Performance Analysis for Training-Based Multi-Pair Two-Way Full-Duplex Relaying with Massive Antennas

# Performance Analysis for Training-Based Multi-Pair Two-Way Full-Duplex Relaying with Massive Antennas

Zhanzhan Zhang, Zhiyong Chen, Manyuan Shen, Bin Xia, Weiliang Xie, and Yong Zhao
Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, P. R. China
China Telecom Corporation Limited Technology Innovation Center
Email: {mingzhanzhang, zhiyongchen, myshen, bxia}@sjtu.edu.cn, {xiewl, zhaoyong}@ctbri.com.cn
###### Abstract

This paper considers a multi-pair two-way amplify-and-forward relaying system, where multiple pairs of full-duplex users are served via a full-duplex relay with massive antennas, and the relay adopts maximum-ratio combining/maximum-ratio transmission (MRC/MRT) processing. The orthogonal pilot scheme and the least square method are firstly exploited to estimate the channel state information (CSI). When the number of relay antennas is finite, we derive an approximate sum rate expression which is shown to be a good predictor of the ergodic sum rate, especially in large number of antennas. Then the corresponding achievable rate expression is obtained by adopting another pilot scheme which estimates the composite CSI for each user pair to reduce the pilot overhead of channel estimation. We analyze the achievable rates of the two pilot schemes and then show the relative merits of the two methods. Furthermore, power allocation strategies for users and the relay are proposed based on sum rate maximization and max-min fairness criterion, respectively. Finally, numerical results verify the accuracy of the analytical results and show the performance gains achieved by the proposed power allocation.

{keywords}

Massive multiple-input multiple-output (MIMO), full-duplex, two-way relay, pilot scheme, power control.

## I Introduction

Massive multiple-input multiple-output (MIMO), an emerging technology which employs a few hundreds even thousands of antennas, is recently a very hot research topic in wireless communications [1]. By providing great array and spatial multiplexing gains, massive MIMO systems can be of higher spectral and energy efficiencies than conventional MIMO. Besides, the simplest linear precoders and detectors (such as maximum-ratio combining/maximum-ratio transmission (MRC/MRT)) can achieve optimal performance as the nonlinears [2]. Therefore, massive MIMO is widely regarded as one of the cornerstone technologies for next-generation mobile networks. Furthermore, the related issues in terms of the implementation of massive MIMO technologies, such as the channel state information (CSI) acquisition and beamforming techniques, have been being proposed and discussed in recent 3GPP meetings [3].

On the other hand, full-duplex (FD) systems have attracted significant interest [4], due to the provided double spectral efficiency (SE) of traditional half-duplex (HD) systems. However, by receiving and transmitting simultaneously on the same channel, FD systems suffer from a great drawback of the inherent loop interference (LI) due to the signal leakage from the FD node output to input.

To suppress loop interference, many researches have already been done [5, 6, 7, 8, 9, 10]. LI suppression approaches can be categorized as passive cancellation and active cancellation and the active cancellation further includes analog cancellation and digital cancellation. For example, [5] showed that LI can be reduced to within a few dB of the noise floor by combining passive and active cancellations. In [6], the authors proposed the signal inversion and adaptive cancellation, which support wideband and high power systems, thus making it possible to build FD 802.11n devices. In addition, [7] extended the cancellation for single channel case to the FD MIMO relay case and proposed new spatial suppression techniques, such as minimum mean square error (MMSE) filtering. The authors in [8] also studied the spatial processing techniques for a FD MIMO relay, and indicated that LI suppression is preferable to pre-cancellation at the relay transmitter. Then the joint precoding/decoding design with low complexity to mitigate LI in spatial domain for FD MIMO relaying was proposed in [9]. In [10], it was shown that the combination of digital and analog cancellation can sometimes increase the LI. Besides, it has been reported in [10, 11, 12] that 70-110 dB overall suppression of the LI can be realized. In a word, recent achievements in radio frequency (RF)/circuit design have made it feasible to perform full-duplex in certain scenarios. On the other hand, in the 3GPP process, it has been proposed by Huawei, NTT DOCOMO, etc., that new radio (NR) access technology should support of FD in the future in a forward compatible way [3]. Furthermore, in a recent paper [13], the authors utilized the large-scale antennas to eliminate the loop interference due to the large array gain, and this discovery promotes the joint consideration of massive MIMO and full-duplex in subsequent analysis. Nevertheless, the work considered the one-way relay with massive antennas, and similar research in such systems for two-way channels is barely addressed.

