Optimal beamforming-selection spatial precoding using population-based stochastic optimization for massive wireless MIMO communication systems

Accepted Manuscript

Optimal Beamforming-Selection Spatial Precoding Using Population-Based Stochastic Optimization for Massive Wireless MIMO Communication Systems Ju-Hong Lee, Jing-Yen Lee PII: DOI: Reference:

S0016-0032(17)30127-8 10.1016/j.jfranklin.2017.03.002 FI 2934

To appear in:

Journal of the Franklin Institute

Received date: Revised date: Accepted date:

31 May 2016 20 January 2017 4 March 2017

Please cite this article as: Ju-Hong Lee, Jing-Yen Lee, Optimal Beamforming-Selection Spatial Precoding Using Population-Based Stochastic Optimization for Massive Wireless MIMO Communication Systems, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.03.002

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Ju-Hong Leea,∗, Jing-Yen Leeb a Department

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Optimal Beamforming-Selection Spatial Precoding Using Population-Based Stochastic Optimization for Massive Wireless MIMO Communication Systems✩

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of Electrical Engineering, Graduate Institute of Communication Engineering, and Graduate Institute of Biomedical Electronics and Bioinformatics, National Taiwan University No. 1, Sec. 4, Roosevelt Rd., Taipei, 10617, TAIWAN b Graduate Institute of Communication Engineering, National Taiwan University No. 1, Sec. 4, Roosevelt Rd., Taipei, 10617, TAIWAN

Abstract

The beamforming-based spatial precoding (BBSP) method has been proposed to reduce the overheads of the downlink training and the channel state information feedback in the frequency-division duplex (FDD) massive multiple-

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input-multiple-output (MIMO) wireless communication systems. However, the original BBSP method suffers from the interference problem at user equipments (UEs) because of using a set of pre-defined fixed beamforming coefficients. More-

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over, the BBSP method can not deal with the performance degradation due to mutual coupling (MC) effect because of massive antennas deployed at transmitter and receiver. This paper presents a precoding method that incorporates

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a beamforming-selection spatial precoding (BSSP) scheme with a populationbased stochastic optimization algorithm such that the designed beamforming

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coefficients can greatly reduce the severe interference between UEs and alleviate the MC effect on the performance of massive MIMO systems. The proposed method can not only achieve better bit error rate (BER) performance than the

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conventional BBSP method, but also preserves the advantages of the BBSP ✩ This work was supported by the Ministry of Science and Technology of Taiwan under Grant MOST 103-2221-E-002-123-MY3. ∗ Corresponding author Email addresses: [email protected] (Ju-Hong Lee), [email protected] (Jing-Yen Lee)

Preprint submitted to Journal of the Franklin Institute

March 15, 2017

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method having lower overheads of the downlink training and the CSI feedback. In particular, we propose an appropriate fitness function based on an averaged BER formula for the population-based stochastic optimization algorithm to find

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the optimal beamforming coefficients. Numerical simulations are also presented for both the urban-macro and the urban-micro wireless MIMO scenarios to validate the superior BER performance of the proposed precoding method as compared to the existing BBSP method.

Keywords: Massive MIMO, Wireless communication, Spatial precoding, Channel state information, Beamforming, Frequency-division duplex,

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Population-based stochastic optimization, Cooperative coevolutionary particle swarm optimization, Bit error rate.

1. Introduction

Recently, joint spatial precoding and beamforming has been proposed to im-

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prove the performance of multiple-input-multiple-output (MIMO) wireless communications [1]. Due to higher transmission rates and the increasing number 5

of users demanded by modern wireless MIMO communication systems, inter-

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ference has become one of the major obstacles limiting the performance and capacity of MIMO wireless communication systems. Array beamforming has

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been recognized as an effective way for interference mitigation. However, downlink beamformnig for alleviating the interference problem at the mobile users is 10

not an easy task for the frequency-division duplex (FDD) mode of transmission

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in MIMO systems. The main reason is due to limited knowledge of the downlink channel for the base station (BS). An appropriate precoding at the transmit-

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ter is essential for a good downlink transmission since they can improve the quality of spatial multiplexing and reduce the interference between user equip-

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ments (UEs) [2]. In the literature, several useful precoding methods, such as the beamforming-based spatial precoding (BBSP) method [3] and the block diagonalization (BD) scheme [4, 5, 6, 7] have been presented. The main drawback of the existing precoding methods is that the BS requires the feedback of full

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downlink channel coefficients or limited channel state information (CSI) from 20

the UEs to accomplish the precoding. This feedback mechanism of the CSI can be realized by the downlink training in the FDD MIMO systems [3] where the

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BS sends a known training signal to UEs to estimate the CSI of the downlink channel, and then UEs send the estimated CSI back to the BS for precoding.

The overheads required by the downlink training and the CSI feedback may 25

be negligible in conventional MIMO systems where a small number of antennas is used at the BS and UEs. However, when large-scale or massive MIMO systems [8] are considered, the BS with massive antennas is employed to simul-

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taneously serve a large number of UEs within the same frequency band and achieve higher spectral and energy efficiency. In this situation, the overheads 30

of downlink training and CSI feedback will increase significantly such that the bandwidth efficiency inevitably declines. In addition, for FDD systems, due to the large amount of antennas in massive MIMO systems, the duration of downlink training symbol is relatively long and may exceed the coherence time,

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which results in inaccurate CSI and thus degrades the downlink transmission quality [3]. Therefore, efficient CSI estimation is required since the full CSI of

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the downlink channel is prohibitively large in massive MIMO systems. In the literature, several precoding methods have been developed for massive MIMO systems to reduce the CSI overhead [3, 4, 5, 6, 7]. In [4, 5, 6], a technique

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that the BS can orthogonalize the precoders of each UE to other UEs’s channel matrices by using block diagonalization was proposed. However, this work as-

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sumes that the BS uses the entire precoding matrix index (PMI) of UEs. This leads to that the CSI feedback overhead would be still large and the spectral efficiency would be reduced when the number of antennas at the BS is massive.

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In [7], a CSI estimation scheme was proposed to reduce the downlink training

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overhead and CSI feedback overhead. This work applies a novel designed pilot signal and uses feedback of unitary matrices in FDD massive MIMO-OFDM system. Nevertheless, this scheme can only serve small number of UEs and hence, the reduction of the CSI feedback overhead is limited. The BBSP method of [3] employs beamforming to estimate CSI by using a 3

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small number of downlink training symbols for FDD massive MIMO systems. The basic idea is to partition all the antennas of the BS into several beam groups so that each antenna in one beam group can transmit the downlink

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training signal to UEs simultaneously. Since UEs only need to feed the selected beam indices back to the BS, the CSI feedback overhead is drastically reduced. 55

Although the BBSP method can effectively reduce the overheads of downlink training and the CSI feedback, it uses a set of pre-defined fixed beamforming

coefficients which cause severe interference problem between UEs. Consequently, the BER performance of the whole system becomes poor.

