Zheqi Gu＊, Zhongpei Zhang

1 Chengdu Research Institute, HUAWEI Technology Co., Ltd, Chengdu 611731, China

2 National Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China,Chengdu 611731, China

Abstract: This paper proposes a mode selection scheme to improve the spectral efficiency for coordinated multi-point (CoMP) transmission with phase synchronization errors (PSE).Upper bounds of average achievable rate for different CoMP transmission modes, such as coordinated beamforming (CB) and joint processing (JP), are derived by random matrix theory and asymptotic mathematical approximation. According to these upper bounds, the proposed scheme switches CoMP transmission mode between CB and JP adaptively to enhance the average achievable rate. Simulation results show that these upper bounds agree well with the average achievable rates for both JP and CB, and the proposed scheme outperforms traditional single mode CoMP transmission when PSE exist.

Keywords: CoMP transmission; synchronization errors; precoding; mode selection

I. INTRODUCTION

CoMP transmission is a potential solution to improve the cell average throughput and the cell edge throughput [1-3]. Despite the appealing features of CoMP transmission, its application in practical systems brings technical challenges. The basic one is that base stations (BSs) and user equipments (UEs) must keep timing and frequency synchronization in CoMP transmission.

We find that when orthogonal frequency division multiplexing (OFDM) is used, the tiny timing misalignments and frequency offsets between BSs and UEs can cause significant performance degradation of CoMP transmission [4]. Since OFDM is widely adopted in current wireless standards (e.g. IEEE 802.11a/g/n and LTE), it is also assumed throughout this paper. In CoMP transmission, BSs and UEs are located at different sites. Due to different propagation delays between BSs and UEs, the multiple timing offsets (TO) are unavoidable. Furthermore, their carrier frequency and sampling frequency are usually driven by their own local oscillators. The carrier frequency offsets (CFO) and sampling frequency offsets (SFO) between BSs and UEs are also investable. In [5], the quasi timing synchronization is achieved, which is also called as slot alignment. The slot alignment makes sure that the transmitted OFDM symbols from different BSs align within the cyclic prefix (CP) of each UE. In [6] and [7], the problem of frequency synchronization in CoMP transmission has been investigated, where the frequency synchronization algorithms have also been developed to estimate and compensate the multiple SFO and CFO.

After timing and frequency synchronization mentioned in [5-7], only tiny residual TO, SFO and CFO exist between BSs and UEs in CoMP transmission. At that time, the inter symbol interferences (ISI) and inter carrier interferences(ICI) can be ignored. We focus on the phase rotations caused by the residual TO, SFO and CFO. These phase rotations, which are called as PSE in this paper, can destroy the interference canceling effect of CoMP transmission.

In order to take advantage of CoMP transmission, a master-slave synchronization scheme is proposed to enable BSs to maintain phase coherence [8]. In [9], a hierarchical synchronization scheme is proposed to compensate PSE at BSs. Both of the master BS and the anchor BSs in [8-9] have to periodically broadcast extra orthogonal synchronization pilot sequences for PSE estimation. Obviously,it is quite a bit of overhead.

In fact, not all of the CoMP transmission modes are affected by PSE. Synchronization schemes in [8-9] only operate at BSs, do not take advantage of the fact that UEs can estimate PSE through downlink pilot signals. In this paper, we don’t compensate PSE directly at BSs, but mitigate the impact of PSE on CoMP transmission in a new way. First, we derive the upper bounds of the average achievable rate for both CB and JP when PSE exist.And then, we find that CB is immune to PSE while JP is sensitive to PSE. Finally, based on these upper bounds, we design a mode selection scheme, which enable UEs to select the CoMP transmission mode between JP and CB to enhance the average achievable rate. Simulation results show that the proposed scheme outperforms traditional single mode CoMP transmission when PSE exist.

Notation: the boldface uppercase and lowercase letters represent matrices and vectors,respectively. (.)?1denotes the matrix inversion, (.)Hdenotes the hermitian transposition,diag(.) converts a vector into a diagonal matrix, E{.} denotes the expectation. ? denotes the convolution. * denotes the kronecker product. INis the N×N identity matrix.

The authors propose a mode selection scheme, which enables CoMP transmission to switch between JP and CB to enhance the average achievable rate.