Inspired by both ad hoc and infrastructure-based (e.g., cellular and WiFi) networks, two-hop wireless relaying is the most possible use case which can benefit from the FD operation [14, 12], since in wireless relaying, the data traffic is inherently symmetric as far as the relay always froward the received information, and this character can efficiently utilize the FD ability of doubling the SE. Recently, FD wireless relaying has been discussed and included in the 3GPP standard [15]. In the experimental aspect, a FD MIMO relay for LTE-A has been studied in lab and proved to be technologically feasible [12]. Then, in this paper, it is straightforward to generalize the FD wireless relaying model to the multi-pair two-way FD massive MIMO relay system model, by considering two-way relaying to more efficiently utilize the time/frequency resource. And either a mobile terminal or a base station can act as the FD relay.

Over the recent years, much progress has been made on two-way or one-way relaying systems. For example, the authors in [16] studied the performance of a two-way amplify-and-forward (AF) MIMO relay system based on orthogonal space-time block codes (OSTBCs). In [17], a differential modulation based two-way relaying protocol was proposed for two-way AF satellite relaying communication. In [18], the joint beamforming optimization and power control were investigated for a two-way FD MIMO relay system. However, [16, 17, 18] all considered the traditional MIMO with a small number of antennas at the relay. Besides, the power efficiency of a multi-pair AF relaying model with massive MIMO was investigated in [19], and it was shown that massive MIMO could greatly improve the power efficiency while maintaining a given quality-of-service. Nonetheless, it only considered the one-way HD relaying. In addition, [20] studied the spectral efficiency and energy efficiency for a multi-pair two-way massive MIMO relay system, but only the case of infinite number of relay antennas was considered in [20]. Moreover, [21] discussed the achievable ergodic rate with a finite number of relay antennas for the same system as [20], however, both [20] and [21] dealt with the HD relays.

In this paper, we model a multi-pair two-way full-duplex AF relay system where the relay has a large-scale antenna array, in the presence of inter-user interference, and the MRC/MRT technique is considered. For massive MIMO systems, it is a big challenge to acquire the CSI. In our system model, the relay needs to acquire the global CSI to perform MRC/MRT processing. The model involves both the uplink channels (from users to the receive antenna array of the relay) and the downlink channles (from the transmit antennas of the relay to users). In general, the estimations of channels are obtained by transmitting pilot signals. Since the frequency-division duplex (FDD) scheme, where users estimate the downlink CSI based on the pilot signals transmitted by the relay and feedback them to the relay, is prohibitive in massive MIMO relay networks [22], we consider the time division duplex (TDD) system where users transmit pilot signals to both the transmit and receive antennas of the relay, then the CSI estimated by the relay transmit antennas is considered as the downlink CSI based on channel reciprocity.

The contributions of this work are summarized as follows.

• We derive a lower bound and an approximate expression for the ergodic sum rate with a finite and large number of relay antennas based on the statistical CSI, and the results are obtained by utilizing the orthogonal pilot scheme and least square (LS) channel estimation. The approximate sum rate expression is demonstrated to be a tight approximation to the ergodic sum rate. It is also shown that the sum rate can be increased significantly by adding the relay antenna number.

• We also derive the achievable rate expression by employing another pilot transmission scheme, which estimates the composite channel for each user pair. We present the comprehensive theoretical analysis on the achievable rates of the two pilot schemes and provide the valuable insights to show the relationship between the two methods.

• We derive the power allocation for maximizing the achievable sum rate and the minimum signal-to-interference-plus-noise ratio (SINR) of all users, respectively. Furthermore, we present the comparison of our scheme with other schemes and demonstrate that when the relay antenna number is very large, our scheme performs better than the corresponding one-way FD relaying scheme as well as the two-way HD relaying scheme.

