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Transmission of wireless backhaul signal in a cellular system with small moving cells
EURASIP Journal on Wireless Communications and Networking volume 2019, Article number: 238 (2019)
Abstract
Deployment of small moving cells (SMCs) has been considered in advanced cellular systems, where wireless backhaul links are required between base stations and SMCs. In this paper, we consider signal transmission by means of multiuser beamforming in the wireless backhaul link. We generate the beam weight in an eigendirection of weighted combination of short and longterm channel information of the backhaul link. The beam weight can maximize the average signaltoleakageplusnoise ratio (SLNR), while providing the transmission robust to SMC mobility. We analyze the performance of the proposed scheme in terms of the average signaltointerferenceplusnoise ratio (SINR) and optimize the transmit power by iterative waterfilling. Finally, we verify the performance of the proposed scheme by computer simulation.
Introduction
Demands for mobile data traffic have rapidly been increasing with extensive deployment of multimedia services [1]. In particular, the growth rate of data traffic of users in mobility is more than two times that of total mobile data traffic [2]. Deployment of small moving cells (SMCs) has been proposed to provide desired quality of service (QoS) to users in mobility [3], where the base station (BS) of an SMC, referred to as SBS, is installed in a vehicle and serves users in the vehicle. The SBS can serve users without experiencing vehicular penetration loss (VPL), but it may require a wireless backhaul link for communication with a macro cell BS (MBS).
The deployment of SMCs can enhance the system capacity by sharing transmission resource with macro cells [3–7]. It can increase the throughput of users near the cell edge [3]. The transmission resource can be partitioned in a nonorthogonal manner [5, 6], where the macro cell and SMCs can use the same resource for serving their users. Since the interference from the MBS to SMC users can be somewhat reduced due to the effect of VPL, the overall spectral efficiency can be improved [5].
However, most of previous studies did not consider the channel aging effect in the backhaul link. When an SBS is in high mobility, the backhaul link capacity may significantly decrease mainly due to the channel mismatch problem [8]. The backhaul link can be a major bottleneck when it cannot provide the required capacity. Therefore, the channel aging can be a critical issue in the deployment of SMCs. Fullcache mode operation was considered for the deployment of SMCs [7], where an SMC can transmit cached data only. However, it may be applicable to a small portion of mobile traffic [9]. It was reported that the use of relay mode operation may not improve the transmission performance because of poor transmission performance in backhaul links [7].
It may be desirable to employ a transmission scheme to make the backhaul link robust to the channel aging effect. A twostage multiantenna beamforming scheme was proposed [10], where the outer beam weight is determined to suppress the interference among users, while the inner beam weight is determined to maximize the multiplexing gain. The outer beam weight can be determined by a set of eigenvectors of interference channel matrix corresponding to nearzero eigenvalues, while the inner beam weight can be determined to support zeroforcing (ZF) or singular value decomposition (SVD) transmission. However, this scheme is vulnerable to the channel aging since the beam weights are determined by using shortterm channel state information (STCSI). Alternatively, the beam weight can be determined by using longterm (LT) or statistical CSI (e.g., spatial channel correlation matrix (CCM)) [11, 12]. The beam weight can be determined by a set of eigenvectors of CCM corresponding to the largest eigenvalues [11]. The use of LTCSI can provide the transmission quite efficient in high mobility