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Moreover, due to the increasing demand for channel capacity, energy ef-

ficiency, and spectral efficiency, wireless MIMO systems with massive antenna arrays deployed at transmitter and receiver have been realized as key methods of achieving the necessary requirements for developing the next generation wireless mobile communications [8, 9]. However, a phenomenon called mutual coupling effect (MCE) inevitably occurs in massive antenna array. The mutual coupling problem was studied several decades ago. Several research efforts have been

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devoted to tackle the MCE between coupled antennas from many disciplines,

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such as communications where multiple antennas are frequently encountered, like in MIMO systems equipped with massive antennas. At the BS, the mutual coupling effect may be negligible because antenna element spacing can be many wavelengths. However, acquiring low mutual coupling may be difficult due to

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limited volume for UEs. Because of the electromagnetic effects [10] between

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antennas with finite spacing, a part of data in one antenna would outflow to other antennas when the antennas in an antenna array operate simultaneously. Therefore, the performance of MIMO systems can be degraded when the antenna elements are in close proximity. It has been shown in [11, 12, 13] that the MCE

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generally degrades the performance of a wireless MIMO system significantly. Nevertheless, the existing precoding and beamforming techniques [3, 4, 5, 6, 7] proposed for massive MIMO systems have not considered how to deal with the MCE.

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In this paper, we consider the optimal spatial precoding to improve the 4

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BER performance of the BBSP method in the presence of MCE for massive MIMO systems. The main contributions of the paper are briefly described as follows. To alleviate the significant interference between UEs raised by the BBSP

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method and the MCE simultaneously, we present a method which incorporates the cooperative coevolutionary particle swarm optimization (CCPSO) [14] with a beamforming-selection spatial precoding (BSSP) scheme. Based on an approx-

imated average BER (AA-BER) of wireless MIMO systems derived for the fitness function of the CCPSO, the presented method called CCPSO-BSSP method finds the optimal precoding coefficients used in the BBSP method by minimizing an AA-BER of wireless MIMO systems. The proposed method can not only

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achieve better average BER performance than the original BBSP method, but also preserves the advantages of the original BBSP method having lower overheads of the downlink training and the CSI feedback in massive wireless MIMO systems. Simulation examples confirm the effectiveness of the proposed method 95

in the urban-micro and urban-macro wireless scenarios with MCE.

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This paper is organized as follows. In Section 2, we briefly describe the downlink transmission model in FDD systems with/without mutual coupling.

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Section 3 briefly reviews the principle of the existing BBSP method. The fundamentals of particle swarm optimization and cooperative coevolutionary particle 100

swarm optimization are summarized in Section 4. In Section 5, we propose a

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method based on a BSSP scheme and the CCPSO algorithm. A new fitness function based on an averaged BER for implementing the CCPSO algorithm

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is also presented. Section 6 presents the simulation results for illustration and comparison. Finally, we conclude this paper in Section 7. We use the following notation in the paper: Superscript .H denotes the

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transpose-conjugate operation. Superscript .T denotes the transpose operation.

S(:, j) stands for the jth column vector of matrix S. The Frobenius norm of a matrix S is denoted by ||S||.

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2. System Model and Fundamentals 2.1. Frequency-Division Duplex System Model

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Since there is a frequency separation large enough compared to the coherence bandwidth of the channel between uplink and downlink channels in a FDD communication system, uplink and downlink channels fade independently. Hence, 115

the downlink channel and uplink channel are not reciprocal. The uplink channel information may not be applied directly for downlink beamforming. As shown in [15], downlink adaptive arrays can be used to mitigate the limitation in chan-

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nel capacity due to co-channel interference. The goal is achieved by using uplink measurements to optimally estimate the downlink covariance matrices required 120

for obtaining the downlink beamforming coefficients. For FDD systems in massive wireless MIMO communications, the BS transmits the pilot symbols to each UE, then the UE estimates downlink CSI and feeds it back to the BS. Hence, the communication procedure is implemented as follows. The BS sends a known

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training signal to UEs. After each UE calculates its CSI and determines the precoder, it feeds the estimated CSI of downlink channel back to the BS. Then,

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the BS determines the precoders for downlink transmission with the estimated CSI which has been fed back from UEs, and finally the BS transmits the information data to UEs. Based on this communication procedure, we note that the

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downlink training overhead would be too long and hence, exceed the coherence time of the channel when massive antennas are deployed at the BS. Hence, the estimated CSI would be inaccurate that results in a performance degradation in

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downlink transmission. Moreover, to feedback the estimated CSI to the BS requires a considerable overhead since the BS uses massive antennas. As described

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above, the numbers of both the downlink pilots and the channel responses to

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be estimated at each UE are proportional to the number of antennas at the BS, the overhead of the downlink pilots and the feedback of downlink CSI becomes massive and unacceptable in massive MIMO systems [9]. As a result, dealing with the above two problems is essential in order to make the FDD massive MIMO systems realizable.

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2.2. Downlink Training Model Consider the downlink transmission of a massive MIMO channel. Let a BS be equipped with a uniform linear array (ULA) with NB antennas and K UEs

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with NU antennas of a ULA deployed at each UE. The corresponding NU × NB MIMO channel matrix H between the BS and each UE can be written as [16] 

h1,1 .. .

... .. .

  H=  hNU ,1

...



h1,NB .. . hNU ,NB

L  X p  al NU NB er (Ωr,l )et (Ωt,l )H ,  =  l=1

(1)

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where L is the number of dominant paths, al is the attenuation of the lth path,

er (Ωr,l ) denotes the unit spatial signature in the directional cosine Ωr,l at a UE, and et (Ωt,l ) denotes the unit spatial signature in the directional cosine Ωt,l at the BS. The downlink transmission of a massive MIMO channel is in line of sight (LOS) environments when L is equal to one. Moreover, er (Ωr,l ) and 

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er (Ωr,l ) are given by

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and

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er (Ωr,l ) =

p

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et (Ωt,l ) =

p

   NU    

e−j2π∆r Ωr,l .. .

e−j2π(NU −1)∆r Ωr,l



   NB    

1

1 e−j2π∆t Ωt,l .. . e−j2π(NB −1)∆t Ωt,l

       

(2)



   ,   

(3)