The remainder of this paper is organized as follows. In Section II, the system model of the multi-pair massive MIMO two-way FD relay channel is described. In Section III, we derive an approximate sum rate expression based on conventional pilot scheme and LS channel estimation, when the relay antenna number is finite. In Section IV, we consider another pilot scheme and the corresponding achievable rate expression is also obtained. Section V compares the two pilot schemes and addresses the problem of power allocation. Numerical results are provided in Section VI. Finally, Section VII concludes the paper.

Notations: Boldface uppercase and boldface lowercase letters denote matrices and column vectors, respectively. , , , , , stand for the expectation, Euclidean norm, the trace of a square matrix, the conjugate transpose, the transpose and the conjugate of a matrix, respectively. represents the distribution of a circularly symmetric complex Gaussian vector with mean vector and covariance matrix . denotes an identity matrix.

## Ii System Model

As shown in Fig. 1, we consider a multi-pair two-way relaying network, where () user pairs try to exchange information within pair through a relay () which operates in AF protocol. Let () denote one source pair, or , . Besides, all the user equipments and the relay operate in the FD mode, so that all nodes suffer from self-LI due to the simultaneous transmission and reception. Assume that each FD user has one FD antenna [23, 24, 25], and the FD relay is equipped with receive antennas and transmit antennas 111We assume that the FD antenna at each user can be used for transmission and reception simultaneously. Based on this assumption and the orthogonality of the pilot sequences, the length of pilot symbols () consumed in the channel estimation stage would be at least the number of users (), which is half of that () under the scenario where each user has two antennas in which one for transmission and the other for reception. In addition, the transmit antennas and receive antennas at the FD relay are separated, then receiving RF chains, for receiving pilot signals during the channel estimation stage, are needed in the transmit antenna set apart from transmitting RF chains, while only receiving RF chains are needed in the receive antenna set. [13, 26], and let . We consider the scenario where the users with odd subscripts () stay in one area and the other users with even subscripts () stay in another area, thus direct links between and do not exist due to the high path loss and shadow fading, while one user can inevitably receive signals from nearby users in the same area due to FD operation, and we regard this interference as inter-user interference. In addition, we adopt the linear precoder and detector MRC/MRT at the relay in this paper, which is a common technique in the massive MIMO system.

Before further description, we assume that some traditional loop interference cancellation (LIC) techniques have been executed at the users and the relay in this paper, such as applying RF attenuation, time-domain suppression and/or spatial cancellation techniques [27, 24, 7]. Then the residual LI channels can be modeled as Rayleigh fading distribution [13, 7]. Furthermore, the residual LIs due to the imperfection of LIC methods are assumed to be additional Gaussian noise variables [25, 7, 28, 29]. This assumption will be the worst-case scenario regarding the achievable data rate if the residual LI is not Gaussian [28, 30].

### Ii-a Signal Model

At time instant , () transmits the signal to the relay, and at the same time, the relay broadcasts the signal to all source nodes. Here, we consider that each user has the same transmit power and . The transmit power of the relay is restricted by , so we have . Therefore the received signals at the relay and the source node are, respectively

 yR(n)=√PSGx(n)+GRRxR(n)+zR(n), (1)
 yk(n)=fTkxR(n)+∑i∈UkΩk,i√PSxi(n)+zk(n), (2)

where the set if is an odd number or otherwise, and . Let us define , where denotes the uplink channels between the antenna of and the receive antenna array of the relay. Also we define , where denotes the downlink channels from the transmit antenna array of to the antenna of . and are assumed to obey the independent identically distributed (i.i.d.) Rayleigh fading and therefore and . Hence, and can be expressed as and , respectively, where and denote the small-scale fading with i.i.d. random entries, and are diagonal matrices representing the large-scale fading, and the -th diagonal elements of and are denoted as and , respectively. In addition, and denote the self-LI channel coefficients at the relay and user respectively, and the entries of and are i.i.d. and random variables, respectively. () represents the inter-user interference channel coefficient from to , and assume [31]. is an additive white Gaussian noise (AWGN) vector at the relay and means AWGN at .