II. A SIMPLE CHANNEL MODEL FOR OFDM WITH SYNCHRONIZATION ERRORS

Assume fS, fCare the nominal sampling and carrier frequencies for all BSs and UEs in CoMP transmission. As shown in figure 1, the actual sampling and carrier frequencies of BS b and UE i are indicated by fS.b, fC.b, fS.iand fC.irespectively. Since BS b and UE i are driven by their own local oscillator, fS.b, fC.b, fS.iand fC.idiffer from fSand fCby SFO and CFO. As shown in figure 1, the local time axis of BS b and UE i have TO dband diwith respect to the nominal time axis. ?dib= db?didenotes the timing misalignment between BS b and UE i.

Fig. 1. Synchronization errors between BS b and UE i.

Since slot alignment has already been achieved in [5] and [11], the timing misalignment between BS b and UE i is much less than the duration of CP. So ?dibmanifests itself as a phase rotation in frequency domain without causing any ISI and ICI. When the CFO between BS b and UE i is much smaller than the subcarrier separation, the ICI caused by CFO can be neglected [9]. Furthermore, it has been proved that the ICI caused by CFO is much smaller than the inter-user interference(IUI) caused by CFO in CoMP transmission[12]. Since TS.b≠ TS.i≠ TS, where TS.b=1/ fS.b,TS.i=1/ fS.i, TS=1/ fS, the SFO between BS b and UE i produces a contraction or dilation of the time axis. If it accumulates over enough OFDM symbols, it will cause not only phase rotations but also ISI. In the practical system,the time axis drift has to be reset at each radio frame by timing synchronization [9]. So the ISI caused by SFO is also negligible.

Through above analysis, a simple approximation of the effective downlink channel between UE i and BS b is given by

where hib[n,k] denotes downlink channel in the air at nth OFDM symbol and kth subcarrier. The combined effects of the TO, CFO and SFO between BS UE i and BS b are captured by the multiplicative phase rotation eib[n,k ]

Where k ∈ {0,...,N ?1}, N represents the number of subcarriers, μb=db/TS,μi=di/TS, εb= (N + L)TS(fS.b?fS) ,εi= (N + L)TS(fS.i?fS), L represents the length of CP. In the practical system, the carrier and sampling frequency are derived from the same local oscillator. Assume fC.b/ fS.b=κ and fC.i/ fS.i=κ. So ?b=κεband ?i=κεi. According to (1/ N )sin(N x) /sin(x)≈1, the last term in Eq.(2) can be omitted.

The similar simple models have been proposed in [8], [9], [11] and [12]. It also has been validated by the CoMP transmission experimental platform [4] [8] [13]. According to Eq.(2), the effective downlink channel between UE i and BS b can be reworded as and

Since ?i[n,k] left multiplies hib[n,k],?i[n,k] can be recovered individually by UE i in standard coherent communication [14]. So we ignore ?i[n,k], only consider ?b[n,k] in this paper,[n,k] can be simplified as

III. AVERAGE ACHIEVABLE RATE ANALYSIS

We consider a CoMP transmission scenario in figure 2, which is consisting of (b=1,...,NBS)BSs equipped with ntantennas each, serving (i=1,...,NUE) single-antenna UEs in the same time-frequency resources. Because of OFDM, the equivalent downlink channel between BS b and UE i is described by a set of subcarriers[n,k], where n denotes the nth

According to Eq.(3), the effective global downlink channel of UE i can be given by denotes the downlink channel of UE i in the air,Φ= diag{φ1,...,φb,...,φNBS}* Int. The definition of φbhave been given in section II. The diagonal matrix Φ denotes PSE, which are caused by the residual TO, SFO and CFO of BSs. Since Φ right multiplies hDiL, Φ cannot be eliminated at UE i.

The CoMP transmission downlink channel capacity can be computed as,

PNis the noise power and PIis the interference power caused by PSE. According to Eq.(4),we have

It can be proved that

Depending upon the way of BSs cooperation, the CoMP transmission roughly fall into two modes called CB and JP [15-16]. Both CB and JP require the downlink channel state information (CSI). As shown in figure 2, BSs share downlink CSI to the central server (CS)via wired backbone, the CS figures out the beamforming or precoding vectors for CoMP transmission. In next sections, we derive the upper bounds of average achievable rate for CB and JP when PSE exist.

3.1 Upper bound of average achievable rate for CB

For CB, coordinated BSs only need to share CSI. The beams, which are formed by each BS, not only increase signal strength towards the UEs in its own cell, but also reduce interference towards the UEs in its adjacent cells. Assume each BS only serve one UE and the number of antenna ntis large enough to provide degrees of freedom (nt＞ NUE), the received signal at UE i is given by

Fig. 2. CoMP transmission scenario.

has already been obtained by BS i. Based on the zero forcing (ZF) criterion, the precoding matrix for BS i isThe CB beamforming vector for UE i can be expressed aswhere(:,i) denotes the ith column of

For CB, the average achievable rate of UE i is given by

where step (a) is obtained by Jensen's inequality.

