# Tensor Rank Regularization with Bias Compensation for Millimeter Wave Channel Estimation
## 3 Signal Model  
We consider a massive MIMO-OFDM system consisting of a base station and multiple mobile stations operating at mmWave frequencies. To facilitate the hardware implementation, hybrid analog/digital architectures are employed by both the BS and the MS, shown in Figure 1. We assume that the BS is equipped with \(N_{bs}\) antennas and \(M_{bs}\) RF chains (\(N_{bs}\geq M_{bs}\)), and each MS is equipped with \(N_{ms}\) antennas and \(M_{ms}\) RF chains (\(N_{ms}\geq M_{ms}\)). The OFDM system occupies \(K_{0}\) subcarriers for data transmission, among which \(K\) subcarriers are selected for training purposes. In the downlink scenario, we only need to consider a single user system because the channel estimation is conducted by each mobile user [25]. For the \(k\)-th subcarrier, the BS utilizes an analog RF precoder \(\mathbf{F}_{A}\in\mathbb{C}^{N_{bs}\times M_{bs}}\). Similarly, the MS employs an analog combiner \(\mathbf{W}_{A}\in\mathbb{C}^{N_{ms}\times M_{ms}}\) for analog signal processing. The digital processors follow \(\mathbf{F}_{A}\) and \(\mathbf{W}_{A}\) in BS, and MS performs channel estimation and digital precoding [26]. The elements of \(\mathbf{F}_{A}\) and \(\mathbf{W}_{A}\) are assumed to be random phases with a unit amplitude. Both the BS and MS employ uniform linear arrays (ULA) with antenna element spacing of half the signal wavelength. The extended Saleh-Valenzuela model [15, 16], is adopted to characterize the sparse mmWave channel with \(R\) multipath between the MS and the BS. The frequency-domain channel matrix \(\mathbf{H}_{k}\) associated with the \(k\)-th subcarrier can be obtained as  
\[\mathbf{H}_{k}=\sum_{r=1}^{R}\alpha_{r}e^{-\jmath_{2}\pi\frac{\
u_{R}^{I_{k}} \tau_{r}}{\bf a}_{MS}(\varphi_{r}){\bf a}_{BS}^{\mathbf{T}}(\theta_{r})}, \tag{3}\]  
Figure 1: Architecture of the proposed Hybrid-MIMO system. where \(f_{s}\) is the system sampling rate, \(\alpha_{r}\) is the complex gain of the \(r\)th path in the frequency domain, \(\tau_{r}\) is the time delay, the DoD--\(\theta_{r}\) and the DoA--\(\varphi_{r}\). \(\mathbf{a}_{MS}(\varphi_{r})=\left[1,e^{j\pi\sin\left(\varphi_{r}\right)},\cdots,e^{j\pi\left(N_{ms}-1\right)\sin\left(\varphi_{r}\right)}\right]^{T}\) and \(\mathbf{a}_{BS}(\theta_{r})=\left[1,e^{j\pi\sin\left(\theta_{r}\right)},\cdots,e^{j\pi\left(N_{bs}-1\right)\sin\left(\theta_{r}\right)}\right]^{T}\) are the steering vectors of the MS and BS, respectively. The MS employs \(M_{ms}\) combining vectors with all the RF chains to simultaneously combine the \(n\)-th pilot \(\mathbf{s}_{n}\) precoded by a beamforming vector \(\mathbf{f}_{n}\in\mathbb{C}^{N_{bs}}\). The BS switches beamforming vectors at \(M_{bs}\) successive time slots, while the MS maintains the combining matrix \(\mathbf{W}_{A}\). Then, adopting the same strategy of [15], the received signals for the \(k\)-th subcarrier can be represented as  
\[\mathbf{Y}_{k}=\mathbf{W}_{A}^{\mathrm{T}}\mathbf{H}_{k}\mathbf{F}_{A} \mathbf{S}+\mathbf{N}_{k}, \tag{4}\]  
where \(\mathbf{Y}_{k}\in\mathbb{C}^{M_{ms}\times M_{bs}}\), \(\mathbf{S}\triangleq\mathbf{D}\left(\left[\mathbf{s}_{1},\cdots\mathbf{s}_{M _{bs}}\right]^{T}\right)\in\mathbb{C}^{M_{bs}\times M_{bs}}\) is the transmitted symbol matrix for \(M_{bs}\) successive time slots, and \(\mathbf{N}_{k}\in\mathbb{C}^{M_{ms}\times M_{bs}}\) is the equivalent noise after the combining process. We assume that the pilot matrix is \(\mathbf{S}=\mathbf{I}_{M_{bs}}\) for simplicity [15].