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respectively, where ∆r and ∆t denote the normalized inter-element spacings of the antennas at the UEs and the BS, respectively. Fig. 1 depicts the downlink system model of MIMO channels for the BS with NB antennas and each UE with NU antennas. In general, the BS employs FDD with uplink carrier frequency fu and downlink carrier frequency fd . These two carrier frequencies may be different. Moreover, λu and λd denote the corresponding uplink and downlink 7

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wavelengths, respectively. The inter-element spacing d of each ULA is set to half of the wavelength. The BS and the UEs use the same time-frequency resources and NB  NU . Based on the principle of the FDD model described above, the

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BS first sends a known training signal to UEs. Each UE performs the estimation of CSI and feeds the estimated CSI back to the BS. Then, the BS designs the

precoder and transmits the data symbol to UEs after the downlink training and the uplink feedback. Consider the downlink of a multiuser MIMO system with

K UEs. The received downlink training signal vector ykDT at the kth UE can be written by

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ykDT = Hk sk + nk ,

(4)

where sk denotes the NB × 1 training signal vector and nk the NU × 1 background noise vector received by the kth UE. Based on the received downlink training signal, each UE estimates its CSI and feeds the CSI back to the BS. Hence, the downlink training overhead and the CSI feedback overhead increase as NB increases.

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2.3. Downlink Transmission Model

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After the BS obtains the CSI estimated by downlink training and designs the downlink precoder, a data symbol vector xk for the kth UE is first preprocessed by a precoding matrix Wk with size NB × N and then transmitted into the

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MIMO channel. Moreover, we denote the downlink channel matrix corresponding to the kth UE as Hk with size NU × NB . Its (i, j)th entry is the channel

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gain from the jth transmitter antenna at the BS to the ith receiver antenna at the kth UE. Then, the NU × 1 received signal vector ykDL at the kth UE can be

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expressed as

ykDL = Hk Wk xk +

K X

Hk Wi xi + nk ,

(5)

i6=k

where xk denotes the N × 1 data symbol vector and nk the NU × 1 background noise vector received by the kth UE.

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2.4. Downlink Communication in the Presence of Mutual Coupling Consider the mutual coupling effect for a downlink communication, the chan-

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nel matrix Hk associated with the downlink training model (4) and the downlink

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transmission model (5) is replaced by

Cr Hk Ct ,

(6)

where Cr denotes the NU × NU mutual coupling matrix of the receiver antenna array and Ct the NB × NB mutual coupling matrix of the transmitter antenna

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array. According to [10, 17], the Cr and Ct are constructed as follows: C = (ZA + ZT ) + (Z + ZT I) where



... .. .

Z1,M .. .

...

ZM,M

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Z1,1   .. Z= .  ZM,1

−1

,

    

(7)

(8)

and M is the number of antennas. Consider the dipole length is half wavelength

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(l = λ2 ). The elements Za,b of Z are generated as follows:  η η    4π (ζ + ln(2kl) − Ci (2kl)) + j 4π Si (2kl) = ZA      ,a = b         n q  η (2C (kd )) − C (k( d2a,b + l2 + l)) i a,b i Za,b = , 4π  o q    −Ci (k( d2a,b + l2 − l))    n q   η   −j (2S (kd )) − S (k( d2a,b + l2 + l)) i a,b i  4π  o  q    −S (k( d2 + l2 − l)) , a 6= b i

a,b

where

∗ ZT = ZA , Z x sin(τ ) Si (x) = dτ, τ 0 Z ∞ cos(τ ) Ci (x) = − dτ, τ x

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(9)

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η = intrinsic impedance ' 120π(ohms), ζ = ln(1.781) ' 0.5772, k =

2π λ ,

and

da,b is the distance between the ath and the bth antennas (in λ), a and b = 1, 2, ...., M .

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3. Conventional Beamforming-Based Spatial Procoding

In this section, we briefly describe the principle of the conventional beamformingbased spatial procoding (BBSP) method [3]. 3.1. The Beam-Grouping Scheme

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The basic idea of the beam-grouping (BP) scheme is to partition the service region of the BS into B different sectors with different angle-of-arrivals (AOAs).

By using the concept of array beamforming, a beam group is formed to serve a service sector. Each beam group consists of J =

NB B

beams with NB being a

factor of B. For the bth beam group, a unit-norm steering vector is constructed

√

B NB

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pb =

bJ X



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as follows:

l=J(b−1)+1

      

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where T = {1, 2, ..., B} and ∆t =

1 e−j2π∆t Ωl .. .

e−j2π(NB −1)∆t Ωl

D λc



    , b ∈ T,   

(10)

is the normalized transmitter antenna

separation at the BS, where D and λc denote the antenna spacing and the carrier

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wavelength at the BS, respectively. The directional cosine Ωl of the lth beam is given by [3] Ωl = cos φl , l = 1, 2, ..., NB , where φl = cos−1 (1 − (2(l − 1)/NB )).

Accordingly, a BBSP matrix P is set to P = [p1 , p2 ..., pB ]. The downlink

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training signal matrix received at the kth UE is given by

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e k or YDL = H e k S´k + N e k, YkDL = Hk PS´k + N k

(11)

e k = Hk P denotes the NU × B where Hk is the NU × NB channel matrix, H

e k the received NU × B noise matrix and S´k is the virtual channel matrix and N B × B downlink training signal matrix with reduced size. 10

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3.2. The Beam-Selection and Feedback Scheme In order to provide an appropriate precoding and reduce the unacceptable feedback overhead in the downlink transmission of massive MIMO channels,

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each UE selects N beams from the BBSP matrix P = [p1 , p2 ..., pB ] as its precoder and feeds the selected beam indices back to the BS. The UE selects

e k,j be the jth cola beam based on the channel gain of each beam. Let h

e k . The channel gain is defined as umn vector of the virtual channel matrix H

e k,j k, j ∈ S and one sorts the channel gain of each beam according to γk,j = kh

γk,j1 ≥ γk,j2 ≥ ... ≥ γk,jB , where (j1 , j2 , ..., jB ) ∈ S B denotes the indices of

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beams. The kth UE then selects the first N beams as its serving beams and

feeds the selected beam indices (j1 , j2 , ..., jN ) back to the BS. This results in significant reduction in feedback overhead. Accordingly, the precoder for downlink transmission becomes

Wpk = [pj1 pj2 ...pjN ] .