environments, but not in low mobility environments [12]. The beam weight can be determined in a hybrid manner [13–15], where the outer and the inner beam weight are determined using CCM and STCSI, respectively. However, it may not be efficient for users in low or high mobility, compared to the use of STCSI or LTCSI, respectively [13]. Temporal variation of backhaul channel can be estimated by using predictor antennas located in front of the backhaul link antennas of an SMC [16–18]. As the vehicle is moving forward, the location of the backhaul antenna passes through the location of the predictor antenna. This implies that the channel of the backhaul antennas and the predictor antennas may be correlated in spatial and temporal domain. The MBS can predict the STCSI of backhaul antennas by exploiting the STCSI of predictor antennas [16]. However, the correlation between the channels of the predictor antennas and the backhaul antennas may noticeably vary in the presence of mobility [19]. When the correlation is not large, the channel prediction may not be effective for the estimation of STCSI of the backhaul channel.
In this paper, we consider signal transmission in the backhaul link of SMCs. We determine the beam weight by exploiting ST and LTCSI in consideration of SMC mobility. The MBS estimates the signaltoleakageplusnoise ratio (SLNR) of the backhaul link and determines the beam weight to maximize the SLNR in an average sense. The beam weight makes the signal transmission in an eigendirection of weighted combination of ST and LTCSI. We analyze the average signaltointerferenceplus noise ratio (SINR) of the proposed scheme and use it to allocate the transmission power. Joint utilization of ST and LTCSI can provide the transmission performance robust to the variation of SMC mobility.
The remainder of this paper is organized as follows. The system model in consideration is described in Section 2. The proposed beamforming scheme is described in Section 3. The performance of the proposed scheme is verified by computer simulation in Section 4. Finally, conclusions are given in Section 5.
System model
Consider an orthogonal frequency division multiplexing (OFDM) cellular system that employs K SMCs. We consider wireless backhaul links from an MBS to K SBSs, where an MBS and each of SBSs employ N_{T} and N_{R} antennas, respectively, and N_{T}≫N_{R}. Let \({\mathbf {H}_{k}} \in {\mathbb {C}^{{N_{R}} \times {N_{T}}}}\) be the backhaul channel from the MBS to SBS k as illustrated in Fig. 1, and \({\mathbf {R}_{k,r}} \in {\mathbb {C}^{{N_{T}} \times {N_{T}}}}\) be the CCM corresponding to the rth antenna of SBS k in spatial domain where 1≤k≤K. We assume that no correlation exists among the receive antennas (i.e., the CCM is an identity matrix) since the SBS may experience rich scattering compared to the MBS. We also assume that SBSs are sufficiently far away from the MBS, yielding an identical CCM for different receive antennas (i.e., R_{k,r}≈R_{k}). Then, it can be shown that
where \({\tilde {\mathbf {H}}_{k}}\) is an uncorrelated channel matrix whose elements are independent and identically distributed (i.i.d.) zero mean complex normal random variables with unit variance. Since the spatial correlation slowly changes in time and frequency domain [13], we can assume that R_{k} is unchanged during the time interval between the acquisition of STCSI and the data transmission.
Let \({\hat {\mathbf {H}}_{k}} \in {\mathbb {C}^{{N_{R}} \times {N_{T}}}}\) be the estimated STCSI by the MBS, which has the same statistical characteristics as H_{k} in (1). We assume that the MBS can perfectly estimate the STCSI, implying that the channel mismatch occurs only due to temporal variation of CSI. Then, H_{k} can be represented as [20]
where \({\mathbf {Z}_{k}} \in {\mathbb {C}^{{N_{R}} \times {N_{T}}}}\) is a temporal channel variation matrix which follows the same distribution as \({\hat {\mathbf {H}}_{k}}\) and ρ_{k} is temporal correlation between \({\hat {\mathbf {H}}_{k}}\) and H_{k}, represented as [20]
