December 2019 Accepted 23 January 2020 Published 24 February 2020

Academic Editor Michael McGuire

Copyright copy 2020 Van-Khoi Dinh et al +is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In this paper a low-complexity linear precoding algorithm based on the principal component analysis technique in combinationwith the conventional linear precoders called Principal Component Analysis Linear Precoder (PCA-LP) is proposed for massiveMIMO systems+e proposed precoder consists of two components the first one minimizes the interferences among neighboringusers and the second one improves the system performance by utilizing the Principal Component Analysis (PCA) techniqueNumerical and simulation results show that the proposed precoder has remarkably lower computational complexity than its low-complexity lattice reduction-aided regularized block diagonalization using zero forcing precoding (LC-RBD-LR-ZF) and lowercomputational complexity than the PCA-aided Minimum Mean Square Error combination with Block Diagonalization (PCA-MMSE-BD) counterparts while its bit error rate (BER) performance is comparable to those of the LC-RBD-LR-ZF and PCA-MMSE-BD ones

1 Introduction

In order to combat fading phenomenon in wireless com-munication the Multiple-Input Multiple-Output (MIMO)technique has been proposed and already applied in 4thgeneration (4G) cellular networks [1] +e MIMO systemscan significantly improve the channel capacity the BERperformance and the reliability of wireless systems by in-creasing the number of antennas at both transmitter andreceiver sides +e signal processing techniques for theuplink and downlink of both single-user MIMO ie point-to-point MIMO and multiuser MIMO (MU-MIMO) sys-tems have been extensively researched in recent yearsHowever in practice the number of antennas at the basesStation (BS) side in the MU-MIMO system is limited(normally fewer than 10) [2] +erefore the spectrum ef-ficiency and system capacity are still relatively modest

In order to cope with these issues Massive MIMOsystems have recently been proposed [1 3ndash5] In the Massive

MIMO the number of antennas at the BS can be up tohundreds (or even thousands) to simultaneously servedozens of users using the same frequency resource By in-creasing the number of antennas at the BS side the MassiveMIMO systems can significantly improve the channel ca-pacity and enhance the spectrum utilization efficiency andthe BER performance of the system [4] However theMassive MIMO systems also face many challenges such ashardware complexity power consumption and system costdue to a large number of antennas deployed at the BS [1 2]Basically the Massive MIMO system can work in TimeDivision Duplex (TDD) or Frequency Division Duplex(FDD) mode However the TDD operation is preferable tothe FDD operation because the TDD system can increase thenumber of antennas at the BS side to expand the systemcapacity without being affected by the coherence interval [1]It is expected that Massive MIMO will be a key and brightcandidate for the next generation wireless networks (eg 5Gnetwork) [1 4 6]

HindawiWireless Communications and Mobile ComputingVolume 2020 Article ID 3425952 9 pageshttpsdoiorg10115520203425952

Undoubtedly the Massive MIMO systems will becomemore complex as the number of antennas at the BS side getsvery large +erefore reducing the complexities of the signalprocessing algorithms for both uplink and downlink inMassive MIMO systems is necessary In Massive MIMOsystems the complex signal processing is performed at theBS side +erefore the precoding algorithms with lowcomplexity such as Zero Forcing (ZF) Minimum MeanSquare Error (MMSE) and Maximum Ratio Transmission(MRT) are considered as suitable solutions for the downlinkin the Massive MIMO system [7ndash9] Besides to eliminateinterference from neighboring users and hence improve thesystem performance the Block Diagonalization algorithm isadopted [10] However the computational complexity of theBD algorithm is very high In this context there exist dif-ferent proposals to reduce the computational complexitybased on the BD algorithm For example the QR decom-position based on Block Diagonalization (QR-BD) algo-rithm and the Pseudoinverse Block Diagonalization (PINV-BD) algorithms are proposed in [11 12] Nevertheless theapplication of these algorithms to Massive MIMO systemsremains a challenge task due to their excessive complexities

In [13] Wang et al proposed the precoder consisting oftwo components that utilize the LQ decomposition andSingular Value Decomposition (SVD) of the channel matrixBased on the proposed approach in [14] in [15] the authorsproposed the low-complexity Lattice Reduction- (LR-) aidedprecoding algorithms for theMU-MIMO system referred toas LC-RBD-LR-ZF and LCR-BD-LR-MMSE In the LC-RBD-LR-ZF and LC-RBD-LR-MMSE algorithms the firstprecoding matrix is created by applying the QR decom-position to the extended channel matrix +e second pre-coding matrix is obtained using the conventional ZF andMMSE precoding algorithms in combination with the LRtechnique to provide the corresponding LC-RBD-LR-ZFand LC-RBD-LR-MMSE precoders It was shown in [15]that the precoders significantly improved the system per-formance while reducing the computational complexitywhen compared to the one in [14] In [16] the authorsproposed a low-complexity linear precoding scheme forMU-MIMO systems based on the principal componentanalysis technique However the computational complex-ities of the precoders in [13ndash16] are still very high due to theQR and LQ decomposition operations +erefore thesealgorithms could hardly be applied to the Massive MIMOsystems Moreover the systems are investigated under theassumption that the perfect channel state information (CSI)is available at the BS side

Based on the principal component analysis techniqueand the linear precoding algorithms the paper proposes alow-complexity precoder for MassiveMIMO systems In ourproposed method the precoding matrix is designed toconsist two components +e first one is created by theMMSE precoding algorithm to minimize the interferencesamong neighboring users +e second one is designed basedon the principal component analysis technique to improvethe system performance Numerical and simulation resultsshow that the proposed precoder has remarkably lowercomputational complexity than the LC-RBD-LR-ZF in [15]

and lower computational complexity than the PCA-MMSE-BD in [16] while its BER performance is comparable tothose of the LC-RBD-LR-ZF and PCA-MMSE-BD precodersin both perfect and imperfect CSI scenarios In additionsimulation results show that the channel estimation erroradversely affects the system performance no matter whichprecoder is adopted +e system performance decreases asthe channel estimation error increases and vice versa for allprecoders

+e rest of this paper is organized as follows In Section2 we present the Massive MIMO system model with im-perfect CSI +e principal component analysis technique isreviewed in Section 3 In Section 4 we propose the linearprecoding algorithm for the Massive MIMO systems thatadopts the principal component analysis technique Simu-lation results are shown in Section 5 Finally conclusions aredrawn in Section 6

11 Notation +e notations are defined as follows Matricesand vectors are represented by symbols in bold (middot)T and (middot)H

denote the transpose and conjugate transpose respectivelyINR

denotes the NR times NR identity matrix and trace middot is thetrace of a square matrix

2 Downlink Channel Model in MassiveMIMO System

Let us consider a TDD-based Massive MIMO system withNT antennas at the BS to simultaneously serve K users asillustrated in Figure 1 Each user is equipped with Nu an-tennas Let NR be the total number of antennas for the Kusers then we have NR KNu

Let xu isin CNutimes1 u 1 K represent the transmittedsignal vector for the uth user +e received signal at the uthuser yu isin CNutimes1 can be expressed as follows

yu Hu 1113944

K

k1Wkxk + nu

HuWuxu + 1113944K

k1kneuHuWkxk + nu

(1)

where Hu isin CNutimesNT Wu isin CNTtimesNu and nu isin CNutimes1 are thechannel matrix from the BS to the uth user the precodingmatrix and the noise vector for the uth user respectively+e entries of the channel matrix are assumed to be identicalindependent distributed (iid) random variables with zeromean and unit variance