(12)

= Hk Wpk xk +

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ykDL

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As a result, the received signal vector in the downlink transmission is given by K X

Hk Wpi xi + nk ,

(13)

i6=k

where xk denotes an N × 1 data symbol vector and K the number of UEs. 4. Cooperative Coevolutionary Particle Swarm Optimization - A Population-

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Based Stochastic Optimization

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In this section, we briefly review the principle of the CCPSO developed

by [14]. Population-based stochastic optimization algorithms like particle swarm optimization (PSO) algorithms have been shown to be effective in solving many real-world optimization problems [18]. However, their capabilities degrade con-

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siderably as the dimensionality of an optimization problem increases [19]. Based on the concept of divide-and-conquer strategy, an approach called cooperative coevolutionary algorithm (CCEA) was proposed in [20] to deal with highdimensional optimization problems. However, the effectiveness of the CCEA

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greatly depends on the separability of the high-dimensional optimization problems to be solved. In [21], the model of CCEA was incorporated with the conventional PSO to create a cooperative PSO model. This model was then

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employed for serving optimization problems with dimensions up to 30 [21] and 190 [22]. Nevertheless, this model still suffers from its inability in handling the 180

optimization problems of significantly higher dimensions. Recently, a cooperative coevolutionary PSO termed as CCPSO was developed by integrating a random grouping scheme, an adaptive scheme, and a new PSO with an up-

dated rule for particle positions in [14]. This CCPSO demonstrates its potential

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for effectively solving multimodal optimization problems with dimensions up to 2000 [14]. In the following, we briefly describe the principle of the CCPSO. The basic idea of a PSO algorithm introduced by [23] is to utilize a swarm with multiple particles. It has become an efficient algorithm in solving difficult multidimensional optimization problems. Using the information of social interaction between independent agents (called particles) and the concept of an objective function or fitness, a PSO algorithm searches the optimum solution.

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It estimates each particle at each moment t (t = 0, 1, 2, ...) by using the value of

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the objective function at the ith particle’s current position xi (t) ∈ D, where D denotes the search space. The velocity vi (t) of the ith particle at the moment t depends on the velocity at the moment (t − 1), the position xi (t), the best position pi (t) found up to now, and the best position g(t) found up to now by

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the swarm. The update formulas for vi (t) and xi (t) are given by [24]

vj,i (t + 1) =w(t)vj,i (t) + c1 U1 × (pj,i (t) − xj,i (t)) + c2 U2 × (gj (t) − xj,i (t))

(14)

xj,i (t + 1) =xj,i (t) + vj,i (t + 1),

where vj,i (t), xj,i (t), pj,i (t) denote the jth dimension of the velocity vi (t), position xi (t), and the personal best position pi (t), respectively; and gj (t) denotes the jth dimension of the global best position g(t) of the swarm. U1 and U2 are the random variables uniformly distributed in [0, 1]. The purpose of using U1 12

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and U2 is to maintain randomness during the search process. The parameters w(t), c1 , and c2 are called the inertial weight, cognitive attraction, and social attraction, respectively. In general, c1 and c2 are set to 2 for simulation. More-

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over, the inertial weight w(t) provides a dampening effect to the oscillation size of a particle over time and prevents each particle from overexploration in the

search space. From [25], it is appropriate to set the time-varying inertial weight w(t) to a value between [0, 1] according to the following equation w(t) = wstart −

wstart − wend × t, tmax

(15)

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where wstart = 0.9 and wend = 0.4 denote two preset values related to w(t). tmax

is the time limit or the maximum number of iterations during the optimization process. This linear decreasing scheme for adjusting the inertial weight w(t) 200

enables the PSO to have more global search ability at the beginning of the iteration process and more local search ability near the end of the optimization. The above time-varying inertial weight leads to that the optimization process

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finds a satisfactory solution within a reasonable convergence speed according to our experience [26]. Moreover, vmax is set to constrain the velocity in (14). As shown in [14], it is generally set to 10% ∼ 20% of the search space. If the

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velocity exceeds vmax , one makes it equal to vmax . The framework of PSO is summarized in Algorithm 1, where f (xm ) denotes the fitness related to the

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mth particle at the current position xm , pm denotes the personal best position of the mth particle and g the global best position.

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As to the basic concept of cooperative coevolution (CC), we regard an ecosystem of two or more sympatric species as a society with an ecological relationship of mutualism. A rule is set to reward the species based on how well they coop-

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erate with one or another in solving a problem. For solving a high dimensional and more structured optimization problem, one has to define a multi-population architecture to represent the problem to be solved and then partition the dimensions/variables space into certain groups and evolve the current group’s variables using the best solutions obtained from the other groups [21]. By appropriately incorporating the CC concept with PSO, a cooperative coevolutionary particle 13

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Algorithm 1: Pseudocode for the PSO algorithm Initialization : For d dimension PSO problem, initialize M particle positions :

repeat for each particle m ∈ [1, . . . , M ] do if f (xm ) ≤ f (pm ) then pm = xm

if f (pm ) ≤ f (g) then g = pm end

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end

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xm = [x1 , . . . , xd ] and velocity vm = [v1 , . . . , vd ] , m = 1, . . . , M

update velocity and position by using (14) end

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until tmax is reached ;

swarm optimization (CCPSO) has been proposed in [14]. Instead of using one swarm of particles for finding the optimal solution vector, the CCPSO parti-

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tions the searching space by splitting the solution vectors into smaller vectors and then CCPSO employs a separate PSO to search each of the smaller search spaces. Hence, each swarm of particles represents a subcomponent of a po-

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tential solution. Finally, the CCPSO combines the solutions found by each of PSOs to obtain the complete solution. To maximize the performance of us-

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ing CC concept, correlated variables are partitioned into the same group, and uncorrelated variables into different groups. The CCPSO presents a grouping scheme as follows. At each generation, a new group size s is chosen randomly

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in uniform distribution from a set S only if there is no performance improvement to the global best g. The variables are then divided into V = d/s groups. The effectiveness of the CCPSO for solving large-scale unconstrained continuous optimization problem has been demonstrated by competitive results in [14]. Instead of using the conventional update law for particle positions as shown in

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(14), the CCPSO proposes a new update law as follows:   p (t) + C(1)|p (t) − pb (t)|, ρ ≤ Z j,i j,i j,i xj,i (t + 1) = ,  pb (t) + ϕ|p (t) − pb (t)|, else j,i

(16)

j,i

where C(1) denotes a number generated by following a Cauchy distribution, ϕ

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j,i

a standard Gaussian random number of N (0, 1) with mean equal to zero and variance equal to one, and pj,i (t) the personal best position. Moreover, pbj,i is a

local neighborhood best for the ith particle in the jth dimension. It is chosen

among all three particles including the current ith particle and its immediate 215

left and right neighbors. Z is a parameter to be preset for implementation. ρ is

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a user-specified probability value for Cauchy sampling to occur. The framework of CCPSO is summarized in Algorithm 2.

