Hybrid and Full-Digital Beamforming in mmWave Massive MIMO Systems: A Comparison Considering Low-Resolution ADCs

Wence Zhang,Xiaoxuan Xia,Yinkai Fu,Xu Bao,＊

1 School of Computer Science and Communications Engineering,Jiangsu University,Zhenjiang 212013,China

2 National Mobile Communications Research Lab.,Southeast University,Nanjing 210089,China

Abstract: In a millimeter-wave (mmWave)Massive multiple-input multiple-output(MIMO) systems,full-digital beamforming(i.e.,connecting each antenna with a specificradio-frequency (RF) chain) becomes inefficient due to the hardware cost and power consumption.Therefore,hybrid analog and digital transceiver where the number of RF chains are much smaller than that of the antennas has drawn great research interest.In this work,we investigate the use of low-resolution analog-to-digital converters (ADCs) in the uplink of multi-user hybrid and full-digital mmWave Massive MIMO systems.To be specific,we compare the performance of full-digital minimum mean square error (MMSE) and hybrid MMSE beamforming in both sum rates and energy efficiency.Accurate approximations of sum rates and energy efficiency are provided for both schemes,which captures the dominant factors.The analytical results show that full-digital beamforming outperforms hybrid beamforming in terms of sum rates and requires only a small portion (γ) of antennas used by hybrid beamforming to achieve the same sum rates.We given sufficient condition for full-digital beamforming to outperform hybrid beamforming in terms of energy efficiency.Moreover,an algorithm is proposed to search for the optimal ADC resolution bits.Numerical results demonstrate the correctness of the analysis.

Keywords: hybrid; massive MIMO; mmWave;MMSE; low-resolution ADCs

I.INTRODUCTION

The millimeter-wave (mmWave) frequency band (typically 30-300 GHz) has recently attracted much attention due to the fact that current cellular spectrum below 3 GHz is under severe shortage and the under-utilized mmWave band can provide much more spectrum resource to meet the extensive needs in the coming 5G era [1].Due to the shorter wavelength,the antenna arrays at the mmWave frequencies occupy much smaller physical dimension (as compared to the antenna arrays at current 3G or LTE frequencies)[2].This enables the use of massive MIMO systems[3]-[7],for beamforming to combat the higher path-loss and absorption at higher frequencies.However,the use of traditional full-digital beamforming for massive MIMO communications is not practical.This is because traditional beamforming is performed at baseband,which requires not only phase and amplitude signal control,but also the use of a dedicated radio frequency (RF) chain for each antenna element.Due to the high cost and power consumption of the RF chains [2],such full-digital beamforming solution is not viable for implementation for massive MIMO systems at mmWave frequencies.

In this work,we have compared full-digital and hybrid beamforming with low-resolution ADCs in mmWave Massive MIMO systems.

Due to the above reasons,hybrid analog and digital beamforming schemes for mmWave massive MIMO systems was first proposed in [8].The hybrid architecture uses a few RF chains,the number of which is much smaller than that of the antennas.A phase shifter network is used to connect the antennas and the RF chains and enables analog beamforming.In [8],it concludes that hybrid beamforming can approach the performance of full-digital beamforming provided that at least two RF chains per antenna.In [9],the phase of the conjugate transpose of the downlink channel is used for analog beamforming to achieve large array gain.Then,zero-forcing (ZF)digital detection is performed on the effective channel obtained by multiplying the analog beamformer and the actual channel matrix.In [10],the authors propose an HBF scheme,which achieves near-to-all-digital beamforming performance by switching the phase shifters on and off.In [11],the authors consider the downlink communication in the multi-user mmWave massive MIMO system and propose an energy-efficient weighted minimum mean square error based hybrid beamforming algorithm to maximize the achievable sum rate.In [12],a low-complexity hybrid beamforming and combining design for the multi-user mmWave massive MIMO downlink is proposed,which applies to both fully-connected structures and sub-connected structures.

Besides hybrid structure,reducing the resolution of the ADC is also an effective way to reduce system power consumption and improve energy efficiency.There are many works on Massive MIMO systems with low-resolution ADCs (see [13]and references therein.)Different models exist for studying the effect of low-resolution ADCs.The authors of [14]proposed pseudo quantization noise (PQN)model which assumes the quantization error is uniformly distributed and uncorrelated with the input signal.Another model is additive quantization noise model (AQNM)[15]-[18],in which the output of the ADCs is a scaled version of the input signal plus an uncorrelated Gaussian noise.Although AQNM is an approximation,it is convenient for analysis and provides insights to the effect of low-resolution ADCs.Moreover,comparison between AQNM and PQN indicates there is no difference in the fundamental conclusions[14].