Let y yT1 yT

1 middot middot middot yTK1113960 1113961

T isin CNRtimes1 be the overall re-ceived signal vector for all users +en based on (1) y isgiven by

y HWx + n (2)

whereW isin CNTtimesNR is the precoding matrix for all users x

xT1 xT

2 middot middot middot xTK1113960 1113961

Tis the transmitted signal vector for K

users and n isin CNRtimes1 is the noise vector at the K users +eentries of n are assumed to be identical independent

2 Wireless Communications and Mobile Computing

distributed (iid) random variables with zero mean andvariance σ2n

In reality it is almost impossible for the BS to fully getthe CSI due to the effects of thermal noise and pilotcontamination In other words the system has to operateunder imperfect CSI conditions +e accuracy of the CSIavailable at the BS side depends on the channel estimatorsto be used +e imperfect channel matrixH HT

1 HT2 middot middot middot HT

K1113960 1113961T isin CNRtimesNT obtained by using the

MMSE channel estimation can be modeled as follows[17 18]

H

1 minus ϕ21113969

1113957H + ϕEerr (3)

where 1113957H 1113957HT

11113957HT

2 middot middot middot 1113957HT

K1113960 1113961

T isin CNRtimesNT is the perfectRayleigh fading channel from the BS to all users in whichentries 1113957hij are assumed to be complex Gaussian randomvariables with zero mean and unit variance Eerr isin CNRtimesNT isthe channel estimation error matrix in which entries areassumed to be normalized iid zero mean complex Gaussianrandom variables ϕ isin [0 1] is a parameter that indicates theaccuracy of the channel estimator From (3) we can see thatϕ 0 means that there is no channel estimation error andthe CSI at the BS is perfect Conversely ϕ 1 indicates acomplete failure of the channel estimator

3 Review of the Principal ComponentAnalysis Technique

+e Principal Component Analysis technique was presentedin [16 19ndash21] It is a mathematical tool that uses an or-thogonal transformation to convert a set of observations ofpossibly correlated variables into a set of values of linearlyuncorrelated variables which are called principal compo-nents Let U isin CMtimesN be the original data set and Y isin CMtimesN

be the rerepresentation of that data set Based on the ei-genvalue decomposition of UUT in linear algebra the re-lationship between Y and U is given by [16 19]

Y BU (4)

Herein B isin CMtimesM denotes the principal components ofU +en (4) can be represented as follows

y1 y2 middot middot middot yN1113858 1113859

b1b2⋮

bM

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

u1 u2 middot middot middot uN1113858 1113859 (5)

where bi isin C1timesM (i 1 2 M) is the row of B which isbasic vectors correspond to eigenvectors of covariancematrixUUT for representing the columns ofU+e Bmatrixis defined in such a way that the first principal component b1has the largest variance each succeeding component in turnhas the highest variance under the constraint that it is or-thogonal to the previous components

According to [16 19 20] the PCA technique based onthe eigenvalue decomposition of UUT is summarized asfollows

First each row of U is normalized to have zero mean asfollows

1113957U U minus Umean (6)

where Umean isin CMtimesN represents the mean of rows of thematrixU Defining vk (v1 v2 vm) isin CMtimes1 as the set ofeigenvectors associated with the eigenvalues μk of thesymmetric matrix 1113957U1113957UT then we have

1113957U1113957UT1113874 1113875vk μkvk (7)

where μk (μ1 μ2 μm) isin RMtimes1 Applying the QR de-composition to 1113957U we obtain

1113957U QR (8)

whereR isin CMtimesN is an upper triangularmatrix andQ isin CMtimesM

is a unitary matrix +erefore we can write1113957U1113957UT

QR(QR)T (9)

By using the SVD decomposition the RH matrix can beexpressed as follows

RH UΣVH

(10)

From (9) and (10) it follows that

1113957U1113957UT Q UΣVH

1113872 1113873TUΣVT

1113872 1113873QT QVΣ2

(QV)T (11)

Or equivalently we have

1113957U1113957UT1113874 1113875(QV) (QV)Σ2

(12)

From (7) and (12) we can see that the eigenvectors andeigenvalues of the covariance matrix ( 1113957U1113957UT

) are contained inthe matrices QV and Σ2 respectively +erefore the prin-cipal component matrix B is given by

B QV (13)

NT antennas-BS

Nuantennas-user

Figure 1 +e downlink channel model in the Massive MIMOsystem

Wireless Communications and Mobile Computing 3

4 Proposed Precoder

41 Proposed PCA-LPPrecoder In this section we constructa linear precoding algorithm based on the principal com-ponent analysis technique called PCA-LP precoder In ourproposal the precoding matrix is given by

WPCA LP βWaWb (14)

whereWa isin CNTtimesNR Wb isin CNRtimesNR and β is the normalizedpower factor which is given by

β

NR

trace WPCA LPWHPCA LP( 1113857

1113971

(15)

+e first precoding matrix Wa is obtained using theconventional MMSE technique as follows

Wa HH HHH+ σ2INR

1113872 1113873minus 1

W1aW2

a WKa1113960 1113961

(16)

where σ2 σ2nEs Es is the transmit symbol energy andWk

a isin CNTtimesNu (k 1 2 K) is the precoding matrix for

the kth user+e second precodingmatrixWb is constructed based on

the PCA transformation as followsFirst using Wk

a in (16) the effective channel matrix forthe kth user is computed to be

Hkeff HkW

ka (17)

whereHk isin CNutimesNT is the channel matrix from BS to the kthuser

Next the channel matrix Hkeff is normalized to give

Hknor isin C

NutimesNu with zero mean as follows

Hknor Hk

eff minus Hkmean (18)

where Hkmean isin C

NutimesNu denotes the mean matrix whoseentries are the means of the rows of the matrix Hk

eff Applying the QR decomposition to Hk

nor we obtain

Hknor Qk

norRknor (19)

where Rknor isin C

NutimesNu is an upper triangular matrix andQk

nor isin CNutimesNu is a unitary matrix with orthogonal columns

After that the SVD operation is applied to (Rknor)

H togive

Rknor1113872 1113873

H Uk

norΣknor Vk

nor1113872 1113873H

(20)

where Uknor isin C

NutimesNu and Vknor isin C

NutimesNu are unitary ma-trices with orthogonal columns and Σk

nor isin RNutimesNu is a di-

agonal matrix+e principal component factor matrix Ak

PCA isin CNutimesNu

for the kth user is obtained as

AkPCA Qk

norVknor (21)

Using AkPCA and Hk

eff the combination channel matrixHk

com isin CNutimesNu for the kth user is computed as follows

Hkcom Ak

PCAHkeff (22)

From (22) the precoding matrix Wkb isin C

NutimesNu for thekth user based on the conventional ZF algorithm is given by

Wkb Hk

com1113872 1113873H

Hkcom Hk

com1113872 1113873H

1113876 1113877minus 1

(23)

Finally the second precoding matrix Wb and theprincipal component factor matrix APCA isin CNRtimesNR for allusers are represented as follows

Wb

W1b 0 0

0 W2b 0

⋮ ⋮ ⋱ ⋮

0 0 WKb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

APCA

A1PCA 0 00 A2

PCA 0⋮ ⋮ ⋱ ⋮0 0 AK

PCA

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (25)

+e proposed algorithm PCA-LP is summarized inAlgorithm 1

At the user side the received signal vector for all userscan be expressed as

y HWPCA LPx + n (26)

Using y in (26) the estimated signal vector is given by

1113954x APCAy

β APCA

HWPCA LPx + n( 1113857

β1113890 1113891 (27)