Let () denote the processing delay of the relay. At time instant (), the relay amplifies the previously received signal and broadcasts it to the sources. We thus have

 xR(n)=αWyR(n−τd), (3)

where is the relay processing matrix, and denotes a power constraint factor at the relay.

Due to the processing delay of the relay, we assume that the transmitted signal of the relay is uncorrelated with the received signal [13, 32, 33]. In addition, after performing some LIC techniques, let represent the residual LI at the relay. And because the amount of LI is mainly decided by the transmit power , we have according to the previous assumption of the residual LI. Then, substituting (1) into (3) and owing to the power constraint of the relay, i.e. , we have222We consider that the relay adopts the statistical CSI instead of instantaneous CSI to derive , known as the “fixed gain relay” [34]. Then the relay has a long-term power constraint where the expectation is taken over the channel realizations as well as the signal and the noise. Note that the “fixed gain relay” has lower complexity and is easier to deploy than the “variable gain relay” using instantaneous CSI to derive .

 α=√PRPS⋅Δ1+(PRσ2LI+σ2nr)⋅Δ2, (4)

in which

 Δ1=E[Tr(WGGHWH)], (5) (6)

Then, substituting (1) and (3) into (2), we can get the received signal at in detail as represented by (7) (see top of next page333Note that the self-interference () which stems from and is amplified and forwarded to itself by the relay due to two-way relaying is not included in (7) after applying the self-interference cancellation (SIC) technique. We will introduce the SIC briefly in the following derivation.), where the time labels are omitted, we also omit the time labels hereinafter for convenience. It is seen that the first term of the right hand side of (7) is the desired signal. The second term denotes the inter-pair interferences which are transmitted by other source pairs and then are amplified and forwarded to by the relay. The third and fourth terms indicate that the residual LI due to the FD operation of the relay and the noise at the relay are also forwarded to the user by the relay, respectively. The fifth term consists of the inter-user interferences () which are caused by nearby users and the self-LI () arising from the FD operation of the user itself. And the last term is the local noise.

## Iii Achievable Rate Analysis with individual channel estimation

In this section, we derive the achievable rate for the multi-pair two-way FD relay system, when the number of relay antennas is finite.

### Iii-a Individual Channel Estimation (ICE)

In this paper, we consider the flat block-fading channel, i.e. the channels during a block keep constant and vary independently across different blocks. The coherence interval (in symbols) of a block is denoted by . Suppose that symbols of the coherence interval are consumed in the pilot transmission phase. All users transmit deterministic pilot sequences (, ) to the relay simultaneously, where and denotes the transmit power of each pilot symbol. Then the received pilot signals at the receive and transmit antenna arrays of the relay are represented by, respectively

 Yrp=√τPpGΦ+Zrp, (8)
 Ytp=√τPpFΦ+Ztp, (9)

where is the transmitted pilot signal matrix, and denote the AWGN matrices with their elements are all random variables. The goal of channel estimation is to obtain individual CSI for each user, thus all pilot sequences need to be orthogonal to each other, i.e. , which requires .

In this paper, the popular LS channel estimation444 The reason we adopt the LS estimation method here is that it has the lowest complexity than other estimation methods and the main focus of this paper is to study and compare the impact of different pilot schemes on the system performance. Although its estimation error is a little larger than other methods, such as MMSE channel estimation, the estimation error will be cancelled out by the large array gain offered by massive MIMO. Thus we consider the popular LS approach. [35] is applied at the relay, and the LS estimations of the matrices and are given by

 ^G=1√τPpYrpΦH=G+Zr, (10)
 ^F=1√τPpYtpΦH=F+Zt, (11)

respectively, where and indicate the relevant estimation error matrices and their entries have zero means and variances of . Apparently, the actual channel matrices and are independent with the error matrices and , hence the large-scale fading matrices are estimated as

 ^Du=Du+σ2nrτPpI2K, (12)
 ^Dd=Dd+σ2nrτPpI2K, (13)

where the -th diagonal elements of and are denoted by and , respectively.

