Abstract
This paper considers the channel estimation problem for massive multipleinput multipleoutput (MIMO) systems that use onebit analogtodigital converters (ADCs). Previous channel estimation techniques for massive MIMO using onebit ADCs are all based on singleshot estimation without exploiting the inherent temporal correlation in wireless channels. In this paper, we propose an adaptive channel estimation technique taking the spatial and temporal correlations into account for massive MIMO with onebit ADCs. We first use the Bussgang decomposition to linearize the onebit quantized received signals. Then, we adopt the Kalman filter to estimate the spatially and temporally correlated channels. Since the quantization noise is not Gaussian, we assume the effective noise as a Gaussian noise with the same statistics to apply the Kalman filtering. We also implement the truncated polynomial expansionbased lowcomplexity channel estimator with negligible performance loss. Numerical results reveal that the proposed channel estimators can improve the estimation accuracy significantly by using the spatial and temporal correlations of channels.
Introduction
Massive multipleinput multipleoutput (MIMO) systems are one of the promising techniques for nextgeneration wireless communication systems [2–5]. By using a large number of antennas at base stations (BSs), it is possible to support multiple users simultaneously to boost network throughput and improve the energyefficiency by beamforming techniques [4]. Due to the large number of antennas at the BS, high implementation cost and power consumption could be major problems for implementing massive MIMO in practice.
Using lowresolution analogtodigital converters (ADCs) is an effective way of mitigating the power consumption problem in massive MIMO systems because the ADC power consumption exponentially decreases as its resolution level [6]. However, symbol detection and channel estimation in massive MIMO systems with lowresolution ADCs become difficult tasks because the quantization process using lowresolution ADCs becomes highly nonlinear. Recent works have revealed that it is possible to implement practical symbol detectors and channel estimators for massive MIMO even with lowresolution ADCs. For the symbol detection, a massive spatial modulation MIMO approach based on sumproductalgorithm was developed in [7], a convex optimizationbased multiuser detection for massive MIMO with lowresolution ADC was considered in [8], a mixedADC massive MIMO detector was proposed in [9], and a blind detection technique was developed by exploiting supervised learning [10]. Also, an iterative detection and decoding scheme based on the messagepassing algorithm and lowresolution aware (LRA) minimum mean square error (MMSE) receive filter was presented in [11], a lowcomplexity maximum likelihood detection (MLD) algorithm called onebitspheredecoding was developed in [12], and a successive cancelation softoutput detector by exploiting a previous decoded message was proposed in [13].
For the channel estimation, a nearmaximum likelihood channel estimator based on the convex optimization was developed in [14], and a Bayesoptimal joint channel and data estimator was proposed in [15]. To reduce the complexity, the generalized approximate messagepassing algorithm was applied in [16], and the hybrid architectures were considered in [17]. Moreover, an oversampling based LRAMMSE channel estimator that exploits the correlation of filtered noise for a given channel was proposed in [18]. However, up to the authors’ knowledge, the previous channel estimators with lowresolution ADCs have not considered the temporal correlation in channels, which is inherent in communication channels.
In this paper, we develop a channel estimator taking both spatial and temporal correlations into consideration for massive MIMO systems with onebit ADCs. We first discuss how to estimate the spatial correlation matrix for the channel estimation. Then, we reformulate the nonlinear onebit quantizer to the linear operator based on the Bussgang decomposition[19]. To exploit both the spatial and temporal correlations, we implement the Kalman filterbased (KFB) estimator [20] assuming the statistically equivalent quantization noise after the Bussgang decomposition follows a Gaussian distribution with the same mean and covariance matrix. The numerical results demonstrate that the normalized mean square error (NMSE) of the proposed KFB estimator decreases as the time slot increases. Moreover, as channels are more correlated with space and time, it is possible to track the channels more accurately. To reduce the complexity of KFB estimator, which comes from the large size matrix inversion, we also exploit a truncated polynomial expansion (TPE) approximation for the matrix inversion in the Kalman gain matrix. We analytically show that, with some moderate assumptions, the NMSE of the TPEbased estimator also keeps decreasing with the time slots. The numerical results show that the lowcomplexity TPEbased estimator gives approximately the same performance as the KFB estimator even with low approximation orders.
