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Channel estimation for FDD massive MIMO system by exploiting the sparse structures in angular domain
EURASIP Journal on Wireless Communications and Networking volume 2019, Article number: 32 (2019)
Abstract
Massive multipleinput multipleoutput (MIMO) is a powerful supporting technology to meet the energy/spectral efficiency and reliability requirement of Internet of Things (IoT) network. However, the gain of massive MIMO relies on the availability of channel state information (CSI). In this paper, we investigate the channel estimation problem for frequency division duplex (FDD) massive MIMO system. By analyzing the sparse property of the downlink massive MIMO channel in the angular domain, a structured priorbased sparse Bayesian learning (SPSBL) approach is proposed to estimate the downlink channels between base station (BS) and users reliably. The scheme can be implemented without the knowledge of channel statistics and angular information of users. The simulation results show that the proposed scheme outperforms the reference schemes significantly in terms of normalized mean square error (NMSE) for a variety of scenarios with different lengths of pilot sequence, transmit signaltonoise ratios (SNRs), and angular spreads.
Introduction
As an emerging technology that aims to allow everything to connect, interact, and exchange data, the Internet of Things (IoT) has attracted extensive attention in various areas such as governments, industry, and academia [1]. The massive connectivity, strict energy limitation, and requirement of ultrareliable transmission are often termed as the most distinct features of IoT [2, 3]. These features make massive multipleinput multipleoutput (MIMO) [4] a natural supporting technology for IoT since through coherent processing over the signals of largescale antenna array, massive MIMO can produce tremendous spectral/energy efficiency gain and improve the reliability of IoT transmission significantly.
However, the coherent processing depends on the reliable estimation of channel state information (CSI) between base station (BS) and users [5–7]. In time division duplex (TDD) massive MIMO system, by exploiting the channel reciprocity, both the uplink and downlink channels can be obtained using the simple least square (LS) approach [8]. The consumption of pilot resource scales with the number of users. However, there is no channel reciprocity in frequency division duplex (FDD) massive MIMO system. In this case, the downlink channel estimation becomes particularly challenging since the downlink training and corresponding CSI feedback yield an unacceptably high overhead. For example, with the traditional LS channel estimation scheme, it is wellknown that the length of the required pilot sequence must be larger than the number of antennas at BS [8], which will degrade the system performance greatly.
In the practical system, the BS is usually elevated at a relatively high altitude with few surrounding scatters. Therefore, the scattering process often occurs in vicinity of user, which results in very narrow signal angle of departure (AoD) at BS [8, 9] for farfield transmission. Based on a physical channel model, several works have shown that the signals from different antennas of BS exhibit high correlation, and hence, the channel between BS and user can be represented by a sparse vector in some alternate domain (usually called angular domain) [10, 11]. This property can be utilized to design costefficient channel estimation schemes [12–14]. The main idea behind is to transform the channel to the angular domain by exploiting the correlations between the channel elements and then estimate the effective lowdimension angulardomain channel with the LS or minimum mean square error (MMSE) method. In this way, the training consumption can be reduced greatly. However, these schemes require the knowledge of channel covariance matrix or angulardomain information, such as the AoDs of the signals, which is not always available. Another path to reduce the training overhead is to formulate the channel estimation as a sparse recovery problem and estimate the angulardomain channel using the algorithms in compressive sensing, such as orthogonal matching pursuit (OMP) [15, 16] and sparse Bayesian learning (SBL) [17]. These schemes do not need the channel covariance matrix and angular information; however, suffer from performance loss when the length of pilot sequence is small. To improve the quality of channel estimation, novel channel estimation schemes based on block l_{1}/l_{2} optimization [18] and variational SBL [19] were proposed recently. These schemes exploit the structures in the channel sparsity to give a robust channel estimation.
In this paper, we investigate the downlink channel estimation problem for FDD massive MIMO system considering the above challenges. The main contributions are as follows:

1
The sparse property of downlink massive MIMO channel in angular domain is analyzed theoretically. The results show that the angulardomain channel exhibits two kinds of sparse structures, namely the joint sparsity and burst sparsity.