Here J_{0} is the 0thorder Bessel function of the first kind, v_{k} is the velocity of SBS k, f_{c} is the carrier frequency, T_{k} is the time interval between the STCSI estimation and the data transmission of SBS k, and c is the speed of light.
The signal received by SBS k can be represented as
where β_{k} is the largescale fading coefficient, \({\mathbf {B}_{k}} \in {\mathbb {C}^{{N_{T}} \times {s_{k}}}}\) is the transmit beamforming matrix from the MBS to SBS k, \({\mathbf {P}_{k}} \in {\mathbb {C}^{{s_{k}} \times {s_{k}}}}\) is a diagonal matrix for power allocation of B_{k}, \({\mathbf {s}_{k}} \in {\mathbb {C}^{{s_{k}} \times 1}}\) is the data vector, and \({\mathbf {n}_{k}} \in {\mathbb {C}^{{N_{R}} \times 1}}\) is an additive noise vector whose elements are i.i.d. zero mean complex normal random variables with variance \(\sigma _{n}^{2}\). The SBS k can estimate H_{k}B_{k}P_{k} from demodulation reference signal (DMRS) transmitted from the MBS. It can decouple multiple streams of s_{k} by means of zeroforcing reception with a weight determined by
where the superscript † denotes the pseudoinverse of a matrix. The corresponding received signal can be represented as
Proposed transmission scheme
Proposed beam design
The SINR of the ith stream received by SBS k can be represented as
where \({\mathbf {T}_{k}} \equiv \sum \limits _{l \ne k} {{\beta _{k}}{\mathbf {H}_{k}}{\mathbf {B}_{l}}{\mathbf {P}_{l}}{\mathbf {s}_{l}}}\), [·]_{ii} denotes the ith diagonal term of a matrix, and (·)^{H} denotes the Hermitian. It may not be easy to determine B_{k} maximizing the sumrate \(\sum \limits _{k,i} {{{\log }_{2}}\left ({1 + {\gamma _{k,i}}} \right)} \) since B_{k} and B_{l} should jointly be determined by (7). This problem can be alleviated by means of SLNRbased beamforming that minimizes leakage power instead of interference power T_{k} [21], where the leakage refers to the interference from the signal of a desired SMC to the other SMCs.
For ease of description, we assume that each signal transmission is equally powered, i.e., \({\mathbf {P}_{k}} = \sqrt {{P_{k}}} {\mathbf {I}_{{s_{k}}}}\), where \({\mathbf {I}_{{s_{k}}}} \in {\mathbb {C}^{{s_{k}} \times {s_{k}}}}\) denotes an identity matrix. The desired signal power of the ith stream received by SBS k can be represented as [22]
The SLNR of SBS k can be represented as [21]
where tr(·) denotes the trace of a matrix.
We consider temporal variation of the backhaul channel. The average of leakage L_{k} can be represented as
where U_{k} and D_{k} are the eigenvectors and the eigenvalues of \(\sum \limits _{j \ne k} {\beta _{j}^{2}\left \{ {\rho _{j}^{2}\hat {\mathbf {H}}_{j}^{H}{\hat {\mathbf {H}}_{j}} + {N_{R}}\left ({1  \rho _{j}^{2}} \right){\mathbf {R}_{j}}} \right \}}\), respectively. The two terms in the parenthesis in the last equation of (10) can be removed by B_{k} when
where \({\mathbf {B}_{k,2}} \in {\mathbb {C}^{{N_{T}} \times {s_{k}}}}\). The corresponding leakage power can be represented as
The desired signal power in (9) can be represented as
where \({\tilde {\mathbf {R}}_{k}} = \mathbf {B}_{k,1}^{H}{\mathbf {R}_{k}}{\mathbf {B}_{k,1}}\) and \({\mathbf {M}_{k}} = {\hat {\mathbf {H}}_{k}}{\mathbf {B}_{k,1}}\). B_{k,2} can be represented in an SVD form as
where \({\mathbf {U}_{B}} \in {\mathbb {C}^{{N_{T}} \times {N_{T}}}}\) and \({\mathbf {V}_{B}} \in {\mathbb {C}^{{s_{k}} \times {s_{k}}}}\) are the left and the right unitary matrix, respectively, \({\mathbf {D}_{B}} \in {\mathbb {C}^{{s_{k}} \times {s_{k}}}}\) is a diagonal matrix, 0 denotes a zero matrix, and the SMC index is omitted for simplicity of description. Then, (12) and (13) can be rewritten as, respectively,
where \({{\bar {\mathbf {S}}}_{k}} \equiv \beta _{k}^{2}\bar {\mathbf {D}}_{B}^{H}\mathbf {U}_{B}^{H}\left ({{\rho _{k}}\mathbf {M}_{k}^{H} + \sqrt {1  \rho _{k}^{2}} \tilde {\mathbf {R}}_{k}^{1/2}\mathbf {Z}_{k}^{H}} \right) \cdot \left ({{\rho _{k}}{\mathbf {M}_{k}} + \sqrt {1  \rho _{k}^{2}} {\mathbf {Z}_{k}}\tilde {\mathbf {R}}_{k}^{1/2}} \right){\mathbf {U}_{B}}{{\bar {\mathbf {D}}}_{B}}\).