From (27) with WPCA WaWb the error covariancematrix can be obtained as

Q E trace (1113954x minus x)(1113954x minus x)H

1113960 11139611113966 1113967

trace Es APCAHWPCA minus INR1113872 1113873 APCAHWPCA minus INR

1113872 1113873H

1113896

+σ2nAPCAAH

PCA

β21113897 (28)

+e Empirical Cumulative Distribution Functions(ECDFs) of the maximum diagonal element of Q in (28) forthe PCA-LP and PCA-MMSE-BD precoders at differentaccuracy levels of the channel estimator are illustrated inFigure 2 It can be observed from the figure that the largestelements on the diagonals of the error covariance matricesfor both precoders increase as the channel estimation errorincreases In addition the PCA-MMSE-BD precoder pro-vides slightly smaller maximum errors than the PCA-LP onein all cases

Figure 3 shows the ECDFs of the sums of diagonal el-ements of the error covariancematrices for the PCA-MMSE-BD precoder and the proposed PCA-LP precoder As can beseen from the figure the sums of the diagonal elements forall the precoders experience exactly the same behavior as the


maximum diagonal elements Similar to the results inFigure 2 the mean square errors for both precoders increasewhen the channel estimation error increases Besides thePCA-MMSE-BD precoder provides slightly smaller sum-mation errors than the PCA-LP one in all cases

Since the diagonal elements of error covariance matricesdetermine the mean square errors (MSEs) between thetransmitted symbols and the recovered ones BER perfor-mances of the PCA-MMSE-BD and the proposed PCA-LPprecoders decrease as the channel estimation error increasesFortunately the slightly higher MSE of the proposed pre-coder does not incur significant BER performance degra-dation as compared to the PCA-MMSE-BD precoder

42 Computational Complexity Analysis In this sectionthe computational complexity of the proposed PCA-LPprecoder is evaluated and compared with those of the

LC-RBD-LR-ZF in [15] and the PCA-MMSE-BD in [16]+ecomplexities are evaluated by counting the necessaryfloating point operations (flops) We assume that each realoperation (an addition a multiplication or a division) iscounted as a flop Hence a complex multiplication and adivision is equal to 6 flops and 11 flops respectively It isworth noting that the QR decomposition of an m times n

complex matrix requires 6mn2 + 4mn minus n2 minus n flopsAccording to [22] the SVD of a m times n (mge n) complexmatrix requires (4m2n + 8mn2 +9n3) flops Based on theabovementioned assumptions the computational com-plexity of the proposed algorithm PCA-LP is computed to be

FPCAminus LP F1 + F2 + F3(flops) (29)

where F1 and F2 are the number of flops to find the matricesWa and Wb and F3 is the number of flops for the multi-plication two matrices Wa and Wb

005 01 015 02 025 03 035 04 045Max (diag(Q))

0

01

02

03

04

05

06

07

08

09

1

ECD

F

PCA-MMSE-BD with ϕ = 0PCA-LP with ϕ = 0PCA-MMSE-BD with ϕ = 05

PCA-LP with ϕ = 05PCA-MMSE-BD with ϕ = 07PCA-LP with ϕ = 07

ϕ = 05

ϕ = 07

ϕ = 0

Figure 2 Empirical CDF of the maximum of diagonal elements ofQ for NT 64 Nu 2 and K 32

5 6 7 8 9 10 11 12 13 14Sum (diag(Q))

0

01

02

03

04

05

06

07

08

09

1

ECD

F



ϕ = 0

ϕ = 05

ϕ = 07

Figure 3 Empirical CDF of the sum of diagonal elements of Q forNT 64 Nu 2 and K 32

(1) Input NT NRH(2) Generate the matrix Wa [W1

aW2a WK

a ] as in (16)(3) Compute the matrices Hk

eff HkWka and Hk

nor Hkeff minus Hk

mean(4) Apply QR decomposition to Hk

nor Hknor Qk

norRknor

(5) Apply the SVD to (Rknor)

H (Rknor)

H UknorΣ

knor(V

knor)

H(6) Create the matrices Ak

PCA QknorV

knor and Hk

com AkPCAH

keff

(7) Generate the precoding matrix Wkb (Hk

com)H[Hkcom(Hk

com)H]minus 1(8) Repeat Step 3 to Step 7 until the precoding matrices Wk

b for all users are obtained(9) Create the matrices Wb and APCA by arranging Wk

b and AkPCA to the main diagonals of Wb and APCA as in (24) and (25)

respectively(10) Output Wa [W1

aW2a WK

a ] Wb diag(W1b W2

b WKb ) APCA diag(A1

PCAA2PCA AK

PCA)β

NRtrace(WPCA LPWHPCA LP)

1113969

ALGORITHM 1 +e PCA-LP precoding algorithm


+e number of flops to find the matrix Wa is equal to

F1 8N3R + 16N

2RNT minus N

2R minus 2NRNT + NR + 1(flops)

(30)

+e number of flops to find Wb is expressed as follows

F2 F4 + F5 + F6 + F7 + F8 + F9(flops) (31)

where F4 is the number of flops for the multiplication twomatrices Hk and Wk

a F5 is the number of flops for the QRdecomposition ofHk

nor F6 is the number of flops for SVD of(Rk

nor)H F7 is the number of flops for the multiplication two

matrices Qknor and Vk

nor F8 is the number of flops of themultiplication two matrices Ak

PCA and Hkeff Finally F9 is

number of flops to find the matrices Wkb +ese items are

given by

F4 K 8N2uNT minus 2N

2u1113872 1113873(flops)

F5 K 6N3u + 3N

2u minus Nu1113872 1113873(flops)

F6 21KN3u(flops)

F7 K 8N3u minus 2N

2u1113872 1113873(flops)

F8 K 8N3u minus 2N

2u1113872 1113873(flops)

F9 K 24N3u minus 4N

2u1113872 1113873(flops)

(32)

Besides F3 is calculated as

F3 8N2RNT minus 2N

2R(flops) (33)

+erefore the number of flops for the proposed PCA-LPprecoder is represented as follows

FPCAminus LP F1 + F2 + F3

8N3R + 16N

2RNT minus N

2R minus 2NRNT + 1

+ K 8N2uNT minus 2N

2u1113872 1113873 + K 6N

3u + 3N

2u minus Nu1113872 1113873

+ 21KN3u + 2K 8N

3u minus 2N

2u1113872 1113873 + K 24N

3u minus 4N

2u1113872 1113873

+ 8N2RNT minus 2N

2R(flops)

sim O NTN2R1113872 1113873

(34)

Using similar complexity analysis steps we are able toobtain the complexities of both the LC-RBD-LR-ZF and thePCA-MMSE-BD precoders which are summarized in Ta-ble 1 in the next page

5 Simulation Results

In this section we compare both the computational com-plexity and the system performance of the proposed PCA-LPprecoder with those of its LC-RBD-LR-ZF and PCA-MMSE-BD counterparts

Figure 4 demonstrates the computational complexities ofthe PCA-LP LC-RBD-LR-ZF and PCA-MMSE-BD pre-coders In this scenario NT NR is varied from 40 to 100transmit antennas Nu 2 and K NR2 Numerical re-sults show that the computational complexity of the pro-posed PCA-LP precoder is significantly lower than thecomplexity of the LC-RBD-LR-ZF precoder and lower thanthat of the PCA-MMSE-BD precoder For example at NR

NT 80 antennas the complexity of the proposed PCA-LP

Table 1 Computational complexity comparison

Precoding algorithms Complexity (flops) Complexitylevel

LC minus RBD minus LR minus ZF[15]

