Performance Analysis of Uplink Distributed Massive MIMO System with Cross-Layer Design over Rayleigh Fading Channel

，，2*，，，

1.College of Electronic and Information Engineering，Nanjing University of Aeronautics and Astronautics，Nanjing 211106，P.R.China；2.Key Laboratory of Wireless Sensor Network and Communication Shanghai Institute of Microsystem and Information Technology，Chinese Academy of Sciences，Shanghai 200030，P.R.China

（Received 9 April 2021；revised 29 September 2021；accepted 28 October 2021）

Abstract:The performance of uplink distributed massive multiple-input multiple-output（MIMO）systems with crosslayer design（CLD）is investigated over Rayleigh fading channel，which combines the discrete rate adaptive modulation with truncated automatic repeat request.By means of the performance analysis，the closed-form expressions of average packet error rate（APER）and overall average spectral efficiency（ASE）of distributed massive MIMO systems with CLD are derived based on the conditional probability density function of each user’s approximate effective signal-to-noise ratio（SNR）and the switching thresholds under the target packet loss rate（PLR）constraint.With these results，using the approximation of complementary error functions，the approximate APER and overall ASE are also deduced.Simulation results illustrate that the obtained theoretical ASE and APER can match the corresponding simulations well.Besides，the target PLR requirement is satisfied，and the distributed massive MIMO systems offer an obvious performance gain over the co-located massive MIMO systems.

Key words：distributed massive multiple-input multiple-output（MIMO）system；zero-forcing detection；cross-layer design；spectral efficiency；quality of service；packet error rate

0 Introduction

As the age of the fifth generation（5G）communication is coming，distributed massive multiple-input multiple-output（DM-MIMO）technology is being paid more and more attention［1］.Different from co-located massive multiple-input multiple-output（CM-MIMO）systems，DM-MIMO systems can improve the quality of service（QoS）by reducing the distance between the user and the access points（APs），and distributed APs can increase the coverage and bring sufficient diversity［2］.

In DM-MIMO，the distributed antenna system（DAS）is a reliable technology that maximizes cellular coverage in cellular networks［3-5］.DAS has gained lots of interest as a promising technology to satisfy rapidly growing demands for next generation wireless systems，because it is energy-saving and achieves more capacity over conventional co-located antenna systems（CASs）［6-8］.In DAS，a large number of single-antenna distributed APs are distributed in the cell，and all APs are connected to the central processing unit（CPU）through the backhaul link，where the received signal is processed.In addition to the above architecture，DM-MIMO benefits from channel hardening，which plays a vital role in the design of CM-MIMO.However，different from CMMIMO，the distributed structure allows DM-MIMO to obtain larger gain relying on macro diversity.In addition，since the APs are relatively simple and cheap，the deployment of DM-MIMO is comparatively flexible，and applicable in various scenarios［9-11］.

In the existing research，most papers focus on performance analysis and optimization on the physical layer of DM-MIMO systems.The uplink spectral efficiency（SE）of DM-MIMO systems was analyzed considering pilot pollution and channel aging in Ref.［12］.In Ref.［13］，the SE of DM-MIMO downlink systems with hardware and channel damage was studied，and a power allocation algorithm was proposed to maximize the minimum user rate.A random pilot multiplexing scheme was proposed in Ref.［14］，and the system SE expression was given.In addition，a method of sparse matrix was proposed in Ref.［14］to simplify signal processing and analyze the bit error ratio（BER）.

To improve the SE and throughput of the system and satisfy the given QoS requirements at the same time，cross-layer design（CLD）has received much attention［15-17］.Usually，CLD combines adaptive modulation（AM）at the physical layer to enhance the throughput and truncated automatic retransmission request（ARQ）protocol at the data link layer to ensure the reliability of the system.Ref.［18］analyzed the performance of the co-located MIMO systems using CLD scheme under the Rayleigh channel.Furthermore，the CLD scheme of MIMO system with feedback delay over Nakagamim fading channel was presented in Ref.［19］.In Ref.［20］，the performance analysis of CLD scheme of multiuser MIMO system with antenna selection was presented.The effectiveness of CLD scheme in CM-MIMO systems over composite Rayleigh fading channel was verified in Ref.［21］.In addition，Ref.［22］investigated the effect that fading correlation had on the performance of CLD scheme in the MIMO system on Nakgami-m fading channels.In Ref.［23］，a CLD scheme which considered packet combining was presented，where MIMO Nakagami fading channels were assumed，and retransmitted packets were not necessarily modulated using the same modulation format as in the initial transmission.

However，there are few studies on the performance of DM-MIMO systems，especially those with CLD schemes.Performance analysis of DMMIMO is a challenging problem compared with CMMIMO，which can be considered as a special case of DM-MIMO.This is because channels between a user and the distributed APs are characterized by different path-loss and shadowing fading，which makes the regular analytical methods inapplicable.

