Energy-efficiency based downlink multi-user hybrid beamforming for millimeter wave massive MIMO system

The Journal of China Universities of Posts and Telecommunications August 2016, 23(4): 53–62 www.sciencedirect.com/science/journal/10058885

http://jcupt.bupt.edu.cn

Energy-efficiency based downlink multi-user hybrid beamforming for millimeter wave massive MIMO system Jiang Jing (

), Cheng Xiaoxue, Xie Yongbin

School of Communication and Information Engineering, Xi’an University of Posts and Telecommunications, Xi’an 710061, China

Abstract The fifth generation mobile communication (5G) systems can provide Gbit/s data rates from massive multiple-input multiple-output (MIMO) combined with the emerging use of millimeter wavelengths in small heterogeneous cells. This paper develops an energy-efficiency based multi-user hybrid beamforming for downlink millimeter wave (mmWave) massive MIMO systems. To make better use of directivity gains of the analog beamforming and flexible baseband processing of the digital beamforming, this paper proposes the analog beamforming to select the optimal beam which can maximize the power of the objective user and minimize the interference to all other users. In addition, the digital beamforming maximizes the energy efficiency of the objective user with zero-gradient-based approach. Simulation results show the proposed algorithm provide better bit error rate (BER) performance compared with the traditional hybrid beamforming and obviously improved the sum rate with the increase in the number of users. It is proved that multi-user MIMO (MU-MIMO) can be a perfect candidate for mmWave massive MIMO communication system. Furthermore, the analog beamforming can mitigate the inter-user interference more effectively with the selection of the optimal beam and the digital beamforming can greatly improve the system performance through flexible baseband processing. Keywords 5G systems, mmWave, massive MIMO, MU-MIMO, analog beamforming, digital beamforming, hybrid beamforming

1 Introduction MmWave spectrum, typical values 30 GHz~300 GHz, would leverage the large bandwidths of the underutilized frequency spectrum which is desirable for Gbit/s data rates, so is a promising candidate for future cellular systems in Refs. [1–3]. Owing to shorter wavelengths of mmWave frequencies, the large-scale antenna arrays can be packed into small form factors [4]. Higher MIMO array gains could combat the increased path losses and are essential to provide a suitable link budget. In conventional lower frequency systems, the precoding is usually performed in the digital baseband which requires dedicated baseband and radio frequency (RF) hardware for each antenna element. For mmWave large-scale MIMO, the full digital MIMO is difficult to be implemented due to Received date: 21-03-2016 Corresponding author: Jiang jing, E-mail: [email protected] DOI: 10.1016/S1005-8885(16)60045-6

the high cost and power consumptions of RF chains [4–5]. The trade-off between performance and simplicity drives the need for the hybrid beamforming architectures, especially when a multitude of antennas are required as in the mmWave bands. In Refs. [6–10], Ayach et al. put forward single-user hybrid beamforming algorithms. In Refs. [6–7], El Ayach et al. suggest a hybrid architecture for single user massive MIMO systems, and utilize the matching pursuit algorithm to formulate the design of hybrid RF/baseband (BB) precoders as a sparsity constrained matrix reconstruction problem. In Ref. [9], Kim et al. exploit the effective channels formed by RF beamforming at both transmitter and receiver, then formulates the BB precoder design problem that meets a certain criterion, e.g. maximizing effective signal noise ratio (SNR). For multi-user hybrid beamforming algorithms, in Ref. [11], Choi studies multi-user beam selection exploiting a certain sparsity of mmWave channels. This

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The Journal of China Universities of Posts and Telecommunications