### Iii-B Achievable Rate: A Lower Bound

From (7), we can obtain the ergodic sum rate of the multi-pair two-way FD relay system with massive MIMO processing as represented by

 C=Tc−τTcE{2K∑k=1log2(1+SINRk)}, (14)

where denotes the received instantaneous SINR at the user node .

However, it is extremely difficult to derive a closed-form expression of the system capacity from (14). Therefore, instead of calculating (14) directly, we refer to the technique from [36] which is widely used in the regime of massive MIMO [2, 13, 37, 38, 39]. This technique utilizes the statistical channels to detect the received signals. With this technique, the received signal expression (7) can be rewritten as

 (15)

where is defined as the effective noise at , and is given by

 ~zk ≜α√PS(fTkWgk′−E{fTkWgk′})xk′ +α√PS2K∑j=1j≠k,k′fTkWgjxj+αfTkWGRR~xR+αfTkWzR +√PS∑i∈UkΩk,ixi+zk. (16)

Fortunately, it is easy to verify that the expected desired signal () and the effective noise () are uncorrelated. Based on the Theorem 1 in [36] which states that the worst case uncorrelated additive noise is independent Gaussian noise with the same variance in terms of the mutual information, we arrive at an achievable data rate of the system shown as

 R=Tc−τTc2K∑k=1log2(1+γk). (17)

where the statistical SINR is given by (18) based on (15) and (III-B).

###### Remark 1

Since the worst case uncorrelated Gaussian noise property is used to derive , it is expected that the rate expression (17) is a lower bound of the ergodic rate, i.e. (). And it will be demonstrated via the numerical results that the performance gap between the lower bound and the achievable ergodic rate is very small, which verifies that the lower bound is a good predictor of the achievable rate.

### Iii-C An Approximate Rate Expression

According to [20], the MRC/MRT processing matrix is given by

 W=^F∗T^GH=(F+Zt)∗T(G+Zr)H, (19)

where is the diagonal permutation matrix indicating the exchange of information between each user pair, and , for any .

Substituting (19) into (5) and (6), we have

 Δ1=Nt2K∑i=1^βdi(N2rβ2ui′+Nr^βui′2K∑j=1βuj), (20) Δ2=NtNr2K∑i=1^βdi^βui′. (21)

Equations (20) and (21) are proved in Appendix A. Thus we can obtain the power constraint factor by substituting (20) and (21) into (4).

In the following theorem, we derive an approximate closed-form expression of the achievable lower bound given by (17).

###### Theorem 1

With a fixed value of , when the number of relay antennas is finite and , an approximate closed-form expression for the SINR of user under MRC/MRT processing is represented by

 γk≈NtAk+MPk+LIRk+NR% k+MUk+ANk, (22)

where

 Ak=κ^βuk′βuk′+^βdkβdk, (23) MPk=2K∑j=1j≠k,k′⎛⎝κβuj^βuk′β2uk′+^βdj′β2ujβdkβ2uk′⎞⎠, (24) LIRk=PRσ2LIPSκ^βuk′β2uk′, (25) NRk=σ2nrPSκ^βuk′β2uk′, (26) MUk=1β2dkβ2uk′Δ3∑i∈Ukσ2k,i, (27) ANk=σ2nPSβ2dkβ2uk′Δ3, (28)

and555Considering the power-scaling law: and where and are fixed, we have . .

{proof}

See Appendix B.

Theorem 1 provides an approximate achievable rate expression when the number of relay antennas is large and finite. We observe that the small-scale fading is averaged out and the achievable rate is decided by the large-scale fading coefficients, which is the advantage of using the statistical channels for signal detection. On the other hand, since only the average effective channel is utilized for detection, there will be a deviation from the instantaneous channel, which is denoted by . In addition, it is easy to discover that represents the inter-pair interference; and denote LI and noise from the relay, respectively; signifies the inter-user interference and self-LI; indicates the additive noise at . Furthermore, (22) indicates that increasing the transmit antenna number of the relay can greatly enhance the sum rate, and approximately logarithmically in very large .