The rest of the paper is organized as follows. In Section 2, we explain a system model with onebit ADCs. In Section 3, we first discuss how to estimate the spatial correlation matrix. Then, we review the singleshot channel estimator based on the Bussgang decomposition [21]. After, we explain our proposed successive channel estimator based on the Bussgang decomposition and the Kalman filter. We also propose the lowcomplexity TPEbased channel estimator and analyze the complexities of competing estimators. After explaining the data transmission with onebit ADCs in Section 4, we evaluate numerical results in Section 5. Finally, we conclude the paper in Section 6.
Notation: Lower and upper boldface letters represent column vectors and matrices. A^{T},A^{∗},A^{H}, and A^{†} denote the transpose, conjugate, conjugate transpose, and pseudo inverse of the matrix A. \(\mathbb {E}\{\cdot \}\) represents the expectation, and ℜ{·},I{·} denote the real part and imaginary part of the variable. 0_{m} is used for the m×1 all zero vector, and I_{m} denotes the m×m identity matrix. ⊗ denotes the Kronecker product. diag(·) returns the diagonal matrix. vec(·) denotes the columnwise vectorization. \({\mathbb {C}}^{m \times n}\) and \({\mathbb {R}}^{m \times n}\) represent the set of all m×n complex and real matrices. · denotes the amplitude of the scalar, and ∥·∥ represents the ℓ_{2}norm of the vector. \(\mathcal {C} \mathcal {N}(m,\sigma ^{2})\) denotes the complex normal distribution with mean m and variance σ^{2}. tr(·) represents the trace operator. \(\mathcal {O}\) denotes the BigO notation.
System model
As in Fig. 1, we assume a singlecell massive MIMO system with M BS antennas and K singleantenna users with M≫K. Each BS antenna is connected to two onebit ADCs, one for the inphase component and the other for the quadrature component of received signals. We consider the blockfading channel with the coherence time of T. The received signal at the ith fading block is given by
where ρ is the signaltonoise ratio (SNR), \(\mathbf {H}_{i}=[\mathbf {h}_{i,1},\mathbf {h}_{i,2},...,\mathbf {h}_{i,K}]\in {\mathbb {C}}^{M\times K}\) is the channel matrix, h_{i,k} is the channel between the BS and the kth user in the ith fading block, s_{i} is the transmit signal, and \(\mathbf {n}_{i}\sim \mathcal {C} \mathcal {N}(\boldsymbol {0}_{M},\mathbf {I}_{M})\) is the complex Gaussian noise. We consider the spatially and temporally correlated channels by assuming h_{i,k} follows the firstorder GaussMarkov process,
where η_{k} is the temporal correlation coefficient, \(\mathbf {R}_{k}={\mathbb {E}}\left \{\mathbf {h}_{i,k}\mathbf {h}_{i,k}^{\mathrm {H}}\right \}\) is the spatial correlation matrix, and \(\mathbf {g}_{i,k}\sim \mathcal {C} \mathcal {N}(\boldsymbol {0}_{M},\mathbf {I}_{M})\) is the innovation process of the kth user in the ith fading block. Note that η_{k} and R_{k} do not have the time index i since both are longterm statistics that are static for multiple coherence blocks.
Although other models are also possible, to have concrete analyses, we adopt the exponential model for the spatial correlation matrix R_{k},
where \(\phantom {\dot {i}\!}r_{k}=re^{j\theta _{k}}\) satisfying 0≤r<1 and 0≤θ_{k}<2π. We assume all users experience the same spatial correlation coefficient r since it is dominated by the BS antenna spacing while each user has an indifferent phase θ_{k} since it is more related to the user position [22].