algorithm shows that beam selection can be carried out without explicit channel estimation using the notion of compressive sensing (CS). In Ref. [12], Adhikary et al. analyzes the performance of joint spatial division and multiplexing (JSDM) algorithm that takes into account the partial overlap of the angular spectra from different users, and formulates the problem of user grouping for two different objectives, namely maximizing spatial multiplexing and maximizing total received power in a graph-theoretic framework. In Ref. [13], Stirling et al. present different strategies for supporting MU-MIMO transmission with code book based RF beamforming and non-code book based digital precoding for hybrid beam-forming communication systems. It has been shown that instead of just performing MU-MIMO digital precoding with single-user MIMO (SU-MIMO) RF beam selection, selecting the RF beams to optimize MU-MIMO transmission yields a significant performance benefit. In Ref. [14], Bogale et al. design the hybrid beamforming indirectly by considering a weighed sum mean square error (WSMSE) minimization problem incorporating the solution of the detected signals which is obtained from the block diagonalization technique. The resulting WSMSE problem is solved by applying the orthogonal matching pursuit (OMP) algorithm. In Ref. [15], Alkhateeb et al. indicate that interference management in multi-user mmWave systems is required even when the number of antennas is large. It designs RF beamforming vectors only considering single-user performance with partial channel and solves the baseband bemforming vectors using zero forcing (ZF) algorithm to suppress the multi-user interference. Many multi-user hybrid beamforming algorithms [14– 15], suppress the multi-user interference only through digital beamforming. Other multi-user hybrid beamforming algorithms [11–13], focus on multi-user beam selection without considering the further optimization for the digital beamforming. In this paper, an efficient hybrid analog/digital beamforming algorithm for downlink multi-user mmWave systems is developed. As everyone knows, mmWave links are inherently directional. By controlling the phase of RF chains through the analog beamforming, the antenna array steers its beam towards any direction electronically and achieves a high gain at this direction, while offering a very low gain in all other directions [11–13]. Once the base station (BS) transmits signals to multiple users with same or adjacent beams, the

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multi-user interference is hard to mitigate by digital beamforming. On the other hand, digital beamforming provides a higher degree of freedom and offers better performance at the expense of increased complexity. To make better use of the directivity gains of the analog beamforming and the flexible calculation of the digital beamforming, the paper proposes that the analog beamforming selects the optimal beam which can maximize the power of the objective user and minimize the interference to all other users. Furthermore, digital beamforming maximizes the energy efficiency of the objective user with flexible baseband processing. This paper is organized as follows. Sect. 2 discusses the system and channel models. In Sect. 3, the proposed hybrid beamforming design is presented. In Sect. 4, computer simulations are used to evaluate the performance of the proposed hybrid beamforming in different parameter conditions. Conclusions are drawn in Sect. 5. Notations: In this paper, upper/lower-case boldface letters denote matrices/column vectors. x (i, j ) , x F ,

tr ( x ) ,

xT ,

xH

and

E( x)

denote the

( i, j ) th

element, frobenius norm, trace, transpose, conjugate transpose and expected value of x , respectively. I n is the identity matrix of size n × n , ℂ M × M represent spaces of M×M matrices with complex entries. The acronym null, s.t. and i.i.d. denote ‘(right) null space’, ‘subject to’ and ‘independent and identically distributed’, respectively.

2 System model 2.1

System model

Consider a downlink mmWave massive multiuser MIMO system as shown in Fig. 1. It is assumed that perfect channel state information (CSI) is achieved by simple time division duplex (TDD) training method as in Ref. [16]. Each BS serves exactly K users and each user configures N r receiving antenna. The BS is equipped with

Nt

antennas driven by

N RF

RF chains, and

communicating Ns data streams to K users, such that

N s ≤N RF ≤N t . As shown in Fig. 1, Ns data streams map to N RF RF chains through a digital beamforming vector FBB , followed by inverse fast Fourier transform (IFFT), parallel-serial transform, digital to analog converters (DAC) and a N t × N RF analog beamforming vector f RF using phase shifters. Since f RF is implemented using

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Jiang Jing, et al. / Energy-efficiency based downlink multi-user hybrid beamforming for…

analog phase shifters, its entries are constrained to satisfy

(

H

[ f RF ]:,i [ f RF ]:,i

)

ℓ,ℓ

= N t−1 , where

Fig. 1

Simplified hardware block diagram of mmWave MU-MIMO system

time domain will be fast Fourier transform (FFT) transformed to FRF of frequency domain. Hence, received signals Yk ∈ ℂ

of the kth user can be

expressed as k p H k FRF FBB sk +

∑

the widely used geometric channel model with Lk scatters between the BS and the kth user [17–18]. Hence, the channel matrix hk can be expressed as

hk ( t ) =

Nt Nr ρk Lk

Lk

∑ a (ϕ k R

r i

(

,θ ir )βik ( t ) aTk (ϕit ,θ it )

i =1

)

Τ

(2)

where ρ k is path-loss and βik ( t ) is the complex gain of

K

Yk =

diagonal element, i.e., all elements are of equal norm.