2
By exploiting the two kinds of sparsity, a structured priorbased SBL (SPSBL) approach is proposed to estimate the downlink channel between the BS and user reliably. The scheme does not require the channel statistics and AoD information.

3
Extensive numerical simulations are presented to validate the effectiveness of the proposed scheme. The results show that the proposed scheme outperforms the reference schemes significantly in terms of normalized mean square error (NMSE) for a variety of scenarios with different lengths of pilot sequence, transmit signaltonoise ratios (SNRs), and angular spreads.
Notations: We use B^{∗}, B^{T}, B^{H}, B, and ∥B∥ to denote conjugate, transpose, conjugate transpose, determinant, and Frobenius norm of matrix B, respectively. \(\mathbf {B} \in \mathbb C^{N \times M}\) means B is an N×M complexvalued matrix. \({\mathcal {C}N}\left (\mathbf {b} \left  \mathbf {m},\mathbf {C}\right. \right)\) means that b is a complex Gaussian variable with mean m and covariance matrix C. [B]_{i,j} denotes the {i,j}th element of matrix B. \(\mathbb E(\cdot)\) denotes the expectation. ∇_{b}(f(b)) denotes the gradient of function f(b) w.r.t. the vector b.
Methods
The rest of the paper is organized as follows: Section 3 describes the system model of the FDD massive MIMO system. Section 4 analyzes the sparse property of the angulardomain channel and presents the SPSBLbased channel estimation scheme. Section 5 presents the simulation results to validate the effectiveness of the proposed scheme. Section 6 draws the conclusions.
System model
Consider the massive MIMO system with a BS and K users. It is assumed that both the BS and users are equipped with uniform linear arrays (ULAs) with half wavelength antenna spacing. The numbers of antennas at BS and user are N and M, respectively, which satisfy M≪N. According to the raytracing model [8], the downlink channel between BS and user k can be expressed as:
where θ_{k} denotes AoD w.r.t. the array of BS, and φ_{k} denotes the angle of arrival (AoA) w.r.t. the array of user k. Δ_{d} and Δ_{a} denote the corresponding angular spreads. r_{k}(θ,φ) denotes the complex channel gain for angle {θ,φ}. a_{BS}(θ) and a_{U}(φ) denote the array steering vectors, which are given by:
In practice, the BS is often deployed at a high place such as the top of high building. As discussed in the introduction, the limited number of scatterers around BS will result in very narrow angular spread Δ_{d} in farfield propagation. Conversely, the waves arriving at the user are usually uniformly distributed in AoA. Therefore, it is reasonable to assume Δ_{a} is close to π [9]. Note that the proposed channel estimation scheme in the next section is very general, which is valid for arbitrary Δ_{a}.
Let S denote the N×T pilot matrix of BS which is broadcasted in T successive symbol times. The power of each pilot symbol is [S]_{i,j}^{2}=P. For a practical training scheme, T should be much smaller than N. The received training signal at user k can be expressed as:
where \(\mathbf {N} \in \mathbb C^{M\times T}\) denotes the additive white Gaussian noise (AWGN) matrix whose elements have zero mean and variance σ^{2}.
Note that for the fullduplex massive MIMO system with separate antenna configuration [11], the channel reciprocity is also not available which makes the downlink channel estimation very challenging. Since the downlink transmission model of fullduplex massive MIMO is similar with the FDD counterpart in the above, the proposed scheme can be utilized to estimate the downlink channel directly to reduce the pilot consumption.
Channel estimation by exploiting the sparse structures
In this section, we first analyze the property of channel sparsity in the angular domain. Based on the theoretical results, we propose a novel SPSBL approach to estimate the downlink channel between BS and user.
Joint and burst sparsity in the angular domain
According to [11], the angulardomain channel matrix between BS and user k can be expressed as:
where A_{N} is a shifted discrete Fourier transform (DFT) matrix of dimension N, that is:
Note that the mth column of X_{k} (denoted by x_{k,m}) is the angulardomain channel vector between the BS and the mth antenna of user k. The angulardomain channel is the projection of channel onto the space spanned by the DFT bases. Since a DFT basis is in fact equivalent to an array steering vector with specific AoD, the angulardomain channel can be viewed as the channel response in the angular domain, and the amplitude of each angulardomain channel element indicates the channel strength for the path with specific AoD. Moreover, due to the limited number of scatterers around the BS, the spread of AoD will be very narrow in farfield propagation scenario. As a result, only a small fraction of elements in angulardomain channel matrix has significant amplitude. This results in the sparsity in angulardomain channel.
Mathematically, following a similar analytic approach as that in [10, 11], it can be shown that the angulardomain channel matrix has the following two kinds of sparse structures.