The quantized signal by the onebit ADCs is
where \(\mathcal {Q}(\cdot)\) is the elementwise onebit quantization operator, i.e., \({\mathcal {Q}}(\cdot)=\frac {1}{\sqrt {2}}(\text {sign}({\Re }\{\cdot \})+j~\text {sign}({\Im }\{\cdot \})).\)
Channel estimation with onebit ADCs
In this section, we first discuss how to estimate the spatial correlation matrix. Then, we explain the Bussgang linear minimum mean square error (BLMMSE) estimator, which is the baseline of the proposed estimator. The BLMMSE estimator is a singleshot channel estimator based on the Bussgang decomposition without exploiting any temporal correlation [21]. Then, we propose the KFB estimator, which is a successive channel estimator, for massive MIMO with onebit ADCs exploiting the temporal correlation. Also, we propose the lowcomplexity TPEbased estimator to reduce the complexity of the proposed KFB estimator.
Spatial correlation matrix estimation
In this subsection, we discuss how to estimate the spatial correlation matrix since all the channel estimators in this paper exploit the spatial correlation of channel. We omit the user index k since the BS can estimate the spatial correlation of each user separately.
When the BS does not have any prior channel information, it can use the least square (LS) estimate of the quantized signal r_{i}, which is given by
where Φ_{i} is the pilot matrix. The LS estimator for onebit quantized signal performs well when the number of antennas at the BS is large, as shown in [23]. The BS then can obtain a sampled spatial correlation matrix as
where N_{s} is the number of samples. We evaluate the performance loss by using the sampled correlation matrix in Fig. 3 in Section 5. After this subsection, we assume that the true spatial correlation matrices and the temporal correlation coefficients of all users are known to the BS to derive analytical results.
BLMMSE estimator
In this subsection, we omit the time slot index i since the singleshot channel estimator does not use any temporal correlation. To estimate the channel at the BS, K users transmit the length τ pilot sequences to the BS,
where \(\mathbf {Y}\in {\mathbb {C}}^{M\times \tau }\) is the received signal, \({\boldsymbol {\Phi }} \in {\mathbb {C}}^{\tau \times K}\) is the pilot matrix, and \(\mathbf {N}=[\mathbf {n}_{1},\mathbf {n}_{2},...,\mathbf {n}_{\tau }]\in {\mathbb {C}}^{M\times \tau }\) is the complex Gaussian noise. We assume that the pilot sequences are columnwise orthogonal, i.e., Φ^{T}Φ^{∗}=τI_{K}, and all the elements of the pilot matrix have the same magnitude. For the sake of simplicity, the receive signal is vectorized as
where \(\bar {\boldsymbol {\Phi }}=(\boldsymbol {\Phi }\otimes \sqrt {\rho }\mathbf {I}_{M}), \underline {\mathbf {h}}=\text {vec}(\mathbf {H})\), and \(\underline {\mathbf {n}} = \text {vec}(\mathbf {N})\). The quantized signal by onebit ADCs is
Assuming independent spatial correlations across the users, the aggregated spatial correlation matrix \(\mathbf {R}={\mathbb {E}}\{\underline {\mathbf {h}} \underline {\mathbf {h}}^{\mathrm {H}}\} \) is given by
The Bussgang decomposition of quantized signal is given by
where A denotes the linear operator and q represents the statistically equivalent quantization noise. The linear operator A is obtained from [21],
where \(\mathbf {C}_{\underline {\mathbf {y}}}\) is the autocovariance matrix of the received signal. In (12), (a) is derived in Appendix Appendix 1: Proof of (12). Substituting (8) into (11), \(\underline {\mathbf {r}}\) is represented as
where \(\tilde {\boldsymbol {\Phi }}=\mathbf {A}\bar {\boldsymbol {\Phi }}\in {\mathbb {C}}^{M\tau \times {MK}}\) and \( \tilde {\underline {\mathbf {n}}}=\mathbf {A}\underline {\mathbf {n}}+\mathbf {q}\in {\mathbb {C}}^{M\tau \times 1}\).
After adopting the Bussgang decomposition, a linear MMSE estimator, which is denoted as the BLMMSE channel estimator [21], is given as
where \(\mathbf {C}_{\underline {\mathbf {h}}}\) is the autocovariance matrix of the channel \(\underline {\mathbf {h}}\), and \(\mathbf {C}_{\underline {\mathbf {r}}}\) is the autocovariance matrix of the quantized signal \(\underline {\mathbf {r}}\). In (14), \(\mathbf {C}_{\underline {\mathbf {r}}}\) is obtained by the arcsin law [24],
where \(\Sigma _{\underline {\mathbf {y}}}=\text {diag}(\mathbf {C}_{\underline {\mathbf {y}}})\).
