Yao Zhang,Meng Zhou,Haitao Zhao,Longxiang Yang,Hongbo Zhu

The Department of Wireless Communication Key Lab of Jiangsu Province,Nanjing University of Posts and Telecommunications,Nanjing 210003,China

Abstract:In this paper,the spectral efficiency(SE)of an uplink hardware-constrained cell-free massive multi-input multi-output(MIMO)system with maximal ratio combining(MRC)receiver filters in the context of superimposed pilots(SPs)is investigated.Tractable closed-form SE expressions for the considered system are derived,which share us with opportunities to explore the impacts of the hardware quality coefficient,the length of coherence interval,and the power balance factor between pilot and data signals.Numerical results indicate that the achievable SE deteriorates as the hardware quality decreases and is more susceptible to the hardware impairments at the user equipments(UEs).Besides,we observe that SPs outperform regular pilots(RPs)in terms of SE and this performance gain is heavily dependent on the values of power balance factor and coherence interval.However,the superiorities of SPs over RPs have vanished when severe hardware imperfections are considered.

Keywords:cell-free massive MIMO;hardware impairments;superimposed pilots;spectral efficiency

I.INTRODUCTION

As an up-to-date evolution of cellular distributed massive multi-input multi-output(MIMO)and network MIMO systems,cell-free massive MIMO system,where a large number of access points(APs)equipped with single or multiple antennas provide services to all user equipments(UEs)at the same time/frequency resource blocks by exploiting phenomenons such as favorable propagation with channel hardening,has gained plenty of attentions over the past five years[1–6].The distributed dense topology of cell-free massive MIMO enriches the huge macro-diversity and aggressive spatial multiplexing of the UEs,thus promising unprecedented levels of spectral efficiency(SE)and energy efficiency(EE)in rural and urban scenarios[7–9].With these merits,cell-free,massive MIMO system is regarded as an indispensable technology for the beyond fifth-generation(B5G)networks.

In cell-free massive MIMO systems,the accuracy of spatial data detection heavily relies on the quality of the channel state information(CSI),which is generally estimated by the UEs sending orthogonal pilot sequences to the APs[1–3,6–8].However,a common issue in such systems is that when the number of UEs exceeds the number of available pilots(i.e.,a heavilyloaded network),there is no choice but to share the same pilot sequence among different UEs.As a result,different UE channels become correlated,which gives rise to the pilot contamination[10–13].This interference constitutes a bottleneck for the channel estimation and system capacity and does not vanish even when the number of APs tends to infinity[1].

To alleviate the impact of pilot contamination in cell-free large-scale MIMO systems,many researchers have shifted their attentions to pilot assignment(PA)strategies,which can reduce the pilot contamination by ingeniously assigning pilots to different UEs without changing the pilot structure[1,10–13].More specifically,a greedy PA approach was proposed in[1],where the UE with the lowest system capacity iteratively updated its pilot signal based on a specific standard.In[10],Liuet al.presented a Tabu-search-based PA scheme to enhance the 5%-outage achievable rate by constructing a Tabu list to avoid local optimums.Additionally,Zhanget al.[11]focused on the UEs’locations and the same pilot sequence was prohibited from being assigned to adjacent UEs,and in[12],Attarifaret al.gave a structured PA strategy to maximize the minimum geographical distance between UEs that shared the same pilot symbols.Note that both[11]and[12]aimed to prevent the adjacent UEs from reusing the identical pilot,but with different design criteria.Moreover,in[13],Liuet al.put forward a graph coloring based PA scheme with the aid of AP selection techniques.

Although the above contributions showed their effectiveness in resisting the pilot contamination,the proposed PA schemes entailed a much higher computational complexity than the random PA strategy in[1].Besides,they all supposed that the pilot training and data transmission are performed separately within the entire coherence interval,which is referred to as regular pilots(RPs)transmission.In contrast to RPs,an alternative pilot architecture,called superimposed pilots(SPs),has exhibited remarkable features in enhancing the achievable SE in cellular massive MIMO systems[14–17].In general,SPs and data symbols are simultaneously transmitted throughout the entire coherence interval,which makes the length of SPs is equal to the duration of the coherence interval.Consequently,the number of available orthogonal SPs is more than that of RPs,thus alleviating the pilot contamination by reducing the possibility of pilot-reusing among different UEs.However,this training strategy will make SPs and data symbols correlated,which ineluctably hinders the quality of the channel estimation and degrades the strength of the received signal-tointerference-plus-noise ratio(SINR).To be specific,the pilot contamination problem in the RP case is transferred into the pilot-data cross-interference issue in the SP counterpart.In this regard,Verenzuelaet al.[14]derived achievable closed-form SE and EE expressions for an uplink massive MIMO system in the context of SPs and Upadhyaet al.[15]presented a non-iterative SPs-aided channel estimation approach for reducing the correlation that arises from the training process.A hybrid pilot structure that combining SPs and RPs was also proposed in[15].Besides,Zhanget al.[16]turned their focus to the asymptotic analysis with SPs by assuming that the number of antenna branches at the base station grows without bound and Teetiet al.[17]analyzed the impact of 1-bit analog-to-digital converters for an uplink massive MIMO system in the presence of SPs.Only very recently,Zhanget al.[18]explored the superiorities of SPs over RPs in cell-free massive MIMO systems.All the above works demonstrated that the SPs-aided training method can harvest significant SE gains than the RPs counterpart.