K[6(NR minus Nu)(NR + NT minus Nu)2 + 4(NR minus Nu)(NR + NT minus Nu) minus (NR + NT minus Nu)2

minus (NR + NT minus Nu)] +K(8N2TNu minus 2NTNu) + K(16N2

uNT minus 2NuNT + 8N3u minus 2N2

u

+ Fupdateminus LLL) + K(8N3u + 16N2

uNT minus 2N2u minus 2NuNT) + 8KN2

TNR minus 2NTNR

O(N4R)

PCA minus MMSE minus BD[16]

[8N3R + 16N2

RNT minus N2R minus 2NRNT + NR + 1] + K(6N2

TNu + 4NTNu minus N2u minus Nu)

+ K(8N2uNT minus 2N2

u) + K(24N3u + 4N2

u minus 2Nu) + 41KN3u + K(8N3

u minus 2N2u)

+ K(16N3u minus 4N2

u) + K(56N3u minus 8N2

u) + 8N2RNT minus 2N2

R

O(NTN2R)

PCA minus LP8N3

R + 16N2RNT minus N2

R minus 2NRNT + 1 + K(8N2uNT minus 2N2

u)

+ K(6N3u + 3N2

u minus Nu) + 21KN3u + 2K(8N3

u minus 2N2u) + K(24N3

u minus 4N2u)

+ 8N2RNT minus 2N2

R

O(NTN2R)

40 50 60 70 80 90 100Number of the transmit antennas

106

107

108

109

1010

Num

ber o

f flo

ps

LC-RBD-LR-ZFPCA-MMSE-BDPCA-LP

Figure 4 Comparison of the complexity of the proposed precodingalgorithm with the LC-RBD-LR-ZF and PCA-MMSE-BDalgorithms


precoder is approximately equal to 258 and 8407 of thecomplexities of the LC-RBD-LR-ZF and PCA-MMSE-BDprecoders respectively We can see that the complexity ofthe LC-RBD-LR-ZF precoder is very large for two reasonsfirstly the number of the QR operations applied to theextended channel matrix is too large Secondly the sizes ofthe precoding matrices Pa isin CNTtimesKNT and 1113957Pb isin CKNTtimesNR

increase proportionally to NR and NT +erefore thenumber of flops required for the multiplications of the twomatrices Pa and 1113957Pb also increase

+e BER curves of the proposed PCA-LP as well as thoseof the LC-RBD-LR-ZF and PCA-MMSE-BD precoders areillustrated in Figures 5ndash8 In Figure 5 the system is assumedto work in the perfect CSI condition (ie ϕ 0 and H 1113957H)with the following parameters NR NT 64 Nu 2

K NR2 and 4-QAM modulation It can be seen fromFigure 5 that the BER curves of the three precoders arealmost identical when the SNRle 27 dB For larger SNR theLC-RBD-LR-ZF precoder outperforms the remaining ones

In Figure 6 we simulate the system performance in theimperfect CSI condition (ieH

1 minus ϕ21113969

1113957H + ϕEerr) for ϕ

05 and ϕ 07 Other parameters are the same as those usedto generate Figure 5 Similar to the results in Figure 5 theresults in Figure 6 show that the three precoders providenearly the same system performance As the SNR is suffi-ciently large the LC-RBD-LR-ZF precoder is able to providebetter BER performance +e results from Figures 4ndash6 showthat the LC-RBD-LR-ZF precoder can marginally outper-form the proposed PCA-LP and PCA-MMSE-BD precodersin the sufficiently large SNR regions However it suffersfrom noticeably higher computational complexities

In Figure 7 the BER performances of the system withNT 128 Nu 2 andK 64 are illustrated for the threeprecoders in the perfect and imperfect CSI conditions at the

BS side Here 4-QAM modulation is also adopted +esimulation results in Figure 7 show that the BER curves ofthe PCA-LP proposed precoder almost coincide with thoseof the remaining precoders in all scenarios Moreover onecan observe from Figures 5ndash7 that as NT increases from 64to 128 the LC-RBD-LR-ZF precoder no longer outper-forms the proposed PCA-LP one in the high SNR regionsClearly the larger NT is deployed at the BS the better

0SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER

LC-RBD-LR-ZF with ϕ = 0PCA-MMSE-BD with ϕ = 0PCA-LP with ϕ = 0LC-RBD-LR-ZF with ϕ = 05PCA-MMSE-BD with ϕ = 05

PCA-LP with ϕ = 05LC-RBD-LR-ZF with ϕ = 07PCA-MMSE-BD with ϕ = 07PCA-LP with ϕ = 07

21

66

68

times10ndash4

238

2

22

times10ndash4 times10ndash5

268

859

95

ϕ = 07

ϕ = 05

ϕ = 0

5 10 15 20 25 30

24 242 27

Figure 7 +e system performance with NT NR 128 K 64

andNu 2 in the perfect and imperfect CSI conditions at the BSside

0 5 10 15 20 25SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER


236 238 24 242 24412

141618

2times10ndash4

Figure 5 +e systemperformancewithNT NR 64 K 32 Nu

2 in the perfect CSI condition at the BS side

0 5 10 15 20 25 30SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER

LC-RBD-LR-ZF with ϕ = 05PCA-MMSE-BD with ϕ = 05PCA-LP with ϕ = 05


266 27 27445

555

665 times10ndash5

265 27 275

12

14

1618

times10ndash4

ϕ = 07

ϕ = 05


Nu 2 ϕ 05 and ϕ 07 in the imperfect CSI condition at theBS side


system performance the proposed PCA-LP precoder canachieve

Figure 8 illustrates the BER curves of the three precodersas functions of ϕ at SNR 24 dB and 27 dB Other simulationparameters are kept the same as the ones used for Figure 5ie NT NR 64 K 32 Nu 2 and 4-QAM modula-tion We again see that for the same parameters the threeprecoders provide nearly the same BERs particularly when ϕbecomes larger Furthermore the channel estimation errorhas an adverse effect on the system performance no matterwhich precoder is employed +e larger values of ϕmake thesystem performance deteriorate more rapidly

It is noteworthy that the simulation results inFigures 5ndash8 are obtained for the worst cases whenNT K times Nu ie when the systems work in full-loadconditions As the number of user gets smaller the systemperformance definitely becomes better

6 Conclusions

+is paper proposes the PCA-LP precoder with low com-plexity in Massive MIMO systems with imperfect CSI at theBS side +e proposed precoder consists of two component+e first one is designed to eliminate interference fromneighboring users and the second one is created based on thePrincipal Component Analysis technique and the linearprecoding algorithm to improve the system performanceNumerical and simulation results show that the computa-tional complexity of the proposed PCA-LP precoder issignificantly lower than the complexity of the LC-RBD-LR-ZF precoder and lower than that of the PCA-MMSE-BDprecoder while they all provide nearly the same systemperformance in all simulation scenarios Simulation resultsalso show that the channel estimation error has an adverseeffect on the system performance no matter which precoderis adopted For all precoders the system performance

decreases as the channel estimation error increases and viceversa Moreover the larger the channel estimation error isthe more rapidly the system performance deterioratesTaking both system performance and complexity intoconsideration the proposed PCA-LP precoder can be apotential digital beamforming technique for the downlinksof Massive MIMO systems

Data Availability

All results included in this paper are generated by simulationin Matlab

Conflicts of Interest

+e authors declare that they have no conflicts of interest

References

[1] H Q Ngo Massive MIMO Fundamentals and System De-signs Vol 1642 Linkoping University Electronic PressLinkoping Sweden 2015