5. The Proposed Optimal Beamforming-Selection Spatial Procoding In this section, to deal with the drawback of the original BBSP method [3] and the mutual coupling effect, we present an optimal beamforming-selection

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spatial procoding (BSSP) scheme for massive MIMO channels. The presented

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method incorporates the BSSP scheme with the CCPSO algorithm to determine the optimal beamforming coefficients instead of the predefined beamforming

225

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coefficients used by the original BBSP method [3]. 5.1. The Optimal BSSP using CCPSO: Proposed CCPSO-BSSP Here, we develop the optimal BSSP by utilizing the CCPSO to find the

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optimal spatial precoding matrix for downlink communication in massive MIMO channels. Instead of setting a preset fixed beamforming coefficients as shown in (10) [3] for the B groups of the NB antennas at the BS, we set a unit-norm

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steering vector of the bth beam group as follows:  vb = √

1 ab,1 + ab,2 + ... + ab,NB

15

      

ab,1 ejθb,1 ab,2 ejθb,2 .. .

ab,NB ejθb,NB



   ,   

(17)

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Algorithm 2: Pseudocode for the CCPSO algorithm Initialization : Create and initialize V swarms, each with s dimensions, (where s is

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randomly chosen from a set S, and d = V × s always being true), the jth swarm is denoted as Pj , j ∈ [1, . . . , V ], The function b(j, z) returns a vector (P1 .g, P2 .g, . . . , Pj−1 .g, z, Pj+1 .g, . . . , PV .g) repeat if f (g) has not improved then randomly choose s from S and let V = d/s

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Randomly permutate all d dimension indices, Construct V swarms, each with s dimensions. end

for each swarm j ∈ [1, . . . , V ] do

for each particle m ∈ [1, . . . , M ] do

end

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if f (b(j, Pj .xm )) ≤ f (b(j, Pj .pm )) then Pj .pm = Pj .xm

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if f (b(j, Pj .pm )) ≤ f (b(j, Pj .g)) then Pj .g = Pj .pm end end

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for each particle m ∈ [1, . . . , M ] do Pj .b pm = localbest(Pj .pm−1 , Pj .pm , Pj .pm+1 )

end

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if f (b(j, Pj .g)) ≤ f (g) then jth part of g is replaced by Pj .g

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end

end

for each swarm j ∈ [1, . . . , V ] do for each particle m ∈ [1, . . . , M ] do Perform position update for the mth particle in swarm Pj using (16) end end until tmax is reached ;

16

where Pj .pm denotes the personal best position of the mth particle in the jth swarm and Pj .g the global best position of in the jth swarm.

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where b = 1, 2, · · · , B, ab,k and θb,k , k = 1, 2, · · · , NB , are variables to be adjusted by the CCPSO. Therefore, when considering the feedback, the feedback overhead in bits required by the proposed method is almost the same as that of

230

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the original BBSP method of [3]. It can be written as Nf = K × N × log2 B. During the optimization process, each particle is treated as a point in the D(= 2BNB )-dimensional optimization problem space and a swarm consists of

D random particles. Then, the CCPSO searches for the best position (solution or optimum) by updating generations until obtaining a relatively steady position or reaching the limit of a preset iteration number. At the tth mo-

ment or iteration, the position vector of the ith particle is given by xi (t) =

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235

[a1,1 (t), a2,1 (t), · · · , aB,1 (t), a1,2 (t), · · · , aB,NB (t)]. The personal best position vector pi (t) and the global best position vector g(t) are obtained by evaluating the performances in terms of the fitness values associated with the current population of particles. Moreover, we update the position and velocity charac240

terizing a particle status on the search space according to the formulas given by

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(16) and convert a particle position vector into a candidate solution vector by employing an appropriate mapping. The performance of the mapped solution

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vector evaluated by a massive MIMO channel is viewed as the fitness of the corresponding particle. At the end of iteration, the global best position vector 245

g(tmax ) is regarded as the optimal beamforming coefficient vector. In general,

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the optimization process will be terminated if the specified maximum iteration number tmax is reached or the best particle position of the whole swarm remains

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almost static for a significantly large number of successive iterations. 5.2. Fitness Function for CCPSO

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Here, we develop an appropriate fitness function based on the bit error rate

(BER) performance for implementing the CCPSO to search the optimal spatial precoding matrix for downlink communication in massive MIMO channels. For practical applications, massive MIMO channels with linear receivers, such as the minimum mean square error (MMSE) receiver have been considered in several emerging standards, e.g., IEEE 802.11n and 802.16e. We consider the MMSE 17

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receiver r at each UE. Using the downlink transmission given by (13) and letting N = 1, we assume that the kth UE selects the jth beam group as its precoder and the other B −1 beam groups are chosen by the other B −1 UEs. The MMSE have E[(YkDL )H ] = 0, where e k (:, j)xk +  = r(H

K X i6=k

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equalizer r can be obtained as follows. Using the orthogonality principle, we

e k (:, i)xi + nk ) − xk H

(18)

(19)

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e k (:, j) represents the jth column vector of H e k . Moreover, assume that and H

xk , xi , and nk are independent of each other and their expectations are 0. From (18) and (19), we can get r

B X i=1

e k (:, i)H e k (:, i)H + r(1/Γ) − H e k (:, i)H = 0, H

(20)

receiver r is given by

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where Γ denotes the signal to noise power ratio (SNR). As a result, the MMSE

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B X e k (:, j)H ( e k (:, i)H e k (:, i)H + (1/Γ)IN )−1 . r=H H U

(21)

i=1

Based on the downlink transmission with the MMSE receiver, the received signal

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at the kth UE becomes

= r(Hk vj xk +

B X

Hk vi xi + nk ),

(22)

i6=j

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ykDL

where xk is the modulated signal of the kth UE, xi is the modulated signal

of other UEs (interference signals). After the demodulation of ykDL , we then

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compare the demodulated bit and the source bit to decide the BER as follows. Consider the binary phase-shift keying (BPSK) for modulation and the maximum likelihood (ML) criterion for data decision after demodulation. Assume that the bit value of each UE is 0 or 1 with equal probability and the noise has complex Gaussian distribution. The downlink transmission with the MMSE receiver is given by (22). Let rH Hq vj(q) = sq and rH Hq vc = iv,q , where 18

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c = 1, · · · , B and c 6= j(q), and v = 1, · · · , (B − 1), the subscript q denotes the qth Monte Carlo random channel realization for the MIMO channel matrix by using the 3GPP for simulation. When the number of UEs is B and all of their

where

Z

1 Q(a) = √ 2π

∞

e−

u2 2

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bit values are 1, the estimated BER can be written as [27] ! !s B−1 X 2Γ Q < sq + , iv,q 2 krk v=1 du

(23)

(24)

a

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and <(x) represents the real part of x. There are 2B conditions to be concerned.