Proposed KFB estimator
Although effective, the BLMMSE estimator does not exploit any inherent temporal correlation in wireless channels. We now propose a simple, yet effective, channel estimator based on the Bussgang decomposition and the Kalman filtering. We recover the time slot index i to explicitly use the temporal correlation. We first reformulate the channel model in (2) by vectorization,
where \(\underline {\mathbf {g}}_{i}\) is the vectorized innovation process, which is expressed as
The temporal correlation matrices η and ζ in (16) are given by the Kronecker product,
where η_{k} denotes the kth user temporal correlation coefficient and \(\zeta _{k}=\sqrt {1\eta _{k}^{2}}\).
Following the same steps as in Section 3.2, the onebit quantized signal can be represented using the Bussgang decomposition as
where A_{i} is the linear operator, q_{i} is the statistically equivalent quantization noise, \(\tilde {\boldsymbol {\Phi }}_{i}=\mathbf {A}_{i}\bar {\boldsymbol {\Phi }}_{i}\in {\mathbb {C}}^{M\tau \times {MK}}\), and \( \tilde {\underline {\mathbf {n}}}_{i}=\mathbf {A}_{i}\underline {\mathbf {n}}_{i}+\mathbf {q}_{i}\in {\mathbb {C}}^{M\tau \times 1}\).
The Kalman filter guarantees the optimality when the noise is Gaussian distributed [20]; however, the effective noise \(\tilde {\underline {\mathbf {n}}}_{i}\) in (21) is not Gaussian because of the onebit quantization noise q_{i}. Although the noise is not Gaussian, it is still possible to apply the Kalman filter using the same covariance matrix \(\mathbf {C}_{\tilde {\underline {\mathbf {n}}}_{i}}\). The proposed KFB channel estimator is summarized in Algorithm 1.
Remark 1
Assuming the effective noise is Gaussian distributed may result in inaccurate channel estimation. This effect becomes more dominant as SNR increases, which is shown in Fig. 4 in Section 5. In the high SNR regime, the noise \(\underline {\mathbf {n}}_{i}\) in (1) becomes negligible, and the effective noise \(\tilde {\underline {\mathbf {n}}}_{i}\) in (21) is dominated by the quantization noise q_{i}, which would severely violate the Gaussian assumption of \(\tilde {\underline {\mathbf {n}}}_{i}\). In the low SNR regime, however, the effective noise \(\tilde {\underline {\mathbf {n}}}_{i}\) is more like Gaussian, and the proposed KFB estimator is nearly optimal.
Lowcomplexity TPEbased estimator
The BLMMSE estimator is a singleshot estimator, which returns a new channel estimate while the KFB estimator is a successive channel estimator, which tracks the channel based on a previous channel estimate at each time slot. Thus, the complexity of both channel estimators is the same at each time slot.
The matrix inversion has the most dominant computation complexity among matrix operations. The large channel dimensions in massive MIMO systems even exacerbate the complexity of matrix inversion. Therefore, when comparing the complexity of algorithms, we only consider the complexity of the matrix inversion. To reduce the complexity of KFB estimator, the truncated polynomial expansion [25] can be used to approximate the matrix inversion at the Kalman gain matrix K_{i} in Step 4 of Algorithm 1.
The L^{th}order TPE approximation of the inversion of N×N matrix X is expressed as
In (22), α is the convergence coefficient, which can be set as \(0<\alpha <\frac {2}{\max _{n} \lambda _{n}(\mathbf {X})}\) where λ_{n}(X) is the nth eigenvalue of the matrix X[25].
The complexity of TPE approximation in (22) is \(\mathcal {O}(LN^{2})\) since it has only the matrix multiplication with the L^{th}order. This is a large complexity reduction as compared to \(\mathcal {O}(N^{3})\) for the complexity of the N×N matrix inversion when L is much smaller than N. In Table 1, we summarize the complexity of three competing estimators. The TPEbased estimator has much lower computational complexity than the other estimators because L≪Mτ in practice.