[2] L Lu G Y Li A L Swindlehurst A Ashikhmin andR Zhang ldquoAn overview of massive mimo benefits andchallengesrdquo IEEE Journal of Selected Topics in Signal Pro-cessing vol 8 no 5 pp 742ndash758 2014

[3] T L Marzetta ldquoNoncooperative cellular wireless with un-limited numbers of base station antennasrdquo IEEE Transactionson Wireless Communications vol 9 no 11 pp 3590ndash36002010

[4] T L Marzetta ldquoMassive MIMO an introductionrdquo Bell LabsTechnical Journal vol 20 pp 11ndash22 2015

[5] H Y T L Marzetta E G Larsson and H Q Ngo Fun-damentals of Massive MIMO Cambridge University PressCambridge UK 2016

[6] E G Larsson O Edfors F Tufvesson and T L MarzettaldquoMassive MIMO for next generationC wireless systemsrdquo IEEECommunications Magazine vol 52 no 2 pp 186ndash195 2014

[7] V P Selvan M S Iqbal and H S Al-Raweshidy ldquoPerfor-mance analysis of linear precoding schemes for very largemulti-user MIMO downlink systemrdquo in Proceedings of theFourth Edition of the International Conference on the Inno-vative Computing Technology (INTECH 2014) pp 219ndash224London UK 2014

[8] H Q Ngo E G Larsson and T L Marzetta ldquoMassive MU-MIMO downlink TDD systems with linear precoding anddownlink pilotsrdquo in Proceedings of the 51st Annual AllertonConference on Communication Control and Computing(Allerton) pp 293ndash298 Monticello IL USA 2013

[9] Y S Cho J Kim W Y Yang and C G KangMIMO-OFDMWireless Communications with MATLAB JohnWiley amp SonsHoboken NJ USA 2010

[10] Q H Spencer A L Swindlehurst and M Haardt ldquoZero-forcing methods for downlink spatial multiplexing in mul-tiuser mimo channelsrdquo IEEE Transactions on Signal Pro-cessing vol 52 no 2 pp 461ndash471 2004

[11] J Wu S Fang L Li and Y Yang ldquoQR decomposition andgram Schmidt orthogonalization based low-complexity multi-user MIMO precodingrdquo in Proceedings of the 10th Interna-tional Conference on Wireless Communications Networkingand Mobile Computing (WiCOM 2014) pp 61ndash66 BeijingChina September 2014

0ϕ

10ndash4

10ndash5

10ndash3

10ndash2

10ndash1

100

BER



SNR = 24dB

SNR = 27dB

02 04 06 08 1

Figure 8 +e system performance according to ϕ at SNR 24 dBand 27 dB with NT NR 64 K 32 andNu 2


[12] W Li and M Latva-aho ldquoAn efficient channel block diago-nalization method for generalized zero forcing assisted mimobroadcasting systemsrdquo IEEE Transactions on Wireless Com-munications vol 10 no 3 pp 739ndash744 March 2011

[13] H Wang L Li L Song and X Gao ldquoA linear precodingscheme for downlink multiuser mimo precoding systemsrdquoIEEE Communications Letters vol 15 no 6 pp 653ndash6552011

[14] V Stankovic and M Haardt ldquoGeneralized design of multi-user mimo precoding matricesrdquo IEEE Transactions onWireless Communications vol 7 no 3 pp 953ndash961 2008

[15] K Zu and R C de Lamare ldquoLow-complexity lattice reduc-tion-aided regularized block diagonalization for mu-mimosystemsrdquo IEEE Communications Letters vol 16 no 6pp 925ndash928 2012

[16] S B M Priya and P Kumar ldquoDesign of low complex linearprecoding scheme for mu-mimo systemsrdquo Wireless PersonalCommunications vol 97 no 1 pp 1097ndash1116 2017

[17] I Frigyes J Bito and P Bakki Advances in Mobile andWireless Communications Views of the 16th IST Mobile andWireless Communication Summit Springer Berlin Germany2008

[18] F Rusek D Persson B K Buon Kiong Lau et al ldquoScaling upMIMO opportunities and challenges with very large arraysrdquoIEEE Signal Processing Magazine vol 30 no 1 pp 40ndash602013

[19] A Sharma K K Paliwal S Imoto and S Miyano ldquoPrincipalcomponent analysis using qr decompositionrdquo InternationalJournal of Machine Learning and Cybernetics vol 4 no 6pp 679ndash683 2013

[20] J Shlens ldquoA tutorial on principal component analysisrdquo 2005httpswwwcscmueduelawpaperspcapdf

[21] C M Bishop Pattern Recognition and Machine LearningSpringer Berlin Germany 2006

[22] G H Golub and C F Van LoanMatrix Computations JohnsHopkins University Press Baltimore MD USA 1996


Undoubtedly the Massive MIMO systems will becomemore complex as the number of antennas at the BS side getsvery large +erefore reducing the complexities of the signalprocessing algorithms for both uplink and downlink inMassive MIMO systems is necessary In Massive MIMOsystems the complex signal processing is performed at theBS side +erefore the precoding algorithms with lowcomplexity such as Zero Forcing (ZF) Minimum MeanSquare Error (MMSE) and Maximum Ratio Transmission(MRT) are considered as suitable solutions for the downlinkin the Massive MIMO system [7ndash9] Besides to eliminateinterference from neighboring users and hence improve thesystem performance the Block Diagonalization algorithm isadopted [10] However the computational complexity of theBD algorithm is very high In this context there exist dif-ferent proposals to reduce the computational complexitybased on the BD algorithm For example the QR decom-position based on Block Diagonalization (QR-BD) algo-rithm and the Pseudoinverse Block Diagonalization (PINV-BD) algorithms are proposed in [11 12] Nevertheless theapplication of these algorithms to Massive MIMO systemsremains a challenge task due to their excessive complexities

In [13] Wang et al proposed the precoder consisting oftwo components that utilize the LQ decomposition andSingular Value Decomposition (SVD) of the channel matrixBased on the proposed approach in [14] in [15] the authorsproposed the low-complexity Lattice Reduction- (LR-) aidedprecoding algorithms for theMU-MIMO system referred toas LC-RBD-LR-ZF and LCR-BD-LR-MMSE In the LC-RBD-LR-ZF and LC-RBD-LR-MMSE algorithms the firstprecoding matrix is created by applying the QR decom-position to the extended channel matrix +e second pre-coding matrix is obtained using the conventional ZF andMMSE precoding algorithms in combination with the LRtechnique to provide the corresponding LC-RBD-LR-ZFand LC-RBD-LR-MMSE precoders It was shown in [15]that the precoders significantly improved the system per-formance while reducing the computational complexitywhen compared to the one in [14] In [16] the authorsproposed a low-complexity linear precoding scheme forMU-MIMO systems based on the principal componentanalysis technique However the computational complex-ities of the precoders in [13ndash16] are still very high due to theQR and LQ decomposition operations +erefore thesealgorithms could hardly be applied to the Massive MIMOsystems Moreover the systems are investigated under theassumption that the perfect channel state information (CSI)is available at the BS side