Motivated by the analysis above，this paper will study the cross-layer design of uplink DM-MIMO systems over Rayleigh fading channel.First of all，based on the first item of the accurate BER formulas，approximate packet error rate（PER）formulas are given.Then the switching thresholds in physical layer are introduced based on the approximate formulas under the instantaneous packet loss rate（PLR）constraint.Furthermore，based on the conditional probability density function（PDF）of each user’s approximate effective signal-to-noise ratio（SNR），the closed expressions of overall average SE and average PER of cross-layer design scheme in DM-MIMO system are inferred with the thresholds，which provides theoretical basis for the evaluation of its performance.Finally，with the approximation of complementary error functions，the approximate average PER and overall average SE are deduced to simplify the expressions.

1 Signal Model

As shown in Fig.1，we consider uplink DMMIMO systems，in whichLAPs andKusers are randomly distributed in the cell，andL?K.The users are equipped with a single antenna.All APs are connected to the CPU via the backhaul network，assuming that the backhaul link is perfect.

Fig.1 DM-MIMO system model

Suppose that the APs have perfect channel state information（CSI）.The channel model contains small-scale fading that obeys Rayleigh fading and large-scale fading that includes path loss.The channel fading coefficient between thelth AP and thekth user is recorded as

wherehlkrepresents the small-scale fading between thelth AP and thekth user，andhlkobeys the independent standard Gaussian distribution.βlk=(dlk/d0)－vrepresents the large-scale fading，wheredlkrepresents the distance between thelth AP and thekth user，d0the minimum distance between all users and all APs，andvthe path loss exponent.

The signal received by thelth AP is

All APs transmit their received signals to the CPU over the backhaul，and the signal matrix received by the CPU is

wherey=[y1，…，yL]T，andx=[x1，…，xK]Trepresents the signal vector sent by all users.G=[g1，…，gK]represents the channel between the APs and the users，wheregk=[g1k，…，gLk]Trepresents the channel between thekth user and the APs，andglk～CN(0，βlk).n=[n1，…，nL]Tis the additive Gaussian noise at APs.

After linear detection with linear detector matrixA，the received signal at the CPU becomes

In order to eliminate multi-user interference，zero-forcing（ZF）detection is used，i.e.，A=G(GHG)－1.So the received signal can be expressed as

Then thekth user’s signal is

Thus the SNR for thekth user is

The SNR in Eq.（7）can be converted as follows

In massive MIMO systems，there are a large number of antennas，so.Hence，we can get the approximate SNR for thekth user［24］

whereulkis thelth element of the vectoruk.Because ulkobeys the standard complex Gaussian distribution，obeys the exponential distribution.From the characteristic of the exponential distribution，the approximate SNR can be seen as the sum of exponential variables with different parameters.

2 Performance Analysis of Cross-Layer Design in DM-MIMO System

In this section，the performance of CLD scheme for DM-MIMO systems is studied，which combines adaptiveMn-ary quadrature amplitude modulation（QAM）at the physical layer and truncated ARQ at the data link layer.At the physical layer，the receiver estimates the channel to obtain CSI and sends it to the transmitter through the feedback channel.The transmitter chooses the best modulation mode for the next transmission based on the received CSI.At the data link layer，when the receiver receives the data，if the packet is wrong，a negative acknowledgement is sent back to request retransmission.Otherwise，an acknowledgement is sent.The packet which fails after the maximum number of retransmissionsNmaxr，will be abandoned.

2.1 Switching thresholds

In the cross-layer design scheme of DM-MIMO systems，Ntypes of QAM modulation modes with different constellation sizes are selectable in the physical layer，and the stop-and-wait ARQ protocol is used in the data link layer，and the maximum number of retransmissions is set asNmaxr.Given the system target packet loss rate PLR0，the target PER is

Assuming that each bit in the packet is not related with others，the instantaneous PER for thenth mode of modulation is

whereNbis the number of bits in each packet and BERn(χ) the BER of thenth mode of modulation.

The accurate BER formula consists of a number of complementary error functions.Furthermore，the BER formulas of 2QAM and 4QAM have only one item，and the first item of the higher-order modulation accounts for the most.Therefore，selecting the first complementary error function of the exact BER，the approximate PER can be expressed as

Considering instantaneous PER constraints，i.e.，PERn(χ)=PER0，the switching thresholds for the system are

2.2 Performance analysis

With PDF of thekth user’s SNR given in Eq.（11），the average SE at the physical layer of thekth user based on the switching thresholds in Eq.（15）is

whereRnis the rate corresponding to the modulation mode whose size of constellation graph isMn，andRn=log2Mn.

The average PER of thekth user in a DM-MIMO system with CLD scheme is defined as

The second item in the brace in Eq.（18）is recorded as，which is too complex to be derived as a closed expression.Hence，numerical integration is used.

whereQis the order of the numerical integral，xq=.Substituting Eq.（19）into Eq.（18），we can obtain

Therefore，substituting Eqs.（16，20）into Eq.（17），the average PER of thekth user is

The average PER of the system is expressed as

When a packet still fails after＋1 times transmissions，the packet is discarded，so the average number of transmission for the packet of thekth user is

Combined with the average SE of the physical layer in Eq.（16），the overall average SE in DM-MIMO systems with CLD scheme is presented at the top of next page.