In the presence of unknown Z_{k}, it may be desirable to determine U_{B}, V_{B}, and D_{B} to maximize the average SLNR, which can be represented as
However, it may not be easy to calculate the mean of the numerator of (17) since \(\bar {\mathbf {S}}_{k}^{ 1}\) is the inverse of a noncentral Wishart distributed matrix with N_{R} degrees of freedom whose expectation may not exist [23].
We first consider a case when the MBS has perfect information on Z_{k}. Let \(\mathbf {U}_{B}^{*}\), \(\mathbf {V}_{B}^{*}\), and \(\mathbf {D}_{B}^{*}\) be a solution of this case. Then, \(\mathbf {V}_{B}^{*}\) can be determined by
Since \({\bar {\mathbf {S}}_{k}}\) is positive definite, it can be shown that \(\sum \limits _{i} {\left [ {\mathbf {V}_{B}^{H}\bar {\mathbf {S}}_{k}^{ 1}{\mathbf {V}_{B}}} \right ]_{ii}^{ 1}} \le {\text {tr}}\left ({{\mathbf {D}_{S}}} \right)\), where D_{S} is a diagonal matrix comprising eigenvalues of \({\bar {\mathbf {S}}_{k}}\) and the equality holds when V_{B} is an eigenvector matrix of \({\bar {\mathbf {S}}_{k}}\), denoted by \(\mathbf {V}_{B}^{*} = {\mathbf {U}_{S}}\) (refer to the Appendix). The corresponding SLNR can be represented as
Let \({\tilde {\mathbf {U}}_{k}}\) and \({\tilde {\mathbf {D}}_{k}}\) be the matrix of eigenvectors and eigenvalues of \(\beta _{k}^{2}\mathbf {B}_{k,1}^{H}\mathbf {H}_{k}^{H}{\mathbf {H}_{k}}{\mathbf {B}_{k,1}}\). With \({\mathbf {U}_{B}} = {\tilde {\mathbf {U}}_{k}}{\tilde {\mathbf {U}}_{B}}\), \(\left \{ {\beta _{k}^{2}\mathbf {B}_{k,1}^{H}\mathbf {H}_{k}^{H}{\mathbf {H}_{k}}{\mathbf {B}_{k,1}}} \right \}\) in (19) can be represented in a diagonal form. Then, (19) can be rewritten as
Since \({\mathbf {B}_{k,3}} = {\left [ {\begin {array}{*{20}{c}} {{{\tilde {\mathbf {U}}}_{B}}{\mathbf {D}_{B}}}&{\mathbf {0}} \end {array}} \right ]^{H}} = {\left [ {\begin {array}{*{20}{c}} {{\mathbf {I}_{{s_{k}}}}}&{\mathbf {0}} \end {array}} \right ]^{H}}\) maximizes φ_{k} [21], we can determine \(\mathbf {U}_{B}^{*} = {\tilde {\mathbf {U}}_{k}}{\tilde {\mathbf {U}}_{B}} = {\tilde {\mathbf {U}}_{k}}\) and \(\mathbf {D}_{B}^{*} = {\mathbf {I}_{{s_{k}}}}\).
Consider a case when Z_{k} is unknown to the MBS. Although the MBS cannot jointly determine \(\mathbf {U}_{B}^{*}\) and \(\mathbf {V}_{B}^{*}\), it can determine \(\mathbf {U}_{B}^{*}\) from the SLNR in an average sense if \(\mathbf {V}_{B}^{*}\) is known (and vice versa). Assuming that \(\mathbf {V}_{B}^{*}\) is known to the MBS, the average SLNR can be represented as
Using the same approach as in the case when Z_{k} is known to the MBS, it can be shown that the average SLNR can be maximized with \(\hat {\mathbf {U}}_{B}^{*} = {\bar {\mathbf {U}}_{k}}\) and \(\hat {\mathbf {D}}_{B}^{*} = {\mathbf {I}_{{s_{k}}}}\), where \({\bar {\mathbf {U}}_{k}}\) is an eigenvector matrix of \(\rho _{k}^{2}\mathbf {M}_{k}^{H}{\mathbf {M}_{k}} + {N_{R}}\left ({1  \rho _{k}^{2}} \right){\tilde {\mathbf {R}}_{k}}\). On the other hand, when \(\mathbf {U}_{B}^{*}\) is known to the MBS, \(\bar {\mathbf {S}}_{k}^{ 1}\) becomes a diagonal matrix. Then, it can be shown that \(\hat {\mathbf {V}}_{B}^{*} = {\mathbf {I}_{{s_{k}}}}\) is a solution of (18). The beam weight can be determined by
Instead of the proposed approach, a suboptimal solution of (18) can be obtained by searching a unitary matrix that diagonalizes \(\bar {\mathbf {S}}_{k}^{ 1}\) in an average sense when Z_{k} is unknown. The solution can be obtained by means of approximated joint diagonalization (AJD) that searches for a unitary matrix by jointly diagonalizing a large number of samples of \(\bar {\mathbf {S}}_{k}^{ 1}\) (or \({\bar {\mathbf {S}}_{k}}\)) in a MonteCarlo manner [24]. However, the computational complexity may become extremely large as the number of samples increases.
Performance analysis
We determine the transmit power based on the SINR by means of waterfilling. We may separate the power of desired signal from the interference plus noise term in (7). The SINR can be represented as