Based on the principal component analysis techniqueand the linear precoding algorithms the paper proposes alow-complexity precoder for MassiveMIMO systems In ourproposed method the precoding matrix is designed toconsist two components +e first one is created by theMMSE precoding algorithm to minimize the interferencesamong neighboring users +e second one is designed basedon the principal component analysis technique to improvethe system performance Numerical and simulation resultsshow that the proposed precoder has remarkably lowercomputational complexity than the LC-RBD-LR-ZF in [15]

and lower computational complexity than the PCA-MMSE-BD in [16] while its BER performance is comparable tothose of the LC-RBD-LR-ZF and PCA-MMSE-BD precodersin both perfect and imperfect CSI scenarios In additionsimulation results show that the channel estimation erroradversely affects the system performance no matter whichprecoder is adopted +e system performance decreases asthe channel estimation error increases and vice versa for allprecoders

+e rest of this paper is organized as follows In Section2 we present the Massive MIMO system model with im-perfect CSI +e principal component analysis technique isreviewed in Section 3 In Section 4 we propose the linearprecoding algorithm for the Massive MIMO systems thatadopts the principal component analysis technique Simu-lation results are shown in Section 5 Finally conclusions aredrawn in Section 6

11 Notation +e notations are defined as follows Matricesand vectors are represented by symbols in bold (middot)T and (middot)H

denote the transpose and conjugate transpose respectivelyINR

denotes the NR times NR identity matrix and trace middot is thetrace of a square matrix

2 Downlink Channel Model in MassiveMIMO System

Let us consider a TDD-based Massive MIMO system withNT antennas at the BS to simultaneously serve K users asillustrated in Figure 1 Each user is equipped with Nu an-tennas Let NR be the total number of antennas for the Kusers then we have NR KNu

Let xu isin CNutimes1 u 1 K represent the transmittedsignal vector for the uth user +e received signal at the uthuser yu isin CNutimes1 can be expressed as follows

yu Hu 1113944

K

k1Wkxk + nu

HuWuxu + 1113944K

k1kneuHuWkxk + nu

(1)

where Hu isin CNutimesNT Wu isin CNTtimesNu and nu isin CNutimes1 are thechannel matrix from the BS to the uth user the precodingmatrix and the noise vector for the uth user respectively+e entries of the channel matrix are assumed to be identicalindependent distributed (iid) random variables with zeromean and unit variance

Let y yT1 yT

1 middot middot middot yTK1113960 1113961

T isin CNRtimes1 be the overall re-ceived signal vector for all users +en based on (1) y isgiven by

y HWx + n (2)

whereW isin CNTtimesNR is the precoding matrix for all users x

xT1 xT

2 middot middot middot xTK1113960 1113961

Tis the transmitted signal vector for K

users and n isin CNRtimes1 is the noise vector at the K users +eentries of n are assumed to be identical independent







H

1 minus ϕ21113969

1113957H + ϕEerr (3)

where 1113957H 1113957HT

11113957HT


K1113960 1113961





Y BU (4)



b1b2⋮

bM

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦







1113957U1113957UT1113874 1113875vk μkvk (7)


1113957U QR (8)



QR(QR)T (9)


RH UΣVH

(10)


1113957U1113957UT Q UΣVH

1113872 1113873TUΣVT

1113872 1113873QT QVΣ2

(QV)T (11)


1113957U1113957UT1113874 1113875(QV) (QV)Σ2

(12)



B QV (13)

NT antennas-BS

Nuantennas-user



4 Proposed Precoder


WPCA LP βWaWb (14)


β

NR


1113971

(15)


Wa HH HHH+ σ2INR

1113872 1113873minus 1

W1aW2

a WKa1113960 1113961

(16)






Hkeff HkW

ka (17)



Hknor isin C


Hknor Hk


where Hkmean isin C



nor we obtain

Hknor Qk

norRknor (19)

where Rknor isin C




H togive

Rknor1113872 1113873

H Uk

norΣknor Vk

nor1113872 1113873H

(20)

where Uknor isin C





PCA isin CNutimesNu


AkPCA Qk

norVknor (21)

Using AkPCA and Hk



Hkcom Ak

PCAHkeff (22)



Wkb Hk

com1113872 1113873H

Hkcom Hk

com1113872 1113873H

1113876 1113877minus 1

(23)


Wb

W1b 0 0

0 W2b 0

⋮ ⋮ ⋱ ⋮

0 0 WKb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

APCA

A1PCA 0 00 A2

PCA 0⋮ ⋮ ⋱ ⋮0 0 AK

PCA

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (25)





1113954x APCAy

β APCA


β1113890 1113891 (27)



1113960 11139611113966 1113967


1113872 1113873H

1113896

+σ2nAPCAAH

PCA

β21113897 (28)











005 01 015 02 025 03 035 04 045Max (diag(Q))

0

01

02

03

04

05

06

07

08

09

1

ECD

F



ϕ = 05

ϕ = 07

ϕ = 0


5 6 7 8 9 10 11 12 13 14Sum (diag(Q))

0

01

02

03

04

05

06

07

08

09

1

ECD

F



ϕ = 0

ϕ = 05

ϕ = 07



aW2a WK


eff HkWka and Hk

nor Hkeff minus Hk


nor Hknor Qk

norRknor


H (Rknor)

H UknorΣ

knor(V

knor)


PCA QknorV

knor and Hk

com AkPCAH

keff


com)H[Hkcom(Hk





aW2a WK

a ] Wb diag(W1b W2


PCAA2PCA AK

PCA)β


1113969




F1 8N3R + 16N

2RNT minus N


(30)


F2 F4 + F5 + F6 + F7 + F8 + F9(flops) (31)









given by


2u1113872 1113873(flops)

F5 K 6N3u + 3N

2u minus Nu1113872 1113873(flops)

F6 21KN3u(flops)

F7 K 8N3u minus 2N

2u1113872 1113873(flops)

F8 K 8N3u minus 2N

2u1113872 1113873(flops)

F9 K 24N3u minus 4N

2u1113872 1113873(flops)

(32)


F3 8N2RNT minus 2N

2R(flops) (33)



8N3R + 16N

2RNT minus N

2R minus 2NRNT + 1

+ K 8N2uNT minus 2N

2u1113872 1113873 + K 6N

3u + 3N

2u minus Nu1113872 1113873

+ 21KN3u + 2K 8N

3u minus 2N

2u1113872 1113873 + K 24N

3u minus 4N

2u1113872 1113873

+ 8N2RNT minus 2N

2R(flops)

sim O NTN2R1113872 1113873

(34)












u



TNR minus 2NTNR

O(N4R)


[8N3R + 16N2




u) + K(24N3u + 4N2


u minus 2N2u)

+ K(16N3u minus 4N2



R

O(NTN2R)

PCA minus LP8N3



u)

+ K(6N3u + 3N2



u minus 4N2u)

+ 8N2RNT minus 2N2

R

O(NTN2R)


106

107

108

109

1010

Num

ber o

f flo

ps









1 minus ϕ21113969





0SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER



21

66

68

times10ndash4

238

2

22


268

859

95

ϕ = 07

ϕ = 05

ϕ = 0

5 10 15 20 25 30

24 242 27



0 5 10 15 20 25SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER


236 238 24 242 24412

141618

2times10ndash4



0 5 10 15 20 25 30SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER



266 27 27445

555

665 times10ndash5

265 27 275

12

14

1618

times10ndash4

ϕ = 07

ϕ = 05







6 Conclusions



Data Availability




References












0ϕ

10ndash4

10ndash5

10ndash3

10ndash2

10ndash1

100

BER



SNR = 24dB

SNR = 27dB

02 04 06 08 1




















H

1 minus ϕ21113969

1113957H + ϕEerr (3)

where 1113957H 1113957HT

11113957HT


K1113960 1113961





Y BU (4)



b1b2⋮

bM

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦







1113957U1113957UT1113874 1113875vk μkvk (7)