A crucial issue in realizing cell-free systems in practice is how to handle the increasing financial deployment expenditures since the growing demand for wireless communications urgently requires more and more service antennas to be implemented at the APs and UEs[19].To ensure economic viability,a simple but efficacious method is to advocate low-quality transceivers.Nevertheless,this approach inevitably introduces the interferences called hardware impairments,which not only decrease the accuracy of the channel estimation but also deteriorate the strength of the received SINR[20].In particular,hardware impairments,such as phase noise,amplifier nonlinearities,quantization distortions,and I/Q imbalance are decisive factors affecting the system throughput in the high-power regime[21].Although part of hardware imperfections can be canceled by exploiting compensation algorithms[22–24],the residual hardware distortions still manifest themselves in transmission and reception phases and dominate the system performance.The achievable SE of cell-free massive MIMO systems in the presence of hardware and channel imperfections was analyzed in[25]and the impact of hardware impairments on the receiver design was studied in[26].Besides,Masoumiet al.[27]proposed a low-complexity fronthaul rate allocation algorithm for the hardware-constrained systems and Zhanget al.[28]established a hardware scaling law when considering the secure transmission in cell-free large-scale MIMO systems.However,all the above works were based on RPs.With hardware distortions,we want to know whether SPs still outperform RPs in terms of the SE.But to the best of authors’knowledge,the research on cell-free massive MIMO systems with hardware impairments and SPs has not been done in prior work.It should deserve special attentions.

In this paper,an uplink cell-free massive MIMO system with hardware impairments in the presence of SPs is considered.In the context of SPs,no additional time slots are reserved for training and all UEs send pilots at a reduced power alongside the data symbols throughout the entire coherence interval.Leveraging on a well-established transceiver distortion model from[23–28],closed-form linear minimum mean square error(LMMSE)channel estimation and achievable SE expressions with maximal ratio combining(MRC)receiver filters are derived,assuming that the SP interferences can or cannot be completely canceled from the received data symbols.The tractability of the closed-form results facilitates analyzing the effects of key design system parameters,such as the hardware quality coefficient,the length of coherence interval,and the power balance factor between pilot and data signals.Note that our work is not a straightforward extension of[18,23–28]since both the pilot-data cross-interferences and hardware imperfections need to be considered,which makes the derivation of the SE expressions more complicated than that of[18,23–28].That is why Theorem 2 is the main contribution of this paper.Numerical results indicate that the achievable SE degrades as the hardware quality decreases and is more susceptible to the hardware imperfections at the UEs.Besides,SPs outperform RPs in terms of achievable SE when middle- and high-quality transceivers are considered and it is recommended to transmit data symbols with 40%-60% of the total power to highlight the superiorities of SPs.However,for a heavily-hardwaredeteriorated cell-free massive MIMO system,the advantages of SPs over RPs have disappeared and the RPs-based training strategy is preferable.

The remainder of this paper is summarized in the following order.Section II briefly describes the hardware-constrained cell-free massive MIMO system and Section III presents the SPs-aided LMMSE channel estimation.In addition,Section IV derives closedform achievable SE expressions with MRC receiver filters and Section V exhibits simulation results and discussions.At last,Section VI concludes this paper.Table 1 exemplifies the notations that appeared in this paper,shown at the top of this page.

Table 1.Notations.

Table 2.System Parameters.

II.CELL-FREE MASSIVE MIMO SYSTEM

Considered an uplink cell-free large-scale MIMO system,whereLAPs(each possessesNantennas)provide service forKsingle-antenna UEs in the timedivision duplex(TDD)mode with the same time/frequency units[1–3,6].All APs are supposed to be connected to a central server(CS)through the lossless fronthaul networks for collaborations.Besides,we advocate SPs in this work since they can provide remarkable SE gains than RPs in cellular massive MIMO systems[14–18].In the contexts of SPs,all UEs transmit the pilots at a decreased power alongside the data symbols within the entire coherence interval,then the received signal at APl(before hardware impairments are added)can be written as

Next,we will separately explain the meaning of the symbols in(1).