Joint sparsity among the columns of X_{k}: The nth element of x_{k,m} has a significant amplitude only when n∈Ω_{k}, where Ω_{k} is given by \({\Omega _{k}} = \left \{ n\left  {  \frac {1}{2} + \frac {{n  1}}{N} \in } \right.\left [ {\frac {1}{2}\sin \left ({{\theta _{k}}  {\Delta _{d}}} \right),\frac {1}{2}\sin \left ({{\theta _{k}} + {\Delta _{d}}} \right)} \right ], \right. \left.n \in {\mathbb N^ +} \right \}\). Since Ω_{k} is independent of the column index m, the indexes of the significant elements in each column of X_{k} are the same.

Burst sparsity in each column of X_{k}: The indexes of significant elements in x_{k,m} appear in block.
As discussed in the above, Δ_{d} is commonly small. Thus, only a small number of elements in x_{k,m} has a significant value. Physically, the joint sparsity of angulardomain channel is due to the fact that the size of the user’s array can be neglected in farfield propagation, and thus, the channels between the BS and all antennas of user can be considered to have similar property in the angular domain. Moreover, it is noted that the AoD of downink signal varies in a continuous interval [θ_{k}−Δ_{d},θ_{k}+Δ_{d}]. This causes the burst sparsity nature in the angulardomain channel. The two kinds of sparse structures are shown in Fig. 1 for an example with N=64 and M=4.
Here, we shall point out that, for the practical system with finite N, the nth element of x_{k,m} with n∉Ω_{k} is small but not zero as shown in Fig. 1. Therefore, x_{k,m} is in fact approximately sparse. Note that [16, 18] assumed that the angulardomain channel element is exactly zero if n∉Ω_{k}, which we call as exactly sparse channel. Although this assumption simplifies the model, it may change the real system performance in the high SNR region greatly as will be shown in the simulation of Section 5.
Structured priorbased Bayesian channel estimation
By substituting (4) in (3), the received training signal at user k can be rewritten as:
where \({\Phi } = {\textbf {S}^{H}}{\textbf {A}_{N}} \in {\mathbb C^{T \times N}}\). Note that as long as X_{k} is known, the channel matrix H_{k} can be recovered directly based on (4), that is, H_{k}=AX_{k} (which is called basis expansion model).
By treating Φ as the sensing matrix, X_{k} can be estimated using the conventional SBL [17] or OMP [15] approach from (6). However, the performance will be poor if the number of pilot symbols (i.e., T) is small as will be seen in the simulations. In this paper, we propose a SPSBL approach to estimate the channel reliably by exploiting the two kinds of sparse structures discussed in the last subsection.
Different from the conventional SBL where a i.i.d. Gaussian prior is utilized, we propose a structured prior for X_{k} as follows:
where γ_{k}=[γ_{k,1},⋯,γ_{k,M}]^{T} and α_{k}=[α_{k,1},⋯,α_{k,N}]^{T}. D_{k} an N×N diagonal matrix whose nth diagonal element is modeled as:
For the prior model in (7) and (8), the mth column of angulardomain channel matrix X_{k} has precision matrix γ_{k,m}D_{k}. Thus, precision matrices for different columns of X_{k} differ only with a scaling factor γ_{k,m}. This property captures the joint sparsity among the columns of X_{k}. The parameter γ_{k,m} is utilized to model the relative difference between the amplitudes of \(\phantom {\dot {i}\!}\left \{ {{\textbf {x}_{k,m}}} \right \}_{m = 1}^{M}\). By exploiting the joint sparsity, the recovery errors which predict zero for one element of x_{k,m} and predict nonzero for the element of \(\phantom {\dot {i}\!}{\textbf {x}_{k,m'}}\) (m≠m^{′}) in the same position can be mitigated effectively.
Additionally, the precision of the nth element of x_{k,m} can be expressed as γ_{k,m}(α_{k,n−1}+α_{k,n}+α_{k,n+1}). Thus, the precisions of adjacent elements in x_{k,m} are mutually coupled. When α_{k,n}→∞, the estimations of nth, (n−1)th, and (n+1)th elements of x_{k,m} will be driven to zero simultaneously. Therefore, the zero elements (and hence nonzero elements) in the estimation of x_{k,m} will appear in block, which just captures the block sparsity structure. By exploiting burst sparsity, the situation that a significant element of x_{k,m} is predicted as zero (or near zero), isolated in x_{k,m}, can be reduced greatly. Note that the precisions of the first and the last elements of x_{k,m} are also coupled. This structure captures a basic property of angulardomain channel, i.e., the elements at the beginning and that at the end of x_{k,m} tend to be close to zero or have large amplitude simultaneously.