(⋅)ℓ,ℓ denotes the ℓth

For simplicity, the system model will be described in frequency domain. Analog beamforming vector f RF in

Nr × N t

55

i p H k FRF FBB si +nk

(1)

i =1, i ≠ k

2 the ith path with E  βik ( t )  = 1 . aRk (ϕir ,θ ir )  

and

where p is the average transmit power, H k ∈ ℂ Nr × N t is

aTk (ϕit ,θ it ) represent the normalized transmit and receive

the frequency domain channel from the BS to the kth user. k FBB (k = 1, 2,..., K ) indicates the digital weighting vector

array response vectors at the azimuth angles ϕit , ϕir and

for the kth user, The data streams are mapped N RF RF chains through digital beamforming vector FBB and FBB k represents the set of all FBB . The total power constraint is

enforced by letting

k FRF FBB

2 F

= N k , transmitted symbols

N k × 1 vector containing N k

sk is the

transmitted

streams for the kth users, which satisfies Ε  skΗ sk  = 1 and Ε  skΗ si  = 0，( i ≠ k ) . The interference from the ith K

user is

∑

i i p H k FRF FBB si

the elevation angles θit , θir , respectively. The variables

ϕit and θit are the azimuth and elevation angles of the departure direction. ϕir and θir are the azimuth and elevation angles of the arrival direction. Considering the BS configures the large-scale MIMO, the planar antenna arrays with N row × N col are assumed. aT (ϕit ,θit ) can be defined as [19] 1 aT (ϕi ,θi ) = vec a N row ( µ ) a NT col ( υ ) = Nt

(

and nk is the additive

1  jυ j  µ + ( N −1) υ  j N −1 υ 1, e ,..., e ( col ) , e j( µ + υ ) ,..., e  col  ,  Nt

i =1, i ≠ k

Gaussian noise vector at the kth receiver with i.i.d. entries of mean zero and variance 1. 2.2

Channel model

)

j N row −1) µ

j ( N row −1) µ + ( N col −1) υ 

,..., e 

a N row ( µ ) = 1, e j µ ,..., e ( 

j N row −1) µ

Considering that the high path-loss at mmWave frequencies leads to limited scattering [9–13], and tightly packed arrays lead to high antenna correlation, we adopt

T

 (3)  where µ = (2 π d x cos ϕ sin θ ) λ , υ = (2πd y sin ϕ sin θ) λ , ..., e (

e(

j Ncol −1)υ

T

T

 and a N ( υ ) = 1,e j υ ,..., col  

 . λ is the wavelength. The scalars d x and 

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The Journal of China Universities of Posts and Telecommunications

d y are the spacings between the array elements parallel to the row and column, respectively. The operator vec ( ⋅) stretches N row × N col matrix to N t × 1 vector by stacking

2016

beamforming problem has become a low-dimensional digital beamforming problem. Then, the optimal baseband beamforming vector FBB can be solved based on the

the columns of the matrix. The index N row and N col

criterion maximizing the energy efficiency.

refer to the antenna element located at N row th row and

3.1 Analog beamforming scheme based on codebooks and signal-to-leakage plus-noise ratio (SLNR)

N col th column of the planar antenna arrays. The user equipment configures uniform linear arrays (ULA), and aR (ϕir ,θir ) can be defined as a N r × 1 vector:

aR (ϕi , θi ) = 1, e j υ ,..., e j(Nr − 1)υ 

Τ

In this section, the optimal analog beamforming vector FRF will be designed. Owning to that the independent beams for each user is selected, FRF can be represented

(4)

by

1 FRF = [ FRF , FRF2 ,..., FRFK ]

k k , N RF RF

and

k k ,1 k ,2 = [ FRF FRF , FRF ,..., FRF

k N t × N RF

k RF

indicates FFT transformation FRF [ FRF , FRF ,..., F ] , where F ∈ ℂ 3 Hybrid beamforming design for the mmWave k multi-user channel of the kth user’s RF beamforming vector f RF ∈ ℂN ×N t