3)ρp=(1−λ)ρandρu=λρrepresent the normalized signal-to-noise ratios(SNRs)of each pilot and data symbols withρ=P/N0andλ∈(0,1)being the power scaling factor which aims to strike a power balance between pilot and data signals.HerePandN0are the total transmit power of UEs and the noise power.

Considering the implementation of high-quality hardware equipments in massive access scenarios poses severe cost and energy consumption issues,lowquality transceivers are advocated in this work.The combined impact of different non-linear hardware imperfections(such as multiplicative phase-drifts,additive distortion noise,noise amplification,and intercarrier interference)can be described by the wellestablished model from[23–28].Specifically,the signal power for the transceiver is weighted by a scalar factorκand the removed signal power is manifested itself in the additive noise.With this insight,Ylin(1)becomes

whereκt∈[0,1]andκr∈[0,1]describe the hardware quality of the transmitter and receiver,respectively.In particular,κt=κr=1 means the perfect transceiver whileκt=κr=0 stands for the useless transceiver.Besides,wk,tpand wk,turefer to the transmitter distortions associated with the pilot sequence and data symbols of UEkand Wl,rdenotes the receiver distortion at APl.On the basis of[23–28],we have

where(6)is the conditional distribution given the set of channel vectors{glk}in a coherence interval.

In this work,we consider the TDD transmission format,and channel estimation is given first.

III.SPS-AIDED LMMSE CHANNEL ESTIMATION

In the context of SPs,the CSI can be estimated over the whole coherence interval of durationτc.The standard LMMSE technique is advocated to acquire highquality CSI,which is used to construct the receiver filters in the uplink decoding phase.Specifically,to estimate glk,∀l,∀k,APlfirst multipleswithto de-spread the received signal,and we have

According to[1],ylkis not a sufficient statistic for estimating glkdue to arbitraryϕk.However,one can still use this term to conduct a sub-optimal channel estimation.With this insight,the SPs-aided LMMSE channel estimation for an arbitrary UE is shown in the following Theorem 1.

Theorem 1.The SPs-aided LMMSE channel estimation in the presence of hardware impairments is given

as

where clk is given in(9).Besides,the channel estimationand the corresponding estimation errorrespectively satisfy

where.

Proof:See Appendix A.

Remark 1.We stress that,ylk is,in general,not Gaussian distributed since the second term of the righthand-side(RHS)ofylk is the inner product of two independent Gaussian random vectors,which suggests thatis uncorrelated with εlk,not independent,which is different from RPs[1–3,6–8,10–13].Besides,the longer the SPs duration,the more accurate channel estimate will be since the possibility of pilotreusing among different UEs is reduced.However,this comes at the price that the interference which arises from the correlation between pilots and data symbolswill deteriorate the qualitiesof the channel estimation and achievable SE,particularly for a larger λ.Moreover,we find that the accuracy of the channel estimate is a decreasing function with respect to κrand κt,which is a natural consequence of non-ideal hardware transceivers and cannot be eliminated.

IV.ACHIEVABLE SE

In this work,the MRC receiver filters are adopted to detect sk,since the inverse operations required by zero-forcing(ZF)and MMSE receiver filters entail a much higher computational complexity,particularly for heavily-loaded cell-free networks.Specifically,APlprojects Ylontoand then delivers the obtainedto the CS through the fronthaul links.After receiving the signal from all APs,the CS sees

In cell-free large-scale MIMO systems,the use-andthen-forget methodology[1–3,6–8],which makes full use of the channel hardening property,is always introduced to derive achievable SE expression when exact CSI is unavailable at the APs.In this context,the expectationis assumed to be known a priori.In addition,since the pilots to a communication system are known at the APs and CS,the pilot signals with estimated channelscan be canceled from,but the residual pilot interference that arises from imperfect CSI still remains in(Hereafter,it is referred to as imperfect pilot removal(IPPR)).Moreover,sinceis correlated with,we isolate the term that contains the desired information and recast the signalwith IPPR as

where

Theorem 2.A closed-form achievable uplink SE expression for UE k with MRC receiver filters and transceiver hardware impairments in the presence of SPs is formulated as

where

Proof:See Appendix B.

The sum-SE of the considered system with SPs and IPPR is thus.