According to Eq.(9), the signal to interference plus noise ratio (SINR) of UE i can be expressed as

where γ=p/σ2denotes the signal to noise ratio (SNR), step (a) comes bystep (b) uses these facts thatBecausethe SINR of UE i can be expressed as

According to the random matrix theory,is a complex inverse Wishart matrix. The expectation of

Considering Rayleigh fading channel here,each entry of a channel matrix Hiis independent and identically distributed (i.i.d), also follow a complex Gaussian distribution with zero mean and unit variance. According to a property of Wishart matrices, using the Lemma 2.10 in [17], we can obtain E{1/λii} =(nt? NUE)?1.A similar mathematical result can be also found in [18]. The average SINR of UE i for CB is

According to Eq.(13), we find that PSE have no impact on the performance of CB.

3.2 Upper bound of average achievable rate for JP

For JP, coordinated BSs not only need to share downlink CSI, but also need to share the data for each UE. The cooperative BSs, which behave like a large virtual BS, form beams towards all UEs in their coverage area and turn interference into useful signals. The received signal at UE i is

Assume

In order to derive the upper bound of average achievable rate for JP, the JP beamforming vector for UE i is rewritten asis an extended channel matrix that excludes hi. Qiis a projection matrix, which is used to project vectors into the null space of Gi.

For JP, the average achievable rate upper bound of UE i is also given by

According to Eq.(14), the SINR of UE i can be expressed as

where step (a) is obtained by putting Eq.(14)into the expression of. Step (b) comes by hiΦ = hi+ei, where ei=hi(Φ ? INBSnt), and if i≠j,=0.In order to obtain an approximation of E{SINRiJP}, we assume the number of antenna at each cooperative BS tends to be infinity(nt→∞), and begin with two useful lemmas.

Lemma (1)

Proof. According to the definition of eiand hj, we rewrite eDue to the strong law of large number, we can get

where hib(k) and hjb(k) denote the kth entry of hiband hjb(the value of k can be chosen arbitrarily). Considering Rayleigh fading channel here, if i≠j,

Thus, we get the result in Eq.(17b). The proof of Eq.(17a) and Eq.(17c) are similar to the proof of Eq.(17b).

Lemma (2)

Proof. According to definition of Qi, we can obtiansimilar manner as proving lemma (1), it is not hard to get

According to the definition of Gi, we can ob-hus, we get the result in lemma (2).

Based on the lemma (1) and (2), we can obtain the proposition:

Proposition 1. As nt→∞, the average value of SINRiJPcan be approximated as

Proof. See Appendix I.

According to proposition 1,if PSE do not exist (Φ=INBSnt),= γ(NBSnt?NUE). Compared to=γ( nt? NUE) in Eq.(13), JP performs better than CB when synchronization is ideal. If PSE exist, proposition 1 also reveals that E{} increase with ntand Mφ, but decreases with NBSand NUE. Since Mφ?PSE degrade the performance of JP significantly. Because hiand hjare Rayleigh fading channel, the instantaneous values of hiand Ehave no impact on E{} in proposition 1.

3.3 Numerical validation

In this section, we evaluate and verify the upper bounds of average achievable rate for CB and JP by simulations.

The simulation parameters are specified in accordance with the typical 3GPP LTE standard [19]. The number of subcarriers is N=2048. The interval between adjacent subcarriers is η=15kHz. The length of CP is L=144. The nominal sampling and carrier frequencies are set as fS=30.72 MHz ,fC=2.65G Hz, κ=fC/fS≈86.

A CoMP transmission scenario is consid-ered, where NBS=3 BSs equipped with nt=4 antennas each, serve NUE=3 UEs cooperatively on the same time and frequency resource.The downlink channels from BSs to UEs in the air are all set as Rayleigh fading channels.

According to the LTE standard requirements, the TO μbis assumed to be uniformly distributed over [0,L/2].The residual SFO and CFO of BSs are specified in parts per million (ppm). The SFO ( fS.b? fS) is assumed to be uniformly distributed over[?1 .5 Hz, 1.5 Hz] (about 0.05 ppm). The CFO( fC.b? fC) =κ( fS.b? fS). Sincethe TO, SFO and CFO of BSs change slowlyin each radio frame (10ms), they are considered static with respect to the Rayleigh fading channels In simulations, the residual TO, SFO and CFO are randomly assigned to BSsin each radio frame.