According to (6), the likelihood function of the received training signal Y_{k} can be expressed as:
Using (7), (9), and the property of complex Gaussian distribution, the posterior distribution of x_{k,m} can be derived as:
where the posterior mean and covariance matrix of x_{k,m} are given by:
To give a fully Bayesian treatment, similar to conventional SBL, we introduce a gamma hyperprior for α_{k}:
where a and b are fixed parameters. Commonly, a and b are set as small values to impose a noninformative prior. In this paper, a and b are chosen as a=b=10^{−4}. Moreover, we introduce a Dirichlet hyperprior for γ_{k}:
where u=[u_{0},⋯,u_{M}] is a fixed parameter with \(u_{0} = \sum _{m=1}^{M} u_{m}\). \(C\left (\textbf {u} \right) = \frac {{\Gamma \left ({{u_{0}}} \right)}}{{\Gamma \left ({{u_{1}}} \right) \cdots \Gamma \left ({{u_{M}}} \right)}}\) is the normalization constant. Since the expectation of γ_{k,m} with respect to the distribution (13) is given by \(\mathbb E\left [ {{\gamma _{k,m}}} \right ] = \frac {{{u_{m}}}}{{{u_{0}}}}\), we can interpret u as the parameter which gives an initial guess on the relative difference between the amplitudes of \(\left \{\mathbf {x}_{k,m}\right \}_{m=1}^{M}\).
As in the convention of sparse Bayesian learning framework, the hyperpriors are utilized to imbed the prior knowledge of the parameters into the estimation algorithm. Moreover, as shown in [17], the utilization of gamma hyperprior is helpful to produce a sparse solution which is desired in our problem.
As long as α_{k} and γ_{k} are obtained, the maximum posterior (MAP) estimation of X_{k} can be given by its posterior mean in (11). Therefore, in the following, we focus on finding the optimal α_{k} and γ_{k} by solving the MAP problem:
Different from the conventional SBL, solving the above problem directly is quite challenging due to the utilization of the structured prior. To address this problem, we resort to the expectation maximization (EM) [20] algorithm to find a computationally efficient solution.
Solving the SPSBL using expectation maximization
Instead of solving the MAP problem in (14) directly, the EM algorithm tries to find the optimal α_{k} and γ_{k} by maximizing the expected completedata loglikelihood function, that is:
where the expectation is w.r.t. the posterior distribution of X_{k} given by (10). \({\hat {\alpha }}_{k}^{{\text {old}}}\) and \({\hat {\gamma }}_{k}^{{\text {old}}}\) denote the latest estimations of α_{k} and γ_{k}, respectively. Each iteration of the algorithm consists of an expectation step (Estep) and a maximization step (Mstep).
In the Estep of the ith iteration, the posterior distribution of X_{k} is computed approximately using the estimations of α_{k} and γ_{k} in the (i−1)th iteration \(\left (\text {denoted by}\ {\hat {\alpha }}_{k}^{(i1)}\ \text {and}\ {\hat {\gamma }}_{k}^{(i1)}\right)\) based on (10), which is then used to evaluate the expected completedata loglikelihood function.
In the Mstep of the ith iteration, the estimation of α_{k} and γ_{k} is updated to maximize the expected completedata loglikelihood function obtained in Estep.
Update of γ _{k}
By substituting (10) and (13) into (15) and discarding the terms irrelevant to γ_{k}, it can be shown that the expected completedata loglikelihood function reduces to"
Note that α_{k} has been fixed at its estimation after the last iteration, i.e., \({ \alpha }_{k} = \hat {\alpha }_{k}^{(i1)}\). \({{\hat {\mathbf {D}}_{k}}}\) can be computed using (8) by replacing α_{k} with \( \hat {\alpha }_{k}^{(i1)}\). \({{{\hat {\Sigma } }_{k,m}}}\) and \( {{\hat {{\mu }}}_{k,m}}\) can be computed using (11) by replacing {α_{k},γ_{k}} with \(\left \{{{{\hat {\alpha } }_{k}^{(i1)}}}, {{{\hat {\gamma } }_{k}^{(i1)}}}\right \}\). The firstorder derivative of \(Q\left ({{{\hat {\alpha }}_{k}^{(i1)}},{{\gamma }_{k}}} \right)\) w.r.t γ_{k,m} can be expressed as:
By setting (17) to zero, we can obtain the estimation for γ_{k,m} in the ith iteration:
Update of α _{k}