Low-resolution ADCs can also be combined with hybrid beamforming.In [19],the authors propose a generalized hybrid architecture with low-resolution ADCs,the achievable rate of which is comparable to that obtained by full-resolution ADC receivers at low and medium signal-to-noise ratio (SNR).There are many low-resolution ADCs studies that consider only one-bit quantization [20],[21],but for a hybrid architecture with one-bit quantization,its performance is not superior to an full-digital architecture [22].

In this paper we compare the hybrid and full-digital beamforming schemes considering low-resolution ADCs with multiple quantized bits.Minimum mean square error (MMSE)design criterion is considered because it generally outperforms maximum ratio combining(MRC) and zero-forcing (ZF) schemes.Regarding the ADC quantization model,AQNM is adopted since it facilities the performance analysis and provides insights.This work differs form existing work,such as [16]-[18],[22]-[24].The work in [17]and [16]investigate performance of different beamforming schemes with low-resolution ADCs,while no closed-form expression for data rates are given.In [23],a performance analysis of mixed-ADC massive MIMO systems is provided.However,the results are valid for MRC detection,not for MMSE that we considered here.In [24],the authors carried out performance analysis for massive MIMO with low-reso-lution ADCs using a channel model different from our work.Moreover,[24]does not consider hybrid beamforming.The most related work is [22],where the achievable rates and energy efficiency for hybrid beamforming with low-resolution ADCs is analyzed,while the results therein are only for channel inversion and singular value decomposition (SVD)based schemes and no results are provided for MMSE.

The main contribution of this work are summarized as follows:

· In order to compare hybrid and full-digital beamforming,we derive the MMSE hybrid beamforming scheme with low resolution ADCs and carry out performance analysis.The approximate SINR of both beamforming methods are provided with simple and elegant expression by taking advantage of law of large numbers and focusing on the dominant effective factors.

· Regarding the sum rates,we prove that full-digital beamforming is superior to hybrid beamforming.To achieve the same sum rates as hybrid beamforming,full-digital scheme only requires a proportion (γ) of antennas,wherefor ADCs with high resolutions ADCs andfor low resolution ADCs withN,Kbeing the number of antennas and the number of users,respectively.

· The energy efficiency analysis of hybrid and full-digital beamforming is given and the sufficient condition when full-digital beamforming outperforms hybrid beamforming is provided.We also proposed an algorithm to search for the optimal quantization bits that maximizes the energy eff iciency for both schemes.

· Simulations are carried out and the analysis are verified by numerical results

The rest of the paper is organized as follows.Section II introduces the system model and the design of hybrid/full-digital beamforming with low-resolution ADCs for comparison.In Section III the analysis on the performance of hybrid and full-digital beamforming is provided in terms of sum rate and energy efficiency.The numerical results are presented in Section IV,and conclusions are drawn in Section V.

Notation:Boldface lower and upper case symbols represent vectors and matrices,respectively.(·)*,(·)T,(·)Hdenote the conjugate,the transpose and Hermitian transpose of a vector or a matrix,respectively.E{·} denotes the expectation operator.IMand IKis the size-Mand size-Kidentity matrix.Tr(·) denote the trace of a matrix.diag[X]denote the diagonal matrix with the same diagonal elements as matrix X.var(·) denote the variance of a random variable.denotes the covariance of random vector s.

II.PROBLEM FORMULATION

In this section,we will describe the problem of interest.

2.1 Signal model considering lowresolution ADCs

Consider the uplink of a multi-user mmWave massive MIMO system consisting of a BS which has an array ofNantennas and servingK(KN＜) single-antenna user terminals in the same time-frequency resource.We consider the hybrid beamforming design for a system as shown in figure 1,where theKRF chains use low-resolution ADCs.The receivedNdimensional vector y at the BS can be expressed as

Fig.1.A diagram of hybrid architecture in mmWave massive MIMO systems.

where H represents theNK× channel matrix between BS and users,x denotes theK×1 vector of symbols transmitted by all users andis the average transmitted power of each user,and n ～ C N ( 0,I ) is the additive white Gaussian noise vector.

In order to model the mmWave propagation environment,we consider that each propagation path between the BS and user is related to a scatterer.Assuming the scatterers seen by different users are independent,this model is referred to as independent multipath channel model in [25].