This section discusses the proposed hybrid beamforming algorithm for the downlink multiuser mmWave massive MIMO system. MmWave channel possesses the strong directivity, the analog beamforming can acquire most of performance gains with constructively forming a beam(s) to a max-power direction. For multi-user hybrid beamforming, when the BS transmits the multiplexing users with adjacent beams, the multi-user interference is hard to mitigate only by digital beamforming. Thus the analog beamforming is proposed to select the optimal beam which can maximize the power of the objective user and minimize the interference to all other users. Further, digital beamforming maximizes the energy efficiency of the objective user with flexible baseband processing. Unfortunately, the hybrid beamforming is often intractable to directly solve a joint optimization over the two matrix variables ( FBB , FRF ) [20–21]. To simplify the transceiver design, the joint transmitter-receiver optimization problem is temporarily decoupled. First we focus on the design of analog beamforming vector FRF , and then consider the design of the optimal baseband beamforming vector FBB . Thus the proposed design will be divided into two steps: Firstly, analog beamforming vectors are designed indirectly to satisfy the objective function which maximizes the performance of the objective user and minimize the energy leakage to the other multiplexing users. At this step, assuming that the digital beamforming vector FBB is fixed and the data streams are mapped directly onto each of the RF chains. After analog beamforming vectors are solved, the hybrid

k RF

k and N RF is the RF chains number corresponding to the

kth user,

(k = 1, 2,..., K ) . Assume that the digital

k k is fixed as N RF × N k matrix beamforming vector FBB T

k =  I N k , I Nk ,..., J r  , where I N k is the identity matrix FBB of size N k × N k , J r is the first r columns of I N k and r k k is the remainder of N RF N k . Then let sk′ = FBB sk .

Rewrite the receive signal of the kth user as K

Yk =

k k FBB sk + p H k FRF

∑

p

i i H k FRF FBB si + nk =

i =1,i ≠ k K

k p H k FRF sk′ +

p

∑

i H k FRF si′ + nk

(5)

i =1,i ≠ k

in which the second term represents the co-channel interference caused by the other users. SLNR criterion is a simple and effective MU-MIMO algorithm, which can maximize the power of the objective user and minimize the power leaked from the kth user to all other users. Moreover, SLNR criterion leads to a decoupled multi-user optimization problem and admits an analytical closed form solution. So the analog beamforming vector based on SLNR criterion is solved. From Eq. (5), the SLNR at the kth user is k p H k FRF sk′

RSLNR ( k ) =

K

p

∑

2

2 k RF k

Hi F s′

=

+σ

2 k

i =1,i ≠ k k k pN RF H k FRF K

k pN RF

∑

k H i FRF

2

(6) 2

+ σ k2

i =1,i ≠ k k Η k k where Ε  sk′Η sk′  = Ε  skΗ FBB FBB sk  = N RF . The power of

]

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Jiang Jing, et al. / Energy-efficiency based downlink multi-user hybrid beamforming for…

k k the objective user k is pN RF H k FRF

2

leaked

other

from

user

K

k pN RF

∑

k H i FRF

2

k

to

all

, respectively.

and the power users

is

f RF is implemented

i =1,i ≠ k

using analog phase shifters, so its entries are of constant modulus. Further, the angles of the phase shifters are limited by factors of the RF hardware and quantized to a finite set of possible values. With these assumptions, f RF is chosen form F and F is the beam codebook of

the

RF

F = aT (Ψ , Θ ) ,

beamforming.

Ψ ∈（ Ψ 1 ,Ψ 2 ,… ,Ψ M ） and

where

Θ ∈（ Θ1 , Θ2 ,… , ΘN） denote

the set of quantized azimuth and elevation angles, respectively. M and N are the number of quantized azimuth and elevation angles, respectively. And aT (Ψ , Θ ) can be defined as

aT (Ψ , Θ ) =

1 Nt

(

)

vec α Nrow (U ) a NT col (V ) =

1  jV j U + ( N col −1)V  j N −1 V 1, e ,..., e ( col ) , e j(U + V ) ,..., e  , N  t

 (7)  where U = (2 π d x cosΨ sin Θ) λ , V = (2π d y sin Ψ sin Θ) λ , ..., e (

j N row −1)U

j ( N row −1)U + ( N col −1)V 

,...,e 

a Nrow (U ) = 1, e jU ,...,e ( 

j N row −1)U

T

 and a N (V ) = 1, e jV ,..., e col  

T

1, e ,..., e j( Ncol −1)V  . FFT transformation of F is defined as   Fcodebook = AT (Ψ , Θ ) , which can be the RF codebook of frequency domain. According to the SLNR criterion, the most optimal k is designed based on the beamforming vector FRF following metric