First, when the bit value of the kth UE is 1, its BER can be approximately estimated by Pe,1 (q) =

1 2B−1

1 1 X X

a1 =0 a2 =0

Q(<(sq +

B−1 X

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b=1

250

···

1 X

aB−1 =0

s

(−1)av × iv,q )

(25)

2Γ

2 ).

krk

The summation in (25) means the summation of all cases because the signals of

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other UEs are viewed as interference for the kth UE and the values of their sigPB−1 nals can be +1 or −1 due to BPSK. v=1 (−1)av ×iv,q denotes the permutation of these interference signals and 1/2B−1 denotes the prior probability.

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On the other hand, when the bit value of the kth UE is 0, the estimated BER is Pe,0 (q), and Pe,0 (q) = Pe,1 (q). Consequently, the total estimated BER

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for the kth UE is written as 1 1 1 X 1 1 1 X X Pe (q) = Pe,1 (q) + Pe,0 (q) = B−1 ··· 2 2 2 a1 =0 a2 =0 aB−1 =0 s B−1 X 2Γ Q(<(sq + (−1)av × iv,q ) 2 ). krk v=1

(26)

Finally, we compute the average of the approximated BERs (AA-BER) as follows:

Q

Fitness =

1 X Pe (q), Q q=1 19

(27)

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where Q denotes the number of the Monte Carlo random channel realizations 255

for the MIMO channel matrices by using the 3GPP for simulation. The Fitness

5.3. Computational Complexity of the Proposed Method

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of (27) is set to be the fitness function to be minimized by the CCPSO.

From the procedure described above for implementing the proposed CCPSOBSSP method, we note that the proposed CCPSO-BSSP method requires the

additional computational complexity due to using the CCPSO for finding the optimal precoding coefficients without costing additional feedback overhead as

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compared with the original BBSP method. According to the CCPSO [14] em-

ployed by the proposed method, its computational complexity is dominated by the number tmax of iterations, the number s of the group sizes, the number M of particles, the computational complexity of the fitness function, and the computational complexity of the particles’ position updating rule. Therefore, the computational complexity of the CCPSO can be expressed as follows:

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U OCCP SO = O M × s × tmax × OCCP SO ,

(28)

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U where OCCP SO denotes the total computational complexity of the particles’

position updating rule shown by (16) and the fitness function given by (27). 260

As a result, the proposed method requires the additional computational com-

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plexity given by (28) for implementation as compared with the original BBSP

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method [3].

6. Simulation Results This section presents several simulation examples for illustration and com-

parison. In the first example, a simple multiple-input-single-output (MISO)

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channel model is employed to demonstrate the severe interference problem at the UEs due to using the existing BBSP method. In contrast, the proposed CCPSO-BSSP method can effectively mitigate the interference problem. The other 4 examples are presented to compare the average BER versus SNR in the

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270

urban macro and micro scenarios with/without mutual coupling effect under different number of antennas used.

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6.1. Example 1: The beam pattern of a LOS wireless channel Here, we consider the line of sight (LOS) channel and NU is set to 1, i.e, the multiple-input-single-output (MISO) channel model. We plot the beam

patterns of using the BBSP method according to (10) and the proposed CCPSOBSSP method based on an estimated average signal-to-noise plus interference

ratio (SINR) to evaluate the interference between the B beams. The estimated

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average SINR is derived as follows. Let the angle between the BS and the kth

UE be θ. From the BS to the kth UE, the channel vector hk (θ) is then given by



M

   j2πd   ) hk (θ) = aexp( λc    

1

e−j2π∆t cos θ

e−j2π2∆t cos θ .. .

e−j2π(NB −1)∆t cos θ



     .    

(29)

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For this example, we set d = λc , a = 1, and the normalized transmitter antenna separation ∆t at the BS to 0.5. For the BBSP method with N = 1, 275

the kth UE would select the beam vj(θ) from the B beams as its precoder, i.e.,

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Wpk = vj(θ) . Assume that other B − 1 beams are selected by the other UEs.

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Hence, the downlink transmission of the kth UE becomes

ykDL = hH k (θ)vj(θ) xk +

B X

hH k (θ)vi xi + nk .

(30)

i6=j(θ)

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Accordingly, the SINR at the kth UE is calculated by 2

SIN Rk =

ΓkhH k (θ)vj(θ) k . PB 2 1 + Γ i6=j(θ) khH k (θ)vi k

(31)

Further, we assume that θ ∈ {0o , 1o , ..., 180o } and the probabilities of the kth UE appearing in this 181 angles are equal. Then, we compute the estimated

21

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average SINR at the kth UE as follows: 2

180

E [SIN Rk ] =

ΓkhH 1 X k (θ)vj(θ) k . PB 2 181 1 + Γ i6=j(θ) khH k (θ)vi k

(32)

θ=0

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This estimated average SINR is employed as the fitness function for PSO-based optimization and the interference between the B beams is evaluated based on

this value. The parameters used are as follows: NB = 4, NU = 1, B = 2, and Γ = 10. Based on (17), there are totally 16 variables, namely ab,k and

θb,k , b = 1, 2, k = 1, · · · , 4. We employ the conventional PSO of Algorithm

1 as described in Section 4 since the considered channel with NB = 4 and

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NU = 1 is not a massive case. The particle speed limit is set to 0.1π for θb,k

and 0.05 for ab,k , tmax = 1000. The number of particles is 500 , wstart = 0.4, and wend = 0.9. After performing the PSO, we obtain the following optimal beamforming coefficients for precoding:

θ1,1 = 4.299, θ1,2 = 1.155, θ1,3 = 4.293, θ1,4 = 1.148,

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θ2,1 = 0.018, θ2,2 = 0.012, θ2,3 = 0.005, θ2,4 = 6.283,

(33)

a1,1 = 0.755, a1,2 = 1.461, a1,3 = 1.461, a1,4 = 0.755,

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a2,1 = 0.535, a2,2 = 1.291, a2,3 = 1.291, a2,4 = 0.535. Fig. 2 shows the beam patterns associated with the two beam groups of 2

using (10) given by [3]. The power gain is given by khH k (θ)gb k , b = 1, 2. The estimated average SINR given by (32) of these two beam patterns is 8.176.