Fig. 3. Average achievable rate for CB and JP over SNR.

Figure 3 indicates that the average achievable rate upper bounds for CB and JP agree well with the results of Monte Carlo simulation(NBS=3, nt=4, NUE=3, N=2048, k=10, n=15,TO=[38.6989 50.9893 31.2096], SFO=[0.4391 1.2435 -0.5522], CFO=κSFO). The numerical result in figure 3 also infers that the asymptotic mathematical approximation in proposition 1 is still valid for the realistic BS antenna configuration. If there is no PSE, the average achievable rates for CB and JP grow linearly with SNR.We find that CB with PSE and without PSE has the same average achievable rate in figure 3. It verifies that PSE have no impact on the performance of CB. However, we also find that the average achievable rates for JP is greatly reduced by PSE in figure 3.

According to the definition of ?b, ?bis a function of subcarrier index k. Sincethe average achievable rate for JP is also a function of subcarrier index k.Essentially, different subcarrier index k corresponds to different subcarrier frequency. So the average achievable rate for JP is sensitive to the subcarrier frequency when PSE exist.As shown in figure 4(NBS=3, nt=4, NUE=3,N=2048, SNR=10dB, n=15, TO=[38.6989 50.9893 31.2096], SFO=[0.4391 1.2435-0.5522], CFO=κSFO), when k＞40, the average achievable rate for JP is lower than the average achievable rate for CB. It implies that JP suffers from unequal PSE on different subcarriers. Since PSE have no impact on the performance of CB, the average achievable rate for CB keeps constant over k in figure 4.

Figure 5 illustrates the average achievable rate for JP as a function of OFDM symbol index n(NBS=3, nt=4, NUE=3, N=2048,SNR=10dB, k=10, TO=[38.6989 50.9893 31.2096], SFO=[0.4391 1.2435 -0.5522],CFO=κSFO). The length of one OFDM symbol is (144 + 2048)/ fS≈72μs .Obviously, the relationship between the average achievable rate for JP and n is not linear in the time domain. As shown in figure 5, when n＞50, the average achievable rate for JP is lower than the average achievable rate for CB. Meanwhile, the average achievable rate for CB still keeps constant over n in figure 5. Although the upper bounds of the average achievable rate for JP and CB are not tight in figure 3, figure 4 and figure 5, they can reveal the performance difference between JP and CB.

IV. MODE SELECTION FOR COMP TRANSMISSION WITH PSE

Because JP can take full advantage of spatial dimension between BSs and UEs, theoretically it is more spectrum efficient than CB. However, CB is immune to PSE, JP may not always be superior to CB when PSE exist.

Meanwhile, PSE can be estimated at UEs by utilizing the downlink pilot signals, such as cell-specific reference signals (CRS), which have already been specified in LTE standard[19]. Regretfully, the knowledge of PSE at UEs has not been exploited in [8] and [9].

Based on these facts, we propose a mode selection scheme to mitigate the performance degradation of CoMP transmission caused by PSE.

4.1 PSE estimation at UEs

The effective downlink channelcan be easily obtained at UE i by standard pilot-assisted channel estimation [14]. According to Eq.(4),. Within the channel coherent bandwidth and the channel coherent time, the phases of hibremain unchanged. The phase rotation ofonly depends on φb. Through the phase rotation of, the phase slope of φbcan be acquired by least square (LS) estimation.After that, the value of φbon every subcarrier and OFDM symbol can be calculated.

For example, the frequency interval between subcarrier k=k1and k=k2does not exceed the channel coherent bandwidth. Fix the OFDM symbol indexThe relationship between

where ./ denotes the vector scalar division,which means each element of the dividend vector is divided by the corresponding element of the divisor vector, ∠(.)denotes the phase of each element in the complex vec-, and?k12=(k1?k2)× [1 1 ...1]T∈Rnt×1. The LS estimation of (μb?n1εb) is

Fig. 4. Average achievable rate for CB and JP over subcarrier k.

Fig. 5. Average achievable rate for CB and JP over OFDM symbol n.