  F = arg max = … ; k 1, 2, , K K k  FRF 2 k k 2 pN RF ∑ H i FRF + σ k  i =1, i ≠ k   s.t.  k ,m k  FRF ∈ Fcodebook ; m = 1, 2,… , N RF  2  FRF F = N RF  k k H k FRF pN RF

k RF

2

(8) k ,m where FRF is the mth chain’s analog beamforming vector sent to kth user in frequency domain. Search in the codebook Fcodebook = AT (Ψ , Θ ) to find the corresponding analog beamforming vector maximum,

then

get

the

k FRF

whose SLNR is

corresponding

analog

57

k beamforming vector f RF from codebook F of the time

domain. To calculate the RF beamforming vector of all the users in turn, thus the analog beamforming matrix 1 2 K  is obtained. , f RF ,… , f RF f RF =  f RF 3.2

Zero-gradient (ZG) beamforming in digital domain

After the optimal analog beamforming vector FRF has been obtained, the design of the optimal baseband beamforming vector FBB is considered. Our objective is maximize the energy efficiency of the transmission at the BS while guaranteeing a specified rate for each user. Based on this objective function, the optimal digital weighting vectors are then given by: opt  FBB = argmax ηEE FBB   s.t.: (9)  SSINR ( k )≥γ k ; k = 1,..., K   Pt ≤Pmax  where γk is the required signal to interference plus noise ratio (SINR) at user k, and Pmax is the maximum transmit power at the BS. The total signal power consumption at the BS can be divided into two main parts: one in the power amplifiers  PPA , and the other in all other circuit blocks PC .

PPA = ρ Pt represents the power consumed by the power amplifiers, where ρ is the reciprocal of the power amplifier efficiency and Pt is the sum of the transmit power. The circuit power PC consists of the digital signal processor (DSP), frequency synthesizer, mixer, etc. Here we assume that PC is fixed. Thus, the energy efficiency (in bit/(s ⋅ Hz ⋅ J) ) of the BS can be expressed as K

B ∑ rk ηEE =

i =1

(10)

K

ρ ∑ Pk + PC i =1

where B is the bandwidth, and

Pk

is the power

K

transmitted to user k.

∑r

k

in Eq. (10) is the sum-rate

i =1

achieved by all users, and the denominator is the total power consumption transmitted to the K users. The achievable rate at the kth user is (11) rk = lb (1 + SSINR ( k ) ) where the SINR for the kth user is

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The Journal of China Universities of Posts and Telecommunications

k k FBB p H k FRF

SSINR ( k ) =

K

∑

p

i i H k FRF FBB

2

(12) 2

+ σ k2

in which σ

2 k

K

is the additive noise power at the kth

receiver. So rk can be rewritten as

  k k 2   p H k FRF FBB  (13) rk = lb 1 + K  i i 2 2  p H F F + σ ∑ k RF BB k   i =1, i ≠ k   The normalized (with respect to bandwidth) energy efficiency (in bit/ (s ⋅ Hz ⋅ J) ) can be written as  k k 2  p H k FRF FBB  lb 1 + ∑ K 2  i i k =1 p FBB H k FRF + σ k2 ∑  i =1,i ≠ k  = K k k 2 FBB ρ ∑ p FRF + PC K

ηEE

The interference and noise power of the kth user in Eq. (16) is represented by Γ k in Eq. (18).

∑

Γk = p

i =1, i ≠ k

      ; k = 1, 2,..., K

2

Due to the non-concavity of the energy efficiency as a function of the weighting vectors, traditional methods of solving constrained optimization problems, such as using Lagrange multipliers, are difficult to apply here. Instead, a zero-gradient-based approach is used to iteratively find an optimal digital beamforming vector. Define the objective function as   k k 2   K p H k FRF FBB  ln 1 + ∑ K  i i 2 2  k =1 p ∑ H k FRF FBB + σ k   i =1, i ≠ k   k 1 2 f ( FBB , FBB ,..., FBB ) = K 2 k k ρ∑ p FRF FBB + PC k =1

(15) Since Eq. (15) is not convex/concave in FBB , it is difficult to obtain a closed-form solution. Denote the numerator and the denominator as rˆk and D in Eqs. (16) and (17), respectively.  k k  p H k FRF FBB  rˆk = ln 1 + K  i i p ∑ H k FRF FBB  i =1,i ≠ k  K

k k D = ρ ∑ p FRF FBB k =1

2

+ PC

   2 2  + σk   2

(16)

(17)

+ σ k2

(18)

k is giving by (Proof in the Appendix A) FBB

∂f 1 k = 2 ( εk ϕ k − M k ) FBB k ∂ ( FBB ) D

(19)

where

εk =

D

(Γ + p H F k

k

k RF

k FBB

2

(20)

)

H

ϕ k = 2 p ( FRFk ) H kH H k FRFk

(21)

and K

∑ i =1, i ≠ k

is the transmit power for user k.