Since the numerical integration in Eq.（19）is complex，we adopt the following approximation of complementary error functions to simplify performance analysis.

Substituting Eq.（25）intoΞik，n，we can infer that the approximate expression ofΞik，nis

As a result，the approximate system average PER is

Substituting Eq.（28）into Eq.（23），the corresponding average number of packet transmissions for thekth user is

Based on Eqs.（16，29），the approximate overall average SE in DM-MIMO systems with crosslayer design is given as follows

3 Simulation Results and Analysis

We will analyze and evaluate the performance of the CLD scheme in DM-MIMO systems，and compare the theory and simulation results of the average SE and average PER of the system to verify the validity of the theoretical formulas.We assume that the number of bits contained in each packet isNb=1 080，and the sizes of QAM with graymapped constellation are{2，4，8，16，32，64}.Unless otherwise stated，the target packet loss rate of the system is PLR0=10－3，the cell radius isR=1 000 m，the reference distance isd0=100 m，the number of randomly distributed APs isL=128，the number of users isK=5，and the path loss exponent isv=4.0.

In Fig.2，we plot the overall average spectral efficiency（ASE）of CLD scheme in DM-MIMO systems under different maximum numbers of retransmissions.As shown in Fig.2，the ASE of the system under the same SNR increases asincreases.The reason is that with the increase of，the probability of a packet discarded becomes lower，so more packets are successfully transmitted and the ASE of the system increases accordingly.Moreover，it should be noted that the ASE of=1，2，3 is higher than that of=0，and=0 means that the CLD scheme is not used.It proves that the CLD scheme improves the system performance.In addition，the approximate results are close to the accurate results.This illustrates the validity of the approximate formula（30）.Hence，the approximate formulas can be used to effectively evaluate the system.

Fig.2 ASE of CLD scheme in DM-MIMO systems under different

Fig.3 shows the average packet error rate（APER）performance of the DM-MIMO systems when=0，1，2，3，respectively.We can find that the APER performance of the system deteriorates with theincreases.This is because that，under given target PLR，the target PER in Eq.（12）will increase with the increase of，namely，the limit on the system PER is gradually relaxed.In addition，combined with Fig.2，it is found that the ASE of=3 increases little than that of=2，but the APER performance of=3 is worse.Consequently，it does not hold that largerleads to better system performance，which also verifies that packets cannot be retransmitted indefinitely.At last，the approximate performance is close to the accurate performance.As the result，the approximate analysis can be used to effectively evaluate the system when the system has strict requirements for complexity.

Fig.3 APER of CLD scheme in DM-MIMO systems under different

Figs.4，5 compare the ASE and APER of CMMIMO and DM-MIMO systems when=2 and=0，respectively.First of all，it can be found from the figures that the DM-MIMO system can obtain higher ASE and lower APER.This is because that the APs of the DM-MIMO system are dispersed in the cell to shorten the distance between the receiver and the user，thereby reducing the signal fading.Consequently，a higher-order modulation can be selected，so the ASE becomes higher.In CM-MIMO systems，the distance between the base station and users at the edge of cell is long，so the system performance is limited，namely，the ASE is lower.In summary，Figs.4，5 illustrate the superiority of the DM-MIMO system.In addition，the system ASE of=2 is higher than that of=0，which is consistent with the previous analysis.Finally，we can observe that the theoretical curves and the simulation curves are basically consistent，which shows the effectiveness of the performance analysis.

Fig.5 Comparison of APER between CM-MIMO and DMMIMO systems under different

4 Conclusions

By combining discrete-rate AM with truncated ARQ protocol，the performance of uplink DM-MIMO systems with CLD scheme is investigated over Rayleigh fading channels.The APER and overall ASE of the system are analyzed.Firstly，using the ZF detection，we derive effective SNR and the corresponding conditional PDF.Secondly，under the target PLR constraint，the adaptive switching thresholds for CLD are obtained based on the approximate expression of PER.According to these results，the closed-form ASE at the physical layer is attained.Then with the help of the numerical integration，we deduce the theoretical APER and overall ASE，respectively，and the closed-form expressions are achieved.Finally，with the approximation of complementary error functions，the approximate APER and overall ASE are derived.It can be seen from the simulation results that the theoretical ASE and APER can match the corresponding simulation results well，proving the effectiveness of the theoretical analysis.Thus，the performance of DM-MIMO with CLD scheme can be assessed well.Besides，the impact of the number of the maximum retransmissions on system performance is also analyzed.The ASE performance can be improved through increasing the maximum retransmission number，but the improvement will become less for larger number of the maximum retransmission.Besides，the DMMIMO can obtain superior performance over the CM-MIMO according to Figs.4，5.