To verify the effectiveness of TPE approximation, we evaluate the minimum NMSE of TPEbased estimator. For a tractable analysis, we assume R=I_{MK} and τ=K as in [21], which results in \(\mathbf {C}_{\underline {\mathbf {r}}}=\mathbf {I}_{MK}\) in (15) since \(\mathbf {C}_{\underline {\mathbf {y}}}=(K\rho +1)\mathbf {I}_{MK}\). Then the NMSE of BLMMSE estimator in [21], which is a performance baseline of the proposed estimators, is represented as
where \(\beta =\frac {2}{\pi }\frac {K\rho }{K\rho +1}\).
To derive the NMSE of TPEbased estimator, we first expand the covariance matrix of q_{i} as
where we define
In (24), (a) comes from \(\mathbf {A}_{i}=\sqrt {\frac {2}{\pi }}\sqrt {\frac {1}{K\rho +1}}\mathbf {I}_{MK}\) and \(\mathbf {C}_{\underline {\mathbf {y}}_{i}}=(K\rho +1)\mathbf {I}_{MK}, (b)\) is from the arcsin law in [24], and (c) is derived by substituting \(\mathbf {C}_{\underline {\mathbf {y}}_{i}}=(K\rho +1)\mathbf {I}_{MK}\) into (25).
The firstorder TPE approximation of matrix inversion in Kalman gain matrix is given by
Thus, the Kalman gain matrix is approximated as
We define the normalized trace of M_{ii−1} and M_{ii} as
where we denote m_{ii−1} as the prediction NMSE and m_{ii} as the minimum NMSE.
We assume that the temporal correlation coefficient is identical for all users, i.e., η_{k}=η for all k. With this assumption, we can further expand m_{ii−1} and m_{ii} as
and
where (a) is derived by the Kalman gain matrix approximation in (27), (b) is from
(c) comes from
and (d) is derived by the fact that M_{ii−1} and M_{ii} are diagonal matrix based on the mathematical induction with M_{00}=R=I_{MK}.
Now, we will show that m_{ii}<m_{i−1i−1}, i.e., the minimum NMSE decreases as the time slot index i increases. It is enough to show that m_{ii−1} is a monotonic decreasing sequence since m_{ii} and m_{ii−1} has a linear relationship in (29),
First, we can reformulate (29)
We define f(x) as
Then, in Appendix Appendix 2: Proof of (36), we prove
where γ is the root of f(x)=x. In (36), we exploited the condition 0<α<2 that is proved in Appendix Appendix 3: Proof of 0<α<2. Thus, we conclude
which is equivalent to m_{ii}<m_{i−1i−1}. Furthermore, we prove
in Appendix Appendix 4: Proof of (38). Therefore, the prediction NMSE m_{ii−1} decreases as the time slot index i increases and converges to γ. Also, we can easily check that m_{11}=1−(2α−α^{2})β=1−β=NMSE_{BLM} with m_{10}=1 and α=1. After many time instances, we will have
and the TPEbased estimator would outperform the BLMMSE estimator.
So far, we assume that R=I_{MK}, i.e., spatially uncorrelated channels, to derive the NMSE of the TPE estimator. Even for spatially correlated channels, the numerical results in Section 5 show that the TPEbased estimator outperforms the BLMMSE estimator.
Uplink data transmission
In this section, we derive the achievable sumrate of massive MIMO with onebit ADCs following similar steps as in [21] for the sake of completeness. The K users transmit data symbols to the BS. Based on the Bussgang decomposition, the quantized signal in the ith time slot can be represented as
where s_{i} is the transmit signal satisfying \({\mathbb {E}}\{s_{i,k}^{2}\}=1\), and the subscript d denotes the data transmission. The linear operator in (40) can be approximated as
In (41), (a) is from the channel hardening effect in massive MIMO systems as in [21]. After applying the receive combiner for the quantized signal, we have
where W_{i} is the receive combining matrix, \(\hat {\mathbf {H}}_{i} =\text {unvec}({\hat {\underline {\mathbf {h}}}_{i}})\) is the unvectorized channel estimation matrix, and \(\boldsymbol {\mathcal {E}}_{i}=\mathbf {H}_{i}\hat {\mathbf {H}}_{i}\) is the estimation error matrix. The kth element of \(\hat {\mathbf {s}}_{i}\) can be represented as
where \(\mathbf {w}_{i,k},\hat {\mathbf {h}}_{i,k}\) and ε_{i,k} represent the kth columns of \(\mathbf {W}_{i},\hat {\mathbf {H}}_{i}\), and \(\boldsymbol {\mathcal {E}}_{i}\), respectively.