2

1 2 k The gradient of f ( FBB , FBB ,..., FBB ) with respect to

k =1

k k where p FRF FBB

i i H k FRF FBB

i =1, i ≠ k

Mk = D (14)

2016

i i 2 p 2 H i FRF FBB

(

2

k H RF

(F )

i i p H i FRF FBB

2

k H iH H i FRF

)

+

+ Γi Γi

  2  ∑ rˆi  ρpI N k RF  i =1  K

(22)

1 2 k The gradient of f ( FBB , FBB ,..., FBB ) is expressed as

follows H H    ∂f  ∂ f  ,  ,..., ∇f ( F , F ,..., F ) =   ∂ ( F 1 )   ∂ ( F 2 )  BB BB     1 BB

2 BB

k BB

H

  (23)   for an unconstrained optimization problem, the stationary 1 2 k points of f ( FBB , FBB ,..., FBB ) satisfy the zero-gradient

 ∂f    K   ∂ ( FBB )  

H

k condition ∇f ( FBB ) = 0 . If FBB is a stationary point, the

following equation must be satisfied k k εk φk FBB = M k FBB ; 1≤k≤K

(24)

Since the objective function is not convex/concave and the constrained condition in this optimization problem, solve the duality problem by traditional methods cannot guarantee the optimal solution. An iterative algorithm can be given by choosing the step size appropriately converges to a local maximum [22]. Detailed process of this iterative algorithm is as follows: 1) Select F（BB0） as an arbitrary initial value for the beamforming matrix. 2) At iteration n=1, 2, … for all k=1,2,…,K.

Compute

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Jiang Jing, et al. / Energy-efficiency based downlink multi-user hybrid beamforming for…

(

ak( n ) = εk( n −1) M k( n −1)

)

−1

( n −1)

k φk( n −1) ( FBB )

(25)

Linear search Eq. (26) find the optimal µk( n ) with an suitable step size of ν from 0 to 1.

(

( n −1)

k µk( ) = arg max f νak( ) + (1 − ν ) ( FBB ) n

ν∈[ 0,1]

n

)      

s.t. ( ) RSINR ≥γk ; k = 1, 2,… , K k n

Pt ( ) ≤Pmax n

(26)

and let

(F ) k BB

( n)

(

= µk( n ) ak( n ) + 1 − µk( n )

) ( F )( k BB

n −1)

(27)

3) Update εk( n ) and M k( n ) with Eqs. (20) and (22), respectively. When

k f ( FBB )

( n)

k − f ( FBB )

( n -1)

2

≤δ , stop

the search. Otherwise, return to step 2).

4 Simulation and analysis In this section, the paper demonstrates the performance of the proposed hybrid analog-digital beamforming algorithms and provides some simulation results of various transceiver designs under IEEE 802.11ad mmWave MIMO channel model. Simulation parameters are listed in Table 1. Table 1

Simulation parameters

Parameters Number of cell Number of user K Channel type Number of transmit antenna Nt Number of receive antenna Nr Inter-element spacing in BS antenna arrays Number of RF chains NRF Carrier frequency/GHz Bandwidth/GHz Transmission power p/dB Power amplifier efficiency ρ /(%) Modulation and coding scheme Simulation frames/frame