Similarly, the time interval between OFDM symbol n=n1and n=n2does not exceed the channel coherent time. Fix the subcarrier index k== hib[n2,k1]. The relationship between[n1,k1] and[n2,k1] is

Since UE i already knows the value of N,n1and k1in Eq.(22) and Eq.(24) and ?b=κεb,UE i can derive μb, εband ?bthrough the estimation of (μb?n1εb) and (k1εb+ N ?b). So UE i can calculate ?bfor every subcarrier and OFDM symbol. By analogy, the rest of PSE Φ are also available at UE i. Though UE i can feed μb, εband ?bback to BSs for PSE correction, how to design the feedback codebook is beyond the scope of this paper. Meanwhile, it suffers from the feedback delay and quantization error. So we propose a mode selection scheme, in which UEs only have to feed the mode selection results back to BSs.

4.2 Mode selection scheme

Recently, a lot of mode selection schemes have been proposed for different downlink transmission scenarios [20-23]. However,none of mode selection schemes have taken the synchronization errors of BSs into account.In other words, these mode selection schemes are not suitable for CoMP transmission with PSE.

Assume UE i has perfect knowledge of PSE Φ. According to Eq.(10) and Eq.(15),UE i selects JP when. Conversely,

From Eq.(25), we find that the value of the decision threshold is just related to the system parameters, and has nothing to do with the instantaneous channel fading coefficient of UEs.We also find that the decision threshold increases with NBSand γ. It implies that the more BSs are cooperated, the more accurate time and frequency synchronization is required. It also implies that the higher power is used for CoMP transmission, the greater performance loss of JP is caused by PSE.

According to the definition of ?b, the decision variable Mφis a function of the residual TO, SFO and CFO of BSs. As shown in figure 6(NBS=3, nt=4, NUE=3, N=2048, SNR=10dB, TO=[38.6989 50.9893 31.2096],SFO=[0.4391 1.2435 -0.5522], CFO=κSFO),if μb, εband ?b(b=1,2...NBS) stay constant, the value of Mφchanges with the subcarrier index k and the OFDM symbol index n. It indicates that different CoMP transmission mode should be applied to different subcarriers or OFDM symbols. Since the pilot-aided PSE estimator suffers estimation errors, the UE, which has the best channel quality, is chosen to select transmission mode and feedback the result to BSs. The channel quality of a UE can be judged by its channel quality indicator (COI)in LTE [19].

According to above analysis, we propose the mode selection scheme for CoMP Transmission with PSE. Firstly, BSs choose a UE,whose COI is largest. If the largest CQI corresponds to more than one candidate UEs, BSs just choose the UE randomly from the candidate UEs. Then, the chosen UE estimates PSE and calculates the decision variable Mφfor every subcarrier and OFDM symbol. Assume the chosen UE already knows all the system parameters, it also can calculate the decision threshold. According to the Eq.(25), the chosen UE selects downlink transmission mode(JP or CB) for every subcarrier and OFDM symbol. Finally, the chosen UE feeds the mode selection results back to BSs. After receiving the feedback from the chosen UE, BSs carry out the downlink transmission according to the selected mode. Since PSE are time-varying,the mode selection should operate periodically. In algorithm 1, we give a brief description of the proposed mode selection scheme.

Since the decision variable Mφis a continuous function of the subcarrier index k and the OFDM symbol index n, the chosen UE can just feedback the mode switch point and the mode selection indicator to reduce the feedback redundancy. As shown in figure 7(NBS=3, nt=4, NUE=3, N=2048, SNR=10dB, TO=[38.6989 50.9893 31.2096],SFO=[0.4391 1.2435 -0.5522], CFO=κSFO),when SNR γ=10dB, there are two mode switch points (k=10and k=220) for the OFDM symbol n=50. If 10 ≤k＜ 220, JP is selected. If k＜10 or k≥220, CB is selected.

4.3 Numerical results

In this section, we verify our analysis and evaluate the performance of the proposed mode selection scheme by numerical results.

For simplicity, simulation conditions maintain unchanged, which have been used in section 3.3. The value of TO μb, SFO ( fS.b?fS)and CFO ( fC.b? fC) are traversed in this section, which are randomly chosen only once and keep constant in section 3.3. So we can't fix the value of TO, SFO and CFO in figure 8, figure 9. The mode selection interval is assumed as the length of half radio frame.

Fig. 6. The decision variable for subcarriers and OFDM symbols

Algorithm 1. Proposed mode selection scheme.Operation procedure Operation procedure at the chosen UE 1) For b=1:NBS estimate the PSE φb;End 2) calculate the decision variable Mφ;3) calculate the decision threshold;4) select the transmission mode based on the decision criterion in Eq.(25);5) feed the mode selection results back to BSs.Operation procedure at BSs 1) choose a UE by CQI 2) carry out the downlink transmission according to the mode selection feedback.