We can obtain a lower bound on the achievable rate of the kth user by treating the uncorrelated interuser interference (IUI) and the quantization noise (QN) q_{d,i} as a Gaussian noise [26], and assuming the Gaussian channel input as in [21],
where
The autocovariance matrix of q_{d,i} is given by
where we define
In (46), \(\phantom {\dot {i}\!}\mathbf {C}_{\mathbf {r}_{d,i}}\) can be obtained by the arcsin law in (15), and (a) comes from the approximation of the low SNR as in [21]. This approximation holds even in correlated channels, which is different from (24) that is based on the assumption R=I_{MK}. We define the achievable sumrate as
To reduce the interference, we adopt the zeroforcing (ZF) combiner,
for numerical studies.
Results and discussion
In this section, we verify the proposed channel estimator by MonteCarlo simulation. We define the NMSE as the performance metric,
where \(\underline {\hat {\mathbf {h}}}\) is the channel estimate and \(\underline {\mathbf {h}}\) is the true channel. We adopt the pilot matrix Φ by the discrete Fourier transform (DFT) matrix, which satisfies the assumptions in Section 3.2, and select K columns of τ×τ DFT matrix with τ≥K to obtain the pilot sequences. We adopt the Jakes’ model for the temporal correlation, which is given as η_{k}=J_{0}(2πf_{D,k}t) where J_{0}(·) denotes the 0th order Bessel function, f_{D,k} is the Doppler frequency, and t is the channel instantiation interval. For simulations, we set f_{D,k}=v_{k}f_{c}/c with the user speed v_{k}, the carrier frequency f_{c}=2.5 GHz, and the speed of light c=3×10^{8} m·s^{−1}. We also set t=5 ms [27]. We denote NMSE(h_{i}) as the NMSE of KFB estimator at the ith time slot and \({\text {NMSE}}(\mathbf {M})=\frac {1}{MK}\text {tr}(\mathbf {M}_{ii})\) as the theoretical NMSE of Kalman filtering with the Gaussian noise, not the quantization noise. Therefore, NMSE(M) gives the performance limit of Kalman filtering with the Gaussian noise. We depict the “BLMMSE” as the NMSE performance of the singleshot channel estimator discussed in Section 3.2.
In Fig. 2, we compare the NMSEs of BLMMSE estimator and KFB estimator with the time slot i for r=0.5 or r=0.8 with SNR = −5 dB. We assume the BS antennas M=128, the users K=8, and the symbols τ=8. We set the temporal correlation coefficient η_{k}=0.988, which corresponds to v=3 km/h. As the time slot increases, the proposed KFB estimator outperforms the BLMMSE estimator. By comparing NMSE(h_{i}) and NMSE(M), the loss from using onebit ADCs is around 1.5 dB. As the amount of spatial correlation increases from 0.5 to 0.8, all estimators perform better since it becomes easier to estimate channels as the channels become more correlated in space [28–30].
In Fig. 3, we compare the NMSEs of the channel estimators with and without the perfect spatial correlation knowledge. All the parameters are the same as in Fig. 2 with r=0.8. Without the spatial correlation knowledge, we use N_{s} samples to estimate the spatial channel correlation by the LS estimates; then, we estimate the channel. When we use N_{s}=500,1000, the performance loss is about 4, 2 dB compared to the case of perfect correlation knowledge. Although the performance degradation due to the imperfect knowledge of spatial correlation is nonnegligible, the loss is inevitable for the channel estimators, including the BLMMSE estimator, that exploit the spatial correlation. The KFB estimator outperforms the BLMMSE estimator even with the sample correlation matrix, and as time slot increases, the KFB estimator using the sample correlation matrix achieves lower NMSE than the BLMMSE estimator using the true spatial correlation matrix.