Value 1 2,3,4 IEEE 802.11ad, living room 64, 128, 256, Planar antenna arrays 1

59

azimuth and elevation domain. M beams cover the target area of 360° in the direction of azimuth and N beams cover the target area of 180° in the direction of elevation. Therefore, the number of quantized azimuth angles M=18 and the number of quantized elevation angles N=9. Exhaustive search is used for finding desired analog beams. It is well known that the digital beamforming is often performed at baseband, which enables controlling both the signal’s phase and amplitude. However, in analog/digital hybrid beamforming, the use of analog phase shifters places a constant modulus constraint on the elements of the RF beamforming vector. Therefore, the performance of the hybrid beamforming is inferior to that of the digital one [15]. Firstly, the performance of different algorithms is compared when K=2, Ns =2 and N t =64. The orthogonal matching pursuit (OMP) hybrid beamforming algorithm [14] and the full digital beamforming based on block diagonalization (BD) algorithm are used for comparison to show the performance of the proposed analog-digital hybrid beamforming algorithm. In Fig. 2, SNR vs. BER curve is evaluated with all of the three beamforming algorithms. Simulation results show that the curve of the proposed hybrid beamforming algorithm is close to full digital beamforming and 2 dB~4 dB better than OMP hybrid beamforming. BER of the proposed algorithm is close to 0 when SNR is bigger than − 14 dB.

0.5λ Equal to Ns 60 2.56 0 38 QPSK, 1/2 Turbo codes 5 000

Considering that a cell with only one BS and K users connected to one BS. The cell radius at the BS is a typical value for a microcellular system. At the BS a separate queue is maintained for each user. Assuming that the CSI is updated once per frame. Five orthogonal frequency division multiplexing (OFDM) symbols are grouped into a frame and one OFDM symbol has 512 subcarriers including 352 data subcarriers. The analog beamforming codebook is designed as Eq. (7). In Eq. (7), the quantized angles are assumed to be spacing 20° in both the

Fig. 2 SNR vs. BER curve with different algorithms when K=2, Ns =2 and N t =64

In the following two simulations, the sum-rate performance of the proposed algorithm in different parameter conditions is considered. First, the performance of the proposed hybrid beamforming algorithm with different number of transmit antennas is discussed. Fig. 3 shows the sum-rate achieved by the proposed

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The Journal of China Universities of Posts and Telecommunications

beamforming algorithm with N t =64, N t =128, N t =256. The parameters K and Ns are the same as in Fig. 2. It can be found in Fig. 3 that once increasing the number of transmitting antennas greatly improves the system performance when SNR is low. When the SNR is close to − 15 dB, the sum-rate of three cases approximately are the same and achieve to the maximum value. It is obvious that performance will be better with more antennas.

2016

The energy-efficiency of the proposed hybrid beamforming algorithm and the OMP hybrid beamforming algorithm are compared in Fig. 5 with N t =64, K =

N s = 2 and PC = 1 W . It shows the energy-efficiency as a function of the SNR. When SNR is − 24 dB, the energyefficiency value is 0.49 bit/(s ⋅ Hz ⋅ J) for the proposed hybrid beamforming algorithm and 0.48 bit/(s ⋅ Hz ⋅ J) for the OMP hybrid beamforming algorithm. When the SNR is up to − 14 dB, the proposed algorithm achieve the maximum value 0.78 bit/(s ⋅ Hz ⋅ J) and the OMP algorithm is 0.73 bit/(s ⋅ Hz ⋅ J) . Simulation results show that, the gap between the proposed hybrid algorithm and the OMP hybrid beamforming increased by increasing the SNR. Obviously, the proposed algorithm achieves much higher energy efficiency than the traditional OMP scheme.

Fig. 3 Sum-rate performance of the proposed hybrid beamforming algorithm with K=2, Ns =2 and with different N t value

In third simulation, the performance of the proposed algorithm with different numbers of MU-MIMO users is illustrated in Fig. 4. The number of transmit antennas is fixed as N t =64 and the number of user is set to K=2, K=3, K=4 corresponding to three curves in Fig. 4. The number of streams Ns in each line is equal to the number of users. As can be seen from this figure, with the increase in the number of users the sum rate is also improved accordingly. And the three lines have risen sharply at SNR is − 25 dB ~ − 15 dB and reach their maximum at point about SNR is − 15 dB.