The channel vector of thek-th user is expressed as [25]

whereLkis the number of propagation paths,gkl～ CN ( 0,1) is the complex gain of thel-th path,andθkl∈ [0,2π]denotes the AoA of thel-th path.Assume that a uniform linear array(ULA) is equipped at the BS,then the steering vector aN(θ) can be modeled as

with Δ the antenna spacing normalized by carrier wavelengthλ.

Denote

we have

where Acis chosen to be orthogonal to Ak(kK=…1,,).

In this paper,we also consider the use of low-resolution ADCs to reduce the cost and power consumption.Let s be the input signal.

TableI.The value of ρ for different ADC quantization bits b.

The quantized output is modeled as [26]

in which,αρ=-1 whereρis the inverse of the signal-to-quantization-noise ratio,and nqis the additive Gaussian quantization noise with zero mean and covariance given by

Note that the quantization noise is uncorrelated with s.

Letbbe the number of quantization bits and assume that the input to the quantizer is Gaussian.For the non-uniform scalar minimum mean-square-error quantizer of a Gaussian random variable,the values ofρare listed in TableI forb≤5 and can be approximated

In this work,we will focus on comparison between hybrid and full-digital beamforming for mmWave Massive MIMO systems.For the purpose of fair comparison,we will introduce the MMSE-based full-digital beamforming with low-resolution ADCs and then derive its hybrid counterpart.

2.2 Full-digital beamforming

For full-digital beamforming,the phase shifter network in figure 1 is removed and each antenna is connected to a separate RF chain with a low-resolution ADC.The quantized signal yqis obtained from (4) and described as

where the covariance of quantization noise nq,FDis

The corresponding full-digital beamforming matrix iswhere

Therefore,we have the signal after detection given by

2.3 Hybrid beamforming

As illustrated in figure 1,hybrid beamforming involves two steps: analog beamforming and baseband beamforming.The overall beamforming matrix is denoted as VH= VBVP,where VP∈CK×Mand VB∈CK×Krepresent the analog and baseband beamforming matrix,respectively.

1) Analog Beamforming:The signal after the phase shifters is given by

where VPis designed by extracting the phases of HH[11],i.e.,whereφijis the phase of the (,)i jth element of HH.

2) Baseband Beamforming:The output of the low-resolution ADCs is

where the covariance of quantization noise nq,His derived as

Here,VBis designed based on the MMSE criterion,i.e.,where

Therefore,the received signal for hybrid beamforming is given by

Here,we derive the Hybrid MMSE beamforming scheme for fair comparison between full-digital and hybrid receiver structure with low-resolution ADCs.This work is different from [22],where channel inversion and singular-value-decomposition (SVD) based beamforming are considered,while we considered MMSE based methods here.Therefore,the analysis are completely different.

III.PERFORMANCE ANALYSIS

In this section,we will derive the sum rates and energy efficiency of hybrid and full-digital beamforming and carry out comparison between them.In order to make the results simple and elegant,we will capture the dominant factors by approximations.

3.1 Sum rates

As can be seen from (6) and (8),the exact analysis on the sum rates is very complicated.Therefore,we will try to focus on the dominant factors that affect the sum rates.The main results are summarized in Theorem 1.

Theorem 1:The uplink sum rateRFDof full-digital beamforming with low-resolution ADCs is well-approximated by

where

and that of hybrid beamforming is well-approximated by

where

with

Proof:See Appendix A.

It can be seen from Theorem 1 that,both the SINR of full-digital and hybrid beamforming are are related toN,K,αandpu.In general,the sum rates grows withNdue to increased diversity gain,while decreases withKbecause of increased inter-user interference.Note that,the above results can also be applied to cases with full-resolution ADCs by settingα=1.

Proposition 1:The sum rates of full-digital beamforming is always higher than hybrid beamforming.

Proof:The sum rates of both beamforming schemes are determined by the SINR of each.Therefore,we only need to prove that SINRFDis greater than SINRH,which is not straightforward.The following proof would be a little tricky by construction a special function.

From (9) and (10),it can be seen that both SINRFDand SINRHhave a form ofw h e r e f o r SINRFDwe have

and for SINRHwe have

Note that whena,b,c,puare non-negative,f(a,b,c,pu) decreases withaandb,while increases withc.It is not dif ficult to find out thataFD＜aH,bFD＜bHandcFD＞cH.Therefore,we haveand thus SINRFD＞ S INRH.

Proposition 1 shows that full-digital beamforming outperforms hybrid beamforming in terms of sum rates.This is obvious since full-digital structure has more degree of freedoms.The following proposition shows fewer antennas are needed for full-digital structure in order to achieve the same sum rates as hybrid beamforming.