1113957U QR (8)



QR(QR)T (9)


RH UΣVH

(10)


1113957U1113957UT Q UΣVH

1113872 1113873TUΣVT

1113872 1113873QT QVΣ2

(QV)T (11)


1113957U1113957UT1113874 1113875(QV) (QV)Σ2

(12)



B QV (13)

NT antennas-BS

Nuantennas-user



4 Proposed Precoder


WPCA LP βWaWb (14)


β

NR


1113971

(15)


Wa HH HHH+ σ2INR

1113872 1113873minus 1

W1aW2

a WKa1113960 1113961

(16)






Hkeff HkW

ka (17)



Hknor isin C


Hknor Hk


where Hkmean isin C



nor we obtain

Hknor Qk

norRknor (19)

where Rknor isin C




H togive

Rknor1113872 1113873

H Uk

norΣknor Vk

nor1113872 1113873H

(20)

where Uknor isin C





PCA isin CNutimesNu


AkPCA Qk

norVknor (21)

Using AkPCA and Hk



Hkcom Ak

PCAHkeff (22)



Wkb Hk

com1113872 1113873H

Hkcom Hk

com1113872 1113873H

1113876 1113877minus 1

(23)


Wb

W1b 0 0

0 W2b 0

⋮ ⋮ ⋱ ⋮

0 0 WKb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

APCA

A1PCA 0 00 A2

PCA 0⋮ ⋮ ⋱ ⋮0 0 AK

PCA

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (25)





1113954x APCAy

β APCA


β1113890 1113891 (27)



1113960 11139611113966 1113967


1113872 1113873H

1113896

+σ2nAPCAAH

PCA

β21113897 (28)











005 01 015 02 025 03 035 04 045Max (diag(Q))

0

01

02

03

04

05

06

07

08

09

1

ECD

F



ϕ = 05

ϕ = 07

ϕ = 0


5 6 7 8 9 10 11 12 13 14Sum (diag(Q))

0

01

02

03

04

05

06

07

08

09

1

ECD

F



ϕ = 0

ϕ = 05

ϕ = 07



aW2a WK


eff HkWka and Hk

nor Hkeff minus Hk


nor Hknor Qk

norRknor


H (Rknor)

H UknorΣ

knor(V

knor)


PCA QknorV

knor and Hk

com AkPCAH

keff


com)H[Hkcom(Hk





aW2a WK

a ] Wb diag(W1b W2


PCAA2PCA AK

PCA)β


1113969




F1 8N3R + 16N

2RNT minus N


(30)


F2 F4 + F5 + F6 + F7 + F8 + F9(flops) (31)









given by


2u1113872 1113873(flops)

F5 K 6N3u + 3N

2u minus Nu1113872 1113873(flops)

F6 21KN3u(flops)

F7 K 8N3u minus 2N

2u1113872 1113873(flops)

F8 K 8N3u minus 2N

2u1113872 1113873(flops)

F9 K 24N3u minus 4N

2u1113872 1113873(flops)

(32)


F3 8N2RNT minus 2N

2R(flops) (33)



8N3R + 16N

2RNT minus N

2R minus 2NRNT + 1

+ K 8N2uNT minus 2N

2u1113872 1113873 + K 6N

3u + 3N

2u minus Nu1113872 1113873

+ 21KN3u + 2K 8N

3u minus 2N

2u1113872 1113873 + K 24N

3u minus 4N

2u1113872 1113873

+ 8N2RNT minus 2N

2R(flops)

sim O NTN2R1113872 1113873

(34)












u



TNR minus 2NTNR

O(N4R)


[8N3R + 16N2




u) + K(24N3u + 4N2


u minus 2N2u)

+ K(16N3u minus 4N2



R

O(NTN2R)

PCA minus LP8N3



u)

+ K(6N3u + 3N2



u minus 4N2u)

+ 8N2RNT minus 2N2

R

O(NTN2R)


106

107

108

109

1010

Num

ber o

f flo

ps









1 minus ϕ21113969





0SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER



21

66

68

times10ndash4

238

2

22


268

859

95

ϕ = 07

ϕ = 05

ϕ = 0

5 10 15 20 25 30

24 242 27



0 5 10 15 20 25SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER


236 238 24 242 24412

141618

2times10ndash4



0 5 10 15 20 25 30SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER



266 27 27445

555

665 times10ndash5

265 27 275

12

14

1618

times10ndash4

ϕ = 07

ϕ = 05







6 Conclusions



Data Availability




References












0ϕ

10ndash4

10ndash5

10ndash3

10ndash2

10ndash1

100

BER



SNR = 24dB

SNR = 27dB

02 04 06 08 1















4 Proposed Precoder


WPCA LP βWaWb (14)


β

NR


1113971

(15)


Wa HH HHH+ σ2INR

1113872 1113873minus 1

W1aW2

a WKa1113960 1113961

(16)






Hkeff HkW

ka (17)



Hknor isin C


Hknor Hk


where Hkmean isin C



nor we obtain

Hknor Qk

norRknor (19)

where Rknor isin C




H togive

Rknor1113872 1113873

H Uk

norΣknor Vk

nor1113872 1113873H

(20)

where Uknor isin C





PCA isin CNutimesNu


AkPCA Qk

norVknor (21)

Using AkPCA and Hk



Hkcom Ak

PCAHkeff (22)



Wkb Hk

com1113872 1113873H

Hkcom Hk

com1113872 1113873H

1113876 1113877minus 1

(23)


Wb

W1b 0 0

0 W2b 0

⋮ ⋮ ⋱ ⋮

0 0 WKb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

APCA

A1PCA 0 00 A2

PCA 0⋮ ⋮ ⋱ ⋮0 0 AK

PCA

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (25)





1113954x APCAy

β APCA


β1113890 1113891 (27)



1113960 11139611113966 1113967


1113872 1113873H

1113896

+σ2nAPCAAH

PCA

β21113897 (28)











005 01 015 02 025 03 035 04 045Max (diag(Q))

0

01

02

03

04

05

06

07

08

09

1

ECD

F



ϕ = 05

ϕ = 07

ϕ = 0


5 6 7 8 9 10 11 12 13 14Sum (diag(Q))

0

01

02

03

04

05

06

07

08

09

1

ECD

F



ϕ = 0

ϕ = 05

ϕ = 07



aW2a WK


eff HkWka and Hk

nor Hkeff minus Hk


nor Hknor Qk

norRknor


H (Rknor)

H UknorΣ

knor(V

knor)


PCA QknorV

knor and Hk

com AkPCAH

keff


com)H[Hkcom(Hk





aW2a WK

a ] Wb diag(W1b W2


PCAA2PCA AK

PCA)β


1113969




F1 8N3R + 16N

2RNT minus N


(30)


F2 F4 + F5 + F6 + F7 + F8 + F9(flops) (31)









given by


2u1113872 1113873(flops)

F5 K 6N3u + 3N

2u minus Nu1113872 1113873(flops)

F6 21KN3u(flops)

F7 K 8N3u minus 2N

2u1113872 1113873(flops)

F8 K 8N3u minus 2N

2u1113872 1113873(flops)

F9 K 24N3u minus 4N

2u1113872 1113873(flops)

(32)


F3 8N2RNT minus 2N

2R(flops) (33)