Figure 4 depicts the NMSEs of the KFB estimator when each user experiences different temporal fading. We set r=0.8 and the temporal correlation coefficient of user 1 to 4 as η_{k}=0.872,0.936,0.967, and 0.988, which correspond to v_{k}=10 km/h,7 km/h,5 km/h, and 3 km/h. All other settings are the same as in Fig. 2. As expected, the users with high temporal correlations benefit more from the KFB estimator. Even the user with the moderate velocity of 10 km/h also has the gain more than 1 dB.
In Figs. 5 and 6, we compare the NMSEs of the KFB estimator according to SNR with different time slots when M=128, K=8, τ=8, η_{k}=0.988,0.724 (correspond to v_{k}=5,15 km/h), and r=0.5. When the temporal correlation is high, the NMSEs of the KFB estimator decreased as the SNR increased in low SNR regime. In the low SNR regime, NMSE(h_{i}) is almost the same as the theoretical NMSE of NMSE(M) after 10 successive estimations. In the high SNR, however, the NMSE of KFB estimator suffers from the saturation effect, which is referred as the stochastic resonance due to onebit quantization noise [31]. In the proposed KFB estimator, the loss also comes from the Gaussian model mismatch in the onebit quantization as explained in Remark 1 in Section 3.3. When the temporal correlation is low, the NMSEs of the KFB estimator decreased as the SNR increased in all SNR regime. This is because the channel estimation error comes mostly from the large temporal channel variation, not from the onebit quantization.
In Figs. 7 and 8, we compare the achievable sumrates of the BLMMSE estimator and KFB estimator according to the time slot when M=128, K=8, τ=8, r=0.8, and SNR = 0 and 10 dB. We assume all users experience the same η. In both scenarios, the achievable sumrate of the KFB estimator outperforms the BLMMSE estimator as the time slot increases.
In Figs. 9 and 10, we compare the NMSEs of the KFB estimator and the lowcomplexity TPEbased estimator with the time slot. We set M=128, K=8,τ=8,r=0.5,η_{k}=0.988,0.872, and SNR = −5,10 dB. We numerically optimize α=0.5 for the TPEbased estimator. In the high temporal correlation and low SNR case (Fig. 9), the NMSE gap between the KFB and TPEbased estimators is negligible and already quite small even with L=1. In the low temporal correlation and high SNR case (Fig. 10), the performance is degraded but the gap becomes small with L=2. Therefore, in practice, the lowcomplexity TPEbased estimator can be used with negligible performance loss.
Conclusion
In this paper, we proposed the Kalman filterbased (KFB) channel estimators that exploit both the spatial and temporal correlations of channels for massive MIMO systems using onebit ADCs. We adopted the Bussgang decomposition to linearize the nonlinear effect from onebit quantization. Based on the linearized model and assuming the effective noise as Gaussian, we exploited the Kalman filter to estimate the channel successively. The proposed KFB estimator has a remarkable gain compared to the previous estimator in [21], which does not exploit any temporal correlation in channels. To resolve the complexity issue of the KFB estimator due to the largescale matrix inversion, we also implemented the truncated polynomial expansion (TPE)based estimator. We analytically derived the minimum NMSE based on the firstorder TPE approximation, and the numerical results showed that the lowcomplexity TPEbased estimator gives nearly the same accuracy as the KFB estimator even with lower approximation orders.
Methods
This paper studies the channel estimation problem for massive MIMO systems using onebit ADCs. The performance of proposed algorithm was evaluated by different settings and metrics, which are NMSE and achievable sumrate. We use MATLAB R2018a to simulate the algorithm.
Appendix 1: Proof of (12)
We first expand \(\bar {\boldsymbol {\Phi }}{\mathbf {R}}\bar {\boldsymbol {\Phi }}^{\mathrm {H}}\) as,
where (a) comes from the independent spatial correlation matrix R in (10) and (b) follows by assuming that the pilot sequences are columnwise orthogonal with the same magnitude for all elements. Since the diagonal term of R_{k} is 1 for all k, we have
which finishes the proof.