Proposition 2:AssumingNHandNFDantennas are used by hybrid beamforming and full-digital beamforming,respectively.In order to achieve the same sum rates,NFDis approximately given byNFD≈NHγ,where

Proof:Whenα→1,we haveandSinceKN≪ in massive MIMO systems,it gives SINRH≈NHpuπ/4.Setting SINRFD=SINRHyieldsSimilarly forα→0,we can obtain

Proposition 2 shows that whenα→0,i.e.,the number of ADC resolution bits is rather low,hybrid structure is not very efficient.Since by use only a small portion of antennasfull-digital beamforming could save a lot of implementation cost.

3.2 Energy efficiency

The energy efficiency of the receiverηEEis defined as

whereBis the bandwidth andPTOTdenotes the total power consumption,which are given for full-digital and hybrid beamforming,respectively,by

wherePRFandPPSare the power consumption of the RF link and phase shifters,respectively;CW2bdenotes the power consumption of ADCs,whereCis the power cost per conversion andWis the sampling rate.For the RF links,it consists of mixer,low path filter,and base-band amplifier and etc.The typical values forPRF,PPSandCare chosen to be 39 mW,2 mW and 495× 1 0-15J /Step according to[27]-[29].

The energy efficiency for the hybrid and full-digital beamforming are given,respectively,by

whereRFDandRHare given by Theorem 1.

Although the comparison between hybrid and full-digital beamforming in terms of energy efficiency can be obtained from (13) and (14),it is still not easy to get more insights.Therefore,we will focus on some typical scenarios.

Proposition 3:The energy efficiency of full-digital beamforming is larger than hybrid beamforming provided that

Proof:DenotePGapas total power consumption difference between the full-digital archi-tecture and the hybrid architecture,and we have

SettingPGAP＜0 yields (15),which indicates full-digital beamforming consumes less power than hybrid beamforming.According to (13),(14) and Proposition 1,we haveηEE,FD＞ηEE,H.

Figure 2 shows the condition whenηEE,FD＞ηEE,Haccording to (15).In general,hybrid beamforming works well with small number ofK,while full-digital beamforming outperforms its counterpart whenKis large,e.g.K＞20 forb＜8.Note that the condition in Proposition 3 is sufficient but not necessary.It can be seen from Figure 2 that with increased value ofN,Wandb,this condition becomes stricter,which implies the advantage of full-digital beamforming over hybrid beamforming becomes weak.

As can be seen from (14) and (13),ηEE,HandηEE,FDare functions ofb.The optimal choice ofbis of great interest for practice.However,this is too complicated to handle in analytical way.Therefore,in this subsection we propose an algorithm to compute the optimal ADC resolution bits for hybrid and full-digital beamforming in order to maximize the energy efficiency.

In practice,ηEE,HandηEE,FDfirst increase withband then decrease.This motivates to use methods based on bisection search.The algorithm is summarized in TableII.The optimalb*for different scenarios are given in TableIII.In general,the optimal resolution bits of hybrid beamforming is a little higher than that of full-digital beamforming.

IV.NUMERICAL RESULTS

TableII.Algorithm to Find the optimal ADC Quantization Bits b*.Input: b b← ;1 While b b L min← ,b b R max LR＜-1;2 2.1 b1← round [( )/2]b b 1 R+ ;2.2 Compute ηEE 1()b and ηEE 2( )b according to (13) or (14);2.3 If η η L R+ ;b2← round [( )/2]b b bb＞If b b EE 1 EE 2() ()2 R= ,then b b R2←;End of if;2.4 else If b b RR←-1;else b b L1←;End of if;2.5 End of if;End of while.3 If η η LL←+1;else b b 1 L= ,then b b bb＞,then bb*←L;4 else b b*← R;5 End of if;Output: b*;EE L EE R( ) ( )

TableIII.Optimal b* for different scenarios with N=100.

Fig.2.The suf ficient condition for ηE E ,FD ＞ η EE,H.

In this section,we present numerical results to show the performance of the hybrid and the full-digital beamforming in terms of sum data rates,and energy efficiency.In all the simulations,the number of userK=10,the number of antennasM=100,the number of multi-path isL=5 and the carrier frequency isf=28 GHz.In the plots,the number of quantization bits are set to be 14 for full-resolution ADCs.

Fig.3.Uplink sum rate versus different pu,where N=100,K=10.

Fig.4.Sum rates versus the number of quantization antennas,where K=10,pu=10 dB.