Next, we investigate the best relation between and with which the sum rate will achieve its peak value.

For simplicity of analysis, we consider the case where all large-scale fading coefficients are normalized to be 1, i.e., . Without loss of generality, consider perfect CSI with no channel estimation error, , ( and ), and . Then, based on (22) (28), the SINR for any user is given by

 γk=NtaK−b,∀k, (29)

where and . Let , and obviously indicates the transmit signal-to-noise ratio (SNR) of users. Thereby, the lower bound in (17) is represented as

 R=Tc−τTc2Klog2(1+NtaK−b). (30)

By taking the first order derivative of with respect to , and letting , we have

 ln(1+NtaK−b)=NtaK(aK−b+Nt)(aK−b), (31)

which shows the best relation between and . The “best relation” means that the sum rate will achieve its peak value when the number of users satisfies (31) here. However, it’s nontrivial to obtain some meaningful insights from (31). Indeed, in massive MIMO case and when , we can obtain the following expression from (31)

 Nt≈(aK−b)eaKaK−b. (32)

By differentiation with respect to , we have

 N′t(K)=a(aK−2b)aK−beaKaK−b. (33)

Note that with , thus we have . Therefore, for satisfying the best relation, the required transmit antenna number is increasing with respect to the optimal .

Furthermore, (32) implies that with fixed , the sum rate will increase with the number of user pairs. But when the number of user pairs is larger than the optimal which satisfies the best relation, the sum rate will decline. And this insight will be verified by the simulation results in Fig. 2.

###### Remark 2

About the self-interference , when and are very large and based on the law of large numbers (Lemma 1 in [20]), we have

 fTkWgk =fTk[2K∑i=1(f∗i+z∗ti)(gHi′+zHri′)]gk ≈∥fk∥22^gHk′gk+fTk^f∗k′∥gk∥22, (34)

where and are the -th columns of and , respectively. We see that only the CSI of the user pair (, ) is required for to perform SIC when and are large. In addition, when is fixed and , we get , while , thus SIC is needless when .

## Iv Achievable Rate Analysis with composite channel estimation

In the previous section, every user’s CSI can be estimated by pilot-based channel estimation at the cost of at least pilot symbols, and only () symbols are left for payload transmission. When is small, the achievable data rate would be very little. Motivated by [22] in which the scheme, where all users in a cell exploit the same pilot sequence and different cells use orthogonal pilot sequences, is proposed to eliminate the inter-cell interference, we are interested in investigating the performance for our system model when two users in each user pair employ the same pilot sequence and different user pairs adopt orthogonal pilot sequences. With this pilot scheme, the minimum pilot sequence length can be reduced to a half, i.e. only pilot symbols are required at least. As a result, the relay can only estimate the composite channels for each user pair instead of each user’s CSI.

In addition, this pilot scheme was also employed in [40], where the performance was evaluated for the multi-pair two-way relay system when the number of relay antennas went to infinity. However, [40] only evaluated the performance when was little ( therein), and the performance in the regime of large coherence interval is worth exploring. Besides, only the HD relay and the infinite relay antenna number were considered in [40].

### Iv-a Composite Channel Estimation (CCE)

Assume that all users transmit pilot signals simultaneously and the two users in the -th user pair transmit the same pilot sequence (, ), the received signal matrices of the receive and transmit antenna array of the relay are shown as

 =√τcPpGcΦc+¯Zrc, (35) Ytc=√τcPpFcΦc+¯Ztc, (36)

respectively, where and (). Let , , and , thus and . Besides, and denote the AWGN matrices with each element’s variance of .