Considering the multiplications,divisions,squares,and logarithms as the dominated computational complexity,the number of arithmetic operations required for obtainingin Theorem 2 is 38+77K+72KL.Besides,Theorem 2 indicates that hardware proportionality coefficientsκr,κt,and coherence intervalτchave effects on the achievable SE.Largerκrandκtalways make contributions to improvingsince the hardware distortion is infinitesimal.Besides,is also an increasing function with respect toτc.On the one hand,forτc

Remark 2.An assumption for deriving Theorem 2 is that the pilot interference with estimated channels is removed from.However,the remaining pilot signal due to imperfect CSI(i.e.,still has neg-ative effects on.If perfect pilot removal(PPR)is considered,the pilot interference no longer makes negative effects on the achievable SE and the resul-tant SEis obtained by substituting Dk=0into(15).in some sense,can be regarded as an upperbound ofIn[29],He et al.stated that iterativedecoding algorithms can be implemented to bridge thegap betweenandbut this comes at the costof high computational complexity.In Section IV,we will numerically show that there is no need to resort to high-complexity iterative decoding algorithms in some certain scenarios.

Remark 3.We stress that the derived closed-form SE in(15)is generalized and covers many special cases.Specifically,a closed-form achievable SE for RPs and MRC receiver filters can be obtained by letting Ck=Dk=0and performing some necessary substitutions,which yields

where

and where τpis the size of additional time slots re-served for RPs.Compared withthere is no pre-log factor insince the data transmission contin-ues throughout the entire coherence interval,thus preserving transmission efficiency.Besides,if we further substitute κr=κt=1into(15)and(22),then the achievable SE expressions under ideal hardware and MRC receiver filters for SPs and RPs are respectively obtained.Moreover,considering the multiplications,divisions,squares,and logarithms as the dominated computational complexity,the number of arithmeticoperations required for obtainingin Remark 3is8+26K+31KL.Although SPs can degrade the transmission reliability and increase the processing complexity at the APs,they have exhibited remarkable merits in enhancing the achievable SE than the RPs,see Section V.

V.NUMERICAL RESULTS

We consider a square area of size 1 km×1 km,in which allLAPs andKUEs are equipped with nonideal hardware transceivers and randomly scattered.All UEs transmit pilots and data symbols with full power,which entailsηk=ξk=1,∀k.The sum-SEs for SPs with PPR and RPs are respectively given as.The training power for RPs is set toP.Unless otherwise specified,Table 2 exemplifies the key parameters enumerated in this paper,which are taken from[1,Table I]and shown at the top of this page.

Figure 1.Uplink sum-SE versus the number of UEs K for different hardware quality factors κr and κt with L=100 and N=2.

Figure 2.Uplink sum-SE versus the hardware quality factor κ for different number of antennas of each AP N with L=100 and K=120.

Figure 3.Uplink sum-SE versus the number of APs L for different hardware quality factors κr and κt with K=120 and N=2.

Figure 2 displays the behaviors of the sum-SEs,andvarying withκrandκtfor differentN.Note that we focus onκr/=κtand utilizeκto refer to the hardware quality factor with variable values.It can be readily observed that all sum-SEs deteriorate with the decreasing ofκ.With the sameκ,the sum-SE degradation that arises from the transmitter distortions is more significant than that of receiver impairments,which indicates that the achievable SEs are dominated by the hardware imperfections at the UEs.This can be explained as the transmitter imperfections appear as disturbers with the same effective channels as the desired signal,which cannot be eliminated by filtering.To approach a better trade-off between achievable SE and hardware costs,it is recommended to implement low-quality hardware at the APs rather than UEs.Another interesting finding from Figure 2 is that the merits of SPs with IPPR over RPs in terms of sum-SE deteriorate asκdecreases and have vanished whenκ≤0.91.This is because the non-ideal transceivers may produce more hardware distortions when simultaneously transmitting and receiving SPs and data symbols.Consequently,RPs are preferable when extremely low-quality transceivers are considered.

Figure 4.Uplink sum-SE versus the power scaling coefficient λ for different hardware quality factors κr and κt with L=100,K=120,and N=2.

Figure 3 describes the sum-SEsandforLranging from 20 to 200 against differentκrandκt.As expected,deploying more APs in cellfree networks always reaps more macro-diversity,thus providing a better sum-SE.Therefore,it is reasonable to increaseLto compensate for the SE degradation that arises from hardware impairments.Once again,we find that for a higherκ(κr=κt≥0.975),SPs with PPR yield the best sum-SE while RPs possess the worst sum-SE as previously discussed.However,this is not the case forκr=κt=0.95,in which the simultaneous transmission of SPs and data signals could accelerate the SE loss since severe transceiver imperfections are involved.This finding further validates the insights that drawn from Figure 2.