The simulation results for uplink sum rate of both architectures with different resolution ADCs are given in figure 3.It can be seen that the sum rates of all the schemes improve with respect topuandb.As concluded in Proposition 1,the sum rates of full-digital beamforming is higher than that of hybrid beamforming,especially when the number of ADC resolution bits are low.Whenbbecomes large,the gap between full-digital and hybrid beamfomring in terms of sum rates becomes negligible.The numerical results demonstrate the correctness of the analytical results given in Theorem 1,with an approximation error within 8%.As can been in the plots,the analysis are actually upper bounds of the sum rates.This is due to that the inter-user interference is omitted during the derivation.It is worth to point out that the sum rate saturates with low-resolution ADCs in the high SNR region as a result of the existing quantization error.

Figure 4 illustrates the simulation results for sum rates of full-digital and hybrid beamforming with respect toN.As can be seen in the figure,the sum rates of both schemes increase withN.As described in Proposition 2,however,full-digital beamforming requires less number of antennas to achieve the same sum rates as hybrid beamforming.Whenb=2,i.e.,αis close to 0,full-digital beamforming requires only 10 antennas to reach the performance of hybrid beamforming withN=100.The ratio is aroundWhenb=14,i.e.,αis close to 1,this ratio becomes around

Figure 5 shows the relationship between energy efficiency and the number of quantization bits.The energy efficiency of the hybrid and full-digital beamforming first increase and then decline with respect to the number of quantization bits.The reason is that asbincreases,the achievable rates first grow and then saturate,while the power consumption keeps growing.When the sufficient condition proposed in Proposition 3 is satisfied,i.e.,K=20 forb＜8,the energy efficiency of full-digital beamforming outperforms that of hybrid beamforming.However,whenK=10,with optimalb* obtained using the proposed algorithm in TableII,hybrid beamforming performs significantly better than full-digital beamforming.

V.CONCLUSIONS

In this work,we have compared full-digital and hybrid beamforming with low-resolution ADCs in mmWave Massive MIMO systems.Closed-form analysis are given for both sum rates and energy efficiency.We have derived sufficient conditions when full-digital beamforming outperforms hybrid beamforming in terms of energy efficiency.Both analysis and numerical results reveal that hybrid beamforming works well when the number of users is small.Regarding the optimal choice of ADC resolution bits,have we proposed an algorithm to search the optimal solution based on bisection methods.It can be drawn from the results,the optimal ADC resolution bits of hybrid beamforming is slightly higher than that of full-digital beamforming.

ACKNOWLEDGEMENT

This work was supported in part by the Key Research & Development Plan of Jiangsu Province (No.BE2018108),National Nature Science Foundation of China (Nos.61701198& 61772243),Nature Science Foundation of Jiangsu Province (No.BK20170557),Nature Science Foundation for Higher Education Institutions of Jiangsu Province of China (No.17KJB510009),the open research fund of National Mobile Communications Research Laboratory,Southeast University(No.2018D13),Young Talent Project of Jiangsu University and Postgraduate Research& Practice Innovation Program of Jiangsu Province (No.SJCX18_0742).

APPENDIX A Proof of Theorem 1

The derivation of SINRFDcan be divided into two steps: 1) to simplify the expression of the received signal,especially VFD; 2) derivation of the signal power and noise power.

Fig.5.Energy efficiency versus the ADC resolution bits,where N=100,pu=10 dB.

Let us start with simplification of the received signal expression.The diagonal elements of HHHis given by where ()afollows law of large numbers which shows the average value of a series of samples approaches the statistical mean.From (7),we have

Regarding VFD,it is obtained that

In order to get the approximate SINR,we first derive the signal part as

where

which follows the law of large numbers.Then,the signal power of thek-th user is given by

To handle the noise part,we first observe

where (c) follows that AHA = I.Therefore,we have

Then,the noise power of thek-th user is given by

Divide (17) by (18) gives (9).

Now let us derive the SINR of the hybrid beamforming scheme by starting with

It is easy to find out that

Denotehkias the (,)k i-th element of H.Since

hkican be approximated by a complex Gaussian random variable with zero mean and unit variance.Therefore,we have

where ()dand ()efollows law of large number.Then it is straightforward to get

Similar technique can be applied toand it yields

The covariance of the quantization noise is approximated by and the baseband beamforming matrix is approximated by

where

Therefore,we are able to calculate the signal and noise power of thek-th user,respectively,as

and

Divide (19) by (20) gives (10) and thus completes the proof of Theorem 1.