where \({\mathbf {W}_{k}} = \beta _{k}^{ 1}{\left ({\mathbf {B}_{k}^{H}\mathbf {H}_{k}^{H}{\mathbf {H}_{k}}{\mathbf {B}_{k}}} \right)^{ 1}}\mathbf {B}_{k}^{H}\mathbf {H}_{k}^{H}\). The interference power term in the denominator of (23) can be represented as
Since W_{k} and H_{k}B_{l} are correlated to each other, it may be desirable to split H_{k}B_{l} into two terms: one term that includes \(\mathbf {W}_{k}^{\dag } \) and the other term that is independent of W_{k}. The term Z_{k}B_{l}P_{l} can be represented as
where Ψ_{l} is a constant matrix and X_{l} is a matrix comprising independent normal random row vectors, while satisfying
Here, \({\bar {\mathbf {B}}_{l}} \equiv {\mathbf {B}_{l}}{\mathbf {P}_{l}}\). Then, it can be shown that
where \({\mathbf {Q}_{l}} \equiv {\rho _{k}}{\hat {\mathbf {H}}_{k}}{\bar {\mathbf {B}}_{l}}  {\rho _{k}}{\hat {\mathbf {H}}_{k}}{\mathbf {B}_{k}}{\mathbf {\Psi }_{l}}\) and Ψ_{l} and Q_{l} are known to the MBS. Since X_{l} and Z_{k} are independent of each other, C_{l,3} can be represented as
where
As described in the previous section, we cannot represent the expectation of \({\mathbf {W}_{k}}\mathbf {W}_{k}^{H}\) in a closed form. Instead, we consider the SINR when ρ_{k}→0 and ρ_{k}→1 where the diagonal elements of \({\mathbf {W}_{k}}\mathbf {W}_{k}^{H}\) can be properly approximated. When ρ_{k}→0, C_{l,1} and C_{l,2} can be ignored, and the SINR of the ith stream of SBS k can be represented as
The ith diagonal element of \({\mathbf {W}_{k}}\mathbf {W}_{k}^{H}\) can be represented as [25]
where G_{k}≡H_{k}B_{k}, \(\mathbf {G}_{k}^{\left ({i } \right)}\) denotes G_{k} without the ith column and · denotes the determinant of a matrix. It can be shown that when ρ_{k}→0, the column vectors of Z_{k}B_{k} are independent of each other since the beam weight of the proposed scheme converges to eigenvectors of R_{k}. Thus, (31) can be rewritten as
The average determinant of the noncentral Wishart matrix, \({\mathbf {G}_{k}^{H}{\mathbf {G}_{k}}}\), can be written as [26]
where \({\bar {\mathbf {R}}_{k}} = \mathbf {B}_{k}^{H}{\mathbf {R}_{k}}{\mathbf {B}_{k}}\), \({\mathbf {\Omega }_{k}} = \bar {\mathbf {R}}_{k}^{ 1}\mathbf {B}_{k}^{H}\hat {\mathbf {H}}_{k}^{H}{\hat {\mathbf {H}}_{k}}{\mathbf {B}_{k}}\), \({\Gamma _{{s_{k}}}}\left (\cdot \right)\) denotes the multivariate gamma function and _{1}F_{1}(·) denotes the hypergeometric function of matrix arguments. Similarly, it can be shown that
where \(\bar {\mathbf {R}}_{k}^{\left ({i } \right)} = {\left ({\mathbf {B}_{k}^{\left ({i } \right)}} \right)^{H}}{\mathbf {R}_{k}}\mathbf {B}_{k}^{\left ({i } \right)}\) and \(\mathbf {\Omega }_{k}^{\left ({i } \right)} = \bar {\mathbf {R}}_{k}^{ 1}{\left ({\mathbf {B}_{k}^{\left ({i  1} \right)}} \right)^{H}} \hat {\mathbf {H}}_{k}^{H}{\hat {\mathbf {H}}_{k}}\mathbf {B}_{k}^{\left ({i } \right)}\). When ρ_{k}→0, the SINR can be approximated as
On the other hand, when ρ_{k}→1, C_{l,1} and C_{l,2} can be approximated as, respectively,
It can be shown that C_{l,3} is the same as that when ρ_{k}→0. When ρ_{k}→1, the SINR, denoted by \(\hat \gamma _{k,i}^{\left (1 \right)}\), can approximately be represented as
where \({\mathbf {A}_{l}} \equiv {\mathbf {\Psi }_{l}}\mathbf {\Psi }_{l}^{H} + {\hat {\mathbf {C}}_{l,1}} + {\hat {\mathbf {C}}_{l,2}}\).
When 0<ρ_{k}<1, the numerator and the denominator of (31) are not independent of each other, making it difficult to get an approximated representation. However, simulation results show that \(\hat \gamma _{k,i}^{\left (1 \right)}\) and \(\hat \gamma _{k,i}^{\left (0 \right)}\) are valid for 0<ρ_{k}<1. When 0<ρ_{k}<1, the SINR of the ith stream of SBS k can approximately be represented as
where the threshold value \(\sqrt {0.5} \) is determined from \({\rho _{k}} = \sqrt {1  \rho _{k}^{2}} \) in (2).
Finally, we allocate the transmit power by means of iterative waterfilling [27] with the use of average SINR by (39). The procedure is summarized in Table 1, where p_{k,i} is the ith diagonal element of P_{k} and N_{p} is the number of iterations for the waterfilling.
Complexity of the proposed scheme