By substituting (10) and (12) into (15) and discarding the terms irrelevant to α_{k}, it can be shown that the expected completedata loglikelihood function reduces to:
where γ_{k} has been fixed at its estimation after the last iteration, i.e., \({ \gamma }_{k} ={\hat {\gamma }}_{k}^{(i1)}\). The firstorder derivative of \(Q\left ({{{{\alpha }}_{k}},{{\hat {\gamma } }_{k}^{(i1)}}} \right)\) w.r.t α_{k,n} can be expressed as:
with
In (21), the derivative w.r.t. α_{k,n} is correlated with α_{k,n−2}, α_{k,n−1}, α_{k,n+1}, and α_{k,n+2}. This makes the closedform solution for α_{k,n} unavailable. Nevertheless, we can obtain a valid solution using the gradientbased algorithm as follows:
where δ is the stepsize, and the gradient can be directly computed using (20).
By virtue of the EM’s properties, the above algorithm will always converge since each iteration is guaranteed to reduce the target function. However, the update formula of α_{k} in (22) is still complex due to the lack of analytic expressions. Moreover, (22) gives little insight into the basic mechanism of SPSBL. In the following, we derive a closedform solution for α_{k} by adding some heuristic assumptions.
When updating α_{k,n}, we temporarily assume that the nth element of x_{k,m} has the same variance with its neighbors, i.e., the (n−1)th and (n+1)th elements of x_{k,m}. In this case, we will have α_{k,n−2}=α_{k,n−1}=α_{k,n}=α_{k,n+1}=α_{k,n+2}. This assumption is reasonable if we do not have much prior knowledge about α_{k}. Note that this does not mean we will obtain an estimation of α_{k} with \(\hat {\alpha }_{k,n2}^{(i)}=\hat {\alpha }_{k,n1}^{(i)}=\hat {\alpha }_{k,n}^{(i)}=\hat {\alpha }_{k,n+1}^{(i)}=\hat {\alpha }_{k,n +2}^{(i)}\) because, with the incoming training signal, the estimation of α_{k,n} is rectified and is expected to get close to the true value. Under this assumption, (20) becomes:
By setting (23) to zero, we can obtain:
The numerical results show that the update formula in (24) gives rise to the similar performance with that using gradientbased update in (22). Moreover, the solution in (24) provides a clear insight into the difference between SPSBL and conventional SBL. With conventional SBL, the update of the precision for the nth element of x_{k,m} is only related to the posterior mean and variance of itself. In contrast, in SPSBL, the update of α_{k,n} is effected by the posterior means and variances of (n−1)th, nth, and (n+1)th elements of x_{k,m} for all m=1,⋯,M due to the utilization of joint and burst sparsity. Therefore, the SPSBL is expected to achieve better performance.
After the downlink channel estimation, the estimated CSI should be sent back to the BS in order to perform the downlink beamforming. In this process, additional error can be introduced by quantization, noise, and feedback delay. However, the results in [21] showed that the error due to the imperfect feedback can be made much smaller than that due to the estimation error in downlink training phase. Therefore, as in [11], we optimistically neglect the additional error due to the feedback in this paper.
Computation complexity analysis
The main computation load in each iteration is due to the N×N matrix inversion when updating the posterior mean and covariance matrix of x_{k,m} in Estep. By using the matrix inversion lemma, the calculation of each matrix inversion has complexity \({\mathcal {O}}\left (T^{3}\right)\). Therefore, the overall computational complexity of the algorithm scales with \({\mathcal {O}}\left (GKMT^{3}\right)\), where G is the number of EM iterations. Since we consider the situation that T is much smaller than N, the computation complexity will not pose a significant problem.
Extension to FDD massive MIMO system with hybrid analogdigital processing
In the practical system, the hybrid analogdigital processing may be utilized at the BS to reduce the implementation complexity. In the following, we will show that the proposed scheme can still be applied in this case. With hybrid analogdigital processing at the BS, the received training signal at user k can be expressed as:
where \(\mathbf {W}\in \mathbb C^{N \times F}\) is the arbitrary analog beamforming matrix utilized by BS, and F≤N denotes the number of radio frequency chain at the BS. The pilot signal S is an F×T matrix in this case. Note that we have reuse some notations to avoid introducing too many new definitions. Using the definition of angulardomain channel in (4), we can rewrite (25) as:
Note that, by treating (WS)^{H}A_{N} as the sensing matrix, (26) is in the same form with (6). Therefore, the proposed scheme can be utilized directly to estimation the channel matrix from (26).
Simulation results and discussion
This section presents the simulation results to validate the proposed scheme. Without loss of the generality, the largescale fading and noise variance are normalized to 1. For illustration, the number of users is set to K=8. The central frequency is set to 2.4 GHz, and the antenna spacing is equal to the half of wavelength. Similar to [22], the rows of sensing matrix Φ are designed as the lengthN ZadoffChu sequence [23] with shifting step 7. In particular, the first row of Φ is given by \(\frac {1}{{\sqrt N }}\left [ {1,{e^{j\frac {{\nu \pi {1^{2}}}}{N}}},{e^{j\frac {{\nu \pi {2^{2}}}}{N}}}, \cdots,{e^{j\frac {{\nu \pi {{\left ({N  1} \right)}^{2}}}}{N}}}} \right ]\). The second row of Φ is obtained by cyclically right shifting the first row with a step of 7, which is given by \(\frac {1}{{\sqrt N }}\left [ {{e^{j\frac {{\nu \pi {{\left ({N  7} \right)}^{2}}}}{N}}},{e^{j\frac {{\nu \pi {{\left ({N  6} \right)}^{2}}}}{N}}}, \cdots,{e^{j\frac {{\nu \pi {{\left ({N  1} \right)}^{2}}}}{N}}},1,{e^{j\frac {{\nu \pi {1^{2}}}}{N}}},{e^{j\frac {{\nu \pi {2^{2}}}}{N}}},} \right. \left. { \cdots,{e^{j\frac {{\nu \pi {{\left ({N  8} \right)}^{2}}}}{N}}}} \right ]\). Moreover, ν=7 is used in the simulations. The rest rows of Φ are generated in a similar way. The AoD θ_{k} and AoA φ_{k} for different users are generated randomly from the interval [−90°,90°]. It is assumed that the waves arriving at the user are uniformly distributed in AoA [12]. Therefore, Δ_{a} is equal to π, and the complex channel gain reduces to r_{k}(θ,φ)=r_{k}(θ). It is assumed that the r_{k}(θ) for different θ are uncorrelated. For each sample of θ, r_{k}(θ) is the complex Gaussian distributed with zero mean and variance U_{k}(θ), where U_{k}(θ) is the power angle spectrum. We model U_{k}(θ) as the truncated Laplacian distribution centered at θ_{k} [12]. The simplified update rule for α_{k} in (24) is utilized to reduce the complexity. The parameters in the hyperprior models (12) and (13) are set as a=b=10^{−4}, u_{1}=⋯=u_{M}, and u_{0}=M.
We consider four reference schemes in the simulations, i.e., the conventional SBL based on the original algorithm in [17], OMP [15], block l_{1}/l_{2} minimization [18], and variational SBL [19]. Note that the block l_{1}/l_{2} minimization and variational SBL can also exploit the joint sparsity among the angulardomain channel vectors between BS and different antennas of user.
We consider the NMSE performance which is defined as follows:
All figures are obtained by averaging the results for 10^{3} independent channel realizations.
Figure 2 shows the NMSE performance for different lengths of pilot sequence T, where the transmit SNR of BS is \(\frac {NP}{\sigma ^{2}} = 15\) dB and the angular spread is Δ_{d}=5°. It is seen that the performances of conventional SBL and OMP are poor when the length of pilot sequence is small. By exploiting the joint sparsity among the angulardomain channel vectors between BS and different antennas of user, the block l_{1}/l_{2} minimization and variational SBL can provide better performance. Moreover, the proposed scheme based on SPSBL achieves the best NMSE performance since it exploits the joint sparsity and burst sparsity simultaneously.