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280

This figure shows that the beam groups interfere each other seriously in a wide

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range of directional angles. In contrast, Fig. 3 plots the beam patterns of using the proposed BBSP scheme with the optimal beamforming coefficients given by (33). The estimated average SINR given by (32) of these two beam patterns is 22.407. Clearly, we observe that the proposed optimal BSSP scheme reduces

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285

the interference between the beam groups significantly. 6.2. Example 2: The BER performance of 3GPP wireless channel with NU = 2 In this example, we utilize the 3rd Generation Partnership Project (3GPP) spatial channel model presented by [28] for simulation. The 3GPP has intro22

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290

duced the Spatial Channel Model (SCM). Unlike traditional channel models, SCM incorporates not only a random power delay profile (PDP) but also a random angular profile (AP). It represents scatterers through statistical param-

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eters without having a real physical location. The SCM belongs to the class of geometric stochastic model and separately defines large-scale parameters (e.g., 295

shadow fading, delay spread, and angular spreads) and small-scale parameters

(e.g., delays, cluster powers, and arrival and departure angles). Both parameter sets are randomly drawn from tabulated distributions. The MATLAB implementation of the 3GPP SCM can be downloaded from [29]. We employ this

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package to generate 40 links with each link sampled 10 times for obtaining a set of 400 channel matrices under urban macrocell and urban microcell environments for illustration and comparison. The parameters used in these two scenarios are presented in APPENDIX as shown in [28]. Moreover, we com-

multipath component. The other channel specifications are given as follows.

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bine all of the L multipath components produced by sampling each link to get PL the qth channel matrix Hq = l=1 Hl for simulation, where Hl denote the lth

B = K = 4, Γ = 8, (N = 1), the number of bits is 107 . All B precoders

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are selected by K different UEs, the modulation used for signal transmission is BPSK, and the noise is complex Gaussian distribution with zero mean. As to the parameters required for implementing the CCPSO to find the best coefficients for ab,k and θb,k , b = 1, ..., B, k = 1, · · · , NB as shown in (17) are given as

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310

follows. The dimension s ∈ S = {2, 4, 8, 64, 128, 256}, swarm number V =

512 s ,

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tmax = 100, the particle number of each swarm is 450, and the fitness function is given by (27) with Q = 400. The value for the parameter Z is set to 0.5. Fig. 4 plots the average BER versus SNR of using the existing BBSP

method [3] and the proposed CCPSO-BSSP method in the urban macro sce-

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315

nario [29]. The proposed CCPSO-BSSP method can achieve better BER performance than the existing BBSP method for both cases of NB = 8 and NB = 64. For the BBSP method, increasing NB from 8 to 64 cannot provide significant BER performance improvement. This is because using the fixed beam groups in

320

the BBSP method can cause severe interference between UEs. In contrast, the 23

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proposed CCPSO-BSSP method can significantly reduce the interference between UEs by using the CCPSO to find the optimal beamforming coefficients. As a result, the proposed CCPSO-BSSP method can greatly improve the BER

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performance particularly when the number of antennas at the BS is large, e.g., the situation of NB = 64 in massive MIMO systems. Fig. 5 depicts the aver-

age BER versus SNR of using the existing BBSP method [3] and the proposed CCPSO-BSSP method in the situation of the urban micro scenario [29]. In the urban micro scenario, with the increase of NB from 8 to 64, the BER per-

formance of the existing BBSP method has much less BER improvement than that in the situation of macro scenario. This is because the interference between

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UEs for the BBSP method may become more significant when UEs are located in a smaller area. However, the proposed CCPSO-BSSP method still provides considerable BER improvement for the cases of NB = 8 and NB = 64 in the micro scenario. Especially, the improvement in average BER is more significant 335

when NB = 64.

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Moreover, comparing Figs. 4 and 5, we observe that both of the existing BBSP method and the proposed CCPSO-BSSP method have better performance

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in the urban macro scenario than in the urban micro scenario. The main reason is that the interference between UEs becomes more significant when UEs are located in a smaller area under the urban micro scenario.

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6.3. Example 3: The BER performance of 3GPP wireless channel with NU = 2 and mutual coupling

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Here, we consider the mutual coupling effect on BER performance of 3GPP

wireless channel. The specifications and parameters used are the same as those used by Example 2 except that all channel matrices in Example 2 are replaced

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345

by (6). Fig. 6 depicts the average BER performance versus SNR for the urban macro scenario with mutual coupling and Z = 0.5. The BER performance of

the existing BBSP method is degraded when the mutual coupling is considered. However, the proposed CCPSO-BSSP method with 8 antennas and 64 anten-

350

nas deployed at the BS still outperforms the existing BBSP method under the 24

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influence of antenna mutual coupling. Especially, the improvement in average BER is more significant when NB = 64. Fig. 7 shows the average BER performance versus SNR for the urban micro scenario with mutual coupling and

355

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Z = 0.7. In the urban micro scenario with mutual coupling, the average BER performance of the proposed CCPSO-BSSP method has much better BER per-

formance than the existing BBSP method. From the simulation results, we note that the proposed CCPSO-BSSP method is effective in dealing with the mutual

coupling problem. Again, the improvement in average BER is more significant by utilizing the proposed CCPSO-BSSP method when NB = 64.

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Moreover, comparing Figs. 6 and 7, we observe that both of the existing

360

BBSP method and the proposed CCPSO-BSSP method have better performance in the urban macro scenario than in the urban micro scenario. Again, the main reason is that the interference between UEs becomes more significant when UEs are located in a smaller area under the urban micro scenario with mutual coupling.

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6.4. Example 4: The BER performance of 3GPP wireless channel with NU = 4

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The specifications and parameters used here are the same as those used by Example 2 except that NU = 4. Fig. 8 plots the average BER versus SNR of using the existing BBSP method and the proposed CCPSO-BSSP method for the urban macro scenario with Z = 0.5 when NU = 4. From Fig. 4 and

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Fig. 8, we observe that both the BBSP method and the proposed CCPSOBSSP method upgrade the BER performance as the number of the antennas at

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each UE increases from 2 to 4. However, the proposed CCPSO-BSSP method still provides much better average BER performance than the existing BBSP method in this case. Fig. 9 depicts the average BER versus SNR of using the

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existing BBSP method and the proposed CCPSO-BSSP method in the urban micro scenario with NU = 4 and Z = 0.5. Comparing Fig. 5 and Fig. 9, we find

that the improvement in average BER performance by using the existing BBSP method is not obvious as the number of antennas at the BS increases from 8 to

380

64 when NU = 4. However, the improvement in average BER performance by 25

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using the proposed CCPSO-BSSP method is very significant as the number of antennas at the BS increases from 8 to 64 when NU = 4. Moreover, we observe from Examples 2 and 4 that both of the existing BBSP

385

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method and the proposed CCPSO-BSSP method have significant improvement in performance as NU increases from 2 to 4 for both of the urban macro and urban micro scenarios. This is due to that using more antennas at UEs enhances the capabilities of UEs’ signal reception.