Appendix 2: Proof of (36)
First, we define g(x) as
Then, we have
where the inequality of g(1) is due to 0<α<2. The roots of g(x) are \(\gamma _{0^{}}\in (\infty,0), \gamma \in (0,1), \gamma _{1^{+}}\in (1,\infty)\) since g(x) is the thirdorder polynomial, and g(−∞)g(0)<0,g(0)g(1)<0,g(1)g(∞)<0 based on the intermediate value theorem. Therefore, γ is the unique solution of g(x)=0 on x∈(0,1).
Now, we will show g(x)<0 for 0<γ<x<1. The derivative of g(x) is given by
Then, we have
since 0<α<2. This result implies g^{′}(x)<0 for 0<x<1 because g^{′}(x) is the secondorder polynomial with the positive leading coefficient 3η^{2}α^{2}β^{2}. Since g^{′}(x)<0 for 0<x<1 and g(γ)=0, then g(x)<g(γ)=0 for 0<γ<x<1, which finishes the proof.
Appendix 3: Proof of 0<α<2
We first reformulate \(\max _{n}\lambda _{n}\Big (\mathbf {C}_{\tilde {\underline {\mathbf {n}}}_{i}}+\tilde {\boldsymbol {\Phi }}_{i}\mathbf {M}_{ii1}\tilde {\boldsymbol {\Phi }}_{i}^{\mathrm {H}}\Big)\) as,
where (a) comes from the fact that M_{ii−1} is a diagonal matrix, and (b) is from 0<γ≤m_{ii−1}≤1. By plugging (57) into the bound on α,
we have the tightened bound
Appendix 4: Proof of (38)
Based on the mathematical induction, we assume
First, we proof that f(x) is the increasing function on x∈(γ,1). The derivative of f(x) is
Then, we have
which comes from \(0<\beta <\frac {2}{\pi }<\frac {2}{3}\). Thus, f(x) is the increasing function on x∈(γ,1). Finally, we get f(x)>f(γ)=γ, which implies m_{i+1i}=f(m_{ii−1})>f(γ)=γ. Thus, m_{ii−1}>γ for all i>0 due to the mathematical induction. Since m_{ii−1} is the monotonic decreasing and bounded sequence, m_{ii−1} converges by the monotone convergence theorem [32].
Thus, we can define \(L_{m}={\lim }_{i\rightarrow \infty }m_{ii1}\),
which implies L_{m} is also a root of f(x)=x. Since m_{ii−1} converges L_{m}, and γ is the unique solution of f(x)=x on \(x\in (0,1), L_{m}={\lim }_{i\rightarrow \infty } m_{ii1}=\gamma \), which finishes the proof.
Availability of data and materials
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
Abbreviations
 ADC:

Analogtodigital converter
 BLMMSE:

Bussgang linear minimum mean square error
 BS:

Base station
 DFT:

Discrete Fourier transform
 IUI:

Interuser interference
 KFB:

Kalman filterbased
 LRAMMSE:

Lowresolution aware minumum mean square error
 LS:

Least square
 MIMO:

Multipleinput multipleoutput
 MLD:

maximum likelihood detection
 NMSE:

Normalized mean square error
 QN:

Quantization noise
 SNR:

Signaltonoise ratio
 TPE:

Truncated polynomial expansion
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Acknowledgements
This work was partly supported by Institute for Information & communications Technology Promotion(IITP) grant funded by the Korea government(MSIT) (No. 2016000123, Development of IntegerForcing MIMO Transceivers for 5G & Beyond Mobile Communication Systems) and by the National Research Foundation (NRF) grant funded by the MSIT of the Korea government (2019R1C1C1003638).
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Kim, H., Choi, J. Channel estimation for spatially/temporally correlated massive MIMO systems with onebit ADCs. J Wireless Com Network 2019, 267 (2019). https://doi.org/10.1186/s136380191587x
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Keywords
 Massive MIMO
 Channel estimation
 Onebit ADC
 Kalman filter
 Spatial and temporal correlations
 Truncated polynomial expansion