In Figure 4,the impacts ofλon the sum-SEsandagainst differentκrandκtare investigated.Note that the sum-SEis also plotted for comparison.Focusing on SPs with PPR,we observe that increasingλhas negative effects onregardless of the values ofκrandκt.The lowerλ,the higher?will approach since the correlation that arises from SPs and data signals is diminished,thus reducing the estimation distortion caused by the data interference.However,different insights can be obtained when we shall focus on IPPR,in which all curves belonging to IPPR are unimodal functions and possess unique optimums.In particular,the optimalλthat achieves the highestincreases with the decreasing of hardware quality and varies in the range of 0.4–0.6,which suggests that for smallerκrandκt,the UEs should allocate more power for data signals to maintain the highest transmission efficiency.

Figure 5.Uplink sum-SE versus the length of coherence interval τc for different hardware quality factors κr and κt with L=100,K=120,and N=2.

Figure 5 displays the sum-SEs,andforτcranging from 60 to 240 against differentκrandκt.As expected,all sum-SEs monotonically grow withτcsince increasingτcbenefits from a more accurate channel estimate for SPs and a higher pre-log factor for RPs.Besides,it is clear that the relative gap betweenandbecomes less pronounced asτcgrows,which means that IPPR can provide comparable sum-SE to that of PPR whenτcis asymptotically large.Note that forκr=κt=0.95,exceedswhenτc≥150.This is because in the context of SPs,the positive effect of suppressing pilot contamination is outweighed by the negative impact of introducing data interferences and hardware distortions under such condition.Therefore,RPs are recommended in the scenarios with low-mobility UEs and low-quality transceivers.

VI.CONCLUSIONS

In this paper,closed-form achievable SE expressions for an uplink cell-free massive MIMO system with SPs and MRC receiver filters in the presence of transceiver hardware imperfections are derived,assuming that the SP interferences can or cannot be completely canceled from the received data symbols.Based on the tractable results,we observe that low-quality transceivers entail significant SE loss and the SE deterioration is primarily dominated by the hardware distortions at the UEs.Besides,SPs are able to enhance the achievable SE than RPs and it is recommended to transmit data symbols with 50%–60%of the total power to highlight the superiorities of SPs when low-quality transceivers are implemented at the APs and UEs.Moreover,RPs are preferable to achieve a higher SE when severe hardware impairments are involved.

APPENDIX A

From[1],the LMMSE channel estimation is defined as

Similarly,the second-order moment of ylkequals

Substituting(27)and(29)into(26)and performing the conjugate transposition operation,(8)is proved.Besides,sinceandεlkare uncorrelated under the LMMSE property,the covariance matrices ofandεlkare calculated as

APPENDIX B

The derivation of(15)needs to calculate all expectations in(14).

The first expectation in the RHS of(32)equals

Substituting(31)and(33)into(32)yields

Then,prior to conducting the detailed derivation for the variance of the rest of the effective noise,we define the following symbols for notational conciseness.

With the above definitions,the variance ofcan be decomposed into

For the first expectation in the RHS of(43),we note that

where in(a)we useand.

For the second expectation in the RHS of(43),we have

1)ComputeJ1:

2)ComputeJ2:By means of(4),we obtain

3)ComputeJ3:By substituting(5)intoJ3,it yields

4)ComputeJ4:

where(b)follows[Wl,r]i=and.

5)ComputeJ5:Since Nland glkare uncorrelated and both have zero means,we obtain

Substituting(44)–(50)into(43)and performing some linear algebra operations,we can find the value of

Next,we focus on the derivation ofThese expectations can be also computed by isolating the term that contains the related part and performing the same methodology in deriving(51).For the sake of convenience,we only give the final results for.

Note that in(56),Ψlkk′is defined in(21).

Until now,the last three terms in(42)have not yet been computed.The square of the expectation ofυik,1 ≤i≤7 can be expressed as

Moreover,the remaining un-calculated terms in(42)can be simplified as

As a result,substituting(31),(34),(42),(51)–(65)into(14)and performing some linear algebra operations,the proof of Theorem 2 is concluded.

ACKNOWLEDGEMENT

This work was supported in part by the National Natural Science Foundation of China under Grants 62071246,61771252,61861039,and 61427801,in part by the National Key Research and Development Program of China under Grants 2020YFB1806608 and 2018YFC1314903,in part by the Jiangsu Province Special Fund Project for Transformation of Scientific and Technological Achievements under Grant BA2019058,in part by the Major Natural Science Research Project of Jiangsu Higher Education Institutions under Grant 18KJA510005,and in part by the Postgraduate Research&Practice Innovation Program of Jiangsu Province under Grants SJKY19_0740 and KYCX20_0709.