6.5. Example 5: The BER performance of 3GPP wireless channel with NU = 4

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and mutual coupling

Again, we consider the mutual coupling effect on BER performance of 3GPP

390

wireless channel. The specifications and parameters used are the same as those used by Example 4 except that all channel matrices in Example 4 are replace by (6). Fig. 10 depicts the average BER performance versus SNR for the urban macro scenario with mutual coupling and Z = 0.5. We note from the figure that the BER performance of the existing BBSP method deteriorates

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when the mutual coupling is considered. However the proposed CCPSO-BSSP

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method with 8 antennas and 64 antennas deployed at the BS still outperforms the existing BBSP method in this case. Especially, the improvement in average BER is more significant when NB = 64. Fig. 11 shows the average BER performance versus SNR for the urban micro scenario with mutual coupling and

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400

Z = 0.5. In the urban micro scenario with NB = 64 and NU = 4, the average BER performance of the proposed CCPSO-BSSP method has much better BER

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performance than the existing BBSP method under the situations with mutual coupling. From the simulation results, we note that the proposed CCPSO-BSSP method is very effective in dealing with the mutual coupling problem in massive

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405

wireless MIMO communications. Moreover, we observe from Examples 3 and 5 that both of the existing BBSP

method and the proposed CCPSO-BSSP method have significant improvement in performance as NU increases from 2 to 4 for both of the urban macro and

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urban micro scenarios with mutual coupling. Again, the main reason is that 26

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using more antennas at UEs enhances the capabilities of UEs’ signal reception.

7. Conclusion

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In this paper, we have presented an optimal spatial precoding by incor-

porating the beamforming-selection scheme and a population-based stochastic 415

optimization for massive wireless MIMO communication systems. Based on the

concept of beamforming selection, the proposed precoding method can preserve

the advantage of reducing the overheads for downlink training and feedback of

channel state information between the base station (BS) and user equipments

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(UEs). Instead of using fixed beamforming coefficients for procoding as proposed by the existing method, the proposed precoding method utilizes the cooperative coevolutionary particle swarm optimization (CCPSO) to find the optimal beamforming coefficients. As a result, the proposed method can significantly mitigate the interference problem between UEs, particularly when massive an-

425

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tennas are deployed at the BS. Moreover, an appropriate fitness function based on an estimated average bit error rate (BER) is proposed for implementing the CCPSO. Simulation results show that the proposed method can indeed achieve

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much better BER performance than the existing method in both urban-macro and urban-micro scenarios with massive antennas and antenna mutual coupling.

430

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Hopefully, the proposed method offers a more appropriate spatial precoding scheme for future massive wireless MIMO communications.

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8. Appendix

Table 1 is the same as that of Table 5.1 presented by [28] which provides

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the parameters used in urban macro and urban micro scenarios, where the Aod denotes the angle of departure, AoA denotes the angle of arrival, the RMS

435

denotes the root mean square, the NLOS denotes not line of sight, σAS denotes

the angle spread or azimuth spread (AS), σDS denotes the delay spread, σAoD is the standard deviation of the AoD distribution, σAoA is the standard deviation

of the AoA distribution, Pr is the relative power of the nth path, σdelay is 27

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a weighting for random delay in urban macro scenario, N (a, b) represents a 440

Gaussian distribution with mean a and variance b, U(a,b) represents a uniform

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distribution with the minimum value a and the maximum value b.

References

[1] R. Doostnejad, T. J. Lim, E. Sousa, Joint precoding and beamforming

design for the downlink in a multiuser MIMO system, in: IEEE International Conference on Wireless and Mobile Computing, Networking and

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Communications, Montreal, Canada, Vol. 1, Aug. 2005, pp. 153–159.

[2] D. Gesbert, M. Kountouris, R. Heath, C.-B. Chae, T. Salzer, Shifting the MIMO paradigm, IEEE Signal Processing Magazine 24 (5) (2007) 26–37. [3] M.-F. Tang, M.-Y.Lee, B. Su, C.-P. Yen, Beamforming-based spatial precoding in FDD massive MIMO systems, in: IEEE 48th Asilomar Confer-

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2014, pp. 2073–2077.

M

ence on Systems and Computers, Pacific Grove, CA, USA, Vol. 1, Nov.

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[4] Q. Spencer, A. Swindlehurst, M. Haardt, Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels, IEEE Transactions on Signal Processing 52 (2) (2004) 461–471.

455

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[5] C. Peel, Q. Spencer, A. Swindlehurst, B. Hochwald, Downlink transmit beamforming in multiuser MIMO systems, in: IEEE Sensor Array and

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Multichannel Signal Processing Workshop, Barcelona, Spain, July 2004, pp. 43–51.

[6] D. Love, R. Heath, Limited feedback unitary precoding for spatial multi-

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plexing systems, IEEE Transactions on Information Theory 51 (8) (2005) 2967–2976.

[7] R. Kudo, S. Armour, J. McGeehan, M. Mizoguchi, A channel state information feedback method for massive MIMO OFDM, Journal of Commu-

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nications and Networks 15 (4) (2013) 352–361. 28

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Table 1: Environment Parameters for Simulations

Channel Scenario

Urban Macro

Urban Micro

Number of paths

6

6

Number of sub-paths per-

20

20

E(σAS ) = 8◦ , 15◦

NLOS:E(σAS )

Mean AS at BS

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path

=

19◦

rAS = σAoD /σAS Per-path

AS

at

BS

(Fixed)

1.3

N/A

2◦

5◦

BS per-path AoD Distribution standard distribu-

σAoD = rAS σAS

where

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tion

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Per-path AS at UE (fixed)

UE Per-path AoA Distribution

spread

σDS

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10

(εDS x+µDS )

=

, x ∼ N (0, 1)

U(−40◦ ,40◦ )

35◦

35◦

2 N (0, σAoA (Pr ))

2 N (0, σAoA (Pr ))

µDS

=

−6.18,

N/A

εDS = 0.18 0.65 µs

0.251 µs(output)

rDS = σdelays /σDS

1.7

N/A

Distribution for path de-

N/A

U(0,1.2µs)

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Mean total RMS Delay Spread

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and

NLOS)

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Delay

(LOS

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Figure 1: Downlink block diagram of MIMO channels for the BS with NB antennas and each UE with NU antennas

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