For the generation of beam weight by (22), the eigendecomposition in (11) and (21) may require a computational complexity of \(O\left ({N_{T}^{3}} \right)\), which is similar to that of conventional STCSIbased SVD transmission scheme [10]. The only difference comes from the linear combination of STCSI and LTCSI in (11) and (21), which may require a computational complexity of \(O\left ({N_{T}^{2}} \right)\).
For the power allocation, the MBS calculates \(\hat \gamma _{k,i}^{\left (1 \right)}\) (or \(\hat \gamma _{k,i}^{\left (0 \right)}\)) and the maximum number of beams for each SMC is s_{k}=N_{R}. The SINR can be initialized with a computational complexity of \(O\left ({{N_{T}}N_{R}^{2}} \right)\). It may require a computational complexity of \(5N_{T}^{2}{N_{R}} + 10{N_{T}}N_{R}^{2} + 21N_{R}^{3} + 9N_{R}^{2} + {N_{R}} + O\left ({P_{{N_{h}}{N_{R}}}^{2}{N_{R}}} \right)\) for the estimation of \(\hat \gamma _{k,i}^{\left (1 \right)}\), where \(O\left ({P_{{N_{h}}{N_{R}}}^{2}{N_{R}}} \right)\) is the computational complexity for the hypergeometric function in (33) and (34), \(P_{{N_{h}}{N_{R}}}^{2} \sim O \left (\exp \left (2\pi \sqrt {2N_{h} \left /\right. 3} \right) \right)\) and N_{h} is the number of iterations [28]. The power matrices are only required to be updated for each iteration, not the beam weight and the hypergeometric function. The procedure may require a computational complexity of \(O\left ({N_{R}^{2}} \right)\) for each iteration, which is relatively small compared to that of the generation of beam weight.
Since N_{T}≫N_{R} in the system model, the proposed scheme may require an additional computational complexity of \(O\left ({N_{T}^{2}} \right)\). This means that the computational complexity of the proposed scheme mainly depends upon the generation of beam weight requiring a computational complexity of \(O\left ({N_{T}^{3}} \right)\), which is almost the same as that of the conventional scheme. However, the computational complexity for the hypergeometric function increases subexponentially proportional to N_{h}. The hypergeometric function is only used in the power allocation, not in the beam weight calculation. It was shown that the performance degradation of the power allocation is small when the interference terms in SINR is approximated to zero [10]. Although it did not consider temporal channel variation [10], it can be applied to estimate the SINR without using the hypergeometric function terms for the power allocation.
Simulation results and discussions
We verify the performance of the proposed scheme by computer simulation. The main simulation parameters are summarized in Table 2. We estimate the SINR by (39), where we approximately calculate the hypergeometric function by means of finiteseries approximation [28]. For fair comparison, we apply the same waterfilling algorithm to the conventional schemes in the simulation.
Figure 2 depicts the average spectral efficiency of the backhaul link according to temporal channel correlation ρ_{k} when the number of SBS antennas is 4 or 8, where “AJD” refers to the proposed scheme when \(\hat {\mathbf {V}}_{B}^{*}\) is determined by the AJD of (22), “perfect CSI” refers to the proposed scheme with \(\mathbf {V}_{B}^{*}\) determined using perfect Z_{k}, and “analysis” refers to the analysis by (39). It can be seen that the proposed and the AJD scheme may suffer from performance degradation due to unknown Z_{k}. However, the proposed scheme with approximation by (22) provides performance similar to AJD, significantly reducing the computational complexity. The AJD requires large computational complexity to generate samples of the channel matrix and determines \(\hat {\mathbf {V}}_{B}^{*}\) by an iterative algorithm [24], but the proposed scheme simply determines \(\hat {\mathbf {V}}_{B}^{*}\) by an identity matrix. It can also be seen that the analysis agrees quite well with the simulation results.