8N3R + 16N

2RNT minus N

2R minus 2NRNT + 1

+ K 8N2uNT minus 2N

2u1113872 1113873 + K 6N

3u + 3N

2u minus Nu1113872 1113873

+ 21KN3u + 2K 8N

3u minus 2N

2u1113872 1113873 + K 24N

3u minus 4N

2u1113872 1113873

+ 8N2RNT minus 2N

2R(flops)

sim O NTN2R1113872 1113873

(34)












u



TNR minus 2NTNR

O(N4R)


[8N3R + 16N2




u) + K(24N3u + 4N2


u minus 2N2u)

+ K(16N3u minus 4N2



R

O(NTN2R)

PCA minus LP8N3



u)

+ K(6N3u + 3N2



u minus 4N2u)

+ 8N2RNT minus 2N2

R

O(NTN2R)


106

107

108

109

1010

Num

ber o

f flo

ps









1 minus ϕ21113969





0SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER



21

66

68

times10ndash4

238

2

22


268

859

95

ϕ = 07

ϕ = 05

ϕ = 0

5 10 15 20 25 30

24 242 27



0 5 10 15 20 25SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER


236 238 24 242 24412

141618

2times10ndash4



0 5 10 15 20 25 30SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER



266 27 27445

555

665 times10ndash5

265 27 275

12

14

1618

times10ndash4

ϕ = 07

ϕ = 05







6 Conclusions



Data Availability




References












0ϕ

10ndash4

10ndash5

10ndash3

10ndash2

10ndash1

100

BER



SNR = 24dB

SNR = 27dB

02 04 06 08 1






















005 01 015 02 025 03 035 04 045Max (diag(Q))

0

01

02

03

04

05

06

07

08

09

1

ECD

F



ϕ = 05

ϕ = 07

ϕ = 0


5 6 7 8 9 10 11 12 13 14Sum (diag(Q))

0

01

02

03

04

05

06

07

08

09

1

ECD

F



ϕ = 0

ϕ = 05

ϕ = 07



aW2a WK


eff HkWka and Hk

nor Hkeff minus Hk


nor Hknor Qk

norRknor


H (Rknor)

H UknorΣ

knor(V

knor)


PCA QknorV

knor and Hk

com AkPCAH

keff


com)H[Hkcom(Hk





aW2a WK

a ] Wb diag(W1b W2


PCAA2PCA AK

PCA)β


1113969




F1 8N3R + 16N

2RNT minus N


(30)


F2 F4 + F5 + F6 + F7 + F8 + F9(flops) (31)









given by


2u1113872 1113873(flops)

F5 K 6N3u + 3N

2u minus Nu1113872 1113873(flops)

F6 21KN3u(flops)

F7 K 8N3u minus 2N

2u1113872 1113873(flops)

F8 K 8N3u minus 2N

2u1113872 1113873(flops)

F9 K 24N3u minus 4N

2u1113872 1113873(flops)

(32)


F3 8N2RNT minus 2N

2R(flops) (33)



8N3R + 16N

2RNT minus N

2R minus 2NRNT + 1

+ K 8N2uNT minus 2N

2u1113872 1113873 + K 6N

3u + 3N

2u minus Nu1113872 1113873

+ 21KN3u + 2K 8N

3u minus 2N

2u1113872 1113873 + K 24N

3u minus 4N

2u1113872 1113873

+ 8N2RNT minus 2N

2R(flops)

sim O NTN2R1113872 1113873

(34)












u



TNR minus 2NTNR

O(N4R)


[8N3R + 16N2




u) + K(24N3u + 4N2


u minus 2N2u)

+ K(16N3u minus 4N2



R

O(NTN2R)

PCA minus LP8N3



u)

+ K(6N3u + 3N2



u minus 4N2u)

+ 8N2RNT minus 2N2

R

O(NTN2R)


106

107

108

109

1010

Num

ber o

f flo

ps









1 minus ϕ21113969





0SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER



21

66

68

times10ndash4

238

2

22


268

859

95

ϕ = 07

ϕ = 05

ϕ = 0

5 10 15 20 25 30

24 242 27



0 5 10 15 20 25SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER


236 238 24 242 24412

141618

2times10ndash4



0 5 10 15 20 25 30SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER



266 27 27445

555

665 times10ndash5

265 27 275

12

14

1618

times10ndash4

ϕ = 07

ϕ = 05







6 Conclusions



Data Availability




References












0ϕ

10ndash4

10ndash5

10ndash3

10ndash2

10ndash1

100

BER



SNR = 24dB

SNR = 27dB

02 04 06 08 1
















F1 8N3R + 16N

2RNT minus N


(30)


F2 F4 + F5 + F6 + F7 + F8 + F9(flops) (31)









given by


2u1113872 1113873(flops)

F5 K 6N3u + 3N

2u minus Nu1113872 1113873(flops)

F6 21KN3u(flops)

F7 K 8N3u minus 2N

2u1113872 1113873(flops)

F8 K 8N3u minus 2N

2u1113872 1113873(flops)

F9 K 24N3u minus 4N

2u1113872 1113873(flops)

(32)


F3 8N2RNT minus 2N

2R(flops) (33)



8N3R + 16N

2RNT minus N

2R minus 2NRNT + 1

+ K 8N2uNT minus 2N

2u1113872 1113873 + K 6N

3u + 3N

2u minus Nu1113872 1113873

+ 21KN3u + 2K 8N

3u minus 2N

2u1113872 1113873 + K 24N

3u minus 4N

2u1113872 1113873

+ 8N2RNT minus 2N

2R(flops)

sim O NTN2R1113872 1113873

(34)












u



TNR minus 2NTNR

O(N4R)


[8N3R + 16N2




u) + K(24N3u + 4N2


u minus 2N2u)

+ K(16N3u minus 4N2



R

O(NTN2R)

PCA minus LP8N3



u)

+ K(6N3u + 3N2



u minus 4N2u)

+ 8N2RNT minus 2N2

R

O(NTN2R)


106

107

108

109

1010

Num

ber o

f flo

ps









1 minus ϕ21113969





0SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER



21

66

68

times10ndash4

238

2

22


268

859

95

ϕ = 07

ϕ = 05

ϕ = 0

5 10 15 20 25 30

24 242 27



0 5 10 15 20 25SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER


236 238 24 242 24412

141618

2times10ndash4



0 5 10 15 20 25 30SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER



266 27 27445

555

665 times10ndash5

265 27 275

12

14

1618

times10ndash4

ϕ = 07

ϕ = 05







6 Conclusions



Data Availability




References












0ϕ

10ndash4

10ndash5

10ndash3

10ndash2

10ndash1

100

BER



SNR = 24dB

SNR = 27dB

02 04 06 08 1




















1 minus ϕ21113969





0SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER



21

66

68

times10ndash4

238

2

22


268

859

95

ϕ = 07

ϕ = 05

ϕ = 0

5 10 15 20 25 30

24 242 27



0 5 10 15 20 25SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER


236 238 24 242 24412

141618

2times10ndash4



0 5 10 15 20 25 30SNR (dB)

10ndash4

10ndash3

10ndash2

10ndash1

100

BER



266 27 27445

555

665 times10ndash5

265 27 275

12

14

1618

times10ndash4

ϕ = 07

ϕ = 05







6 Conclusions



Data Availability




References












0ϕ

10ndash4

10ndash5

10ndash3

10ndash2

10ndash1

100

BER



SNR = 24dB

SNR = 27dB

02 04 06 08 1


















6 Conclusions



Data Availability




References












0ϕ

10ndash4

10ndash5

10ndash3

10ndash2

10ndash1

100

BER



SNR = 24dB

SNR = 27dB

02 04 06 08 1















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