Figure 8 shows the number of time-frequency resource element selected for JP and CB in one radio frame for different CoMP transmission scenarios. A time-frequency resource element is the smallest unit for resource scheduling [19]. We can find that the number of time-frequency resource element selected for JP decreases sharply with the number of cooperative BSs and the SNR. This simulation result confirms our analysis on the proposed scheme in section 4.2.

Since the decision variable and the decision threshold have nothing to do with UEs,each UE is equivalent for the mode selection scheme. And the UE’s downlink channel is i.i.d here. So each UE has the same average achievable rate. We just show the average achievable rate of UE 1 in figure 9.

Fig. 7. The JP region and the CB region for subcarriers and OFDM symbols.

Fig. 8. The number of time-frequency resource element for JP and CB in one radio frame.

As shown in figure 9, the proposed mode selection scheme predicts the switch point,and adaptively switches the CoMP transmission mode between CB and JP. In figure 9(a), the switch point is about γ=15dB and γ=12dB on the subcarrier k=10 and k=15, OFDM symbol n=15. The switch point changes with the subcarrier k. In figure 9(b), the switch point is about γ=17dB and γ=15dB on the subcarrier k=10, OFDM symbol n=10 and n=15. The switch point also changes with the OFDM symbol n. In any case,the proposed mode selection scheme outperforms the traditional single mode JP and CB.

The robust JP mode algorithm has been proposed in reference [9] to alleviate the performance degradation caused by PSE. It can achieve about 2dB achievable rate gain when SNR is low. However, it will be almost invalid when SNR is high.

Since CB is immune to PSE, Our proposed scheme selects CB to resist PSE when SNR is high. Conversely, the proposed scheme selects JP to take advantage of the spatial dimensions when SNR is low. As shown in figure 9, Our proposed scheme achieves performance gain compared to the robust JP mode algorithm proposed in reference [9] when SNR is larger than 16dB on the subcarrier k=10 and k=15,OFDM symbol n=10 and n=15.

V. CONCLUSION

It is really difficult to achieve ideal timing and frequency synchronization between BSs in CoMP transmission. So the residual TO, SFO and CFO of cooperative BSs give rise to PSE.

In this paper, we derive the upper bounds of the average achievable rate for both CB and JP when PSE exist. Based on these upper bounds,we find that PSE degrade the performance of JP significantly while PSE have no influence on the performance of CB. Therefore, we propose a mode selection scheme, which enable CoMP transmission to switch between JP and CB to enhance the average achievable rate.Simulation results show that the proposed scheme predicts the switching point well, and outperforms the traditional single mode CoMP transmission when PSE exist.

Fig. 9. Average achievable rate of UE 1 for mode selection.

The proposed scheme estimates PSE by effective downlink channels. In fact, the effective downlink channels suffer from estimation errors. The impact of the estimation errors on the mode selection scheme should be analyzed in future work. In addition, there is still a large performance difference between the mode selection scheme with PSE and the JP with ideal synchronization. So how to design the codebook, which is suitable for UEs to feed the estimated PSE back to BSs for PSE correction,is also worthy for further research.

APPENDIX I

As nt→∞, the random variableconverges to a determinate value,can be expressed as

where step (a) comes by using lemma (1) and(2). Similarly, we rewrite the interference term of UE i as

Thus, E{IiJP} can be expressed as

As nt→∞, we can assume NUEis large enough to makebe independent of each other. Thus, the average SINR of UE i can be approximated by its lower bound [25].

Substituting Eq.(A.2) and Eq.(A.4) into Eq.(A.5), we get the result in proposition 1.

APPENDIX II

The above formula can be regarded as the trace ofis normalized to NUE.We assume matrix A=HDLW and therefore

It can be proved that (A.8) is maximized when all the eigen values of AAHare equal to 1, and the maximal value isΛ=and λi≥ 0 due to AAHis positive semi-definite. Then

Lagrange multipliers method can be utilized here to find the maximum solution. Letmaximum solution should satisfy

Which is equivalent to

The solutions of the above equation array are λ1= λ2= … = λNUE=1, and the maximum value is (

ACKNOWLEDGEMENTS

Part of the material in this paper was presented at IEEE international conference on communications (ICC 2015), London, UK, 2015.This work is supported in part by National High Technology Research and Development Program Of China (863 Program) under Grant No. 2014AA01A704; and National Natural Science Foundation Of China under Grant No.61101092.