We compare the spectral efficiency of the proposed scheme with that of twostage beamforming [10], statistical beamforming [11] and hybrid beamforming scheme [13], referred to as “STCSIbased,” “LTCSIbased,” and “hybrid,” respectively. The twostage beamforming only exploits STCSI, where the outer beam weight is determined by a set of eigenvectors of interference channel matrix and the inner beam weight as a set of eigenvectors of the effective channel equal to the product of the original channel and the outer beam weight. This scheme is identical to the proposed scheme when ρ_{k}=1. The statistical beamforming only exploits LTCSI, where the beam weight is determined by a set of eigenvectors of CCM corresponding to the largest eigenvalue. Since the conventional scheme only considers users equipped with a single antenna, it determines the beam weight in a twostage manner, which is the same as the proposed scheme when ρ_{k}=0. The hybrid beamforming exploits ST and LTCSI to determine the inner and the outer beam weights, which are equal to B_{k,2} with ρ_{k}=1 and B_{k,1} with ρ_{k}=0 in (22), respectively.
Figure 3 depicts the performance when each SMC is moving at a speed of uniformly distributed in a range of (0, 60) km/h, and Fig. 4 depicts the performance when each SMC is moving at a fixed velocity of 30 km/h. The cumulative distribution function (CDF) of spectral efficiency of an SMC is empirically depicted when the average SNR is 20 dB. The delay between the channel estimation and the signal transmission is set to 5 ms [30] and N_{R}=4. It can be seen from Fig. 3 that the proposed scheme outperforms the other schemes in a wide range of SNR. The LTCSIbased scheme shows the worst performance since achievable beamforming gain in low mobility dominates the average transmission performance as seen in Fig. 2. By the same reason, the STCSIbased scheme shows better performance even though the SMCs suffer from the channel aging effect in the presence of high SMC mobility. It can be seen from Fig. 4 that the hybrid beamforming outperforms STCSI and LTCSIbased schemes. The STCSIbased schemes provide the worst performance mainly due to the channel mismatch problem, while the LTCSIbased schemes provide poor performance since they do not exploit beamforming gain from STCSI. The proposed and the hybrid schemes yield empirical CDF similar to each other. However, the proposed scheme can provide higher spectral efficiency than the hybrid scheme, by adjusting the combine weight of ST and LTCSI according to the SMC mobility.
Conclusions
We have considered signal transmission in the wireless backhaul links for SMCs deployed in a cellular system. We have determined the beam weight to maximize the average SLNR by exploiting ST and LTCSI. By transmitting data in a weighted eigendirection of ST and LTCSI, the proposed scheme can provide transmission performance robust to the variation of SMC mobility. It adjusts the transmit power based on the estimated SINR. The simulation results show that the proposed scheme is quite efficient in the backhaul link.
Appendix
Let \(\mathbf {U} \in {\mathbb {C}^{N \times N}}\) be a unitary matrix, \(\mathbf {D} \in {\mathbb {C}^{N \times N}}\) be a diagonal matrix with distinct positive elements, u_{ij} be an element of U corresponding to the ith row and the jth column, and d_{i} be the ith diagonal element of D. Assume that the elements in D are sorted in an ascending order. It can be shown that
where u_{ij} is constrained by
Let \(\hat {\mathbf {U}} = {\mathbf {I}_{N}}\) with corresponding element \({\hat u_{ij}}\). It can be shown that
where \({\hat \eta _{i}} = \sum \limits _{j = 1}^{N} {{d_{i}}{{\left  {{{\hat u}_{ij}}} \right }^{2}}}\). By the convexity of (40), it can be shown that
When D is not a diagonal matrix, it can be diagonalized by representing \(\mathbf {U} = \bar {\mathbf {UU}}_{D}^{H}\), where U_{D} is an eigenvector matrix of D and \(\bar {\mathbf {U}}\) is a unitary matrix. Then, it can be shown that the equality in (43) holds when \(\mathbf {U} = \mathbf {U}_{D}^{H}\).
Availability of data and materials
The first author has the data and source codes.
Abbreviations
 BS:

Base station
 CCM:

Channel correlation matrix
 CDF:

Cumulative distribution function
 CSI:

Channel state information
 DMRS:

Demodulation reference signal
 LTCSI:

Longterm CSI
 MIMO:

Multiinput multioutput
 NLOS:

Nonlineofsight
 QoS:

Quality of service
 SBS:

Small moving cell BS
 SINR:

Signaltointerferenceplusnoise ratio
 SLNR:

Signaltoleakageplusnoise ratio
 SMC:

Small moving cell
 SNR:

Signaltonoise ratio
 STCSI:

Shortterm CSI
 SVD:

Singular value decomposition
 VPL:

Vehicular penetration loss
 ZF:

Zeroforcing
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Acknowledgements
This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2019R1F1A1063171).
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HK conceived and designed the study. HK, YB, and GJ performed experiments and analyzed the experimental results. YL directed the academic research. HK wrote the paper, and YL revised the manuscript. All authors read and approved the manuscript.
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Correspondence to YongHwan Lee.
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Kim, H., Byun, Y., Jung, G. et al. Transmission of wireless backhaul signal in a cellular system with small moving cells. J Wireless Com Network 2019, 238 (2019). https://doi.org/10.1186/s1363801915519
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Keywords
 MIMO
 Moving cell
 Backhaul