## 3 System Model  
In this work, a single-cell massive MIMO FDD system equipped with \(N_{\rm{t}}\gg 1\) antennas at BS and \(N_{\rm{r}}\) antennas at UE is considered. For simplicity, \(N_{\rm{r}}\) is set to 1. During downlink transmission, we apply orthogonal frequency division multiplexing (OFDM) and denote the number of subcarriers as \(N_{\rm{c}}\). The received signal on the \(n\)-th subcarrier is given by  
\[y_{n}=\hat{\mathbf{h}}_{n}^{\rm{H}}\mathbf{p}_{n}x_{n}+z_{n} \tag{1}\]  
where \(\hat{\mathbf{h}}_{n}\in\mathbb{C}^{N_{\rm{t}}\times 1}\) is the channel vector in frequency domain, \(\mathbf{p}_{n}\in\mathbb{C}^{N_{\rm{t}}\times 1}\) represents the precoding vector, \(x_{n}\) represents the transmitted data symbol, and \(z_{n}\) denotes the additive white Gaussian noise. We denote the channel matrix as \(\hat{\mathbf{H}}=\left[\hat{\mathbf{h}}_{0},\hat{\mathbf{h}}_{2},...,\hat{ \mathbf{h}}_{N_{\rm{t}}-1}\right]^{\rm{H}}\in\mathbb{C}^{N_{\rm{c}}\times N_{ \rm{t}}}\). We assume perfect channel estimation. At the receiver side, UE estimates the channel matrix and feeds this information back to BS. Based on the accurate CSI, BS can design the corresponding precoding vector, which will help improve the transmission quality.  
According to the CSI matrix, the number of elements needed to be fed back is \(2N_{\rm{c}}N_{\rm{t}}\) with the real and imaginary parts separated. Due to the large feedback overhead in massive MIMO systems, we need to design feedback schemes intentionally. First of all, we can take advantage of the CSI sparsity in the angular-delay domain resulting from short delay spread due to multipath effects [14]. \(\tilde{\mathbf{H}}\) can be converted to \(\mathbf{H}\in\mathbb{C}^{N_{\rm{c}}\times N_{\rm{t}}}\) in angular-delay domain by performing 2D discrete Fourier transform. The element of \(\mathbf{H}\) at the \(k\)-th row and \(l\)-th column can be calculated by  
\[h_{k,l}=\frac{1}{N_{\rm{c}}\times N_{\rm{t}}}\sum_{m=0}^{N_{\rm{c}}-1}\sum_{n =0}^{N_{\rm{t}}-1}\tilde{h}_{m,n}e^{-j2\pi\left(\frac{k}{N_{\rm{c}}}m+\frac{l} {N_{\rm{t}}}n\right)} \tag{2}\]  
where \(k=0,1,2,\ldots,N_{\rm{c}}-1,l=0,1,2,\ldots,N_{\rm{t}}-1\). After the processing, we find that most of the elements of \(\mathbf{H}\) are close to zero except for the first \(N_{\rm{a}}\) rows [17]. So, \(\mathbf{H}\) can be replacedby a matrix \(\mathbf{H_{a}}\) consisting of the first \(N_{a}\) rows of itself without losing much information. Consequently, the number of parameters is reduced to \(2N_{a}N_{t}\).  
To further reduce the feedback overhead, we are supposed to do compression on \(\mathbf{H_{a}}\) before feedback and recover it after feedback. The whole feedback scheme can be derived as  
\[\mathbf{\hat{H}_{a}}=f_{\text{re}}(f_{\text{com}}(\mathbf{H_{a}},\Theta_{\text {c}}),\Theta_{\text{r}}) \tag{3}\]  
where \(f_{\text{com}}(\cdot)\) and \(f_{\text{re}}(\cdot)\) denote the compression and recovery process, and \(\Theta_{\text{c}}\) and \(\Theta_{\text{r}}\) represent the parameters involved in the compression and recovery scheme. In our work, we assume a perfect feedback link where the compressed CSI can be transmitted to the BS without loss.  
The aim of optimization is to obtain \(\Theta_{\text{c}}\) and \(\Theta_{\text{r}}\) by training to minimize the distance between \(\mathbf{H_{a}}\) and \(\mathbf{\hat{H_{a}}}\), which can be concluded by  
\[\left(\hat{\Theta}_{\text{c}},\hat{\Theta}_{\text{r}}\right)=\underset{\Theta _{\text{c}},\Theta_{\text{r}}}{\arg\min}\|\mathbf{H_{a}}-f_{\text{re}}(f_{ \text{com}}(\mathbf{H_{a}},\Theta_{\text{c}}),\Theta_{\text{r}})\|_{2}^{2} \tag{4}\]