Fig. 4 Sum-rate performance of the proposed hybrid beamforming algorithm with N t =64, Ns = K

Fig. 5 Energy-efficiency comparison of the two hybrid beamforming algorithms

5 Conclusions In this paper, an efficient hybrid analog/digital beamforming algorithm for downlink multiuser mmWave systems is proposed. First, the analog beamforming is designed to select the optimal beam on SLNR criterion, which leads to a decoupled multi-user optimization problem and can maximize the power of the objective user and minimize the power leaked from the objective user to all other users. Then, the digital beamforming is solved to maximize the energy efficiency of the BS through flexible baseband processing. The resulting maximum energy efficiency problem is solved by using the zero-gradientbased approach, thereby iteratively finding an optimal digital beamforming vector. Simulation results show that the proposed algorithm performs surprisingly well in mmWave channels and greatly increases the sum rate with the increases in the

Issue 4

Jiang Jing, et al. / Energy-efficiency based downlink multi-user hybrid beamforming for…

number of MU-MIMO users. It is proved that multi-user massive MIMO can be a perfect candidate for mmWave communication system. In mmWave multi-user massive MIMO system, the analog beamforming can mitigate the inter-user interference more effectively with the selection of the optimal beam. Furthermore, the digital beamforming can improve the system performance through flexible baseband processing.

K

∂ ⌢ ri = − k ∂FBB

i i p H i FRF FBB 2 i BB

2 p Hi F F

k

i i 2 p H i FRF FBB

⋅ l l H i FRF FBB

2

+ σ i2

k H RF

p (F

)

2

k k H iH H i FRF FBB 2

 +σ  

2

=

2 i

H

k p ( FRF ) H iH H i FRFk FBBk

2

(A.5)

the numerator of Eq. (29) can be expressed as ∂  K ⌢  K ⌢ ∂ r D −  ∑ ri  D= k ∑ i  k ∂FBB  i =1   i =1  ∂FBB H

D

k 2 p ( FRF ) H kH H k FRFk FBBk 2

k k Γ k + p H k FRF FBB

∑ rˆ

−

k

i =1

(A.1) D k The partial derivative of f ( FBB ) with respect to FBBk

is as follows ∂  K ⌢  K ⌢ ∂ − r D ∑ i   ∑ ri  ∂F k D k  ∂FBB ∂f i =1    i =1  BB = k D2 ∂FBB

(A.2)

(A.3)

K

∑

i i H k FRF FBB

2

+ σ k2

i =1,i ≠ k k RF

k p H k F FBB

2

⋅

K

∑

+p

i i 2 p H i FRF FBB

2

H

k p ( FRF ) H iH H i FRFk FBBk 2

=

H

 K ⌢ k and the calculation of ∂  ∑ ri  ∂FBB can be divided  i =1  into two parts : when i = k ,

p

 K ⌢ k H 2  ∑ ri  ρ p ( FRF ) FRFk FBBk −  i =1  K Γi D ∑ ⋅ 2 i i i =1, i ≠ k p H F F + Γi i RF BB

( Γi )

where ∂ k H D = 2 ρ p ( FRF ) FRFk FBBk k ∂FBB

∂ ⌢ rk = k ∂FBB

+ σ i2

K

∑

+p

( Γi )

Proof In this section we prove the Eq. (19). We plugged Eqs. (16) and (17）into the Eq. (15) get

)=

2

 K l l  p ∑ H i FRF FBB l = l ≠ i 1,  Γi − ⋅ i i 2 p H i FRF FBB + Γi

Development Program of China (2014AA01A705).

Appendix A Proof of Eq. (19)

2

l =1, l ≠ i

This work was supported by the Hi-Tech Research and

f (F

l l H i FRF FBB

l =1,l ≠ i

i RF

Acknowledgements

k BB

∑

p

61

i i H k FRF FBB

2

+ σ k2

D

k 2 p ( FRF ) H kH H k FRFk FBBk

K

D

2

k k Γ k + p H k FRF FBB

∑ i =1, i ≠ k

i i 2 p H i FRF FBB

H

2

k p ( FRF ) H iH H i FRFk FBBk

(p H F i

−

i RF

i FBB

2

)

−

+ Γi Γi

 K ⌢ k H 2  ∑ ri  ρ p ( FRF (A.6) ) FRFk FBBk  i =1  Let εk , φk and M k as Eqs. (20)–(22), then we get the Eq. (19).

i =1,i ≠ k H

k 2 p ( FRF ) H kH H k FRFk FBBk K

∑

p

i i H k FRF FBB

2

References =

+ σ k2

i =1, i ≠ k H

k 2 p ( FRF ) H kH H k FRFk FBBk

k k Γ k + p H k FRF FBB

when i ≠ k ,

2

(A.4)

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(Editor: Wang Xuying)