Figure 3 shows the NMSE performance for different transmit SNRs of BS, where the length of pilot sequence is T=16 and the angular spread is Δ_{d}=5°. It is seen that the performances of all schemes converge to NMSE floors for large SNR. The reason is that the angulardomain channel is approximately sparse as discussed in Section 4.1. That is, the nth element of x_{k,m} with n∉Ω_{k} is small but not zero. With small T, recovering all these small elements of x_{k,m} with high accuracy is difficult. As a result, with the increasing of the SNR, the effect of the mismatch between the true values of these small elements and their estimates becomes significant when compared with AWGN, which incurs NMSE floor in the high SNR region^{Footnote 1}. Note that the NMSE floor does not occur in the simulations of [16, 18]. This is because these works assume that there is no channel power leakage outside Ω_{k}. Although this assumption simplifies the model, the resultant simulations cannot reflect the real performance in the high SNR region very well. For illustration, we also present the NMSE performance for all schemes in Fig. 4 under exactly sparse angulardomain channel model considered in [16, 18], where the nth element of x_{k,m} is set to zero as long as n∉Ω_{k} when generating the channel matrix. From the figure, we can see that the performance floors disappear as expected.
Then, we consider the performance for different angular spreads Δ_{d} ranged from 3 to 18°. This corresponds to the scenarios with scattering ring of 30 m and the distance between BS and user varying from about 100 to 500 m. Figure 5 shows the number of significant elements in the angulardomain channel x_{k,m}, i.e., the number of elements that contain 90% of the channel power, varies from 3 to 17 when the angular spread increases from 3 to 18°. Figure 6 shows the NMSE performance for different angular spreads, where the transmit SNR of BS is \(\frac {NP}{\sigma ^{2}} = 15\) dB and the length of pilot sequence is T=16. Again, the proposed scheme based on SPSBL achieves the best performance. Moreover, we note that the NMSEs of all schemes degrade when the angular spread increases. This is because the number of significant elements in angulardomain channel becomes larger as shown in Fig. 5. In this case, to maintain the estimation performance, the capacity of all schemes must be enhanced by increasing the training samples.
Figure 7 shows the NMSE performance for different numbers of antennas at user, i.e., M. The length of pilot sequence is T=16. The transmit SNR of BS is \(\frac {NP}{\sigma ^{2}} = 15\) dB. The angular spread is Δ_{d}=5°. It is seen that the NMSEs of conventional SBL and OMPbased schemes are independent of M. In contrast, the performances of block l_{1}/l_{2} minimization, variational SBL, and SPSBLbased schemes are improved gradually with the increasing of M, which is just the benefit of exploiting the joint sparsity.
Figure 8 shows the NMSE performance for different numbers of iterations. The length of pilot sequence is T=16. The transmit SNR of BS is \(\frac {NP}{\sigma ^{2}} = 15\) dB. The angular spread is Δ_{d}=5°. It is seen that the NMSE performance converges after 40 EM iterations. Moreover, the number of iterations required for SPSBL is smaller than that of block l_{1}/l_{2} minimization and greater than that of conventional SBL and variational SBL. The computation complexities in each iteration for SPSBL, block l_{1}/l_{2} minimization, conventional SBL, and variational SBL are \({\mathcal {O}}\left (KMT^{3}\right)\), \({\mathcal {O}}\left (KMNT\right)\) [18], \({\mathcal {O}}\left (KMT^{3}\right)\) [17], and \({\mathcal {O}}\left (KMT^{3}\right)\) [19], respectively. Note that the OMP needs no iteration and its total computation complexity is \({\mathcal {O}}\left (\eta KMNT\right)\) [15], where η denotes the number of significant elements in x_{k,m}. Therefore, from the analysis above, the SPSBL has the highest computation complexity but converges to best NMSE.
Conclusions
This paper proposes a SPSBL approach for downlink channel estimation in FDD massive MIMO system. By exploiting the two kinds of sparse structures, the scheme can substantially reduce the pilot resource assumption while maintaining the estimation performance. Through numerical simulations, it is shown that the proposed scheme outperforms the reference schemes significantly in terms of NMSE for a variety of scenarios with different lengths of pilot sequence, transmit SNRs, and angular spreads.
Notes
 1.
To address this problem, a possible solution is to exploit other inherent structures of angulardomain channel in the prior model, for example, the additional correlation between the phases and amplitudes of the elements in angulardomain matrix if they exist. In general, the problem is quite challenging and will be considered as future study.
Abbreviations
 AoA:

Angle of arrival
 AoD:

Angle of departure
 AWGN:

Additive white Gaussian noise
 BS:

Base station
 CSI:

Channel state information
 DFT:

Discrete Fourier transform
 EM:

Expectation maximization
 FDD:

Frequency division duplex
 IoT:

Internet of Things
 LS:

Least square
 MIMO:

Multipleinput multipleoutput
 MMSE:

Minimum mean square error
 NMSE:

Normalized mean square error
 OMP:

Orthogonal matching pursuit
 SBL:

Sparse Bayesian learning
 SNR:

Signaltonoise ratio
 SPSBL:

Structured priorbased sparse Bayesian learning
 ULA:

Uniform linear array
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Funding
This work is supported by National Natural Science Foundation of China (No. 61671472) and Jiangsu Province Natural Science Foundation (BK20160079).
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XX provided the main idea and is responsible for the manuscript writing. YW conducted the simulations. KX, WX, and MW assisted with the revision of the manuscript. All authors have read and approved the final manuscript.
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Wang, Y., Xia, X., Xu, K. et al. Channel estimation for FDD massive MIMO system by exploiting the sparse structures in angular domain. J Wireless Com Network 2019, 32 (2019). https://doi.org/10.1186/s1363801913530
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Keywords
 FDD massive MIMO
 Channel estimation
 Structured prior
 Sparse Bayesian learning
 Normalized mean square error