Then we obtain the LS estimations of and as

 ^Gc=1√τcPpYrcΦHc=Gc+Zrc, (37) ^Fc=1√τcPpYtcΦHc=Fc+Ztc, (38)

respectively, where and signify the error matrices and their elements are all random variables. We observe that , , and are pairwise independent. Besides, we can easily get that

 E[GHcGc]=Nr(Du1+Du2), (39) E[FHcFc]=Nr(Dd1+Dd2), (40)

where , , , and . Therefore, the covariance matrices of the rows of and are denoted as

 ^Duc=E[^GHc^Gc]Nr=Du1+Du2+σ2nrτcPpIK, (41) ^Ddc=E[^FHc^Fc]Nt=Dd1+Dd2+σ2nrτcPpIK, (42)

where the -th diagonal elements of and are represented as and , respectively.

With respect to the training length, [36] shows that the optimal training length equals the minimum possible, i.e., and , assuming that the training power and data power can vary. However, when the training power and the data power are equal and very low, the optimal number of training symbols may be larger. Without loss of generality, we use in the following.

###### Corollary 1

When , based on (12), (13) and (41), (42), we can easily get that

 ^βucn=^βu(2n−1)+^βu(2n), (43) ^βdcn=^βd(2n−1)+^βd(2n), (44)

for any user pair ().

### Iv-B Achievable Rate with CCE

With the estimated composite channels, the relay takes the following MRC/MRT matrix

 Wc=^F∗c^GHc=(Fc+Ztc)∗(Gc+Zrc)H. (45)

Similar to (20) and (21), substituting (45) into (5) and (6), we get

 Δ1=NtK∑n=1^βdn[N2r(β2u(2n−1)+β2u(2n))+Nr^βun2K∑j=1βuj], (46) Δ2=NtNrK∑n=1^βdn^βun. (47)

Then the power limiting factor with CCE is achieved by substituting (46) and (47) into (4).

###### Theorem 2

Without loss of generality, consider user () in user pair . When is fixed and , the SINR of user for a finite number of relay antennas under CCE is approximated as

 γck≈NtAck+MPck+LIRck+NRck+MUck+ANck, (48)

where

 Ack=κ^βucmβu(2m)+^βdcmβd(2m−1), (49) MPck=K∑n=1,n≠mκ^βucm(βu(2n−1)+βu(2n))β2u(2m) +K∑n=1,n≠m^βdcn(β2u(2n−1)+β2u(2n))βd(2m−1)β2u(2m), (50) LIRck=PRσ2LIPSκ^βucmβ2u(2m), (51) NRck=σ2nrPSκ^βucmβ2u(2m), (52) MUck=1β2d(2m−1)β2u(2m)Δc3∑i∈Ukσ2k,i, (53)
 ANck=σ2nPSβ2d(2m−1)β2u(2m)Δc3, (54)

and666Similar to the fourth footnote, considering the power-scaling law, we have . .

{proof}

The proof is similar with Theorem 1.

We observe that the SINR of user with CCE is similar to that with ICE. By comparing (48) with (22) and based on (43) and (44), we obtain . Particularly, consider that the system is symmetric777It is known that the large-scale fading is closely related to the distance, hence symmetry here can be interpreted as the same distance from the two users in each user pair to the relay., i.e. and , then we can easily obtain , .

###### Remark 3

As to the self-interference under CCE, according to the law of large numbers (Lemma 1 in [20]) and in the regime of very large and , we get

 fTkWcgk =fTkK∑n=1(f∗2n−1+f∗2n+z∗tcn)(gH2n−1+gH2n+zHrcn)gk ≈∥fk∥22∥gk∥22≈NtNrβdkβuk, (55)

where and are the -th columns of and , respectively. Note that the individual CSI for each user cannot be acquired under CCE. But based on the law of large numbers, we can approximate the self-interference as . It is meant that the self-interference is only related to the large-scale fading coefficients and then can be cancelled out.

## V Performance Evaluation

In this section, we evaluate the system performance with different pilot schemes. First, we analytically compare the performance under ICE with that under CCE. Then, the power control of the users and the relay is derived based on sum rate maximization and max-min fairness criterion, respectively.

### V-a Performance Comparison Between ICE and CCE

The previous analysis shows that in the symmetric system (, ), we have , . Then we can obtain the following corollary.

###### Corollary 2

Consider the symmetric traffic and . Let